Handbook of Pseudo-riemannian Geometry and Supersymmetry (IRMA Lectures in Mathematics and Theoretical Physics)

IRMA Lectures in Mathematics and Theoretical Physics 16 Edited by Christian Kassel and Vladimir G. Turaev Institut de R...

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IRMA Lectures in Mathematics and Theoretical Physics 16 Edited by Christian Kassel and Vladimir G. Turaev

Institut de Recherche Mathématique Avancée CNRS et Université de Strasbourg 7 rue René-Descartes 67084 Strasbourg Cedex France

IRMA Lectures in Mathematics and Theoretical Physics Edited by Christian Kassel and Vladimir G. Turaev This series is devoted to the publication of research monographs, lecture notes, and other material arising from programs of the Institut de Recherche Mathématique Avancée (Strasbourg, France). The goal is to promote recent advances in mathematics and theoretical physics and to make them accessible to wide circles of mathematicians, physicists, and students of these disciplines. Previously published in this series: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Deformation Quantization, Gilles Halbout (Ed.) Locally Compact Quantum Groups and Groupoids, Leonid Vainerman (Ed.) From Combinatorics to Dynamical Systems, Frédéric Fauvet and Claude Mitschi (Eds.) Three courses on Partial Differential Equations, Eric Sonnendrücker (Ed.) Infinite Dimensional Groups and Manifolds, Tilman Wurzbacher (Ed.) Athanase Papadopoulos, Metric Spaces, Convexity and Nonpositive Curvature Numerical Methods for Hyperbolic and Kinetic Problems, Stéphane Cordier, Thierry Goudon, Michaël Gutnic and Eric Sonnendrücker (Eds.) AdS/CFT Correspondence: Einstein Metrics and Their Conformal Boundaries, Oliver Biquard (Ed.) Differential Equations and Quantum Groups, D. Bertrand, B. Enriquez, C. Mitschi, C. Sabbah and R. Schäfke (Eds.) Physics and Number Theory, Louise Nyssen (Ed.) Handbook of Teichmüller Theory, Volume I, Athanase Papadopoulos (Ed.) Quantum Groups, Benjamin Enriquez (Ed.) Handbook on Teichmüller Theory, Volume II, Athanase Papadopoulos (Ed.) Michel Weber, Dynamical Systems and Processes Renormalization and Galois Theory, Alain Connes, Frédéric Fauvet and Jean-Pierre Ramis (Eds.)

Volumes 1–5 are available from Walter de Gruyter (www.degruyter.de)

Handbook of Pseudo-Riemannian Geometry and Supersymmetry Vicente Cortés Editor

Editor: Vicente Cortés Department Mathematik und Zentrum für Mathematische Physik Universität Hamburg Bundesstraße 55 20146 Hamburg Germany

2000 Mathematical Subject Classification (primary; secondary): 53-00; 53C26, 53C50, 81T60, 83E30, 83E50, 53C25, 53C29, 53C35, 53C10

ISBN 978-3-03719-079-1 Key words: Pseudo-Riemannian manifold, supersymmetry, special geometry, quaternionic Kähler manifold, c-map, generalized geometry, sigma-model, skew-symmetric torsion, nearly Kähler manifold, Euclidian supersymmetry, black hole, para-geometries, pluriharmonic map, holonomy theory, symmetric space, conformal geometry, Lorentzian manifold, D-brane The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

© 2010 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum FLI C4 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the author’s TEX files: I. Zimmermann, Freiburg Printed in Germany 987654321

To the memory of Krzysztof Galicki

Preface The project for this special volume on pseudo-Riemannian geometry and supersymmetry grew out of the 77th “Encounter between Mathematicians and Theoretical Physicists” at the Institut de Recherche Mathématique Avancée in Strasbourg in 2005. A number of authors of this volume participated in that meeting, including our friend Kris Galicki, who died as a consequence of a tragic hiking accident in 2007. He was always interested in the differential geometric structures occurring in physical theories and was one of the rare scientists who mastered the language of mathematics as well as that of physics. The aim of this handbook is to cover recent developments in the field in a language comprehensible to both, mathematicians and theoretical physicists. The intended audience consists of advanced students and researchers working in differential geometry, string theory and related areas. It includes a chapter about each of the following subjects: special geometry and supersymmetry, generalized geometry, geometries with torsion, para-geometries, holonomy theory, symmetric spaces and spaces of constant curvature, conformal geometry and other topics of recent interest. The contents of each chapter is briefly summarised in the introduction. Acknowledgements. I am grateful to the authors for their beautiful expositions and results. I would like to thank Vladimir Turaev for inviting me to realise this book project and Manfred Karbe and Irene Zimmermann from the EMS Publishing House for the good cooperation. Finally, I would like to thank Eva Kuhlmann for assisting me in the editorial process. Hamburg, January 2010

Vicente Cortés

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Part A. Special geometry and supersymmetry Chapter 1. Quaternion-Kähler spaces, hyper-Kähler cones, and the c-map by Martin Roˇcek, Cumrun Vafa, and Stefan Vandoren . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Chapter 2. Differential forms on quaternionic Kähler manifolds by Gregor Weingart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Chapter 3. Sasakian geometry, holonomy, and supersymmetry by Charles P. Boyer and Krzysztof Galicki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Chapter 4. Special geometry for arbitrary signatures by María A. Lledó, Óscar Maciá, Antoine Van Proeyen, and Veeravalli S. Varadarajan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Chapter 5. Special geometry, black holes and Euclidean supersymmetry by Thomas Mohaupt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Part B. Generalized geometry Chapter 6. Generalized geometry – an introduction by Nigel Hitchin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Chapter 7. Generalizing geometry – algebroids and sigma models Alexei Kotov und Thomas Strobl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Chapter 8. A potential for generalized Kähler geometry by Ulf Lindström, Martin Roˇcek, Rikard von Unge, and Maxim Zabzine . . . . . . . . 263 Part C. Geometries with torsion Chapter 9. Non-integrable geometries, torsion, and holonomy by Ilka Agricola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

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Chapter 10. Connections with totally skew-symmetric torsion and nearly-Kähler geometry by Paul-Andi Nagy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 Chapter 11. Homogeneous nearly Kähler manifolds by Jean-Baptiste Butruille . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 Chapter 12. Nearly pseudo-Kähler and nearly para-Kähler six-manifolds by Lars Schäfer and Fabian Schulte-Hengesbach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 Chapter 13. Quaternionic geometries from superconformal symmetry by Andrew Swann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 Part D. Para-geometries Chapter 14. Twistor and reflector spaces of almost para-quaternionic manifolds by Stefan Ivanov, Ivan Minchev, and Simeon Zamkovoy . . . . . . . . . . . . . . . . . . . . . . . 477 Chapter 15. Para-pluriharmonic maps and twistor spaces by Matthias Krahe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 Chapter 16. Maximally homogeneous para-CR manifolds of semisimple type by Dmitri V. Alekseevsky, Costantino Medori, and Adriano Tomassini . . . . . . . . . . 559 Part E. Holonomy theory Chapter 17. Recent developments in pseudo-Riemannian holonomy theory by Anton Galaev and Thomas Leistner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581 Chapter 18. Geometric applications of irreducible representations of Lie groups by Antonio J. Di Scala, Thomas Leistner, and Thomas Neukirchner . . . . . . . . . . . . 629 Chapter 19. Surface holonomy by Konrad Waldorf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653

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xi

Part F. Symmetric spaces and spaces of constant curvature Chapter 20. Classification results for pseudo-Riemannian symmetric spaces by Ines Kath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685 Chapter 21. Pseudo-Kähler and para-Kähler symmetric spaces by Dmitri V. Alekseevsky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703 Chapter 22. Prehomogeneous affine representations and flat pseudo-Riemannian manifolds by Oliver Baues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731 Part G. Conformal geometry Chapter 23. The conformal analog of Calabi–Yau manifolds by Helga Baum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 821 Chapter 24. Nondegenerate conformal structures, CR structures and quaternionic CR structures on manifolds by Yoshinobu Kamishima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863 Part H. Other topics of recent interest Chapter 25. Linear wave equations on Lorentzian manifolds by Christian Bär . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897 Chapter 25. Survey of D-branes and K-theory by Daniel S. Freed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915 List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 927 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 931

Introduction The present handbook focuses on recent developments in pseudo-Riemannian geometry and supersymmetry. In this introduction we give a short overview of the material contained in the various parts of the volume.

Part A. Special geometry and supersymmetry A classical field theory is usually specified by a Lagrangian L. The scalar fields of the theory are functions 1 ; : : : ; n on space-time M. They can be interpreted as the components of a map W M ! M from space-time into a target manifold M with respect to some system of local coordinates x 1 ; : : : ; x n on M . The kinetic 1 P term 2 gij ./@ i @ j for the scalars in the Lagrangian L defines a pseudoP Riemannian or even Riemannian metric g D gij dx i dx j on M , provided that the symmetric matrix .gij / of the scalar couplings is nondegenerate or even positive definite. Since the discovery of the first supersymmetric field theories, physicists have found that supersymmetry is often reflected in geometric properties of the target metric g. The specific restrictions imposed by supersymmetry depend on the dimension and signature of space-time, as well as on the field content of the theory. When the number of supercharges increases, the allowed target geometry is more and more restricted and becomes finally locally symmetric. The most interesting case is that of eight (real) supercharges. The corresponding geometry is called special geometry. In four-dimensional supergravity theories the matter fields are assembled in vector and hyper multiplets, which are coupled to the supergravity multiplet. The special geometry of supergravity coupled to vector mutiplets is (projective) special Kähler geometry. That of supergravity coupled to hyper mutiplets is quaternionic Kähler geometry. Both geometries are related by the so-called c-map, which associates a quaternionic Kähler manifold N of real dimension 4n C 4 (and negative scalar curvature) to any special Kähler manifold M of real dimension 2n. Any quaternionic Kähler manifold N (of non-zero scalar curvature) is the base of an H =f˙1g-bundle U .N / ! N called the Swann bundle of N . The Swann bundle is a hyper-Kähler cone. In particular, it carries a hyper-Kähler metric, the signature of which depends on the sign of the scalar curvature of the quaternionic Kähler base N . The hyper-Kähler metric can be locally derived from a (real valued) hyper-Kähler potential f , which effectively encodes also the geometry of N . Similarly, the geometry of the special Kähler manifold M is locally encoded in a holomorphic function F (called the holomorphic prepotential). In Chapter 1 by Martin Roˇcek, Cumrum Vafa and Stefan Vandoren, the hyperKähler potential f is determined in terms of the holomorphic prepotential F . The c-map is thus reduced to the correspondence F 7! f . In Gregor Weingart’s contribution, the bundle of differential forms on a quaternionic Kähler (or hyper-Kähler) manifold is decomposed into parallel subbundles. In

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particular, the multiplicities of the corresponding irreducible Sp.n/Sp.1/-representations (or Sp.n/-representations in the hyper-Kähler case) are explicitly calculated. Charles Boyer and Kris Galicki discuss Sasakian manifolds and their relation to special holonomy groups and supersymmetry in Chapter 3. Sasakian manifolds are intimately related to Kähler manifolds, which are fundamental objects in mathematics and theoretical physics. In fact, the metric cone over a Sasakian manifold is Kähler and the geometry transversal to the Sasakian vector field is also Kähler. Similarly, 3-Sasakian manifolds are intimately related to hyper-Kähler and quaternionic Kähler manifolds. In the chapter by María A. Lledó, Oscar Maciá, Antoine Van Proeyen and Veeravalli S. Varadarajan the space-time signature remains Lorentzian but the signature of the special (pseudo-)Kähler target metric is arbitrary. In Chapter 5, Thomas Mohaupt explains the role of special geometry in the theory of supersymmetric black holes. In particular, he shows how Euclidian supersymmetry in three dimensions can be used to study stationary black hole solutions in four dimensions. As one can see already from this example, the geometric structure of the target manifold of a supersymmetric theory can change significantly when the space-time signature changes from Lorentzian to Euclidian. Here not only the target metric changes from Riemannian to neutral, but from quaternionic Kähler to paraquaternionic Kähler. Such para-geometries are further discussed below.

Part B. Generalized geometry Mirror symmetry relates deformations of complex structures to deformations of symplectic structures (on the mirror manifold). Nigel Hitchin’s notion of a generalized complex structure provides a superordinate conceptual framework in which complex and symplectic structures can be treated symmetrically. The chapter by Hitchin is an introduction to the rapidly developing subject of generalized geometry, which incorporates central concepts of supergravity and string theory. In particular, the B-field and the 3-form gauge field H occur naturally in the twisting of the generalized tangent bundle by a gerbe. Moreover, the three-form H plays also the role of the torsion of a metric connection on the base manifold. Alexei Kotov and Thomas Strobl focus on the role of such ’generalized’ geometries encoded in some algebroid structure as targets of supersymmetric sigma models. In particular, generalized Kähler manifolds occur as such targets. Ulf Lindström, Martin Roˇcek, Rikard von Unge and Maxim Zabzine show in Chapter 8 that generalized Kähler structures can be derived from a generalized Kähler potential.

Introduction

xv

Part C. Geometries with torsion Let G O.n/ be a closed subgroup. A G-structure on an n-dimensional manifold admits a torsion-free connection only if the holonomy group of the Levi-Civita connection is a subgroup of G. Therefore, for a given G-structure, there may be no torsion-free connection at all. One is led to allow connections with non-zero torsion and to look for conditions on the torsion which ensure the uniqueness of the connection. It turns out that complete skew-symmetry of the torsion provides such a condition for certain G-structures. Moreover, connections with totally skew-symmetric torsion occur naturally in string theory and supergravity, as explained in the broad survey about geometries with torsion by Ilka Agricola. A beautiful example of a G-structure admitting a unique connection with totally skew-symmetric torsion is provided by the class of nearly Kähler manifolds, which are almost Hermitian manifolds such that the Levi-Civita covariant derivative of the almost complex structure is completely skew-symmetric. General almost Hermitian structures admitting a unique connection with totally skew-symmetric torsion are discussed by Paul-Andi Nagy, whereas Jean-Baptiste Butruille explains the classification of homogeneous nearly Kähler manifolds. Nearly Kähler structures with indefinite metric are considered by Lars Schäfer and Fabian Schulte-Hengesbach in Chapter 12. In particular, they prove that SL.2; R/ SL.2; R/ admits a unique left-invariant nearly pseudo-Kähler structure. As mentioned in Part A of the introduction, the Swann bundle provides a fundamental correspondence M 7! U.M /, which associates a hyper-Kähler cone U.M / with any quaternionic Kähler manifold M . The inverse construction, which associates (at least locally) a quaternionic Kähler manifold M.U / with any hyper-Kähler cone U , is known as the superconformal quotient in the physics literature. It relates a superconformal field theory with scalar manifold U to a Poincaré supergravity theory with target M.U /. It turns out that geometric and field theoretic constructions are often much simpler when described in terms of hyper-Kähler geometry. In Chapter 13 Andrew Swann explains how these results extend to the framework of geometries with torsion. The underlying superconformal algebra is now the one-parameter family of simple Lie superalgebras D.2; 1I ˛/, which occurs, for instance, in the work of Michelson and Strominger on superconformal quantum mechanics.

Part D. Para-geometries A complex structure on a (smooth) manifold M can be defined as an endomorphism field J 2 .End.TM // such that J 2 D Id and such that the eigendistributions T 1;0 M; T 0;1 M TM ˝ C are involutive. Similarly, a para-complex structure on a manifold is an endomorphism field J such that J 2 D Id with involutive eigendistributions T C M; T M TM of the same dimension. In virtue of the Frobenius theorem, a para-complex structure is simply a local product structure with factors of equal dimension. Nevertheless, it is helpful to make use of the analogy between complex

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and para-complex manifolds. There is a useful para-holomorphic calculus in which the role of the field of complex numbers C D RŒi , i 2 D 1, is played by the ring of para-complex numbers RŒe, e 2 D 1. Many interesting structures in Riemannian geometry have natural “para-analogues”. In particular, there is a notion of para-Kähler (or bi-Lagrangian), special para-Kähler, para-hyper-Kähler (or hypersymplectic) and para-quaternionic Kähler manifold. These manifolds carry pseudo-Riemannian metrics of split signature. Remarkably, these structures occur as special geometries of supersymmetric field theories, when the Lorentzian space-time metric is replaced by a positive definite metric, see the chapter by Thomas Mohaupt. In Chapter 14, Stefan Ivanov, Ivan Minchev and Simeon Zamkovoy discuss twistor spaces of general almost para-quaternionic manifolds. Matthias Krahe establishes a Darboux theorem for para-holomorphic symplectic and contact structures. This fundamental result can be applied, for instance, in the twistor theory of para-quaternionic Kähler manifolds. His contribution develops the twistor theory of para-pluriharmonic maps into symmetric spaces. Dmitri V. Alekseevsky, Constantino Medori and Adriano Tomassini classify maximally homogeneous para-CR manifolds of semisimple type.

Part E. Holonomy theory The holonomy group of a pseudo-Riemannian manifold M of signature .p; q/ at a point x 2 M is the subgroup Holx O.Tx M / Š O.p; q/ generated by parallel transports along loops based at x. For connected manifolds this yields a subgroup Hol O.p; q/ well defined up to conjugation in the pseudo-orthogonal group O.p; q/. Holonomy groups were introduced by Élie Cartan in the twenties for the study of Riemannian symmetric spaces and became a powerful tool in Riemannian geometry with Berger’s classification of holonomy groups of complete simply connected Riemannian manifolds in the fifties. Anton Galaev and Thomas Leistner review recent developments in the holonomy theory of pseudo-Riemannian manifolds. These include their classification of Lorentzian holonomy groups and Anton Galaev’s classification of holonomy groups which are subgroups of U.1; n/. The general classification problem for holonomy groups of pseudo-Riemannian manifolds of arbitrary signature remains unsolved. It includes the classification of pseudo-Riemannian symmetric spaces of arbitrary signature, which is already too complicated a problem to expect a simple solution. Chapter 18 by Antonio J. Di Scala, Thomas Leistner and Thomas Neukirchner contains proofs of some facts about irreducible representations of Lie groups and applications of these results in holonomy theory. In the chapter by Konrad Waldorf, the notion of holonomy of a line bundle (endowed with a connection) around a loop is extended to the holonomy of a gerbe along a closed oriented surface, which corresponds to the interaction of a string with a three-form gauge field.

Introduction

xvii

Part F. Symmetric spaces and spaces of constant curvature A pseudo-Riemannian manifold M is called a symmetric space if every point x 2 M is an isolated fixed point of an involutive isometry. This includes the complete simply connected pseudo-Riemannian manifolds of constant curvature. Ines Kath reviews the state of the art in the classification of pseudo-Riemannian symmetric spaces. Like for the classification of pseudo-Riemannian holonomy groups, little is known beyond metrics of index 2. She explains various approaches and partial results, for instance under the assumption of additional geometric structures. Dmitry Alekseevsky discusses in Chapter 21 the classification problem for pseudoKähler and para-Kähler symmetric spaces. In particular, he describes some classes of Ricci-flat examples. Oliver Baues develops the theory of flat pseudo-Riemannian manifolds in the general context of flat affine structures and prehomogeneous affine representations. Flat Riemannian manifolds are well understood by Bieberbach’s theorems, but there are still many long standing open problems concerning flat pseudo-Riemannian manifolds of arbitrary signature. For instance, it is not known whether every compact flat pseudo-Riemannian manifold of signature .p; q/ (with p q 2) is a quotient of the pseudo-Euclidian space Rp;q and also cocompact properly discontinuous groups of pseudo-Euclidian motions are scarcely understood.

Part G. Conformal geometry A conformal structure of signature .p; q/ on a smooth manifold M is a ray subbundle L S 2 T M such that any local section of L defines a pseudo-Riemannian metric of signature .p; q/. In particular, any pseudo-Riemannian metric g on M defines a conformal structure L D RC g. Conformal geometry is concerned with properties which do not depend on the choice of a section g 2 .L/. The holonomy group of a pseudo-Riemannian manifold .M; g/, for instance, is not a conformal invariant. Helga Baum’s contribution is a survey on the holonomy theory of Cartan connections. This theory applies, in particular, to conformal geometry. As explained in her exposition, the conformal holonomy group contains important information about a pseudo-Riemannian manifold. The knowledge of the holonomy group allows one to decide, for example, whether the pseudo-Riemannian manifold admits conformal Killing spinors or an Einstein metric in the conformal class. She describes Lorentzian manifolds with conformal holonomy in SU.1; n/. Chapter 24 by Yoshinubo Kamishima is also concerned with conformal and related geometric structures. It provides a unified treatment of conformal, CR and quaternionic CR-structures. In the positive definite case, the corresponding model spaces are the boundaries at infinity of the hyperbolic spaces over the real, complex and quaternionic numbers, respectively.

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Part H. Other topics of recent interest The chapter by Christian Bär summarises the analytic theory of linear wave equations on globally hyperbolic Lorentzian manifolds, as developed in his book with Nicolas Ginoux and Frank Pfäffle. In the final chapter, Dan Freed explains the relation between D-branes in string theory and K-theory.

Part A

Special geometry and supersymmetry

Chapter 1

Quaternion-Kähler spaces, hyper-Kähler cones, and the c-map Martin Roˇcek, Cumrun Vafa, and Stefan Vandoren

Contents 1 2 3 4

Introduction . . . . . . . . . . . . . . . . . . . . The c-map . . . . . . . . . . . . . . . . . . . . Hyper-Kähler cones and the Legendre transform Hyper-Kähler cones from the c-map . . . . . . . 4.1 Gauge fixing and the contour integral . . . 4.2 The hyper-Kähler potential . . . . . . . . . 4.3 Twistor space . . . . . . . . . . . . . . . . 4.4 The quaternionic metric . . . . . . . . . . . 5 Summary and conclusion . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

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3 4 5 7 7 8 9 9 10 12

1 Introduction The moduli spaces of Kähler and complex structure deformations of Calabi–Yau manifolds are naturally related to special Kähler (SK) and quaternion-Kähler (QK) geometry. Consequently, these types of manifolds arise in the low energy effective action of string theory compactifications on Calabi–Yau three-folds. SK manifolds were discovered in the context of N D 2 supergravity1 theories coupled to vector multiplets [1]. They are described by a holomorphic function F .X/ that is homogeneous of degree two in complex coordinates X I . A more mathematically precise and intrinsic formulation of this special geometry was given in [2], [3]. QK manifolds arose in the context of N D 2 supergravity coupled to hypermultiplets [4]. QK manifolds are also described by a single function. This follows from the construction of the Swann bundle [5] over the QK space. This bundle is hyper-Kähler; locally, its metric is determined by a hyper-Kähler potential ./, where are local coordinates on the space. In [6], we called such spaces hyper-Kähler cones (HKC’s) because they have a homothety arising from the underlying conformal symmetry. One therefore also uses the terminology conformal hyper-Kähler manifolds, as in [7], [8]. 1 The

supersymmetry transformations have eight real components.

4

Martin Roˇcek, Cumrun Vafa, and Stefan Vandoren

In this note, we review the construction of quaternion-Kähler (QK) manifolds from special Kähler (SK) geometry, along the lines of our recent work [9], but with more emphasis on the mathematical structure. One constructs QK spaces from SK manifolds by using the c-map [10]. This maps extends Calabi’s construction of hyperKähler metrics on cotangent bundles of Kähler manifolds [11], [12] – a construction well known in the mathematics community – to quaternionic geometry. As we shall see, the hyper-Kähler cones arising from the c-map have additional symmetries: they have an equal number of commuting triholomorphic isometries as their quaternionic dimension. Hyper-Kähler manifolds with such isometries were classified in [13] by performing a Legendre transform on the (hyper-)Kähler potential [14] and writing the result in terms of a contour integral of a meromorphic2 function H . In our case, because of the conformal symmetry of the HKC, this function H is homogeneous of degree one. As a result, the c-map induces a map from the holomorphic function F , which characterizes the SK geometry and is homogeneous of degree two, to a function H , which characterizes the QK geometry and is homogeneous of degree one. Following [9], we now describe this construction.

2 The c-map In this section, we introduce our notation and review the c-map [10], [15]. Consider an affine (or rigid) special Kähler manifold3 of dimension 2.n C 1/. It is characterized by a holomorphic prepotential F .X I /, which is homogeneous of degree two (I D 1; : : : ; n C 1). The Kähler potential and metric of the rigid special geometry are given by x D i.Xx I FI X I FxI /; ds 2 D NIJ dX I dXx J ; NIJ D i.FIJ FxIJ /; (2.1) K.X; X/ where FI is the first derivative of F , etc. The projective (or local) special Kähler geometry is then of real dimension 2n, with complex inhomogeneous coordinates XI D f1; Z A g; X1 where A runs over n values. Its Kähler potential is given by ZI D

x I /: x D ln.Z I NIJ Z K.Z; Z/

(2.2)

(2.3)

We further introduce the matrices [1] .NX /I .NX /J ; NIJ D i FxIJ .XNX / where .NX /I NIJ X J , etc. 2Actually, 3 We

certain branch cut singularities sometimes arise. use the language of local coordinates, but a coordinate free description can be found in [2], [3].

(2.4)

Chapter 1. Quaternion-Kähler spaces, hyper-Kähler cones, and the c-map

5

The c-map defines a 4.n C 1/-dimensional quaternion-Kähler metric as follows: One builds a G-bundle, with 2.n C 2/ dimensional fibers coordinatized by , , AI and BI , over the projective special Kähler manifold; the real group G is a semidirect product of a Heisenberg group with R, and acts on the coordinates by AI ! eˇ .AI C I /;

BI ! eˇ .BI C I /; ! e2ˇ . C ˛ 12 I AI C 12 I BI /;

! C ˇ;

(2.5)

Then the explicit G-invariant QK metric is [15]

S J C e2 d 1 .AI dBI BI dAI / 2 ds 2 D d 2 e .N C Nx /IJ W I W 2 x Bx : 4KABx dZ A dZ

(2.6)

x is positive and hence The metric is only positive definite in the domain where .ZN Z/ x KABx is negative definite. One can then show that N C N is negative definite [16]. The one-forms W I are defined by W I D .N C Nx /1 IJ 2NxJK dAK i dBJ : (2.7) As shown in [15], such metrics are indeed quaternion-Kähler; they were further studied in [17], [18], including an analysis of their isometries. There are always the 2.n C 4/ manifest isometries (2.5), of which n C 2 are commuting, e.g., BI ! BI C I ;

! C ˛ 12 I AI :

(2.8)

This is one isometry more than the quaternionic dimension of the QK.

3 Hyper-Kähler cones and the Legendre transform The Swann bundle over a QK geometry, i.e., the hyper-Kähler cone (HKC), is a hyper-Kähler manifold with one extra quaternionic dimension. As for special Kähler manifolds, the geometry of the HKC is again affine. In physics terminology, this arises because one adds a compensating hypermultiplet. Adding the compensator to the original hypermultiplets that parametrize the 4.n C 1/-dimensional QK space, one obtains a cone with real dimension 4.n C 2/. This space is hyper-Kähler and admits a homothety as well as an SU.2/ isometry group that rotates the three complex structures. The metric on the HKC can be constructed from a hyper-Kähler potential [5], which is a Kähler potential with respect to any of the complex structures. In real local coordinates A , the metric and the hyper-Kähler potential ./ are related by gAB D DA @B ./;

(3.1)

where DA is the Levi-Civita connection. As for all Kähler manifolds, in complex coordinates, the hermitian part of (3.1) defines the metric in terms of the complex

6


hessian of the potential; however, in this case, the vanishing of the holomorphic parts of the metric is an additional constraint on the geometry. Any QK isometry can be lifted to a triholomorphic isometry on the HKC. In the physics literature, this was shown in [6], [19]. Using the notation of the previous section, we thus have an HKC of real dimension 4.nC2/ together with nC2 commuting triholomorphic isometries determined by (2.8). As mentioned before, hyper-Kähler manifolds of this type were classified in [13]. It is convenient to introduce complex O coordinates v I and wIO , in such a way that the isometries act as imaginary shifts in wIO . Notice that IO D 0; 1; : : : ; n C 1. The hyper-Kähler potential is then a function .v; v; N w C w/, N and can be written as a Legendre transform of a function L.v; v; N G/ O of 3.n C 2/ variables, The Legendre transform with respect to G I is defined by O

.v; v; N w; w/ N L.v; v; N G/ .w C w/ N IO G I ;

wIO C wN IO D

@L

: (3.2) @G IO The constraints from hyper-Kähler geometry can be solved by writing L in terms of a contour integral [20], [21], [13] I d H.; /; (3.3) L.v; v; N G/ Im C 2 i with

O

vI O O C G I vN I : IO

(3.4)

These objects have an interpretation in twistor space as sections of an O.2/ bundle. In physics terminology, these are N D 2 tensor multiplets. Furthermore, the conditions for a homothetic Killing vector and SU.2/ isometries imply that H is a function homogeneous of first degree4 (in ) and without explicit dependence [6]. Since H is homogeneous of first degree in , it follows that the hyper-Kähler potential is also homogeneous of first degree in v and v: N . v; v; N w; w/ N D .v; v; N w; w/: N

(3.5)

In addition to a homothety, hyper-Kähler cones also have an SU.2/ isometry group that rotates the sphere of complex structures. Under infinitesimal variations with respect to an element of the Lie algebra "C TC C " T C "3 T3 , with " D ."C / , these act as [6] @L O O O N w; w/; N ı"E wIO D "C O ; (3.6) ı"E v I D i"3 v I C " G I .v; v; @vN I O

N w; wN obtained where G I has to be understood as the function of the coordinates v; v; by the Legendre transform defined in (3.2). The coordinates wI do not transform under "3 . One can now explicitly check that the hyper-Kähler potential is SU.2/R quasihomogeneity up to terms of the form ln./ is sufficient [6], but such terms do not seem to arise in the c-map. 4Actually,


7

invariant, O

O

O

ı"E D LvIO ı"E v I C LvN IO ı"E vN I ı"E .wIO C wN IO / G I D 0:

(3.7)

(The ıG terms cancel identically because is a Legendre transform.) For the generators "˙ this is immediately obvious; for variations proportional to "3 one needs to use O O the invariance of L, i.e., v I LvIO D vN I LvN IO .

4 Hyper-Kähler cones from the c-map The quaternion-Kähler space in the image of the c-map has dimension 4.n C 1/. The hyper-Kähler cone above it has dimension 4.n C 2/. It therefore needs to be described by n C 2 twistor variables, say I and 0 , where I D 1; : : : ; n C 1. As we shall show, the result for the tree level c-map is given by H.I ; 0 / D

F .I / ; 0

(4.1)

where F is the prepotential of the special Kähler geometry, now evaluated on the twistor variables . This is our main result. Note that H does not depend explicitly on and, since F is homogeneous of degree two, H is homogeneous of degree one, as required by superconformal invariance. We now give a detailed proof of (4.1) by explicit calculation [9]. To be precise, we prove that (4.1) leads to (2.6).

4.1 Gauge fixing and the contour integral As explained in the previous section, any hyper-Kähler cone has an SU.2/ symmetry and a homothety. The generators of the homothety and U.1/ SU.2/ give a natural complexified action on the HKC; the remaining two generators of the SU.2/ combine to give the roots T˙ . To evaluate the contour integral (3.3), it is convenient to make use of the isometries. In physics terminology, one can impose gauge choices. Mathematically, the isometries fiber the quotient space by the orbits, and a gauge choice is just a choice of section. For the symmetries generated by T˙ , whose action is given by (3.6), we choose v 0 D 0:

(4.2)

In this gauge, we have that 0 D G 0 and this simplifies the pole structure in the complex -plane. Then, using as well the homogeneity properties of F , the contour integral (3.3) simplifies to I d F .I / 1 ; (4.3) L.v; v; N G/ D 0 Im G 2 i 3

8


with

I D v I C G I 2 vN I ;

I D 1; : : : ; n C 1;

(4.4)

which, for nonzero values of v, has no zeroes at D 0. Therefore, assuming F is regular at D 0, F ./ has no poles (in ) inside the contour around the origin. It is now easy to evaluate the contour integral, because the residue at D 0 replaces all the I by v I . The result is 1 L.v; v; N G/ D N ; (4.5) NIJ G I G J 2K.v; v/ 0 4G where K.v; v/ N is the Kähler potential of the rigid special geometry given in (2.1), with FI .v/ now the derivative with respect to v I , etc. Notice that the function L satisfies the Laplace-like equations [20], [21], [13] LG I G J C LvI vN J D 0:

(4.6)

The equation is not satisfied for the components LG 0 G 0 and LG 0 G I , because we have chosen the gauge v 0 D 0. It would be interesting to compute L for arbitrary values of v 0 . For a special case, this was done in [22].

4.2 The hyper-Kähler potential To compute the hyper-Kähler potential , we have to Legendre transform L, N I GI ; .v; v; N w; w/ N D L.v; v; N G/ C .w C w/ N 0 G 0 .w C w/

(4.7)

The hyper-Kähler potential , computed by extremizing (4.7) with respect G ; G I completely determines the associated hyper-Kähler geometry. In general, it is a function of the 2.n C 2/ complex coordinates v 0 ; v I and w0 ; wI , but we work only on the (Kähler but not hyper-Kähler) submanifold v 0 D 0. The geometry of the HKC only depends on w through the combination w C wN which makes manifest the n C 2 commuting isometries. The Legendre transform of (4.5) gives 5

GI D 2N IJ .w C w/ N J; G0 K : .G 0 /2 D IJ 2 .w C w/ N I N .w C w/ N J .w C w/ N 0

0

(4.8)

Up to an irrelevant overall sign we find, using (4.5), K.v; v/ N v; v; N G.v; v; N w; w/ N D ; G0

(4.9)

where G 0 is determined by (4.8). More explicitly, in terms of the HKC coordinates, q p p .v; v; N w; w/ N D 2 K.v; v/ N .w C w/ N I N IJ .w C w/ N J .w C w/ N 0 : (4.10) 5 The

relative minus signs between the last two terms in (4.7) is purely a matter of convention.


9

4.3 Twistor space The twistor space above a 4.n C 1/ dimensional QK has dimension two higher, and is Kähler. It can be seen as a CP1 bundle over the QK. It can also be obtained from the HKC by a Kähler quotient with respect to U.1/ SU.2/. Equivalently, we define inhomogeneous coordinates, e.g., vI D f1; Z A g; (4.11) v1 where A runs over n values. As we show below, these inhomogeneous coordinates will be identified with (2.2). The Kähler potential on the twistor space, denoted by KT , is given by the logarithm of the hyper-Kähler potential restricted to the submanifold given by v 1 D 1 [6]: p 1 x N w; w/ .wC w/ N I N IJ .wC w/ N D K.Z; Z/Cln N J .wC w/ N 0 Cln. 2/; KT .Z; Z; 2 (4.12) x where K.Z; Z/ is the same as the special Kähler potential (2.3). On the twistor space, there always exists a holomorphic one-form X which can be constructed from the holomorphic two-form that any hyper-Kähler manifold admits. In our case this one-form is obtained from the holomorphic HKC two-form D dwI ^ dv I . Without going into details, it is given by [6] ZI D

X D 2Z I dwI X˛ dz ˛ ;

(4.13)

where the index ˛ D 1; : : : ; 2.n C 1/ runs over the complete set of holomorphic coordinates wI ; w0 ; Z A on the submanifold6 of the twistor space given by v 0 D 0. In total this gives 2.n C 1/ C 2 C 2n D 4.n C 1/ – the (real) dimension of the QK. The metric on the QK manifold can then be computed7 : x N: G˛ˇN D KT; ˛ˇN e2KT X˛ X ˇ

(4.14)

4.4 The quaternionic metric We now compute the QK metric that follows from the c-map using (4.14). To compare with (2.6) we only need to identify the coordinates wI ; w0 with those of (2.6), since the Z A coordinates of the special Kähler manifold can be identified with the ones above. We define 1 I I J w0 D iA A FIJ i C A BI e ; 2 (4.15) i J wI D iFIJ A BI : 2 6 This 7 Note

submanifold can be thought of invariantly as a quotient of the original HKC. that the constant term in KT (4.12) enters in (4.14).

10


The metric can be written in these coordinates; after considerable calculation [9], up to an irrelevant overall all factor of 1=8, we obtain precisely the result (2.6)! From (4.8), we find the following relations between the QK coordinates and the twistor O variables G I (see (3.4)): 2AI D

GI ; G0

4e D

K.v; v/ N : 0 .G /2

(4.16)

This concludes the proof of (4.1).

5 Summary and conclusion We have constructed the Swann bundle over the quaternion-Kähler manifolds that arise in the c-map. The corresponding hyper-Kähler potential was given in (4.10), and was first derived in [9]. Introducing coordinates vI ; N w; w/ N p X I .v; v; G 0 .v; v; N w; w/ N

(5.1)

we can conveniently rewrite the hyper-Kähler potential as follows: .v; v; N w; w/ N D K X I .v; v; N w; w/; N Xx I .v; v; N w; w/ N

(5.2)

x D i.Xx I FI X I FxI / of the affine special Here K is the Kähler potential K.X; X/ geometry. The special hyper-Kähler cones given by the c-map have as many (nC2) commuting triholomorphic isometries as their quaternionic dimension. As explained before, this implies the hyper-Kähler potential can be Legendre transformed to a function L that can be written in terms of a contour integral over a function H./; equivalently, the twistor space of the HKC can be described in terms of sections of .nC2/ O.2/-bundles. These twistor variables were defined in (3.4) and the function H was determined in (4.1). Defining I (5.3) X I ./ p ; 0 we can write H as follows: H.I ; 0 / D F X I .I ; 0 /

(5.4)

The function F is well known to be related to the topological string amplitude [23], [24]. Typical examples that appear in the context of Calabi–Yau compactifications are of the form X AX B X C ; (5.5) F .X I / D dABC X1


11

where X I D fX 1 ; X A g and the constants dABC are related to the triple intersection numbers of the Calabi–Yau. To give an explicit example, one can choose specific values for these coefficients such that the local (projective) special Kähler geometry is the symmetric space SO.n 1; 2/ SU.1; 1/ : U.1/ SO.n 1/ SO.2/

(5.6)

After the c-map, the hyper-Kähler cone is based on the function O

H.I / D dABC

A B C : 0 1

(5.7)

This corresponds to a homogeneous quaternion-Kähler manifold of the form (see e.g. appendix C in [10], and references therein) SO.n C 1; 4/ : SO.n C 1/ SO.4/

(5.8)

Other examples were recently given in [25], were quantum effects were taken into account. The connection of these geometries with topological strings is very profound, and has important physical implications. For instance, it was recently shown that the topological string amplitude F appears in the study of supersymmetric black holes in string theory [26], [27]. More precisely, the Legendre transform of F is related to the entropy of the black hole. It would be interesting to see if this Legendre transform is related to the one described here; speculations along these lines can be found in [9]. To make progress on this issue, one needs to evaluate the contour integral (3.3) without making use of the special coordinate system in which we can set v 0 D 0. We leave this for future research. Acknowledgements. This work was presented at the 77th Rencontre entre Physiciens Théoriciens et Mathématiciens on “Pseudo-Riemannian Geometry and Supersymmetry”, Strasbourg. SV thanks V. Cortés for the invitation and kind hospitality. Most of this work has been initiated and completed during the 2004 and 2005 Simons Workshops in Physics and Mathematics. SV and CV thank the C.N. Yang Institute for Theoretical Physics and the Department of Mathematics at Stony Brook University for hosting the workshops and for partial support. MR thanks the Institute for Theoretical Physics at the University of Amsterdam for hospitality. MR is supported in part by NSF grant no. PHY-0354776, by the University of Amsterdam, and by FOM. CV is supported in part by NSF grants PHY-0244821 and DMS-0244464.

12


References [1]

B. de Wit and A. Van Proeyen, Potentials and symmetries of general gauged N D 2 supergravity-Yang–Mills models. Nuclear Phys. B 245 (1984), 89–117. 3, 4

[2]

D. S. Freed, Special Kähler manifolds. Commun. Math. Phys. 203 (1999), 31–52. 3, 4

[3]

V. Cortés, Special Kähler manifolds: a survey. Proceedings of the 21st Winter School “Geometry and Physics” (Srní, 2001), Rend. Circ. Mat. Palermo (2) Suppl. 2002, no. 69, 11–18. 3, 4

[4]

J. Bagger and E. Witten, Matter couplings in N D 2 supergravity. Nuclear Phys. B 222 (1983), 1–10. 3

[5]

A. Swann, Hyperkähler and quaternionic Kähler geometry. Math. Ann. 289 (1991), 421–450. 3, 5

[6]

B. de Wit, M. Roˇcek, and S. Vandoren, Hypermultiplets, hyperkähler cones and quaternion-Kähler geometry. J. High Energy Phys. 2001, no. 2, 039. 3, 6, 9

[7]

E. Bergshoeff, S. Cucu, T. de Wit, J. Gheerardyn, S. Vandoren, and A. Van Proeyen, The map between conformal hypercomplex hyper-Kähler and quaternionic(-Kähler) geometry. Commun. Math. Phys. 262 (2005), 411–457. 3

[8]

E. Bergshoeff, S. Vandoren, and A. Van Proeyen, Internat. J. Geom. Methods Mod. Phys. 3 (5-6) (2006), 913–932. 3

[9]

M. Roˇcek, C. Vafa, and S. Vandoren, Hypermultiplets and topological strings. J. High Energy Phys. 2006, no. 2, 062. 4, 7, 10, 11

[10] S. Cecotti, S. Ferrara, and L. Girardello, Geometry of type II superstrings and the moduli of superconformal field theories. Internat. J. Modern Phys. A 4 (10) (1989), 2475–2529. 4, 11 [11] E. Calabi, Métriques kähleriennes et fibrés holomorphes. Ann. Sci. École Norm. Sup. 12 (1979), 269–294. 4 [12] V. Cortés, C. Mayer, T. Mohaupt, and F. Saueressig, Special geometry of Euclidean supersymmetry. II: Hypermultiplets and the c-map. J. High Energy Phys. 2005, no. 6, 025. 4 [13] N. J. Hitchin, A. Karlhede, U. Lindström, and M. Roˇcek, Hyperkähler metrics and supersymmetry. Commun. Math. Phys. 108 (1987), 535–589. 4, 6, 8 [14] U. Lindström and M. Roˇcek, Scalar tensor duality and N D 1, N D 2 nonlinear sigma models. Nuclear Phys. B 222 (1983), 285–308. 4 [15] S. Ferrara and S. Sabharwal, Quaternionic manifolds for type II superstring vacua of Calabi–Yau spaces. Nuclear Phys. B 332 (1990), 317–332. 4, 5 [16] E. Cremmer, C. Kounnas, A. Van Proeyen, J. P. Derendinger, S. Ferrara, B. de Wit, and L. Girardello, Vector multiplets coupled to N D 2 supergravity: superhiggs effect, flat potentials and geometric structure. Nuclear Phys. B 250 (1985), 385–426. 5 [17] B. de Wit and A. Van Proeyen, Symmetries of dual-quaternionic manifolds. Phys. Lett. B 252 (1990), 221–229. 5 [18] B. de Wit, F. Vanderseypen, and A. Van Proeyen, Symmetry structure of special geometries. Nuclear Phys. B 400 (1993), 463–521. 5


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[19] B. de Wit, M. Roˇcek, and S. Vandoren, Gauging isometries on hyperKähler cones and quaternion-Kähler manifolds. Phys. Lett. B 511 (2001), 302–310. 6 [20] S. J. Gates, Jr., C. Hull, and M. Roˇcek, Twisted multiplets and new supersymmetric nonlinear sigma models. Nuclear Phys. B 248 (1984), 157–186. 6, 8 [21] A. Karlhede, U. Lindström, and M. Roˇcek, Self-interacting tensor multiplets in N D 2 superspace. Phys. Lett. B 147 (1984), 297–300. 6, 8 [22] L. Anguelova, M. Roˇcek, and S. Vandoren, Quantum corrections to the universal hypermultiplet and superspace. Phys. Rev. D 70 (2004), 066001. 8 [23] I. Antoniadis, E. Gava, K. S. Narain, and T. R. Taylor, Topological amplitudes in string theory. Nuclear Phys. B 413 (1994), 162–184. . 10 [24] M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa, Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes. Commun. Math. Phys. 165 (1994), 311–427. 10 [25] D. Robles Llana, F. Saueressig, and S. Vandoren, String loop corrected hypermultiplet moduli spaces. J. High Energy Phys. 2006, no. 3, 081. 11 [26] H. Ooguri, A. Strominger, and C. Vafa, Black hole attractors and the topological string. Phys. Rev. D (3) 70 (10) (2004), 106007. 11 [27] H. Ooguri, C. Vafa, and E. P. Verlinde, Hartle-Hawking wave-function for flux compactification: the entropic principle. Lett. Math. Phys. 74 (3) (2005), 311–342. 11

Chapter 2

Differential forms on quaternionic Kähler manifolds Gregor Weingart

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . 2 The Lie algebra so4;1 .R/ of operators on forms 3 Quaternionic Kähler decomposition of forms . . References . . . . . . . . . . . . . . . . . . . . . . .

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15 17 29 37

1 Introduction Quaternionic Kähler manifolds and hyper-Kähler manifolds have long been studied for their rich geometry and their marvellous implications on the topology of the underlying manifold. In the context of these studies the holonomy decomposition of the bundle of complex valued differential forms into minimal parallel subbundles has been studied extensively for hyper-Kähler manifolds generalizing ideas developed in Kähler geometry. Interestingly there exists an intimate relationship between this construction and a strange property of general representation theory, the existence of a dual Lie algebra governing the decomposition of tensor products of exterior or symmetric algebras. For the complex valued differential forms on hyper-Kähler manifolds this dual Lie algebra is isomorphic to so4;1 .R/ ˝R C. In this chapter we will define the action of the Lie algebra so4;1 .R/ ˝R C in Section 2 entirely from the representation theoretic point of view with only a passing reference on how this algebra can be constructed using wedge products and contractions with Kähler forms. One merit of this approach is that it highlights the close relationship to Weyl’s construction of the irreducible representations of the classical matrix groups showing in particular that for a generic hyper-Kähler manifold the resulting decomposition is minimal. Moreover the representation theoretic approach allows us to use the comparatively simple branching formulas from so4;1 .R/ to R ˚ sp.1/ instead of the branching formulas from so4n .R/ to sp.1/ ˚ sp.n/ to calculate the parallel decomposition of differential forms on quaternionic Kähler and hyper-Kähler manifolds. Before presenting the final decomposition formula let us recall that a quaternionic Kähler manifold M carries quite a few geometric vector bundles besides the usual tensor bundles. Eventually this plentitude is due to the fact that the complexified tangent

16

Gregor Weingart

bundle of a quaternionic Kähler manifold M of dimension 4n factorizes parallely into a tensor product TM ˝R C Š HM ˝ EM of two (locally defined) complex symplectic vector bundles HM and EM of dimensions 2 and 2n respectively. All minimal parallel subbundles of the differential forms bundle ƒ T M ˝R C on a generic quaternionic Kähler manifold M are of the form rN Syms HM ˝ ƒr; B EM rN with n r rN 0 and s r C rN modulo 2 compare [3], where ƒr; B EM is the joint kernel of all possible contractions with the parallel symplectic form on S W ƒr EM ˝ ƒrN EM ! EM inside the Schur functor bundle ƒr;rN EM WD ker.Pl rC1 r1 N ƒ EM ˝ ƒ EM /:

Theorem 1.1 (Parallel decomposition of differential forms). On a generic quaternionic Kähler manifold M of dimension 4n the holonomy decomposition of the bundle of complex valued differential forms into minimal parallel subbundles reads M rN ms;r;rN .k/Syms HM ˝ ƒr; ƒ k T M ˝R C D B EM; nrr0 N s0

where the multiplicities ms;r;rN .k/ vanish unless k s r C rN modulo 2 and s 2n r r; N

rN

ks 2n r; 2

r

kCs 2n r: N 2

A precise and ready to use formula for the multiplicities ms;r;rN .k/ is given in the specification Theorem 3.1 of Theorem 1.1 in Section 3, it appears inadequate for an introduction to reproduce the threefold case distinction necessary. Suffice it to say at this point that under the conditions of Theorem 1.1 the multiplicities ms;r;rN .k/ > 0 are strictly positive unless minfr r; N sg D 0 and r 6 ks modulo 2. It is useful to think of 2 2n

2n

@

@

@

@

@

r

n

kCs 2

@

@

@

@

@ @ ks p ppp p ppp p ppp p pp p p p p p p p p p p p p p p p p p p p p p p p p pr@ ppp @ 2 ppp @ ppp p ppp p ppp p ppp p @ pp p p r rN ppp @ pp pp ppp @

0

@

@

@ @

n

@

2n

or

n

@ @

0

r n

@

@

@ ks p p p p p@ p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p@ p p p p p p p p p p prpp 2 @ @ ppppp pp @ @ pppp p p p p p p p p p p p p p p p p p p p p @ p p p p p r ppp @ pppp rN p p @ kCs 2

(1)

2n

ms;r;rN .k/ as depending on the position of the point . kCs ; ks / in the hexagonal pattern 2 2 determined by the point .r; r/ N the different shapes correspond to 3r rN 2n and

Chapter 2. Differential forms on quaternionic Kähler manifolds

17

3r rN 2n. Essentially the multiplicities ms;r;rN .k/ vanish outside the hexagon and increase from the value 1 on the edges towards the central triangle by steps of 1 every second parallel to their maximal value d r2 rN e or 1 C n r respectively taken in the triangle, of course only points . kCs ; ks / satisfying k s r C rN modulo 2 ought 2 2 to be considered. If r and rN share the same parity, then this geometric interpretation of the formula for ms;r;rN .k/ given in Theorem 3.1 is not entirely correct, the actual multiplicities may be one less than their geometrically deduced values.

2 The Lie algebra so4;1 .R/ of operators on forms Usually an orthogonal quaternionic structure on a euclidian vector space T with scalar product g is defined as the choice of three anticommuting, skew-symmetric complex structures I; J and K on T . In this exposition however we prefer an alternative weaker notion, an orthogonal quaternionic structure on T is to be the choice of a subalgebra Q End T containing the identity of T and isomorphic to H as an associative algebra N / for all A 2 Q and all X; Y 2 T . with unit over R, such that g.AX; Y / D g.X; AY Fixing an isomorphism Q Š H brings us back to the standard notion of orthogonal quaternionic structures, this innocuous difference for euclidian vector spaces becomes important for Riemannian manifolds. In order to discuss orthogonal quaternionic structures on euclidian vector spaces in more detail we recall that a positive quaternionic structure on a complex symplectic vector space E with symplectic form is a conjugate linear map C W E ! E satisfying C 2 D 1 as well as .C e1 ; C e2 / D .e1 ; e2 / and .C e; e/ > 0 for all e ¤ 0. Quite remarkably the complexification T ˝R C of a euclidian vector space T of dimension 4n with scalar product g and orthogonal quaternionic structure Q End T turns out to be the tensor product (2) H ˝ E Š T ˝R C of two complex symplectic vector spaces H and E of dimension 2 and 2n endowed with positive quaternionic structures, such that every A 2 Q acts as a Kronecker product A ˝ idE on H ˝ E, while the complex bilinear extension of g and the real structure on T ˝R C agree with ˝ and C ˝ C respectively. One way to justify this convenient description of T ˝R C as a tensor product is to use the representation theory of the unitary symplectic group Sp.n/, n 1; which can be interpreted alternatively as the group of orthogonal maps T ! T commuting with Q or as the group of symplectic maps E ! E commuting with C . A somewhat more direct approach is to note that Q Š ClR2;0 is a Clifford algebra with a unique complex module H , which is a complex symplectic vector space of dimension 2 with a positive quaternionic structure, forcing the multiplicity space E WD Hom Q .H; T ˝R C/ to have a symplectic form and positive quaternionic structure as well [5]. A particular merit of the description of T ˝R C D H ˝ E is that the algebra Q acts on T essentially via its action on its module H . In particular the choice of a

18

Gregor Weingart

complex structure in Q corresponds to the choice of decomposition H D L ˚ CL into two conjugated complex lines L and CL, in fact for any such decomposition we can find I 2 Q such that L and CL are the eigenspaces for the eigenvalues i and i respectively. In the same vein every unit vector q 2 H with .C q; q/ D 1 determines a unique algebra isomorphism 'q W H ! Q such that the images I; J of i; j 2 H satisfy I q D iq and J q D C q, more precisely the matrices of the images I; J; K 2 Q with respect to the canonical basis C q; q of H , read as follows: i 0 0 C1 0 i I D ; J D ; KD : (3) 0 Ci 1 0 i 0 Consider now a Riemannian manifold M with scalar product g together with a smooth choice of orthogonal quaternionic structures Qp M End Tp M on every tangent space Tp M; p 2 M . Such a Riemannian manifold M is called a quaternionic Kähler manifold, if the subalgebra bundle QM End TM is parallel with respect to the Levi-Civita connection r, in other words if the parallel transport along every curve from p to q conjugates Qp M with Qq M . Hyper-Kähler manifolds are special quaternionic Kähler manifolds, their subalgebra bundle QM End TM is not only parallel, but trivial for the Levi-Civita connection. Hence we can choose a parallel isomorphism HM Š QM to obtain a smooth choice of an orthogonal quaternionic structure on every tangent space Tp M in the stronger standard sense. In order to discuss the decomposition of the bundle of differential forms on a hyper-Kähler or quaternionic Kähler manifold M into minimal parallel subbundles we fix once and for all a point p 2 M and decompose the vector space ƒ T ˝R C of complex-valued alternating forms on the tangent space T WD Tp M in p into irreducible subspaces under the holonomy group of parallel transports of curves beginning and ending in p, parallel transport along arbitrary curves will deliver this decomposition in p to all of M . The holonomy group in p is a group of orthogonal transformations normalizing the given orthogonal quaternionic structure Q End T of the euclidian vector space T . In terms of decomposition (2) the group of all such orthogonal transformations agrees with Sp H Sp E SO T , where Sp E Š Sp.n/ say is the group of all symplectic transformations of E commuting with C . Of course on a hyperKähler manifold the holonomy group actually centralizes the subalgebra Q End T and the holonomy group becomes a subgroup of Sp E. Choosing a unit vector q 2 H satisfying .C q; q/ D 1 we get a canonical basis p, q of H with p WD C q, the resulting algebra isomorphism Q Š H allows to replace Sp H by Sp.1/ in the holonomy group and the complex linear forms on T by E ˚ E via the isomorphism Š

E ˚ E ! T ˝R C;

˚ Q 7! .dq ˝ / C .dp ˝ /: Q

This isomorphism extends to an Sp E-equivariant decomposition of the alternating forms ˆq W

Š

ƒ E ˝ ƒ E ! ƒ .T ˝R C/ Š ƒ T ˝R C

(4)


19

on T (with the product grading on the left indicated by the repeated grading symbol) via ˆq .1 ^ ^ k ˝ Q 1 ^ ^ Q kN / ´ .dq ˝ 1 / ^ ^ .dq ˝ k / ^ .dp ˝ Q 1 / ^ ^ .dp ˝ Q kN /: Evidently the isomorphism ˆq is an isomorphism of algebras for the twisted multiplib ƒE . On the other hand the algebra cation on ƒE ˝ƒE sometimes denoted ƒE ˝ isomorphism Q Š H coming along with the choice of unit vector q is characterized by I q D i q and J q D p. The finer bigrading on ƒ E ˝ ƒB E thus agrees with the Hodge bigrading ƒ;B T ˝R C with respect to the complex structure I , in other N N N words ˆq maps ƒk E ˝ ƒk E to the space ƒk;k T ˝R C of .k; k/-forms on T . Interestingly the natural real structure on the algebra ƒ T ˝R C of complex valued alternating forms on T by conjugation of values x 1 ˝ e1 ; : : : ; hr ˝ er / WD .C h1 ˝ C e1 ; : : : ; C hr ˝ C er / .h on real arguments interchanges ƒ;B T ˝R C with ƒ;B T ˝R C. In order to describe its precise relation to ˆq we extend the quaternionic structures on E and H to quaternionic structures on E and H via .C /.e/ WD .C e/. The musical isomorphism ] W E ! E ; e 7! .e; /; and its inverse [ are then both real due to C.e ] / D .e; C / D .C e/] and the identity .C ˛ ˝ C /.h ˝ e/ D ˛.C h/.C e/ D .˛ ˝ /.C h ˝ C e/ shows that the real structure C ˝ C on H ˝ E coincides with the natural real N structure. In consequence Cdp D dq and Cdq D dp tell us that on ƒk E ˝ ƒk E we have as expected: N

Q D .1/k.kC1/ C Q ˝ C : ˆ1 q .ˆq . ˝ // Remark 2.1 (Real Structure on ƒ E ˝ ƒB E ). The bigraded algebra isomorphism ˆq W ƒ E ˝ ƒB E ! ƒ;B T ˝R C is real with respect to ƒ E ˝ ƒB E ! ƒB E ˝ ƒ E ;

Q ˝ Q 7! .1/jj.jjC1/ C Q ˝ C :

For the moment we want to quit discussion of ƒ E ˝ƒB E and study the simpler algebra ƒ E instead. Note first that for every pair fe g and fde g of dual bases for ] [ E and E respectively the pairs fC e g and fCde g as well as fde g and fe g are dual bases, too. Bilinear sums over pairs of dual bases are independent of the pair, so we can replace one of these pairs by another in such sums whenever convenient. P P ] [ For example the symplectic form D 12 de ^ e and [ D 12 de ^ e are real, in consequence the two operators 1X 1X ] [ de ^ e ^; [³ WD e ³ de ³ ^ WD 2 2

20

Gregor Weingart

on ƒ E of bidegree C2 and 2 respectively both commute with C . The calculation 1 X [ ] ] ] ^; e ³ g fde ^; e ³ ge ^ D e ] ^ .de / ^ fe Œ ^; e ³ D 2 and its analogue for Œ [ ³; ^ provide us with the fundamental commutation relations: Œ ^; ^ D 0;

Œ [ ³; ^ D [ ³;

Œ^; e ³ D e ] ^;

Œ [ ³; e ³ D 0:

(5)

On a symplectic vector space E of dimension 2n the commutator of ^ and [ ³ reads 1 X [ [ ³ C e ³ Œ^; de ³ Œ ^; e ³ de Œ^; [ ³ D 2 1 X de ^ e ³ C e ³ de ^ DW N n D 2 P where N WD de ^ e ³ is the so called (fermionic) number operator, which multiplies forms in ƒk E by k. In other words there is a canonical sl2 -triple of real operators on ƒ E H WD n N;

X WD [ ³;

Y WD ^

(6)

satisfying the classical commutation relations ŒH; X D 2X , ŒH; Y D 2Y , and ŒX; Y D H . Coming back to the description of the alternating forms we note that the factor Sp H of the quaternionic Kähler holonomy group Sp H Sp E does not act on the source ƒ E ˝ ƒB E of the isomorphism ˆq , although it acts on the target ƒ;B T ˝R C. Nevertheless there is a well-defined action of the group Sp.1/ H of unit quaternions on ƒ E ˝ ƒB E such that for all unit vectors q 2 H the isomorphism ˆq is actually equivariant over the isomorphism Š

'q id W Sp.1/ Sp E ! Sp H Sp E induced by the algebra isomorphism 'q W H ! Q. The characterization 'q .i /q D iq and 'q .j /q D C q of the isomorphism 'q implies that the isomorphism 'Aq associated to another unit vector Aq 2 H with A 2 Sp H Q is conjugated to 'q in the sense 'Aq WD A'q A1 . On the other hand the automorphisms A 2 GL T of T act on the differential forms by their inverse adjoint A . For the special isomorphisms A 2 Sp H this representation satisfies A ˆq . ˝ / Q D Âq . ˝ / Q because C.Aq/ D A.C q/ D Ap and fA dp; A dqg is the basis dual to Ap; Aq. In consequence the representation ? of Sp.1/ on ƒ E ˝ ƒB E defined to make ˆq equivariant ˆq . ˝ / Q (7) z ? . ˝ / Q WD ˆ1 q 'q .z/


21

over 'q id is actually independent of the choice of the unit vector q 2 H , because 1 1 Âq Q D ˆ1 A ˆq . ˝ / Q 'Aq .z/ Âq . ˝ / q A .A'q .z/A / and .A'q .z/A1 / D A 'q .z/ A . For the time being we will only make the infinitesimal representation associated to (7) explicit. Recall that the infinitesimal representation for the representation of GL T on alternating forms by inverse adjoints lets the endomorphism A 2 End T act by minus the derivation extension of the adjoint endomorphism A with .DerA /.X1 ; X2 ; : : : ; Xr / D .AX1 ; X2 ; : : : ; Xr / C C .X1 ; X2 ; : : : ; AXr / For the images I , J and K of the imaginary unit quaternions i; j; k 2 H under 'q we find the special values I dp D idp; I dq D idq; as well as J dp D dq; J dq D dp; and K dp D idq; K dq D idp; using (3) and conclude for the infinitesimal representation x i? WD ˆ1 q B . DerI / B ˆq D i N iN; S j ? WD ˆ1 q B . Der J / B ˆq D Pl C Pl;

(8)

S k? WD ˆ1 q B . DerK / B ˆq D i Pl i Pl

where the Pücker differential Pl say is defined by (as usual ys denotes an omitted factor): Pl.1 ^ ^k ˝ Q 1 ^ ^ Q kN / D

k X .1/ks 1 ^ ^ ys ^ ^k ˝s ^ Q 1 ^ ^ Q kN : sD1

S reverses the role of the two sides while keeping .1/ks , equivaThe definition of Pl lently, X X S WD .1/N B e ³ ˝ de ^ ; Pl de ^ ˝ e ³ : Pl WD .1/N B

Defnition 2.2 (Natural operators on ƒ E ˝ ƒB E ). Consider the tensor product ƒ E ˝ƒB E of two copies of the exterior algebra the alternating forms on a complex symplectic vector space E of dimension 2n with symplectic form and a pair of dual x of the two tensor factors bases fe g and fde g. Using the number operators N and N we can define ten natural, bigraded operators on ƒ E ˝ ƒB E , namely two copies x WD n N x; H WD n N; H X X 1 1 [ [ X WD e ³ de ³ ˝ id; Xx WD id ˝ e ³ de ³; 2 2 1X 1X ] ] de ^ e ^ ˝ id; Yx WD id ˝ de ^ e ^ Y WD 2 2

22

Gregor Weingart

of sl2 .C/ acting on the left and right tensor factor respectively, and four diagonal operators: X X ] e ³ ˝ de ^ ; ^ WD .1/N B de ^ ˝ e ^ ; Pl WD .1/N B

S WD .1/N B Pl

X

de ^ ˝ e ³ ; [ ³ WD .1/N B

X

[ de ³ ˝ e ³ :

In due course we are going about to prove that the subspace of endomorphisms of ƒ E ˝ ƒB E spanned by these ten natural operators is actually a Lie algebra isomorphic to so5 .C/ or equivalently sp4 .C/ with maximal torus spanned by H and x . At this point let us simply point out that H and H x are linear in the number operators H x x . In N and N so that every bigraded endomorphism is an eigenvector for ad H and ad H x terms of the basis f"; "Ng dual to the basis fH; H g of the maximal torus the eigenvalues N of the other operators and the weight spaces ƒk E ˝ ƒk E of the representation ƒ E ˝ ƒB E can be read off from the diagram below, where the dashed region 2n

@

q

@

@ q

N

ƒ k E ˝ ƒk E

@

q @ tYx

@

@q @t t ^ Pl @ t "@u g tY n X [ t @ S ? "N tPl ³ @ q t q @ Xx @ @ @ @ p p p p p @ p p pps q @ kN p p p p p pp @ p @

0

@

k

n

(9)

2n

indicates a preferred Weyl chamber and the octagon the support of the character of the irreducible representation of so5 .C/ of highest weight Lemma 2.3 (Action of so4;1 .R/˝R C on ƒ E ˝ƒB E ). The subspace of operators on ƒ E ˝ ƒB E spanned linearly over C by the ten operators of Definition 2.2 is closed under brackets and thus a complex Lie algebra isomorphic to so5 .C/. The real structure on ƒ E ˝ ƒB E induces the real structure on this Lie algebra so5 .C/ of operators indicated by the choice of notation, both ^ D ^ and [ ³ D [ ³ are imaginary. The real subalgebra fixed by this real structure is isomorphic to so4;1 .R/.


23

Proof. According to the calculation .6/ of the commutators of the operators ^ and x W Xx W Yx i span commuting [ ³ on ƒ E the two sl2 -triples hH W X W Y i and hH sl2 .C/-subalgebras of operators acting on different tensor factors. All operators in the direct sum sl2 .C/ ˚ sl2 .C/ preserve the parity of the bigrading and so commute with the operator .1/N , hence brackets with this subalgebra leaves the subspace S ^ and [ ³ invariant. More precisely the fundamental commutation spanned by Pl; Pl; relations (5) tell us that ŒH; Pl D Pl;

ŒX; Pl D 0;

ŒY; Pl D ^;

S D Pl; S ŒH; Pl

S D [ ³; ŒX; Pl

S D 0; ŒY; Pl

ŒH; ^ D ^; [

[

ŒH; ³ D ³;

ŒX; ^ D Pl; [

ŒX; ³ D 0;

ŒY; ^ D 0; S ŒY; [ ³ D Pl:

x ; Xx and Yx directly we Instead of verifying the missing commutation relations with H N x can infer these relations using the reality ŒA; B D ŒA; B of the commutator. For this reason we skip this point and proceed to determine the real structure induced by the real structure on ƒ E ˝ ƒB E made explicit in Remark 2.1. This real structure interchanges the tensor factors and hence interchanges the corresponding x , X $ Xx , Y $ Yx , say: endomorphisms H $ H Q X. ˝ / Q D .1/jj.jjC1/ C. [ ³ / Q ˝ C D Xx . ˝ /: Q

S ^ and [ ³ is that The main point of the argument concerning the operators Pl; Pl; these operators change the parity of both (sic!) factors, with this in mind we find for example X Q ] ^ . ˝ / Q D .1/jjjjC1 .Cde / ^ .C / Q ˝ .C e / ^ .C / D ^ . ˝ /: Q

S ^ and Last but not least we have to calculate the commutators of the operators Pl, Pl, [ ³. To begin with we note that these operators anticommute with .1/N , hence the .1/N -factor of square 1 conveniently turns into a minus sign in all the calculations: X S D ŒPl; Pl fe ³; de ^g ˝ de ^ e ³ de ^ e ³ ˝fde ^; e ³g

x C N ˝ id D H x H: D id ˝ N We omit the analogous calculation of the remaining commutators and tabulate the result: x H; S DH ŒPl; Pl x C H; Œ^; [ ³ D H

ŒPl; ^ D 2Yx ;

ŒPl; [ ³ D 2X;

S ^ D 2Y; ŒPl;

S [ ³ D 2Xx : ŒPl;

In order to prove that the algebra of operators is isomorphic to so5 .C/ we observe that the ten operators of Definition 2.2 all occur in at least one sl2 .C/-subalgebra

24

Gregor Weingart

of operators, more precisely the relevant sl2 .C/-subalgebras are spanned by the sl2 triples of operators: hH W X W Y i;

x W Xx W Yx i; hH

x W Pl W Pli; S hH H

x W [ ³ W î: hH C H

On the other hand the trace form of every finite-dimensional representation R of sl2 .C/ satisfies tr R .H 2 / D 2tr R .XY /, it is actually sufficient to verify this for the symmetric powers Symk C 2 of the defining representation C 2 of sl2 .C/. Consulting the weight diagram (9) for the eigenvalues of the diagonalizable operators ad H and x we find immediately ad H x; H x /; B.H; H / D 12 D B.H

x/ D 0 B.H; H

for the trace form B of the adjoint representation, the so called Killing form, and x; H C H x / D 24 D B.H H x; H H x /. The remaining non-zero conclude B.H C H values of B are B.X; Y / D 6 D B.Xx ; Yx / and

S D 12 D B. [ ³ ^/; B.Pl; Pl/

all other tuples of basis vectors are orthogonal/isotropic by equivariance alone. Having a non-degenerate Killing form the Lie algebra of operators on ƒ E ˝ƒB E spanned by the ten operators of Definition 2.2 is semisimple and the weight diagram (9) tells us that it is isomorphic to so5 .C/. An interesting twist to this argument allows us to determine the signature of the Killing form B restricted to the real subalgebra and thus its isomorphism type as well. Evidently the real structure preserves the orthogonal decomposition x g ˚ spanC fX; Xx ; Y; Yx g ˚ spanC fPl; Plg S ˚ spanC f ^; [ ³g spanC fH; H into subspaces of signature .1; 1/, .2; 2/, .0; 2/ and .1; 1/. Restricted to the real subalgebra the Killing form thus has signature .4; 6/ characterizing the real form so4;1 .R/ of so5 .C/. Remark 2.4 (Explicit isomorphism with so4;1 .R/). Consider the realization sor .C/ D fA 2 Mat rr C j AT S C SA D 0g of the complex Lie algebra sor .C/ associated to a non-degenerate, real, symmetric r r-matrix S. The signature of S determines the N given by isomorphism type of the standard real structure sor .C/ ! sor .C/; A 7! A; conjugation of coefficients. In particular the symmetric matrix of signature .4; 1/ 0 1 1 0 0 0 0 B0 1 0 0 0C B C C 0 0 1 0 0 S WD B B C @0 0 0 0 1A 0 0 0 1 0

25


defines a real structure with real form so4;1 .R/ on so5 .C/. Sending the operators H , X, Y to 0 1 0 1 0 1 0 i 0 0 0 0 0 0 0 1 0 0 0 1 0 Bi 0 0 0 0 C B0 0 0 0 i C B0 0 0 i 0C B C B C B C B 0 0 0 0 0 C ; p1 B0 0 0 0 0 C ; p1 B 0 0 0 0 0C B C B C B C; 2 @1 i 0 0 0 A 2@0 @0 0 0 1 0A 0 0 0 0A 0 0 0 0 1 0 0 0 0 0 1 i 0 0 0 respectively, and the operators Pl; ^ and [ ³ similarly to the matrices 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 i 0 0 B0 0 0 0 0C B0 0 0 B0 C 0 1 0 0 C p B B C p B C Bi 1 0 0 0C ; 2B 2B B0 0 0 i 0C ; B0 0 0 B C @0 0 0 0 0A @0 0 i @0 0 0 0 0A 0 0 i 0 0 0 0 0 0 0 0 0 0

1 0 0 0 0C C 0 i C C; 0 0A 0 0

defines a real isomorphism between the algebra of operators on ƒ E ˝ ƒB E and so5 .C/. There are other ways to construct the algebra so4;1 .R/ besides Definition 2.2 closer in spirit to the standard procedure in Kähler geometry. Research on this idea and the resulting algebra goes back at least to the extensive work of Bonan [1], [4], who introduced the algebra so4;1 .R/ by considering all ways to wedge and contract with Kähler forms. An orthogonal quaternionic structure Q End T gives rise to three linearly independent Kähler forms !I D g.I ; / etc. associated to the images I , J , K of the unit quaternions i; j; k 2 H under an isomorphism H ! Q. Every such isomorphism is induced by the choice of a unit vector q 2 H satisfying .C q; q/ D 1 and is in this way related to a canonical basis p; q of H with p WD C q. In terms of this basis the scalar product on T ˝R C Š H ˝ E can be written in the form X ] ] / .dq ˝ de / ˝ .dp ˝ e / : .dp ˝ de / ˝ .dq ˝ e g D ˝ D

The equivalent formulation !I D .I ˝ id/.g/ of the definition of the Kähler form together with the explicit values I dp D idp; I dq D idq as well as J dp D dq; J dq D dp and K dp D idq; K dq D idp allows us to expand the three Kähler forms according to X ] .dp ˝ de / ^ .dq ˝ e /; !I D i

1 X ] ] / C .dq ˝ de / ^ .dq ˝ e / ; .dp ˝ de / ^ .dp ˝ e !J D 2 i X ] ] / .dq ˝ de / ^ .dq ˝ e / : .dp ˝ de / ^ .dp ˝ e !K D 2

26

Gregor Weingart

Some additional considerations based on these expansions lead to the description ˆ1 q B .!I ^/ B ˆq D i ^;

[ [ ˆ1 q B .!I ³/ B ˆq D i ³;

x ˆ1 q B .!J ^/ B ˆq D Y C Y;

[ x ˆ1 q B .!J ³/ B ˆq D X C X ;

1 [ x x ˆ1 q B .!J ^/ B ˆq D i Y iY; ˆq B .!K ³/ B ˆq D iX i X

of wedge products and contractions with Kähler forms in terms of the Lie algebra so4;1 .R/, conversely these six operators evidently suffice to generate all so4;1 .R/. Recall now that the isomorphism ˆq conjugates the representation of Sp H SO T on the alternating forms ƒ;B T ˝R C on the euclidian vector space T to a representation of the group Sp.1/ on ƒ E ˝ ƒB E , whose infinitesimal generators we have calculated in equation (8): x i? D ˆ1 q B . DerI / B ˆq D iH i H ; S j ? D ˆ1 q B . Der J / B ˆq D Pl C Pl; S k? D ˆ1 q B . DerK / B ˆq D i Pl i Pl: Somewhat more general the infinitesimal representation of the Lie algebra so4;1 .R/ of operators on ƒ E ˝ ƒB E integrates to an actual representation of the unique connected and simply connected Lie group SpinC 4;1 .R/ with Lie algebra so4;1 .R/. The weight diagram (9) tells us that the central element 1 2 SpinC 4;1 .R/ acts as N

N

N

.1/kCk on ƒk;k T ˝R C Š ƒk E ˝ ƒk E , hence this representation does not ev descend to SOC 4;1 .R/. Nevertheless the invariant subspace ƒ T ˝R C of forms of C even total degree is a genuine representation of SO4;1 .R/. Besides the subgroup Sp.1/ SpinC 4;1 .R/ conjugated to the factor Sp H SO T of the holonomy group we want to mention two other interesting subgroups C of SpinC 4;1 .R/, namely the double cover SL2 .C/ Š Spin3;1 .R/ of the Lorentz group and the maximal compact subgroup Spin4 .R/. Interestingly there seems to be no natural choice for the latter, and it is tempting to try to relate this oddity to geometrical and topological properties of hyper-Kähler or quaternionic Kähler manifolds. Concerning the former we note that the involutive automorphism of so5 .C/ given by conjugation with .1/N fixes the subalgebra sl2 .C/˚sl2 .C/ so5 .C/ spanned by the operators x , X, x Yx . The resulting symmetric pair of real Lie algebras H , X, Y and H so4;1 .R/ D so3;1 .R/ ˚ R3;1 so3;1 .R/

defines a Lorentz symmetric space, the de Sitter space SpinC 4;1 .R/= SL2 .C/. Incidentally we note that the inclusion of the factor .1/N in the definition of the operators S ^ and [³ has the same effect on the symmetric pair so4;1 .R/ so3;1 .R/ as Pl; Pl; the multiplication by i in the eigenspace of eigenvalue 1 in the classical construction of the dual symmetric pair.

27


Interestingly the construction of the algebra so4;1 .R/ acting on ƒ E ˝ ƒB E is a special case of a general property of representation theory related to Howe’s Theory of dual pairs. On the tensor product of r copies of the exterior or symmetric algebra of a complex vector space V lives a natural simple Lie algebra of operators isomorphic to glr .C/, consisting of the r.r 1/ possible generalizations of the Plücker x . If either the S and r shifted number operators like H or H differentials Pl and Pl factors are exterior algebras and the vector space V is symplectic or the factors are symmetric algebras and V is euclidian, then we can extend this algebra by r.r C 1/ additional operators like X or ^ and the resulting algebra is isomorphic to sp2r .C/ D sp.r/ ˝R C. The remarkable thing about this construction is that Weyl’s construction of the irreducible representations of the classical matrix groups using Schur functors [2] tells us that under rather general assumptions on r and dim V the multiplicity spaces for the decomposition of the tensor product into irreducible representations of glr .C/ or sp2r .C/ are irreducible representations of the automorphism group of V ! The algebra so4;1 .R/ ˝R C Š sp4 .C/ acting on ƒ E ˝ ƒB E is a special example of this construction. The space of highest weight vectors of so4;1 .R/ in N " C .n k/" in the Weyl chamber ƒ E ˝ ƒB E given a dominant weight .n k/N indicated in diagram (9) is by definition the intersection of the kernels of the four S corresponding to the positive roots with the weight space operators X, [³, Xx and Pl kN k ƒ E ˝ƒ E . Denoting the irreducible representation of so4;1 .R/ of highest weight so .R/ we can rewrite this definition in the following way by R 4;1 Hom so4;1 .R/ .R

so4;1 .R/

N "C.nk/" .nk/ N

N

S \ .ƒk E ˝ ƒk E / ; ƒE ˝ ƒE / D ker X \ ker Pl

S corresponding to the simple roots generate the subalgebecause the operators X and Pl bra of operators corresponding to positive roots. According to Weyl’s construction of the irreducible representations of the symplectic Lie groups as trace-free Schur functors kN k;kN however the right hand side is exactly the irreducible representation ƒk; B E Š ƒB E C C "k , where ˙"1 ; : : : ; ˙"n are of Sp E of highest weight 2"1 C C 2"kN C "kC1 N the weights of the defining representation E of Sp E. In consequence the complete decomposition of the tensor product ƒE ˝ ƒE of two copies of the exterior algebra of a symplectic vector space E into irreducible representations reads for either member of the Howe dual pair Sp E and so4;1 .R/: M so .R/ rN ƒ;B T ˝R C D ƒ E ˝ ƒB E D R.n4;1 ˝ ƒr; (10) B E: r/ N "C.nr/" N nrr0 N

A detailed discussion of Weyl’s construction is outside the scope of this exposition, the reader is referred to [2] instead, nevertheless we want to give a very brief sketch S ƒk E ˝ ƒkN E as a of Weyl’s argument for the irreducibility of ker X \ ker Pl representation of Sp E. The most difficult aspect of Weyl’s argument concerns proving the surjectivity of the restriction N

N

E ˝ ƒk1 E [ ³ W ƒkB E ˝ ƒkB E ! ƒk1 B B

28

Gregor Weingart N

N

of the operator [ ³ to the joint kernel ƒkB E ˝ ƒkB E ƒk E ˝ ƒk E of the two commuting operators X and Xx in bidegrees n k; kN 0. Because of surjectivity we can calculate the dimension of the joint kernel ƒ B E ˝B ƒB B E of the three N operators X, Xx and [ ³ from the known dimension of ƒkB E ˝ ƒkB E . In turn the surjectivity of the restriction N S W ƒkB E ˝B ƒkBN E ! ƒkC1 E ˝B ƒk1 E Pl B B

S to the joint kernel of X , Xx and [ ³ for bidegrees k kN is a direct of the operator Pl consequence of representation theory of sl2 so that we eventually end up calculating kN x [ S the dimension of the joint kernel ƒk; B E of all four operators X, X, ³ and Pl in bidegrees n k kN 0. Choosing on the other hand a complex basis 1 ; : : : ; n 2 L of a maximal isotropic or Lagrangian subspace L E we note that for all n k kN 0 the form kN 1 ^ ^ k ˝ 1 ^ ^ kN 2 ƒk; B E N

k is a highest weight vector in ƒk; C C "k for a N B E of weight 2"1 C C 2"kN C "kC1 suitable choice of a maximal torus of Sp E and a suitable ordering of weights. In conkN x [ S sequence the joint kernel ƒk; B E of X , X , ³ and Pl contains at least the irreducible representation of Sp E of this specific highest weight, by the dimension part of Weyl’s Character Formula however the dimension of this irreducible summand agrees with kN the dimension of ƒk; B E we have calculated before proving the irreducibility of the latter under Sp E. In concluding this section we want to formulate some straightforward consequences of the decomposition (10) into irreducible representations. Say the maximal torus x C H and i H x iH , which take the values R ˚ iR of so4;1 .R/ is generated by H N N kN C k and i kN ik on the weight space ƒk E ˝ ƒk E Š ƒk;k T ˝R C. Branching N from so4;1 .R/ to R ˚ i R thus calculates the decomposition of ƒk;k T ˝R C under the holonomy group Sp.n/ of hyper-Kähler manifolds:

Lemma 2.5 (Differential forms on hyper-Kähler manifolds). Under the action of N N Sp E SO T the space ƒk;k T ˝R C of complex .k; k/-forms with respect to the rN complex structure I decomposes into a sum of representations ƒr; B E with multiplicities M N so .R/ rN ƒk;k T ˝R C Š dim Hom R˚i R RR˚iNR ; R.n4;1 ƒr; B E r/ N "C.nr/" N nrr0 N

.nk/"C.nk/" N

N "C given by the multiplicity of the irreducible representation of highest weight .n k/N .n k/" for R ˚ iR in the irreducible representation of so4;1 .R/ of highest weight .n r/N N " C .n r/". On quaternionic Kähler manifolds we are interested instead in decomposing ƒ T ˝R C under the holonomy group Sp H Sp E or equivalently ƒ E ˝ƒB E un


29

der Sp.1/ Sp E using the isomorphism ˆq , where the factor Sp.1/ SpinC 4;1 .R/ has x , Pl C Pl S and i Pl i Pl, S compare Lie algebra sp.1/ so4;1 .R/ generated by iH i H equation (8). It is convenient to consider the central extension RSp.1/ SpinC 4;1 .R/ x of Sp.1/ with Lie algebra generated by sp.1/ and H C H in order to keep track of x the total degree of a differential form. To wit e t .H CH / 2 R SpinC 4;1 .R/ acts by t.2nk/ multiplication with e on forms ƒE ˝ ƒE of total degree k similar to its action on the irreducible representation C.n k /."C"/ ˝ Syms H of R Sp.1/ of highest N weight .n k2 /.N" C "/ C 2s .N" "/ D .n

2

ks /N" 2

C .n

kCs /": 2

Lemma 2.6 (Differential forms on quaternionic Kähler manifolds). Under the action of Sp H Sp E SO T the space of complex valued differential forms on a euclidian vector space T of dimension 4n with orthogonal quaternionic structure Q End T rN decomposes into a sum of irreducible representations Syms H ˝ ƒr; B E with multiplicities ƒk T ˝R C M dim Hom R˚sp.1/ RR˚sp.1/ Š ks s0 nrr0 N

.n

2

kCs /"C.n N 2 /"

so

.R/

rN ; R.n4;1 Syms H ˝ ƒr; B E r/ N "C.nr/" N

given by the multiplicity of the irreducible representation of R ˚ sp.1/ of highest weight .n ks /N" C .n kCs /" in the irreducible representation of so4;1 .R/ for 2 2 .n r/N N " C .n r/".

3 Quaternionic Kähler decomposition of forms Of course neither Lemma 2.5 nor Lemma 2.6 are really explicit decomposition formulas, but they reduce the problem of decomposing the differential forms on a hyperKähler manifold or quaternionic Kähler manifold of quaternionic dimension n under the holonomy group Sp.n/ or Sp.1/ Sp.n/ respectively to finding the branching rules from so4;1 .R/ to R˚iR or R˚sp.1/. In this section we will describe a rather general strategy to solve this standard problem in representation theory for a pair g h of real reductive Lie algebras. Using this strategy we will readily turn Lemma 2.5 and Lemma 2.6 into effective decomposition formulas. Consider for a moment the Lie algebra g of a compact group G and for each X 2 g the adjoint endomorphism ad X W g ! g, Y 7! ŒX; Y . With G being compact there exists a G-invariant negative definite scalar product B on g, for example we can take the Killing form B.X; Y / WD tr g .ad X B ad Y / if G is semisimple. On the Zariski open subset of regular elements X 2 g the rank of ad X becomes maximal, in turn its kernel becomes minimal and defines a maximal abelian subalgebra of g also known as a maximal torus t D fH 2 g j ŒX; H D 0g. With g being the Lie algebra of a compact group G the kernel of the exponential map exp W g ! G, X 7! e X , restricted

30

Gregor Weingart

to t is a lattice in t, whose dual lattice ƒ WD f 2 it j .X / 2 2 i Z for all X 2 t with e X D 1g i t is called the weight lattice of G. In general the weight lattice ƒ becomes finer if we Q in other words the weight lattice encodes replace G by a compact covering group G, the global structure of the Lie group G and does not only depend on its Lie algebra g. A very useful tool for the calculations to come is the group ring Zƒ of the weight lattice ƒ considered as an additive group with coefficients in Z. Elements of the group ring Zƒ are appropriately thought of as finite formal sums of terms ce with c 2 Z and 2 ƒ, because the naive multiplication of such formal sums agrees with the standard convolution product in the group ring Zƒ. In particular the character of every finite-dimensional representation V of the compact group G finds a natural home in the ring Zƒ X .dim E /e ch V WD 2ƒ

where E is the generalized complex eigenspace of the action of the maximal torus on V : E WD fv 2 V ˝R C j H ? v D .H /vg The character ch defined this way is a ring homomorphism from the representation ring RG of the group G to the group ring Zƒ of the weight lattice, in other words the equalities ch V ˚ W D ch V C ch W and ch V ˝ W D .ch V /.ch W / hold true. For connected G the character ch is injective, moreover its image is precisely the subring ŒZƒW of elements invariant under the Weyl group W D NormT =T of G acting on Zƒ by the linear extension of .w; e / 7! e w with .w/.X / WD .Ad1 w X /. Incidentally we note that the automorphism W Zƒ ! Zƒ, e 7! e ; is useful to “dualize” the character ch.V / D .ch V / of a representation V . For group rings like Zƒ the analogue of the residue of complex analysis ev W Zƒ ! Z;

e 7! ıD

can be introduced to pick up the coefficient of the weight 2 ƒ in a given element of Zƒ. The weights (in the support of the character) of the adjoint representation g of G are of special importance and are called roots. Choosing once and for all a regular element X 2 t in a fixed maximal torus t we can classify a root ˛ according to the value of i˛.X/ 2 R n f0g as positive or negative. In turn an imaginary valued linear form 2 it on t is called dominant, if it has non-negative scalar product B.; ˛/ 0 with every positive root ˛. The basic example of a dominant weight is the half sum of positive roots: X 1 ˛ 2 ƒ: WD 2 ˛positive root


31

For every connected compact Lie group G the dominant weights are in bijective correspondence to the isomorphism classes of irreducible representations, up to isomorphism there is thus a unique representation R of G of “highest” dominant weight 2 ƒ. The character of this representation is determined by a very useful formula called Weyl’s Character Formula X A ch R D .1/jwj e w.C/ (11) w2W

where the denominator A can be defined in two ways due to Weyl and Kostant respectively: Y X ˛ ˛ .1/jwj e w D e 2 e 2 (12) A´ ˛positive root

w2W

A direct consequence of Weyl’s Character Formula is the following formula for the multiplicity m .R/ WD dim Hom G .R ; R/ D evC ŒA ch R of the irreducible representation R in an arbitrary finite-dimensional representation R. Coming back to our branching problem we consider a pair g h of Lie algebras associated to a pair G H compact Lie groups. In this case we can find a regular element X 2 h for g in h and a fortiori get a pair t s of maximal tori for h and g respectively. By construction the restriction map i s ! i t preserves positivity of linear forms in the sense i .X / > 0 and maps the weight lattice ƒg i s to the weight lattice ƒh of h. In consequence the restriction map induces a ring homomorphism res W Zƒg ! Zƒh , which maps the characters of representations of G to the characters of their restriction to H . For example the adjoint representations g and h of G and H both induce characters in Zƒh satisfying X res.ch g/ ch h D ch.g=h/ D .dim s dim t/ C e ˛ C e ˛ ˛positive weight ofg=h

where the right hand side is to be understood as a summation with multiplicities. Using Kostant’s formula (12) for Weyl’s denominator and the restriction homomorphism we find Y Y ˛ res Ag ˛ e h res g D e h res g e 2 e 2 D 1 e ˛ Ah ˛positive ˛positive weight ofg=h

weight ofg=h

and this expression can be inverted in a suitable completion of the group ring Zƒh to define Y Ah D (13) 1 C e ˛ C e 2˛ C e 3˛ C Bg=h WD e res g h res Ag ˛positive weight ofg=h

the universal branching formula for the pair g h. The standard branching problem h to find the multiplicity of R in Rg is solved in terms of this universal branching

32

Gregor Weingart

formula via h dim Hom H .R ; Rg / D evCh Ah ch Rg h i Ah X D evCres g e res g h .1/jwj e res w.Cg / res Ag D

X

w2Wg

.1/

jwj

evCres g res w.Cg / e

w2Wg

res g h

Ah : res Ag

For later reference we will write this branching formula in the following way: X h ; Rg / D .1/jwj evres.w.Cg /g / Bg=h : dim Hom H .R

(14)

w2Wg

According to the results of Section 2 we need to solve the branching problems for the pairs so4;1 .R/ R ˚ i R and so4;1 .R/ R ˚ sp.1/ respectively in order to turn Theorem 2.5 and Theorem 2.6 into effective decomposition formulas. Of course none of the algebras involved is the Lie algebra of a compact group, however the compact pairs Sp.2/ S 1 S 1 and Sp.2/ S 1 Sp.1/ have on the level of Lie algebras isomorphic complexifications sp.2/ ˝R C .i R ˚ i R/ ˝R C

sp.2/ ˝R C .i R ˚ sp.1// ˝R C;

and

and this is sufficient to ensure that we can apply the branching formula (14) for the problems at hand as well. The maximal torus S 1 S 1 S 1 Sp.1/ Sp.2/ is evidently the same for all groups and so the restriction is simply the identity. The root d d d negative PP P 2" t " PtfPP d PP PP t ? "N t

PP P

positive

"N C "

t

"N "

2N"

diagram (9) of so4;1 .R/ tells us that the positive roots of the pair so4;1 .R/ R˚sp.1/ S 2 sp.1/ ˝R C. Hence are 2N", "N C " and 2", the positive root "N " corresponds to Pl the branching formula reads as follows: Bso4;1 .R/=R˚sp.1/ N N D 1 C e "C" C e 2"C2" C 1 C e 2" C e 4" C 1 C e 2"N C e 4"N C X N and r k kN mod 2ge k"CkN "N : ]fr j 0 r minfk; kg D N k;k0


33

Evaluation of this power series in e " ; e "N in the weight kN "N C k" defines the function Bso4;1 .R/=R˚sp.1/ Bk;k N ´ evkN "Ck" N ´ N 1 C b min¹2k;kº c if k; kN 2 N0 and k kN mod 2; D 0 for all other arguments. Consider now a tuple .r; r/ N of integers satisfying 0 rN r n with associated dominant weight WD .n r/N N " C .n r/" in the Weyl chamber indicated in diagram (9). For this choice of Weyl chamber the half sum of positive roots is so4;1 .R/ D 2N" C" and the affine Weyl orbit of under the Weyl group Wso4;1 .R/ D S2 ËZ22 of SO4;1 .R/ is readily calculated: .1/jwj

w

w. C so4;1 .R/ / so4;1 .R/

C1

id

.Cn r/N N " C .Cn r/"

1

1

.Cn r 1/N" C .Cn rN C 1/"

1

2

.Cn r/N N " C .n C r 2/"

C1

1 2

.n C r 3/N" C .Cn rN C 1/"

C1

2 1

.Cn r 1/N" C .n C rN 3/"

1

2 1 2

.n C r 3/N" C .n C rN 3/"

1

1 2 1

.n C rN 4/N" C .Cn r/"

C1

1 2 1 2

.n C rN 4/N" C .n C r 2/"

Under the rather superficial assumption 0 k 2n on the degree of the differential /N" C.n kCs /" of the corresponding forms considered the highest weight D .n ks 2 2 s representation C.n k /."C"/ ˝ Sym H of R ˚ sp.1/ has " N -coefficient and coefficient 2 N sum positive or zero. For this reason the five Weyl group elements w, which have "N-coefficient or coefficient sum of w. C so4;1 .R/ / so4;1 .R/ negative, can not contribute to the summation in the branching formula (14) and we are left with the three summands corresponding to id and the reflections 1 W "N 7! ", " 7! "N and

2 W "N 7! "N, " 7! " along the simple roots:

so

dim Hom R˚sp.1/ RR˚sp.1/ ks .n

2

kCs /"C.n N 2 /"

.R/

; R.n4;1 r/ N "C.nr/" N

B ks r;N kCs Cr2n2 : D B ks r;N kCs r B ks r1; kCs rC1 N 2

2

2

2

2

(15)

2

The other branching problem from so4;1 .R/ to R ˚ i R can be treated similarly, the main difference is the positive root "N ". The evaluation of the universal branching formula N N Bso4;1 .R/=R˚i R D 1 C e "" C e 2"2" C Bso4;1 .R/=R˚sp.1/

34

Gregor Weingart

at a weight kN "N C k" however defines a significantly more involved function 8 N 2 ˆ b .kC2/ c if k kN 0 and k kN mod 2; ˆ 4 ˆ ˆ 2 ˆ N N N 0: 4

kCs 2

r, depending on whether Second case. The second case is characterized by ks 2 the common parity of k s r rN mod 2 is even or odd the multiplicities in this

35


case are given by ms;r;rN .k/ D

minfr r; N sg minfr r; N sg C 1 C ı ks r.2/ or ms;r;rN .k/ D 2 2 2

respectively, where the ı-summand equals 1 if ks r mod 2 and 0 otherwise. These 2 multiplicities are strictly positive unless k is even, s D 0 or r D rN and ks 6 r 2 mod 2. Third case. Characterized by kCs 2nr s C rN the third case has strictly positive 2 multiplicities: ms;r;rN .k/ D 1 C n r C

minfr r; N sg s > 0: 2

The theorem is a good example of how a seemingly simple sum (15) of three terms may lead to a messy case distinction. Although it is natural to believe that there is a different, simpler way to present the result, no such simplification has been found by the author and the hexagonal pattern (1) of the multiplicities ms;r;rN .k/ suggests that no simpler presentation exists. The reader interested in the details of the proof is invited to study the diagram first which gives a geometric interpretation of the inequalities in Theorem 3.1. The black points represent the images of N " C .n k/" under the Weyl group elements id; 1 the highest weight D .n k/N n

@ aa @ a a @ aaaaa aa a aa s Dp 2nr rN a a @ pp aa a Second @ aa aa pp a a p ppppppppppppp a t a r C1 pp pp

@ pp pp 1 p @ pp a a @ pp aa@ pp ks pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp d aaa@ p p 2 pp pp aaaa@ pp pp aaa aThird p p First a p p p pppppppppppppppppppppppppppppppppppppppppppppppp p a@ a t b t rN pp

pp pp pp @ pp 2 pp pp pp id @ pp pp pp pp

0

rN 1

r

n

kCs 2

2nr C2

(17)

@

2n

and 2 contributing to (15). The support of the corresponding B-summand considered as functions of . kCs ; ks / is a translate of the first quadrant based at this point. The 2 2 case distinction thus arises from sectors, where some parts of formula (15) vanish. Proof of Theorem 3.1. To begin with we note that the arguments of the B-summands in formula (15) are integers of the same parity if and only f k s r C rN modulo 2, hence this condition is certainly necessary to have a positive multiplicity. Fixing a ; ks / lies in degree 0 k 2n and s 0 we note that the corresponding point . kCs 2 2

36

Gregor Weingart

the big triangle of diagram (17) or below, hence this point lies in the support of some r. N The stronger r and ks of the summands of formula (15) if and only if kCs 2 2 kCs assumptions of the first case r 2 < 2n r C 2 and rN ks r correspond 2 exactly to the rectangle denoted “First” in diagram (17). In this rectangle only the summand corresponding to id contributes to (15) and we obtain

B ks r;N kCs r 2

2

minfk s 2r; N k C s 2rg D1C 4

k s 2r C 2 minfr r; N sg D1C : 4

> r Under our standing assumptions 0 k 2n and s 0 the assumption ks 2 characterizing the second case corresponds to the triangle denoted “Second” in diagram (17), which lies in the support of the two summands corresponding to id and 1 in formula (15): B ks r;N kCs r B ks r1; kCs rC1 N 2

2

2

2

k s 2r 2 2ı ks k s 2r C 2 minfr r; N sg 2 r.2/ 4 4 minfr r; N sg C 1 C ı ks r.2/ 2 D : 2

D

Evidently the ı-term will have no bearing on the result in case s r rN is odd, for even s r rN on the other hand it will take us to the next even integer or not. Changing the assumptions of the remaining third case slightly we observe that the 2n r C 2 characterize points in the triangle conditions 0 k 2n and kCs 2 denoted “Third” or below. In this region the two non-trivial summands in formula (15) will cancel each other in points below the diagonal s D 2n r r. N According ks ; / on or above to the branching formula (15) the multiplicities in the points . kCs 2 2 this diagonal s 2n r rN can be calculated as B ks r;N kCs r B ks r;N kCs 2nCr2 2

2

2

2

k C s 4n C 2r 4 2ı kCs k s 2r C 2 minfr r; N sg 2 6r.2/ D 4 4 minfr r; N sg s C ı kCs 6r.2/ 2 DnC1r C 2

However s r rN modulo 2 by our standing assumption, hence minfr r; N sg s is even and we can safely trade the ı-summand for skipping to round down. Of course the stated positivity of the multiplicities in the different cases of Theorem 3.1 is of particular importance in applications, because in this way it is possibly rN to locate exactly those degrees, in which a representation Syms H ˝ ƒr; B E definitely


37

does occur in the differential forms. In particular the exception k even, s D 0 or 6 r mod 2 in the second case is relevant for the Betti numbers of r D rN and ks 2 quaternionic Kähler manifolds, in that it allows the Betti numbers to increase in steps of 4 only instead of the expected 2. More precisely it has been shown in [3] that on compact quaternionic Kähler manifolds of positive scalar curvature > 0 every harmonic form is a sum of harmonic forms of types ƒr;r B E with n r 0, while on a compact quaternionic Kähler manifold with negative scalar curvature < 0 the basic harmonic forms can be 2nrrN rN H ˝ ƒr; of the two different types ƒr;r B E; n r 0; and Sym B E with n r rN 0. In this context Theorem 3.1 confirms the central conclusion of [3] about the degrees and the multiplicities of general differential forms and thus about the 2nrrN rN H ˝ƒr; degrees of harmonic forms of type ƒr;r B E and Sym B E in the differential r;r forms. The representation ƒB E occurs with multiplicity 1 in the forms of degrees k D rN 2r; 2r C4; 2r C8; : : : ; 4n2r, while the representation Sym2nrrN H ˝ƒr; B E occurs with multiplicity 1 in the forms of degrees k D 2nr Cr; N 2nr CrN C2; : : : ; 2nCr r. N Acknowledgement. This work was supported by PAPIIT (UNAM) through research project IN115408 Geometría Riemanniana Global.

References [1] E. Bonan, Décomposition de l’algèbre extérieure d’une variété hyperkählerienne. C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 4, 457–462. 25 [2] W. Fulton and J. Harris, Representation theory: a first course. Grad. Texts in Math. 131, Readings in Math., Springer-Verlag, New York 1991. 27, 34 [3] U. Semmelmann and G. Weingart, Vanishing theorems for quaternionic Kähler manifolds. J. Funct. Anal. 173 (2000), no. 1, 214–255. 16, 37 [4] M. Verbitsky, Cohomology of compact hyper-Kähler manifolds and its applications. Geom. Funct. Anal. 6 (1996), no. 4, 601–611. 25 [5] G. Weingart, Rozansky–Witten invariants for quaternionic Kähler manifolds. In preparation. 17

Chapter 3

Sasakian geometry, holonomy, and supersymmetry Charles P. Boyer and Krzysztof Galicki

Contents 1 2 3 4 5 6 7 8

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cones, holonomy, and Sasakian geometry . . . . . . . . . . . . Sasakian and Kählerian geometry . . . . . . . . . . . . . . . . Sasaki–Einstein and 3-Sasakian geometry . . . . . . . . . . . . Toric Sasaki–Einstein 5-manifolds . . . . . . . . . . . . . . . . The Dirac operator and Killing spinors . . . . . . . . . . . . . Real Killing spinors, holonomy and Bär’s correspondence . . . Geometries associated with 3-Sasakian 7-manifolds . . . . . . 8.1 Nearly parallel G2 -structures and Spin.7/ holonomy cones 8.2 Nearly Kähler 6-manifolds and G2 holonomy cones . . . . 9 Geometries associated with Sasaki–Einstein 5-manifolds . . . . 10 Geometric structures on manifolds and supersymmetry . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

39 41 44 49 51 54 58 61 62 66 70 73 77

1 Introduction Supersymmetry has emerged in physics as an attempt to unify the way physical theories deal with bosonic and fermionic particles. Since its birth around the early ’70s it has come to dominate theoretical high energy physics (for a historical perspective see [86] with the introduction by Kane and Shifman, and for a mathematical treatment see [132]). This dominance is still ongoing in spite of the fact that after almost 40 years there is no single experimental evidence that would directly and convincingly “prove” or “discover” the existence of supersymmetry in nature. On the other hand, especially in the last 20 years, supersymmetry has given birth to many beautiful mathematical theories. Gromov–Witten theory, Seiberg–Witten theory, Rozansky–Witten theory as well as the Mirror Duality Conjecture are just a few of the more famous examples of important and deep mathematics having its origins in the physics of various supersymmetric theories. Various supersymmetric field theories naturally include both Riemannian and pseudo-Riemannian manifolds. The latter is necessary in order to incorporate the physical space-time into the picture while the former typically describes the geometry associated with ‘invisible’ extra dimensions. It is mainly in such a context that Sasaki–

40

Charles P. Boyer and Krzysztof Galicki

Einstein manifolds appear in physics: they are compact Einstein manifolds of positive scalar curvature that occur in abundance in the physically interesting dimensions five and seven. Moreover, when they are simply connected they admit real Killing spinors. It is this last property that vitally connects them to supergravity, superstring, and M theory. The main purpose of this chapter is to describe geometric properties of Sasaki– Einstein manifolds which make them interesting in modern theoretical physics. In spite of the fact that it is supersymmetry that connects Sasaki–Einstein spaces to physics, it is not the purpose of this review to describe what this concept really means to either physicists or mathematicians. There have been many recent attempts to frame these important notions of theoretical physics in precise mathematical terms. This enormous task is far beyond the scope of this chapter, so we refer the reader to recent monographs and references therein [50], [132], [81], [2]. Here we content ourselves with providing the main theorems and results concerning Killing spinors. It is most remarkable that, even though Sasaki–Einstein manifolds always have holonomy SO.TM /, i.e., the holonomy of any generic Riemannian metric, they are far from being generic. In fact, the most interesting thing about this geometry is that it naturally relates to several different Riemannian geometries with reduced holonomies. It is this point that we will try to stress throughout this chapter. For more detailed exposition we refer the interested reader to our recent monograph on Sasakian geometry [26]. The key to understanding the importance of Sasakian geometry is through its relation to Kählerian geometry. Before we define Sasakian manifolds and describe some of their elementary properties in Section 3 let us motivate things in the more familiar context of contact and symplectic manifolds. These two provide the mathematical foundations of Lagrangian and Hamiltonian Mechanics. Let .M; ; / be a contact manifold where is a contact form on M and is its Reeb vector field. It is easy to see that the cone .C.M / D RC M , ! D d.t // is symplectic. Likewise, the Reeb field defines a foliation of M and the transverse space Z is also symplectic. When the foliation is regular the transverse space is a smooth symplectic manifold giving a projection called Boothby–Wang fibration, and D d relates the contact and the symplectic structures as indicated by .C.M /; !/ o

? _ .M; ; /

.Z; /

We do not have any Riemannian structure yet. It is quite reasonable to ask if there is a Riemannian metric g on M which “best fits” into the above diagram. As the preferred metrics adapted to symplectic forms are Kähler metrics one could ask for

Chapter 3. Sasakian geometry, holonomy, and supersymmetry

41

the Riemannian structure which would make the cone with the metric gN D dt 2 C t 2 g together with the symplectic form ! into a Kähler manifold. Then gN and ! N Alternatively, one could ask for a Riemannian metric define a complex structure ˆ. g on M which would define a Kähler metric h on Z via a Riemannian submersion. Surprisingly, in both cases the answer to these questions leads naturally and uniquely to Sasakian geometry. Our diagram becomes N o .C.M /; !; g; N ˆ/

? _ .M; ; ; g; ˆ/

.Z; ; h; J /

From this point of view it is quite clear that Kählerian and Sasakian geometries are inseparable, Sasakian geometry being naturally sandwiched between two different types of Kählerian geometry.

2 Cones, holonomy, and Sasakian geometry As we have just described Sasakian manifolds can and will be (cf. Theorem–Definition 10) defined as bases of metric cones which are Kähler. Let us begin with the following more general Definition 1. For any Riemannian metric gM on M , the warped product metric on C.M / D RC M is the Riemannian metric defined by g D dr 2 C 2 .r/gM ; where r 2 RC and D .r/ is a smooth function, called the warping function. If .r/ D r then .C.M /; g/ is simply called the Riemannian cone or metric cone on M . If .r/ D sin r then .C.M /; g/ is called the sine-cone on M . The relevance of sine-cones will become clear later while the importance of metric cones in relation to the Einstein metrics can be summarized by the following fundamental Lemma 2. Let .M; g/ be a Riemannian manifold of dimension n, and consider N the cone on M with metric gN D dr 2 C r 2 g. Then if gN .C.M / D M RC ; g/ is Einstein, it is Ricci-flat, and gN is Ricci-flat if and only if g is Einstein with Einstein constant n 1. Interestingly, there is a similar lemma about sine-cone metrics.

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Lemma 3. Let .M n ; g/ be an Einstein manifold with Einstein constant n 1 and consider .Cs .M / D M .0; /; gN s / the sine-cone on M with metric gN s D dr 2 C .sin2 r/g. Then gN s is Einstein with Einstein constant n. It is well known that one characterization of Kählerian geometry is via the holonomy reduction. We now recall some basic facts about the holonomy groups of irreducible Riemannian manifolds. Let .M; g/ be a Riemannian manifold and consider parallel translation defined by the Levi-Civita connection and its associated holonomy group which is a subgroup of the structure group O.n; R/ (SO.n; R/ in the oriented case). Since this connection r g is uniquely associated to the metric g, we denote it by Hol.g/, and refer to it as the Riemannian holonomy group or just the holonomy group when the context is clear. Indeed, it is precisely this Riemannian holonomy that plays an important role here. Now on a Riemannian manifold .M; g/ there is a canonical epimorphism 1 .M / ! Hol.g/=Hol0 .g/, in particular, if 1 .M / D 0 then Hol.g/ D Hol0 .g/. In 1955 Berger proved the following theorem [16] concerning Riemannian holonomy: Theorem 4. Let .M; g/ be an oriented Riemannian manifold which is neither locally a Riemannian product nor locally symmetric. Then the restricted holonomy group Hol0 .g/ is one of the groups listed in the following table. Table 1. Berger’s Riemannian holonomy groups. Hol0 .g/

dim.M /

Geometry of M

Comments

SO.n/

n

orientable Riemannian

generic Riemannian

U.n/

2n

Kähler

generic Kähler

SU.n/

2n

Calabi–Yau

Ricci-flat Kähler

Sp.n/ Sp.1/

4n

quaternionic Kähler

Einstein

Sp.n/

4n

hyper-Kähler

Ricci-flat

G2

7

G2 -manifold

Ricci-flat

Spin.7/

8

Spin.7/-manifold

Ricci-flat

Originally Berger’s list included Spin.9/, butAlekseevsky proved that any manifold with such holonomy must be symmetric [5]. In the same paper Berger also claimed a classification of all holonomy groups of torsion-free affine (linear) connections that act irreducibly. He produced a list of possible holonomy representations up to what he claimed was a finite number of exceptions. But his classification had some gaps discovered 35 years later by Bryant [34]. An infinite series of exotic holonomies was found in [42] and finally the classification in the non-Riemannian affine case was completed by Merkulov and Schwachhöfer [107]. We refer the reader to [107] for


43

the proof, references and the history of the general affine case. In the Riemannian case a new geometric proof of Berger’s Theorem is now available [115]. An excellent review of the subject just prior to the Merkulov and Schwachhöfer’s classification can be found in [35]. We should add that one of the first non-trivial results concerning manifolds with the exceptional holonomy groups of the last two rows of Table 1 is due to Bonan [22] who established Ricci-flatness of manifolds with parallel spinors. Manifolds with reduced holonomy have always been very important in physics. Partly because Calabi–Yau, hyper-Kähler, quaternionic Kähler, G2 and Spin.7/ manifolds are automatically Einstein. In addition, all of these spaces appear as -model geometries in various supersymmetric models. What is perhaps less known is that all of these geometries are also related, often in more than one way, to Sasakian structures of various flavors. Let us list all such known relations. • SO.n/-holonomy. As remarked this is holonomy group of a generic metric on an oriented Riemannian manifold .M n ; g/. As we shall see Sasaki–Einstein metrics necessarily have maximal holonomy. • U.n/-holonomy and Kähler geometry (i) Metric cone on a Sasakian manifold is Kähler. (ii) Transverse geometry of a Sasakian manifold is Kähler. (iii) Transverse geometry of a positive Sasakian manifold is Fano. (iv) Transverse geometry of a Sasaki–Einstein manifold is Fano and Kähler– Einstein of positive scalar curvature. (v) Transverse geometry of a negative Sasakian manifold is canonical in the sense that the transverse canonical bundle is ample. (vi) Transverse geometry of a 3-Sasakian manifold is a Kähler–Einstein with a complex contact structure, i.e., twistor geometry. • SU.n/-holonomy and Calabi–Yau geometry (i) Metric cone on a Sasaki–Einstein manifold is Calabi–Yau. (ii) Transverse geometry of a null Sasakian manifold is Calabi–Yau. • Sp.n/Sp.1/-holonomy and Quaternionic Kähler geometry (i) Transverse geometry of the 3-dimensional foliation of a 3-Sasakian manifold is quaternionic-Kähler of positive scalar curvature. (ii) 3-Sasakian manifolds occur as conformal infinities of complete quaternionic Kähler manifolds of negative scalar curvature. • Sp.n/-holonomy and hyper-Kähler geometry (i) Metric cone on a 3-Sasakian manifold is hyper-Kähler. (ii) Transverse geometry of a null Sasakian manifold with some additional structure is hyper-Kähler.

44


• G2 -holonomy (i) The ‘squashed’ twistor space of a 3-Sasakian 7-manifold is nearly Kähler; hence, the metric cone on it has holonomy inside G2 . (ii) Sine-cone on a Sasaki–Einstein 5-manifold is nearly Kähler; hence, its metric cone has holonomy inside G2 . • Spin.7/-holonomy (i) The ‘squashed’ 3-Sasakian 7-manifold has a nearly parallel G2 -structure; hence, its metric cone has holonomy in Spin.7/. (ii) Sine-cone on a ‘squashed’ twistor space of a 3-Sasakian 7-manifold has a nearly parallel G2 structure; hence, its metric cone has holonomy inside Spin.7/. (iii) Sine cone on a sine cone on a 5-dimensional Sasaki–Einstein base has a nearly parallel G2 -structure; hence, its metric cone has holonomy inside Spin.7/. Note that Sasakian manifolds are related to various other geometries in two very distinct ways. On one hand we can take a Sasakian (Sasaki–Einstein, 3-Sasakian, etc.) manifold and consider its metric or sine-cone. These cones frequently have interesting geometric properties and reduced holonomy. On the other hand, a Sasakian manifold is always naturally foliated by one-dimensional leaves (three-dimensional leaves in addition to the one-dimensional canonical foliation when the manifold is 3-Sasakian) and we can equally well consider the transverse geometries associated to such fundamental foliations. These too have remarkable geometric properties including reduced holonomy. In particular, Sasakian manifolds are not just related to all of the geometries on Berger’s holonomy list, but more importantly, they provide a bridge between the different geometries listed there. We will investigate some of these bridges in the next two sections.

3 Sasakian and Kählerian geometry Definition 5. A .2nC1/-dimensional manifold M is a contact manifold if there exists a 1-form , called a contact 1-form, on M such that ^ .d/n ¤ 0 everywhere on M . A contact structure on M is an equivalence class of such 1-forms, where 0 if there is a nowhere vanishing function f on M such that 0 D f . Lemma 6. On a contact manifold .M; / there is a unique vector field , called the Reeb vector field, satisfying the two conditions ³ D 1;

³ d D 0:


45

Definition 7. An almost contact structure on a differentiable manifolds M is a triple .; ; ˆ/, where ˆ is a tensor field of type .1; 1/ (i.e., an endomorphism of TM ), is a vector field, and is a 1-form which satisfy ./ D 1

and

ˆ B ˆ D 1 C ˝ ;

where 1 is the identity endomorphism on TM . A smooth manifold with such a structure is called an almost contact manifold. Remark 8. The reader will notice from Definitions 5 and 7 that an almost contact structure actually has more structure than a contact structure! This is in stark contrast to the usual relationship between a structure and its ‘almost structure’; however, we feel that the terminology is too well ensconced in the literature to be changed at this late stage. Let .M; / be a contact manifold with a contact 1-form and consider D D ker TM . The subbundle D is maximally non-integrable and it is called the contact distribution. The pair .D; !/, where ! is the restriction of d to D gives D the structure of a symplectic vector bundle. We denote by J.D/ the space of all almost complex structures J on D that are compatible with !, that is the subspace of smooth sections J of the endomorphism bundle End.D/ that satisfy J 2 D 1;

d.JX; J Y / D d.X; Y /;

d.JX; X / > 0

(1)

for any smooth sections X; Y of D. Notice that each J 2 J.D/ defines a Riemannian metric gD on D by setting gD .X; Y / D d.JX; Y /:

(2)

One easily checks that gD satisfies the compatibility condition gD .JX; J Y / D gD .X; Y /. Furthermore, the map J 7! gD is one-to-one, and the space J.D/ is contractible. A choice of J gives M an almost CR structure. Moreover, by extending J to all of TM one obtains an almost contact structure. There are some choices of conventions to make here. We define the section ˆ of End.TM / by ˆ D J on D and ˆ D 0, where is the Reeb vector field associated to . We can also extend the transverse metric gD to a metric g on all of M by g.X; Y / D gD C .X/.Y / D d.ˆX; Y / C .X/.Y /

(3)

for all vector fields X; Y on M . One easily sees that g satisfies the compatibility condition g.ˆX; ˆY / D g.X; Y / .X /.Y /. Definition 9. A contact manifold M with a contact form , a vector field , a section ˆ of End.TM /, and a Riemannian metric g which satisfy the conditions ./ D 1;

ˆ2 D 1 C ˝ ;

g.ˆX; ˆY / D g.X; Y / .X/.Y / is known as a metric contact structure on M .

46


Definition–Theorem 10. A Riemannian manifold .M; g/ is called a Sasakian manifold if any one, hence all, of the following equivalent conditions hold: (i) There exists a Killing vector field of unit length on M so that the tensor field ˆ of type .1; 1/, defined by ˆ.X / D rX , satisfies the condition .rX ˆ/.Y / D g.X; Y / g.; Y /X for any pair of vector fields X and Y on M . (ii) There exists a Killing vector field of unit length on M so that the Riemann curvature satisfies the condition R.X; /Y D g.; Y /X g.X; Y /; for any pair of vector fields X and Y on M . (iii) The metric cone .C.M /; g/ N D .RC M; dr 2 C r 2 g/ is Kähler. We refer to the quadruple D .; ; ˆ; g/ as a Sasakian structure on M , where is the 1-form dual vector field . It is easy to see that is a contact form whose Reeb vector field is . In particular D .; ; ˆ; g/ is a special type of metric contact structure. The vector field is nowhere vanishing, so there is a 1-dimensional foliation F associated with every Sasakian structure, called the characteristic foliation. We will denote the space of leaves of this foliation by Z. Each leaf of F has a holonomy group associated to it. The dimension of the closure of the leaves is called the rank of . We shall be interested in the case rk./ D 1. We have Definition 11. The characteristic foliation F is said to be quasi-regular if there is a positive integer k such that each point has a foliated coordinate chart .U; x/ such that each leaf of F passes through U at most k times. Otherwise F is called irregular. If k D 1 then the foliation is called regular, and we use the terminology non-regular to mean quasi-regular, but not regular. Let .M; / be a Sasakian manifold, and consider the contact subbundle D D ker . There is an orthogonal splitting of the tangent bundle as TM D D ˚ L ;

(4)

where L is the trivial line bundle generated by the Reeb vector field . The contact subbundle D is just the normal bundle to the characteristic foliation F generated by . It is naturally endowed with both a complex structure J D ˆjD and a symplectic structure d. Hence, .D; J; d/ gives M a transverse Kähler structure with Kähler form d and metric gD defined as in (2) which is related to the Sasakian metric g by g D gD ˚ ˝ as in (3). We have [23] the following fundamental structure theorem: Theorem 12. Let .M; ; ; ˆ; g/ be a compact quasi-regular Sasakian manifold of dimension 2n C 1, and let Z denote the space of leaves of the characteristic foliation.


47

Then the leaf space Z is a Hodge orbifold with Kähler metric h and Kähler form ! 2 .Z; Z/ so that W .M; g/ ! .Z; h/ is an which defines an integral class Œ! in Horb orbifold Riemannian submersion. The fibers of are totally geodesic submanifolds of M diffeomorphic to S 1 . and its converse: Theorem 13. Let .Z; h/ be a Hodge orbifold. Let W M ! Z be the S 1 V-bundle whose first Chern class is Œ!, and let be a connection 1-form in M whose curvature is 2 !, then M with the metric h C ˝ is a Sasakian orbifold. Furthermore, if all the local uniformizing groups inject into the group of the bundle S 1 , the total space M is a smooth Sasakian manifold. Irregular structures can be understood by the following result of Rukimbira [121]: Theorem 14. Let .; ; ˆ; g/ be a compact irregular Sasakian structure on a manifold M . Then the group Aut.; ; ˆ; g/ of Sasakian automorphisms contains a torus T k of dimension k 2. Furthermore, there exists a sequence .i ; i ; î ; gi / of quasiregular Sasakian structures that converge to .; ; ˆ; g/ in the C 1 compact-open topology. p .Z; Z/ were defined by Haefliger [74]. In The orbifold cohomology groups Horb analogy with the smooth case a Hodge orbifold is then defined to be a compact Kähler 2 .Z; Z/. Alternatively, we can develop the orbifold whose Kähler class lies in Horb concept of basic cohomology which works equally well in the irregular case, but only has coefficients in R. It is nevertheless quite useful in trying to extend the notion of Z being Fano to both the quasi-regular and the irregular situation. This can be done in several ways. Here we will use the notion of basic Chern classes. Recall [130] that a smooth p-form ˛ on M is called basic if

³ ˛ D 0;

£ ˛ D 0;

(5)

p and we let ƒB

p denote the sheaf of germs of basic p-forms on M , and by B the set of p p 0 global sections of ƒB on M . The sheaf ƒB is a module over the ring, ƒB , of germs of 0 0 smooth basic functions on M . We let CB1 .M / D B denote global sections of ƒB ,

i.e. the ring of smooth basic functions on M . Since exterior differentiation preserves basic forms we get a de Rham complex d

p pC1 ! B ! ! B

(6)

whose cohomology HB .F / is called the basic cohomology of .M; F /. The basic cohomology ring HB .F / is an invariant of the foliation F and hence, of the Sasakian structure on M . It is related to the ordinary de Rham cohomology H .M; R/ by the long exact sequence [130] jp

ı

! HBp .F / ! H p .M; R/ ! HBp1 .F / ! HBpC1 .F / !

(7)

48


where ı is the connecting homomorphism given by ıŒ˛ D Œd ^ ˛ D Œd [ Œ˛, and jp is the composition of the map induced by ³ with the well-known isomorphism 1 1 H r .M; R/ H r .M; R/S where H r .M; R/S is the S 1 -invariant cohomology de1 fined from the S 1 -invariant r-forms r .M /S . We also note that d is basic even though is not. Next we exploit the fact that the transverse geometry is Kähler. Let DC denote the complexification of D, and decompose it into its eigenspaces with respect to J , that is, DC D D 1;0 ˚ D 0;1 . Similarly, we get a splitting of the complexification 1 of basic one forms on M , namely of the sheaf ƒB 1;0 0;1 1 ˝ C D ƒB ˚ ƒB : ƒB

We let EBp;q denote the sheaf of germs of basic forms of type .p; q/, and we obtain a splitting M p;q r ˝C D EB : (8) ƒB pCqDr

The basic cohomology groups HBp;q .F / are fundamental invariants of a Sasakian structure which enjoy many of the same properties as the ordinary Dolbeault cohomology of a Kähler structure. Consider the complex vector bundle D on a Sasakian manifold .M; ; ; ˆ; g/. As such D has Chern classes c1 .D/; : : : ; cn .D/ which can be computed by choosing a connection r D in D [93]. Let us choose a local foliate unitary transverse frame .X1 ; : : : ; Xn /, and denote by T the transverse curvature 2-form with respect to this frame. A simple calculation shows that T is a basic .1; 1/-form. Since the curvature 2-form T has type .1; 1/ it follows as in ordinary Chern–Weil theory that Theorem 15. The k th Chern class ck .D/ of the complex vector bundle D is represented by the basic .k; k/-form k determined by the formula

1 T det 1n 2 i

D 1 C 1 C C k :

Since k is a closed basic .k; k/-form it represents an element in HBk;k .F / HB2k .F / that is called the basic k th Chern class and denoted by ck .F /. We now concentrate on the first Chern classes c1 .D/ and c1 .F /. We have Definition 16. A Sasakian structure D .; ; ˆ; g/ is said to be positive (negative) if c1 .F / is represented by a positive (negative) definite .1; 1/-form. If either of these two conditions is satisfied is said to be definite, and otherwise is called indefinite. is said to be null if c1 .F / D 0. Notice that irregular structures cannot occur for negative or null Sasakian structures, since the dimension of Aut.; ; ˆ; g/ is greater than one. In analogy with common terminology of smooth algebraic varieties we see that a positive Sasakian structure is


49

a transverse Fano structure1 , while a null Sasakian structure is a transverse Calabi– Yau structure. The negative Sasakian case corresponds to the canonical bundle being ample.

4 Sasaki–Einstein and 3-Sasakian geometry Definition 17. A Sasakian manifold .M; / is Sasaki–Einstein if the metric g is also Einstein. For any 2n+1-dimensional Sasakian manifold Ric.X; / D 2n.X / implying that any Sasaki–Einstein metric must have positive scalar curvature. Thus any complete Sasaki–Einstein manifold must have a finite fundamental group. Furthermore the metric cone .C.M /; g/ N D .RC M; dr 2 C r 2 g/ on M is Kähler Ricci-flat (Calabi– Yau). The following theorem [23] is an orbifold version of the famous Kobayashi bundle construction of Einstein metrics on bundles over positive Kähler–Einstein manifolds [19], [92]. Theorem 18. Let .Z; h/ be a compact Fano orbifold with 1orb .Z/ D 0 and Kähler– Einstein metric h. Let W M ! Z be the S 1 V-bundle whose first Chern class is c1 .Z/ . Suppose further that the local uniformizing groups of Z inject into S 1 . Then Ind.Z/ with the metric g D h C ˝ , M is a compact simply connected Sasaki–Einstein manifold. Here Ind.Z/ is the orbifold Fano index [23] defined to be the largest positive integer c1 .Z/ 2 such that Ind.Z/ defines a class in the orbifold cohomology group Horb .Z; Z/. A very special class of Sasaki–Einstein spaces is naturally related to several quaternionic geometries. Definition 19. Let .M; g/ be a Riemannian manifold of dimension m. We say that .M; g/ is 3-Sasakian if the metric cone .C.M /; g/ N D .RC M; dr 2 C r 2 g/ on M is hyper-Kähler. We emphasize the important observation of Kashiwada [89] that a 3-Sasakian manifold is automatically Einstein. We denote a Sasakian manifold with a 3-Sasakian structure by .M; Ã/, where Ã D .1 ; 2 ; 3 / is a triple or a 2-sphere of Sasakian structures i D .i ; i ; î ; g/. Remark 20. In the 3-Sasakian case there is an extra structure, i.e., the transverse geometry O of the 3-dimensional foliation which is quaternionic-Kähler. In this case, 1 For

a more algebro-geometric approach to positivity and fundamentals on log Fano orbifolds see [26].

50


the transverse space Z is the twistor space of O and the natural map Z ! O is the orbifold twistor fibration [124]. We get the following diagram which we denote by }.M; Ã/ [29], [30]: Hyper-Kähler geometry C.M /

Twistor geometry

@@@ @@ @@ @@ @ o Z@ M @@ }} @@ } @@ }} @@ }} } @ ~}}

3-Sasakian geometry

(9)

O

Quaternion Kähler geometry Remark 21. The table below summarizes properties of the cone and transverse geometries associated to various metric contact structures. Cone geometry of C.M /

M

Transverse geometry of F

Symplectic

Contact

Symplectic

Kähler

Sasakian

Kähler

Kähler

positive Sasakian

Fano, c1 .Z/ > 0

Kähler

null Sasakian

Calabi–Yau, c1 .Z/ D 0

Kähler

negative Sasakian

ample canonical bundle, c1 .Z/ < 0

Calabi–Yau

Sasaki–Einstein

Fano, Kähler–Einstein

Hyper-Kähler

3-Sasakian

C-contact, Fano, Kähler–Einstein

For numerous examples and constructions of Sasaki–Einstein and 3-Sasakian manifolds see [26]. We finish this section with a remark that both the 3-Sasakian metric on M and the twistor space metric on Z admit ‘squashings’ which are again Einstein. More generally, let W M ! B be an orbifold Riemannian submersion with g the Riemannian metric on M . Let V and H denote the vertical and horizontal subbundles of the tangent bundle TM . For each real number t > 0 we construct a one parameter family g t of Riemannian metrics on M by defining g t jV D tgjV ;

g t jH D gjH ;

g t .V ; H / D 0:

(10)


51

So for each t > 0 we have an orbifold Riemannian submersion with the same base space. Furthermore, if the fibers of g are totally geodesic, so are the fibers of g t . We apply the canonical variation to the orbifold Riemannian submersion W M ! O and W Z ! O. Theorem 22. Every 3-Sasakian manifold M admits a second Einstein metric of positive scalar curvature. Furthermore, the twistor space Z also admits a second orbifold Einstein metric which is Hermitian-Einstein, but not Kähler–Einstein.

5 Toric Sasaki–Einstein 5-manifolds Examples of Sasaki–Einstein manifolds are plentiful and we refer the interested reader to our monograph for a detailed exposition [26]. Here we would like to consider the toric Sasaki–Einstein structures in dimension 5 again referring to [26] for all necessary details. Toric Sasaki–Einstein 5-manifolds recently emerged from physics in the context of supersymmetry and the so-called AdS/CFT duality conjecture which we will discuss in the last section. It is known that, in dimension 5, toric Sasaki– Einstein structures can only occur on the k-fold connected sums k.S 2 S 3 / [26]. The first inhomogeneous toric Sasaki–Einstein structures on S 2 S 3 were constructed by Gauntlett, Martelli, Sparks, and Waldram. It follows that S 2 S 3 admits infinitely many distinct quasi-regular and irregular toric Sasaki–Einstein structures [66]. Toric geometry of these examples was further explored in [102], [104], [105]. We will now describe a slightly different approach to a more general problem. Consider the symplectic reduction of C n (or equivalently the Sasakian reduction of S 2n1 ) by a k-dimensional torus T k . Every complex representation of a T k on C n can be described by an exact sequence f

0 ! T k ! T n ! T nk ! 0: The monomorphism f can be represented by the diagonal matrix f . 1 ; : : : ; k / D diag

k Y iD1

ai

i 1 ; : : : ;

k Y iD1

ai

i n ;

where . 1 ; : : : ; k / 2 S 1 S 1 D T k are the complex coordinates on T k , and a˛i 2 Z are the coefficients of a k n integral weight matrix 2 Mk;n .Z/. We have ([26]): Proposition 23. Let X./ D .C n n 0/==T k ./ denote the Kähler quotient of the standard flat Kähler structure on .C n n 0/ by the weighted Hamiltonian T k -action

52


with an integer weight matrix . Consider the Kähler moment map i .z/ D

n X

a˛i jz˛ j2 ;

i D 1; : : : ; k:

(11)

˛D1

If all minor k k determinants of are non-zero then X./ D C.Y .// is a cone on a compact Sasakian orbifold Y ./ of dimension 2.n k/ 1 which is the Sasakian reduction of the standard Sasakian structure on S 2n1 . In addition, the projectivization of X./ defined by Z./ D X./=C is a Kähler reduction of the complex projective space CP n1 by a Hamiltonian T k -action defined by and it is the transverse space of the Sasakian structure on Y ./ induced by the quotient. If X a˛i D 0 for all i D 1; : : : ; k, (12) ˛

then c1 .X.// D c1 .D/ D 0. In particular, the orbibundle Y ./ ! Z./ is anticanonical. Moreover, the cone C.Y.//, its Sasakian base Y ./, and the transverse space Z./ are all toric orbifolds. Remark 24. The conditions on the matrix that assure that Y ./ is a smooth manifold are straightforward to work out. They involve gcd conditions on certain minor determinants of . This proposition is nicely summarized by the ‘reduction’ diagram CP n1 o

S 2n1 o

C n n .0/

Z./ o

Y./ o

(13)

C.Y .//.

Both the toric geometry and the topology of Y ./ depend on . Furthermore, Y ./ comes equipped with a family of Sasakian structures. When n k D 3, assuming that Y./ is simply connected (which is an additional condition on ), we must have m.S 2 S 3 / for some m k. We will be mostly interested in the case when m D k. Gauntlett, Martelli, Sparks, and Waldram [66] gave an explicit construction of a Sasaki–Einstein metric for D .p; p; p C q; p q/, where p and q are relatively prime nonnegative integers with p > q. (The general case for k D 1 was treated later in [49], [102], see Remark 27 below). To connect with the original notation we write Y./ D Yp;q . Then we get: One can check that Y1;0 is just the homogeneous metric on S 2 S 3 which is both toric and regular. The next simplest example is Y2;1 which, as a toric contact (Sasakian) manifold, is a circle bundle over the blow up of CP 2 at one point F1 D CP 2 # CP 2 [103]. As F1 cannot admit any Kähler–Einstein metric, Kobayashi’s bundle construction cannot give a compatible Sasaki–Einstein structure. But there is a choice of a Reeb vector field in the torus which makes it possible to give Y2;1


53

a Sasaki–Einstein metric. The Sasaki–Einstein structure on Y2;1 is not quasi-regular and this was the first such example in the literature. Hence, S 2 S 3 admits infinitely many toric quasi-regular Sasaki–Einstein structures and infinitely many toric irregular Sasaki–Einstein structures of rank 2. We have the following generalization of the Yp;q metrics due to [60], [44]: Theorem 25. Let Y./ be as in Proposition 23. Then Y ./ admits a toric Sasaki– Einstein structure which is unique up to a transverse biholomorphism. This existence of a Sasaki–Einstein metric is proved in [60] although the authors do not draw all the conclusions regarding possible toric Sasaki–Einstein manifolds that can be obtained. They give one interesting example of an irregular Sasaki– Einstein structure which generalizes the Y2;1 example of [102] in the following sense: One considers a regular positive Sasakian structure on the anticanonical circle bundle over the del Pezzo surface CP 2 # 2CP 2 which gives a toric Sasakian structure on 2.S 2 S 3 /. The regular Sasakian structure on 2.S 2 S 3 / cannot have any Sasaki– Einstein metric. However, as it is with Y2;1 Futaki, Ono and Wang [60] show that one can deform the regular structure to a unique irregular Sasaki–Einstein structure. A slightly different version of Theorem 25 is proved in [44] where uniqueness is also established. Cho, Futaki and Ono work with toric diagrams rather than with Kähler (Sasakian) quotients which amounts to the same thing by Delzant’s construction. We should add that the results of [44] apply to the toric Sasaki–Einstein manifolds in general dimension and not just in dimension 5. Corollary 26. The manifolds k.S 2 S 3 / admit infinite families of toric Sasaki– Einstein structures for each k 1. As in the k D 1 case one would expect infinitely many quasi-regular and infinitely many irregular such Sasaki–Einstein structures for each satisfying all the condition. Remark 27. The general anticanonical circle reduction was considered independently in two recent papers, [49], [102]. There it was shown that for D p D .p1 ; p2 ; q1 ; q2 /, with pi ; qi 2 ZC , p1 C p2 D q1 C q2 , and gcd.pi ; qj / D 1 for all i; j D 1; 2, the 5-manifold Y ./ S 2 S 3 admits a Sasaki–Einstein structure which coincides with that on Yp;q when p1 D p2 D p and q1 D p q; q2 D p C q. In [49] this family is denoted by L5 .a; b; c/, where p D .a; b; c; a b C c/ and they write the metric explicitly. However, in this case it appears to be harder (though, in principle, possible) to write down the condition under which the Sasaki–Einstein Reeb vector field D .a; b; c/ is quasi-regular. A priori, it is not even clear whether the quasi-regularity condition has any additional solutions beyond those obtained for the subfamily Yp;q . Moreover, it follows from [44] that the metrics of [49], [102] describe all possible toric Sasaki–Einstein structures on S 2 S 3 . There have been similar constructions of a two-parameter family Xp;q of toric Sasaki–Einstein metrics on 2.S 2 S 3 / [75], and another two-parameter family, called

54


Zp;q , on 3.S 2 S 3 / [117]. All these examples, and many more, can be obtained as special cases of Theorem 25 as they are all Y ./ for some choice of . The Yp;q , L5 .a; b; c/, Xp;q and Zp;q metrics have received a lot of attention because of the role such Sasaki–Einstein manifolds play in the AdS/CFT Duality Conjecture. They created an avalanche of papers studying the properties of these metrics from the physics perspective [8], [117], [118], [116], [90], [77], [41], [18], [40], [126], [75], [37], [15], [14], [119], [76]. The AdS/CFT duality will be discussed in the last section.

6 The Dirac operator and Killing spinors We begin with a definition of spinor bundles and the bundle of Clifford algebras of a vector bundle [98], [55]. Recall that the Clifford algebra Cl.Rn / over Rn can be defined as the quotient algebra of the tensor algebra T .Rn / by the two-sided ideal generated by elements of the form v ˝ v C q.v/ where q is a quadratic form on Rn . Definition 28. Let E be a vector bundle with inner product h; i on a smooth manifold M , and let T .E/ denote the tensor bundle over E. The Clifford bundle of E is the quotient bundle Cl.E/ D T .E/=.E/ where is the bundle of ideals (two-sided) generated pointwise by elements of the form v ˝v Chv; vi with v 2 Ex . A real spinor bundle S.E/ of E is a bundle of modules over the Clifford bundle Cl.E/. Similarly, a complex spinor bundle is a bundle of complex modules over the complexification Cl.E/ ˝ C. As vector bundles Cl.E/ is isomorphic to the exterior bundle ƒ.E/, but their algebraic structures are different. The importance of Cl.E/ is that it contains the spin group Spin.n/, the universal (double) covering group of the orthogonal group SO.n/, so one obtains all the representations of Spin.n/ by studying representations of Cl.E/. We assume that the vector bundle E admits a spin structure, so w2 .E/ D 0. We are interested mainly in the case when .M; g/ is a Riemannian spin manifold and E D TM in which case we write S.M / instead of S.TM /. The Levi-Civita connection r on TM induces a connection, also denoted r, on any of the spinor bundles S.M /, or more appropriately on the sections .S.M //. Definition 29. Let .M n ; g/ be a Riemannian spin manifold and let S.M / be any spinor bundle. The Dirac operator is the first order differential operator D W .S.M // ! .S.M // defined by n X Ej rEj ; D D j D1

where fEj g is a local orthonormal frame and denotes Clifford multiplication. The Dirac operator, of course originating with the famous Dirac equation describing fermions in theoretical physics, was brought into mathematics by Atiyah and Singer


55

in [9]. Then Lichnerowicz [99] proved his famous result that a Riemannian spin O manifold with positive scalar curvature must have vanishing A-genus. An interesting question on any spin manifold is: what are the eigenvectors of the Dirac operator. In this regard the main objects of interest consists of special sections of certain spinor bundles called Killing spinor fields or just Killing spinors for short. Specifically (cf. [13], [55]): Definition 30. Let .M; g/ be a complete n-dimensional Riemannian spin manifold, and let S.M / be a spin bundle (real or complex) on M and a smooth section of S.M /. We say that is a Killing spinor if for every vector field X there is ˛ 2 C, called Killing number, such that rX

D ˛X :

Here X denotes the Clifford product of X and . We say that is imaginary when ˛ 2 Im.C /, is parallel if ˛ D 0 and is real2 if ˛ 2 Re.C /. We shall see shortly that the three possibilities for the Killing number ˛: real, imaginary, or 0, are the only possibilities. The name Killing spinor derives from the fact that if is a non-trivial Killing spinor and ˛ is real, the vector field X D

n X

g. ; Ej

/Ej

(14)

j D1

is a Killing vector field for the metric g (which, of course, can be zero). If Killing spinor on an n-dimensional spin manifold, then D

D

n X j D1

Ej rEj

D

n X

˛Ej Ej

D n˛ :

is a

(15)

j D1

So Killing spinors are eigenvectors of the Dirac operator with eigenvalue n˛. In 1980 Friedrich [54] proved the following remarkable theorem: Theorem 31. Let .M n ; g/ be a Riemannian spin manifold which admits a non-trivial Killing spinor with Killing number ˛. Then .M n ; g/ is Einstein with scalar curvature s D 4n.n 1/˛ 2 . A proof of this is a straightforward curvature computation which can be found in either of the books [13], [55]. It also uses the fact that a non-trivial Killing spinor vanishes nowhere. It follows immediately from Theorem 31 that ˛ must be one of the three types mentioned in Definition 30. So if the Killing number is real then .M; g/ must be a positive Einstein manifold. In particular, if M is complete, then it is compact. On the other hand if the Killing number is pure imaginary, Friedrich shows that M must be non-compact. 2 Here the standard terminology real and imaginary Killing spinors can be somewhat misleading.

spinor

The Killing is usually a section of a complex spinor bundle. So a real Killing spinor just means that ˛ is real.

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The existence of Killing spinors not only puts restrictions on the Ricci curvature, but also on both the Riemannian and the Weyl curvature operators [13]. Proposition 32. Let .M n ; g/ be a Riemannian spin manifold. Let be a Killing spinor on M with Killing number ˛ and let R; W W ƒ2 M ! ƒ2 M be the Riemann and Weyl curvature operators, respectively. Then for any vector field X and any 2-form ˇ we have W .ˇ/

D 0I

.rX W /.ˇ/

D 2˛ X ³ W .ˇ/ 2

(16) I

D 0I D 2˛ X ³ R.ˇ/ C 4˛ 2 ˇ.X/ :

(17)

.R.ˇ/ C 4˛ ˇ/

(18)

.rX R/.ˇ/

(19)

These curvature equations can be used to prove (see [13] or [55]) Theorem 33. Let .M n ; g/ be a connected Riemannian spin manifold admitting a nontrivial Killing spinor with ˛ 6D 0. Then .M; g/ is locally irreducible. Furthermore, if M is locally symmetric, or n 4, then M is a space of constant sectional curvature equal to 4˛ 2 . Friedrich’s main objective in [54] was an improvement of Lichnerowicz’s estimate in [99] for the eigenvalues of the Dirac operator. Indeed, Friedrich proves that the eigenvalues of the Dirac operator on any compact manifold satisfy the estimate 1 ns0 ; (20) 4n1 where s0 is the minimum of the scalar curvature on M . Thus, Killing spinors are eigenvectors that realize equality in equation (20). Friedrich also proves the converse that any eigenvector of D realizing the equality must be a Killing spinor with r s0 1 : (21) ˛D˙ 2 n.n 1/

2

Example 34 (Spheres). In the case of the round sphere .S n ; g0 / equality in equation (20) is always attained. So normalizing such that s0 D n.n 1/, and using Bär’s Correspondence Theorem 38 below the number of corresponding real Killing spinors equals the number of constant spinors on RnC1 with the flat metric. The latter is well known (see the appendix of [120]) to be 2bn=2c for each of the values ˛ D ˙ 12 , where bn=2c is the largest integer less than or equal to n=2. Remark 35. Actually (without making the connection to Sasakian geometry) already in [54] Friedrich gives a non-spherical example of a compact 5-manifold with a real Killing spinor: M D SO.4/= SO.2/ with its homogeneous Kobayashi–Tanno Sasaki– Einstein structure.

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We now wish to relate Killing spinors to the main theme of this chapter, Sasakian geometry. First notice that if a Sasakian manifold M 2nC1 admits a Killing spinor, Theorem 31 says it must be Sasaki–Einstein, so the scalar curvature s0 D 2n.2n C 1/, and equation (21) implies that ˛ D ˙ 12 . We have the following result of Friedrich and Kath [57] Theorem 36. Every simply connected Sasaki–Einstein manifold admits non-trivial real Killing spinors. Furthermore, (i) if M has dimension 4m C 1 then .M; g/ admits exactly one Killing spinor for each of the values ˛ D ˙ 12 , (ii) if M has dimension 4m C 3 then .M; g/ admits at least two Killing spinors for one of the values ˛ D ˙ 12 . Outline of proof. (Details can be found in [57] or the book [13].) Every simply connected Sasaki–Einstein manifold is known to be spin, so M has a spin bundle S.M /. Given a fixed Sasakian structure D .; ; ˆ; g/ we consider two subbundles E˙ ./ of S.M / defined by E˙ ./ D f rX˙

2 S.M / j .˙2ˆX C £ X / 1 X . 2

D 0 for all X 2 .TM /g.

(22)

˙

Set D rX ˙ A straightforward computation shows that r preserves the subbundles E˙ and defines a connection there. Moreover, by standard curvature computations it can be shown that the connection r ˙ is flat in E˙ ./. So it has covariantly constant sections which are precisely the Killing spinors. One then uses some representation theory of Spin.2n C 1/ to compute the dimensions of EC ./ and E ./ proving the result. We have the following: Corollary 37. Let .M; g/ be a Sasaki–Einstein manifold of dimension 2m C 1. Then .M; g/ is locally symmetric if and only if .M; g/ is of constant curvature. Moreover, Hol.g/ D SO.2m C 1/ and .M; g/ is locally irreducible as a Riemannian manifold. z . This is a compact simply connected Proof. If necessary, go to the universal cover M Sasaki–Einstein manifold; hence, it admits a non-trivial Killing spinor by Theorem 36. The first statement then follows from Theorem 33. The second statement follows from the Berger Theorem 4. Since M has dimension 2mC1 the only possibilities for Hol.g/ are SO.2mC1/ and G2 . But the latter is Ricci flat, so it cannot be Sasaki–Einstein. Friedrich and Kath began their investigation in dimension 5 [56] where they showed that a simply-connected compact 5-manifold which admits a Killing spinor must be Sasaki–Einstein. In dimension 7 they showed that there are exactly three possibilities: weak G2 -manifolds, Sasaki–Einstein manifolds which are not 3-Sasakian, and 3-Sasakian manifolds [57]. Later Grunewald gave a description of 6-manifolds admitting Killing spinors [73]. We should add an earlier result of Hijazi who showed

58


that the only 8-dimensional manifold with Killing spinors must be the round sphere [78]. By 1990 a decade of research by many people slowly identified all the ingredients of a classification of such manifolds in terms of their underlying geometric structures. The pieces of the puzzle consisting of round spheres in any dimension, Sasaki–Einstein manifolds in odd dimensions, nearly Kähler manifolds in dimension 6, and weak G2 -holonomy manifolds in dimension 7 were all in place with plenty of interesting examples to go around [13]. What remained at that stage was to show that in even dimensions greater than 8 there is nothing else but the round spheres, while in odd dimensions greater than 7 the only such examples must be Sasaki–Einstein. The missing piece of the puzzle was finally uncovered by Bär: real Killing spinors on M correspond to parallel spinors on the cone C.M / [12]. A bit earlier Wang [133] had shown that on a simply connected complete Riemannian spin manifold the existence of parallel spinors corresponds to reduced holonomy. This led Bär to an elegant description of the geometry of manifolds admitting real Killing spinors (in any dimension) in terms of special holonomies of the associated cones. We refer to the correspondence between real Killing spinors on M and parallel spinors on the cone C.M / (equivalently reduced holonomy) as Bär’s correspondence. In particular, this correspondence not only answered the last remaining open questions, but also allowed for simple unified proofs of most of the theorems obtained earlier.

7 Real Killing spinors, holonomy and Bär’s correspondence As already mentioned the Bär correspondence relates real Killing spinors on a compact Riemannian spin manifold .M; g/ to parallel spinors on the Riemannian cone .C.M /; g/. N We now make this statement precise. N Theorem 38. Let .M n ; g/ be a complete Riemannian spin manifold and .C.M n /; g/ be its Riemannian cone. Then there is a one to one correspondence between real N Killing spinors on .M n ; g/ with ˛ D ˙ 12 and parallel spinors on .C.M n /; g/. N implies that gN is Ricci flat Proof. The existence of a parallel spinor on .C.M n /; g/ by Theorem 31. Then by Lemma 2 .M n ; g/ is Einstein with scalar curvature s D n.n 1/. So any Killing spinors must have ˛ D ˙ 12 by equation (21). As in the proof of Theorem 36, rX˙ D rX ˙ 12 X defines a connection in the spin bundle S.M /. The connection 1-forms ! ˙ of r ˙ are related to the connection 1-form ! of the Levi-Civita connection by ! ˙ D ! ˙ 12 ˇ, where ˇ is a 1-form called the soldering form. This can be interpreted as a connection with values in the Lie algebra spin.n C 1/ D spin.n/ ˚ Rn , and pulls back to the Levi-Civita connection in the N So parallel spinors on the cone correspond spin bundles on the cone .C.M n /; g/. to parallel spinors on .M; g/ with respect to the connection r ˙ which correspond precisely to real Killing spinors with respect to the Levi-Civita connection.


59

Now we have the following definition: Definition 39. We say that a Riemannian spin manifold .M; g/ is of type .p; q/ if it carries exactly p linearly independent real Killing spinors with ˛ > 0 and exactly q linearly independent real Killing spinors with ˛ < 0. The following theorem has an interesting history. As mentioned above it was Bär [12] who recognized the correspondence between real Killing spinors on .M; g/ and parallel spinors on the Riemannian cone .C.M /; g/. N The relation between parallel spinors and reduced holonomy was anticipated in the work of Hitchin [79] and Bonan [22], but was formalized in the 1989 paper of Wang [133]. It has also been generalized to the non-simply connected case in [134], [110]. Theorem 40. Let .M n ; g/ be a complete simply connected Riemannian spin manifold, and let Hol.g/ N be the holonomy group of the Riemannian cone .C.M /; g/. N Then n admits a non-trivial real Killing spinor with .M ; g/ of type .p; q/ if and .M n ; g/ only if dim M; Hol.g/; N .p; q/ is one of the 6 possible triples listed in the table below. Table 2 dim.M /

Hol.g/ N

type .p; q/ .2

bn=2c

; 2bn=2c /

n

id

4m C 1

SU.2m C 1/

.1; 1/

4m C 3

SU.2m C 2/

.2; 0/

4m C 3

Sp.m C 1/

.m C 2; 0/

7

Spin.7/

.1; 0/

6

G2

.1; 1/

Here m 1, and n > 1. Outline of proof. Since .M; g/ is complete and has a non-trivial real Killing spinor, it is compact by Theorem 31. It then follows from a theorem of Gallot [64] that if the Riemannian cone .C.M /; g/ N has reducible holonomy it must be flat. So we can apply Berger’s Theorem 4. Now Wang [133] used the spinor representations of the possible irreducible holonomy groups on Berger’s list to give the correspondence between these holonomy groups and the existence of parallel spinors. First he showed that the groups listed in Table 4 that are not on the above table do not admit parallel spinors. Then upon decomposing the spin representation of the group in question into irreducible pieces, the number of parallel spinors corresponds to the multiplicity of the trivial representation. Wang computes this in all but the first line of the table when .C.M /; g/ N is flat. In this case .M; g/ is a round sphere as discussed in Example 34, so the number of linearly independent constant spinors is .2bn=2c ; 2bn=2c /. By Bär’s Correspondence

60


Theorem 38 real Killing spinors on .M; g/ correspond precisely to parallel spinors on .C.M /; g/. N Note that the hypothesis of completeness in Wang’s theorem [133] is not necessary, so that the correspondence between the holonomy groups and parallel spinors holds equally well on Riemannian cones. However, the completeness assumption on .M; g/ guarantees the irreducibility of the cone .C.M /; g/ N as mentioned above. Let us briefly discuss the types of geometry involved in each case of this theorem. As mentioned in the above proof the first line of the table corresponds to the round spheres. The next three lines correspond to Sasaki–Einstein geometry, so Theorem 40 generalizes the Friedrich–Kath Theorem 36 in this case. The last of these three lines corresponds precisely to 3-Sasakian geometry by Definition 19. Finally the two cases whose cones have exceptional holonomy will be discussed in more detail in Section 8.1 below. Suffice it here to mention that it was observed by Bryant and Salamon [36] that a cone on a nearly parallel G2 manifold has its own holonomy in Spin.7/. It is interesting to note that Theorem 40 generalizes the result of Hijazi in dimension eight mentioned earlier as well as part of the last statement in Theorem 33, namely Corollary 41. Let .M 2n ; g/ be a complete simply connected Riemannian spin manifold of dimension 2n with n ¤ 3 admitting a non-trivial real Killing spinor. Then M is isometric to the round sphere. We end this section with a brief discussion of the non-simply connected case. Here we consider two additional cases for Hol.g/, N namely SU.2mC2/ÌZ2 and Sp.2/Zd . See [134], [110] for the list of possibilities. Example 42. Hol.g/ N D SU.2m/ Ì Z2 . Consider the .4m 1/-dimensional Stiefel 2mC1 manifold V2 .R / with its homogeneous Sasaki–Einstein metric. The quotient manifold M4m1 of V2 .R2mC1 / by the free involution induced from complex conjugation has an Einstein metric which is “locally Sasakian”. The cone C.M4m1 / is not Kähler and its holonomy is Hol.g/ N D SU.2m C 2/ Ì Z2 . According to Wang [134] C.M4m1 / admits a spin structure with precisely one parallel spinor if and only if m is even, and according to Moroianu and Semmelmann [110] C.M4m1 / admits exactly two spin structures each with precisely one parallel spinor if m is even. Thus, by Theorem 38 M4m1 admits exactly two spin structures each with exactly one Killing spinor if and only if m is even. Example 43. Consider a 3-Sasakian manifold .M 4n1 ; Ã/ and choose a Reeb vector field ./. Let Cm be the cyclic subgroup of order m > 2 of the circle group generated by ./. Assume that m is relatively prime to the order .Ã/ of Ã and that the generic fibre of the fundamental 3-dimensional foliation FQ is SO.3/, so that Cm acts freely on M 4n1 . This last condition on the generic fibre is easy to satisfy; for example, it holds for any of the 3-Sasakian homogeneous spaces other than the standard round sphere, as well as the bi-quotients described in [30]. (To handle the case when the


61

generic fibre is Sp.1/ we simply need to divide m by two when it is even). Since Cm is not in the center of SO.3/, the quotient M 4n1 =Cm is not 3-Sasakian. However, Cm does preserve the Sasakian structure determined by ./, so M 4n1 =Cm is Sasaki– Einstein. The cone C.M 4n1 =Cm / has holonomy Sp.n/ Zm , and admits precisely nC1 parallel spinors if and only if m divides n C 1 [134], [110]. Thus, by Theorem 38 m 4n1 M =Cm admits precisely nC1 Killing spinors when m divides n C 1. m

8 Geometries associated with 3-Sasakian 7-manifolds It is most remarkable that to each 4n-dimensional positive QK metric .O; gO / (even just locally) one can associate nine other Einstein metrics in dimensions 4n C k, k D 1; 2; 3; 4. Alternatively, one could say that each 3-Sasakian metric .M; g/ canonically defines an additional nine Einstein metrics in various dimensions. We have already encountered all of these metrics. First there are the four geometries of the diamond diagram }.M; Ã/. Then M and Z admit additional “squashed” Einstein metrics discussed in Theorem 22. Thus we get five Einstein metrics with positive Einstein constants: .O; gO /, .M; g/, .M 0 ; g 0 /, .Z; h/, .Z0 ; h0 /. Of course M ' M 0 and Z ' Z0 as smooth manifolds (orbifolds) but they are different as Riemannian manifolds (orbifolds), hence, the notation. Let us scale all these metrics so that the Einstein constant equals the dimension of the total space minus 1. Note that any 3-Sasakian metric already has this property. In the other four cases this is a choice of scale which is quite natural due to Lemma 2. However, note that this is not the scale one gets for .Z; h/, and .O; gO / via the Riemannian submersion from .M; g/. Now, in each case one can consider its Riemannian cone which will be Ricci-flat by Lemma 2. We thus obtain five Ricci-flat metrics on the corresponding Riemannian cones. In addition, one can also take (iterated) sine-cone metrics defined in (1) on the same five bases. These metrics are all Einstein of positive scalar curvature (cf. Lemma 3). Let us summarize all this with the following extension of }.M; Ã/: C.Z0 / o

C.Z/ o

M0 x x xx xx x |xx / C.O/ O bFF }> F } F FF }} FF }} } F } _? o M Z

? _ Z0 AA AA AA AA

/ C.M 0 /

(23)

/ C.M /.

There would perhaps be nothing special about all these 10 (and many more by iterating the sine-cone construction) geometries beyond what has already been discussed in the previous sections. This is indeed true when dim.M / > 7. However, when dim.M / D 7, or, alternatively, when O is a positive self-dual Einstein orbifold

62


metric (more generally, just a local metric of this type) some of the metrics occurring in diagram (23) have additional properties. We shall list all of them first. For the moment, let us assume that .M; g/ is a compact 3-Sasakian 7-manifold, then the following hold: (1) .O; gO / is a positive self-dual Einstein manifold (orbifold). We will think of it as the source of all the other geometries. (2) .C.O/; dt 2 C t 2 gO / is a 5-dimensional Ricci-flat cone with base O. (3) .Z; h/ is the orbifold twistor space of O. (4) .Z0 ; h0 / is a nearly-Kähler manifold (orbifold). (5) .M; g/ is the 3-Sasakian manifold. (6) .M 0 ; g 0 / is a 7-manifold with weak G2 structure. (7) .C.Z0 /; dt 2 C t 2 h0 / is a 7-manifold with holonomy inside G2 . (8) .Cs .Z0 /; dt 2 C .sin2 t /h0 / is a 7-manifold with weak G2 structure. (9) .C.Z/; dt 2 C t 2 h/ is a 7-dimensional Ricci-flat cone with base Z. (10) .C.M /; dt 2 C t 2 g/ is hyper-Kähler with holonomy contained in Sp.2/. (11) .C.M 0 /; dt 2 C t 2 g 0 / has holonomy contained in Spin.7/. The cases (2) and (8) do not appear to have any special properties other than Ricciflatness. The cases (1), (3), (5), and (10) are the four geometries of }.M; Ã/. The five remaining cases are all very interesting from the point of view of the classification of Theorem 40. Indeed Z0 and C.Z0 / are examples of the structures listed in the last row of the table while C2 .Z0 /, M 0 and C.M 0 / give examples of the structures listed in the fifth row. In particular, our diagram (23) provides for a cornucopia of orbifold examples in the first case and smooth manifolds in the latter.

8.1 Nearly parallel G2 -structures and Spin.7/ holonomy cones Recall, that geometrically G2 is defined to be the Lie group acting on the imaginary octonions R7 and preserving the 3-form ' D ˛1 ^ ˛2 ^ ˛3 C ˛1 ^ .˛4 ^ ˛5 ˛6 ^ ˛7 / C ˛2 ^ .˛4 ^ ˛6 ˛7 ^ ˛5 / C ˛3 ^ .˛4 ^ ˛7 ˛5 ^ ˛6 /;

(24)

where f˛i g7iD1 is a fixed orthonormal basis of the dual of R7 . A G2 structure on a 7-manifold M is, by definition, a reduction of the structure group of the tangent bundle to G2 . This is equivalent to the existence of a global 3-form ' 2 3 .M / which may be written locally as (24). Such a 3-form defines an associated Riemannian metric, an orientation class, and a spinor field of constant length.


63

Definition 44. Let .M; g/ be a complete 7-dimensional Riemannian manifold. We say that .M; g/ is a nearly parallel3 G2 structure if there exist a global 3-form ' 2 3 .M / which locally can be written in terms of a local orthonormal basis as in (24), and d' D c ? ', where ? is the Hodge star operator associated to g and c ¤ 0 is a constant whose sign is fixed by an orientation convention. The case c D 0 in Definition 44 is somewhat special. In particular, it is known [123] that the condition d' D 0 D d ?' is equivalent to the condition that ' be parallel, i.e., r' D 0 which is equivalent to the condition that the metric g has holonomy group contained in G2 . The following theorem provides the connection with the previous discussion on Killing spinors [12] Theorem 45. Let .M; g/ be a complete 7-dimensional Riemannian manifold with a N of the metric cone .C.M /; g/ N nearly parallel G2 structure. Then the holonomy Hol.g/ is contained in Spin.7/. In particular, C.M / is Ricci-flat and M is Einstein with positive Einstein constant D 6. Remark 46. The sphere S 7 with its constant curvature metric is isometric to the isotropy irreducible space Spin.7/=G2 . The fact that G2 leaves invariant (up to constants) a unique 3-form and a unique 4-form on R7 implies immediately that this space has a nearly parallel G2 structure. Definition 47. Let .M; g/ be a complete 7-dimensional Riemannian manifold. We say that g is a proper G2 -metric if Hol.g/ N D Spin.7/. We emphasize here that G2 is the structure group of M , not the Riemannian holonomy group. Specializing Theorem 40 to dimension 7 gives the following theorem due to Friedrich and Kath [57]. Theorem 48. Let .M 7 ; g/ be a complete simply-connected Riemannian spin manifold of dimension 7 admitting a non-trivial real Killing spinor with ˛ > 0 or ˛ < 0. Then there are four possibilities: (i) .M 7 ; g/ is of type .1; 0/ and it is a proper G2 -manifold. (ii) .M 7 ; g/ is of type .2; 0/ and it is a Sasaki–Einstein manifold, but .M 7 ; g/ is not 3-Sasakian. (iii) .M 7 ; g/ is of type .3; 0/ and it is 3-Sasakian. (iv) .M 7 ; g/ D .S 7 ; gcan / and is of type .8; 8/. 3 It had become customary to refer to this notion as ‘weak holonomy G ’, a terminology introduced by Gray 2 [71]. However, it was pointed out to us by the anonymous referee that this terminology is misleading due to the fact that Gray’s paper contains errors rendering the concept of weak holonomy useless as discovered by Alexandrov [3]. Hence, the term ‘nearly parallel’ used in [58] is preferred.

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Conversely, if .M 7 ; g/ is a compact simply-connected proper G2 -manifold then it carries precisely one Killing spinor with ˛ > 0. If .M 7 ; g/ is a compact simply-connected Sasaki–Einstein 7-manifold which is not 3-Sasakian then M carries precisely two linearly independent Killing spinors with ˛ > 0. Finally, if .M 7 ; g/ is a 3-Sasakian 7-manifold, which is not of constant curvature, then M carries precisely three linearly independent Killing spinors with ˛ > 0. Remark 49. The four possibilities of Theorem 48 correspond to the sequence of inclusions Spin.7/ SU.4/ Sp.2/ 1: All of the corresponding cases are examples of nearly parallel G2 metrics. If we exclude the trivial case when the associated cone is flat, we have three types of nearly parallel G2 geometries. Following [58] we use the number of linearly independent Killing spinors to classify these geometries, and call them type I, II, and III corresponding to cases (i), (ii), and (iii) of Theorem 48, respectively. We are now ready to describe the G2 geometry of the M 0 ,! C.M 0 / part of the diagram (23) [63], [58]: Theorem 50. Let .M; Ã/ be a 7-dimensional 3-Sasakian manifold. Then the 3-Sasakian metric g is a nearly parallel G2 metric. Moreover, the second Einstein metric g 0 given by Theorem 22 and scaled so that the Einstein constant D 6 is a nearly parallel G2 metric; in fact, it is a proper G2 metric. 0 Proof. p For the secondpEinstein metric p g 3we have three mutually orthonormal 1-forms 1 1 2 2 3 ˛ D t , ˛ D t , ˛ D t , where t is the parameter of the canonical variation. Let f˛ 4 ; ˛ 5 ; ˛ 6 ; ˛ 7 g be local 1-forms spanning the annihilator of the vertical subbundle V3 in T such that

N 1 D 2.˛ 4 ^ ˛ 5 ˛ 6 ^ ˛ 7 /; ˆ N 2 D 2.˛ 4 ^ ˛ 6 ˛ 7 ^ ˛ 5 /; ˆ N 3 D 2.˛ 4 ^ ˛ 7 ˛ 5 ^ ˛ 6 /: ˆ Then the set f˛ 1 ; : : : ; ˛ 7 g forms a local orthonormal coframe for the metric g 0 . Let X X Na D a ^ ˆ a ^ da C 6‡ (25) ‡ D 1 ^ 2 ^ 3 ; ‚ D a

a

p p3 In terms of the 3-forms ‡ and ‚ we have ' D 12 t ‚ C t ‡ . One easily sees that this is of the type of equation (24) and, therefore, defines a compatible G2 -structure. Moreover, a straightforward computation gives p 1 1 1p t C t .t C 1/d ‡; ?' D t d ‡ : d' D 2 2 24


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p p p Thus, d' D c ? ' is solved with t D 1= 5, and c D 12= 5. So g 0 is nearly parallel. That g 0 is a proper G2 metric is due to [58]. The idea is to use Theorem 48. Looking at the four possibilities given in that theorem, we see that it suffices to show that g 0 is not Sasaki–Einstein. The details are in [58]. Example 51. 3-Sasakian 7-manifolds are plentiful [26]. All of them give, by Theorem 50, examples of type I and type III geometries. Examples of simply connected type I geometries that do not arise via Theorem 50 are the homogeneous Aloff–Wallach 7 , .m; n/ 6D .1; 1/ which, as special cases of Eschenburg bi-quotients [42], spaces Mm;n [13], are together with an isotropy irreducible homogeneous space defined as follows: Consider the space H2 of homogeneous polynomials of degree 2 in three real variables .x1 ; x2 ; x3 /. As dim.H2 / D 5 it gives rise to the embedding SO.3/ SO.5/. We take M D SO.5/= SO.3/. This example was used by Bryant to get the first 8-dimensional metric with holonomy Spin.7/ [33]. Examples of type II geometries (Sasaki–Einstein) are equally rich [26]. In particular, there are hundreds of examples of type II nearly parallel G2 metrics on each of the 28 homotopy spheres in dimension 7. 7 has three Einstein Remark 52. According to [42] the Aloff–Wallach manifold M1;1 metrics. One is the homogeneous 3-Sasakian metric. The second is the proper G2 metric of Theorem 50. The third Einstein metric is also nearly parallel most likely being of type I, but we could not positively exclude type II as a possibility.

Open Problem 53. Classify all compact 7-manifolds with nearly parallel G2 structures of type I, II, or III, respectively. The classification of type III consists of the classification of all compact 3-Sasakian 7-manifolds. This is probably very hard. The case of 3-Sasakian 7-manifolds with vanishing aut.M; Ã/ appears quite difficult. The type II classification (7-dimensional Sasaki–Einstein manifolds which are not 3-Sasakian) is clearly completely out of reach at the moment. A classification of proper nearly parallel G2 structures on a compact manifold that do not arise via Theorem 50 would be very interesting and it is not clear how hard this problem really is. Remark 54. The holonomy Spin.7/ cone metrics are plentiful but never complete. However, some of these metrics can be deformed to complete holonomy Spin.7/ ones on non compact manifolds. The first example was obtained by Bryant and Salamon who observed that the spin bundle over S 4 with its canonical metric carries a complete metric with holonomy Spin.7/ [36]. Locally the metric was later considered also in [69]. More generally, spin orbibundles over positive QK orbifolds also carry such complete orbifold metrics as observed by Bryant and Salamon in [36]. Other complete examples were constructed later by physicists [47], [48], [87], [88]. Finally, the first compact examples were obtained in 1996 by Joyce [82], [85]. See Joyce’s book [84] for an excellent detailed exposition of the methods and the discussion of examples.

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Open Problem 55 (Complete metrics on cones). Let .M 7 ; Ã/ be any 3-Sasakian 7-manifold and let .M 7 ; g 0 / be the associated proper nearly parallel G2 squashed metric. Consider the two Riemannian cones for these metrics. (i) When does the metric cone .C.M /; dt 2 C t 2 g 0 / admit complete holonomy Spin.7/ deformations? (ii) When does the metric cone .C.M /; dt 2 C t 2 g/ admit complete holonomy Sp.2/ (hyper-Kähler) deformations? In other dimensions one also could ask the following more general questions: (iii) Let .M 4nC3 ; Ã/ be a compact 3-Sasakian manifold. When does the metric cone .C.M /; dt 2 C t 2 g/ admit complete hyper-Kähler (or just Calabi–Yau) deformations? (iv) Let .M 2nC1 ; / be a compact Sasaki–Einstein manifold. When does the metric cone .C.M /; dt 2 C t 2 g/ admit complete Calabi–Yau deformations? (v) Let .M 7 ; g/ be a compact nearly parallel G2 -manifold. When does the metric cone .C.M /; dt 2 C t 2 g/ admit complete holonomy Spin.7/ deformations? (vi) Let .M 6 ; g/ be a compact strict nearly Kähler manifold. When does the metric cone .C.M /; dt 2 C t 2 g/ admit complete holonomy G2 deformations? The metric on the spin bundle S.S 4 / by Bryant and Salamon is a deformation of the Spin.7/ holonomy metric on the cone over the squashed metric on S 7 [47], [48], so there are examples of such deformations regarding question (i). Regarding (ii), we recall that every compact 3-Sasakian 3-manifold is isometric to S 3 = and the metric cone is the flat cone C 2 = . Hence, one could think of (ii) as a 7-dimensional analogue of a similar problem whose complete solution was given by Kronheimer [96]. There are non-trivial examples also in the higher dimensional cases. The metric cone on the homogeneous 3-Sasakian manifold .1; 1; 1/ of [30] admits complete hyper-Kähler deformations, namely the Calabi metric on T CP 2 . We do not know of any other example at the moment. In case (iv) it is not known whether there are any complete Calabi–Yau deformations. However, in [59] Futaki showed that there are deformations to complete constant scalar curvature metrics with scalar curvature either zero or negative.

8.2 Nearly Kähler 6-manifolds and G2 holonomy cones In this section we explain the geometry of the Z0 ,! C.Z0 / part of the diagram (23). Before we specialize to dimension 6 we begin with a more general introduction. Nearly Kähler manifolds were first studied by Tachibana in [129] and they appear under the name of almost Tachibana spaces in Chapter VIII of the book [138]. They were then rediscovered by Gray [70] and given the name nearly Kähler manifolds which by now is the accepted name.


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Definition 56. A nearly Kähler manifold is an almost Hermitian manifold .M; g; J; !/ such that .rX J /X D 0 for all tangent vectors X, where r is the Levi-Civita connection and J is the almost complex structure. One says that a nearly Kähler manifold is strict if it is not Kähler. This definition is equivalent to the condition .rX J /Y C .rY J /X D 0

(26)

for all vector fields X, Y , which is to say that J is a Killing tensor field. An alternative characterization of nearly Kähler manifolds is given by Proposition 57. An almost Hermitian manifold .M; g; J; !/ is nearly Kähler if and only if 1 r! D d!: 3 In particular, a strict nearly Kähler structure is never integrable. Any nearly Kähler manifold can be locally decomposed as the product of a Kähler manifold and a strict nearly Kähler manifold. Such a decomposition is global in the simply connected case [112]. Hence, the study of nearly Kähler manifolds reduces to the case of strict ones. In addition every nearly Kähler manifold in dimension 4 must be Kähler so that the first interesting dimension is six. The following theorem establishes relationship between the twistor space Z ! O of a quaternionic Kähler manifold (orbifold) and nearly Kähler geometry. Theorem 58. Let W .Z; h/ ! .O; gO / be the twistor space of a positive QK manifold with its Kähler structure .J; h; !h /. Then Z admits a strict nearly Kähler structure .J1 ; h1 ; !h1 /. If TM D V ˚ H is the natural splitting induced by then hjV D 2h1jV ;

hjH D h1jH D .gO /;

JjV D J1jV ;

JjH D J1jH :

(27) (28)

Theorem 58 is due to Eells and Salamon [51] when O is 4-dimensional. The higher dimensional analogue was established in [4] (see also [112]). Remark 59. Observe that the metric of the nearly Kähler structure of Theorem 58, in general, is not Einstein. In particular, h1 is not the squashed metric h0 introduced in the diagram (23), unless dim.Z/ D 6. In six dimensions, we can scale h1 so that it has scalar curvature s D 30 and then indeed h1 D h0 as one can easily check. Definition 60. Let M D G=H be a homogeneous space. We say that M is 3-symmetric if G has an automorphism of order 3 such that G0 H G , where G is the fixed point set of and G0 is the identity component in G0 .

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We have the following two theorems concerning nearly Kähler homogeneous Riemannian manifolds. The first is due to Wolf and Gray in all dimensions but 6 [136], [137]. They also conjectured that the result is true for strict nearly Kähler 6-manifolds. The Wolf–Gray conjecture was proved quite recently by Butruille [38], [39] which is the second theorem below. Theorem 61. Every compact homogeneous strict nearly Kähler manifold M of dimension different than 6 is 3-symmetric. Theorem 62. Let .M; g/ be a strict nearly Kähler 6-dimensional Riemannian homogeneous manifold. Then M is isomorphic as a homogeneous space to a finite quotient of G=H , where G and H are one of the following: (1) G D SU.2/ SU.2/ and H D fidg; (2) G D G2 and H D SU.3/, where metrically G=H D S 6 the round sphere; (3) G D Sp.2/ and H D SU.2/U.1/, where G=H D CP 3 with its nearly Kähler metric determined by Theorem 58; (4) G D SU.3/ and H D T 2 , where G=H is the flag manifold with its nearly Kähler metric determined by Theorem 58. Each of these manifolds carries a unique invariant nearly Kähler structure, up to homothety. In every dimension, the only known compact examples of nearly Kähler manifolds are 3-symmetric. On the other hand, Theorem 58 can be easily generalized to the case of orbifolds so that there are plenty examples of compact inhomogeneous strict nearly Kähler orbifolds in every dimension. Theorem 63. Let M be a compact simply-connected strict nearly Kähler manifold. Then, in all dimensions, as a Riemannian manifold M decomposes as a product of (1) 3-symmetric spaces, (2) twistor spaces of positive QK manifolds Q such that Q is not symmetric, (3) 6-dimensional strict nearly Kähler manifold other than the ones listed in Theorem 62. This theorem is due to Nagy [111], but our formulation uses the result of Butruille together with the fact that the twistor spaces of all symmetric positive QK manifolds are 3-symmetric. The LeBrun–Salamon conjecture can now be phrased as follows Conjecture 64. Any compact simply connected strict irreducible nearly Kähler manifold .M; g/ of dimension greater than 6 must be a 3-symmetric space. In particular, the Conjecture 64 is automatically true in dimensions 4n because of Nagy’s classification theorem and also true in dimensions 10 and 14 because all positive QK manifolds in dimension 8 and 12 are known. The third case leads to an important


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Open Problem 65. Classify all compact strict nearly Kähler manifolds in dimension 6. Dimension 6 is special not just because of the rôle it plays in Theorem 63. They have several remarkable properties which we summarize in the following theorem. Theorem 66. Let .M; J; g; !g / be a compact strict nearly Kähler 6-manifold. Then (i) The metric g is Einstein of positive scalar curvature. (ii) c1 .M / D 0 and w2 .M / D 0. (iii) If g is scaled so that it has Einstein constant D 5 then the metric cone .C.M /; dt 2 t C t 2 g/ has holonomy contained in G2 . In particular, C.M / is Ricci-flat. The first property is due to Matsumoto [106] while the second is due to Gray [72]. The last part is due to Bär [12]. In fact, nearly Kähler 6-manifolds is the geometry of the last row of the table of Theorem 40. More precisely we have the following theorem proved by Grunewald [73]: Theorem 67. Let .M 6 ; g/ be a complete simply connected Riemannian spin manifold of dimension 6 admitting a non-trivial Killing spinor with ˛ > 0 or ˛ < 0. Then there are two possibilities: (i) .M; g/ is of type .1; 1/ and it is a strict nearly Kähler manifold, (ii) .M; g/ D .S 6 ; gcan / and is of type .8; 8/. Conversely, if .M; g/ is a compact simply-connected strict nearly Kähler 6-manifold of non-constant curvature then M is of type .1; 1/. Compact strict nearly Kähler manifolds with isometries were investigated in [108] where it was shown that Theorem 68. Let .M; J; g; !g / be a compact strict nearly Kähler 6-manifold. If M admits a unit Killing vector field, then up to finite cover M is isometric to S 3 S 3 with its standard nearly Kähler structure. Remark 69. The first example of a non-trivial G2 holonomy metric was found by Bryant [33], who observed that a cone on the complex flag manifold U.3/=T 3 carries an incomplete metric with G2 -holonomy. The flag U.3/=T 3 is the twistor space of the complex projective plane CP 2 and as such it also has a strict nearly Kähler structure. As explained in this section, this therefore is just one possible example. One gets such non-trivial metrics also for the cones with bases CP 3 and S 3 S 3 with their homogeneous strict nearly Kähler structures. Interestingly, in some cases there exist complete metrics with G2 holonomy which are smooth deformations of the asymptotically conical ones. This fact was noticed by Bryant and Salamon [36] who constructed complete examples of G2 holonomy metrics on bundles of self-dual

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2-forms over CP 2 and S 4 . Replacing the base with any positive QK orbifold O gives complete (in the orbifold sense) metrics on orbibundles of self-dual 2-forms over O. Locally some of these metrics were considered in [125]. More complete examples of explicit G2 holonomy metrics on non-compact manifolds were obtained by Salamon [122]. G2 holonomy manifolds with isometric circle actions were investigated by Apostolov and Salamon [7]. The first compact examples are due to the ground breaking work of Joyce [83].

9 Geometries associated with Sasaki–Einstein 5-manifolds Like 3-Sasakian manifolds Sasaki–Einstein 5-manifolds are naturally associated to other geometries introduced in the previous section. Of course, each such space N and, if the Sasaki–Einstein struc.M 5 ; / comes with its Calabi–Yau cone .C.M /; g/ ture is quasi-regular, with its quotient log del Pezzo surface .Z; h/. But as it turns out, there are two more Einstein metrics associated to g. The examples of this section also illustrate how Theorem 40 and Bär’s correspondence break down when .M; g/ is a manifold with Killing spinors which is, however, not complete. We begin by describing a relation between 5-dimensional Sasaki–Einstein structures and six-dimensional nearly Kähler structures which was uncovered recently in [52]. This relation involves the sine-cones of Definition 1. We use the notation gN s to distinguish the sine-cone metric from the usual Riemannian cone metric g. N Of course this metric is not complete, but one can compactify M obtaining a very tractable stratx D N Œ0; with conical singularities at t D 0 and t D . Observe ified space M the following simple fact which shows that the Riemannian cone on a sine cone is always a Riemannian product. Lemma 70. Let .M; g/ be a Riemannian manifold. Then the product metric ds 2 D dx 2 C dy 2 C y 2 g on R C.M / can be identified with the iterated cone metric on C.Cs .M //. Proof. Consider the map RC .0; / ! R RC given by polar coordinate change .r; t / 7! .x; y/ D .r cos t; r sin t /, where r > 0 and t 2 .0; /. We get ds 2 D dx 2 Cdy 2 Cy 2 g D dr 2 Cr 2 dt 2 Cr 2 sin2 tg D dr 2 Cr 2 .dt 2 Csin2 tg/: So the iterated Riemannian cone .C.Cs .M //; ds 2 / has reducible holonomy 1 Hol.C.M //. This leads to Corollary 71. Let .N; g/ be a Sasaki–Einstein manifold of dimension 2n C 1. Then the sine-cone Cs .N / with the metric gN s D dr 2 C .sin2 r/g is Einstein with Einstein constant 2n C 1. We are particularly interested in the case n D 2. Compare Lemma 70 with the following result in [84], Propositions 11.1.1-2:


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Proposition 72. Let .M 4 ; g4 / and .M 6 ; g6 / be Calabi–Yau manifolds, and let .R3 ; ds 2 D dx 2 C dy 2 C dz 2 / and .R; ds 2 D dx 2 / be the Euclidean spaces. Then (1) .R3 M 4 ; g D ds 2 C g4 / has a natural G2 structure and g has holonomy Hol.g/ 13 SU.2/ G2 , (2) .R M 6 ; g D ds 2 C g6 / has a natural G2 structure and g has holonomy Hol.g/ 1 SU.3/ G2 . As long as .M 4 ; g4 / and .M 6 ; g6 / are simply connected then the products R3 M 4 and R M 6 are simply connected G2 -holonomy manifolds with reducible holonomy groups and parallel Killing spinors. Note that this does not violate Theorem 40 as these spaces are not Riemannian cones over complete Riemannian manifolds. Using (ii) of Proposition 72 we obtain the following corollary of Theorem 3 first obtained in [52] Corollary 73. Let .N 5 ; g/ be a Sasaki–Einstein manifold. Then the sine cone Cs .N 5 / D N 5 .0; / with metric gN s is nearly Kähler of Einstein constant D 5. Furthermore gN s approximates pure SU.3/ holonomy metric near the cone points. Using Corollary 73 we obtain a host of examples of nearly Kähler 6-manifolds with conical singularities by choosing N 5 to be any of the Sasaki–Einstein manifolds constructed in [32], [31], [24], [95], [94], [66], [65], [49], [60], [44]. For example, in this way we obtain nearly-Kähler metrics on N .0; / where N is any Smale manifold with a Sasaki–Einstein metric such as S 5 or k.S 2 S 3 /, etc. Note that every simply connected strict nearly Kähler manifold has exactly two real Killing spinors. So as long as N 5 is simply connected Cs .N 5 / will have two real Killing spinors. Using Theorem 3 the Sasaki–Einstein metrics constructed in [23], [27], [28], [68], [25] in all odd dimensions also give new Einstein metrics on Cs .N 2nC1 /. For example, one obtains many positive Einstein metrics on †2nC1 .0; / where †2nC1 is any odd dimensional homotopy sphere bounding a parallelizable manifold. Of course, there are no Killing spinors unless n D 2. Returning to the case of dimension 6, a somewhat more general converse has been obtained in [52], namely Theorem 74. Any totally geodesic hypersurface N 5 of a nearly Kähler 6-manifold M 6 admits a Sasaki–Einstein structure. The method in [52] uses the recently developed notion of hypo SU.2/ structure due to Conti and Salamon [45]. The study of sine cones appears to have originated in the physics literature [20], [1], but in one dimension higher. Now recall the following result of Joyce (cf. [84], Propositions 13.1.2-3) Proposition 75. Let .M 6 ; g6 / and .M 7 ; g7 / be Calabi–Yau and G2 -holonomy manifolds, respectively. Let .R2 ; ds 2 D dx 2 C dy 2 / and .R; ds 2 D dx 2 / be Euclidean spaces. Then

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(1) .R2 M 6 ; g D ds 2 C g6 / has a natural Spin.7/ structure and g has holonomy Hol.g/ 12 SU.3/ Spin.7/, (2) .R M 7 ; g D ds 2 C g7 / has a natural Spin.7/ structure and g has holonomy Hol.g/ 1 G2 Spin.7/. Again, if .M 6 ; g6 / and .M 7 ; g7 / are simply connected so are the Spin.7/-manifolds R M 6 and R M 7 so that they have parallel spinors. Not surprisingly, in view of Lemma 70 and Proposition 75, the sine cone construction now relates strict nearly Kähler geometry in dimension 6 to nearly parallel G2 geometry in dimension 7. More precisely [20]: 2

Theorem 76. Let .N 6 ; g/ be a strict nearly Kähler 6-manifold such that g has Einstein constant 6 D 5. Then the manifold Cs .N / D N 6 .0; / with its sine cone metric gN s has a nearly parallel G2 structure with Einstein constant 7 D 6 and it approximates pure G2 holonomy metric near the cone points. Proof. Just as before, starting with .N 6 ; g6 / we consider its metric cone C.N 6 / with the metric gN D dy 2 C y 2 g6 and the product metric g8 on R C.N 6 /. With the above choice of the Einstein constant we see that g8 D dx 2 C dy 2 C y 2 g6 must have holonomy Hol.g8 / 1 G2 Spin.7/. By Lemma 70 g8 is a metric cone on the metric g7 D dt 2 C sin2 tg6 , which must, therefore, have weak G2 holonomy and the Einstein constant 7 D 6. Again, any simply connected weak G2 -manifold has at least one Killing spinor. That real Killing spinor on Cs .N 6 / will lift to a parallel spinor on C.Cs .N 6 // D R C.N 6 / which is a non-complete Spin.7/-manifold of holonomy inside 1 G2 . One can iterate the two cases by starting with a compact Sasaki–Einstein 5-manifold N 5 and construct either the cone on the sine cone of N 5 or the sine cone on the sine cone of N 5 to obtain a nearly parallel G2 manifold. We list the Riemannian manifolds coming from this construction that are irreducible. Proposition 77. Let .N 5 ; g5 / be a compact Sasaki–Einstein manifold which is not of constant curvature. Then the following have irreducible holonomy groups: (1) the manifold C.N 5 / with the metric g6 D dt 2 C t 2 g5 has holonomy SU.3/; (2) the manifold Cs .N 5 / D N 5 .0; / with metric g6 D dt 2 C sin2 t g5 is strict nearly Kähler; (3) the manifold Cs .Cs .N 5 // D N 5 .0; / .0; / with the metric g7 D d˛ 2 C sin2 ˛.dt 2 C sin2 t g5 / has a nearly parallel G2 structure. In addition we have the reducible cone metrics: C.Cs .N 5 // D R C.N 5 / has holonomy in 1 SU.3/ G2 and C.Cs .Cs .N 5 /// D R C.Cs .N 5 // D R R C.N 5 / has holonomy 12 SU.3/ 1 G2 Spin.7/. If N 5 is simply connected then g5 , g6 and g7 admit two Killing spinors. For a generalization involving conformal factors see [109].


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Remark 78. Recall Remark 49. Note that when a nearly parallel G2 metric is not complete then the type I-III classification is no longer valid. The group Spin.7/ has other subgroups than the ones listed there and we can consider the following inclusions of (reducible) holonomies Spin.7/ G2 1 SU.3/ 12 SU.2/ 13 18 : According to the Friedrich–Kath Theorem 48 the middle three cannot occur as holonomies of Riemannian cones of complete 7-manifolds with Killing spinors. But as the discussion of this section shows, they most certainly can occur as holonomy groups of Riemannian cones of incomplete nearly parallel G2 metrics. These metrics can be still separated into three types depending on the holonomy reduction: say the ones that come from strict nearly Kähler manifolds are generically of type Is while the ones that come from Sasaki–Einstein 5-manifolds via the iterated sine cone construction are of type IIs and of type IIIs when H SU.3/ is some proper non-trivial subgroup. On the other hand, it is not clear what is the relation between the holonomy reduction and the actual number of Killing spinors one gets in each case.

10 Geometric structures on manifolds and supersymmetry The intricate relationship between supersymmetry and geometric structures on manifolds was recognized along the way the physics of supersymmetry slowly evolved from its origins: first globally supersymmetric field theories (’70s) arose, later came supergravity theory (’80s), which evolved into superstring theory and conformal field theory (late ’80s and ’90s), and finally into M-theory and the supersymmetric branes of today. At every step the “first” theory would quickly lead to various generalizations creating many different new ones: so it is as if after discovering plain vanilla ice cream one would quickly find oneself in an Italian ice cream parlor confused and unable to decide which flavor was the right choice for the hot afternoon. This is a confusion that is possibly good for one’s sense of taste, but many physicists believe that there should be just one theory, the Grand Unified Theory which describes our world at any level.4 An interesting way out of this conundrum is to suggest that even if two theories appear to be completely different, if both are consistent and admissible, they actually do describe the same physical world and, therefore, they should be dual to one another in a certain sense. This gave rise to various duality conjectures such as the Mirror Symmetry Conjecture or the AdS/CFT Duality Conjecture. 4Actually, string theory of today appears to offer a rather vast range of vacua (or possible universes). Such possible predictions have been nicknamed the string landscape [127]. This fact has been seen as a drawback by some, but not all, physicists (see more recent discussion on landscape and swampland in [131], [114]). The insistence that the universe we experience, and this on such a limited scale at best, is the only Universe, is largely a matter of ‘philosophical attitude’ towards science. See the recent book of Leonard Susskind on the anthropic principle, string theory and the cosmic landscape [128].

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The first observation of how supersymmetry can restrict the underlying geometry was due to Zumino [139] who discovered that globally N D 1 supersymmetric models in d D 4 dimensions require that the bosonic fields (particles) of the theory are local coordinates on a Kähler manifold. Later Alvarez-Gaumé and Friedman observed that N D 2 supersymmetry requires that the -model manifold be not just Kähler but hyper-Kähler [6]. This relation between globally supersymmetric -models and complex manifolds was used by Lindström and Roˇcek to discover the hyper-Kähler quotient construction in [100], [80]. The late seventies witnessed a series of attempts to incorporate gravity into the picture which quickly led to the discovery of various supergravity theories. Again the N D 1 supergravity-matter couplings in d D 4 dimensions require bosonic matter fields to be coordinates on a Kähler manifold with some special properties [135] while N D 2 supergravity demands that the -model manifold be quaternionic Kähler [11]. The quaternionic underpinnings of the matter couplings in supergravity theories led to the discovery of quaternionic Kähler reduction in [61], [62]. At the same time manifolds with Killing spinors emerged as important players in the physics of the supergravity theory which in D D 11 dimensions was first predicted by Nahm [113] and later constructed by Cremmer, Julia and Scherk [46]. The well-known Kaluza–Klein trick applied to a D D 11 supergravity model is a way of constructing various limiting compactifications which would better describe the apparently fourdimensional physical world we observe. The geometry of such a compactification is simply a Cartesian product R3;1 M 7 , where R3;1 is the Minkowski space-time (or some other Lorentzian 4-manifold) and M 7 is a compact manifold with so small a radius that its presence can only be felt and observed at the quantum level. Many various models for M 7 were studied in the late seventies which by the eighties had already accrued into a vast physics literature (cf. the extensive three-volume monograph by Castellani, D’Auria and Fré [43]). Most of the models assumed a homogeneous space structure on M 7 D G=H (see Chapter V.6 in [43], for examples). Two things were of key importance in terms of the required physical properties of the compactified theory. First, the compact space M 7 , as a Riemannian manifold, had to be Einstein of positive scalar curvature. Second, although one could consider any compact Einstein space for the compactification, the new theory would no longer be supersymmetric unless .M 7 ; g/ admitted Killing spinor fields, and the number of them would be exactly the number of residual supersymmetries of the compactified theory. For that reason compactification models involving .S 7 ; g0 / were quite special as they gave the maximally supersymmetric model. However, early on it was realized that there are other, even homogeneous, 7-manifolds of interest. The Sp.2/-invariant Jensen metric on S 7 , or as physicists correctly nicknamed it, the squashed 7-sphere is one of the examples. Indeed, Jensen’s metric admits exactly one Killing spinor field since it has a nearly parallel G2 structure. Of course, any of the Einstein geometries in the table of Theorem 40 can be used to obtain such supersymmetric models. The D D 11 supergravity theory only briefly looked liked it was the Grand Theory of Einstein’s dream. It was soon realized that there are difficulties with getting from


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D D 11 supergravity to the standard model. The theory which was to solve these and other problems was superstring theory and later M-theory (which is yet to be constructed). With the arrival of superstring theory and M-theory, supersymmetry continues its truly remarkable influence on many different areas of mathematics and physics: from geometry to analysis and number theory. For instance, once again five-, six-, and seven-dimensional manifolds admitting real Killing spinors have become of interest because of the so called AdS/CFT duality. Such manifolds have emerged naturally in the context of p-brane solutions in superstring theory. These so-called p-branes, “near the horizon” are modelled by the pseudo-Riemannian geometry of the product AdSpC2 M , where AdSpC2 is the .p C 2/-dimensional anti-de-Sitter space (a Lorentzian version of a space of constant sectional curvature) and .M; g/ is a Riemannian manifold of dimension d D D p 2. Here D is the dimension of the original supersymmetric theory. In the most interesting cases of M2-branes, M5branes, and D3-branes D equals either 11 (Mp-branes of M-theory) or 10 (Dp-branes in type IIA or type IIB string theory). String theorists are particularly interested in those vacua of the form AdSpC2 M that preserve some residual supersymmetry. It turns out that this requirement imposes constraints on the geometry of the Einstein manifold M which is forced to admit real Killing spinors. Depending on the dimension d , the possible geometries of M are as follows: d

Geometry of M

.; / N

any

round sphere

.1; 1/

7

nearly parallel G2

. 18 ; 0/

Sasaki–Einstein

. 14 ; 0/

3-Sasakian

. 38 ; 0/

6

nearly Kähler

. 18 ; 18 /

5

Sasaki–Einstein

. 14 ; 14 /

where the notation .; /, N which is common in the physics literature, represents the ratio of the number of real Killing spinors of type .p; q/ to the maximal number of real Killing spinors that can occur in the given dimension. This maximum is, of course, realized by the round sphere of that dimension. So this table is just a translation of the table of Theorem 40 for the special dimensions that occur in the models used by the physicists. Furthermore, given a p-brane solution of the above type, the interpolation between AdSpC2 M and Rp;1 C.M / leads to a conjectured duality between the supersymmetric background of the form AdSpC2 M and a .p C 1/-dimensional superconformal field theory of n coincident p-branes located at the conical singularity of the Rp;1 C.M / vacuum. This is a generalized version of the Maldacena or AdS/CFT Conjecture [101]. In the case of D3-branes of string theory the relevant near

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horizon geometry is that of AdS5 M , where M is a Sasaki–Einstein 5-manifold. The D3-brane solution interpolates between AdS5 M and R3;1 C.M /, where the cone C.M / is a Calabi–Yau threefold. In its original version the Maldacena conjecture (also known as AdS/CFT duality) states that the ’t Hooft large n limit of N D 4 supersymmetric Yang-Mills theory with gauge group SU.n/ is dual to type IIB superstring theory on AdS5 S 5 [101]. This conjecture was further examined by Klebanov and Witten [91] for the type IIB theory on AdS5 T 1;1 , where T 1;1 is the other homogeneous Sasaki–Einstein 5-manifold T 1;1 D S 2 S 3 and the Calabi–Yau 3-fold C.T 1;1 / is simply the quadric cone in C 4 . Using the well-known fact that C.T 1;1 / is a Kähler quotient of C 4 (or, equivalently, that S 2 S 3 is a Sasaki–Einstein quotient of S 7 ), a dual super Yang–Mills theory was proposed, representing D3-branes at the conical singularities. In the framework of D3-branes and the AdS/CFT duality the question of what are all the possible near horizon geometries M and C.M / might be of importance. Much of the interest in Sasaki–Einstein manifolds is precisely due to the fact that each such explicit metric, among other things, provides a useful model to test the AdS/CFT duality. We refer the reader interested in the mathematics and physics of the AdS/CFT duality to the recent book in the same series [21]. In particular, in this context, Sasaki–Einstein geometry is discussed in one of the articles there [67]. Remark 79 (G2 holonomy manifolds, unification scale and proton decay). Until quite recently the interest in 7-manifolds with G2 holonomy as a source of possible physical models was tempered by the fact the Kaluza–Klein compactifications on smooth and complete manifolds of this type led to models with no charged particles. All this has dramatically changed in the last few years largely because of some new developments in M-theory. Perhaps the most compelling reasons for reconsidering such 7-manifolds was offered by Atiyah and Witten who considered the dynamics on manifolds with G2 holonomy which are asymptotically conical [10]. The three models of cones on the homogeneous nearly Kähler manifolds mentioned earlier are of particular interest, but Atiyah and Witten consider other cases which include orbifold (quotient) singularities. Among other things they point to a very interesting connection between Kronheimer’s quotient construction of the ALE metrics [96], [97] and asymptotically conical manifolds with G2 -holonomy. To explain the connection, consider Kronheimer’s construc1 ' U.1/ K.ZnC1 / D U.1/n tion for D ZnC1 . Suppose one chooses a circle Sk;l and then one considers a 7-manifold obtained by performing Kronheimer’s HK quotient construction with zero momentum level . D 0/ while “forgetting” the three moment map equations corresponding to this particular circle. An equivalent way of looking at this situation is to take the Kronheimer quotient with nonzero momentum 1 (such is never D a 2 sp.1/ but only for the moment map of the chosen circle Sk;l in the “good set”) and then consider the fibration of singular Kronheimer quotients over a 3-dimensional base parameter space. Algebraically this corresponds to a partial 1 , resolution of the quotient singularity and this resolution depends on the choice of Sk;l hence . This example was first introduced in [10]. It can be shown that the 7-manifold is actually a cone on the complex weighted projective 3-space with weights .k; k; l; l/,


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where k C l D n C 1. It then follows from the physical model considered that such a cone should admit a metric with G2 holonomy. However, unlike the homogeneous cones over the four homogeneous strict nearly Kähler manifolds of Theorem 62, the metric in this case is not known explicitly. This construction appears to differ from all previous geometric constructions of metrics with G2 holonomy. One can consider similar constructions for other choices of S 1 K. / [17]. In [53] using specific models of M-theory compactifications on manifolds with G2 holonomy, Friedman and Witten address the fundamental questions concerning the unification scale (i.e., the scale at which the Standard Model of SU.3/ SU.2/ U.1/ unifies in a single gauge group) and proton decay. The authors point out that the results obtained are model dependent, but some of the calculations and conclusions apply to a variety of different models.

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Chapter 4

Special geometry for arbitrary signatures María A. Lledó, Óscar Maciá, Antoine Van Proeyen, and Veeravalli S. Varadarajan

Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kähler manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Generalities on quasicomplex and complex manifolds . . . . . . . 2.2 Hermitian metrics and Kähler metrics . . . . . . . . . . . . . . . 2.3 Hermitian line bundles and fiber metrics . . . . . . . . . . . . . . 3 Rigid special Kähler manifolds . . . . . . . . . . . . . . . . . . . . . 3.1 Some geometric preliminaries . . . . . . . . . . . . . . . . . . . 3.2 Definition of rigid special Kähler manifolds . . . . . . . . . . . . 3.3 The signature of the metric . . . . . . . . . . . . . . . . . . . . . 3.4 The prepotential . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 The pseudo-Kähler case . . . . . . . . . . . . . . . . . . . . . . . 3.6 A special pseudo-Kähler manifold . . . . . . . . . . . . . . . . . 3.7 The holomorphic cubic form . . . . . . . . . . . . . . . . . . . . 4 Projective Kähler (Kähler–Hodge) manifolds . . . . . . . . . . . . . . 4.1 Affine transformations, isometries and homothetic Killing vectors 4.2 Definition of projective Kähler manifolds . . . . . . . . . . . . . 4.3 The Levi-Civita connection on a Kähler–Hodge manifold . . . . . 4.4 Examples of Kähler–Hodge manifolds . . . . . . . . . . . . . . . 5 Conformal calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Real manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Kähler manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Projective special Kähler manifolds . . . . . . . . . . . . . . . . . . . 6.1 Definition of projective special Kähler manifolds . . . . . . . . . 6.2 Examples of projective special Kähler manifolds . . . . . . . . . 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Some technical results . . . . . . . . . . . . . . . . . . . . . . . . . . B Connection on a principal bundle and covariant derivative . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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María A. Lledó, Óscar Maciá, Antoine Van Proeyen, and Veeravalli S. Varadarajan

1 Introduction Special Kähler geometry is the geometry of the manifold spanned by the scalars of vector multiplets of D D 4, N D 2 supersymmetry. The rigid version, that occurs in supersymmetry without gravity, appeared first in the references [1] and [2]. The construction for N D 2 supergravity appeared in [3], and it is called projective special Kähler geometry. It played an important role in several developments of string theory. These first formulations offered a local point of view. For the rigid case, the condition for a Kähler geometry to be ‘special’ is the existence of a preferred set of holomorphic coordinates z i , called special coordinates, in which the Kähler potential K can be expressed in terms of a holomorphic function, the prepotential F ,

@K @F kN gi |N D i |N ; K D 2= zN : (1) @z @zN @z k It is seen then as a further restriction on the metric, compatible with the complex structure. For the projective case, the original construction was based on superconformal tensor calculus and involves a projectivization of the manifold due to the extra vector field, the graviphoton, which does not have associated a scalar. In simple words, one has a rigid special manifold with a dilation symmetry and a non physical scalar, which is projected out by fixing the symmetry. The property of being a special Kähler manifold is then a purely geometrical one, and can be formulated independently of supersymmetry. It is given though in terms of a preferred set of coordinates. Although this local formulation is not incomplete (there has to exists an open cover of the manifold by special coordinates), it remains the intriguing question if there is a way of defining what is a special Kähler manifold with global statements, independent of coordinates. The first attempts were made in references [4], [5], [6]. A set of equivalent definitions was found in [7], and later on, a mathematical formulation appeared in [8]. One fundamental ingredient in the global approach is the existence of a certain flat symplectic bundle. Peculiar to Freed’s formulation [8] is that the symplectic bundle is recognised as the tangent bundle, so the construction is intrinsic. In fact, the rigid case (see Definition 3.1) comes out very elegantly, and for this part we will follow closely Freed’s work (with the exception of the pseudo-Riemannian case, which we will mention later). The projective case is much more involved. We define a projective special manifold in terms of a rigid special manifold with a homothetic Killing vector (see Definition 4.2). In this way, the definition is not only intrinsic but directly related to the way in which it is obtained in supergravity [3]. The point of contact of this definition with Freed’s work is in his Proposition 4.6. So far as for Riemannian, special Kähler manifolds. Pseudo-Riemannian special Kähler manifolds1 are very relevant in supergravity. A physically sensible supergravity 1 Note that all discussions on the signature in this work concern the signature of the Kähler manifold, i.e. the target manifold of the supergravity theory. This is unrelated to the signature of spacetime, which we keep Minkowskian to have the standard special geometries. Discussions on generalizations to Euclidean spacetime signature are in references [9], [10], [11].

Chapter 4. Special geometry for arbitrary signatures

87

theory must have a positive definite target-space metric. From the conformal calculus approach it is known that in order to get such positive definite metric the rigid Kähler manifold before projection has to have signature .2; 2n/. But pseudo-Riemannian special Kähler manifolds present an additional complication. Special coordinates are complex coordinates constructed from a set of flat Darboux coordinates .q i ; pi / by taking the holomorphic extension of the q i ’s (or, alternatively, of the pi0 s). They have then the prepotential property (1). When the signature of the metric is indefinite, this holomorphic extension does not always result in a set of n independent holomorphic coordinates. There is a subclass of Darboux systems that have this property. It is important thought that one can always make a constant symplectic rotation to coordinates .q 0 i ; pi0 / such that the q 0 i ’s extend to special coordinates, so there is still a covering of the manifold by special coordinates. But not all flat Darboux systems are suitable to obtain special coordinates. As a consequence, the structure group of the bundle is reduced to a subgroup of the symplectic group. This was first observed in [12]. Nevertheless, flat Darboux coordinates which do not lead to special coordinates nor prepotential are very relevant. They were used to prove that one can break N D 2 supersymmetry partially to N D 1 [13] and not necessarily to N D 0, as it was thought before. This is an extremely important property for phenomenological applications. It is then one of the main motives of this work (which was missing in [8]) to generalize the construction of special geometry to arbitrary signatures. In another context it has been recently shown ([14], [15], [16]) that relating flat Darboux coordinates with the real central charges and attractor equations would have a simplifying role in the description of the attractor mechanism of black holes in N D 2 supergravity (see references [17], [18], [19] for the attractor mechanism). Pseudo-Riemannian, projective special Kähler manifolds appear also as dimensional reductions of supergravity theories in eleven dimensions and exotic signatures, obtained by duality transformations from the standard Minkowskian signature. These are the theories M and M 0 , in signatures (9,2) and (6,5) proposed in [20]. The pseudo-Riemannian special manifolds arising in D D 4 are discussed in [21]. An important part of the work in dealing with projective special Kähler manifolds concerns in fact a more general class of Kähler manifolds, the so-called Kähler–Hodge manifolds.2 It was found in [22] that the Kähler geometries of N D 1 supergravity should be Kähler–Hodge. We propose an intrinsic definition of projective Kähler manifolds (see Definition 4.2), inspired in the conformal calculus approach used in physics. Then we show that they have integer Kähler cohomology class, so they are Kähler–Hodge. This chapter is as much self-contained as possible, so we have included vast review material. On the other hand, having in mind the connection to physics, we have tried to work everything out in coordinates, as to have the sometimes difficult translation between two languages, the physicist’s and the mathematician’s one, each of them with its own advantages. We have also taken time in explaining some examples, which may clarify the abstract definitions. 2A

Kähler–Hodge manifold is a Kähler manifold with integer Kähler cohomology class.

88


The chapter is organized as follows. In Section 2 we review basic material on complex and Kähler manifolds and Hermitian bundles. It is used extensively in the exposition, so with it we set the basic notation. The reader can also skip it and come back to it punctually when some concept is called for. Section 3 is devoted to the rigid case. We start with some geometric preliminaries not included in Section 2 and then we take up the definition of rigid special Kähler manifolds. We follow the lines of reference [8], explaining carefully how the formulas in coordinates are obtained from the abstract definition. Then we treat the pseudoRiemannian case, giving some clarifying simple examples. We come back to Freed’s treatment for the holomorphic cubic form, which will be used later on. Section 4 is dedicated to projective Kähler manifolds as a previous step towards projective special geometry. We introduce some concepts on affine transformations and homothetic Killing vectors and derive some of its properties. This material is not new, but perhaps not so widely known, so it is fundamental to have it at hand. Then we z (with arbitrary define projective Kähler manifolds starting from a Kähler manifold M signature) which has an action of C (as well as other properties, see Definition 4.2). z and construct the symplectic and the line bundles over We then consider M D M=C z (here it. The line bundle has a Hermitian metric induced from the Kähler metric on M the importance of the intrinsic approach), whose Ricci form turns out to be closed and non degenerate, defining then a symplectic structure (actually, a Kähler one). Since it is the first Chern class of a line bundle, the manifold is Kähler–Hodge. We then propose an alternative and beautiful way of understanding the Levi-Civita z It is a bit involved, but it really connection in M, directly induced from the one in M. gives precious insight into the geometry of M. Section 5 is a brief excursion on the origin of projective Kähler geometry as it is seen from a model in physics. It is the simplest one to consider, and it does not include supersymmetry. Indeed, the ideas of conformal calculus are more general than their applications to supergravity. z the condition to be rigid special Kähler, then M will In Section 6 we impose on M be a projective special Kähler manifold. The precise definition is Definition 6.1, and the consequences are analysed in the sequel. In particular, we obtain the holomorphic cubic form and then the formula for the curvature. We conclude with some examples, in particular the pseudo-Riemannian space SU.1; 2/ : SU.1; 1/ U.1/

2 Kähler manifolds This first section recapitulates the basic definitions on complex manifolds and Kähler manifolds in particular. It is essentially a summary of part of Chapter IX in [23].


89

It can be skipped by readers familiar with Kähler manifolds or used just to set the conventions.

2.1 Generalities on quasicomplex and complex manifolds Let M be a quasicomplex (or almost complex) manifold of dimension 2n, with J W TM 7! TM the quasicomplex structure, J 2 D 1. Remark 2.1. Suppose that M is a complex manifold and that .z 1 ; : : : ; z n / are complex coordinates on an open set U M, z j D x j C iy j . Then .x 1 ; : : : ; x n ; y 1 ; : : : ; y n / is a coordinate system in U and we have that

J

@ @x j

D

@ ; @y j

J

@ @y j

D

@ ; @x j

j D 1; : : : ; n:

Let Tmc M denote the complexification of the tangent space at m 2 M. We denote by Tm1;0 M and Tm0;1 M the eigenspaces of J at m with eigenvalues i and i respectively. Then x D X C iJX 2 Tm0;1 M Z D X iJX 2 Tm1;0 M; Z x is a real linear endomorphism for any real vector X 2 Tm M. The operation Z 7! Z called complex conjugation. From now on we will denote the (complexified) tangent space simply as Tm D Tm M. Let Tm c denote the complexification of the cotangent space at m and T c the complexified cotangent bundle of M. Let ! 2 T . The pull back, at each point m, of ! through J , J !m .X/ D !m .JX / for all X 2 Tm ; defines an endomorphism

JW T ! T

with .J /2 D 1, which extends in the obvious way to the complexified cotangent space. The eigenspaces of eigenvalues i and i of J at m are denoted as 1;0 Tm 1;0 m Dƒ

c

and

0;1 0;1 Tm m Dƒ

c

respectively. One has that c

1;0 0;1 m D f!m 2 Tm j !m .Z/ D 0 for all Z 2 Tm g; c

0;1 1;0 m D f!m 2 Tm j !m .Z/ D 0 for all Z 2 Tm g: P 0;0 1;0 r c Since the exterior product space, m D 2n rD0 ƒ Tm , is generated by m , m and 0;1 m , m has a bigrading

m D

n X p;qD0

p;q m ;

90


and so has the space of complex forms D

n X

p;q :

p;qD0

Remark 2.2. If M is a complex manifold as in Remark 2.1, then ² ³ 1 @ @ ˇˇ n @ ˇˇ 1;0 i j ˇ ; Tm D spanC ˇ D m j D1 @z j m 2 @x j @y ² ³ 1 @ @ ˇˇ n @ ˇˇ Tm0;1 D spanC D C i : ˇ ˇ @Nz |N m 2 @x j @y j m j D1 For the complex forms we have

˚ j n j j 1;0 m D spanC dz jm D .dx C idy /jm j D1 ; ˚ |N n 0;1 N jm D .dx j idy j /jm j D1 : m D spanC d z

The set of forms ˚ j N N N dz 1 ^ dz j2 ^ ^ dz jp ^ d zN k 1 ^ d zN k 2 ^ ^ d zN k q with 1 j1 jp n, 1 kN 1 kN q n is a local basis of p;q . For a complex manifold one can prove [23] that the differential dp;q pC1;q C p;qC1 : Then we can define @ W p;q ! pC1;q and @N W p;q ! p;qC1 as N d D @ C @; and since d2 D 0 we have @2 D 0;

@N 2 D 0;

@ B @N C @N B @ D 0:

N D 0. A form ! 2 0;p is said A form ! 2 p;0 is said to be holomorphic if @! to be antiholomorphic if @! D 0. A function is holomorphic if @ f D 0; j D 1; : : : ; n @Nz |N (respectively, antiholomorphic). A holomorphic vector field Z is a complex vector field of type .p; 0/ such that Zf is holomorphic for every holomorphic f . Locally, ZD

n X j D1

with all the f j holomorphic.

fj

@ @z j


91

2.2 Hermitian metrics and Kähler metrics A Hermitian metric on a quasicomplex manifold M with quasicomplex structure J is a Riemannian metric g such that g.JX; J Y / D g.X; Y /

for all X; Y 2 TM:

Every paracompact quasicomplex manifold admits a Hermitian metric. This is because for a given Riemannian metric h and a quasicomplex structure J we can obtain a Hermitian metric by setting g.X; Y / D h.X; Y / C h.JX; J Y /

for all X; Y 2 TMI

g is extended to T c by linearity. It is easy to check that 1. g.Z; W / D 0 for Z; W of type .1; 0/, x > 0, 2. g.Z; Z/ x W x / D g.Z; W /. 3. g.Z; The fundamental 2-form of a Hermitian metric is ˆ.X; Y / D g.X; J Y /

for all X; Y 2 TM:

It is non degenerate at each point of the manifold. Remark 2.3 (Almost complex linear connections). The torsion of a quasicomplex structure J is the tensor field (1-covariant, 2-contravariant) N.X; Y / D 2fŒJX; J Y ŒX; Y J ŒX; J Y J ŒJX; Y g: A quasicomplex structure is said to be integrable if it has no torsion. This is equivalent to saying that the commutator of two vector fields of type .1; 0/ (alternatively .0; 1/) is a vector field of type .1; 0/ (alternatively .0; 1/). To see this, let Z; W be such that J Z D iZ and J W D iW , then if N.Z; W / D 0 it is immediate that J ŒZ; W D iŒZ; W . In the other direction, a real vector field can be always written x where Z is .1; 0/ and Z x is .0; 1/. Let also Y D W C W x. as the sum X D Z C Z, Then it is immediate to prove that N.X; Y / D 0. A quasicomplex structure is a complex structure if and only if it is integrable. This is the Newlander–Nirenberg theorem [24]. We say that a linear connection is quasicomplex if the covariant derivative of the quasicomplex structure is zero (which is equivalent to being a connection in the bundle of complex linear frames). Every quasicomplex manifold admits a quasicomplex affine connection whose torsion T is proportional to the torsion N of the quasicomplex structure. In general, the Riemannian connection associated to a Hermitian metric is not quasicomplex. If it is so, then the quasicomplex structure has no torsion and the fundamental form is closed. The converse is also true: for a complex manifold,

92


the Riemannian connection of a Hermitian metric is quasicomplex if and only if the fundamental 2-form ˆ is closed. (The proof of these statements can be read in [23], Chapter IX.) A quasicomplex manifold, with a Hermitian metric is a quasi-Kähler (or almost Kähler) manifold if the fundamental form is closed. Let M be a differential manifold. A symplectic structure on M is a 2-form ˆ such that i. it is closed, dˆ D 0, ii. it is non degenerate: for every X 2 TM, there exists Y 2 TM such that ˆ.X; Y / ¤ 0. The couple .M; ˆ/ is a symplectic manifold, and M has always even dimension. In any symplectic manifold, we have local Darboux coordinates defined by the following theorem. Theorem 2.4 (Darboux). If M is a symplectic manifold, dim M D 2n, for each m 2 M there is a chart .U; ' W U ! R/ such that '.m/ D 0 and for u 2 U , 1 n '.u/ D x .u/; : : : ; x .u/; y1 .u/; : : : ; yn .u/ and ˆ on the open set U is n X ˇ ˇ dx i ^ dyi : Û D

iD1

Notice that a quasi-Kähler manifold is a symplectic manifold, since the fundamental 2-form is non degenerate. If, in addition, the manifold is complex then it is a Kähler manifold. Moreover, let D be a Riemannian connection, so DX g D 0 for every vector field X on M. We have that DX ˆ.Y; Y 0 / D DX g.Y; J Y 0 / C g.Y; .DX J /Y 0 / D 0;

(2)

which means that the Riemannian connection is trivially a symplectic connection. The holonomy of a Kähler manifold of complex dimension n is a subgroup of U.n/ ' O.2n/ \ GL.n; C/, since the Riemannian connection is quasicomplex. Here GL.n; C/ is taken in its real representation A B A C iB ! : B A One can prove that if the manifold is Ricci flat then the restricted holonomy group (that is, considering only parallel displacements along paths that are homotopic to a point) is contained in SU.n/. Kähler manifolds in coordinates. On a quasicomplex manifold, we can consider the principal bundle of unitary frames, that is the bundle of complex frames that are orthonormal with respect to the Hermitian metric. Its structural group is U.n/. We will denote this bundle by U.M/. We want to give the metric, connection and curvature of a Kähler manifold in coordinates.

93


Let M be a complex manifold, with Hermitian metric g and complex structure J . We use the notation of Remarks 2.1, and 2.2 and denote Zi D

@ ; @z i

x i D Z{N D @ : Z @Nz i

It is easy to see that gij D g.Zi ; Zj / D 0;

xi ; Z xj / D 0; g{N|N D g.Z

(3)

xj / D g|Ni , so3 and the only non zero components are of the form gi |N D g.Zi ; Z g D gi |N .dz i ˝ d zN |N C d zN |N ˝ dz i / D 2gi |N dz i d zN |N : x {N / D iZ x {N , the fundamental 2-form is Since J.Zi / D iZi and J.Z ˆ D 2igi |N dz i ^ d zN |N : If M is a Kähler manifold, the fundamental form is closed, so

dˆ D 2i

(4)

gi |N k gi |N N dz ^ dz i ^ d zN |N C N d zN k ^ dz i ^ d zN |N D 0; k @z @Nz k

which implies gk |N gi |N i D 0; k @z @z

gi |N @Nz kN

gi kN D 0: @Nz |N

(5)

These equations are the integrability condition for the existence of a real valued function K such that @K gi |N D i |N : @z zN For any real function K, the tensor gi |N satisfies .gi |N / D gj {N and property (5). If it is positive definite, then it is a Kähler metric on M. So any Kähler metric can be written locally in this way. Notice also that K is defined modulo a holomorphic function f , K ! K C f .z/ C f .z/: N K is the Kähler potential We will denote by I an arbitrary index in f1; 2; : : : ; 2ng and by fx I g arbitrary coordinates in M. Let Y D Y I @x@I D Y I @I be a vector field on M. The covariant derivative of Y with respect to a linear connection can be written as .DJ Y /I D

@Y I I C JK Y K; @x J

where the Christoffel symbols are I D .DJ @K /I : JK

the conventions with factors for symmetric products and for forms. A symmetric product of forms ˛ˇ ˝ ˇ C ˇ ˝ ˛/. Similarly a wedge product is taken as ˛ ^ ˇ D 12 .˛ ˝ ˇ ˇ ˝ ˛/.

3 Note

is

1 2 .˛

94


The Levi-Civita connection is the only torsionfree connection satisfying Dg D 0. The Christoffel symbols are

1 @gJL @gJI @gLI D g KL C : (6) J 2 @x @x I @x L For a complex manifold, we can extend the covariant differentiation to complex vector fields by linearity. We can then consider I D i for I D 1; : : : ; n and I D n C {N for I D n C 1; : : : ; 2n. The Christoffel symbols become complex, and it is easy to see that N I xJK D IN N ; K JI

JK

where we have denoted IN D I C n for I D i and IN D I n for I D n C i . A linear connection is quasicomplex if the complex structure is parallel. For J D idz j ˝ @j id zN |N ˝ @|N this means N

d B D D B .DA J /B ) AdcN D Ac D 0: C D AD JC AC JD D 0 H

If the connection is torsionfree we have, A A D CB ; T .X; Y / D DX Y DY X ŒX; Y D 0 H) BC

so the only non zero Christoffel symbols of the quasicomplex connection are i ; ji k D kj

|{NNkN D k{NN |N :

(7)

If the Riemannian connection is quasicomplex then the manifold is a Kähler manifold, and we have from (6) and (5) N

ji k D g i ` @j g`k N ;

|{NNkN D g {N` @|N g`kN :

(8)

The curvature tensor associated to a linear connection is a 3-contravariant 1covariant tensor given by R.X; Y /Z D ŒDX ; DY Z DŒX;Y Z;

X; Y; Z 2 TM;

and in components R

I

JKL

I I @KJ @LJ D @x K @x L

C

X

M I M I KM KJ LM LJ :

M

It satisfies RI JKL D RI JLK . If the connection is torsionfree, the curvature tensor satisfies the Bianchi identities R.X; Y /Z C R.Z; X /Y C R.Y; Z/X D 0; DX R.Y; Z/ C DZ R.X; Y / C DY R.Z; X/ D 0:

(9) (10)

(If the torsion is not zero, then the Bianchi identities are modified by terms containing the torsion, see [23], volume I, page 135.)

95


It is immediate to see that for a quasicomplex connection we have Ri |Nk` D 0;

Ri |NkN `N D 0;

Ri j kN `N D 0;

Ri |Nk `N D 0:

From (8) one finds that for a Kähler metric Ri j k` D 0; and the only components that can be different from zero are i ; Ri j k `N D @`N kj

i Ri j k` N D @kN j ` ;

{N R{N |Nk` N D @` kN |N ;

R{N |Nk `N D @k |{NN`N ;

(11)

and those obtained using the symmetry property RI JKL D RI JLK . The upper and lower line are related by complex conjugation. The Ricci tensor is the contraction RAB D RC ACB . We have that Rij D R{N|N D 0;

N

x{Nj D @|N k D @|N .g k ` @i g N /: Ri |N D R ik k`

Let G D det gi |N , then N

@i G D Gg k ` @i gk `N H) Ri |N D @i @|N log jGj:

(12)

Example 2.5 (The complex projective space CP1 ). We consider the complex projective space of 1-dimensional subspaces in C 2 . Let z 1 ; z 2 be the natural coordinate system in C 2 , z i W C 2 ! C. They are complex linear maps. Let U1 be the set of subspaces S such that z 1 jS ¤ 0. Then z 1 jS spans the dual space to S , so we may write (13) z 2 jS D t 1 z 1 jS ; t 1 2 C: Each equation as (13) defines a subspace in U1 , so t 1 is a complex coordinate in U1 . In the same way we can define U2 as the set of subspaces S such that z 2 jS ¤ 0. Then we have that z 1 jS D t 2 z 2 jS ; t 2 2 C and t 2 is a complex coordinate in U2 . f.U1 ; t 1 /; .U2 ; t 2 /g is a complex atlas of CP1 . In the intersection U1 \ U2 the gluing condition is t2 D

1 : t1

We want to define a Kähler metric on CP1 . On U1 and U2 we consider, respectively, the following real-valued functions: f1 D .t 1 tN1 C 1/;

f2 D .t 2 tN2 C 1/:

It is easy to see that the two 2-forms defined by 1 ˆ1 D 4i@@N ln f1 D 4i 2 dt 1 ^ d tN1 ; f1

1 ˆ2 D 4i@@N ln f2 D 4i 2 dt 2 ^ d tN2 f2

96


coincide in the intersection, so they define globally a closed 2-form ˆ. The Kähler metric is then g.X; Y / D ˆ.JX; Y /: One can see that it is positive definite by computing it in an open set: d s2 D

4 dt 1 d tN1 : .1 C t 1 tN1 /2

2.3 Hermitian line bundles and fiber metrics Hermitian fiber metrics are introduced here and will be used later, in Section 4. The definitions and statements in this section can be found in references [23], [25]. Let E ! M be a rank k complex vector bundle over the complex manifold M. Then the fiber at m 2 M, Em , is a complex vector space of complex dimension k. Let us assume that the total space E has a complex structure, that the projection W E ! M is a holomorphic map4 between complex manifolds and that there is a local trivialization fUA gA2I such that the maps 1 .UA / ! UA C k and their inverses are holomorphic with 1 .m/ D Em C k . Then we say that E is a holomorphic vector bundle over M. Example 2.6 (The tangent bundle of a complex manifold). Let M be a complex manifold. Let .x j ; y j /, j D 1; : : : ; n be coordinates on a neighbourhood U of m 2 M such that the complex structure on M is given by

@ J @x j

@ D j; @y

@ J @y j

D

@ ; @x j

j D 1; : : : ; n:

(14)

A vector on m is of the form j Vm D Xm

@ @ C Ymj j : @x j @y

j The components Xm ; Ymj are coordinates on Tm .M/. On 1 .U/ we have coordinates .x j ; y j ; X j ; Y j /, and a quasicomplex structure on TM is given by (14) and

@ J @X j

@ D ; @Y j

@ J @Y j

D

@ ; @X j

j D 1; : : : ; n:

The quasicomplex structure is integrable and we have complex coordinates on 1 .U/: .z j D x j C iy j ; Z j D X j C iY j /: TM is a complex manifold and a holomorphic vector bundle over M. 4 That

is, a map preserving the complex structures.


97

A fiber metric on a vector bundle E ! M is a smooth assignment of an inner product on each fiber hm W Em Em ! R: If the fiber has a complex structure Jm we require that the inner product is Hermitian, h.JX; J Y / D h.X; Y /

for all X; Y 2 Em ;

and we say that E is a Hermitian vector bundle. A connection r on E is metric if rh D 0 (the connection is extended to E ˝E ). If the bundle is holomorphic, we can ask the covariant derivative of a holomorphic section to be holomorphic, rs 2 1;0 .E/

for s holomorphic:

(15)

There is a unique metric connection satisfying (15); it is the Hermitian connection of the Hermitian vector bundle (see for example [23]). Let fsa gaD1;:::;k be a holomorphic frame of the bundle E on a neighbourhood U of m 2 M (that is, k independent local sections) and f˛ a g the dual coframe. The connection 1-form on U, for a connection satisfying (15), is .rsb / D a b sa D i a b dz i ˝ sa ;

i D 1; : : : ; n; a; b D 1; : : : ; k;

so the covariant derivative of a holomorphic section s D aa sa is rs D .@i aa C i a b ab /dz i ˝ sa : The hermiticity of the fiber metric means h D 2habN ˛ a ˛N b ; and the condition for the connection to be metric is @i habN i c a hc bN D 0 H) i a b D hacN @i hcb N ;

(16)

where hacN is the inverse matrix of hacN . The curvature is then Ra bi |N D hacN @i @|N hb cN C hacN hd eN @i hb eN @|N hd cN : We can define the Ricci form of the Hermitian bundle as the trace of the curvature tensor, D 2iRa ai |N dz i ^ d zN j : If E is a Hermitian line bundle, that is, it has rank 1, then the metric is just h D .z; z/˛ N ˛; N and the Ricci form becomes N log jj D 2i@i @|N log jjdz i ^ d zN |N : D 2i @@

(17)

98


3 Rigid special Kähler manifolds In this section we will deal with rigid special Kähler geometry, or simply special Kähler geometry, as opposed to projective special Kähler geometry, which will be the subject of Section 6.

3.1 Some geometric preliminaries This part is inspired in the second chapter of [26]. Let E be a vector bundle over M with a connection r. For every vector field X of M (section of TM), r sends sections of E to sections of E, rX W .E/ ! .E/: Let pM .E/ D ƒp .M/ ˝ .E/ be the space of E-valued p-forms on M, with 0M .E/ D .E/. We are going to define the covariant differential dr W pM .E/ ! pC1 M .E/. For 0-forms we define dr

0M .E/ ! 1M .E/; F 7! dr F;

such that dr F .X/ D rX F; X 2 TM:

This definition can be extended to pM .E/, dr

pM .E/ ! pC1 M .E/; F 7! dr F; assuming the condition dr .˛ ^ F / D d˛ ^ F C .1/p ˛ ^ dr F for ˛ 2 ƒp .M/ and F 2 qM .E/. For example, if F 2 1M .E/, locally F D dx i ˝ ˛i D dx i ^ ˛i with ˛i 2 0M .E/, then dr F .X; Y / D dx i ^ dr ˛i .X; Y / D X i rY ˛i Y i rX ˛i : If AIJ D AIJ dx is the 1-form connection matrix in an open set U M, then .dr F /I D dF I C AIJ ^ F J ; from which it is easy to deduce the standard transformation rule under a local fiber (gauge) transformation F 0 D UF;

.dr F /0 D U dr F H) A0 D dU U 1 C UAU 1 ;


99

2 where .dr F /0 dF 0 C A0 F 0 . Differently than for the ordinary differential, dr is not zero in general. In fact, 2 I J I .dr F /I D .dAK C AIJ ^ AK / ^ F K D RK ^ F K;

where R is the (Lie algebra valued) curvature 2-form associated to the connection. A flat connection then defines a complex. The de Rham complex is associated to the trivial connection on the trivial bundle E D M V . It is easy to check the Bianchi identity dr R D 0. In the associated bundle with typical fiber the Lie algebra, the group acts with the adjoint representation. The covariant differential in such bundle is then I I dr RJI D dRJI C AK ^ RJK AK J ^ RK D 0:

3.2 Definition of rigid special Kähler manifolds Here we follow the first section of reference [8]. Definition 3.1. Let M be a Kähler manifold with Kähler form ˆ and complex structure J . A special Kähler structure on M is a real, flat, torsionfree, symplectic connection r satisfying dr J D 0: (18) J is seen here as a 1-form with values in the tangent bundle TM, and the covariant differential must be interpreted in the sense described in Section 3.1. As we have seen, a Kähler manifold is always symplectic, being the Kähler form ˆ its symplectic form. On a symplectic manifold, a linear connection r is said to be symplectic if rˆ D 0: (19) We want to see what is the meaning of the ingredients in this definition. We first examine the implications of the existence of a flat, torsionfree connection. Let U be an open set, with coordinates fx I gI2nD1 and a (matrix-valued) connection K K 1-form AL D AML dx M . 2 D 0, so a dr -closed form is locally dr -exact. Due to the flatness condition, dr Let 1 be the identity endomorphism in TM. It can be seen as a TM valued 1-form, 1 D @I ˝ dx I . The torsionfree condition can be expressed as K K K dr 1 D .dr 1/K @K D ALM dx L ^ dx M ˝ @K D 0 H) ALM D AML :

(20)

˛J @J g2n ˛D1

A local frame on TM on U open M is a set f˛ D of 2n local sections of T U TM that are linearly independent for each point x 2 U . Since the connection is flat (the curvature tensor is zero), there exists a flat frame, that is, rI ˛ D 0 for I D 1; : : : ; 2n;

or equivalently,

dr ˛ D 0:

(21)

100 María A. Lledó, Óscar Maciá, Antoine Van Proeyen, and Veeravalli S. Varadarajan This is because the integrability condition of (21) is 2 I ˛ D RK ^ ˛K @I D 0 dr I D 0. for the 2n independent sections ˛ , which implies necessarily that RK ˛ ˛ I ˛ ˛ Let D I dx be the dual coframe, that is .ˇ / D ıˇ . We have that

I˛ ˇI D ı˛ˇ () ˛J I˛ D ıIJ : Then we can express

1 D ˛ ˝ ˛ D ˛ ^ ˛ ;

and dr 1 D 0 H) dr ˛ ^ ˛ ˛ ^ dr ˛ D 0 H) dr ˛ D d ˛ D 0: This means that ˛ D dt ˛ for some functions t ˛ . Then ˛ D @=@t˛ and t ˛ are local coordinates on U called flat coordinates. Up to here, we used the fact that the connection is real, flat and torsionfree. We introduce now the additional condition that the connection is symplectic, that is, rI ˆ D 0. We denote the symplectic matrix as 0 1nn P D : (22) 1nn 0 The coordinates t ˛ are Darboux coordinates if 1 P˛ˇ dt ˛ ^ dt ˇ : 2 It is possible to choose the flat coordinates t ˛ in such a way that they are Darboux. This is because ˆ.˛ ; ˇ / D P˛ˇ ;

so

ˆD

@I .ˆ.˛ ; ˇ // D rI .ˆ/.˛ ; ˇ / ˆ.rI ˛ ; ˇ / ˆ.˛ ; rI ˇ / D 0; so ˆ.˛ ; ˇ / is a constant (antisymmetric, non degenerate) matrix which can always be brought to the form (22) by a linear change of coordinates. We see that the existence of a flat, torsionfree, symplectic connection on M is equivalent to having a covering by flat Darboux coordinates (it is also said that M has a flat symplectic structure). If fq ˛ g2n ˛D1 are also flat Darboux coordinates, we have that the transition functions satisfy dq ˛ D since This implies

@q ˛ ˇ dt ; @t ˇ

r t dq ˛ D

@q ˛ dt ˇ D 0 @t ˇ @t

r t dt ˇ D r t ˇ D 0:

@q ˛ D 0 H) q ˛ D A˛ˇ t ˇ C c ˛ ; @t ˇ @t with A˛ˇ and c ˛ constant. It follows that A 2 Sp.2n/.

101


Let us now consider the condition (18). In arbitrary coordinates fx I gI2nD1 it becomes J D J I @I D JKI @I ˝ dx K ;

I .dr J /I D dJ I C AL ^ J L D 0;

(23)

which in components reads 1 I 1 I L / C .AML JNL AINL JM / D 0: .@M JNI @N JM 2 2 (The factor 1=2 appears when M and N are arbitrary, so each strict component is counted twice). This implies, assuming that the connection is torsionfree, that AIi|N D AI{Nj D 0:

(24)

The connection r is a linear connection (a connection on the frame bundle of M), so one can compute K K M K .rI J /L D @I JLK C AIM JLM AIL JM :

(25)

The condition rI J D 0 together with the torsionfreeness implies, in addition to (24) that N D 0: AimN nN D A{mn If the connection is torsionfree and rI J D 0 then we have that dr J D 0, but the converse is not necessarily true. Then the flat symplectic connection is not necessarily complex. The complex structure can be written locally in terms of the complex coordinates fz j gjnD0 as J D i.@z j ˝ dz j @zN |N ˝ d zN |N / D i. .1;0/ .0;1/ /; where

.1;0/ D @z j ˝ dz j

and

.0;1/ D @zN |N ˝ d zN |N

(26)

are the projectors onto the TM .1;0/ and TM .0;1/ spaces respectively. The condition dr J D 0, together with the torsionfreeness, is equivalent to dr .1;0/ D 0: Indeed, one can also write

1 D .1;0/ C .0;1/ ;

(27)

and the torsionfree condition was expressed as dr 1 D 0. Using the Poincaré lemma, dr .1;0/ D 0 implies that locally there exists a complex vector field such that r D dr D .1;0/ ; which is unique up to a flat complex vector field. Let fx j ; yj gjnD1 be a flat Darboux coordinate system, that is, ˆ D dx j ^ dyj ;

and

dr

@ @x j

D 0;

dr

@ @yj

D 0:

(28)

102 María A. Lledó, Óscar Maciá, Antoine Van Proeyen, and Veeravalli S. Varadarajan In this coordinate system we denote

@ 1 i @

j ; D i 2 @x @yj

(29)

where j ; j are complex functions ( is a complex vector field). Taking the covariant differential and using (28) we obtain .1;0/ D dr D

1 @ @ d j ˝ j dj ˝ : 2 @x @yj

(30)

.1;0/ is a (1,0) tensor, so it follows that j and j are holomorphic functions. Taking the real part of this equation we have, using (27), indicates the usual transpose). From the set of 2n functions . j ; j / we can always select n independent holomorphic functions that form a set of holomorphic coordinates. From the fact that the symplectic form ˆ D dx i ^ dyj is of type .1; 1/ and using (32) and (35) in (28), and compare with (4), we obtain as conditions for ˛ and ˇ ˛ > ˇ C ˇ > ˛ D 0; ˇ > ˛N ˛ > ˇN D 8ig;

(39)

where g is the n n matrix gi |N . These are the equations that we can obtain in general where we have not used any information on the signature of the metric.

3.3 The signature of the metric Let us first assume that the metric is positive definite (Riemannian metric). We want to show that ˛ itself has rank n. Suppose that rank.˛/ < n. Then there exists a holomorphic vector c such that ˛ j l .z/c l .z/ D 0: But then ˇj l c l .z/ ¤ 0, since otherwise the total rank of the matrix (38) would be lower than n. This means that there exists a non zero, holomorphic linear combination of the vectors f@=@y k g, namely X @ D cQk .z/ ¤ 0; cQj .z/ D ˇj k c k : @yk Then, as has only y-components, 0 D ˆ. ; N / D g. ; J N / D g. ; i N / D ig. ; N /: For a Kähler manifold (with positive definite metric) g. ; N / D 0 () D 0;

(40)

so we have a contradiction and ˛ must have rank n. Looking now to (35) we can conclude that f j g is a set of holomorphic coordinates. In the same way we can prove that ˇ has rank n so fj g is also a set of holomorphic coordinates. Remark 3.2 (Symplectic transformations (Riemannian case)). One can see independently that a real symplectic transformation cannot change the rank of ˛ and ˇ, provided they satisfy the following conditions: 1. ˛ and ˇ have rank n. 2. ˛ > ˇ C ˇ > ˛ D 0,

104 María A. Lledó, Óscar Maciá, Antoine Van Proeyen, and Veeravalli S. Varadarajan 3. ˇ > ˛N ˛ > ˇN D 8ig, where g is the n n matrix gi |N . In our case, conditions 2. and 3. were obtained in (39). Let us first introduce the vielbein for the metric 0 g e 0 0 1 > ; DE E ; ED 0 eN 1 0 g> 0

g D e eN > :

We can define ˛ D 2˛ 0 e > ;

ˇ D 2ˇ 0 e > ;

so we can express 2. and 3. as 2. ˛ 0 > ˇ 0 C ˇ 0 > ˛ 0 D 0, 3. ˇ 0 > ˛N 0 ˛ 0 > ˇN 0 D 2i1. Let us denote ˛ 0 D ˛00 C i˛10 , ˇ 0 D ˇ00 C iˇ10 with ˛i0 and ˇi0 real. We define the matrix 0 ˛0 ˛10 S0 D : ˇ00 ˇ10 Then properties 2. and 3. are equivalent to 0 1 0 1 : S0 D S0> 1 0 1 0

(41)

Equation (41) means that S0 is a symplectic matrix, S0 2 Sp.2n; R/. We are ready now to prove the statement above. We have that 0 1 ˛ D : S0 ˇ0 i1 We assume that rank.˛ 0 / D rank.ˇ 0 / D n and that (41) holds. We want to prove that for Ay By Sy D y y 2 Sp.2n; R/; C D the matrix

0 ˛O y ˛0 D S ˇ ˇO

O D n. Let us write is such that rank.˛/ O D rank.ˇ/ 0 0 1 ˛ 1 ˛ y y D S S0 S0 DS ; S ˇ0 ˇ0 i1

with S D Sy S0 :

S is an arbitrary matrix in Sp.2n; R/, so all we have to prove is that 1 A B 1 A C iB S D D i1 C D i1 C C iD


105

is such that rank.A C iB/ D rank.C C iD/ D n. We consider the matrices M D i.A C iB/;

N D .C C iD/:

We have that A M N D 1 iAH ;

AH D A> C C B > D;

since S is a symplectic matrix, A> C D C > A;

B > D D D > B;

A> D C > B D 1:

The matrix AH is therefore also symmetric, and can be diagonalized such that A is diagonalized with eigenvalues of the form .1 C ia/ ¤ 0. The determinant of A is the product of its eigenvalues, so it is different from zero. This implies that det M ¤ 0, det N ¤ 0, so our statement is proven. Remark 3.3 (Symplectic transformations (pseudo-Riemannian case)). If g has pseudo-Riemannian signature, there are symplectic transformation changing the rank of ˛ and ˇ satisfying 1. to 3. in Remark 3.2. It is enough to give one of such symplectic matrices. First we realize that, as before, conditions 2. and 3. can be put as 2. ˛ 0 > ˇ 0 C ˇ 0 > ˛ 0 D 0, 3. ˇ 0 > ˛N 0 ˛ 0 > ˇN 0 D 2i , where is the flat pseudo-Riemannian metric. For definiteness, let us assume that the signature of is .n 1; 1/ (the other cases can be obtained in the same way). We take

in the standard form 0 1 1 0 0 0 B0 1 0 0 C B C B C

D B ::: C; B C @0 0 0 1 A 0 0 1 0

2 D 1 and the vielbein is defined accordingly. We have that the matrix 0 ˛0 ˛10

S0 D ˇ00 ˇ10

is a symplectic matrix, condition that is equivalent to 2 and 3. Also, we have that 0 1 ˛ S0 D ; ˇ0 i

0 so we can bring ˇ˛ 0 to the standard form i1 with the symplectic transformation

106 María A. Lledó, Óscar Maciá, Antoine Van Proeyen, and Veeravalli S. Varadarajan S01 . It is enough to consider the n D 2 case. The symplectic matrix 0 1 1 0 0 0 B0 0 0 1C C S DB @0 0 1 0 A 0 1 0 0 has the property

0

1 0 1 1 0 1 0 00 B0 1C Bi 0C 1 CDB C D ˛ 00 ; DSB S @0 i A @ 0 i A ˇ i

i 0 0 1

with the property that det ˛ 0 D det ˇ 0 D 0, as we wanted to show. The proof in the remark 3.2 is not valid here because the real part of A, determined by .A C iB / and .C C iD /, would be zero. We consider now a pseudo-Riemannian metric (pseudo-Kähler manifold). Notice that in this case (40) is not true since we can have null vectors. In fact, assume that we have a holomorphic vector field such that ˆ. ; N / D 0 and let us consider the vectors N N˙ D ˙ : Since g is of type (1,1) and ˆ. ; N / D ig. ; N / D 0, each one of the terms below is separately 0, g.N˙ ; N˙ / D g. ; / C g. ; N N / ˙ 2g. ; N / D 0; and then NC and iN are null, real vectors. On the other hand, if N is a null, real vector, its holomorphic and antiholomorphic extensions D N iJN satisfy g. ; N / D 0 and thus ˆ. ; / N D 0. We will treat the case of special pseudo-Kähler manifolds in Section 3.5. We have thus proven in this section that for a positive definite metric, the matrices ˛ and ˇ are each of rank n. When the metric is not positive definite, this proof breaks down due to null vectors that may be zero modes of these matrices. However, these matrices might even then still be invertible (see the example in Section 3.6). In fact, in [7] it is proven that with a symplectic rotation we can always bring ˛ to be non degenerate. A sketch of the proof is given in Appendix A.

3.4 The prepotential We come back to the positive definite metric, or, at least that ˛ and ˇ are invertible. Then f j gjnD1 and fj gjnD1 are called conjugate coordinate systems. Equations (26) and (35) then imply @ .1;0/ D d j ˝ j ; @


from which, comparing with (30),

@ @ 1 @ j k ; D @yk @ j 2 @x j

j k D

@k : @ j

107

(42)

The Kähler form is 1 ˆ D dx j ^ dyj D .d j C d N |N / ^ . j k d k C |NkN d N k /: 4 Since it is of type (1,1), it follows that

ij D j i ;

(43)

so

1 i d ^ d N |N . ij N{N|N /: 4 Comparing to (4), we see the metric and the Kähler form become ˆD

1 1 gi |N D =. ij /; ˆ D i=. ij /d i ^ d N |N : (44) 4 2 Because of (43), there exists a local holomorphic function, determined up to a constant, such that @F @2 F (45) j D 8 j ; ij D 8 i j : @

@ @

F is called the holomorphic prepotential. In terms of it, the Kähler potential becomes

1 @F k K D =.k N k / D 2=

N : 4 @ k

(46)

The coordinate system f j gjnD1 is a special coordinate system. In the particular case in which j k D iıj k , then j D x j C iyj , and i i d ^ d N {N ; 2 so the manifold is locally isometric to C n . ˆD

Recovering the flat connection. A structure of special geometry can be given, in an open set, by a holomorphic function F . / such that =. ij / with ij as in (45), is a non singular, negative definite matrix. The holomorphic coordinates are declared to be special coordinates. From the knowledge of F we can recover the flat symplectic coordinates @F x i D /1 form a subgroup, and this subgroup is maximal (we relegate the proof to the Appendix, Lemma A.3). For matrices of this form, we have that det A ¤ 0, so the rank of ˛ is preserved. On the other hand, as we proved in Remark 3.3, there exists always a symplectic transformation that does not preserve the rank of ˛. The conclusion is that the flat symplectic structure of the tangent bundle is reduced to the subgroup of matrices ³ ² A 0 Sp.n; R/: C .A> /1

3.6 A special pseudo-Kähler manifold Let .z 1 ; z 2 / be holomorphic coordinates on C 2 and consider the prepotential 1 F D iz 1 z 2 ; 8 then

ij D 8

@2 F 0 i 0 1 D D i ; i 0 1 0 @z i @z j

111


1 1 0 1 ; gi |N D =. ij / D 4 4 1 0 1 1 d s 2 D 2gi |N dz i d zN j D =. ij /dz i d zN j D dz 1 d zN 2 C dz 2 d zN 1 ; 2 2 which clearly has signature .2; 2/ (null vectors always come in pairs, one holomorphic and one antiholomorphic). The Kähler form is ˆ D 2igi |N dz i ^ d zN j D

i i =. ij /dz i ^ d zN j D .dz 1 ^ d zN 2 C dz 2 ^ d zN 1 /: 2 2

Let us denote z 1 D x 1 C iy2 ;

z 2 D x 2 C iy1 ;

(50)

the real and imaginary parts of the complex holomorphic coordinates. These are the Darboux coordinates of (28). Then the Kähler form takes the standard form ˆ D dx 1 ^ dy1 C dx 2 ^ dy2 ; so .x i ; yi / are symplectic coordinates. For these symplectic coordinates, there is associated a special holomorphic system of coordinates, just as in the Riemannian case. We want to show now that not all the symplectic coordinate systems have this property when the metric is pseudo-Riemannian. Let us make the following symplectic change of coordinates: x 01 D x 1 ;

x 02 D y2 ;

z 1 D x 01 C ix 02 ;

with We have

y10 D y1 ;

y20 D x 2 ;

(51)

z 2 D y20 C iy10 :

@ 1 @ @ 1 @ @ D dz 1 ˝ i 2 C dz 2 ˝ i 1 i 1 2 @z 2 @x @y 2 @x @y 1 @ @ @ @ D dz 1 ˝ 1 C dz 2 ˝ 2 idz 2 ˝ 1 idz 1 2 : 2 @x @x @y @y

.1;0/ D dz i ˝

(52)

Comparing this equation with (30)

and thus, following (35),

d 1 D dz 1 ;

d1 D idz 2 ;

d 2 D dz 2 ;

d2 D idz 1 ;

1 0 ˛ D ; 0 1 >

0 1 ˇ Di : 1 0 >

We can use the new variables .x 0 ; y 0 / defined in (51) to calculate . 0 ; 0 /: .1;0/ D dz 1 ˝

1 @ @ 1 @ @ i 02 C dz 2 ˝ 0 i 0 : 01 2 @x @x 2 @y2 @y1

(53)

112 María A. Lledó, Óscar Maciá, Antoine Van Proeyen, and Veeravalli S. Varadarajan Comparing this equation with (30) d 01 D dz 1 ;

d01 D idz 2 ;

d 02 D idz 1 ; d02 D dz 2 ; then ˛ 0> D

1 i ; 0 0

ˇ 0> D

0 0 : i 1

We compute now the null vector, following the general case explained at the beginning of Section 3.3. Let c D .c1 ; c2 / be such that 1 i c˛ 0> D .c1 ; c2 / D .0; 0/ H) c1 D 0: 0 0 For any c D .c1 ; c2 / we have that cˇ

0>

0 0 D .c1 ; c2 / D c2 .i; 1/; i 1

so .0; c2 / is not a null vector of ˇ 0> . The vector and its complex conjugate N are then @ @ @ @ D i 1 C 2 and N D i 1 C 2 : @z @z @Nz @Nz Then 1 ˆ. ; / N D i.dz 1 ^ d zN 2 C dz 2 ^ d zN 1 /. ˝ N / D 0: (54) 2

3.7 The holomorphic cubic form Let M be a rigid special Kähler manifold. We want to compute the difference between the Levi-Civita connection D and the flat connection r. Using the same notation as in reference [8], we define the tensor BR as BR 2 1 .M; EndR TM/:

BR r D;

Since both connections are symplectic, Dˆ D 0 and rˆ D 0 (see (2) and (19)), we have that @u .ˆ.v; w// D ru .ˆ.v; w// D ˆ.ru .v/; w/ C ˆ.v; ru .w//; @u .ˆ.v; w// D Du .ˆ.v; w// D ˆ.Du .v/; w/ C ˆ.v; Du .w//; 0 D ˆ..BR /u .v/; w/ C ˆ.v; .BR /u .w//:

(55)

This says that the endomorphism .BR /u , for arbitrary u, is in the Lie algebra sp.2n; R/ defined by ˆ. In components, using (7) and (24), we get N

BR D r D D .Akij ijk /d i ˝ d j ˝ @k C Akij d i ˝ d j ˝ @kN N

N

C .Ak{N|N {Nk|N /d N {N ˝ d N |N ˝ @kN C Ak{N|N d N {N ˝ d N |N ˝ @k :


113

Let u, v, w vectors of type (1,0). Then .BR /u .w/ N D 0, and the last line of (55), with N D 0. In components, this means w replaced by w, N implies that ˆ..BR /u .v/; w/ .Akij ijk / D 0;

N

N

.Ak{N|N {Nk|N / D 0;

where the second one follows by complex conjugation. One can define an element B in 1;0 .Hom.TM; TM// such that N

B D Akij d i ˝ d j ˝ @kN ;

x so BR D B C B:

Lowering the antiholomorphic index with the metric, we can define locally a holomorphic 3-tensor: N

„ij k D 2igi `N Aj`k () Aj{N k D

1 {N` ig „`j k : 2

(56)

Using (47) and the fact that g |N` D 8iˇ |N` we get „ij k D

1 @ ij @3 F D 2 ; 4 @ k @ i @ j @ k

(57)

from which it follows that „ is holomorphic and symmetric. In [8] the following global definition is given for this tensor: „.X; Y; Z/ D ˆ. .1;0/ X; .rY .1;0/ /Z/:

(58)

In fact, since D .1;0/ D 0, we can substitute r by BR in (58) so „.X; Y; Z/ D ˆ. .1;0/ X; Œ.BR /Y ; .1;0/ Z/ x Y ; .1;0/ Z/ D ˆ. .1;0/ X; Œ.B C B/ D ˆ. .1;0/ X; BY .1;0/ Z/; which in components, using (4), gives (56). It is then clear that given the flat connection r we can determine the cubic form „. Conversely, assume that we are given a holomorphic symmetric cubic form „ on a Kähler manifold. We can determine a tensor BR D B C Bx from (56). Then a new connection is defined by r D D C BR . The symmetry of „ guarantees that r is torsionfree, symplectic and satisfies (18), as it follows straightforwardly from (56). 2 D 0, The flatness condition imposes some restrictions on „. We have to impose dr with dr F D dD F C B ^ F C Bx ^ F; for F 2 pM .TM/. Then, if R is the curvature of the Levi-Civita connection, 1 I R JKL dx K ^ dx L , then 2 2 D 0 () R C dD B C dD Bx C B ^ Bx C Bx ^ B D 0: dr

114 María A. Lledó, Óscar Maciá, Antoine Van Proeyen, and Veeravalli S. Varadarajan Analysing the holomorphic and antiholomorphic components in this equation, we obtain that the following expressions have to cancel separately: R C B ^ Bx C Bx ^ B D 0; dD B D 0; dD Bx D 0.

(59) (60) (61)

Equations (60) and (61) are the complex conjugate of each other. (59) imposes a constraint on the curvature of the Kähler manifold. It should be expressed solely in terms of the holomorphic cubic form. In coordinates this means 1 ``N0 p0 pN x N0 N : g g „p0 j i „ (62) pN ` k 4 (60) imposes a constraint on the metric and the cubic tensor. In components we have R` ij kN D AjpNi A`pN kN D N

N

Bjk D Akij d i D so

N

N

1 k` N g „ìj d i ; 2i N

2idD Bjk D dD g k` ^ „ìj d i C g k` dD .„ìj d i / D 0: N

Here, g k` is understood as the components of an element of 0 .M; TM ˝TM/, so its covariant differential is the covariant derivative and it is 0. Only the last term survives, so we have 0

0

j ` „`0 ij kj „ìj 0 / d k ^ d i D 0: dD .„ìj d i / D .@k „lij k`

(63)

It is easy to see that for a metric as in (44) and a cubic form as in (58) this equation is satisfied identically. What this argument proves is that (59) and (60), or the equivalent statements (62) and (63), are sufficient conditions to have a flat symplectic connection satisfying the requirements of a rigid special Kähler structure. Indeed, given a Hermitian metric and an arbitrary holomorphic cubic form with components „ij k D „.ij k/ , one can construct a torsionfree, symplectic connection as r D D C BR , where the connection coefficients for BR are determined by (56). This connection, by construction, satisfies dr J D 0. Then (59), (60) are equivalent to the 2 statement that r is flat, (dr D 0). So given a Hermitian metric and a holomorphic cubic form, they will in this case define a special Kähler structure.

4 Projective Kähler (Kähler–Hodge) manifolds 4.1 Affine transformations, isometries and homothetic Killing vectors For the results in this section see [23], Chapter VI. An affine transformation of a manifold M with linear connection r is a diffeomorphism f W M ! M whose tangent map Tf W TM ! TM maps any parallel


115

vector field along a curve into a parallel vector field along the curve f . /. The push-forward by f of a vector field X on M is f .X/ D Tf B X B f 1 ; or, in components,

@f I J 1 X .f .x//@I : (64) @x J If Y and Z are two vector fields on M and f is an affine transformation, then f X.x/ D

.f rY Z/ D rf Y .f Z/:

(65)

Let K be a vector field on M and let ' t W U ! M be the flow of K on a neighbourhood U of x 2 M, t 2 ; Œ . ' t is a local uniparametric group of transformations, and for each x 2 M, ' t .x/ is an integral curve of K: d' t .x/ D K.' t .x//: dt We say that K is an infinitesimal affine transformation of M if ' t is an affine transformation of U (the connection being the restriction of r to U ). Specifying f D ' t in (65) and taking a derivative with respect to t and putting t D 0 one obtains5 LK B rY rY B LK D rŒK;Y ;

for every vector field Y on M;

(66)

which characterizes K as an infinitesimal affine transformation. (Here LK stands for the Lie derivative with respect to K). In components, this condition reads (67) K J @J rI Z L r K J @J Z L C Z J rI @J K L C .@I K J /rJ Z L D 0: The infinitesimal affine transformations form a subalgebra of the Lie algebra of vector fields on M. For torsionfree connections, (66) reduces to R.K; Y /Z C rY rZ K rrY Z K D 0;

(68)

or, in components (as Z is arbitrary), K J RJIK L C rI rK K L D 0;

(69)

which was used in [27] as the definition of symmetry of the physical sigma model, independently of the action (in fact, such action may not exist). A vector field X on M is complete if each integral curve ' t .x/ extends to t 2 1; C1Œ. This means that the local uniparametric group extends to a global uniparametric group R M ! M; .t; x/ 7! ' t .x/: ˇ 5 One uses here d ' X ˇ D ŒK; X D LK X , where ' t .x/j tD0 D x. The first can be derived d t t t D0 from (64) with f D ' t .

116 María A. Lledó, Óscar Maciá, Antoine Van Proeyen, and Veeravalli S. Varadarajan We say that r is a complete linear connection if every geodesic extends to t 2 1; C1Œ. The Lie algebra of the Lie group6 of affine transformations consists of all infinitesimal affine transformations that are complete. Moreover, if r is complete then all the infinitesimal affine transformations are complete. Let M be a manifold with Riemannian metric g and Riemannian connection r. An isometry of M is a transformation that leaves the metric invariant. An isometry is necessarily an affine transformation of M with respect to the Riemannian connection. A vector field X is an infinitesimal isometry (or Killing vector) if the uniparametric group of transformations generated by X in a neighbourhood of x 2 M (for arbitrary x) consists of local isometries. An infinitesimal isometry X is characterized by LX g D 0; which implies that the set of infinitesimal isometries is a Lie algebra. As in the case of affine transformations, the Lie algebra of the Lie group of isometries is the Lie algebra of all complete infinitesimal isometries, and if M is complete all the infinitesimal isometries are complete. We say that a transformation of a (pseudo) Riemannian manifold M is homothetic if there is a positive constant a2 (which depends on ) such that . g/x .X; Y / D g.x/ .T .X /; T .Y // D a2 gx .X; Y /; for all X; Y 2 TM and x 2 M:

(70)

Notice that the Christoffel symbols for the metrics g and a2 g are the same, so the covariant derivatives are the same. It is then easy to see that a homothetic transformation is an affine transformation of the Levi-Civita connection. An infinitesimal transformation K of a (pseudo) Riemannian manifold is homothetic if its flow is a homothetic transformation in a neighbourhood of each point x 2 M. Infinitesimal homothetic transformations are also called homothetic Killing vectors and can be characterized as LK g D cg;

(71)

for a constant c. This can be seen by substituting D ' t , the flow of X, in (70) and taking the derivative with respect to t at t D 0. We obtain also da2 ˇˇ : dt t D0 If D is the Levi-Civita connection, then (71) is equivalent to the statement that cD

g.X; DY K/ C g.Y; DX K/ D cg.X; Y /

for all X; Y 2 TM;

since DK X LK X D DX K. In components we have DI KJ C DJ KI D cgIJ : 6 It

is necessary to assume that M has a finite number of connected components.

(72)


117

Let us consider the 1-form gK .X/ D g.K; X /. If r is a torsionfree connection we have that dgK .X; Y / D rX .gK /.Y / rY .gK /.X /: This is true for any 1-form. In our case, rX .gK /.Y / D rX .g/.K; Y / C g.rX K; Y /: If the connection is compatible with the metric, rX .g/ D 0, we have dgK .X; Y / D g.rX K; Y / g.rY K; X /; so dgK D 0 () g.rX K; Y / g.rY K; X / D 0

for all X; Y 2 TM;

(73)

in components rI KJ rJ KI D 0: We say that K is a closed homothetic Killing vector if it is a homothetic Killing vector such that gK is a closed 1-form. If K is a closed, homothetic Killing vector and D is the Levi-Civita connection, then equations (72) and (73) imply that DY K D 12 cY

for all Y 2 TM:

(74)

This condition is also sufficient. In components we have that 1 cıI J : 2 Observe that the statement (74) involves only the connection, so we can use it to generalize the concept of closed homothetic Killing vector to any linear connection. For a torsionfree connection r, we will say that a vector field is a closed homothetic Killing vector if (75) rY K D 12 cY for all Y 2 TM: DI K J D

We would like to see if such a vector is in fact an infinitesimal affine transformation for the linear connection. For an arbitrary torsionfree connection (66) is reduced to (68). Using (75) the last two terms of (68) vanish, so the condition for a closed homothetic Killing vector to be an infinitesimal affine transformation is R.K; Y /Z D 0:

(76)

On the other hand, for an arbitrary connection, the integrability condition of (75) is R.Y; Z/K D 0 for all Y and Z, which implies R.K; Y /Z R.K; Z/Y D 0 by using the Bianchi identity (9). The symmetric combination in Y and Z is not zero in general.

118 María A. Lledó, Óscar Maciá, Antoine Van Proeyen, and Veeravalli S. Varadarajan We conclude that in general for a torsionfree connection, a closed homothetic Killing vector is not necessarily an infinitesimal affine transformation. For a flat, torsionfree connection, (76) is trivial and thus in this case the closed homothetic Killing vector is an infinitesimal affine transformation. For the Levi-Civita connection (not necessarily flat), we have seen that any homothetic transformation is an affine transformation. In fact, because of the symmetries of the curvature tensor R.K; Y /Z D 0 () R.Y; Z/K D 0; so K is an infinitesimal affine transformation, even if the connection is not flat. Let M be a Kähler manifold with complex structure J and let g be the Hermitian metric. Let H be a holomorphic vector field. The equations above can be extended to the complexified tangent space. We assume that H is a homothetic Killing vector. In components this reads LH g˛ˇN D g ˇN D˛ H D cg˛ˇN () DY H D cY;

Y holomorphic:

(77)

As before, the last expression does not involve explicitly the metric and can be used as a generalization of holomorphic homothetic Killing vector for any linear connection. x is homothetic with the same Since the metric is Hermitian, it is easy to see that H 1 x constant c (real). It follows that K D 2 .H C H / is also a homothetic Killing vector 1 x / is a Killing vector, with constant c while Ky D JK D 2i .H H 1 .LH g C LHx g/ D cg; 2 1 LKy g D .LH g LHx g/ D 0: 2i Notice that (79) can be written in components as LK g D

(78) (79)

N g ˇN D˛ H g˛ N DˇN H D 0;

which is just the requirement that K is closed, so if H is a holomorphic, homothetic x / is a closed homothetic Killing vector. Killing vector then K D 12 .H C H The converse is also true: if the metric has a closed homothetic Killing vector K, then JK is a Killing vector. It also implies the presence of the holomorphic homothetic Killing vector H D .1 iJ /K, i.e. satisfying (77). Example 4.1 (Euclidean space). We consider C n with the metric d s 2 D dz ˛ d zN ˛ : We take H D z ˇ @=@z ˇ . Then LH g D g; so H is an holomorphic, homothetic Killing x / D 1 .z ˇ @=@z ˇ C zN ˇ @=@Nz ˇ /: Then vector with c D 1. We define K D 12 .H C H 2 gK D

1 ˛ ˛ .z d zN C zN ˛ dz ˛ / H) dgK D 0: 2


119

4.2 Definition of projective Kähler manifolds z be a complex manifold and let H be a holomorphic vector field. Then ŒH; H x D Let M z x 0, so fH; H g form an integrable distribution on T M. By Frobenius theorem, there is z whose leaves7 are complex submanifolds of M z whose tangent space a foliation of M z x . If H ¤ 0 at all points of M the foliation is regular; then is generated by H and H all the leaves have complex dimension 1. We can define an equivalence relation on z by declaring as equivalent two points if they belong to the same leaf. Then, if M z by this relation (the set of all equivalence the foliation is regular, the quotient of M classes) is a manifold. Let K and Ky be, as above, the real and imaginary parts of H , respectively, so y Let ' and 'O the flows of K and Ky respectively, H D K C iK. d' .x/ D K.' .x//; d

d 'O .x/ y 'O .x//; D K. d

'0 .x/ D 'O0 .x/ D x:

y D 0, K is invariant under the flow of Ky and viceversa. This in turn Since ŒK; K implies that ' B 'O D 'O B ' : Let us define D i and D ' B 'O ; then it is easy to see that d .x/ d .x/ d .x/ D Ci D H. .x//; 0 .x/ D x: d d d is a local, complex, 1-dimensional group of transformations,

(80)

B 0 D C 0 : H is the fundamental vector field of the action of G. z a holomorphic action of C : We consider now on M C M ! M; .b; x/ 7! Rb .x/;

(81)

with b 2 C . Locally, b D exp and Rb .x/ D .x/ with as in (80). Let H be the (holomorphic) fundamental vector field of this action (80). The orbits of the action are the integral submanifolds of the foliation defined by H . We assume also that the action is free, so the orbits are diffeomorphic to C . Since the group is abelian, the z is a principal C -bundle over the orbit space left action is also a right action, so M z M=C . z be a (pseudo) Kähler manifold Definition 4.2 (Projective Kähler manifold). Let M z with metric g. Q We assume that on M there is a free holomorphic action of C such that the fundamental vector field H is a non null, holomorphic homothetic Killing 7 The

leaves of a foliation are disjoint sets whose union is the whole manifold.

120 María A. Lledó, Óscar Maciá, Antoine Van Proeyen, and Veeravalli S. Varadarajan z vector of the metric gQ (or the Levi-Civita connection D), z Y H D cY LH gQ D c gQ () D

z for all Y 2 T 1;0 M;

z is a (pseudo) projective x / ¤ 0. Then we say that M D M=C such that g.H; Q H Kähler manifold. We are going to show that M is a Kähler manifold itself, of a particular class. In order to do that, we need to construct a Hermitian line bundle on M. It is in fact z inherited from the tangent bundle on T M. z has the structure of a The symplectic and line bundles and the fiber metric. M z principal C -bundle over M, W M ! M. As in (81) we denote the finite right action z of C on M z ! M; z M Q m Q 7! Rb .m/;

b 2 C,

Q D m. Q with R1 m z be the tangent bundle. The tangent of the action above gives an action on Let T M z TM z ! T M; z TM Q dRb vmQ /: .m; Q vmQ / 7! .Rb .m/; z is a complex vector space, so we also have an action of C on it. We will simply TmQ M denote it by multiplication, z ! TmQ M; z TmQ M Q bvmQ /: .m; Q vmQ / 7! .m; z using these actions. We identify We are going to define an associated bundle to M z elements in T M related by .m; Q vmQ / .Rb .m/; Q b 1 dRb vmQ /:

(82)

It is easy to see that this is an equivalence relation. The quotient space is a complex vector bundle over M of rank n C 1, with dimC M D n. We will denote it by z . It is a bundle over M associated to the C -principal bundle M z !M H D T M= (see [23]), so there is an action of C on it. Also, the underlying real vector bundle of H (and its complexification) inherit the action of the symplectic group Sp.2n C 2; R/ z from T M. Q is a vertical A vector in the kernel of the projection ker.T jmQ / spanC fH.m/g z consisting of vertical vectors. It is a vector. We can consider the subbundle of T M z and it projects to a line bundle on M. We will denote it by trivial line subbundle of T M, L. Two different trivializations .m; Q H.m// Q and .m Q 0 ; 0 H.m Q 0 // (with .m Q 0 / D .m/) Q


121

are related, according to (82), by .m; Q H.m// Q .m Q 0 ; 0 H.m Q 0 // H) m Q 0 D Rb .m/; Q 0 D b 1 ; since (80) implies for the finite global transformation that dRb H.m/ Q D H.Rb .m//. Q 1 The transition functions of the bundle are of the form b . On L we can define a fiber metric. Let .m; Q H.m// Q be a representative of the Q H.m// Q in .m; um / 2 L. We equivalence class .m; vm / 2 L, and the same for .m; set x .m// x .m//: hm .vm ; uN m / D gQ mQ .H.m/; Q N H Q D .N /gQ mQ .H.m/; Q H Q We remind that H.m/ Q is non null by assumption. We just have to check that this definition is independent of the representatives that we have used, so acting with b 2 C we have 1 x .Rb .m/// H.Rb .m//; Q bN 1 N H Q gQ mb Q .b 1 N gQ R .m/ x .Rb .m/// .H.Rb .m//; Q H Q D ./.b N b/ b Q x .m//; Q H Q D ./ N gQ mQ .H.m/;

as we wanted to show. The last equality follows from (70), taking b bN D a2 . We can now define the Kähler structure on M. The metric and the Kähler potential. Let ˛ be a local basis of L (a coframe) dual to the frame fH g of L, so ˛.H / D 1. Using the formulae from the end of Section 2.3 we have (the index a runs only over one value and can be omitted) h D ˛ ˛; N

x .m//: Q H Q D 2gQ mQ .H.m/;

(83)

z Let z i , i D 1; : : : ; n be We want to compute using convenient coordinates in M. complex coordinates on an open set U M. Let s W U ! 1 .U / be a local section on M. Then we can choose the local trivialization 1 .U / U C given by m Q D .m; ys.m//; O

m D .m/; Q yO 2 C : open

z We define homogeneous coordinates .zO i ; y/ O are local coordinates on 1 .U / M. ˛ 1

O on .U / as

O 0 D y; O

O ˛ D yO zO i for ˛ D i:

(84)

z defined in equation (81), expressed in these coordinates, is The action of C on M simply Rb .zO i ; y/ D .zO i ; b p y/; O

Rb . O ˛ / D b p O ˛ ;

for an arbitrary p 2 R. So the fundamental vector field is H D O ˛

@ @ D yO : @ O ˛ @yO

(85)

122 María A. Lledó, Óscar Maciá, Antoine Van Proeyen, and Veeravalli S. Varadarajan (One may choose a multiple of it, which by the definition (77) amounts to a rescaling of c.) The homothety condition is LH gQ ˛ˇN D O

@ O @ g Q C gQ N D c gQ ˛ˇN : N @ O ˛ˇ @ O ˛ ˇ

We make the change of variables

˛ D . O ˛ /c H) H D c ˛

@ : @ ˛

In these coordinates the homothety condition is LH gQ ˛ˇN D c

@ @ g Q C c gQ N D c gQ ˛ˇN : N @ ˛ˇ @ ˛ ˇ

This condition becomes simply

@ gQ N D 0: @ ˛ˇ

(86)

Together with its complex conjugate, (86) implies the following property of the metric: gQ ˛ˇN . ; N / N D gQ ˛ˇN . ; /: N

(87)

If we choose p D 1=c, the action Rb in the coordinates ˛ is Rb ˛ D b ˛ :

(88)

If we denote

0 D y;

i D yz i ;

(89)

z are also homogeneous then the z i are coordinates on M. The coordinates on M coordinates, which we will further use, and from now on @˛ D @@˛ . We have that H D c ˛ @˛ D cy

@ : @y

(90)

The metric in these coordinates can be written in terms of a Kähler potential gQ ˛ˇN D @˛ @ˇN K: The transformation (87) leads to N C f . / C f 0 . /; N K. ; / N D K 0 . ; / with N N D ./K. ;

/: N K 0 . ; N / Since K is real, f D fN0 and with a Kähler transformation K ! K f fN

(91)

123


we can take K 0 as the Kähler potential. We will denote it as K from now on, so we have N K. ; N / N D ./K. ;

/: N (92) In particular, this implies

@ K D K;

N @N K D K;

N ı @ @ıN K D K;

(93)

so the definition (83) gives x ˇN D c 2 ˛ N ˇN @˛ @N N K D c 2 K: D gQ ˛ˇN H ˛ H ˇ Let us consider the exact (1,1)-form

(94)

1 @K @K 1 @2 K Q D 2Q˛ˇN d ^ d N D 2 2 ˛ ˇ C d ˛ ^ d N ˇ ; K @ @ N K @ ˛ @ N ˇ ˛

ˇ

(95)

and let us denote by i its pull-back by the section s, @ ˛ @ N ˇ i dz ^ d zN |N D i: @z i @Nz |N Using (92), we can see that the result is independent of the section s used. In fact, we have that Q D i , @z i @Nz {N (96) Q˛ˇN D ii jN ˛ ˇ ; @ @ N s Q D 2Q˛ˇN

z ! M in coordinates. where z i . ˛ / is the expression of the projection map W M ˛ ˛ The tensor Q˛ˇN is degenerate. Indeed, H D c is a zero eigenvector due to the identities (93). We want to show that there is no other zero mode, under the assumption that gQ ˛ˇN is non-degenerate. Let us write it as 1 1 gQ ˛ˇN ˛ N ˇ D @˛ @ˇN log jKj; ˛ @˛ K D @˛ log jKj: K K We assume now that there is a vector v ˛ such that v ˛ Q˛ˇN D 0, then we find that Q˛ˇN D

N

v ˛ D K.v /gQ ˇ ˛ N ˇ ; N

N

where gQ ˇ ˛ is the inverse of gQ ˛ˇN . Hence any zero eigenvector is proportional to gQ ˇ ˛ N ˇ , and thus there is only one zero mode. In particular, we also obtain N

H ˛ D N ˇ gQ ˇ ˛ ; for some undetermined function . ; /. N 0 i The vectors @i D @=@ are transversal to H , thus the matrix @ ˛ @ N ˇ @z i @Nz |N is non degenerate. This matrix (or a matrix proportional to it) can therefore be taken to be the metric on M. i |N D i Q˛ˇN

124 María A. Lledó, Óscar Maciá, Antoine Van Proeyen, and Veeravalli S. Varadarajan We define therefore the metric on M gi |N D @i @|N Œ˙ log jKj D ˙ii |N ;

˙ D sign K:

(97)

The reason for the ˙ convention will be explained below. The Ricci form of the Hermitian bundle agrees according to (17) with @2 log jKj i dz ^ d zN |N @z i @Nz |N 1 @K @K 1 @2 K D 2i 2 i |N C dz i ^ d zN |N : K @z @zN K @z i @Nz |N We can also compute the signature of the matrix Q˛ˇN . A vector V D V ˛ @˛ is orthogonal to H if ˛ V ˛ D 0 and the space of such vectors has dimension n. For two such vectors, V and V 0 we have D 2i |N dz i ^ d zN |N D 2i

Q Vx 0 /; g.V; Q Vx 0 / D K .V; so the signature of Q in the space orthogonal to H the same than the signature of gQ in such space up to a sign. Furthermore, the sign in the remaining direction of gQ is the sign of K as it follows from (93). We can choose a section s such that the vectors @i have a lift s @i orthogonal to H . Then i |N D is .@ Q i ; @|N /; which is actually independent of the section. So the signature of the metric g in M, (97), is the same than the signature of gQ in the space orthogonal to H . We conclude that defines a symplectic structure compatible with the complex structure, so M is a Kähler manifold with Kähler metric as in (97). The Kähler form is in the first Chern class of a line bundle. This implies that the Kähler form is integer. Such manifolds are called Kähler–Hodge manifolds in the literature. When M is compact, this condition implies that M is a projective variety, so it is embedded in projective space. This is the Kodaira embedding theorem, see for example [28], page 181. What we have proven here is that a projective Kähler manifold is a Kähler–Hodge manifold.

4.3 The Levi-Civita connection on a Kähler–Hodge manifold The previous part leads to consider (97) as the metric on M. One can compute its Levi-Civita connection. However, there is a natural way of inducing a connection on z which gives the same result. It clarifies the M from the Levi-Civita connection in M, geometrical meaning of the metric in the quotient manifold M. We will perform two z and then to the tangent projections of the connection, first to the bundle H D T M= bundle TM.


125

Projecting down to the symplectic bundle. Let X be a vector field on M and a z ! M and p W T M z ! H the natural projections. Let D z section of H . Let W M z denote a linear connection on T M. z in The idea is to find adequate lifts Xz of X and Q of , both vector fields on M, Q z Q projects through p to the same vector such way that the covariant derivative DXz .m/ on H , independently of the point m Q in the fiber 1 .m/ where it has been computed. This will define immediately a covariant derivative on H as Q Q ; .m/ z z . DX .m/ D p D Q D m: (98) X m/ Let us first define the respective lifts. A local section of H is specified by Q D m. We can associating an equivalence class Œ.m; Q vmQ / to any point m, with .m/ Q m/ z choose an arbitrary m Q 2 1 .m/ and set . Q D vmQ . Then Q is a vector field on M satisfying (see equation (82)) Q m/ Q Q b .m// Q D b 1 dRb . Q () Rb Q D b : .R

(99)

There is a one to one correspondence between the set of local sections of H and the z satisfying (99). So Q is a natural lift of . set of local sections of T M z z Q is a vector field on M z satisfying (99), so it Notice that (98) means just that D X z we defines a section of H . For any affine transformation Rb of the connection D, have that (65) Q DD Q z z / z Rb .D z Rb ; X Rb X

so all we need to complete the definition (98) are lifts satisfying Q Rb Q D b ; Rb Xz D Xz :

(100) (101)

z but (100) is already guaranteed. There are many lifts of the vector field X to T M, z (or on its associated bundle L), so we have a connection on the principal bundle M it is natural to consider the horizontal lift. Horizontal lifts satisfy (101), so this will show the existence of the induced connection on H . Note that for the Levi-Civita connection or for an arbitrary flat connection, Rb are affine transformations, so the result applies for these cases of special interest. To understand the horizontal lift we introduce the definition of connection on a principal bundle as a Lie algebra valued 1-form. The relation with the standard covariant derivative in the associated vector bundles can be found in many places (see for example [23]). For completeness we give a brief outline in the Appendix B. Definition 4.3. Let G be a Lie group and g its Lie algebra. A connection on a principal G-bundle P ! M can be given by a g-valued 1-form ! on P such that: (i) If A is a fundamental vector field, generating the action of G on the fibre, assoy ciated to Ay 2 g then !.A/ D A. (ii) Rb ! D Adb 1 ! D b 1 !b; b 2 G. (Ad is the adjoint representation of G).

126 María A. Lledó, Óscar Maciá, Antoine Van Proeyen, and Veeravalli S. Varadarajan A horizontal vector Xu is a vector in Tu P satisfying !.Xu / D 0. In fact, ! defines a distribution of horizontal spaces on TP , denoted by TP h . At each point u with .u/ D m, the horizontal space is mapped isomorphically to Tm M. If TP v D ker.Tju / is the set of vertical vectors tangent to the fiber, then Tu P D Tu P h ˚Tu P v . Moreover, (ii) implies that the distribution is equivariant, that is TRb .u/ P h D T Rb .Tu P h /:

(102)

Let X be a vector field on TM. One can prove that there is a unique vector field Xz on z D X and X. z m/ TP such that T.X/ Q is horizontal for every m. Q It is the horizontal lift of X. The equivariance of horizontal subspaces, (102), implies (101) as we wanted to z satisfying the invariance show. One can also show that any horizontal vector field on M condition (101) is the lift of a vector field on M. One can prove that if Xz and Yz are horizontal lifts of X and Y respectively, then z is a flat connection (as r) z then the z ŒX; Yz is the horizontal lift of ŒX; Y . So if D induced connection D on H is also a flat connection. z As an example that we will use in the following, Example 4.4 (Horizontal lift in M). z we are going to compute the horizontal lift of a holomorphic vector X on TM to T M for the Hermitian connection. In the coordinates (89) we have Rb .m; y/ D .m; by/;

d Rb D dz i ˝ @i C bdy ˝ @y :

(103)

The connection 1-form and its pull back are ! D !y dy C i dz i ;

s ! D i dz i :

i is determined by the pull back, which from (16) and (94) is i D K 1 @i K D @i log jKj: The other component, !y , is determined by conditions (i) and (ii) in Definition 4.3. Since A D y@y and Ay D 1, (i) implies !y D y 1 . (ii) is then satisfied. The connection 1-form is then ! D y 1 dy C i dz i D y 1 dy C @i log jKj dz i :

(104)

z v D v i @i C v y @y is horizontal if and only if A vector on T M, y 1 v y C i v i D 0: If v is arbitrary, then v D v h C v v with v h D v i @i yi v i @y ; v h is the horizontal projection of v.

v v D .v y C yi v i /@y :

(105)


127

A vector Xz D Xz y @y C Xz i @i is the horizontal lift of X D X i @i if z D X; T.X/ !.Xz / D 0;

i.e., Xz i D X i ; i.e., Xz y D yi Xz i ;

so Xz D X i @i yi X i @y :

(106)

Projecting down to the tangent bundle. Let us consider the subbundle of H formed by equivalence classes Œ.m; Q vmQ / such that vmQ is a horizontal vector. Notice that, due to (102) b 1 dRb vmQ is horizontal if so is vmQ . We will denote this bundle by hor.H /. We have the following lemma: Lemma 4.5. hor.H / L ˝ TM. h Proof. Let Œ.m; Q vm / be an element of hor.H /. We can map it to Tm M with the Q h projection vm D dvm 2 Tm M. If we choose another representative of the same Q 0h 0h h equivalence class, .Rb .m/; Q vR /, with vR D b 1 dRb vm we obtain another Q Q Q b .m/ b .m/ 0 1 vector on Tm M, vm D b vm . The natural projection applied to hor.H / defines then a section of L ˝ TM. In the other direction, let Xm 2 Tm M and 2 L. We consider the horizontal z for some choice of m lift of ˝ Xm to L ˝ T M Q 2 1 .m/ and we denote it by h h Q ˝ Xm / 2 L ˝ hor.H /. ˝ XmQ . Then we consider the equivalence class Œ.m; Q 0 0 Q another choice and D b . Then we have the equivalence Let m Q D Rb .m/ h h h h h class Œ.m Q 0 ; 0 ˝ Xm /, with Xm D dRb Xm . Since .m; Q Xm / .m Q 0 ; b 1 Xm /, then Q0 Q0 Q Q Q0 h h 0 0 .m Q ; ˝ XmQ 0 / .m; Q ˝ XmQ /, as we wanted to show.

Let D be a connection on H and let ph W H ! hor.H / be the natural projection. We can define a connection on hor.H / as yX D ph .DX /; D

with X 2 TM; and a section in hor.H / H :

y in coordinates. As before, let s be a local section of M, z We want to compute D z and let fz i g be local coordinates on M. Then fy; z i g so m Q D .m; ys.m// 2 M z We need to compute the horizontal projection of an arbitrary are coordinates on M. h z satisfying (99). In section of H , D ph . /. The section has a lift Q to T M coordinates, using the action of Rb as in (103), these equations imply the following y-dependence: .y; Q z/ D y 1 i .z/@i C y .z/@y ; and according to (105), the horizontal projection is Q h .y; z/ D i .z/ y 1 @i i .z/@y : Let be a section of hor.H /, so

Q D i .z/ y 1 @i i .z/@y :

(107)

128 María A. Lledó, Óscar Maciá, Antoine Van Proeyen, and Veeravalli S. Varadarajan z we know that We have to compute the horizontal projection of DX . Lifting to M, h Q z .DXz / must be of the same form z z Q h D i y 1 @i i .z/@y ; D X for some i . So we can identify yX i D X j .@j i C y i k /: i D D jk For vectors of the form (107) and (106) z z Q j D y 1 X i @i j C y 1 z j kX i z j X i k k D i0 X ik

z j k k z j k X i i C yi X i C y 1 i X i j 00 0k yX j ; D y 1 D

where we used the coordinates fy; z i g in M and used the index 0 for the components with respect to y. We obtain therefore for the connection coefficients: z j C i ı j : z j y z j k y z j i C y 2 i k yj D 00 i0 ik ik 0k k

(108)

The last term is just the connection on L, while the rest defines a connection on TM, j zj : z j y z j k y z j i C y 2 i k ik D 00 i0 ik 0k

(109)

y as a connection on L ˝ TM. We have then written D We can now compute explicitly the Levi-Civita connection of gQ in terms of the Levi-Civita connection of g, and check that the formula (108) is satisfied in this case. Due to (92), K is y yN times a function that depends only on z and z. N The relation between gQ ˛ˇN and gi |N , given by (97), is gQ i |N D K ˙gi |N C i N |N ; gQ {Nj D ˙

1 {Nj g ; K

gQ {N0 D

gQ 0N{ D

y {Nj g j ; K

K N {N ; y N

gQ 00 D

gQ 00N D

K ; y yN

y yN 1 ˙ i N |N g |Ni ; K

(110)

z : where i D @i log jKj. This leads to the Levi-Civita connection coefficients ˛ˇ

1 1 j @i @mN @k K 2 @mN K@i @k K D ik .g/ C i ıkj C k ıij ; K K z j D y 1 g j mN @mN @i log jKj D y 1 ı j ; i0 i y 0 k z ij D yij .g/k C 2yi j ˙ @i @j K; K 0 0 zj D z0i z00 D D 0; 00 z j D g j mN ik

where .g/ is the Levi-Civita connection of the metric on M We thus find that indeed j ik as determined in (109) are the Christoffel symbols of the Levi-Civita connection on M, as we wanted to show.


129

4.4 Examples of Kähler–Hodge manifolds Example 4.6 (Complex Grassmannian as a Kähler–Hodge manifold). We consider the Grassmannian manifold of complex p-planes in C pCq , denoted by G.p; q/. We take z D fZ j Z is a .p C q/ p matrix of rank pg : M We will write

Z0 ZD Z1

with Z0 a p p matrix and Z1 a q p matrix. Each Z defines a p-plane in C pCq as the span of the column vectors. Taking linear combinations of these vectors gives the z which does not change same plane. Then there is a right action of GL.p; C/ on M z the p-plane. M ! G.p; q/ is a principal bundle with structure group GL.p; C/. The group SL.p C q/ acts transitively on G.p; q/, but also the action of SU.p C q/ is transitive, with little group SU.p/ SU.q/ U.1/, so we have that G.p; q/ is the Hermitian symmetric space G.p; q/ D

SU.p C q/ : SU.p/ SU.q/ U.1/

G.p; q/ is a Kähler manifold and we are going to show that it is in fact a Kähler–Hodge manifold. An open cover of G.p; q/ is given by the open sets with some fixed minor of order p of Z different from zero. Notice that this property is not changed by the right action of GL.p; C/, so it is well defined on the equivalence classes For concreteness, let us fix n

U0 D Z D

Z0 Z1

o

2 G.p; q/ j det Z0 ¤ 0 :

A p-plane in U0 can be characterized by a q p matrix T such that a vector .z1 ; : : : ; zp ; zpC1 ; : : : ; zpCq / satisfies 0 1 0 1 zpC1 z1 B :: C B :: C @ : A D T @ : A: zpCq

zp

In fact, a matrix Z with det Z0 ¤ 0 is a collection of p column vectors satisfying the above property, so Z1 D T Z0 () T D Z1 Z01 : An arbitrary matrix T defines a p-plane in U0 , so we have U0 Mqp .C/ C pq ; and the entries of T are local coordinates on U0 .

130 María A. Lledó, Óscar Maciá, Antoine Van Proeyen, and Veeravalli S. Varadarajan The tautological bundle H ! G.p; q/, is the vector bundle with the fiber at each point of G.p; q/ the plane that it represents. It is a rank p subbundle of the trivial z bundle G.p; q/ C pCq . It is a bundle associated to the principal bundle M. On the trivial bundle there is a fiber metric h; 0 i D 1 N 01 C C pCq N 0pCq

(111)

0

for , vectors at a point in G.p; q/. It induces a fiber metric on the tautological bundle. A local section on H is given by functions 1 ; : : : ; p , so that T determines the plane: 0 1 1 B :: C 0 1 B : C B p C 1 B C 1 B:C C (112) .T / D B B pC1 C D T @ :: A : C B p B :: C @ : A pCq

The Hermitian inner product on the fiber is

0 1 1 1 B:C : h.T /; 0 .T /i D .N 01 ; : : : ; N 0p /.1; T / T @:A p 0 1 1 B :: C 01 0p N N D . ; : : : ; /.1 C T T / @ : A : p

If ˛; ˇ D 1; : : : ; p, then we have the fiber metric h.T /; 0 .T /i D h˛ˇN ˛ N 0ˇ ;

ht D 1 C T T:

(113)

We can write the Hermitian fiber metric as h D h˛ˇN dz ˛ d zN ˇ : We consider now the line bundle ƒp .H / with fiber at a point x 2 G.p; q/ ƒp Hx ƒp C p C; i.e. the determinant. The structure group is GL.p; C/=SL.p; C/ C : Let fs˛ .T /g be a local frame on H , with h˛ˇN D hs˛ ; sˇ i. (To compare with (113) it is enough to take s˛ .T / D .T / as in (112) with ˛ D 1 and the rest 0). Then a local section on ƒp .H / is of the form U.T / D u.T / s1 ^ ^ sp : There is an induced fiber metric on this bundle given by N H D det.h˛ˇN /dud u:


131

As in (17) we get for the Ricci form associated to the Hermitian connection on the line bundle (114) i |N D i@|N @i log det.h˛ˇN / D i@|N @i log det.1 C T T /; where i; j D 1; : : : ; pq run over all the entries of the matrix T . Example 4.7 (Non compact “Grassmannian” as a Kähler–Hodge manifold). In the example above, let us change the fiber metric (111) to a pseudo-Euclidean one with signature .p; q/, h; 0 i D 1 N 01 C C p N 0p pC1 N 0pC1 pCq N 0pCq : Then, instead of (113) we have ht D 1 T T; so on the points where the matrix 1 T T is positive definite we have a positive definite, non degenerate fiber metric. The space of matrices satisfying this property is a domain in C pq . It is the Hermitian symmetric space D.p; q/ D

SU.p; q/ : SU.p/ SU.q/ U.1/

The corresponding expression for the Ricci curvature is proportional, as before, to the standard Kähler metric on this symmetric space.

5 Conformal calculus The ideas described in the previous section originate in physics as a property of certain sigma models of scalar fields coupled to gravity with a scaling symmetry. It is in fact a simplification of what occurs in supergravity (see for example references [29], [30]), but the essential idea can be grasped in this simplification. We first consider the version with real scalars and then we move to Kähler manifolds.

5.1 Real manifold We consider a nonlinear sigma model of n real scalar fields I with Lagrangian p LR;0 D 12 gg GIJ @ I @ J : Here g is the metric of space time, gravitational field, g its inverse, and g D j det g j. The target space is a real Riemannian manifold with coordinates I and GIJ ./ is the Riemannian metric. We will be interested in the case that the Lagrangian has a dilatation symmetry given at the infinitesimal level by a vector K D K I @I . Let D be the dimension of

132 María A. Lledó, Óscar Maciá, Antoine Van Proeyen, and Veeravalli S. Varadarajan space-time. We assume that the vector K is a homothetic Killing vector of the metric GIJ , LK GIJ D K L @L GIJ C @I K L GLJ C @J K L GIL D cGIJ : We fix c D D 2:

(115)

Then the Lagrangian LR;0 is invariant under the infinitesimal transformations ı I D K I ; ı g D 2g ;

(116)

for an infinitesimal parameter independent of the point x in spacetime. A simple example is when the metric on the target space is such that GIJ ./ D D4 GIJ ./;

(117)

which means that the vector K D I @I is a homothetic Killing vector (71) with c D D 2. Then the Lagrangian is invariant under the set of transformations I 7! I ; g 7! 2 g ;

2 R;

(118)

for a constant parameter (independent of x). We obtain the infinitesimal transformations (116) by writing D 1 C C O. 2 /. Let us now consider transformations of the form (116) but with a parameter .x/ depending on the point. Then p ı L D gGIJ .@ /K I @ J g : The Lagrangian LR;0 is not invariant, but this can be remedied if we add an Einstein– Hilbert-like term for the spacetime metric p p LR D 12 gGIJ @ I @ J g 12 a.GIJ K I K J / gR.g/; (119) where R.g/ is the spacetime curvature and aD

1 : .D 1/.D 2/

(120)

In addition, we ought to assume that K is a closed homothetic Killing vector (73). In this case we have that p p p ı gR.g/ D .2 D/ gR.g/ C 2.D 1/@ gg @ ; and using the identity 1 @ .GIJ K I K J /; D2 one can prove that LR transforms into a total derivative. GIJ K I @ J D


133

In order to have a positive definite energy for the gravitational field we must have GIJ K I K J < 0, so one of the scalars is a ghost. One can fix this gauge invariance by taking GIJ K I K J D

1 ; a 2

(121)

where 2 is the gravitational coupling constant. Then the second term of (119) is just the Einstein–Hilbert action and the first term is a sigma model defined now on the surface (121).

5.2 Kähler manifolds We can consider the same kind of model for n C 1 complex scalar fields X ˛ , which are coordinates in a Kähler manifold with metric G˛ˇN . We assume now that this metric has a closed homothetic Killing vector K. As mentioned at the end of Section 4.1, this implies the presence of the holomorphic homothetic Killing vector H D .1 iJ /K. The Lagrangian density has the form p N x ˇN pgR.g/: (122) LC;0 D gg G˛ˇN @ X ˛ @ Xx ˇ 14 aG˛ˇN H ˛ H The dilatation symmetry is generated by x ˛N .z/@ K D 12 H ˛ .z/@˛ C H N ˛N ;

(123)

but this model has rigid symmetry generated by JK, which was not present in the real case. This leads to the infinitesimal transformations ı X ˛ D 12 H ˛ ;

ı' X ˛ D 12 iH ˛ ';

ı g D 2g :

(124)

Assuming (115), the action is invariant under these transformations where can be local, but ' is still a global transformation parameter. A relevant example is the finite transformation X ˛ 7! c=2 X ˛ ; g 7! jj2 g ; for which

D jjei ' D 1 C C i' C 2 C;

H ˛ D cX ˛ ;

X @ G˛ˇN D 0:

(125)

(126)

In that case, the transformations (124) can be integrated to a finite transformation. In order to implement the local invariance under ', we introduce a U.1/ connection A , which transforms as A 7! A C @ '; and we couple it minimally to the scalar fields defining p N x ˇN pgR.g/; LC D gg G˛ˇN D X ˛ D Xx ˇ 14 aG˛ˇN H ˛ H D X ˛ D @ X ˛ 12 iA H ˛ :

(127)

134 María A. Lledó, Óscar Maciá, Antoine Van Proeyen, and Veeravalli S. Varadarajan For shortness, we denote 1 x ˇN : G N H ˛H c 2 ˛ˇ Using (77) for the Levi-Civita connection, we have also that N D

1 x ˇN ; @˛ @ N N D G N : G˛ˇN H ˇ ˛ˇ c Hence, N is the Kähler potential of the manifold described by the X ˛ . The field equation for the auxiliary field A is algebraic and it allows us to solve for A : @˛ N D

i x ˇN @ X @ˇN N @ X ˛ @˛ N : cN The first term of (127), the scalar kinetic term Lscalar , is then as follows:

2 1 x ˇN Lscalar N @ X @ˇN N @ X ˛ @˛ N p D G˛ˇN @ X ˛ @ Xx ˇ g 4N

1 1 ˛ x ˇN (128) @ N @ N D @ X @ X @˛ @ˇN N .@˛ N /.@ˇN N / N 4N

1 N D N @ X ˛ @ Xx ˇ @˛ @N ˇN log jN j @ N @ N : 4N Notice that H ˛ is a zero mode of the quantity in square brackets in the second line. We can fix the dilation gauge freedom (125) by taking as before A D

i

c2N

N x ˇN D G˛ˇN H ˛ @ Xx ˇ @ X ˛ H

1 x ˇN D 2 : G˛ˇN H ˛ H (129) 2 c ac 2 2 The second term of (127) is then the Einstein–Hilbert action. As N gets a fixed value, a function of N is not convenient as a Kähler potential for the restricted manifold. We will show now how to construct a Kähler potential, restricting to the case (126). In that case, we rescale the coordinates X ˛ , introducing ˛ by N D

z ! M; z ˆY W M ˛

! X ˛ D ˛ Y . ; /; N for an arbitrary function Y. ; /. N Notice that this map is not holomorphic with respect z denoted as J 0 , by the to J . However, it induces a new complex structure on M, commutativity of the diagram z TM

T ˆY

J0

z TM

/

z TM J

T ˆY >T

/ z M.


135

The map T ˆY then sends holomorphic vectors with respect to J 0 to holomorphic vectors with respect to J . In this sense, it is a holomorphic map. Defining N ; KD Y Yx then K is a function of . ; /. N The homogeneity properties of N imply N K. ; N / N D K. ;

/; N and therefore also @ @ N D Yx ˛ K; ˛ @X @

@2

N D N

@X ˛ @Xx ˇ

@2 @ ˛ @ N ˇN

K:

Hence, c ˛ @@˛ is a holomorphic homothetic Killing vector with respect to J 0 , and K defines the Kähler potential of a projective Kähler manifold, see Definition 4.2. The action reduces to

1 2 LC 1 @2 N @ log jKj N @ N ac NR.g/: (130) p D N @ ˛ @ N ˇ g 4N 4 @ ˛ @ N ˇN The first term in (130) is proportional to Q˛ˇN in (95), which is the pull back of a 2-form on the quotient manifold Q D i as in (96). If z i , i D 1; : : : ; n are coordinates on the quotient, then similarly as in (97), a metric is defined. The appropriate normalization for the Kähler potential is 2 ac 2 gi |N D @i @|N 2 2 log K : ac 2 On the quotient N is constant and thus @ N D 0, so the action reduces to a sigma model in dimension n coupled to gravity in the standard way, 1p p LC D ggi |N @ z i @ zN j C 2 gR.g/: Note that the D D 4 values of (115) and (120) lead to ac 2 =2 D 1=3. That is also the value that one finds in N D 1 supergravity. For N D 2 supergravity, one has two scalar manifolds, the one of the vector multiplets, and the one of hypermultiplets. There is another auxiliary field, whose origin is beyond our discussion here, such that when one eliminates the hypermultiplets, the effective value of a is 1=2, i.e. ac 2 =2 D 1. We remark that we need here the lower signs in (97) in order to get the positive kinetic energy for gravity, and the other signatures should all be C in order to have positive kinetic terms of the sigma model.

136 María A. Lledó, Óscar Maciá, Antoine Van Proeyen, and Veeravalli S. Varadarajan

6 Projective special Kähler manifolds 6.1 Definition of projective special Kähler manifolds A projective special Kähler manifold M is a Kähler–Hodge manifold such that the z is rigid special Kähler. The flat connection on M z is an extra structure manifold M that also projects to M. Here we have the precise definition. z be a rigid special pseudo-Kähler manifold with complex strucDefinition 6.1. Let M z We assume that z and flat symplectic connection r. ture JQ , metric g, Q Kähler form ˆ z there is a free holomorphic action of C such that the fundamental vector field on M H is a non null, holomorphic homothetic Killing vector for the flat connection, z z Y H D cY for all Y 2 T 1;0 M r

and

x / ¤ 0: g.H; Q H

(131)

z there is a projective special Kähler structure. Then we say that on M D M=C In fact, we will prove that (131) implies that H is also an holomorphic homothetic Killing vector for the Levi-Civita connection, that is, z Y H D cY D

which is equivalent to

LY gQ D c gQ

z for all Y 2 T 1;0 M;

(132)

so it is enough to require it for the flat connection. Let us look at (131) in special coordinates. From (48) we have ˇ z D @H d ˛ ˝ @ 1 @ ˇ H d ˛ ˝ @ D c d ˛ ˝ @ : rH @ ˛ 2 @ ˛ @yˇ @ ˛ @ ˇ

The first term is holomorphic, while in the second there is an holomorphic and an antiholomorphic part, since yˇ is real. To cancel the antiholomorphic part necessarily @ ˇ H D 0; @ ˛

(133)

@H ˇ D cı˛ˇ () H ˇ D c ˇ (up to a constant): @ ˛

(134)

and then

We can always shift ˛ by a constant, it is still a special coordinate. In particular, (134) implies that special coordinates are homogeneous coordinates as defined in (84). (133) and (134) imply that ˛ˇ are homogeneous functions of of degree 0,

˛ˇ . / D ˛ˇ . /;

2 C;

so they depend on the prepotential F as in (45):

˛ˇ D 8

@2 F @ ˛ @ ˇ


137

must be a homogeneous function of of degree 2, F . / D 2 F . /;

2 C:

(135)

For the Levi-Civita connection, we have DH D

@ 1 @ @H ˇ ˛ N @ ı d ˝ ˇ C ig ˇ ı ˛ H d ˛ ˝ ; ˛ @

8 @

@ ˇ @

and using (133) and (134) we get DH D c d ˛ ˝

@ ; @ ˛

which proves (132). Using (134) we can compute the integral surfaces (80) of H in special coordinates, d ˛ d

() c ˛ .x/ D () ˛ .x/ D ec ˛ .x/; d d since 0 .x/ D x and 0˛ .x/ D ˛ .x/. We will denote also by Rb .x/ D .x/ with b D ec 2 C . As we saw in general in (94), D c 2 K. Since r is flat, it descends to H as a flat connection, and then it defines a connection on L ˝ TM as in (109). This connection is not necessarily flat. Next we will see that also the holomorphic cubic form descends to an appropriate bundle over M, and we will compute the curvature tensor in terms of it. H. .x// D

The holomorphic cubic form. We consider the holomorphic cubic form „ defined in Section 3.7. We want to see how it descends to the manifold M. If X is a vector field on M, its horizontal lift is (106) Xz D X i @i yi X i @y D X i .@i ˛ i ˛ /@˛ ;

(136)

so @F z D X i .@i F 2i F / D X i DiL .F / D DXL F ; d ˛ .X/ @ ˛ where we have used the fact that F is homogeneous of degree 2. It is in fact a section of .L /˝2 , and D L denotes the covariant derivative with respect to the Hermitian connection. We can also write 1 L 2 log jKj 2 log jKj 2 Di F D e @i .e F / D K @i F : K2 z are their horizontal lifts respectively, If Y; Z are also vector fields on M and Yz , Z we have @3 F L z Yz ; Z/ z D DXL DYL DZ d ˛ d ˇ d .X; .F /: @ ˛ @ ˇ @

138 María A. Lledó, Óscar Maciá, Antoine Van Proeyen, and Veeravalli S. Varadarajan z „ z descends to a section of .L /˝2 ˝ This shows that the holomorphic cubic form on M, ˝3 .T M/ , z Xz ; Yz ; Z/: z „.X; Y; Z/ „. (137) z gives us also another way of This procedure of lifting the vector fields on M to M computing the metric on M. The metric. As before, let Xz and Yz be the horizontal lifts of X and Y , vector fields on M. Then we have, using (136), (93) and (97), z Yxz / D @˛ @ N K.@i ˛ i ˛ /.@j N ˇ Nj N ˇN /X i Yx j g. Q X; ˇ D K@i @|N .log jKj/X i Yx j D jKjg.X; Yx /:

(138)

The Riemannian curvature on M. From (109) we can compute the curvature tensor z and M are Kähler manifolds we can of the Levi-Civita connection on M. Since M use (11). We have j zj N C @ N .ı j @k log jKj C ı j @i log jKj/: Rj ik `N D @`N ik DR i k` ` i k

These are the components of the curvature tensor in the coordinates z i . To avoid confusion, we will split the coordinates ˛ as . 0 ; a /, a D 1; : : : ; n. In this way the indices i; j; k will always refer to the coordinates z. z We first have to zj N in terms of the cubic form „. We can use (62) to express R ik ` i z in terms of .y; z /. We have write „ d 0 D dy;

d i D z i dy C ydz i :

(139)

z being a third derivative of F as in Due to the homogeneity condition (135) and „ (57), we have z ˛ˇ D 0;

˛ „ (140) and therefore the dy terms in (139) do not contribute if we rewrite z D„ z ˛ˇ d ˛ d ˇ d D y 3 „ z abc ıia ıjb ı c dz i dz j dz k D „ij k dz i dz j dz k ; „ k where „ij k has been defined in (137). This leads to „ij k D 2y 2

@3 F .1; z i / ; @z i @z j @z k

where F .1; z i / is F . / with 0 replaced by 1, and i by z i . On the other hand, using (110), we find 1 j |N0 p0 pN x 0 N ˙ ıj g N ˙ ıj g N: g g „p0 ki „ (141) i k` pN |N ` k i` 4K 2 Notice that in (141) all the dependence in y; yN cancels as K is proportional to y y. N Rj ik `N D


139

6.2 Examples of projective special Kähler manifolds Example 6.2 (Projective space and unit ball as special Kähler manifolds). We consider the complex projective space CPm of lines in the complex space C mC1 . It is a special case of Example 4.6, with p D 1, q D m. We have a covering of CPm by open sets Ui D flines S in C mC1 with z i jS ¤ 0g; (these are the lines that do not lie in the hyperplane z i D 0). Let us take a fixed index i D 0, then we have that z j jS D t j z 0 jS ; j ¤ 0; so .t 1 ; : : : ; t m / is a set of coordinates on U0 . The tautological bundle is already a line bundle, L CPm C mC1 so there is no need of taking the determinant. On CPm C mC1 we have the fiber metric h; 0 i D 0 N 0 C C m N m ;

(142)

which we will restrict to L. On L the fiber metric and the Hermitian connection are m m 1 X X t j tNj ; i D 1 C t j tNj tNi : h.t; tN/ D 1 C j D1

j D1

The Ricci form (114) becomes i |N D igi |N D i@|N @i log h D i@|N @i log.1 C t tN/ D

i i j i tN t ı ij : h2 h

We can define the prepotential as 1 0 0 i. C C m m /; i D t i 0 ; i D 1; : : : ; m: 4 z D C mC1 and in CPm is Then the Kähler potential in M F D

K D 2=

@F N

N D

I @

log jKj D log.1 C t tN/ C log 0 N 0 :

As the third derivative of the prepotential vanishes, the curvature is given by the last two terms of (141), where we have to use the C signs. If we change the fiber metric (142) to h; 0 i D 0 N 0 m N m ; we obtain that h is positive on the unit ball h.t; tN/ D 1 t tN > 0

for t tN < 1;

which is the symmetric space SU.1; m/ : SU.m/ U.1/

140 María A. Lledó, Óscar Maciá, Antoine Van Proeyen, and Veeravalli S. Varadarajan Notice that the metric is gi |N D

1 i j 1 t tN C ı ij D @i @Nj log h: h2 h

This means that we have to start with the negative Kähler potential K. This sign is important in physical applications (supergravity), as we saw in Section 5. In this case, we have to use the signs in the last two terms of (141). Example 6.3 (A pseudo-Riemannian special Kähler manifold). We want to describe now the pseudo-Riemannian symmetric space SU.1; 2/ : SU.1; 1/ U.1/

(143)

We start the construction as for the projective space, on which we try to define a pseudo-Riemannian metric. As we did for the passage to the unit-ball, we will have to restrict to those points where this metric is non degenerate. As a fiber metric on the trivial bundle CP2 C 3 we take h; i D N 1 1 N 2 2 C N 3 3 : The space is covered by the three open sets 80 1 1 9 < ˇ = Ui D @ 2 A ˇ i ¤ 0 ; : 3 ; as before. In the patch U1 , we have 2 D T 11; 3

i D 1; 2; 3

t 21 with T D 31 ; t 1

and a local section of the tautological bundle is given by a function 1 .T 1 /, 0 11 1 1 1 2A @ .T1 / D D 1 .T /: T 3 The inner product becomes h.T /; .T /i D N 1 .1; T 1 /g 1

with

0

1

0

1 01 D N 1 .1 tN21 t 21 C tN31 t 31 / 01 ; T1

1 1 0 0 g D @0 1 0A : 0 0 1

(144)


141

Doing the same computation for U2 ; U3 we obtain h.T 1 /; 0 .T 1 /i D N 1 .1 tN21 t 21 C tN31 t 31 / 01 h.T 2 /; 0 .T 2 /i D N 2 .tN12 t 12 1 C tN32 t 32 / 02

for U1 ,

h.T /; .T /i D N 3 .tN13 t 13 tN23 t 23 C 1/ 03

for U3 :

3

0

3

for U2 ,

(145)

Equations (145) give us the fiber metric on the tautological (line) bundle. If in each Ui , (146) h; 0 i D hi i N 0i .no sum over i /; then h1 .T 1 / D .1 tN21 t 21 C tN31 t 31 /; h2 .T 2 / D .tN12 t 12 1 C tN32 t 32 /; h3 .T 3 / D .tN13 t 13 tN23 t 23 C 1/: In the intersections, the change of coordinates t ij D

1 tji

;

i D t ij j

leaves (146) invariant. We have to restrict ourselves to the space where the fiber metric is positive definite. Let Ui D fT i 2 Ui j hi .T i / > 0g: U1 and U3 are homeomorphic to C 2 , but U2 is U2 minus a ball of radius 1 centered at T 2 D 0. The point T 2 D 0 is the only point in U2 that is not contained in U1 or U3 . So we can safely ignore U2 , since fU1 ; U3 g form a covering of the space of points where the fiber metric is definite positive. Notice that in U1 , 2 D t 21 1 serves as a coordinate and the same in U3 , 2 D 23 3 t , so 2 is a global coordinate and describes C. The other coordinate, t 21 or t 12 respectively in U1 and U3 describe a sphere S 2 , so we have that the topology of (143) is S 2 C. Let us compute the Ricci form in U1 . For simplicity we will denote t 21 D t 2 , 31 t D t 3. i 1 t 3 tN3 t 3 tN2 i |N D i@i @|N log h D 2 : t 2 tN3 1 t 2 tN2 h The metric is gi |N D ii |N ; and it is easy to see that it has one positive and one negative eigenvalue. The prepotential is F D i. 1 1 2 2 C 3 3 /;

142 María A. Lledó, Óscar Maciá, Antoine Van Proeyen, and Veeravalli S. Varadarajan with t2 D

2 ;

1

t3 D

3 ;

1

and then the Kähler potential is K D 4. 1 N 1 2 N 2 C 3 N 3 / D 4 1 N 1 h: We can also see that „ D 0. Then (141) gives Ri |N D Rk ik |N D 3gi |N and R D 6.

7 Conclusions In this exposition we have extended the definition of special Kähler geometry to the case of arbitrary signature of the Kähler metric. For the rigid case, we have extended the definition given in [8], while for the projective case we have given a definition inspired in the conformal calculus framework. We have seen that the non-existence of prepotential in some symplectic coordinates, which was known for projective special geometry [12] (the special geometry that occurs in supergravity), is in fact a characteristic of pseudo Riemannian manifolds, and applies also to the rigid case. This was masked by the fact that in physical applications of rigid special geometry one is only interested in the Euclidean signature, which gives positive definite kinetic energy for the scalar fields. Projective (or ‘local’, referring to the local supersymmetry invariance of supergravity) special geometry is obtained from a rigid special manifold that has a closed homothetic Killing vector K. If K is such vector and J is the complex structure JK is a Killing vector, so the metric has an extra U(1) symmetry. The result is that the existence of a closed homothetic Killing vector is equivalent to the existence of a holomorphic homothetic Killing vector, which we define in (77). This means essentially that there is an action of the group C , and the procedure to obtain the projective special manifold is to take quotient of the rigid ‘mother’ manifold by this action (and from here, the name of ‘projective’ geometry). The positive signatures of the kinetic terms of scalars and gravity in supergravity theories require that the rigid manifold has signature .2; 2n/. We extend, however, projective special geometry to arbitrary signatures. If the signature of the projective manifold is .s; t / (s positive eigenvalues, t negative eigenvalues), then the signature of the ’mother’ rigid manifold is either .s C 1; t / or .s; t C 1/. It is the later case that occurs in supergravity. The standard formula for the curvature is generalized to (141), the lower choice in ˙ being the standard supergravity case. The other possibility allows us also to discuss special geometries with a compact isometry group. In fact, this projectivization can be discussed for general Kähler manifolds, not necessarily special. We develop the formalism in this more general case and, for example, we prove that the projective Kähler manifold is automatically Kähler–Hodge. As this is the method that is used in conformal calculus, it implies that all the Kähler


143

manifolds that are constructed in this way for N D 1 or N D 2 supergravity satisfy the Kähler–Hodge condition that was introduced in reference [22]. We also give an interpretation of the Levi-Civita connection in these projective Kähler manifolds as induced from the connection of the ‘mother’ manifold in a particular way, making use of the line bundle and the Hermitian connection.

Appendices A Some technical results Lemma A.1 (See Lemma A1 in [7]). Let

˛ V D ˇ

be a 2n n matrix of rank n (˛ and ˇ are n n matrices). Then there is a matrix S 2 Sp.2n; R/ such that the transformed matrix 0 ˛ V 0 D SV D ˇ0 has the property that ˛ 0 itself has rank n. Proof. We give an outline of the proof. Let us denote 0 1 0 11 ˇ1 ˛ B :: C B :: C ˛ D @ : A; ˇ D @ : A; ˛n

ˇn

and let r 1 n be the rank of ˛. If r 1 D n we have already the result, so we will take r 1 < n. Without loosing generality, we can assume that ˛ 1 ; : : : ; ˛ r1 are linearly independent. Then r1 X i ˛ i : (147) ˛r D iD1 1

r1

Let ˇk be such that ˛ ; : : : ; ˛ ; ˇk are also linearly independent. For the particular case k D r the symplectic matrix Er;r 1 Er;r SD Er;r 1 Er;r gives an ˛ 0 with rank r. (We have used the standard notation .Ei;j /l k D ıil ıj k .) In the generic case k ¤ r, we consider the symplectic matrix 1 Er;r 1 Ek;r Er;k C Er;r ; SD 1 Er;r 1 Er;r

144 María A. Lledó, Óscar Maciá, Antoine Van Proeyen, and Veeravalli S. Varadarajan where is a parameter ¤ 0. It follows that it is always possible to choose such that the vectors ˛ 0 1 ; : : : ; ˛ 0 r are independent. In fact, the conditions on are that must be different from some fixed quantities. By iterating this procedure we see that transforming V with a finite number of symplectic matrices it is possible to construct a matrix V 0 such that rank.˛ 0 / D n. Remark A.2. When passing from a constant matrix V to a point dependent matrix V .z/, one has first to restrict to a neighbourhood where the same components of V are independent (not only in number). Otherwise the theorem could not be applied. So we may have to enlarge the number of open sets of our covering. Next, we want to consider a constant symplectic transformation in order to have flat Darboux coordinates in each open set. The constraints for (which must be constant) become now point dependent, namely must be different from certain functions of z and z. N This is always possible, but perhaps in an open subset of the original open set. For each point there is a neighbourhood contained in a compact set where the constraints can be satisfied. One can cover the manifold with such neighbourhoods and, assuming that the space is locally compact, one can pick up a subcovering which is locally finite. Lemma A.3. The subgroup of Sp.2n; R/ formed by the matrices of the form ³ ² A 0 C .A> /1 is a maximal subgroup. Before going to the proof let us explain a way of seeing maximality. Let g be a Lie algebra and s a Lie subalgebra with respective groups G and S with S G. We are interested in deciding when s is maximal in g. Note that the adjoint action of S on g leaves s stable and so S acts on a WD g=s. Theorem A.4. If there is a subgroup T S such that the action of T on a is irreducible, then s is maximal in g (We can operate over the complex numbers as maximality over the complexes is stronger than maximality over the reals.) Proof. Let h be a subalgebra such that s h g with h ¤ s. We must show that h D g. Since h is invariant under T , the image b of h in a is stable under T . Since h is strictly larger than s, the space b is not 0 and is stable under T . By the irreducibility of the action of T we must have b D a so that h D g. Let us go back to the proof of Lemma A.3.


145

Proof. In our case S is the lower triangular block group and so we can take a to be the space of matrices 0 b ; b D bt : 0 0 We take T to be the subgroup A 0 1 ; 0 At

A 2 GL.n/:

Then the action of T on a works out to be .A; b/ 7! AbAt which is the representation of GL.n/ on the symmetric tensors of the n-space, which is known to be irreducible. Applying Theorem A.4, we complete our proof.

B Connection on a principal bundle and covariant derivative We will relate now the definition 4.3 of connection on a principal bundle to the covariant derivative in associated bundles. Let E be an associated vector bundle to P , with standard fiber F , and let R W G ! End.F / be the representation of G on F . For simplicity we will consider G R.G/, although this is not necessary. We want to define the covariant derivative of a section of E in terms of the connection 1-form. Let fe1 ; : : : ; ek g be a basis on F . A local frame of E is a set of k D rank.E/ independent local sections of E. We will denote it by .m/ D f1 .m/; : : : ; k .m/g, with m 2 M. It can be interpreted as an invertible map .m/ W F ! Em such that .m/.ea / D a .m/; so it provides with an identification of the fiber Em with F . The set of frames is a principal bundle with structure group GLn . P is a subbundle of the bundle of frames, so a local section s W M ! P is a local frame of E. The pull-back D ! defines a local g-valued 1-form on M. The covariant derivative of a local section of E, D a a , is given by (148) ri D @i a C i a b b a ; so i a b D .ri b /a : (148) relates the definition of connection as a g-valued 1-form on P with the notion of covariant derivative that we have been using through the text. (148) is given in terms of a local section on P , but changing the section gives the usual gauge transformation of the local connection 1-form on M. The coordinate independent description of the covariant derivative can be found for example in reference [23].

146 María A. Lledó, Óscar Maciá, Antoine Van Proeyen, and Veeravalli S. Varadarajan Acknowledgments. We are grateful to L. Andrianopoli and S. Ferrara for interesting and very useful discussions. This work is supported in part by the European Community’s Human Potential Programme under contract MRTN-CT-2004-005104 ‘Constituents, fundamental forces and symmetries of the universe’. The work of A. V. P. is supported in part by the FWO-Vlaanderen, project G.0235.05 and by the Federal Office for Scientific, Technical and Cultural Affairs through the “Interuniversity Attraction Poles Programme – Belgian Science Policy” P5/27. The work of M. A. Ll. and O. M. has been supported in part by research grants from the Spanish Ministerio de Educación y Ciencia (FIS2005-02761 and EU FEDER funds), the Generalitat Valenciana (ACOMP06/187, GV-05/102). A. V. P. thanks the Universitat de València for hospitality during a visit that initiated this work. M. A. Ll. and O. M. want to thank the Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven for its kind hospitality during part of this work.

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Chapter 5

Special geometry, black holes and Euclidean supersymmetry Thomas Mohaupt

Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Euclidean special geometry . . . . . . . . . . . . . . . . . . 2.1 Vector multiplets . . . . . . . . . . . . . . . . . . . . . 2.2 Hypermultiplets . . . . . . . . . . . . . . . . . . . . . . 3 Black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The laws of black hole mechanics . . . . . . . . . . . . 3.2 Quantum aspects of black holes . . . . . . . . . . . . . 3.3 Black holes and strings . . . . . . . . . . . . . . . . . . 3.4 Black holes and supersymmetry . . . . . . . . . . . . . 4 Special geometry and black holes . . . . . . . . . . . . . . . 4.1 Vector multiplets coupled to gravity . . . . . . . . . . . 4.2 BPS black holes and the attractor mechanism . . . . . . 4.3 The black hole variational principle . . . . . . . . . . . 4.4 Quantum corrections to black holes solutions and entropy 4.5 Black hole partition functions and the topological string . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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149 152 152 154 157 157 159 161 162 163 163 165 168 171 173 177

1 Introduction Special geometry was discovered more than 20 years ago [1]. While the term special geometry originally referred to the geometry of vector multiplet scalars in fourdimensional N D 2 supergravity, today it is used more generally for the geometries encoding the scalar couplings of vector and hypermultiplets in theories with 8 real supercharges. It applies to rigidly and locally supersymmetric theories in 6 spacetime dimensions, both in Lorentzian and in Euclidean signature. The scalar geometries occurring in these cases are indeed closely related. In particular, they are all much more restricted than the Kähler geometry of scalars in theories with 4 supercharges, while still depending on arbitrary functions. In contrast, the scalar geometries of theories with 16 or more supercharges are completely fixed by their matter content. Theories with 8 supercharges have a rich dynamics, which is still constrained enough to allow one to answer many questions exactly. Special geometry lies at the heart of

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the Seiberg–Witten solution of N D 2 gauge theories [2] and of the non-perturbative dualities between N D 2 string compactifications [3], [4]. While the subject has now been studied for more than twenty years, there are still new aspects to be discovered. One, which will be the topic of this chapter, is the role of real coordinates. Many special geometries, in particular the special Kähler manifolds of four-dimensional vector multiplets and the hyper-Kähler geometries of rigid hypermultiplets are complex geometries. Nevertheless, they also possess distinguished real parametrizations, which are natural to use for certain physical problems. Our first example illustrates this in the context of special geometries in theories with Euclidean supersymmetry. This part reviews the results of [5], [6], and gives us the opportunity to explore another less studied aspect of special geometry, namely the scalar geometries of N D 2 supersymmetric theories in Euclidean space-time. It turns out that the relation between the scalar geometries of theories with Lorentzian and Euclidean space-time geometry is (roughly) given by replacing complex structures by para-complex structures. One technique for deriving the scalar geometry of a Euclidean theory in D dimensions is to start with a Lorentzian theory in D C 1 dimensions and to perform a dimensional reduction along the time-like direction. The specific example we will review is to start with vector multiplets in four Lorentzian dimensions, which gives, by reduction over time, hypermultiplets in three Euclidean dimensions. This provides us with a Euclidean version of the so-called c-map. The original c-map [7], [8] maps any scalar manifold of four-dimensional vector multiplet scalars to a scalar manifold of hypermultiplets. For rigid supersymmetry, this relates affine special Kähler manifolds to hyper-Kähler manifolds, while for local supersymmetry this relates projective special Kähler manifolds to quaternion-Kähler manifolds. By using dimensional reduction with respect to time rather than space, we will derive the scalar geometry of Euclidean hypermultiplets. As we will see, the underlying geometry is particularly transparent when using real scalar fields rather than complex ones. The geometries of Euclidean supermultiplets are relevant for the study of instantons, and, by ‘dimensional oxidation over time’ also for solitons, as outlined in [5]. In this chapter we will restrict ourselves to the geometrical aspects. Our second example is taken from a different context, namely BPS black hole solutions of matter-coupled N D 2 supergravity. The laws of black hole mechanics suggest to assign an entropy to black holes, which is, at least to leading order, proportional to the area of the event horizon. Since (super-)gravity presumably is the low-energy effective theory of an underlying quantum theory of gravity, the black hole entropy is analogous to the macroscopic or thermodynamic entropy in thermodynamics. A quantum theory of gravity should provide the fundamental or microscopic level of description of a black hole and, in particular, should allow one to identify the microstates of a black hole and to compute the corresponding microscopic or statistical entropy. The microscopic entropy is the information missing if one only knows the macrostate but not the microstate of the black hole. In other words, if a black hole with given mass, charge(s) and angular momentum (which characterise the macrostate) can be in d different microstates, then the microscopic entropy is Smicro D log d . If the

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area of the event horizon really is the corresponding macroscopic entropy, then these two quantities must be equal, at least to leading order in the semi-classical limit. In string theory it has been shown that the two entropies are indeed equal in this limit [9], at least for BPS states (also called supersymmetric states). These are states which sit in special representations of the supersymmetry algebra, where part of the generators act trivially. These BPS (also called short) representations saturate the lower bound set for the mass by the supersymmetry algebra, and, as a consequence, the mass is exactly equal to a central charge of the algebra.1 In this chapter we will be interested in the macroscopic part of the story, which is the construction of BPS black hole solutions and the computation of their entropy. The near horizon limit of such solutions, which is all one needs to know in order to compute the entropy, is determined by the so-called black hole attractor equations [11], whose derivation is based on the special geometry of vector multiplets. The attractor equations are another example where real coordinates on the scalar manifold appear in a natural way. In the second part of the exposition we review how the attractor equations and the entropy can be obtained from a variational principle. When expressed in terms of real coordinates, the variational principle states that the black hole entropy is the Legendre transform of the Hesse potential of the scalar manifold. We also discuss how the black hole free energy introduced by Ooguri, Strominger and Vafa [12] fits into the picture, and indicate how higher curvature and non-holomorphic corrections to the effective action can be incorporated naturally. This part is based on [13] and on older work including [14], [15], [16]. Let us now explain how our two subjects are connected to the second topic of this volume, pseudo-Riemannian geometry. In both parts of the chapter we have two relevant geometries, the geometry of space-time and the geometry of the target manifold of the scalar fields. In the first case, space-time is Euclidean, but, as we will see, the scalar manifold is pseudo-Riemannian with split signature. In the second case the scalar geometry is positive definite, but space-time is pseudo-Riemannian with Lorentz signature. In fact, our two subjects, the c-map and black holes, can be related in a rather direct fashion, as follows: for a static black hole one can perform a dimensional reduction along the time-like direction in complete analogy to the dimensional reduction of flat Minkowski space. Then one can dualize the vector multiplets into hypermultiplets, which gives rise to a ‘local’ version of the c-map.2 This construction can be used to study time-independent four-dimensional geometries from a three dimensional perspective, which has the advantage that all bosonic degrees of freedom (metric, gauge fields and scalars) become scalars in the reduced theory and can then be combined into a non-linear sigma model. This method has been used in Einstein–Maxwell theory already a long time [18], and has been elaborated on both for black holes [19] and brane-type solutions [20]. We refer to [21] for a review. More recently, dimensional reduction to three Euclidean dimensions and the corresponding version of the c-map have been used by [22] to elaborate on the 1 See

[10] Chapter 2. local means that supersymmetry is realized as a local, i.e., space-time dependent symmetry.

2 Here

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ideas of Ooguri, Strominger and Vafa [12] by quantizing static, spherically BPS black hole solutions. The three-dimensional solutions obtained by dimensional reduction for four-dimensional static black hole solutions can also be lifted to four Euclidean dimensions, where they describe wormhole solutions, which generalize the D-instanton of type-IIB string theory [17]. Let us finally mention two contributions to this volume which are closely related to our topics. The article [23] (which is based on [24]) discusses new insights into the geometry of the c-map, which have been obtained by relating vector and hypermultiplets to tensor multiplets. The contribution of [25] discusses new developments in para-quaternionic geometry. While we only discuss the ‘rigid’ version of the Euclidean c-map here, its ‘local’ (supergravity) version maps projective special Kähler manifolds to para-quaternionic manifolds.

2 Euclidean special geometry 2.1 Vector multiplets We start by reviewing the geometry of vector multiplets in rigid four-dimensional N D 2 supersymmetry.3 A vector multiplet consists of a gauge field Am (m D 0; : : : ; 3 is the Lorentz index), two Majorana spinors i (i D 1; 2) and one complex scalar X. We consider n such multiplets, labeled by an index I D 1; : : : ; n. The field equations for the gauge fields are invariant under Sp.2n; R/ rotations which I and the dual field strength GI jmn D ıFıLI , act linearly on the field strength Fmn mn where L denotes the Lagrangian. These symplectic rotations generalize the electricmagnetic duality rotations of Maxwell theory and are in fact invariances of the full field equations. A thorough analysis shows that this has the important consequence that all vector multiplet couplings are encoded in a single holomorphic function of the scalars, F .X I /, which is called the prepotential [1]. In superspace language the general action for vector multiplets can be written as a chiral superspace integral of the prepotential F , considered as a superspace function of n so-called restricted I /, which encode the gauge invariant quantities of the chiral multiplets .X I ; I C ; Fmn n vector multiplets. Here I C are the positive chirality projections of the spinors and I Fmn are the antiselfdual projections of the field strength. To be precise, the Lagrangian is the sum of a chiral and an antichiral superspace integral, the latter depending on the IC /. When working out the Lagrangian complex conjugated multiplets .Xx I ; I ; Fmn in components, all couplings can be expressed in terms of F .X I /, its derivatives, which we denote FI ; FIJ ; : : : and their complex conjugates FxI ; FxIJ ; : : : . For later use we specify the bosonic part of the Lagrangian: VM I J mn D 12 NIJ @m X I @m Xx J 2i .FIJ Fmn F c.c./; L4d bos 3 Some

more background material and references on vector multiplets can be found in [26].

(2.1)

where


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NIJ D @I @JN i.X I FxI FI Xx I /

(2.2)

can be interpreted as a Riemannian metric on the target space MVM of the scalars X I .4 N D 1 supersymmetry requires this metric to be a Kähler metric, which is obviously the case, the Kähler potential being K D i.X I FxI FI Xx I /. As a consequence of N D 2 supersymmetry the metric is not a generic Kähler metric, since the Kähler potential can be expressed in terms of the holomorphic prepotential F .X I /. The resulting geometry is known as affine (also: rigid) special Kähler geometry. The intrinsic characterization of this geometry is the existence of a flat, torsionfree, symplectic connection r, called the special connection, such that .rU I /V D .rV I /U;

(2.3)

where I is the complex structure and U; V are arbitrary vector fields [27]. It has been shown that all such manifolds can be constructed locally as holomorphic Langrangian immersions into the complex symplectic vector space T C n ' C 2n [29]. In this context X I ; FI are flat complex symplectic coordinates on T C n and the prepotential is the generating function of the immersion ˆ W MVM ! T C n , i.e., ˆ D dF . For generic choice of ˆ, the X I provide coordinates on the immersed MVM , while FI D @I F D FI .X/ along MVM . The X I are non-generic coordinates, physically, because they are the lowest components of vector multiplets, mathematically, because they are adapted to the immersion. They are called special coordinates. So far we have considered vector multiplets in a four-dimensional Minkowski space-time. In four-dimensional Euclidean space the theory has the same form, except that the complex structure I , I 2 D 1 is replaced by a para-complex structure J . This is an endomorphism of TMVM such that J 2 D 1, with the eigendistributions corresponding to the eigenvalues ˙1 having equal rank. Many notions of complex geometry, including Kähler and special Kähler geometry can be adapted to the paracomplex realm. We refer to [5], [6] for the details. In particular, it can be shown that the target space geometry of rigid Euclidean vector multiplets is affine special para-Kähler. Such manifolds are the para-complex analogues of affine special Kähler manifolds. When using an appropriate notation, the expressions for the Lagrangian, the equations of motion and the supersymmetry transformation rules take the same form as for Lorentzian supersymmetry, except that complex quantities have to be reinterpreted as para-complex ones. For example, the analogue of complex coordinates X I D x I C iuI , where x I ; uI are real and i is the imaginary unit, are para-complex coordinates X I D x I C euI , where e is the para-complex unit characterized by e 2 D 1 and eN D e, where the ‘bar’ denotes para-complex conjugation.5 While in Lorentzian signature the selfdual and antiselfdual projections of the field strength are related by complex conjugation, in the Euclidean theory one can re-define the selfdual 4 The scalar fields X I might only provide local coordinates. We will work in a single coordinate patch throughout. 5 It has been known for quite a while that the Euclidean version of a supersymmetric theory can sometimes be obtained by replacing i ! e [30].

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and antiselfdual projections by appropriate factors of e such that they are related by para-complex conjugation. One can also define para-complex spinor fields such that the fermionic terms of the Euclidean theory take the same form as in the Lorentzian one. The Euclidean bosonic Lagrangian takes the same form (2.1) as the Lorentzian one, with (2.2) replaced by (2.4) NIJ D @I @JN e.X I FxI FI Xx I / : I Note that the Euclidean Lagrangian is real-valued, although the fields X I and Fmn are para-complex. We also remark that a para-Kähler metric always has split signature. The full Lagrangian, including fermionic terms, and the supersymmetry transformation rules can be found in [5]. There we also verified that it is related to the rigid limit of the general Lorentzian signature vector multiplet Lagrangian [31], [32] by replacing i ! e (together with additional field redefinitions, which account for different normalizations and conventions).

2.2 Hypermultiplets Our next step is to derive the geometry of Euclidean hypermultiplets. This can be done by either reducing the Lorentzian vector multiplet Lagrangian with respect to time or the Euclidean vector multiplet Lagrangian with respect to space [6]. Here we start from the Lorentzian Lagrangian and perform the reduction over space and over time in parallel. This is instructive, because the reduction over space corresponds to the standard c-map and gives us hypermultiplets in three-dimensional Minkowski spacetime, while the reduction over time is the new para-c-map and gives us hypermultiplets in three-dimensional Euclidean space. Before performing the reduction, we rewrite the Lorentzian vector multiplet Lagrangian in terms of real fields. Above we noted that the intrinsic characteristic of an affine special Kähler manifold is the existence of the special connection r, which is, in particular, flat, torsionfree and symplectic [27]. The corresponding flat symplectic coordinates are x I D Re X I ;

yI D Re FI :

(2.5)

Note that since F is an arbitrary holomorphic function, these real coordinates are related in a complicated way to the special coordinates X I . The real coordinates x I ; yI are flat (or affine) coordinates with respect to r, i.e., rdx I D 0 D rdyI , and they are symplectic (or Darboux coordinates), because the symplectic form on MVM is ! D 2dx I ^ dyI . While in special coordinates the metric of MVM can be expressed in terms of the prepotential by (2.2), the metric has a Hesse potential when using the real coordinates q a D .x I ; yI /, where a D 1; : : : ; 2n [27], [28]: gab D

@2 H : @q a @q b

(2.6)


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The Hesse potential is related to the imaginary part of the prepotential by a Legendre transform [44]: H.x; y/ D 2 Im F .x C iu/ 2uI yI : (2.7) The two parametrizations of the metric on MVM are related by ds 2 D 12 NIJ dX I d Xx J D gab dq a dq b :

(2.8)

In order to rewrite the Lagrangian (2.1) completely in terms of real fields, we express I˙ I IC I in terms of the field strength Fmn D Fmn C Fmn the (anti)selfdual field strength Fmn I IC I Q and their Hodge-duals Fmn D i.Fmn Fmn /. The result is VM I I Q J mn F D gab @m q a @m q b 14 NIJ Fmn F J mn C 14 RIJ Fmn ; L4d bos

(2.9)

where RIJ D FIJ C FxIJ ; NIJ D i.FIJ FxIJ / D @I @JN i.X I FxI FI Xx I / :

(2.10)

We now perform the reduction of the Lagrangian (2.9) from four to three dimensions. We treat the reduction over space and over time in parallel. In the following formulae, D 1 refers to reduction over time, which gives a Euclidean three-dimensional theory, while D 1 refers to reduction over space. By reduction, one component of each gauge field becomes a scalar. We define p I D AI j0 for D 1;

p I D AI j3 for D 1:

(2.11)

Moreover, the n three-dimensional gauge fields AI jmO obtained from dimensional reduction6 can be dualized into n further real scalars sI . Denoting the new scalars by .qO a / D .sI ; 2p I /; (2.12) the reduced bosonic Lagrangian takes the following, remarkably simple form: LHM D gab .q/@i q a @i qb C g ab .q/@i qO a @i qO b ;

(2.13)

where g ab .q/ is the inverse of gab .q/. In this parametrization it is manifest that the hypermultiplet target space with metric .gab .q// ˚ .g ab .q// is N D MHM D T MVM . The geometry underlying this Lagrangian was presented in detail in [6] for D 1, and works analogously for D 1. Here we give a brief summary. The special connection r on M D MVM , can be used to define a decomposition T N D Hr ˚ Tv N ' Tq M ˚ TqM;

(2.14)

where 2 N is a point on N (with local coordinates .q a ; qO a /), q D ./ 2 M is its projection onto M , Hr is the horizontal subspace with respect to the connection r and Tv N is the vertical subspace. The identification with Tq M ˚ Tq M is canonical, and the scalar fields q a , qO a obtained by dimensional reduction are adapted to the 6

The three-dimensional vector index takes values m O D 0; 1; 2 for D 1 and m O D 1; 2; 3 for D 1.

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decomposition. One can then define a complex structure J1 on N , which acts on T N ' Tq M ˚ Tq M by multiplication with J 0 r J1 WD J1 D ; (2.15) 0 J where J , J denote the action of the complex structure J of M on TM and T M , respectively. Let us now consider the Euclidean case D 1 for definiteness. Using the Kähler form ! on M , one can further define 0 ! 1 ; (2.16) J2 D ! 0 where ! is interpreted as a map Tq M ! Tq M . This is a para-complex structure, J22 D 1. Moreover, J3 D J1 J2 is a second para-complex structure, and J1 , J2 , J3 satisfy a modified version of the quaternionic algebra known as the para-quaternionic algebra. Thus, .J1 ; J2 ; J3 / is a para-hyper-complex structure on N . When defining, as in (2.13), the metric on N by g 0 gN D ; (2.17) 0 g 1 where g is the metric on M , then J1 is an isometry, while J2 ; J3 are anti-isometries. This means that .J1 ; J2 ; J3 ; gN / is a para-hyper Hermitian structure.7 Moreover, the structures J˛ , ˛ D 1; 2; 3 are parallel with respect to the Levi-Civita connection on N . Thus the metric gN is para-hyper Kähler, meaning that it is Kähler with respect to J1 and para-Kähler with respect to J2 ; J3 . The case D 1 works analogously. Here one finds three complex structures satisfying the quaternionic algebra, and the metric defined by (2.13) is hyper-Kähler. One can introduce (para-)complex fields such that one of the complex or (para-) complex structures becomes manifest in the three-dimensional Lagrangian [7], [6]. In these coordinates the Lagrangian is more complicated, and the geometrical structure reviewed above is less clear. Moreover one has singled out one of the three (para-) complex structures. Thus working in real coordinates has advantages, which should be exploited further in the future. Note in particular that for the c-map in local supersymmetry, the target space of hypermultiplets is quaternion-Kähler for Lorentzian space-time, while it is expected to be para-quaternion-Kähler for Euclidean spacetime. In general, the almost (para-)complex structures of a (para-)quaternion-Kähler manifold need not be integrable. Then combining real scalar fields into (para-)complex fields is not natural, as these fields do not define local (para-)complex coordinates.

7

Also note that J1 ; J2 ; J3 are integrable, which follows from the integrability of J .


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3 Black holes In order to prepare for our second application of special geometry, we now give a brief self-contained introduction to certain aspects of black holes.8 Somewhat surprisingly, one can associate thermodynamic properties to black holes. The so called laws of black hole mechanics, which have been derived in the framework of classical, matter-coupled Einstein gravity, formally have the same structure as the laws of thermodynamics [36]. While this was originally suspected to be a coincidence, the (theoretical) discovery of the Hawking effect [38] strongly suggested to take this observation seriously. More recently, developments in string theory have provided additional insights. Let us now briefly review this, starting with the laws of black hole mechanics in classical gravity.

3.1 The laws of black hole mechanics The zeroth laws of black hole mechanics states that the surface gravity S of a black hole is constant over the event horizon .9 The surface gravity can be defined if the event horizon is a Killing horizon, which is the case for all stationary black hole solutions of matter-coupled Einstein gravity. A Killing horizon is a hypersurface in space-time where a Killing vector field becomes null: D 0. One can show that a Killing horizon is generated by the integral lines of the Killing vector fields, which are null geodesics. There are two natural normal vector fields: the Killing vector field itself and the gradient of its norm-squared, r. /. Both vector fields must be proportional, and the factor of proportionality is defined to be the surface gravity: r . / D 2S :

(3.1)

While this implies that S is a function on the horizon, the zeroth law states that this function is constant. The physical interpretation of the surface gravity is that it measures the force which an observer outside the black hole must apply in order to keep a unit test mass fixed at the horizon. Thus it measures the strength of gravity at the horizon. Since the zeroth law of thermodynamics is that temperature is constant in thermodynamical equilibrium, this suggests to interpret surface gravity as temperature and stationary black holes as equilibrium states. At the classical level this interpretation cannot be defended against the obvious problem that a black hole does not emit radiation, a fact which is explicitly alluded to in the term ‘black’ hole. As we will review below this changes once quantum effects are taken into account. For the time being we focus on the assumptions needed to prove the zeroth law. The classical proof uses the explicit form of the Einstein equations, while the effects of matter are controlled by imposing a suitable condition on the energy-momentum tensor. Moreover 8

See [33], [34], [35] for a detailed discussion. term ‘horizon’ is unfortunately used for two different but closely related concepts. We will use to denote the null hypersurface which is the boundary between the exterior and interior of the black hole in space-time, and H for the space-like surface which is the boundary in space, at given time. Thus H is a spatial section of while is the ‘worldline’ of H . 9 The

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the solution must be stationary. It then follows from the field equations that the event horizon is a Killing horizon. However, it was realized later that the zeroth law does not depend on the details of the gravitational field equations. Instead, it can be proved for any covariant (diffeomorphism invariant) action, including actions which contain higher derivative and in particular higher curvature terms [37]. The relevance of such actions will be discussed below. The prize for not specifying the field equations is that one needs to make the following assumptions: (i) the field equations admit stationary black hole solutions with a Cauchy hypersurface, (ii) the event horizon is a Killing horizon, (iii) if the black hole is stationary but not static, then certain symmetry properties, which in Einstein geometry are consequences of the field equations, need to be imposed.10 Before proceeding, let us explain why it is desirable to admit actions containing higher derivative terms. The reason is that we would like to include so-called effective actions which incorporate quantum effects. In quantum field theory the effective action is defined to be the generating functional of the correlation functions. Since the classical action generates the classical contribution to the correlation functions (the leading part in an expansion in „) the effective action might be considered to be its quantum version. Unfortunately the exact effective action is usually a rather formal and inaccessible object. However, certain approximations can be computed, and string theory provides a framework where quantum corrections to the gravitational action can and have been computed.11 As expected on general grounds, quantum gravity manifests itself in the form of higher derivative terms in the effective action, in particular terms which contain higher powers of the Riemann tensor and its contractions. We will discuss a particular class of such terms in the next section. Let us next turn to the first law of black hole mechanics, which states that for a stationary black hole an infinitesimal change of the mass M is related to infinitesimal changes of the horizon area A, of the angular momentum J and of the electric charge Q by S ıM D ıA C !ıJ C ıQ; (3.2) 8 where ! is the angular velocity and the electrostatic potential. This should be compared to the first law of thermodynamics (for a grand canonical ensemble), ıE D T ıS pıV C ıN;

(3.3)

where E is energy, T is temperature, S is entropy, p is pressure, V is volume, is chemical potential and N is particle number. Given the relation between surface gravity and temperature, this suggests to interprets the area of the event horizon as the entropy of the black hole. This is surprising, since the entropy of normal thermody10A space-time is stationary if it admits a time-like Killing vector field. In a static space-time this Killing vector field is in addition required to be the normal vector field of a family of hypersurfaces. The additional requirements needed if the space-time is stationary but not static are that the black hole is axisymmetric and invariant under simultaneous reflection of time and the angle around the symmetry axis. 11 We refer to [26] for more details.


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namical systems is an extensive property, i.e., proportional to the volume rather than the surface area. Like the zeroth law the first law can be derived for general covariant actions, under the same assumptions as for the zeroth law. Moreover, the mass, entropy, angular momentum and charge of the black hole are defined as surface charges, which are obtained by integrating a so-called Noether two-form over a closed spatial surface [69]. The Noether two-form is constructed out of Killing vector fields according to a certain algorithm. For the special case of Einstein gravity this definition reduces to the usual ones (i.e., the Komar or ADM constructions of mass and angular momentum, and the proportionality of entropy and horizon area). Finally, let us turn to the second law of black hole mechanics, the Hawking area law. In contrast to the zeroth and first law, one does not assume the space-time to be stationary. Rather it can be time-dependent, and include processes such as the formation and fusion of black holes, as long as the time evolution is ‘asymptotically predictable’.12 The second law then states that the total area of event horizons is non-decreasing, ıA 0; (3.4) which is obviously analogous to the second law of thermodynamics, which states the same for the entropy, ıS 0: (3.5) This reinforces the identification of area and entropy suggested by the first law. The second law has been derived using Einstein’s field equation together with conditions on the energy-momentum tensor of matter (plus assuming ‘predictability’ of space-time). So far there is no general proof for the case of general covariant actions. However, examples have been studied, and the integrated Noether form is a good candidate for entropy in non-stationary space-times [40]. One interesting question is whether one should expect that the second law holds for all covariant actions. Since dynamical processes such as collision of black holes are admitted, the contents of the second law appears to be more sensitive to the details of the dynamics as the zeroth and first law. It is not clear whether all possible higher derivative actions give rise to ‘sensible’ physics which respects the second law. But one would certainly expect this to be true for string-effective actions, though this does not seem to have studied so far. Anyway, already the zeroth and first law provide compelling evidence for relating relating the surface gravity to the temperature and the area (integrated Noether two-form) to the entropy.

3.2 Quantum aspects of black holes Let us now review the role of the Hawking effect [38] in making plausible the reinterpretation of geometrical as thermodynamic properties. This effect is derived 12 We

refer to [33] for a precise definition and more details.

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by treating space-time geometry as a classical background, while matter is described by quantum field theory. In this framework it has been shown that black holes emit thermal radiation, even if there is no ingoing radiation or matter. Moreover, the socalled Hawking temperature of this radiation is indeed proportional to the surface gravity: S THawking D : (3.6) 2 In Einstein gravity the factor of proportionality between area and entropy is then fixed by the first law: A SD (3.7) 4 (Newton’s and Planck’s constant and the speed of light serve as natural units, GN D „ D c D 1). When using a covariant higher derivative action, the entropy is given by integrating the Noether two form Q over the horizon H : I S D 2 Q: (3.8) H

It has been shown that the entropy can be expressed in terms of variational derivatives of the Lagrangian with respect to the Riemann tensor [39], [40]: I ıL p 2 S D 2 " " h d ; (3.9) H ıR p where is the normal bivector of H (with a certain normalization), and h d 2

is the induced volume element. If L is the Einstein–Hilbert Lagrangian, this formula reproduces the area law. If further terms containing the Riemann tensor are present in L they induce explicit modifications of the area law. Once the Hawking effect is taken into account, black holes can emit radiation, which implies that they loose mass and shrink, thus violating the second law of black hole mechanics. However, as soon as one takes the idea seriously that black holes carry entropy, one should consider the total entropy obtained by adding black hole entropy and the thermodynamical entropy of the exterior region. The generalized second law of thermodynamics, which states that the total entropy is non-decreasing, is expected to be valid in quantum gravity [41]. So far we have considered black hole entropy from what one might call the macroscopic or thermodynamical perspective. When dealing with many-constituent systems one distinguishes two levels of description. The fundamental or microscopic level of description requires knowledge of the precise state of the system. For a classical gas this would require to specify the positions and momenta (and other quantities if internal excitations exist) of all atoms or molecules. At the thermodynamical or macroscopic level of description one only considers collective properties of the system, such as temperature, volume and pressure. Statistical mechanics asserts that these macroscopic quantities arise by ‘coarse graining’ microscopic quantities. E.g., temperature is the average energy per degree of freedom. Obviously many microstates


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will give rise to the same macrostate, where the latter is characterised by fixing only the macroscopic quantities. The so-called statistical or microscopic entropy measures how many different microstates give rise to the same macrosate. If d.E; : : : / denotes the number of microstates corresponding to the macrostate with energy E, etc., then the corresponding microscopic entropy is Smicro D log d.E; : : : /:

(3.10)

In contrast the so-called macroscopic or thermodynamical entropy Smacro is a purely macroscopic quantity, which can be characterized by its relation to other macroscopic quantities, such as temperature, free energy, etc. Both entropies are expected to be equal in the thermodynamcial limit, i.e., when the number of constituents goes to infinity. The geometrical black hole entropy is analogous to the macroscopic entropy, because it has been defined through relations which only involve collective properties of the black hole, such as mass, charge and angular momentum. Any theory of quantum gravity is expected to provide a corresponding microscopic description of black holes, which in particular allows one to identify its microstates. In particular the microscopic entropy should be equal to the macroscopic one, at least in the limit of large mass, which is analogous to the thermodynamical limit. This is widely regarded as a benchmark test for theories of quantum gravity.

3.3 Black holes and strings In string theory four-dimensional black holes can be interpreted as arising from states in the full ten-dimensional string theory. These states might be string states, or winding states of higher-dimensional membranes (in particular D-branes) [42]. One can test the expected relation between macroscopic and microscopic entropy by counting the ten-dimensional states which give rise to the same four-dimensional black hole. This comparison generically involves the variation of parameters such as the string coupling, and it is not a priori clear whether the number of states is preserved under this interpolation. But for a special subclass of states, the so-called supersymmetric states or BPS states, which we will review below, the interpolation is at least highly plausible. Moreover, both the macroscopic and the microscopic entropy can often be computed to high precision and it has been found that they match [9], even when subleading corrections are included [15]. In particular these tests are sensitive to the distinction between the area law and the generalized formula (3.9), and clearly show that string theory ‘knows’ about the modifications of the area law. In performing these precision tests, special geometry plays a central role. It is the indispensable tool for constructing black hole solutions and extracting the macroscopic entropy from them. This will be the subject of the next section. We will not be able to cover the microscopic side of the story, i.e., the counting of microstates, in this exposition.

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3.4 Black holes and supersymmetry Before turning to the details, let us review the concepts of BPS states and BPS solitons.13 Recall that the supercharges which generate supersymmetry transformations are spinors. If there is more than one such spinor, then the supersymmetry algebra admits central operators, which can be organised into a complex antisymmetric matrix. The skew eigenvalues Z.i/ of this matrix are called the central charges. It can be shown that on any irreducible representation the mass is bounded from below by the absolute values of the charges: M jZ.1/ j jZ.2/ j :

(3.11)

Moreover, when the mass saturates one or several of these bounds, part of the supercharges operate trivially, and the corresponding multiplet is shorter than a generic massive multiplet. Such multiplets are called supersymmetric multiplets or BPS multiplets. When all inequalities are saturated, the resulting BPS multiplet is invariant under half of the supertransformations and has as many states as a massless multiplet. In the case of N D 2 supersymmetry considered here, the algebra has one single complex supercharge Z. Consequently, there are generic massive supermultiplets M > jZj and ‘ 12 -BPS multiplets’ with M D jZj. The concept of BPS state can be applied to solitons. By solitons we refer to solutions of the field equations which can be interpreted as particle-like objects. In particular, these solutions are required to have finite energy, and therefore must approach the ground state asymptotically. Since the energy is localized in a small part of space, such ‘lumps’ can be thought of as ‘extended particles’. One also requires that the solution is static (describing ‘a massive particle in its rest frame’) and free of naked singularities (we admit singularities covered by event horizons in order to include black holes). A soliton is then called supersymmetric or BPS, if it is invariant under part of the supersymmetry transformations. Let us denote the fields of the underlying action collectively by ˆ, the spinorial supersymmetry transformation parameters by , the corresponding supersymmetry transformation by ı and the soliton solution by ˆ0 . Then a solution is BPS if there exists a choice of such that .ı ˆ/jˆ0 D 0:

(3.12)

Particular examples of BPS solitons are provided by supersymmetric black hole solutions of supergravity actions. In supergravity the supersymmetry transformation parameters depend on space-time, D .x/. Therefore the BPS condition implies the existence of a spinor field which generates a supertransformation under which the black hole solution is invariant. This is analogous to a Killing vector field, which generates a diffeomorphism under which the metric (and possibly other fields) are invariant. Therefore such spinor fields .x/ are called Killing spinors (more accurately 13 In the following we use basic facts about supersymmetry algebras and their representations.

[10], Chapter II.

See for example


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Killing spinor fields). The interested reader is referred to the monograph [43] for a detailed discussion of supersymmetric solutions.

4 Special geometry and black holes 4.1 Vector multiplets coupled to gravity We are now in position to discuss BPS black hole solutions in N D 2 supergravity coupled to n vector multiplets. This is the relevant part of the effective action for string compactifications preserving N D 2 supersymmetry. The general N D 2 vector multiplet action was constructed using the superconformal calculus [31].14 The idea of this method is to start with a theory of n C 1 rigidly supersymmetric vector multiplets and to impose that the theory is invariant under superconformal transformations. This implies that the prepotential has to be homogenous of degree 2 in addition to being holomorphic: F .X I / D 2 F .X I /;

2 C;

(4.1)

where now I D 0; 1; : : : ; n. Next one ‘gauges’ the superconformal transformation, that is one makes the Lagrangian locally superconformally invariant by introducing suitable connections. The new fields entering through this process are encoded in the so-called Weyl multiplet.15 Finally, one imposes gauge conditions which reduce the local superconformal invariance to a local invariance under standard (Poincaré) supersymmetry. Through the gauge conditions some of the fields become functions of the others. In particular, only n out of the n C 1 complex scalars are independent. A convenient choice for the independent scalars is XA ; (4.2) X0 where A D 1; : : : ; n. This provides a set of special coordinates for the scalar manifold MVM . In contrast, all n C 1 gauge fields remain independent. While one particular linear combination, the so-called graviphoton, belongs to the Poincaré supergravity multiplet, the other n gauge fields sit in vector multiplets, together with the scalars z A . The Weyl multiplet also provides physical degrees of freedom, namely the graviton and two gravitini. From the underlying rigidly superconformal theory the supergravity theory inherits the invariance under symplectic rotations. For the gauge fields this is manifest, as I ; GI jmn / transforms as a vector under Sp.2.n C 1/; R/.16 In the scalar sector .Fmn zA D

14 Further references on N D 2 vector multiplet Lagrangians and the superconformal calculus include [45], [46], [1], [47]. 15 One also needs to add a further ‘compensating multiplet’, which can be taken to be a hypermultiplet. We won’t need to discuss this technical detail here. See for example [26] for more background material and references. 16 The dual gauge fields G I jmn were introduced at the beginning of Section 2.

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.X I ; FI /, where FI D @I F , also transforms as a vector, while the gravitational degrees of freedom are invariant. To maintain manifest symplectic invariance, it is advantageous to work with .X I ; FI / instead of z A . The underlying geometry can be described as follows [27], [28], [29]: the fields X I provide coordinates on the scalar manifold of the associated rigidly superconformal theory. This manifold has complex dimension n C 1, and can be immersed into T C nC1 ' C 2.nC1/ just as described in the previous section. The additional feature imposed by insisting on superconformal invariance is that the prepotential is homogenous of degree 2. Geometrically this implies that the resulting affine special Kähler manifold is a complex cone. The scalar manifold of the supergravity theory is parametrized by the scalars z A and has complex dimension n. It is obtained from the manifold of the rigidly superconformal theory by gauge-fixing the dilatation and U.1/ symmetry contained in the superconformal algebra. This amounts to taking the quotient of the complex cone with respect to the C -action X I ! X I . Thus the scalar manifold MVM is the basis of the conical affine special Kähler manifold C.MVM / of the rigid theory. For many purposes, including the study of black hole solutions, it is advantageous to work on C.MVM / instead of MVM . In particular, this allows to maintain manifest symplectic covariance, as we already noted. In physical terms this means that one can postpone the gauge-fixing of the dilatation and U.1/ transformations. The manifolds which can be obtained from conical affine special Kähler manifolds by a C -quotient are called projective special Kähler manifolds. These are the target spaces of vector multiplets coupled to supergravity. All couplings in the Lagrangian and all relevant geometrical data of MVM are encoded in the prepotential. In particular, the affine special Kähler metric on C.MVM / has Kähler potential KC .X I ; Xx I / D i.X I FxI FI Xx I /; while the projective special Kähler metric on MVM has Kähler potential x K.z A ; zN B / D log i.X I FxI FI Xx I / ; with corresponding metric

x

(4.3)

(4.4)

@2 K.z A ; zN B / : (4.5) @z a @Nz b In string theory the four-dimensional supergravity Lagrangians considered here are obtained by dimensional reduction of the ten-dimensional string theory on a compact six-dimensional manifold X and restriction to the massless modes. Then the scalar manifold MVM is the moduli space of X. It turns out that the moduli spaces of Calabi– Yau threefolds provide natural realizations of special Kähler geometry [58]. Consider for instance the Calabi–Yau compactification of type-IIB string theory. In this case MVM is the moduli space of complex structures of X, the cone MVM is the moduli space of complex structures together with a choice of the holomorphic top-form, and T C nC1 ' C 2.nC1/ is H 3 .X; C/, see [59]. gabN D


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4.2 BPS black holes and the attractor mechanism Let us then discuss BPS black hole solutions of N D 2 supergravity with n vector multiplets. These are static, spherically symmetric solutions of the field equations, which are asymptotically flat, have regular event horizons, and possess 4 Killing spinors. Since the N D 2 superalgebra has 8 real supercharges, these are 12 -BPS solutions. Let us first have a look at pure four-dimensional N D 2 supergravity, i.e., we drop the vector multiplets, n D 0. The bosonic part of this theory is precisely the Einstein–Maxwell theory. In pure N D 2 supergravity, BPS solutions have been classified [60], [61], [62]. The number of linearly independent Killing spinor fields can be 8, 4 or 0. This can be seen, for example, by investigating the integrability conditions of the Killing spinor equation.17 Solutions with 8 Killing spinors are maximally supersymmetric and therefore considered as supersymmetric ground states. Examples are Minkowski space and AdS2 S 2 . Solutions with 4 Killing spinors are called 12 -BPS, because they are invariant under half as many supersymmetries as the ground state. They are solitonic realisations of states sitting in BPS representations. For static 12 -BPS solutions the space-time metric takes the form [60], [61] E E dt 2 C e 2f .x/ d xE 2 ; ds 2 D e 2f .x/

(4.6)

where xE D .x1 ; x2 ; x3 / are space-like coordinates and the function f .x/ E must be E such that e f .x/ is a harmonic function with respect to x. E The solutions also have E . This a non-trivial gauge field, which likewise can be expressed in terms of e f .x/ class of solutions of Einstein–Maxwell theory is known as the Majumdar–Papapetrou solutions [64], [65]. The only Majumdar–Papapetrou solutions without naked singularities are the multi-centered extremal Reissner–Nordstrom solutions, which describe static configurations of extremal black holes, see for example [66]. If one imposes in addition spherical symmetry, one arrives at the extremal Reissner–Nordstrom solution describing a single charged black hole. In this case the metric takes the form ds 2 D e 2f .r/ dt 2 C e 2f .r/ .dr 2 C r 2 d 2 /;

(4.7)

where r is a radial coordinate and d 2 is the line element on the unit two-sphere. The harmonic function takes the form e f .r/ D 1 C

q2 C p2 ; r

(4.8)

where q, p are the electric and magnetic charge with respect to the graviphoton. The solution has two asymptotic regimes. In one limit, r ! 1, it becomes asymptotically flat: e f ! 1. In the other limit, r ! 0, which is the near-horizon limit, it takes the 17 The classification of supersymmetric solutions has recently moved to the focus of interest. Readers who want to get an idea how the classification of supersymmetric solutions of four-dimensional N D 2 supergravity would work with ‘modern’, systematic methods can consult [63], where all supersymmetric solutions of minimal five-dimensional supergravity were constructed.

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form ds 2 D

r2 q2 C p2 2 2 dt C dr C .q 2 C p 2 /d 2 : q2 C p2 r2

(4.9)

This is a standard form for the metric of AdS2 S 2 . The area of the two-sphere, which is the area of the event horizon of the black hole, is given by A D 4.q 2 C p 2 /. The two limiting solutions, flat Minkowski space-time and AdS2 S 2 are among the fully supersymmetric solutions with 8 Killing spinors that we mentioned before. Thus, the extremal Reissner–Nordstrom black hole interpolates between two supersymmetric vacua [48]. This is a property familiar from two-dimensional kink solutions, and motivates the interpretation of supersymmetric black hole solutions as solitons, i.e., as particle-like collective excitations. Let us now return to N D 2 supergravity with an arbitrary number n of vector fields. We are interested in solutions which generalize the extremal Reissner–Nordstrom solution. Therefore we impose that the solution should be 12 -BPS, static, spherically symmetric, asymptotically flat, and that it should have a regular event horizon.18 More general 12 -BPS solutions have been studied extensively in the literature, in particular in [49] and [16]. Recently, the classification of all 12 -BPS solutions was achieved in [50]. BPS black holes in theories with n vector multiplets depend on n C 1 gauge fields and on n scalar fields. For any 12 -BPS solution, which is static and spherically symmetric, the metric can be brought to the form (4.8) [16]. The condition that the solution is static and spherically symmetric is understood in the strong sense, i.e., it also applies to the gauge fields and scalars. Thus gauge fields and scalars are functions of the radial coordinate r, only. Moreover the electric and magnetic fields are spherically I .r/ has only two independent symmetric, which implies that each field strength Fmn components (see for example Appendix A of [26] for more details). The electric and magnetic charges carried by the solution are defined through flux integrals of the field strength over asymptotic two-spheres: I I 1 I I F ; GI ; (4.10) .p ; qI / D 4 I and their where F I ; GI are the two-forms associated with the field strength Fmn duals GI mn . As a consequence, the charges transform as a vector under symplectic transformations. By contracting the charges with the scalars one obtains the symplectic function (4.11) Z D p I FI qI X I :

This field is often called the central charge, which is a bit misleading because Z is a function of the fields X I and FI and therefore a function of the scalar fields z A , which are space-time dependent.19 Hence, in the class of backgrounds we consider, Z is a function of the radial coordinate r. However, when evaluating this field in the 18 This excludes both naked singularities and null singularities, where the horizon coincides with the singularity

and has vanishing area.


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asymptotically flat limit r ! 1, it computes the electric and magnetic charge carried by the graviphoton, which combine into the complex central charge of the N D 2 algebra [67]. In particular, the mass of the black hole is given by M D jZj1 D M.p I ; qI ; z A .1//:

(4.12)

Thus BPS black holes saturate the mass bound implied by the supersymmetry algebra. Note that the mass does not only depend on the charges, but also on the values of the scalars at infinity, which can be changed continuously. The other asymptotic regime is the event horizon. If the horizon is regular, then the solution must be fully supersymmetric in this limit [11]. Thus, while the bulk solution has 4 Killing spinors, both asymptotic limits have 8. In the near horizon limit, the metric (4.8) takes the form ds 2 D

r2 jZj2hor 2 2 dt C dr C jZj2hor d 2 ; r2 jZj2hor

(4.13)

where jZj2hor is the value of jZj2 at the horizon. As in the extremal Reissner–Nordstrom solution, this is AdS2 S 2 . The area of the two-sphere, which is the area of the event horizon, is given by A D 4jZj2hor . Hence the Bekenstein Hawking entropy is A D jZj2hor : (4.14) 4 A priori, Smacro depends on both the charges and the values of the scalars at the horizon, and one might expect that one can change the latter continuously. This would be incompatible with relating Smacro to a statistical entropy Smicro which counts states. But it turns out that the values of the scalar fields at the horizon are themselves x I determined in terms of the charges. Here, it is convenient to define Y I D ZX x I .X /.20 In terms of these variables, the black hole attractor and FI D FI .Y / D ZF equations [11], which express the horizon values of the scalar fields in terms of the charges, take the following, symplectically covariant form: I I Y Yx I p D i : (4.15) qI FI FxI hor Smacro D

The name attractor equations refers to the behaviour of the scalar fields as functions of the space-time radial coordinate r. While the scalars can take arbitrary values at r ! 1, they flow to fixed points, which are determined by the charges, for r ! 0. This fixed point behaviour follows when imposing that the event horizon is regular. Alternatively, one can show that to obtain a fully supersymmetric solution with geo19 One can analyse BPS solutions without imposing the gauge conditions which fix the superconformal symmetry, and in fact it is advantageous to do so [15], [16]. Then the scalars are encoded in the fields X I .r/, which are subject to gauge transformations. Once gauge conditions are imposed, one can express Z.r/ in terms of the physical scalar fields z A .r/. See [26] for more details. 20 Note that F is homogenous of degree 1. I

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metry AdS2 S 2 the scalars need to take the specific values dictated by the attractor equations [16]. This is due to the presence of non-vanishing gauge fields. The gauge fields in AdS2 S 2 are covariantly constant, so that this can be viewed as an example of a flux compactification. In contrast, Minkowski space is also maximally supersymmetric, but the scalars can take arbitrary constant values, because the gauge fields vanish. In type-II Calabi–Yau compactifications, the radial dependence of the scalar fields defines a flow on the moduli space, which starts at an arbitrary point and terminates at a fixed point corresponding to an ‘attractor Calabi–Yau.’ Since the electric and magnetic charges .p I ; qI /, which determine the fixed point, take discrete values, such attractor threefolds sit at very special points in the moduli space. This has been studied in detail in [51]. Using the fields Y I instead of X I to parametrize the scalars simplifies formulae and has the advantage that the Y I are invariant under the U.1/ transformations of the superconformal algebra. Note that jZj2 D p I FI qI Y I ;

(4.16)

which is easily seen using the homogeneity properties of the prepotential. The diffeomorphism X I ! Y I acts non-holomorphically on the cone C.MVM /, but operates trivially on its basis MVM . Note in particular that zA D

XA YA D : X0 Y0

(4.17)

4.3 The black hole variational principle We now turn to the black hole variational principle, which was found in [14] and generalized in [13], motivated by the observations of [12]. First, we define two symplectic functions, the entropy function †.Y I ; Yx I ; p I ; qI / D F .Y I ; Yx I / qI .Y I C Yx I / C p I .FI C FxI /

(4.18)

and the black hole free energy

F .Y I ; Yx I / D i Yx I FI Y I FxI :

(4.19)

The reason for our choice of terminology will become clear later. Now we impose that the entropy function is stationary, ı† D 0, under variations of the scalar fields Y I ! Y I C ıY I . Using that the prepotential is homogenous of degree two, it is easy to see that the conditions for † being stationary are precisely the black hole attractor equations (4.15). Furthermore, at the attractor point we find that21 Fattr D i Yx I FI Y I FxI attr D qI Y I p I FI attr (4.20) D qI Yx I p I FxI D jZj2attr attr

The relation i Yx I FI Y I FxI D qI Y I p I FI follows from the definitions of Z and Y I together with the homogeneity of the prepotential (once the dilatational symmetry of the fields X I has been gauge fixed). Therefore it holds irrespective of whether the scalar fields take their attractor values or not. 21


and therefore

†attr D jZj2attr D

1 S .p I ; qI /: macro

169

(4.21)

Here and in the following we use the label ‘attr’ (instead of ‘hor’ used previously) to indicate that quantities are evaluated at the attractor point determined by the electric and magnetic charges. Thus, up to a constant factor, the entropy is obtained by evaluating the entropy function at its critical point. Moreover, a closer look at the variational principle shows us that, again up to a factor, the black hole entropy Smacro .p I ; qI / is the Legendre transform of the free energy F .Y I ; Yx I /, where the latter is considered as a function of x I D Re.Y I / and yI D Re.FI /. At this point the real variables discussed in the previous section become important again. Note that the change of variables .Y I ; Yx I / ! .x I ; yI / is well defined provided that Im.FIJ / is non-degenerate. This assumption will be satisfied in general, but breaks down in certain string theory applications, where one reaches the boundary of the moduli space.22 We are therefore led to rewrite the variational principle in terms of real variables. First, recall that the Hesse potential H.x I ; yI / is the Legendre transform of (two times) the imaginary part of the prepotential, see (2.7).23 This Legendre transform replaces the independent variables .x I ; uI /D (Re.Y I /, Im.Y I /) by the independent variables .x I ; yI /D( Re.Y I /, Re.FI /) and therefore implements the change of variables .Y I ; Yx I / ! .x I ; yI /. Using (2.7) we find H.x I ; yI / D 2i .Yx I FI FxI Y I / D 12 F .Y I ; Yx I /:

(4.22)

Thus, up to a factor, the Hesse potential is the black hole free energy. We can now express the entropy function in terms of the real variables: †.x I ; yI ; p I ; qI / D 2H.x I ; yI / 2qI x I C 2p I yI :

(4.23)

If we impose that † is stationary with respect to variations of x I and yI , we get the black hole attractor equations in real variables: @H D qI ; @x I

@H D p I : @yI

Plugging this back into the entropy function we obtain

Smacro D 2 H x I

@H @H yI @x I @yI

(4.24)

:

(4.25)

attr

Thus, up to a factor, the black hole entropy is the Legendre transform of the Hesse potential. This is an intriguing observation, because it relates the black hole entropy, which is a space-time quantity, directly to the special geometry encoding the scalar dynamics. In string theory compactifications this relates the geometry of fourdimensional space-time to the geometry of the compact internal space X. The Hesse 22 See

for example [13] for a discussion of some of the implications.

23 Note that this is the Hesse potential of the affine special Kähler metric on C.M

Kähler metric on MVM is obtained by the C -quotient.

VM /.

The projective special

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potential appears to be closely related to the action functional underlying the geometry of X in Hitchin’s approach to manifolds with special holonomy [52], [53], [54]. We can also relate the black hole free energy to another quantity of special geometry. In terms of complex variables we observe that F .Y I ; Yx I / D KC .Y I ; Yx I / WD i.Yx I FI FxI Y I /:

(4.26)

Comparing to (4.3) it appears that we should interpret KC .Y I ; Yx I / as the Kähler potential of an affine special Kähler metric on C.MVM /. Since the diffeomorphism X I ! Y I is non-holomorphic, this is not the same special Kähler structure as with (4.3). However, we already noted that the diffeomorphism acts trivially on MVM , see (4.17). Moreover it is easy to see that when taking the quotient with respect to the C -action Y I ! Y I , then the resulting projective special Kähler metric with Kähler potential K.Y I ; Yx I / D log KC .Y I ; Yx I / is the same as the one derived from (4.4), because the two Kähler potentials differ only by a Kähler transformation. It appears that in the context of black hole solutions the affine special Kähler metric associated with the rescaled scalars Y I is of more direct importance than the one based on the X I . The same remark applies to the Hesse potential, which depends on the real coordinates associated to Y I . Note that the scalars Y I do not only encode the values of the Calabi–Yau moduli z A via (4.17) but also, via (4.16) the size of the twosphere in the black-hole space-time.24 While variations of the moduli correspond to variations along the basis of the cone C.M /, variations of the radius of the two-sphere correspond to motions along the radial direction of the cone. Note that it is more natural to identify the free energy with the Hesse potential than the Kähler potential. The first reason is that the various Legendre transforms involve the real and not the complex coordinates. The second reason is that, as we will discuss below, we need to generalize the supergravity Lagrangian in order to take into account certain corrections appearing in string theory. We will see that this works naturally by introducing a generalized Hesse potential. Before turning to this subject, we also remark that the terms in the entropy function (4.18) which are linear in the charges, and which induce the Legendre transform, have yet another interpretation in terms of supersymmetric field theory. Namely, the symplectic function W D qI Y I p I FI

(4.27)

has the form of an N D 2 superpotential. The four-dimensional supergravity Lagrangian we are studying does not have a superpotential. However, the near-horizon solution has the form AdS2 S 2 and carries non-vanishing, covariantly constant gauge fields. The dimensional reduction on S 2 is a flux compactification, with fluxes parametrized by .p I ; qI /, and the resulting two-dimensional theory will possess a superpotential. This also provides an alternative interpretation of the attractor mechanism, as the resulting scalar potential will lift the degeneracy of the moduli. 24 This

is not only true at the horizon but throughout the whole black hole solution.

171


4.4 Quantum corrections to black holes solutions and entropy So far we only considered supergravity Lagrangians which contain terms with at most two derivatives. The effective Lagrangians derived from string theory also contain higher derivative terms, which modify the dynamics at short distances. These terms describe interactions between the massless states which are mediated by massive string states. While the effective Lagrangian does not contain the massive string states explicitly, it is still possible to describe their impact on the dynamics of the massless states. In N D 2 supergravity a particular class of higher derivative terms can be taken into account by giving the prepotential an explicit dependence on an additional complex variable ‡, which is proportional to the lowest component of the Weyl multiplet [57], [68]. The resulting function F .Y I ; ‡/ is required to be holomorphic in all its variables, and to be (graded) homogenous of degree two:25 F .Y I ; 2 ‡ / D 2 F .Y I ; ‡/:

(4.28)

Assuming that it is analytic at ‡ D 0 one can expand it as F .Y I ; ‡/ D

1 X

F .g/ .Y I /‡ g :

(4.29)

gD0

Then F .0/ .Y I / is the prepotential, while the functions F .g/ .Y I / with g > 0 appear in the Lagrangian as the coefficients of various higher-derivative terms. These include in particular terms quadratic in the space-time curvature, and therefore one often loosely refers to the higher derivative terms as R2 -terms. In type-II Calabi–Yau compactifications the functions F .g/ .Y I / can be computed using (one of) the topologically twisted version(s) of the theory [56]. They are related .g/ to the partition functions Ztop of the topologically twisted string on a world sheet with .g/ .g/ genus g by F D log Ztop . Therefore they are called the (genus-g) topological free energies. It was shown in [15], [16] that the black hole attractor mechanism can be generalized to the case of Lagrangians based on a general function F .Y I ; ‡/. The attractor equations still take the form (4.15), but the prepotential is replaced by the full function F .Y I ; ‡/. The additional variable ‡ takes the value ‡ D 64 at the horizon. The evaluation of the generalized entropy formula (3.9) for N D 2 supergravity gives [15]: (4.30) Smacro .q I ; pI / D jZj2 C 4 Im.‡F‡ / attr ; where F‡ D @‡ F .26 Note that symplectic covariance is manifest, as the entropy is the sum of two symplectic functions. While the first term corresponds to the area law, the second term is an explicit modification which depends on the coefficients F .g/ , g > 0, of the higher derivative terms. we are interested in black hole solutions, we use rescaled fields Y I ; ‡ . the attractor point, ‡ takes the value ‡ D 64.

25 Since 26At

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It was shown in [13] that the variational principle generalizes to the case with R2 -terms. The black hole free energy F is now proportional to a generalized Hesse x which in turn is proportional to the Legendre transform of potential H.x I ; yI ; ‡; ‡/, the imaginary part of the function F .Y I ; ‡/: x D 2 Im F .x I C iuI ; ‡/ 2yI uI : H.x I ; yI ; ‡; ‡/ In terms of complex fields Y I this becomes x D i .Yx I FI FxI Y I / i.‡F‡ ‡ x Fxx / H.x I ; yI ; ‡; ‡/ ‡ 2 I xI 1 x D F .Y ; Y ; ‡; ‡ /:

(4.31)

2

The entropy function (4.23), the attractor equations (4.24) and the formula for the entropy (4.25), which now includes correction terms to the area law, remain the same, except that one uses the generalized Hesse potential. From (4.31) it is obvious that the black hole free energy naturally corresponds to a generalized Hesse potential (defined by the Legendre transform of the prepotential) and not to a ‘generalized Kähler potential’, which would only give rise to the first term on the right hand side of (4.31). There is a second class of correction terms in string-effective supergravity Lagrangians. Quantum corrections involving the massless fields lead to modifications which correspond to adding non-holomorphic terms to the function F .Y I ; ‡/. The necessity of such non-holomorphic terms can be seen by observing that otherwise the invariance of the full string theory under T-duality and S-duality is not captured by the effective field theory. In particular, one can show that the black hole entropy can only be T- and S-duality invariant if non-holomorphic corrections are taken into account [55].27 From the point of view of string theory the presence of these terms is related to a holomorphic anomaly [56], [57]. As the holomorphic R2 -corrections, the non-holomorphic corrections can be incorporated into the black hole attractor equations and the black hole variational principle x /, [55], [13]. The non-holomorphic terms are encoded in a function .Y I ; Yx I ; ‡; ‡ which is real valued and homogenous of degree two. To incorporate non-holomorphic terms into the variational principle one has to define the generalized Hesse potential as the Legendre transform of 2 Im F C 2 : x D 2 Im F .x I C i uI ; ‡; ‡ x / C 2 .x I ; uI ; ‡; ‡ x / 2yOI uI ; (4.32) H.x I ; yOI ; ‡; ‡/ @

@

where yOI D yI C i. I IN / and I D @Y I and IN D @Y x I . Up to these modifications, the attractor equations, the entropy function, and the entropy remain as in (4.24), (4.23) and (4.25). Also note from (4.32) that if is harmonic, it can be absorbed into Im F , because it then is the imaginary part of holomorphic function. Thus, the non-holomorphic modifications of the prepotential correspond to non-harmonic functions . 27 We are referring to compactifications with exact T- and S-duality symmetry. These are mostly compactifications with N D 4 supersymmetry, which, however, can be studied in the N D 2 framework. We refer to [55], [70], [13] for details.


In terms of the complex variables the attractor equation are I p Y I Yx I Di : qI FI C 2i I FI C 2i IN

173

(4.33)

The modified expressions for the free energy and the entropy function can be found in [13]. At this point it is not quite clear what the R2 -corrections and the non-holomorphic corrections mean in terms of special geometry. Since they correspond to higher derivative terms in the Lagrangian, they do not give rise to modifications of the metric on the scalar manifold, which, by definition, is the coefficient of the scalar two-derivative term.28 It would be very interesting to extend the framework of special geometry such that the functions F .g/ get an intrinsic geometrical meaning.

4.5 Black hole partition functions and the topological string Let us now discuss how the black hole variational principle is related to black hole partition functions and the topological string. We start by relating the variational principle described in the last sections to the variational principle used in [12]. One can start from the generalized Hesse potential and perform partial Legendre transforms by imposing only part of the attractor equations. If this subset of fields is properly chosen one obtains a reduced variational principle, which yields the remaining attractor equations, and, by further extremisation, the black hole entropy. Specifically, one can solve the magnetic attractor equations Y I Yx I D ip I by setting29 Y I D 12 . I C ip I /:

(4.34)

Plugging this back, the new, reduced entropy function is x / qI I ; †.p I ; I ; qI / D FE .p I ; I ; ‡; ‡

(4.35)

where30

x D 4 Im F .Y I ; ‡/ C .Y I ; Yx I ; ‡; ‡ x/ FE .p I ; I ; ‡; ‡/ mgn

(4.36)

Here the label ‘mgn’ indicates that the magnetic attractor equations have been imx / D 2H.x I ; yOI ; ‡; ‡ x / and posed, i.e., Y I D 12 . I C ip I /. Both F .Y I ; Yx I ; ‡; ‡ x are interpreted as free energies, which, however, refer to differFE .p I ; I ; ‡; ‡/ ent statistical ensembles. In the microcanonical ensemble the electric and magnetic charges are kept fixed, while they fluctuate around a mean value in the canonical ensemble. The transition between these two ensemble is made by changing the independent variables, i.e., one eliminates the electric and magnetic charges qI ; p I in 28 See

however [72], where such an interpretation was proposed. I D 2x I . We use I to be consistent with the notation used in [13]. The conventions of [12] are slightly different. 30 We suppressed the dependence of † on ‡ , but indicated it for F in order to make explicit that we included E the higher derivative corrections. 29 Obviously,

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favour of the corresponding chemical potentials, which are the electrostatic and magnetostatic potentials I ; I .31 By virtue of the equations of motion the potentials coincide, up to a factor, with the real coordinates on C.MVM /: I D 2x I , I D 2yOI . In black hole thermodynamics the electrostatic and magnetostatic potentials are evaluated at the horizon. Note that both sets of thermodynamical variables correspond to different real symplectic coordinates on C.MVM /: the charges to the imaginary part, the potentials to the real part of the symplectic vector .Y I ; FI /. As an intermediate step, one can go to the mixed ensemble, where the magnetic charges are kept fixed, while the electric charges fluctuate. Then the independent variables are p I and I . This indicates that F is the free energy with respect to the canonical ensemble, while FE is the free energy with respect to the mixed ensemble. If one imposes that †.p I ; I ; qI / is stationary with respect to variations of I , then one obtains the electric attractor equations .FI 2i I / .FxI C 2i IN / D i qI (4.33). Plugging these back one sees that at the stationary point †attr D 1 Smacro .p I ; qI / and that the macroscopic entropy is the partial Legendre transform of the free energy x FE .p I ; I ; ‡; ‡/. Actually, the black hole free energy introduced in [12] includes the contribution from holomorphic higher derivative terms, but not the non-holomorphic corrections. Let us denote this quantity by FOSV .p I ; I ; ‡/. It is proportional to the imaginary part of the generalized holomorphic prepotential F .Y I ; ‡/. If the model under consideration has been obtained by compactification of type-II string theory on a Calabi–Yau threefold, then the prepotential is in turn proportional to the so-called topological free energy Ftop , which is the logarithm of the all-genus partition function of the topological type-II string, Ztop D e Ftop . In our conventions the precise relation between the free energies is (4.37) FOSV D 4 Im F D 2 Re Ftop : Therefore the free energy FOSV is related to the topological partition function by [12] e FOSV .p;;‡ / D jZtop j2 :

(4.38)

This supports the idea to take the interpretation of FOSV .p; ; ‡/ as the free energy of the black hole seriously. Then it should be related to the partition function of the black hole with respect to the mixed ensemble, which is defined by X d.p; q/e q ; (4.39) Zmixed .p; / D q

where d.p; q/ is the number of BPS microstates with charges p I , qJ , and q WD qI I . This relation is a formal discrete Laplace transform which relates the microscopic partition function, i.e., the state degeneracy, to the mixed partition function. The standard relation between free energy and partition function would imply that Zmixed D e FOSV . However, from our discussion of the black hole variational principle and of the 31 Since the charges play the roles of particle number in non-relativistic thermodynamics, it might appear more logical to call the ‘microcanonical’ ensemble canonical, and the ‘canonical’ ensemble grand canonical. However, we follow the terminology established in the recent literature on the OSV conjecture.


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role of non-holomorphic corrections it appears to be natural to contemplate including non-holomorphic terms, thus replacing FOSV by FE .32 Thus we should leave open the option that there are subleading corrections to the relation between the black hole partition function and the topological string partition function. The weak version of the OSV conjecture [12] is Zmixed .p; / e FOSV .p;/ D jZtop .p; /j2 ;

(4.40)

where means equality in the limit of large charges, which is the semiclassical limit. Evidence for this form of the conjecture will be given below. We will also see that the conjecture needs to be modified as soon as subleading corrections are included. By a formal Laplace transform we can equivalently formulate this conjecture as a prediction of the state degeneracy in terms of the free energy, by Z (4.41) d.p; q/ de ŒFOSV q : Q Here d D I d I , and the I are taken to be complex and integrated along a contour encircling the origin. The relation (4.41) is intriguing, as it relates the black hole microstates directly to the topological string partition function. Note that a saddle point evaluation of the integral gives d.p; q/ e Smacro .p;q/ ;

(4.42)

because at the critical point of the integrand we have ŒFE qI I attr D Smacro .p; q/. Thus the microscopic entropy Smicro .p; q/ D log d.p; q/ and the macroscopic entropy Smacro .p; q/ agree to leading order in the semiclassical limit.33 There are several problems which indicate that the proposal (4.41) must be modified. The number of states d.p; q/ should certainly be invariant under stringy symmetries such as S-duality and T-duality. In the context of compactifications with N 2 supersymmetry, where duality symmetries are realized as symplectic transformation, this also means that d.p; q/ should be a symplectic function. However, in the approach of [12] the electric and magnetic charges are treated differently, so that there is no manifest symplectic covariance. A related issue is how to take into account non-holomorphic corrections. While [12] is based on the holomorphic function F .Y I ; ‡/, it is clear that non-holomorphic terms have to enter one way or another, because they are needed to make d.p; q/ duality invariant. A concrete proposal for modifying (4.41) was made in [13]. It is based on the free energy F D 2H , i.e., on the generalized Hesse potential, instead of FOSV . This allows one to treat electric and magnetic charges on equal footing and to keep symplectic covariance manifest. 32 This makes sense microscopically, because the non-holomorphic corrections to the supergravity effective action are related to the holomorphic anomaly of the topological string [56], [57]. The role of the holomorphic anomaly for the OSV conjecture has also been investigated in [71]. 33 S macro and Smicro are expected to be different, once subleading terms are taken into account, because they refer to different statistical ensembles.

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The covariant version of (4.40) is O Zcan .; / D e F .; / D e 2 H.x;y/

X

d.p; q/e .qp / ;

(4.43)

p;q

where I D 2x I and I D 2yOI are the electrostatic and the magnetostatic potentials, respectively, and Zcan .; / is the partition function of the black hole with respect to the canonical ensemble. By a formal Laplace transform we can reformulate the conjecture (4.43) as a prediction of the state degeneracy: Z O : (4.44) d.p; q/ dxd ye O †.x;y;p;q/ Q In lack of R2 - and non-holomorphic corrections, the measure dxdy D I;J dx I dyJ is proportional to the top power of the symplectic form dx I ^ dyI on C.MVM / and therefore is symplectically invariant. In the presence of R2 - and non-holomorphic corrections, dxd yO is the appropriate generalization. Since † is a symplectic function, we have found a manifestly symplectically covariant version of (4.41). As before, the variational principle ensures that in saddle point approximation we have d.p; q/ exp.Smacro /, as Smacro is the Legendre transform of the Hesse potential and hence the saddle point value of †. In order to compare (4.44) to (4.41), we can rewrite (4.44) in terms of the complex variables and perform the integral over Im Y I in saddle point approximation, i.e., we perform a Gaussian integration with respect to the subspace where the magnetic attractor equations are satisfied. The result is [13] Z p (4.45) d.p; q/ d .p; /e ŒFE q and modifies (4.41) in two ways: first, in contrast to [12] we have included nonholomorphic terms into the free energy FE ; second, the integral contains a measure factor .p; /, whose explicit form can be found in [13]. The measure factor is needed in order to be consistent with symplectic covariance. The proposals (4.41) and (4.44) can be tested by comparing the black hole entropy to the microscopic state degeneracy. There are some cases where these are either known exactly, or where at least subleading contributions are accessible. While this chapter is far from being closed, there seems to be agreement by now that (4.41) needs to be modified by a measure factor [73], [72], [13]. In particular, the measure factors extracted from the evaluation of exact dyonic state degeneracies in N D 4 compactifications [74] are consistent, at the semiclassical level, with the proposal (4.44) [13]. Detailed investigations of microscopical N D 2 partition functions have clarified the origin of the asymptotic holomorphic factorization of the black hole partition function, Zmixed jZtop j2 : it results from simultaneous contributions of branes and anti-branes to the state degeneracy [75], [76], [77], [78]. Recently, the refined analysis of [79] has identified a microscopic measure factor, which agrees with the one found in [72], [13] in the semiclassical limit.


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Acknowledgments. The original results reviewed in this chapter were obtained in collaboration with Gabriel Lopes Cardoso, Vicente Cortés, Bernard de Wit, Jürg Käppeli, Christoph Mayer and Frank Saueressig. The chapter is partially based on a talk given at the ‘Bernardfest’ in Utrecht, and I would like to thank the organisers for the opportunity to speak on the occasion of Bernard’s anniversary. My special thanks goes to Vicente Cortés for inviting me to contribute this chapter to the Handbook on Pseudo-Riemannian Geometry and Supersymmetry. Furthermore, I would like to thank the referee for suggesting to make this work accessible to a larger readership by including an introduction to black holes in general and to supersymmetric black hole solutions in particular. This material has been included as a separate section (Section 3). The extended version of this chapter was written during a stay as Senior Research Fellow at the Erwin Schrödinger for Mathematical Physics in Vienna.

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Part B

Generalized geometry

Chapter 6

Generalized geometry – an introduction Nigel Hitchin

Contents 1 Introduction . . . . . . . . . . . 2 The basic scenario . . . . . . . 3 The Courant bracket . . . . . . 4 Affine connections . . . . . . . 5 Gerbes . . . . . . . . . . . . . 6 Generalized complex structures 7 Generalized Kähler structures . 8 Spinors . . . . . . . . . . . . . 9 Structures defined by forms . . 10 Double structures . . . . . . . . 11 Group actions . . . . . . . . . . 12 Quotients . . . . . . . . . . . . References . . . . . . . . . . . . . .

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1 Introduction “Generalized geometry” is an approach to differential geometric structures which seems remarkably well-adapted to some of the concepts in String Theory and Supergravity, for example: 3-form flux, gauged sigma-models, D-branes, connections with skew torsion. It also incorporates in a natural way the role of the B-field as a symmetry. It began life in [11] as an attempt to understand the meaning of critical points of a certain invariant functional but it has since developed considerably, partially at the hands of the author’s former students M. Gualtieri, G. Cavalcanti and F. Witt. Here we shall offer an introduction to this geometry and a few of its applications, showing how in particular some relatively old results in the physics literature acquire a natural meaning within this new setting.

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2 The basic scenario The fundamental idea is to take a manifold M of dimension n and replace its tangent bundle by the direct sum T ˚ T of the tangent bundle and its dual. This has a natural inner product of signature .n; n/ defined by .X C ; X C / D iX (there is a difference in sign here with [11] but this is immaterial and accords with what is now the common usage). The bundle of skew adjoint transformations from T ˚ T to itself splits as End T ˚ ƒ2 T ˚ ƒ2 T and in particular a 2-form B, a section of ƒ2 T , acts. If we exponentiate it to an orthogonal action on X C 2 T ˚ T , then it is simply X C 7! X C C iX B: The general idea now is to take geometric structures on T ˚T which are analogues of the usual objects one studies in differential geometry. We should define them only using the fact that T ˚ T has an inner product, and then we will be able to transport them not only by the natural action of the diffeomorphism group on T ˚ T but also by the action of a two-form – the B-field. As an example, let us consider how to encode an ordinary Riemannian metric g into this picture. We think of g as an isomorphism g W T ! T defined by X 7! g.X; /. Its graph V T ˚ T is defined as the vectors X C g.X; / and the inner product restricted to this is iX g.X; / D g.X; X/. Now let’s try and define this purely in terms of T ˚ T geometry. We take a subbundle V T ˚ T of rank n such that the inner product restricted to V is positive definite. Since the inner product on T is identically zero, V \ T D 0 which means that V is the graph of a map from T to T . This has a symmetric part g and a skew symmetric part B, and a vector in V looks like X C gX C iX B where we write gX for g.X; /. This is simply the transform of a metric, as encoded above, by the 2-form B. So in this generalization, a metric comes along with a skew-symmetric part B – a B-field. Note that the orthogonal complement V ? has induced inner product which is negative definite, and is the graph of X 7! gX C iX B.

3 The Courant bracket Thus far we have been discussing linear algebra, now comes differentiation. There is an analogue of the Lie bracket ŒX; Y of two vector fields for two sections u D

Chapter 6. Generalized geometry – an introduction

187

X C ; v D Y C of T ˚ T . It is called the Courant bracket [6] and is defined by 1 Œu; v D ŒX C ; Y C D ŒX; Y C LX LY d.iX iY /: 2 Without the last term, this would have a simple interpretation as the bracket on the Lie algebra of the group which is the semi-direct product of Diff.M / with the additive group 1 of 1-forms on M . But the last term is indeed there and as a consequence this is not the Lie bracket for any Lie algebra, for in general it does not satisfy the Jacobi identity. Nevertheless, it is natural within the context here. Its characteristic properties are, for u, v, w sections of T ˚ T and f a function, as follows: Œu; f v D f Œu; v C ..u/f /v .u; v/df;

(1)

.u/.v; w/ D .Œu; v C d.u; v/; w/ C .v; Œu; w C d.u; w//:

(2)

Here .X C / D X . A key feature of the Courant bracket is that it is preserved by the automorphism X C 7! X C C iX B if B is closed.

4 Affine connections We shall now put these two pieces of data together to define a natural connection associated to a metric with B-field. When the B-field vanishes we shall get the LeviCivita connection. We start with V T ˚ T a subbundle on which the inner product is positive definite. Given a vector field X we can lift it to a section X C of V by X C D X C gX C iX B and to a section X D X gX C iX B of V ? . Proposition 1. Let v be a section of V and X a vector field. Let ŒX ; v be the Courant bracket and ŒX ; vC the orthogonal projection onto V . Then rX v D ŒX ; vC defines a connection on V which preserves the inner product. Proof. From (1) Œf X ; v D f ŒX ; v ..v/f /X C .X ; v/df but X 2 V ? is orthogonal to V so the projection onto V is f ŒX ; vC . Thus rf X v D f rX v: From (1) again

ŒX ; f v D f ŒX ; v C .Xf /v

and so, projecting to V , we have rX .f v/ D f rX v C .Xf /v

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and hence a connection. Let v; w be sections of V and take u D X in (2). Then we obtain X.v; w/ D .ŒX ; v; w/ C .v; ŒX ; w/ since X is orthogonal to v and w. Since v; w 2 V the right hand side is .ŒX ; vC ; w/ C .v; ŒX ; wC / D .rX v; w/ C .v; rX w/ which means the connection preserves the metric.

The projection identifies V with T so we have a metric connection on T . We write v D Y C and then ŒX ; Y C C D .rX Y /C . But .ŒX ; Y C / D ŒX; Y , so ŒX ; Y C ŒX; Y is a 1-form which we write as 2gZ for some vector field Z. The orthogonal decomposition of this is 2gZ D .Z C gZ C iZ B/ .Z gZ C iZ B/ D Z C Z

(3)

and hence for a vector field Z, Z C is the V -component of 2gZ. Since .ŒX ; Y C ŒX; Y /C D ŒX ; Y C C D .rX Y /C ; this means that the connection on T is given by 2grX Y D ŒX ; Y C ŒX; Y :

(4)

Theorem 2. The connection (4) has skew torsion dB. Remark. Interchanging the roles of V and V ? , we get a connection with torsion dB. Proof. Consider 2g.rX Y rY X ŒX; Y / D ŒX ; Y C ŒY ; X C 2ŒX; Y 2gŒX; Y : (5) One-forms Courant-commute so ŒX C X ; Y C Y D 0:

(6)

Consider now the Courant bracket of X C iX B; Y C iY B. If the 2-form B were closed, we would get ŒX; Y C iŒX;Y B by invariance of the bracket. In general there is an extra term and we obtain ŒX C iX B; Y C iY B D ŒX; Y C iŒX;Y B iX iY dB: But X C C X D 2.X C iX B/ and so 1 C 1 ŒX C X ; Y C C Y D .ŒX; Y C C ŒX; Y / iX iY dB: 4 2

189


Using (6) this gives ŒX C ; Y C ŒX ; Y C D ŒX; Y C C ŒX; Y 2iX iY dB: Substitute this in the right hand side of (5), using the skew-symmetry of the Courant bracket, to get ŒX; Y C ŒX; Y 2gŒX; Y 2iX iY dB which from (3) is 2iX iY dB. We therefore get that the torsion is dB.

Example. If we take B D 0 we get the familiar formula (using the summation convention) for the Levi-Civita connection:

@ @ @ @ gik dxk ; C gj k dxk ; @xi @xj @xi @xj

C

D

@gj k @gik @gij C dxk @xj @xk @xi

D 2g`k ij` dxk : Note that the extra term in the Courant bracket which causes the Jacobi identity to fail yields the third term in this standard formula. With non-zero B-field we get the formula

@ @ gik dxk C Bik dxk ; C gj k dxk C Bj k dxk @xi @xj @gj k @Bj k @Bik @gik @gij C dxk C dxk : D @xi @xj @xk @xi @xj

Thus if Hij k D

@Bj k @Bik @xi @xj

this connection is related to the Levi-Civita connection r by 1 (7) ri C Hij k g k` : 2 It follows that we have two affine connections: one associated with V and the other with V ? . They are 1 riC D ri C Hij k g k` ; 2

1 r D ri Hij k g k` : 2

5 Gerbes For some reason, the Courant bracket is naturally associated with the differential geometry of gerbes (see [3] and [10] for more information on gerbes and their differential geometry). We adopt the simple-minded point of view on gerbes: we have an open covering fU˛ g where all open sets and intersections are contractible and define

190

Nigel Hitchin

a gerbe by smooth circle-valued functions: g˛ˇ W U˛ \ Uˇ \ U ! U.1/ such that 1 g g 1 D 1 on U˛ \ Uˇ \ U \ Uı . g˛ˇ D gˇ1˛ D and gˇ ı g˛ ı ˛ˇ ı ˛ˇ We define a trivialization of the gerbe by functions h˛ˇ such that g˛ˇ D h˛ˇ hˇ h ˛ Q the expreson threefold intersections of open sets. Given two trivializations h and h, 1 Q sion h˛ˇ h˛ˇ is the transition function for a unitary line bundle so we may say that “the ratio of two trivializations is a line bundle”. In [3] a connection on a gerbe is introduced in two stages. The first stage – the “connective structure” is important for the geometry here. This consists of a collection of 1-forms A˛ˇ on twofold intersections such that A˛ˇ D Aˇ ˛ and which satisfy the relation 1 A˛ˇ C Aˇ C A ˛ D g˛ˇ dg˛ˇ on threefold intersections. A flat trivialization (relative to a connective structure) is defined by the condition A˛ˇ D h1 ˛ˇ dh˛ˇ : is a constant so the line bundle If hQ ˛ˇ and h˛ˇ are two flat trivializations then hQ ˛ˇ h1 ˛ˇ this transition function defines is a flat one. Remark. A circle has a good covering by three open intervals, and furthermore the triple intersections are empty. It follows that any connective structure on a gerbe over the circle has a flat trivialization. Any two flat trivializations differ by a flat line bundle and the holonomy of a flat line bundle takes values in H 1 .S 1 ; U.1// which is U.1/. Thus if we say that two flat trivializations are equivalent if they differ by a flat connection with trivial holonomy, the set of equivalence classes is acted on freely and transitively by U.1/. This way a connective structure on a gerbe defines a principal U.1/-bundle on the loop space of M – the fibre over a single loop is just this set of equivalence classes. Given a connective structure on a gerbe we can produce a twisted version of T ˚T . Taking the exterior derivative of the 1-forms A˛ˇ gives 1 dA˛ˇ C dAˇ C dA ˛ D d Œg˛ˇ dg˛ˇ D 0:

(8)

On each twofold intersection we can patch T ˚ T over U˛ with T ˚ T over Uˇ with the automorphism X C 7! X C C iX dA˛ˇ : The relation (8) is the consistency condition to define a vector bundle E, an extension

0 ! T ! E ! T ! 0: A global section of E consists of a vector field X and locally defined one-forms ˛ such that on U˛ \ Uˇ ˇ D ˛ C iX dA˛ˇ :


191

Mathematically speaking (8) is a cocycle with values in C 1 .Hom.T; T // which will define an extension. But it is more than that since it is skew and then the patching preserves the inner product on the local T ˚ T . It is also closed and so preserves the Courant bracket. Thus E inherits all the structure of T ˚ T that we are using – the inner product and the Courant bracket, where by the latter we mean a bracket on sections of E which satisfies the conditions (1) and (2). We also retain an action of B-fields: u 7! u C iu B. Given this set-up we make a general definition: Definition 1. A generalized metric is a subbundle V E of rank n on which the inner product is positive definite. Locally we have the previous situation – V is the graph of a map X 7! X C g˛ X C iX B˛ but now on the twofold intersections we have gˇ X C iX Bˇ D g˛ X C iX B˛ C iX dA˛ˇ : But B and dA are skew, so gˇ .X; X/ D g˛ .X; X / and we have a well-defined metric g. The remaining relation is Bˇ D B˛ C dA˛ˇ : Locally defined 2-forms with this property form a curving of the connective structure in the language of [3], so a generalized metric gives us an ordinary metric and a curving of the gerbe. The 3-form H D dB˛ D dBˇ is globally defined and is the curvature of the gerbe. With this generalized viewpoint Theorem 2 defines a connection with skew torsion H : the three-form H need not be globally exact.

6 Generalized complex structures One of the most fertile developments of the generalized point of view is the notion of a generalized complex manifold, the basic properties of which can be found in [9]. We retain our twisted setting of replacing T ˚ T by the extension E in what follows. Definition 2. A generalized complex structure on a manifold M of dimension 2m with bundle E is an automorphism J W E ! E such that • J 2 D 1 • .J u; v/ C .u; J v/ D 0 • if J u D iu; J v D iv then J Œu; v D i Œu; v using the Courant bracket.

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Nigel Hitchin

Here we have imitated the definition of a Kähler metric, but replaced T by E, a metric by the natural inner product, and the Lie bracket by the Courant bracket. The linear algebra data consists of a reduction of structure group of the bundle E to U.m; m/ SO.2m; 2m/. The third condition can be replaced by the vanishing of a Nijenhuis-type of tensor: ŒJ u; J v J ŒJ u; v J Œu; J v Œu; v D 0. The interesting feature about this notion is that it includes both symplectic and complex manifolds and indeed the possible link with mirror symmetry for Calabi– Yau threefolds was one motivating force behind the author’s paper [11]. For a complex manifold with E D T ˚ T we take I 0 J D : 0 I In this case the Ci eigenspace of J is Œ: : : @=@zi : : : ; : : : d zNi : : : . For a symplectic manifold 0 ! 1 J D ! 0 P and the Ci eigenspace of J is Œ: : : ; @=@xj C i i;j !j k dxk ; : : : . Another, more interesting, class of examples consists of holomorphic Poisson manifolds. Let M be a complex manifold with a holomorphic bivector field D

X i;j

ij

@ @ ^ @zi @zj

which satisfies Œ; D 0 using the Schouten bracket. Then the Ci eigenspace i h X @ @ ; : : : ; d zN k C N k` ;::: :::; @zj @Nz` `

defines a generalized complex structure. A simple case is to take a complex surface with a holomorphic bivector field – this is a section of the dual K of the canonical bundle (for example take CP2 with a section of O.3/). This is always a holomorphic Poisson manifold, since Œ; is a .3; 0/ vector and these are all zero in dimension two. Although it is a complex manifold, from the generalized complex point of view the picture is rather different. The section vanishes on a curve C (an elliptic curve). Outside C , 1 is a well-defined holomorphic section of K and so is a holomorphic 2-form ' D !1 C i!2 . The generalized complex structure there consists of the symplectic structure !2 transformed by the B-field !1 . On C , the generalized complex structure is just the complex structure. This change of type of the generalized complex structure – from symplectic to complex on the same manifold – is a characteristic feature. In four dimensions it takes place over a 2-torus, if the locus is smooth. In fact, as shown in [5], by doing surgery along an embedded torus in a certain symplectic manifold, it can be shown that the four-manifold 3CP2 # 19CP2 has a generalized complex structure which is


193

complex on the torus and symplectic outside. But by Seiberg–Witten theory, there is no ordinary complex or symplectic structure on it.

7 Generalized Kähler structures One step beyond a generalized complex structure is the idea of a generalized Kähler structure, introduced by Gualtieri. The surprising feature here is that, in a different language, this was a differential geometric structure which appeared in the physics literature over 20 years ago. The starting point is that an ordinary Kähler manifold is a manifold with both a complex structure and a symplectic structure and a compatibility condition between the two. Both of these structures can be encoded as generalized complex structures, and it turns out that compatibility means they commute. Thus our generalized geometry definition is: Definition 3. A generalized Kähler structure on a manifold M with bundle E consists of two commuting generalized complex structures J1 , J2 such that the quadratic form .J1 J2 u; u/ is positive definite. The main theorem of Gualtieri (in our formulation using the bundle E) is: Theorem 3. A generalized Kähler structure on a manifold M defines • a generalized metric, • two integrable complex structures I C , I on M such that the metric g is Hermitian with respect to both, • the connections r C ; r of the generalized metric preserve I C , I respectively. Conversely, up to the action of a closed B-field, this data determines a generalized Kähler structure on M . Such structures form the target spaces for the nonlinear sigma model with .2; 2/ supersymmetry as discussed in [7]. We prove the first part of this theorem below. Proof. Firstly, since J1 and J2 commute, we have .J1 J2 /2 D 1 and so E splits into ˙1 eigenspace bundles V , V ? . Since .J1 J2 u; u/ is positive definite, if u D uC C u is the eigenspace decomposition of u 2 E, then .uC ; uC / .u ; u / is positive definite. So V is a subbundle such that the inner product is positive definite and hence defines a generalized metric. Now J1 and J2 preserve the eigenspaces of J1 J2 so J1 D J2 acts on V . It defines the almost complex structure I C on M by J1 X C D .I C X /C

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Nigel Hitchin

and similarly for I . We first need to show that I C is integrable, so suppose I C X D iX, then J1 X C D iX C . But from the definition of a generalized complex structure if J1 X C D iX C and J1 Y C D iY C , then J1 ŒX C ; Y C D i ŒX C ; Y C . Since Œu; v D Œu; v relates the Courant bracket to the Lie bracket, we have IC ŒX; Y D iŒX; Y and the complex structure is integrable by Newlander–Nirenberg. Next we need to determine the compatibility of I C with the connection with skew torsion coming from the generalized metric. We require r C I C D 0, or equivalently g.rXC I C Y; Z/ D g.rXC Y; I C Z/. But from the definition of r C , 2g.rXC Y; Z/ D .ŒX ; Y C ŒX; Y ; Z C / D .ŒX ; Y C ; Z C /; so we need to show that .ŒX ; J1 Y C ; Z C / D .ŒX ; Y C ; J1 Z C /: We use the Nijenhuis condition ŒJ u; J v J ŒJ u; v J Œu; J v Œu; v D 0 for integrability of a generalized complex structure. For J1 this gives ŒJ1 X ; J1 Y C J1 ŒJ1 X ; Y C J1 ŒX ; J1 Y C ŒX ; Y C D 0 and for J2 (recalling that J1 D J2 on V and J1 D J2 on V ? ) we have ŒJ1 X ; J1 Y C J2 ŒJ1 X ; Y C C J2 ŒX ; J1 Y C ŒX ; Y C D 0: Adding these two gives .J1 C J2 /ŒX ; J1 Y C .J1 J2 /ŒX ; J1 Y C 2ŒX ; Y C D 0: Now take the inner product with J1 Z C : ..J1 C J2 /ŒX ; J1 Y C ; J1 Z C / C ..J1 J2 /ŒX ; J1 Y C ; J1 Z C / C 2.ŒX ; Y C ; J1 Z C / D 0: But J1 D J2 on V , so .ŒX ; J1 Y C ; Z C / C .ŒX ; Y C ; J1 Z C / D 0 which is the required identity.

8 Spinors One of the key features of generalized geometry is the way in which differential forms are regarded as spinors for T ˚ T . We define Clifford multiplication by a vector u D X C 2 T ˚ T on a form ˛ by u ˛ D iX ˛ C ^ ˛ and then u2 D .u; u/1. There is an invariant bilinear form with values in ƒn T – the so-called Mukai pairing – defined by h˛; ˇi D Œ˛ ^ .ˇ/n


195

where .ˇ/ D 1 if the degree of ˇ is 0 or 3 mod 4 and 1 otherwise. (In cohomology this is a familiar operator – the Chern character of a bundle V is related to that of its dual by ch.V / D .ch.V //.) The B-field action of a two-form B in the spin representation is the exponentiation of the exterior product: ˛ 7! e B ˛. The exterior derivative still has a natural role when we regard forms as spinors. In fact it is in some respects dual to the Courant bracket. Recall the expression below which represents the usual formula relating the Lie bracket of vector fields to the exterior derivative: 2iŒX;Y ˛ D d.ŒiX ; iY ˛/ C 2iX d.iY ˛/ 2iY d.iX ˛/ C ŒiX ; iY d˛: Here the algebraic action of vector fields on forms is through the interior product iX . Replacing this by the Clifford action of sections of T ˚ T gives 2Œu; v ˛ D d..u v v u/ ˛/ C 2u d.v ˛/ 2v d.u ˛/ C .u v v u/ d˛ (9) where Œu; v is now the Courant bracket. What happens when we twist T ˚ T to get the bundle E? The bundle E was constructed by patching together T ˚ T with the cocycle dA˛ˇ . Now e B .X C / e B ˛ D .X C C iX B/˛: We construct the spinor bundle S.E/ by taking the exterior forms ƒ T and identifying over twofold intersections via ' 7! e dA˛ˇ ': Clearly this does not preserve degrees, only the parity. But equally, since dA˛ˇ is closed, it preserves the exterior derivative. We thus have a well-defined differential operator d W C 1 .S.E// ! C 1 .S.E//: The cohomology of this, with the Z2 -grading given by odd and even forms, is the twisted cohomology. To relate this to the more normal description, we can choose a curving of the gerbe so that we have local one-forms B˛ with Bˇ B˛ D dA˛ˇ . Then a section of S given by local forms '˛ satisfying 'ˇ D e dA˛ˇ '˛ D e Bˇ B˛ '˛ defines a global form D e B˛ '˛ D e Bˇ 'ˇ : It is no longer closed but instead d where dB˛ D H is the curvature.

CH ^

D0

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Nigel Hitchin

We thus have a choice: use only a connective structure and think of spinors with the operator d ; or choose a curving and think of forms but with operator d C H .

9 Structures defined by forms Generalized geometric structures of interest are often defined by reducing the structure group of E from SO.n; n/ to the stabilizer of a spinor. When we interpret spinors as forms then it sometimes happens that when the form is closed we get an integrability condition for the structure. The first example is that of a generalized Calabi–Yau manifold [11]. If ' is a spinor for T ˚ T , then its annihilator A is the vector space of u 2 .T ˚ T / ˝ C such that u ' D 0. The restriction of the inner product to A is always zero, since 0 D u u ' D .u; u/': A pure spinor is one for which the annihilator has maximal dimension n. Definition 4. A generalized Calabi–Yau structure is a complex closed form which N ¤ 0. is a pure spinor for T ˚ T and satisfies h ; i The annihilator of defines a rank n subbundle A of .T ˚ T / ˝ C: the condition h ; i N ¤ 0 means that A \ AN D 0. We define a generalized complex structure by making A the Ci eigenspace of J . It satisfies the Courant integrability condition because is closed, using equation (9). Clearly we can replace T ˚ T by its twisted version E. This is perhaps the easiest way to find generalized complex manifolds. A symplectic manifold .M; !/ for example is given by D e i! and a B-field transform of it by D e BCi! : In this situation (which is generic when both forms are even) we can see how to solve the equations for a generalized Kähler structure. First (following [13]) consider how to achieve two commuting generalized complex structures: Lemma 4. Let 1 D e ˇ1 ; 2 D e ˇ2 be closed forms defining generalized complex structures J1 ; J2 on a manifold of dimension 4k. Suppose that .ˇ1 ˇ2 /kC1 D 0 D .ˇ1 ˇN2 /kC1 and .ˇ1 ˇ2 /k and .ˇ1 ˇN2 /k are non-vanishing. Then J1 and J2 commute. Proof. Suppose that .ˇ1 ˇ2 /kC1 D 0 and .ˇ1 ˇ2 /k is non-zero. Then the 2form ˇ1 ˇ2 has rank 2k, i.e. the dimension of the space of vectors X satisfying


197

iX .ˇ1 ˇ2 / D 0 is 2k. Since iX 1 C ^ 1 D 0 if and only if D 0, this means that the space of solutions u D X C to u exp.ˇ1 ˇ2 / D 0 D u 1 is 2k-dimensional. Applying the invertible map exp ˇ2 , the same is true of solutions to u exp ˇ1 D 0 D u exp ˇ2 : This is the intersection A1 \ A2 of the two Ci eigenspaces. Repeating for ˇ1 ˇN2 we get A1 \ AN2 to be 2k-dimensional. These two bundles are common eigenspaces of .J1 ; J2 / corresponding to the eigenvalues .i; i / and .i; i/ respectively. Together with their conjugates they decompose .T ˚ T / ˝ C into a direct sum of common eigenspaces of J1 ; J2 , thus J1 J2 D J2 J1 on every element. Now let us reformulate what we have without going through Theorem 3: the 2kdimensional space of vectors satisfying iX .ˇ1 ˇ2 / D 0 form the .0; 1/ vectors of an almost complex structure I C . Since ˇ1 ˇ2 is closed, I C is integrable and moreover ! C D ˇ1 ˇ2 is a holomorphic symplectic form. Similarly ! D ˇ1 ˇN2 is a holomorphic symplectic form for a complex structure I . Locally any two complex structures are equivalent by a diffeomorphism, and together with the holomorphic version of the Darboux theorem we have a local diffeomorphism F such that F ! C D ! . But ! C C !N C D ˇ1 C ˇN1 ˇ2 ˇN2 D ! C !N : So if ! C D ! C i! 0 , then ! D ! C i! 00 and hence F ! D !;

F ! 0 D ! 00 :

Apart from the positivity condition, a generalized Kähler structure is thus determined by a local symplectic diffeomorphism F . We can define one by taking an arbitrary smooth function f and integrating its Hamiltonian vector field Xf – roughly the same freedom as for a Kähler structure. So now let us address the definiteness of .J1 J2 u; u/ in Definition 3. The 1 eigenspace of J1 J2 is V ? D A1 \ A2 ˚ AN1 \ AN2 : If X is a .0; 1/-vector it lies in the 2k-dimensional space defined by iX .ˇ1 ˇ2 / D 0 and then u D X iX ˇ2 satisfies u exp ˇ1 D 0 D u exp ˇ2 , i.e. u 2 A1 \ A2 . But then (10) .u C u; N u C u/ N D iX ˇ2 .Xx / i x ˇN2 .X / D .ˇ2 ˇN2 /.X; Xx / X

so we need .ˇ2 ˇN2 /.X; Xx / to be positive definite, which is i.F ! 0 ! 0 /.X; Xx / 0. In fact ! 0 is of type .2; 0/ C .0; 2/ with respect to I C so ! 0 .X; Xx / D 0 and the x 0. Now if F t is the one-parameter group of symplectic condition is iF ! 0 .X; X/

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Nigel Hitchin

diffeomorphisms obtained from the Hamiltonian f , we have @ 0 F ! j t D0 D LX ! 0 D d.iXf ! 0 /: @t t But .! C i ! 0 /.X; Y / D i.! C i ! 0 /.X; I C Y / and iXf ! D df so that iXf ! 0 D I C df and hence @ 0 F ! j t D0 D dI C df: @t t N for the complex structure I C ) defines a local Kähler Thus if dI C df (this is i@@f form, for t small enough iF t ! 0 .X; Xx / will be positive definite. Generalized Kähler structures of this generic type thus seem to be very like Kähler metrics, but more nonlinear – given two Kähler potentials f; fQ the difference fQ f is the real part of a holomorphic function, but two symplectic diffeomorphisms F; FQ must have the composition F 1 FQ holomorphic to describe the same generalized Kähler structure. Note that the construction above has its origins in one of D. Joyce (see [2]) and also lies behind the potentials in [16].

10 Double structures A generalized metric decomposes E into V ˚ V ? . This (with an orientation) is a reduction of the structure group of E from SO.n; n/ to SO.n/ SO.n/. A generalized Kähler structure gave us independent complex structures I C , I on M which were more naturally described by J1 or J2 on V and V ? . So the picture is: a complex structure on V and another on V ? , or equivalently a reduction to U.m/U.m/. There is a further property in Gualtieri’s theorem, that the holonomy of the natural connections on V and V ? reduces to U.m/. This is an example of what we shall call a double structure: a reduction to G G SO.n/ SO.n/ with corresponding holonomy reduction – the case when G preserves a spinor is important for supersymmetric considerations, as in [8]. We shall consider here how generalized geometry is a suitable setting for this (see [17] for a detailed account.) Let w denote the orthogonal automorphism of the bundle E given by reflection in V . It lifts to ! in the Pin group which acts as an automorphism of the spinor bundle S.E/. We shall consider the equations d D 0 D d.! / for a section of S.E/ satisfying certain algebraic conditions. Since any double structure involves a generalized metric, having decomposed E D V ˚ V ? , it is useful to use the curving of the gerbe that it defines. Recall that this is given by local 1-forms B˛ such that Bˇ D B˛ C dA˛ˇ :


199

We interpret this as a splitting of the sequence

0 ! T ! E ! T ! 0 – the cocycle dA˛ˇ is written as a specific coboundary Bˇ B˛ . Since there are various splittings involved, let us consider these more carefully. A splitting is by definition a homomorphism S W T ! E such that S.X / D X . If S, S 0 are two splittings then with this definition so is .S C S 0 /=2, the average. More geometrically, the image of S is a subbundle isomorphic to T and since S.T /\T D 0 this is equivalent to an isomorphism E Š T ˚ T . The subbundle V of a generalized metric defines one splitting (V D S.T /), and its orthogonal complement V ? defines another splitting S 0 . In the first case S.T / is positive definite, in the second S 0 .T / is negative definite. For the averaged splitting, S 00 D .S C S 0 /=2, S 00 .T / is isotropic: this is the splitting given by the B˛ . Since the B˛ are skew the isomorphism E Š T ˚ T defined by this splitting takes the inner product to the standard one, but the B˛ are not in general closed so the Courant bracket transforms to a twisted version 1 ŒX; Y C LX LY d.iX iY / iX iY H 2 where H D dB˛ . The spinor bundle now becomes identified with the standard bundle of exterior forms but the d -operator is replaced by ˛ 7! d˛ C H ^ ˛, and ! D ˙ where is the Hodge star operator for the metric. Let r be the Levi-Civita connection and X1 ; : : : ; Xn a local orthonormal basis of vector fields. Then because r is torsion-free and 2gX D X C X , we can write the exterior derivative of an arbitrary form ˛ as X 1X C d˛ D gXi ^ ri ˛ D .Xi Xi / ri ˛ (11) 2 i

and H ^˛ D

i

1 X Hij k .XiC Xi / .XjC Xj / .XkC Xk / ˛: 48

(12)

i;j;k

Now, in terms of the orthonormal basis of V , the element ! D X1C X2C XnC . Since the XiC and Xj are orthogonal they anticommute except with themselves, so ! XiC D .1/n1 XiC !;

! Xi D .1/n Xi !:

Thus, since ! is covariant constant with respect to r, 1X C d.! ˛/ D .1/n ! .Xi C Xi / ri ˛ 2

(13)

i

and H ^ .! ˛/ D .1/n !

1 X Hij k .XiC C Xi / .XjC C Xj / .XkC C Xk / ˛: (14) 48 i;j;k

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hence, using (11) and (13), and (12) and (14), the two equations .d C H /˛ D 0 D .d C H /.! ˛/ are equivalent to X i

XiC ri ˛ C

X i

1 X Hij k .XiC XjC XkC C 3XiC Xj Xk / ˛ D 0; 24 i;j;k

1 X Xi ri ˛ C Hij k .Xi Xj Xk C 3Xi XjC XkC / ˛ D 0: 24 i;j;k

Now suppose that G SO.n/ is the stabilizer of a unit length spinor ' (here, as in [17] we shall consider only a real spinor representation for Spin.n/ so that G D G2 or Spin.7/). Then G G is the stabilizer of ' C , a spinor for V and ' , a spinor for V ? . But our decomposition of T ˚ T D V ˚ V ? gives us a tensor product decomposition of spinors for the two orthogonal spaces, so that ' C ˝ ' can be considered as a spinor for Spin.n; n/, or equivalently an exterior form. It, or any non-zero multiple, is stabilized by G G. From this viewpoint a vector X C 2 V acts by the Euclidean Clifford action of X on the left hand factor and X , coming from V ? action, on the right – though we must remember that the inner product on V ? is negative definite and V positive. This difference appears in the spin representation: if a 2 so.n/ is given relative to an orthonormal basis Xi by the skew-symmetric matrix aij , then if Xi Xi D 1, the action on spinors is 1X aij Xi Xj 4 i;j

and if Xi Xi D C1, the opposite. Suppose then that D e f ' C ˝ ' satisfies the equations .d C H / D 0 D .d C H /.! /; then setting ˛ D in the first equation above gives (using the summation convention) 1 Xi ri ' C ˝ ' CXi ' C ˝ ri ' C Xi HiC ' C ˝ ' CXi ' C ˝ Hi ' 3 D df ' C ˝ ' :

(15)

Here HiC D

1X 1 Hij k Xj Xk D ı.Hi / 8 2 j;k

where ı is the spin representation on the Lie algebra (recall that XiC XiC D 1). Similarly Hi D ı.Hi /=2.


201

Considering the right hand spinor factor in (15) we get 1 ri ' ı.Hi /' D i ' ; 2 but ' has unit length so i D 0. We deduce (see (7)) that ri '

1 D ri ı.Hi / ' D 0 2

so that the holonomy of the connection r reduces to the stabilizer G of ' . Applying the same argument to the second equation (and taking account of the change of sign of the metric) we get

1 riC ' C D ri C ı.Hi / ' C D 0: 2 Now put this into the equation (15) and consider the left hand spinor factor: we get the equation 1 X Hij k Xi Xj Xk ' C D 0: df C 12 ij k

Together with a similar one for ' we obtain the Type II supersymmetric geometries described in [8]. This last relation says that the spinor e f ' C is annihilated by the so-called cubic Dirac operator [1]. Remark. In 7 and 8 dimensions, the equations .d C H / D 0 D .d C H /.! / arise in a very natural way. First, in 7 dimensions there is an open set of (even or odd) forms for which the stabilizer in Spin.7; 7/ is G2 G2 . This means that the form itself defines, in a complicated nonlinear fashion, the generalized metric. The equations then appear as variational ones for a natural invariant functional, as in the 6-dimensional case of [11]. This was the starting point for Witt’s work. In 8 dimensions there is a form whose stabilizer is Spin.7/ Spin.7/ and this satisfies D ˙! . Thus we only need it to be closed to satisfy the equations – it is however algebraically special. In fact here there are two cases – when the form is odd or even (see [8] and [17]).

11 Group actions Suppose we have a smooth action of a Lie group G (with Lie algebra g) on M . There is a natural Lie algebra homomorphism from g to the space of vector fields: g ! C 1 .T /: In generalized geometry we replace T , with its Lie bracket, by E, with the Courant bracket, so we can ask what the natural generalization should be.

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The most obvious request is a lift of this homomorphism to sections of E e W g ! C 1 .E/ such that Œe.a/; e.b/ D e.Œa; b/ using the Courant bracket on sections of E. Some things need to be adapted, however. First recall that the Courant bracket does not satisfy the Jacobi identity. In fact 3ŒŒu; v; w C ŒŒw; u; v C ŒŒv; w; u D d..Œu; v; w/ C .Œw; u; v/ C .Œv; w; u//: So if u; v; w D e.a/; e.b/; e.c/ the right hand side must vanish but then ..Œu; v; w/ C .Œw; u; v/ C .Œv; w; u// is a constant element of ƒ3 g which we need to incorporate into the picture. Because this is expressed as inner products, it will vanish if we insist that for each a 2 g, .e.a/; e.a// D 0. Secondly, note that in the normal case of a free group action, the image of g in T spans a subbundle which is at each point the tangent space to the G-orbit through that point. In particular it is an integrable distribution: its sections are closed under the Lie bracket. In the generalized case, let F be the subbundle of E spanned by e.g/. Since e is a homomorphism, F has a basis of sections which are closed under Courant bracket. But from (1), it is not always true that this will hold for a linear combination with C 1 coefficients. However if .e.a/; e.a// D 0 (i.e. F is isotropic), (1) shows that all sections of F are closed under the Courant bracket. It is natural therefore to demand the isotropy condition for e. In more generality one ought to extend the notion of Lie algebra to incorporate an analogue of the Courant bracket and the authors of [4] go some way towards doing that, but we shall stick here to the simplest, most geometrical, situation. Explicitly, e.a/ is a section of E which maps under W E ! T to X (which we sometimes write as Xa ), the vector field generated by a: locally e.a/ D X C ˛ where ˇ ˛ D iX dA˛ˇ :

(16)

We also sometimes write e.a/ D eX , to think of a specific lift of X to E. We now want to extend the natural Lie derivative action of X on T ˚ T to a Lie algebra action of g on E. Now LX .Y C ˛ / D ŒX; Y C LX .ˇ iY dA˛ˇ / D ŒX; Y C LX ˇ iŒX;Y dA˛ˇ iY d.LX A˛ˇ /; so the usual action does not make global sense because of the last term. However, from (16) d ˇ D d ˛ C d.iX dA˛ˇ / D d ˛ C d.LX A˛ˇ /; so we can define a new Lie derivative LX by LX .Y C ˛ / D LX .Y C ˛ / iY d ˛


203

which is globally defined. This is sometimes called the Dorfman bracket of X C ˛ and Y C ˛ : a useful formula is LX u D ŒeX ; u C d.eX ; u/:

(17)

Is LX really an action? We need to show that ŒLX ; LY .Z C ˛ / D LŒX;Y .Z C ˛ / for any Z C ˛ . The vector field part is clear. The one-form part of ŒLX ; LY .Z C ˛ / is LX .LY ˛ iZ d˛ / iŒY;Z d ˛ LY .LX ˛ iZ d ˛ / C iŒX;Z d˛ D LŒX;Y ˛ iZ d.LX ˛ LY ˛ /:

(18)

But eŒX;Y D ŒeX ; eY and ŒeX ; eY D ŒX; Y C LX ˛ LY ˛ d.iX ˛ iY ˛ /=2 D ŒX; Y C !˛ ; and so LŒX;Y .ZC˛ / D LŒX;Y .ZC˛ /iZ d!˛ D LŒX;Y .ZC˛ /iZ d.LX ˛ LY ˛ /; so this checks with (18). Note that the inner product is preserved by the Lie derivative: X.u; v/ D .LX u; v/ C .u; LX v/: This follows directly from formula (17) and (2). The extended Lie derivative defines a Lie algebra action on E, and given suitable global conditions on M and G, we can integrate this to a normal G-action on the vector bundle E. Remark. We can also define a Lie derivative on spinors by imitating the Cartan formula for forms: LX ˛ D d iX ˛ C iX d˛. We define LX ˛ D d.eX ˛/ C eX d˛: This clearly commutes with d and one may check, using the definition of LX u above, that LX .u ˛/ D LX u ˛ C u LX ˛: Having seen how to define an action on E, we can now tackle the question of invariant generalized geometrical structures. Note that the inner product is invariant and also the Courant bracket (since d is invariant and the bracket is defined by this and the inner product), so any structure defined by these is transformed by the G-action to another. First consider a curving of a gerbe. As we saw earlier, this is defined by an isotropic splitting of E – a subbundle S.T / such that W S.T / ! T is an isomorphism. Invariance of a curving under the action of G is just the invariance of S.T /. A local

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section is of the form Y C iY B˛ , so LX .Y C iY B˛ / D ŒX; Y C LX iY B˛ iY d ˛ D ŒX; Y C iŒX;Y B˛ C iY .LX B˛ d ˛ / and for invariance we require LX B˛ D d ˛ :

(19)

Note that ˇ ˛ D iX dA˛ˇ ;

Bˇ B˛ D dA˛ˇ

so that ˛ iX B˛ is a globally defined 1-form c. Moreover dc D d ˛ d iX B˛ D iX dB˛ D iX H

(20)

from (19). There is one such form c for each generator of g so we have a 1-form with values in the dual of the Lie algebra: 2 1 .M; g / analogous to the moment map in symplectic geometry. For a generalized metric, invariance is just the invariance of V E. The generalized metric was given by two quantities – a metric and a curving. For the metric we have just the usual invariance LX g D 0 and we considered the curving above.

12 Quotients One of the simplest constructions in Riemannian geometry is the quotient metric – a free isometric action of G on a manifold defines a natural metric on the quotient space. For example, the quotient of the standard metric on the unit sphere in C nC1 by the S 1 action z 7! e i z gives the Fubini–Study metric. We now ask how to adapt this process for a generalized metric. We shall follow the formalism developed here but in fact it gives what the physicists term the “gauged WZW model” and has been known for some time. Recall first the standard quotient. We have a free action of G with quotient map p W M ! M=G. The image of g spans a subbundle TF TM (the tangent bundle along the fibres) and the tangent space of the quotient is identified with TM=TF by p. Given a G-invariant metric g on M , the orthogonal complement TF? maps isomorphically to T .M=G/ by p. Now if Xx is a tangent vector at a point xN on M=G we lift it to a tangent vector X 2 TF? to M at some point x in the orbit over xN and define x X/ x D gx .X; X/. By G-invariance the definition is independent of choices. gN xN .X; Now consider the generalized version. We let F denote the subbundle of E spanned by e.g/, so W E ! T projects F to TF . Since F is isotropic by assumption, F F ? and so F ? =F is a bundle of rank .2n dim G dim G/ D 2.n dim G/, and with an induced nondegenerate inner product. Since E projects to TM and F is the


205

tangent space of the G-orbits, we have a surjective map from F ? =F to p T .M=G/. Its kernel consists of the 1-forms which annihilate F , and this is p T .M=G/. In particular this is an isotropic subspace of half the dimension, so the signature of the quadratic form is .n dim G; n dim G/. Everything is G-invariant and so descends to a bundle Ex on M=G. More importantly, there is an induced Courant bracket. To see this, note that the sections of Ex on M=G pull back precisely to the G-invariant sections of F ? =F . We saw above that since F is isotropic, all its sections are closed under Courant bracket. Now let u; v be invariant sections of F ? , so that LX u D 0 D LX v and for all eX D e.a/, .eX ; u/ D 0. From (17) we have LX u D ŒeX ; u C d.eX ; u/ D 0 so that ŒeX ; u D 0. Now use (2): .u/.v; w/ D .Œu; v C d.u; v/; w/ C .v; Œu; w C d.u; w// with w D eX . Since .u; eX / D 0 D .v; eX / and ŒeX ; u D 0 we get 0 D .Œu; v; eX / C .d.u; v/; eX / D .Œu; v; eX / C X.u; v/=2 But since LX u D LX v D 0, X.u; v/ D .LX u; v/ C .u; LX v/ D 0 so .Œu; v; eX / D 0 and Œu; v is also a section of F ? . Hence the sections of Ex have a Courant bracket. The twisting bundle E descends to a bundle Ex with the same structure. The issue about whether a gerbe with connective structure descends is more complicated – here we content ourselves with just the bundle Ex – an exact Courant algebroid in other language. Now for the generalized metric: if V E is a positive-definite invariant subbundle, then take the image of V \ F ? to define Vx F ? =F . Since V ? is negative definite and F is null, V ? C F has dimension n C dim G. Thus its orthogonal subspace V \ F ? has dimension 2n n dim G D n dim G. It has a positive definite inner product and injects into F ? =F , and intersects the kernel of the projection to T M=G in zero, again by the definiteness of the metric. It thus defines a generalized metric on the quotient. This procedure has here been obtained from the viewpoint of generalized geometry, but in fact in a different language it appeared in the physics literature [14], [15] many years ago. What we have done here may be considered quite abstract, so let us see what happens to the two concrete fields defined by a generalized metric – the actual metric g and the three-form H . The quotient metric is induced from V \ F ? , which is the subspace of E defined by .Y C gY C iY B˛ ; X C ˛ / D 0 for all e.a/ D X C ˛ . But 2.Y C gY C iY B˛ ; X C ˛ / D g.Y; X/ C iX iY B˛ C iY ˛

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and we defined above for each a 2 g a global one-form ca where ca D ˛ iX B˛ if e.a/ D eX . So V \ F ? D fY C 2 E W g.Y; Xa / C ca .Y / D 0g

(21)

for all a 2 g. This is a modified form of the horizontality condition g.Y; Xa / D 0 which gives the usual quotient metric. Let H TM be the G-invariant subbundle defined by this condition, then the procedure is the same: we identify tangent vectors on M=G with vectors in H and take the induced inner product. To find the three-form on the quotient, we consider the pull-back of Ex D F ? =F as p Ex D V \ F ? ˚ V ? \ F ? ; this decomposition defining the generalized metric on the quotient. So a vector in V \ F ? is locally given by Y C gY C iY B˛ where Y satisfies g.Y; Xa / C ca .Y / D 0, and a vector in V ? \ F ? is Y 0 gY 0 C iY 0 B˛ where Y 0 satisfies the similar linear constraint g.Y 0 ; Xa / ca .Y 0 / D 0. If p.Y 0 / D p.Y /, then X Y0 D Y C bi .Y /Xi

(22)

i

where X1 ; : : : ; Xm are the vector fields generated by a basis ai of g. From g.Y; Xi / C ci .Y / D 0 D g.Y 0 ; Xi / ci .Y 0 / we get 2ci .Y / D

X

bj .Y /.g.Xi ; Xj / ci .Xj //:

(23)

i

Note here that c.X/ D ˛ .X / iX iX B˛ D .X C ˛ ; X C ˛ / D 0 by the isotropy condition, hence ci .Xj / D cj .Xi /: In particular, it follows that g.Xi ; Xj / ci .Xj / is always invertible and (23) can be uniquely solved for bi .Y / for any Y – it defines 1-forms bi by X bi D 2 .G C /1 (24) ij cj j

where Cij D ci .Xj /; Gij D g.Xi ; Xj /. We now adopt the point of view mentioned earlier that the average of the positive definite and negative definite splitting is the curving, whose curvature we are aiming


207

to find. This is

1 .Y C gY C iY B˛ C Y 0 gY 0 C iY 0 B˛ / 2 which from (22) we can write as 1X bi .Y /.Xi C iXi B˛ gXi /: Y C iY B˛ C 2

(25)

i

But c D ˛ iX B˛ and we are working in Ex D F ? =F where F is spanned by the X C ˛ , so in F ? =F , we have ci D Xi iXi B˛ which means we write the splitting (25) as 1X bi .Y /.ci C gXi /: Y C iY B˛ 2 i

From (21), this is

1X Y C iY B˛ bi ^ .ci C gXi / 2 i

and from (24) we have X X 1X bi ^.ci CgXi / D .GC /1 cj ^.GCC /j1 ij cj ^.ci CgXi / D i .ci CgXi / 2 i

i;j

i;j

using the symmetry of G and skew-symmetry of C . Now X .G C C /j1 i .ci C gXi /aj i;j

is a Lie-algebra valued 1-form which annihilates H TM and on Xj takes the value aj – it is therefore the connection form A for the connection on M , considered as a principal G-bundle over M=G, defined by the horizontal distribution H . Thus we have A 2 1 .M; g/ and 2 1 .M; g / so using the duality pairing we write the splitting as Y C iY .B˛ h ; Ai/ : Taking the exterior derivative, this means that the three-form on M=G is H d h ; Ai: Example. The simplest situation is where G D S 1 and H D 0. Then since dc D iX H D 0 from (20), the three-form on the quotient is d.c ^ A/ D c ^ F where F D dA is the curvature of M considered as a principal circle bundle. Here we see that non-trivial three-form fluxes can be created from trivial ones by a quotient.

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References [1]

I. Agricola, Connections on naturally reductive spaces, their Dirac operator and homogeneous models in string theory. Comm. Math. Phys. 232 (2003), 535–563. 201

[2]

V. Apostolov, P. Gauduchon, and G. Grantcharov, Bihermitian structures on complex surfaces. Proc. London Math. Soc. 79 (1999), 414–428. 198

[3]

J.-L. Brylinski, Loop spaces, characteristic classes and geometric quantization. Progr. Math. 107, Birkhäuser, Basel 1993. 189, 190, 191

[4]

H. Bursztyn, G. Cavalcanti and M. Gualtieri, Reduction of Courant algebroids and generalized complex structures. Adv. Math. 211 (2007), 726–765. 202

[5]

G. Cavalcanti and M. Gualtieri, A surgery for generalized complex structures on 4-manifolds. J. Differential Geom. 76 (2007), 35–43. 192

[6]

T. J. Courant, Dirac manifolds. Trans. Amer. Math. Soc. 319 (1990), 631–661. 187

[7]

S. J. Gates, C. M. Hull, and M. Roˇcek, Twisted multiplets and new supersymmetric nonlinear -models. Nuclear Phys. B 248 (1984), 157–186. 193

[8]

J. Gauntlett, D. Martelli, and D. Waldram, Superstrings with intrinsic torsion. Phys. Rev. D (3) 69 (2004), no. 8, 086002, 1–27. 198, 201

[9]

M. Gualtieri, Generalized complex geometry. PhD Thesis, Oxford 2003; arXiv:math.DG/ 0401221. 191

[10] N. J. Hitchin, Lectures on special Lagrangian submanifolds. In Winter school on mirror symmetry, vector bundles and Lagrangian submanifolds (Cumrun Vafa & S.-T. Yau, eds.), AMS/IP Stud. Adv. Math. 23, Amer. Math. Soc., Providence, RI; International Press, Somerville, MA, 2001, 151–182. 189 [11] N. J. Hitchin, Generalized Calabi-Yau manifolds. Q. J. Math. 54 (2003), 281–308. 185, 186, 192, 196, 201 [12] N. J. Hitchin, Brackets, forms and invariant functionals. Asian J. Math. 10 (2006), 541–560. [13] N. J. Hitchin, Instantons, Poisson structures and generalized Kähler geometry. Comm. Math. Phys. 265 (2006), 131–317. 196 [14] C. M. Hull and B. Spence, The gauged nonlinear sigma model with Wess-Zumino term. Phys. Lett. B 232 (1989), 204–210. 205 [15] I. Jack, D. R. Jones, N. Mohammedi, and H. Osborn, Gauging the general nonlinear sigma model with a Wess-Zumino term. Nuclear Phys. B 332 (1990), 359–379. 205 [16] U. Lindström, M. Roˇcek, R. von Unge, and M. Zabzine, Generalized Kähler manifolds and off-shell supersymmetry. Comm. Math. Phys. 269 (2007), 833–849. 198 [17] F. Witt, Generalised G2 -manifolds. Comm. Math. Phys. 265 (2006), 275–303. 198, 200, 201

Chapter 7

Generalizing geometry – algebroids and sigma models Alexei Kotov and Thomas Strobl

Contents 1 Introduction . . . . . . . . . . . . . . . . 2 Lie and Courant algebroids . . . . . . . . 3 Dirac structures . . . . . . . . . . . . . . 4 Generalized complex structures . . . . . 5 Algebroids as Q-manifolds . . . . . . . . 6 Sigma models in the AKSZ-scheme . . . 7 Sigma models related to Dirac structures 8 Yang–Mills type sigma models . . . . . . References . . . . . . . . . . . . . . . . . . .

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209 214 225 230 234 241 248 252 260

1 Introduction In this exposition we want to summarize some ideas in the overlap of differential geometry and mathematical physics. In particular we focus on the interplay of socalled sigma models with geometrical structures being related to algebroids in one way or another. Several traditional geometrical notions received various kinds of generalizations in recent years. Some of them give rise to novel sigma models, while also some sigma models bring attention to possible new geometries. Sigma models are action functionals (variational problems) where the underlying space of fields (maps) has a target space equipped with some geometry; in the most standard case one regards maps X from one given Riemannian manifold .†; h/ to another one .M; g/ and considers the functional Z 1 S ŒX D kdXk2 ; (1.1) 2 † where k˛k2 D .X g/.˛ ^; ˛/ for any ˛ 2 p .†; X TM / with denoting the h-induced Hodge duality operation on †,1 the critical points of which are precisely 1 In local coordinates on † and x i on M this reads more explicitly as ˛ i1 :::p dvol where † i1 :::p ˛ indices ip and i are raised and lowered by means of the metric h and (the pullback by X of) g, respectively, and d dvol† D det.h/d , d denoting the dimension of †. In this chapter we use the Einstein sum convention, i.e., a sum over repeated indices is always understood.

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the maps X which are harmonic. For the special case that M is just Rn equipped with the standard flat metric this functional reduces to Z 1 SŒ i D d i ^ d i ; (1.2) 2 † the action of n “free scalar fields” on † (here i denotes the function on † obtained by restricting X to the i-th coordinate in Rn – it is understood that the index i on the l.h.s. is an “abstract index”, i.e., S depends on all the scalar fields, 1 to n ). Another example of a sigma model is the Poisson sigma model [36], [21], where the source manifold † is necessarily 2-dimensional and the target manifold M carries a Poisson structure instead of a Riemannian one. In fact, one considers a functional on the space of vector bundle morphisms from T † to T M in this case. It is “topological”, which we want to interpret as saying that the space of classical solutions (stationary points of the functional) modulo gauge transformations (invariances or symmetries of the functional) is finite dimensional – besides the fact that it does not depend on geometrical structures of the source like a Riemannian metric h (which in this case is even absent in the definition of the functional). The tangent bundle of any manifold as well as the cotangent bundle of a Poisson manifold give rise to what are called Lie algebroids (whose definition is properly recalled in the body of the chapter below – cf. in particular Examples 3.2 and 2.3 below) and the above functional becomes stationary precisely on the morphisms of these Lie algebroid structures [4]. But this also works the other way around: given a functional of such a form defined by an a priori arbitrary bivector field … (and, in some particular extension of the functional, also a closed 3-form H ) the respective functional becomes topological, iff [35], [23] … defines a (for non-zero H twisted) Poisson structure. In fact, twisted Poisson structures (c.f. also [34]), i.e., bivectors … together with closed 3-forms H defined over a manifold M satisfying Œ…; … D .…\ /˝3 H

(1.3)

where Œ ; denotes the Schouten–Nijenhuis bracket of multi-vector fields and …\ W T M ! TM is the natural operator induced by the bivector … as follows: …\ .˛/ D ….˛; /, were even found first in such a manner [23], [30]. This is typical for the interplay of geometrical notions and sigma models: the former are needed to define the latter, but sigma models sometimes also give indications about (focus on) particularly interesting geometrical notions. In this example, in the space of bivectors on a manifold the ones which are Poisson are singled out by the sigma model, or in the space of pairs .…; H / those satisfying equation (1.3) (which can be seen to define a particular Dirac structure [34] – we introduce to Dirac structures in the main text in detail). Another example for such an interplay are supersymmetric sigma models and bihermitian geometry [14]. The latter geometry received renewed and revived interest recently by its elegant reformulation in terms of so-called generalized complex structures [20].

Chapter 7. Generalizing geometry – algebroids and sigma models

211

What is, on the other hand, meant more specifically by generalizing traditional geometrical notions? In fact, also the generalized notions can usually be expressed in terms of ordinary differential geometrical ones, in which case it just boils down to a different way of thinking about them. In any such case one has usually some particular kind of a so-called algebroid in the game. There are several kinds of an algebroid considered in the literature. All of them have the following data in common, which we thus want to use as a definition of the general term: Definition 1.1. We call an algebroid E D .E; ; Œ ; /, a vector bundle E ! M together with a homomorphism of vector bundles W E ! TM , called the anchor of E, and a product or bracket on the sections of E satisfying the Leibniz rule ( ; 0 2 .E/, f 2 C 1 .M /): Œ ;f

0

DfŒ ;

0

C .. /f /

0

:

(1.4)

Note that the map on sections induced by is denoted by the same letter conventionally. Depending on further conditions placed on the bracket (like its symmetry properties or its Jacobiator) and further structures defined on E, one has different kinds of algebroids: Lie algebroids, Courant algebroids, strong homotopy Lie (or L1 -) algebroids, and so on. Lie algebroids are obtained e.g. by requiring in addition that the bracket is a Lie bracket, i.e., antisymmetric and with vanishing Jacobiator. The definition of a Courant algebroid requires a fiber metric on E controlling the symmetric part of the bracket as well as its Jacobiator. We will come back to all these various kinds of algebroids in more detail in the text below. The philosophy now is that we can do differential geometry by replacing TM ˝p ˝q D TM M by E and or, more generally, also the tensor bundle qp .M /L ˝p ˝ T ˝q ˝p ˝q , respectively. Let us call a section t of p;q E ˝ E an E-tensor E ˝E field. The Leibniz property permits us to define a (Lie) “derivative” of t along any section of E: Indeed, set EL .f / WD . /f , EL . 0 / WD Œ ; 0 , and extend this to powers of E by the Leibniz property w.r.t. tensor multiplication and to E by means of compatibility with contraction: EL .h 0 ; !i/ D hEL . 0 /; !i C h 0 ; EL .!/i, defining the Lie derivative for any ! 2 .E /. This implies that given an algebroid E (not necessarily Lie), i.e., data that at least include those of the definition above, we can define a “Lie derivative” of any E-tensor along sections of E. By construction, this generalizes the notion of an ordinary Lie derivative of ordinary tensor fields: indeed, the usual formulas are reproduced in the case where the algebroid is chosen to be a so-called standard Lie algebroid, i.e., where E is the tangent bundle of the base manifold M , the anchor is the identity map and the bracket is the standard Lie bracket of vector fields. Such type of geometrical notions have properties in common with their prototypes in traditional geometry (like, in this case e.g., by construction, the Leibniz property of the generalized Lie derivatives), but they in general also have pronounced differences, possibly depending on the type of algebroid, i.e., on the additional structures. In the above example one can ask e.g. if the commutator of such

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Lie derivatives is the Lie derivative of the bracket of the underlying sections. In general this will not always be the case. However, for an important subclass, containing Lie and Courant algebroids, it will (cf. Lemma 2.6 below). Another important example of such generalized structures is “generalized geometry” or generalized complex geometry in the sense of Hitchin (cf. also the contribution of N. Hitchin to this volume). It is a particular case of the above viewpoint where E is taken to be TM ˚ T M equipped with projection to the first factor as anchor as well as the so-called Courant or Dorfman bracket (in fact, slightly more general and conceptually preferable, one takes E to be what is called an exact Courant algebroid – cf. Definition 2.14 below). Now, by definition a generalized complex structure is what usually would be a complex structure, just replacing the standard Lie algebroid TM by an exact Courant algebroid. In particular, it is an endomorphism of E Š TM ˚ T M squaring to minus one and satisfying an integrability condition using the bracket on E (cf., e.g., Proposition 4.3 below). It is not difficult to see that this notion generalizes simultaneously ordinary complex structures as well as symplectic ones. In fact, the situation is closely related to (real) Dirac structures (particular Lie subalgebroids of exact Courant algebroids, cf. Definition 3.1 below), mentioned already previously above: these generalize simultaneously Poisson and presymplectic structures on manifolds. In fact, generalized complex structures did not only find at least part of their inspiration from real Dirac structures, but they can be even defined equivalently as imaginary Dirac structures – which is the perspective we want to emphasize in the present note.2 An elegant and extremely useful viewpoint on some algebroids arises within the language of differential graded manifolds, sometimes also called Q-manifolds (Q denoting a homological degree-one vector field on the graded manifold, i.e., its differential). We devote a section to explaining this relation, after having introduced the reader to the above mentioned notions of algebroids, Dirac structures, and generalized complex structures, in the three sections to follow. Together these four sections provide our exposition on algebroids and this kind of generalized geometry. Some of the following sections then deal with the respective sigma models: Given a Q-manifold with a compatible graded symplectic structure on it, one can always associate a topological sigma model to it [1]. We review this construction in some detail and specialize it to lowest dimensions (of the source manifold †), reproducing topological models corresponding to Poisson manifolds (dim † D 2, this is the above mentioned Poisson sigma model), to Courant algebroids (dim † D 3, such models were considered in [31]). Some space is devoted to describing these models, somewhat complementary to what is found in the literature, since they can be used to introduce part of the formalism that is needed for the last section to this contribution. There are also topological models that, at least up to now, have not yet been related to the AKSZ formalism and corresponding e.g. to Dirac structures. We recall these models, called Dirac sigma models [25] and generalizing the Poisson sigma models essentially such as (real) Dirac structures generalize Poisson manifolds, in a separate 2 This

point of view was maybe less known at the time when we this note was started, while in the mean time it has received some attention also elsewhere.


213

section. These as well as the AKSZ models share the property that they are topological and that the solutions to their field equations generalize (only) flat connections to the algebroid setting. The final section is devoted to sigma models in arbitrary spacetime dimension (dimension of † which then is taken to be pseudo-Riemannian) with a relation to algebroids but which are nontopological and which generalize connections that are not necessarily flat but instead satisfy the Yang–Mills field equations.3 This deserves some further motivating explanation already in the introduction: Consider replacing in equation (1.2) the functions on † by 1-forms Aa , a D 1; : : : ; r, yielding Z 1 a dAa ^ dAa : (1.5) SŒA D 2 † For r D 1, i.e., there is just one 1-form field A, this is a famous action functional, describing the electromagnetic interactions (the electric and the magnetic fields can be identified with the components of the “field strength” 2-form dA). Having several such 1-forms in the game, r > 1, one obtains the functional describing r free (i.e., mutually independent) 1-form fields4 . The most standard way of making scalar fields interact is to add some “potential term” to the functional (1.2), i.e., to add the integral over V . i / multiplied by the h-induced volume form on † (where V is some appropriately smooth function on Rn , mostly even only a low degree polynomial so as to not spoil “renormalizability”).5 Turning (1.5) into an interacting theory (without introducing further fields and not spoiling its gauge invariance, at most “deforming” the latter one appropriately) is not so simple. In fact, the result is rather restricted (cf., e.g., uniqueness theorems in the context of the deformation theory of gauge theories [3]) and one is lead to only replace dAa by the expression F D dA C 12 ŒA ^; A

(1.6)

for the curvature of the Lie algebra valued connection 1-form A of a (trivialized) principal bundle for some r-dimensional, quadratic Lie group G. “Quadratic” means that its Lie algebra g admits an ad-invariant inner product which, when starting by deforming (1.5), needs to have definite signature such that a sum over the index a 3 Being nontopological is required for most physical applications so as to host the degrees of freedom necessary to describe realistic interactions. – There are also another type of (nontopological) sigma models than those explained in the last section that are related to algebroids and in particular generalized and bihermitian geometry. These are supersymmetric 2-dimensional sigma models, i.e., string theories. Although also highly interesting, we will not touch this issue here but refer e.g. to the review article [42] and references therein. 4 In the physics language they are often called “vector fields” (as opposed to “scalar fields” used in (1.2)). We avoid this somewhat misleading/ambiguous nomenclature, but we will, however, from time to time refer to them as “gauge fields”, despite the fact that in the mathematical setting they correspond to connections in a principal G-bundle (here trivialized with G D U.1/r ). 5An alternative way of having scalar fields interact is coupling them to 1-form gauge fields so that they start being correlated (i.e., interacting) via these 1-form fields. In fact, both ways of interactions are realized in the standard model of elementary particle physics, where the gauge fields describe interaction particles like the photon and the scalar fields describe “matter”, essentially like electrons (or the – not yet discovered – Higgs particles).

214


results from a term of the form

Z

SYM ŒA D

.F ^; F /

(1.7)

†

after choosing a -orthonormal basis in g. The resulting theory is called a Yang–Mills (gauge) theory and was found to govern all the strong, the weak, and the electromagnetic interactions. If one considers an algebroid E as defined above over a 0-dimensional base manifold M , M degenerating to a point, one is left only with a vector space. For E being a Lie algebroid this vector space becomes a Lie algebra, for E being a Courant algebroid it becomes a quadratic Lie algebra. In fact, this is a second, algebraic part that is incorporated in algebroids: general Lie algebroids can be thought of as a common generalization of the important notions of, on the one hand, a Lie algebra g and, on the other hand, standard geometry (i.e., geometry defined for TM or qp .M /), as we partially explained already above. (A similar statement is true for general Courant algebroids, TM then being replaced by the “standard Courant algebroid” TM ˚ T M and g by a quadratic Lie algebra together with its invariant scalar product). From this perspective it is thus tempting to consider the question if one can define e.g. a theory of principal bundles with connections in such a way that the structural Lie algebra is replaced by (or better generalized to) appropriately specified structural algebroids. Likewise, from the more physical side, can one generalize a functional such as the Yang–Mills functional (1.7) to a kind of sigma model, replacing the in some sense flat Lie algebra g by nontrivial geometry described via some appropriate Lie or Courant algebroid? These questions will be addressed and, at least part of them, answered to the positive in the final section to this chapter. Between the part on sigma models and the one on algebroids it would have been nice to also include a section on current algebras (cf. [2] as a first step), as another link between the two; lack of spacetime, however, made us decide to drop this part in the present contribution.

2 Lie and Courant algebroids In the present section, we recall the notions of Lie and Courant algebroids in a rigorous manner and study some of their properties. We also brielfly introduce some of their higher analogues, like Lie 2-algebroids and vector bundle twisted Courant algebroids. In a later section we will provide another, alternative viewpoint of all these objects by means of graded manifolds, which permits an elegant and concise reformulation. Definition 2.1. A Loday algebroid is an algebroid .E; ; Œ ; / as defined in Definition 1.1 where the bracket defines a Loday algebra on .E/, i.e., it satisfies the Loday (or left-Leibniz) property, Œ

1; Œ

2;

3

D ŒŒ

1;

2 ;

3

CŒ

2; Œ

1;

3 :

(2.1)


215

An almost Lie algebroid is an algebroid E where the bracket is antisymmetric. E becomes a Lie algebroid, if Œ ; defines a Lie algebra structure on .E/, i.e., if E is simultaneously Loday and almost Lie. Here we adapted to the nomenclature of Kosmann-Schwarzbach, who prefers to use the name of Loday in the context of (2.1) so as to reserve the terminus Leibniz for compatibility w.r.t. multiplication of sections by functions, (1.4), for which “Leibniz rule” has become standard. Example 2.2. Obviously, .TM; D Id/ together with the Lie bracket of vector fields is a Lie algebroid; it is called the standard Lie algebroid. If M is a point, on the other hand, a Lie algebroid reduces to a Lie algebra. More generally, if the anchor of a Lie algebroid map vanishes (i.e., maps E to the image of the zero section in TM ), E is a bundle of Lie algebras; in general not a Lie algebra bundle, since Lie algebras of different fibers need not be isomorphic. Example 2.3. A less trivial Lie algebroid is the cotangent bundle T M of a Poisson manifold.6 The anchor is provided by contraction with the Poisson bivector field … and the bracket of exact 1-forms Œdf; dg WD dff; gg is extended to all 1-forms by means of the Leibniz rule (1.4). Example 2.4. An example of a Loday algebroid with a non-antisymmetric bracket is the following one: E D TM ˚ T M with being projection to the first factor and the bracket being given by Œ C ˛; 0 C ˛ 0 D Œ ; 0 C L ˛ 0 { 0 d˛I

(2.2)

here ; 0 and ˛; ˛ 0 denote vector fields and 1-forms on M , respectively, and the bracket on the r.h.s. is the usual Lie bracket of vector fields. This is the so-called Dorfman bracket. Note that if one takes the antisymmetrization of this bracket (the original bracket Courant has introduced [10]), E does not become an algebroid in our sense, since equation (1.4) will not be valid any more (as a consequence of the non-standard behavior of the above bracket under multiplication by a function of the section in the first entry). Lemma 2.5. The anchor map of a Loday algebroid is a morphism of brackets. Proof. First, from (2.1) we find ŒŒ 1 ; 2 ; f 3 D Œ 1 ; Œ 2 ; f 3 . 1 $ 2 /. Using the Leibniz rule for the l.h.s., we obtain .Œ 1 ; 2 /f 3 Cf ŒŒ 1 ; 2 ; 3 . Applying it twice to the first term on the r.h.s., we get . 1 /. 2 /f 3 C f Œ 1 ; Œ 2 ; 3 C 6 By definition, one obtains a Poisson structure on a smooth manifold M , if the space of functions is equipped with a Lie bracket f ; g satisfying ff; ghg D gff; hg C hff; gg for all f; g; h 2 C 1 .M /. The latter condition, together with the antisymmetry of the bracket, is equivalent to the existence of a bivector field … 2 .ƒ2 TM / such that ff; gg D h…; df ˝ dgi.

216


. 1 /f Œ 2 ; 3 C . 2 /f Œ 1 ; 3 . The last two terms drop out upon antisymmetrizing in 1 and 2 . In the remaining equation the terms proportional to f cancel by means of (2.1), and one is left with .Œ valid for all sections

1; i,

2 /f

3

D Œ.

1 /; .

2 /f

3;

(2.3)

i D 1; 2; 3, and functions f . This completes the proof.

Lemma 2.6. The E-Lie derivative provides a representation of the bracket of a Loday algebroid on E-tensors, ŒEL 1 ; EL 2 D ELŒ 1 ; 2 . Proof. This follows from (2.1), the previous lemma, and the extension of the ELie derivative to tensor powers and the dual by means of a Leibniz rule (using that commutators of operators satisfying a Leibniz rule for some algebra – here that of the tensor product as well as that of contractions – are of Leibniz type again). A Lie algebroid permits to go further in extrapolating usual geometry on manifolds to the setting of more general vector bundles (algebroids). In particular, there is a straightforward generalization of the de Rham differential in precisely this case:7 In any almost Lie algebroid we may define the following degree-one map Ed on Edifferential forms E .M / .ƒ E /. For any function f and E-1-form ! we set hEdf; i WD . /f; hEd!;

˝

0

i WD . /h!;

0

i .

0

/h!; i h!; Œ ;

0

i

(2.4)

and extend this by means of a graded Leibniz rule to all of E .M /. Clearly, for the standard Lie algebroid .TM; Id/ this reduces to the ordinary de Rham differential. As one proves by induction, with this one finds in generalization of the Cartan–Koszul formula: E

d!.

1; : : : ;

pC1 /

WD

pC1 X iD1

C

.1/i C1 . X

i /!.: : : ;

.1/iCj !.Œ

i;

Oi ; : : : /

O O j : : : ; i ; : : : ; j ; : : : /;

(2.5)

i<j

valid for any ! 2 pE .M / and explicit proofs:

i

2 .M; E/. This text is in part supposed to contain

Proof of equation (2.5). The property holds for p D 1 by definition. Suppose we proved (2.5) for all forms of order at most p 1. It suffices to see that (2.5) holds for E E E each ! D ˛ ^, where ˛ 2 1E .M /, 2 p1 E .M /. d.!/ D d.˛/^ ˛ ^ d. /, 7 We remark in parenthesis that there is also an option to generalize the de Rham differential different from formula (2.4) below, using the language of graded manifolds; that generalization can be used also for Courant algebroids (cf. e.g. [32]) or even arbitrary L1 algebroids, cf. Section 5 below.


therefore E

d.!/.

1; : : : ;

pC1 /

D

X

.1/i Cj C1 Ed˛.

i;

217

Oi ; : : : ; Oj ; : : : /

j /.: : : ;

i<j

C

X

.1/i ˛.

i/

d.: : : ; Oi ; : : : /

E

i

X D .1/i Cj C1 .

. j /˛. i / ˛.Œ i ; j / .: : : ; Oi ; : : : ; Oj ; : : : / X X .1/j C1 . j /.: : : ; Oj ; : : : ; Oi ; : : : / C .1/i ˛. i / i /˛. j /

i<j

i

C

X

j

.1/j .

Oi ; : : : ; Oj ; : : : /

j /.: : : ;

i<j

C

X

.1/kCl .Œ

k;

l ; : : : ;

1; : : : / i;k;l

k
1 the functions ; 0 ; : : : of degree n 1 can be identified with sections in a vector bundle E. The formulas (5.7) and (5.8), where Q denotes the Hamiltonian for the Q-structure, equip E with the structure of a Loday algebroid. Proof. First we note that the functions of any fixed degree d on an N0 -graded manifold are a locally free module over the functions of degree zero, which in turn are isomorphic to C 1 .M /. This implies the existence of vector bundles for any of those degrees d over M . For d D n 1 we call this bundle E. As remarked above, the vector field Q compatible with the symplectic form is always Hamiltonian (even for n 2 N); since the (graded) canonical Poisson bracket has degree n, the respective Hamiltonian Q has to have degree n C 1. By an elementary computation adding up respective degrees, one then finds that . / indeed maps functions f 2 C 1 .M / to functions and the bracket Œ ; takes again values in the sections of E.16 It is also obvious that . / is a vector field on M , since the r.h.s. of the defining expression (5.7) satisfies an (ungraded) Leibniz rule for f being a product of two functions. To have an algebroid (cf. our Definition 1.1), we need to verify two things: first, the Leibniz property (1.4), which follows at once from the two defining expressions on E above and the graded Leibniz property of the Poisson bracket f ; g. Second, that . / is indeed C 1 .M /-linear in so as to really give rise to a bundle map W E ! TM . The only potentially dangerous term which may violate this condition may arise when has terms quadratic or higher in the momenta pi conjugate to the coordinates x i on M . However, since the coordinates pi necessarily have degree n and Q has degree n C 1, this is not possible for n > 1. We are left with verifying the Loday property (2.1). It is only here where the condition fQ; Qg D 0, resulting from Q2 D 0, comes into the game. We leave the respective calculation, which also makes use of the graded Jacobi identity of the Poisson bracket, as an exercise to the reader. Some remarks: First of all, it is clear from the above proof that any symplectic graded manifold of degree n 2 equipped with an arbitrary function Q of degree n C 1 gives, by the above construction, rise to an algebroid structure (in our sense, cf. Definition 1.1). If in addition fQ; Qg D 0, this algebroid becomes a Loday algebroid. Certainly, the higher in degrees we go, the more additional structures arise. Already for the case of n D 2 the Loday algebroid had the additional structures making it into a Courant algebroid. For higher n, however, there will often be other algebroids out of which the Loday algebroid will be composed. Let us illustrate this for n D 3: In Darboux coordinates the symplectic structure will have the form ! D dx i ^ dpi C d a ^ d a ;

(5.9)

denote the expressions on the graded manifold and those isomorphic to them on E ! M by the same symbols. 16 We


239

where the coordinates x i , a , a , and pi have the degrees 0, 1, 2, and 3, respectively. From this we learn that a PQ-manifold of degree 3 has to be symplectomorphic to T Œ3.V Œ1/ with its canonical symplectic form. Here V ! M is a vector bundle and the brackets indicate shifts of degree in the respective fiber coordinates (note that without a shift the momenta a conjugate to the degree-1 fiber linear coordinates a on V Œ1 have to have degree 1 while they are now shifted so as to have degree 2). With D ' a a C 12 ˛ab a a , we see that the above vector bundle E in this case is isomorphic to V ˚ ƒ2 V . Now let us consider the Hamiltonian for the Q-structure, i.e., a function of degree 4; it thus have to have the form c a b c C 12 ˇ ab a b C Q D ai a 12 Cab

1 a b c d : 24 abcd

(5.10)

Obviously, for ˇ D D 0 this is nothing but the canonical lift of the Q-structure (5.2) corresponding to a Lie algebroid, thus equipping V with the structure of a Lie algebroid. In the general case V still is an almost Lie algebroid and E can be considered as an appropriate extension into a Loday algebroid. (V itself is not a Loday algebroid, having a Jacobiator controlled by a contraction of ˇ and . Adding a V -2-form ˛ to the section ', on which the anchor acts trivially, one can restore the Loday property (2.1).) We know that for n D 1 we also obtained an algebroid, even a Lie algebroid. However, this was defined on T M . The above construction leads to a trivial Rbundle over M instead, the sections of which can be identified with functions on M , and W C 1 .M / ! .TM / is R-linear but no more C 1 .M /-linear, in fact it corresponds (up to a sign being subject of conventions) to the map from functions to their Hamiltonian vector fields. In fact, both equations (5.8) and (5.7) become equivalent in this degenerate case, equipping C 1 .M / with the Poisson bracket f; gM , as already remarked above. (This bracket defines a Lie algebra structure on the sections of M R, but not a Lie algebroid or even a general algebroid structure on this bundle since one does not have an anchor map for it). To obtain the Lie algebroid structure on T M from the .P /Q-manifold T Œ1M we need to proceed differently. In fact, this provides a procedure that is applicable for any Lie algebroid E ! M , corresponding to a degree-.1/ Q-manifold EŒ1. Sections of E are in 1-1 correspondence with vector fields ; 0 ; : : : of degree 1 on EŒ1, and their derived bracket ŒŒ ; Q; 0 , where the brackets denote the (super)commutator of vector fields, is rather easily verified to reproduce the Lie algebroid bracket between the respective sections. This procedure can in fact be considered for a Q-manifold of any degree n. While for general n this leads to what one may call Vinogradov algebroids (Loday algebroids with additional structures, cf. [17]), for n D 2 one obtains a V -twisted Courant algebroid as defined in Section 2 above. The degree-.1/ vector fields define the sections of the bundle E, the degree-.2/ vector fields the sections of the other bundle V ; the derived bracket ŒŒ ; Q; 0 of the degree-.1/ vector fields defines the Loday bracket on E, the ordinary commutator bracket Œ ; 0 , which apparently takes values in the degree-.2/ vector fields, gives the V -valued inner product on E. All the

240


defining properties of a V -twisted Courant algebroid are rather easy to verify in this case. On the other hand, a Q-manifold M of degree 2 gives always rise to V Œ2 ! M ! W Œ1, where the first map is an embedding, setting the degree-one coordinates in M to zero and the second map a projection, forgetting about the degree-two coordinates. (Here V is the same bundle as the one in the previous paragraph and W another vector bundle over the same base manifold). A somewhat lengthy analyses (cf. [17]) shows that after one has chosen an embedding of W Œ1 into M, which composed with the projection giving the identity map, the degree-two Q-manifold is in bijection with a Lie 2-algebroid. The vector bundle E of the V -twisted Courant algebroid picture is then composed of W and V , similarly to the situation of the degree-three PQ-manifold discussed above; one finds easily that E Š W ˚ W ˝ V , and, under only a few more assumptions (like that the rank of V is at least two) on a V -twisted Courant algebroid of this form, also vice versa, the latter arises always from a Lie 2-algebroid in such a way. We finally remark that if M is a Q-manifold of degree n, then T ŒnM is a PQmanifold of the same degree.17 E.g. T Œ1EŒ1, with EŒ1 a Lie algebroid (and Q lifted canonically, certainly), is a degree-1 PQ-manifold, and thus isomorphic to the Lie algebroid of a Poisson manifold. Indeed, the dual bundle of a Lie algebroid is canonically a Poisson manifold,18 in accordance with the easy-to-verify isomorphism T Œ1EŒ1 Š T Œ1E. So, Poisson geometry on M can be viewed as a particular case of Lie algebroid geometry (considering T M ), but also vice versa, a Lie algebroid structure on E as a particular (fiber-linear) Poisson structure (on E ) – and likewise so in higher degrees: A Courant algebroid is a particular case of a V -twisted Courant algebroid and also of a Lie 2-algebroid, but, in an appropriate sense, also Lie 2algebroids and their corresponding V -twisted Courant algebroids, can be viewed as particular Courant algebroids. However, as already the example of a Lie algebroid shows, this is not always the most convenient way of viewing them. We conclude this section by returning to the question of morphisms of Lie algebroids and, more generally, of any algebroid described by a Q- or a PQ-manifold. A morphism of Q-manifolds is a degree-preserving map ' W M1 ! M2 such that its pullback ' W C 1 .M2 / ! C 1 .M1 / is a chain map, i.e., one has Q1 B ' D ' B Q2 . It is a morphism of PQ-manifolds, if in addition it preserves the symplectic form, ' !2 D !1 . is grateful to D. Roytenberg for this remark in the context of a talk on V -twisted Courant algebroids. dual E of any Lie algebroid E becomes a Poisson manifold in the following manner: In order to define a Poisson bracket on E , it is obviously sufficient to do so on the fiber constant and fiber linear functions. The former are functions that arise as pullbacks from functions on the base manifold M , the latter are sections of E . It is then straightforward to verify that 17 T.S. 18 The

ff; f 0 g D 0; valid for all f; f 0 2 C 1 .M /,

;

f ; f g D . /f; 0

f ;

0

gDŒ ;

0

;

2 .E /, defines a Poisson structure on E .

(5.11)


241

6 Sigma models in the AKSZ-scheme In this section we want to discuss a particular class of topological sigma models that can be constructed in the context of algebroids. By topological we want to understand that the space of solutions to the classical field equations (the Euler–Lagrange equations of the functional) modulo gauge transformations does not depend on structures defined on the base manifold † in addition to its topology and that for “reasonable” topology (the fundamental group of † having finite rank etc) it is finite dimensional. In the context of ordinary gauge theories, one such a well-known space is the moduli space of flat connections on †. A functional producing such a moduli space is the Chern–Simons theory Z 1 A ^; .dA C 13 ŒA ^; A/ ; (6.1) SCS ŒA D 2 † defined on the space of connections of a trivialized G-bundle over an orientable 3-dimensional base manifold † when specifying such that .g D Lie.G/; Œ ; ; / gives a quadratic Lie algebra; such a connection is represented by a g-valued 1-form A on †, A 2 1 .†; g/. To find an appropriate generalization of this theory to the present context, let us first reinterpret the fields A and the field equations F D 0 of this model within the present context such that it permits a straightforward generalization. First of all, a g-valued 1-form A on † is evidently equivalent to a degree-preserving map (a morphism) a W T Œ1† ! gŒ1:

(6.2)

If ea denotes a basis of g and a the linear odd coordinates on gŒ1 corresponding to a dual basis, then the 1-forms Aa Aa # in A D Aa ˝ ea are given by the pullback of a with respect to the map a, Aa D a . a /; here # D d are the degree-one coordinates on T Œ1† induced by (local) coordinates on †. This is easy to generalize, given the background of the previous sections: As our generalized gauge fields we will consider morphisms of graded manifolds a W M1 ! M2

(6.3)

keeping M1 D T Œ1† (so these gauge fields will be a collection of differential forms on † of various form degrees) and taking M2 as a more general N0 -graded manifold than gŒ1. In the example, not only M1 D T Œ1† but also M2 D gŒ1 is not any graded manifold, but even a Q-manifold, Q1 is the de Rham differential and Q2 the Chevalley– Eilenberg differential (5.1). Let us reconsider the field equations F D 0 with F D F a ˝ ea the curvature (1.6) of A in this context. Obviously F a D 0, iff a b A ^ Ac . With the identification of the vector fields Q1 and Q2 above, dAa D Cbc this in turn is now seen to be Q1 a . a / D a Q2 . a /. In other words, the map a W C 1 .M2 / ! C 1 .M1 / needs to be a chain map, or, according to the definition of a Q-morphism, the field equations of the Chern–Simons gauge theory express that

242


(6.3) is not only a morphism of graded manifolds but even a Q-morphism (a morphism of differential graded manifolds). We are thus searching a functional defined on morphisms (6.3) such that its Euler– Lagrange equations forces these to become morphisms a W .M1 ; Q1 / ! .M2 ; Q2 /:

(6.4)

Certainly, for a true generalization of the gauge theory defined by means of (6.1), we also need to reinterpret its gauge transformations appropriately, so that we can formulate also the desiderata for the gauge symmetries of the searched for action functional. It turns out that on the solutions of (6.4) the gauge symmetries receive the interpretation of Q-homotopy.19 We are thus searching for a functional defined on (6.3) such that the moduli space of classical solutions modulo gauge transformations is the space of Q-morphisms from .T Œ1†; d/ to the target Q-manifold .M2 ; Q2 / modulo Q-homotopy. A functional can be obtained by the so-called AKSZ-method [1] (cf. also [7] and [31]) for the case that the target carries also a compatible symplectic form, i.e., that the target is a PQ-manifold. However, this method yields in fact already the BV extension of the searched-for (“classical”) functional. We thus briefly recall some basic ingredients of the BV formalism and we will do this at the example of a toy model, where we consider functions instead of functionals, as well as for the Chern– Simons theory (6.1) above. The toy model is the following one: Consider a function Scl on a manifold M invariant with respect to the action of some Lie algebra g: v.Scl /.x/ D 0 for all points x 2 M and all elements v 2 .TM / corresponding to the action of an element of g. We learnt that M g carries the structure of a Lie algebroid, the action Lie algebroid. Let us thus, more generally, consider a Lie algebroid E ! M together with a function Scl 2 C 1 .M / constant along the Lie algebroid orbits on M , i.e., such that . /Scl D 0 for all 2 .E/. Scl .x/ is supposed to mimic a functional invariant w.r.t. some gauge transformations (in the more general sense, cf., e.g., [19], also for general details on the BV formalism). Consider for simplicity first the case of the action Lie algebroid again. The gauge invariance here corresponds to ı Scl D 0 where ı x i D a ai .x/ with some arbitrary g-valued parameters D a ea (and .ea / D ai .x/ @x@ i ). In the BRST–BV approach one first replaces the parameters a by anticommuting variables a so that the action of the BRST operator ı on the original, classical fields (coordinates here) becomes ıx i D a ai .x/. This is completed by the action of ı on the “odd parameters” a , a b c ı a D 12 Cbc rendering ı nilpotent, ı 2 D 0. In fact, this BRST operator is evidently nothing but the operator Q defined on EŒ1, ı Q, cf. equation (5.2), and this works for a general Lie algebroid E with .ea / D ai .x/ 19 We

refer to [4] for the details.

@ ; @x i

c Œea ; eb D Cab .x/ec ;

(6.5)


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c where now ea is a local frame of sections of E and Cab thus became structure functions (instead of just structure constants). To obtain the BV-form of the action, one now turns to the (graded) phase space version of this, i.e., one introduces momenta (shifted in degree, cf. below) for each of the fields (coordinates on EŒ1 in our example), which, conventionally, are called the antifields. In our example xi conjugate to x i and a conjugate to a . Adding the Hamiltonian lift of ı to the classical action we obtain c SBV D Scl .x/ C a ai .x/xi 12 Cab .x/ a b c :

(6.6)

To have this to have a uniform total degree, we need to shift the momenta in degree by minus one, deg xi D 1, deg a D 2. Thus the BV odd phase space we are looking at together with the BV-function are of the form MBV D T Œ1EŒ1; !BV D dxi ^ dx i C da ^ d a ; SBV D Scl C Q; (6.7) where Q is the Hamiltonian of the canonical Hamiltonian lift of the vector field Q of the Lie algebroid EŒ1, the odd Poisson bracket, the so-called BV-bracket f; gBV has degree +1, so that QBV D fSBV ; gBV has degree C1 as well. QBV is a differential, i.e., SBV satisfies the so-called classical master equation fSBV ; SBV gBV D 0;

(6.8)

which is completely obvious from our perspective: Q Poisson commutes with itself since it is the Hamiltonian for the Lie algebroid differential Q (there are no “odd constants”, fQ; QgBV having degree 1), Scl .x/ Poisson commutes with itself since it depends on coordinates only, and fScl .x/; Qg D 0 since it corresponds to the Lie algebroid action on Scl , which is zero by assumption. So, .MBV ; !BV ; QBV D .SBV ; /BV / defines a PQ-manifold. In contrast to the previous Z-graded Q-manifolds, this PQ-manifold also has negative degree generators. In fact, it is a cotangent bundle of an N0 -graded Q-manifold with a shift in the cotangent coordinate degrees such that it is precisely the momenta (antifields) that have negative degrees. The total degree is called the ghost number conventionally; so the classical fields (coordinates x in the toy model) have ghost number zero, the “odd gauge parameters” a , the ghosts, have ghost number one, and the antifields have negative ghost number. We can also just consider the number of momenta or antifields: denoting this number by a subscript, we see that SBV D S0 C S1 here, where S0 D Scl and S1 D Q. In general, the BV formalism is more involved, there can be terms of higher subscript. Still, always S0 is the classical action Scl . Also we see that fxi ; SBV gBV j0 D @Scl . Applying QBV to the classical fields and setting the antifield-less part to zero @x i yields the critical points of Scl , i.e., the classical field equations. We now turn to the BV formulation of the Chern–Simons gauge theory (6.1). As before in (6.6), we add to the classical action (6.1) a term linear in the classical antifields Aa with a coefficient that is the (infinitesimal) gauge transformations, replacing the gauge parameters by odd fields ˇ a (so, naturally, Aa should be a 2-form on †), and we

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complete the expression by a term proportional to the odd (anti)fields ˇa , 3-form on †, containing the structure constants such that the master equation (6.8) is satisfied;20 this yields Z a b c c SCS-BV ŒA; A ; ˇ; ˇ D SCS ŒA C .dˇ a C Cbc A ˇ /Aa C 12 Cab ˇ a ˇ b ˇc (6.9) †

which is in a striking similarity with our toy model (6.6). Remembering that we could rewrite the toy model in a much more elegant form using the Q-language, cf. (6.7), we strive for a similar simplification in the present context. For this purpose we first recall that the quadratic Lie algebra used to define the Chern–Simons theory is a Courant algebroid over a point, which in turn is a degree-2 PQ-manifold over a point (cf. Theorem 5.1): M2 D gŒ1;

! D 12 ab d a ^ d b ;

Q D 16 Cabc a b c :

(6.10)

This is the target of the map (6.3), with the source, M1 D T Œ1†, being a Q-manifold (Q1 D d). The map (6.3) corresponds to the classical fields A, which we amended with further fields ˇ; ˇ ; A above. It is tempting to collect all these fields together into a super field (indices were raised by means of ) Aa D ˇ a C Aa C Aa C ˇa

(6.11)

by adding them up with increasing form degrees. In fact, this corresponds to an extension of the morphism (6.3) to what is called a map aQ from M1 to M2 . But before commenting on this extension on this more abstract level, we want to first convince ourselves that the concrete expression (6.11) is useful. Let us consider the action (6.1) simply replacing A by A – and clearly keeping only the top degree forms for the integration over †. Viewing differential forms on † as graded functions on T Œ1†, we can also write this as a Berezin integral over that graded manifold and we will partially do so in what follows. Let us consider the first part of this action first: Z 1 Ssource ŒA D .A ^; dA/ 2 T Œ1† (6.12) Z 1 D .ˇ; dA / C .A ^; dA/ C .A ^; dˇ/: 2 † We see that taking † to have no boundary for simplicity, with appropriate sign rules (see 6.20, in general), the first and the third term become identical and they reproduce all the terms containing a d in the BV-action above. Similarly, Z Z 1 1 .A ^; ŒA ^; A// Cabc Aa ^ Ab ^ Ac (6.13) Starget ŒA D 6 T Œ1† 6 T Œ1† 20Again, also this example is a relatively simple one for what concerns the BV formalism and the simpler BRST approach would be sufficient to yield the same results. However, already the models generalizing the Chern–Simons theory that we will discuss below, like the Poisson sigma model or the more general AKSZ sigma model, have a more intricate ghost and antifield structure.


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is easily seen to reproduce the remaining terms in (6.9). So we see that SCS-BV ŒA; A ; ˇ; ˇ D SCS ŒA D Ssource ŒA C Starget ŒA

(6.14)

and we will strive at understanding this action in analogy to the toy model situation (6.7). For this purpose we need to identify the BV phase space of this situation. We thus first return to a further discussion of the superfields (6.11). The notion of a smooth map of graded manifolds is an extension of the notion of a morphism:21 aQ 2 Map.M1 ; M2 /;

a 2 Mor.M1 ; M2 / D Map0 .M1 ; M2 /;

P0 aQ D a: (6.15)

In the case of a flat target manifold like gŒ1 the description of a smooth map is rather clear from the example (6.11): we (formally) allow the functional dependence of the target coordinates on arbitrary degrees on the source. Map.M1 ; M2 / is naturally graded: The coefficients in the expansion like (6.11) are coordinates on this map space. Since the coordinates on the target M2 D gŒ1 have degree 1, each term in the expansion (6.11) has degree 1 as well. Correspondingly, the ghost ˇ a has degree 1, Aa . /, the coefficient in Aa D Aa # has degree 0 (since # D d , coordinates on the source T Œ1†, have degree 1), .Aa / . /, the coefficients of the 2-form field Aa , are fields of degree 1, etc. A field (or antifield) is the same as a coordinate on Map.M1 ; M2 /. Its degree-zero part Map0 .M1 ; M2 / is the space of morphisms or the space of the original classical maps a, which correspond to the (classical) fields Aa . /. a results from aQ by projection (formal operator P0 in (6.15)) to its degree-zero part, which are those maps that are degree-preserving: in the example (6.11) this is keeping the second term. Generally, the space of maps between graded manifolds M1 and M2 , denoted as Map.M1 ; M2 /, is uniquely determined by the functorial property (cf. for example [13] or also [32]): for any graded manifold Z and a morphism W Z M1 ! M2 there exists a morphism Q W Z ! Map.M1 ; M2 / such that D e v B . Q Id/, where the evaluation map e v is (formally) defined in the obvious way: e v W Map.M1 ; M2 / M1 ! M2 ; .a; Q Q / 7! a. Q Q /:

(6.16)

Though the map aQ underlying (6.11) is not a morphism of graded manifolds (it is does not even induce an ungraded morphism of the associative algebras of functions!), the evaluation map e v is. So the BV-phase space is MCS-BV D Map.T Œ1†; gŒ1/, or, more generally, MBV D Map.M1 ; M2 /. It is canonically an odd (infinite dimensional, weakly) symplectic manifold. The symplectic form !BV is induced by means of the one on the 21 Strictly speaking, the first and consequently the third formula do not make sense: Map.M ; M / turns out 1 2 to be an infinite dimensional graded manifold and, as any graded manifold, only its degree-zero part, the body, contains points; a graded manifold, like a supermanifold, is not even a set. Still, it is useful to think like this; like everything in supergeometry, the real definitions are to be given algebraically on the dual level.

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target, equation (6.10): !BV D

Z T Œ1†

e v ! D

1 2

Z T Œ1†

ab ıAa ^ ıAb ;

(6.17)

where ı denotes the de Rahm differential on MBV (so as to clearly distinguish it from the vector field d on M1 !). Note that the pullback by means of the evaluation map (6.16) of the 2-form ! produces a (highly degenerate) 2-form on MBV M1 , which, moreover, still has degree 2 since e v is a morphism, as remarked above. The Berezin integration over T Œ1† then reduces the degree of the resulting differential form by 3 (since † is 3-dimensional), so that !BV has degree 1, such as in our toy model (6.7). We could write out the right hand side of (6.17) similarly to (6.12); this then makes it clear that, after the integration, the resulting 2-form is indeed (weakly) non-degenerate. Let us stress at this point that certainly this construction R would not work when sticking to the purely classical fields A: A likewise expression † .ıA ^; ıA/ would be nonzero only for a surface † of dimension two – in fact, this then is the symplectic form of the classical phase space of the Chern–Simons theory. So, also at this point the extension from a 2 Mor.M1 ; M2 / to aQ 2 Map.M1 ; M2 / MBV is essential, because only the latter space is naturally symplectic, and indeed the symplectic form has degree 1 so that the BV-bracket will have degree C1, as it should be. From the above the first steps in the generalization of the Chern–Simons BVaction to a more general setting is clear: We will keep .T Œ1†; d/ as our source Q-manifold .M1 ; Q1 /, with † having a dimension d different from three in general. As target we need to choose at least a symplectic graded manifold, but in fact, like in our guiding example (6.10), we will consider a PQ-manifold .M2 ; !2 ; Q2 /, of degree n in general (for n > 0 we can also replace the symplectic R vector field Q2 by its generating Hamiltonian function Q2 of degree n C 1). Now T Œ1† e v !2 gives a degree-.1/ symplectic 2-form on MBV D Map.M1 ; M2 /, iff d D n C 1 (since the Berezin integration reduces the degree of the 2-form by d ). It remains to rewrite the action (6.9) or (6.14) in a form that will resemble somewhat the BV-function of the toymodel (6.7). In particular, according to our assumptions on source M1 D T Œ1† and target (6.10), we are having a vector field Q1 D d and Q2 D fQ; g on the source and the target, respectively. Both of them give rise to a vector field on MBV D Map.M1 ; M2 /:22 Identifying the tangent space at aQ 2 MBV with .M1 ; aQ T M2 /, the two vector fields give rise to aQ Q1

and

Q2 B a; Q

(6.18)

respectively. These two vector fields on MBV , both of degree 1, are (graded) commuting, as acting on the right and the left of the map a. Q Their difference fQ WD aQ Q1 Q2 B aQ

(6.19)

is nothing but the BV Operator QBV . It turns out that both vector fields are Hamiltonian with respect to the symplectic form (6.17), with the Hamiltonians being given (6.12) 22 Details

for the remaining part of the paragraph and the following one can be found in [1], [7], and [31].


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and (6.13), respectively. In the spirit of (6.7), we can now also bring the BV-action into the form Z {d e v .˛/ C .1/d e v Q; (6.20) SBV-AKSZ D T Œ1†

which defines the general AKSZ sigma model. Here d is the dimension of †, in the example of the Chern–Simons theory thus d D 3 and Q is the Hamiltonian (6.10). In the first term ˛ is a primitive of !, ! D d˛ and {d denotes the contraction with Q1 D d. With † having no boundary, the action (6.20) is independent of the choice of ˛. If q ˛ denote Darboux coordinates of the target PQ-manifold, which, as mentioned, has degree d 1, i.e., ! D 12 !˛ˇ dq ˛ ^ dq ˇ with !ab being constants, we can choose ˛ D 12 !˛ˇ q ˛ ^ dq ˇ . Let A˛ D aQ .q ˛ /. Then we can (somewhat formally) rewrite the AKSZ action “evaluated” at aQ more explicitly as Z 1 ! A˛ dAˇ C .1/d aQ Q : (6.21) SBV-AKSZ ŒA D 2 ˛ˇ T Œ1†

In this form it is very easy to see that we reproduce from this the Chern–Simons theory in its form (6.14) upon the choice (6.13) together with d D 3. In general, the AKSZ sigma model is defined for a degree d 1 PQ-manifold on a d -dimensional base. The classical action results from the BV form of it simply by replacing aQ by its degree-zero part a, i.e., with A˛ D a q ˛ Z 1 SAKSZ ŒA D ! A˛ dAˇ C .1/d a Q: (6.22) 2 ˛ˇ T Œ1†

Again, in the Chern–Simons case we easily find the classical action (6.1) reproduced. We can, however, now also parameterize this sigma model more explicitly by means of the considerations in Section 5 for the lowest dimensions of †: For d D 2, we need to regard degree 1 PQ-manifolds, which we had found to be always of the form M2 D T Œ1M;

! D dpi ^ dx i ;

Q D 12 …ij pi pj

(6.23)

with … a Poisson bivector (cf. Example 3.3); i.e., the target data are uniquely determined by a Poisson manifold .M; …/. The most general AKSZ sigma model for d D 2 is thus seen to be the Poisson sigma model [36], [21]: Z Ai ^ dX i C 12 …ij .X /Ai ^ Aj (6.24) SPSM ŒX i ; Ai D †

i

i

where X D a x are 0-forms on †, Ai D a .pi / 1-forms, and we used a more standard notation of integration over the (orientable) base manifold †. For d D 3 we see that we get a (topological) sigma model for any Courant algebroid, cf. Theorem 5.1, and the corresponding sigma model is easily specified by means of (5.5) and (5.6). We call it the Courant sigma model. It was obtained first by Ikeda in [22] and later by Roytenberg more elegantly by the present method [31].

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For d D 4 the geometrical setting of the target, a degree-three PQ-manifold, has not yet been worked out in detail or given any name. But again we can write down the explicit sigma model in this case using (5.9) and (5.10). How we presented the AKSZ sigma model, its main purpose is to find a topological action functional such that its classical field equations are precisely Q-morphisms, (6.4). While this was proven for the Poisson sigma model explicitly in [4], the present formalism permits a short general proof when using that (6.20) is the Hamiltonian for (6.17) (alternatively one may also directly perform a variation of (6.22)). As the comparison of (6.22) with (6.21) shows, the difference between the classical action and its BV extension is that we merely have to perform the replacement (6.4) to go from one to the other. This is a very specific feature of the present topological models, the BV extension is usually not that simple to obtain for a general gauge theory; here, however, it works like this as we saw above. The BV-AKSZ functional (6.21) becomes stationary precisely when (6.19) vanishes (since this is its Hamiltonian vector field and the symplectic form is non-degenerate). Correspondingly, the variation of (6.22) results into the same equation, but where aQ is replaced by a, so the Euler–Lagrange equations are equivalent to the vanishing of f W M1 ! T Œ1M2 ;

f WD a Q1 Q2 B a;

(6.25)

where we shifted the degree of the tangent bundle to the target so as to have f being degree-preserving like a. This in turn is tantamount to the chain property of a (cf. also Lemma 8.1 below). We finally remark that also Dirac structures and generalized complex structures can be formulated with profit into the language of super geometry (the former ones as particular Lagrangian Q-submanifolds in the degree-two PQ-manifold T Œ2T Œ1M ), which in part can be used also to formulate particular sigma models for them within the present scheme not addressed in the present exposition (and different from those of the following section). We refer the reader for example to [16] and [8].

7 Sigma models related to Dirac structures As we have mentioned before, a Poisson manifold gives a particular example of a Dirac structure, determined by the graph of the corresponding bivector in TM ˚ T M . Similar to the Poisson sigma model (PSM), the target space of which is a Poisson manifold, we now want to consider a topological sigma model associated to any Dirac subbundle of an exact Courant algebroid. This Dirac sigma model (DSM) [25] is supposed to be at least equivalent to the PSM for the special choice of a Dirac structure that is the graph of a Poisson bivector. Also, we want to continue pursuing our strategy that its classical field equations should be appropriate morphisms. In fact, in lack of a good notion of a morphism of a Dirac structure, we will content ourselves with a Q-morphism again, i.e., a Lie algebroid morphism in this case (since any Dirac structure is in particular a Lie algebroid structure). The model will be 2-dimensional.


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As we saw in the previous section that the most general 2-dimensional model obtained by the AKSZ scheme is the PSM, the DSM does not result from this method, at least not by its direct application. The target space of the Dirac sigma model is a manifold together with a Dirac structure D in an exact Courant algebroid twisted by a closed 3-form H . The space-time is a 2-dimensional surface †. We need also some auxiliary structures – a Riemannian metric g on M and a Lorentzian metric h on †. A classical field of the DSM is a bundle map T † ! D. First this corresponds to the base map X W † ! M , which is corresponds to a collection of “scalar fields”. Taking into account that D TM ˚ T M , we represent the remaining field content by a couple of sections V 2 1 .†; X TM /, satisfying the constraint that this couple combines into a section of T † ˝ X D. Here p .N; E/ is by definition .ƒp T N ˝ E/ for any smooth manifold N and vector bundle E ! N . The DSM action is the sum of two terms, one using the auxiliary geometrical structures and one that uses the topological structures only: SDSM ŒX; V; A D Sgeom C Stop , where Z ˛ Sgeom ´ kdX V k2 ; (7.1) 2 Z † Z ^ 1 Stop ´ H: (7.2) hA ; dX 2 V i C †

N3

Here ˛ is a real (non-vanishing, in general) constant23 , the absolute value in the first term corresponds to the canonical pairing induced by h˝g on T †˝TM ,24 the brackets h; i denote the pairing between TM and T M , and the last term in the topological part of the action is the integral of H over an arbitrary map N 3 ! M for @N 3 D † which extends X W † ! M (here we assumed for simplicity that X is homotopically trivial25 , which is e.g. always the case if the second homotopy group of M is zero). This sigma model generalizes the G/G Wess–Zumino–Witten (GWZW) [41], [15] and (H -twisted) Poisson sigma model, simultaneously. We first comment on the relation to the PSM. Let us first choose H D 0 (cf. Example 3.3) and ˛ D 0 (yielding Sgeom 0); with D being the graph of a bivector …, we have V D X ….A; / and comparison with (6.24) shows that we indeed have SDSM D SPSM in that case. For H non-zero we get the twisted version of the Dirac structure of Example 3.3 and by means of equation (7.2) of the corresponding sigma model [23], respectively. (We will comment on non-zero ˛ below). The GWZW model results from a special choice of the Dirac structure in an exact Courant algebroid on a quadratic Lie group M D G, H is the Cartan 3-form, g the biinvariant Riemannian metric on G, and ˛ D 1. In the description of a Dirac structure by an orthogonal operator S 2 End.TM / once a 23 The Lorentzian signature of h is chosen for simplicity. In the Riemannian version of the Dirac sigma model the coupling constant ˛ has to be totally complex, that is, ˛ 2 i R. 24 More explicitely this expression was defined after formula (1.1). 25 For less a topologically less restrictive setting, one permits action functionals up to integer multiplies of 2„ – here we refer to the literature on Wess–Zumino terms, cf. [25] and references therein.

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metric g has been fixed on M , cf. Corollary 3.9, the Dirac structure on G is the one given by the adjoint action S D Adg , where g 2 M is the respective base point. The GWZW model is in so far an important special case as it is not only a well known model in string theory (cf., e.g., [15]), but it is also known to be topological, despite the appearance of auxiliary structures needed to define it. Also, its Dirac structure can be shown to not be a graph, e.g. by employing the characteristic classes described in Section 3. Part of the topological nature for the general DSM can be already verified on the level of the classical field equations: Do they depend on the auxiliary structures like g or h? Here we cite the following result from [25]: Theorem 7.1. Let ˛ ¤ 0, then a field .X; V; A/ is a solution of the equations of motion, if and only if the corresponding bundle map T † ! D induces a Lie algebroid morphism. Since the notion of a Lie algebroid morphism does not depend on auxiliary structures as those mentioned above, we see that at least this condition is satisfied for nonvanishing ˛. The proof of this theorem is somewhat lengthy,26 so that we do not want to reproduce it here; instead we want to prove it for the simplest possible Dirac structure, D D TM , Example 2.2 above, and refer for the general fact to [25]. We start by calculating the field equations of the sigma model; but for D D TM and H D 0 (cf. Example 2.2) the topological part of the action is identically zero. So, it remains to look at the variation of (7.1). Since V is an independent (unconstrained) field in this case, the variation of the quadratic term w.r.t. V yields dX D V

(7.3)

while the X-variation vanishes on behalf of that equation. Mathematically, this equation is tantamount to saying that the vector bundle morphism a W T † ! TM is the push forward of a map X W † ! M , a D X . We obtain the required statement in this special case by use of the following Lemma 7.2. A Lie algebroid morphism from the standard Lie algebroid over a manifold † to the standard Lie algebroid over a manifold M is the push forward of a smooth map X W † ! M . Proof. Recall from Section 5 that the Lie algebra morphism above can be defined best by a degree-preserving map aN W T Œ1† ! T Œ1M such that aN W C 1 .T Œ1M / ! C 1 .T Œ1†/ commutes with the respective differentials characterizing the Lie algebroid, here being just the respective de Rahm differentials. Let us choose local coordinates ; # D d and x i ; i D dx i , respectively. Then X i D aN x i corresponds to 26 One of the complications is that the fields A and V are not independent from one another in general and this has to be taken care of when performing the variations. One way of doing that is by using the orthogonal operator S mentioned in Corollary 3.9, which permits us to express these two fields by means of an independent W 2 1 .†; X TM / according to A D .1 C S /W and V D .1 S /W .


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the base map X of the lemma. On the other hand V i WD aN i D daN x i D dX i , where in the second equality we used that aN commutes with d. V i D dX i or, equivalently, Vi D X i ; is the searched-for equation. From the example we also see that ˛ ¤ 0 is a necessary condition for the theorem to hold. Were ˛ D 0 in that special case, there were no field equations and the vector bundle morphism a W T † ! TM were unrestricted and in general not a Lie algebroid morphism. On the other hand, it is not difficult to see (but it certainly also follows from the theorem) that for the (possibly H -twisted) PSM the field equations do not change when adding the term Sgeom with some non-vanishing ˛. We thus propose to consider SDSM for non-vanishing ˛ in general.27 There are two more important issues which we need to at least mention in this context. First, if the DSM is to be associated to a Dirac structure on M and all other structures used in the definition of the functional are to be auxiliary, one needs to show that it is in the end the cohomology class of H only that enters the theory effectively. In [25] we proved Proposition 7.3. The DSM action transforms under a change of splitting (2.17) according to Z (7.4) SDSM 7! SDSM C X .B/.dX V ^; dX V /: †

Let ˛ ¤ 0 and B be a “sufficiently small” 2-form, then there exists a change of variables Vx D V C ıV and AN D A C ıA, such that Z x N SDSM ŒX; V ; A D SDSM ŒX; V; A C X .B/.dX V ^; dX V /: †

So, a change of the splitting can be compensated for by a (local) diffeomorphism on the space of fields. Secondly, we did not yet touch the issue of the gauge symmetries, neither in the previous section on the AKSZ sigma models nor for the DSMs. Certainly the gauge symmetries are of utmost importance in topological field theories (without them we were never able to arrive at a finite dimensional moduli space of solutions for instance). While the BV formalism produces them by means of the BV operator for the AKSZ sigma models (although possibly in a coordinate dependent way, cf., e.g., [4] addressing this issue), they are less obvious to find for the DSM. In fact, here also all the auxiliary structures do enter, cf. [25] for the corresponding formulas. It is only onshell, i.e., on using the field equations (here only dX D V ), that the gauge 27 In [25] it is conjectured that the theory with ˛ D 0 is (essentially) equivalent to the theory with ˛ ¤ 0 in general. E.g. for D D TM , and ˛ ¤ 0 the moduli space of solutions to the field equations (maps X W † ! M ) up to gauge symmetries (which at least contain the homotopies of this map X ) is 0-dimensional, like the moduli space for the vanishing action ˛ D 0. It is argued that the geometrical part serves as a kind of regulator for the general theory, which also should permit localization techniques on the quantum level.

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transformations obtain a nice geometrical interpretation: they turn out to become “Lie algebroid homotopies” (cf. also [4]) in this case. We do, however, not want to go into further details on this here; in the present work we decided to focus more on the field equations, reassuring the reader in words that the more intricate gauge symmetries fit nicely into the picture as well, completing it in an essential way.

8 Yang–Mills type sigma models In the Introduction we recapitulated the idea of sigma models: one wants to replace the flat target space Rn of usually a collection of n scalar fields (functions on spacetime †) by some geometrical object, like a Riemannian manifold. We posed the question, if, in the context of gauge fields (1-forms on space time), we can replace in a likewise fashion the “flat” Lie algebra Rn (or, more generally, g) by some nontrivially curved geometrical object. In fact, the Poisson sigma model (6.24), or more generally, the AKSZ sigma model (6.20), provides, in some sense, half a step into this direction: Let us consider the 3-dimensional case, where this is most evident. The quadratic Lie algebra g needed for the definition of the Chern–Simons gauge theory can be generalized to a Courant algebroid, associated to which is the Courant sigma model [22], [31], which when specializing to the “flat case” g reproduces the Chern–Simons theory. While these models realize in a geometrically nice way the right (classical) field content, the target algebroid being represented by a PQ-manifold and the Lie algebra valued 1-forms of a Yang–Mills theory being interpreted as and generalized to degree-preserving maps from T Œ1† to the respective target, and also the gauge transformations are generalized in a reasonably looking way, there is, from the physical point of view, a major drawback or “flaw” of these theories: What mathematically is usually considered an advantage of a field theory, namely to be topological, in the context of physics rules out a theory for being feasible to describe the degrees of freedom we see realized in the interacting world around us. Indeed, the space of flat connections, which are the field equations of the Chern– Simons theory, modulo gauge transformations is (for, say, † without boundary and of finite genus) a finite dimensional space and this generalizes in a likewise fashion to the moduli space of solutions modulo gauge transformations for all the AKSZ models, where one considers the space of Q-morphisms modulo Q-homotopy. The moduli space needed to host a physical particle (like a photon, electron, Higgs, etc), on the other hand, is always infinite dimensional (like the space of harmonic functions on † for a Laplacian corresponding to a (d–1,1)–signature metric). There is also another way of seeing that one has gone half way only: Such as we want that when in a sigma model for scalar fields the choice of a “flat background” (i.e., in that case, that the target is a flat Riemannian manifold Rn ) the action reduces to (1.2), we want that when in a Yang–Mills type sigma model the target is chosen to be a Lie algebra Rn or, more generally, g, the gauge theory reduces to (1.5) and (1.7), respectively. So we will pose


253

this condition, maybe adding that there should be a “comparable number” of gauge symmetries in the general model as in the special case. In a similar way we may generalize this condition by extending it to also higher form gauge fields B b , namely that for an appropriate flat choice of the target geometry one obtains from the (higher) Yang–Mills type sigma model Z 1 S ŒB b D dB b ^ dB b : (8.1) 2 † For the case that B’s are 2-forms, this will yield an action functional for nonabelian gerbes. Before continuing we bring the standard Yang–Mills action (1.7) into a form closer to a topological model first. We consider a trivial bundle in what follows, in which case the curvature or field strength is a g-valued 2-form. If d denotes the dimension of spacetime †, we introduce a g -valued d 2 form ƒ in addition to the connection 1-forms A. With 1 denoting the scalar product on g induced by on g, then Z 0 (8.2) SYM ŒA; ƒ WD hƒ; F i C 1 .ƒ ^; ƒ/ †

is easy to be seen as equivalent to (1.7) on the classical level (Euler–Lagrange equations) after elimination of the auxiliary field ƒ. (Equivalence on the quantum level follows from a Gaussian integration over the field ƒ, as it enters the action quadratically only – up to an overall factor, which is irrelevant for the present considerations). The first part of this action is topological, it is only the second term, breaking some of the symmetries of the topological one, that renders the theory physical. Let us first generalize the topological part, a so-called BF-theory, by an appropriate reformulation. In fact, we can obtain the BF-theory from the AKSZ-method by considering M2 D T Œd 1gŒ1 as a target PQ-manifold (as before d is the dimension of spacetime †). Indeed, it is canonically a symplectic manifold with the symplectic form being of degree d 1, as it was required to be in Section 6, and the Q-vector field on gŒ1, equation (5.1), can always be lifted canonically to the cotangent bundle with the following Hamiltonian 28 a b c pa : Q D .1/d 21 Cbc

The AKSZ action (6.22) now just produces Z a b ƒa .dAa C 12 Cbc A ^ Ac / SBF Œƒ; A D

(8.3)

(8.4)

†

where we used Aa D a . a / and ƒa D a .pa /; and this is just equal to the first term in (8.2). It is now easy to find a generalization of this part of the action for a general z2; Q z 2 as a target z 2 /: Just consider M2 D T Œd 1M (N0 -graded) Q-manifold .M 28 We

form.

b

use the following sign convention: fpa ; q b g D .1/mjq j ıab , where m is the degree of a symplectic

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z 2 to the Hamiltonian vector field Q2 (with PQ-manifold with the canonical lift of Q Hamiltonian Q2 D Q). z 2 D EŒ1 and Let us exemplify this at the example of a Lie algebroid, where M c i i i z Q2 is given by means of formula (5.2) (with a D a @=@x / and a .x/, Cab .x/ being i a the Lie algebroid structure functions in the local coordinates x ; on EŒ1). Now the same procedure yields the action functional for a “Lie algebroid BF-theory” [38], [39], [5]

i

a

SLABF Œƒi ; ƒa ; X ; A D

Z †

.ƒi ^ F i C ƒa ^ F a /

(8.5)

where ƒi and ƒa are .d 1/-forms and .d 2/-forms, respectively, and F i D dX i ai .X /Aa

(8.6)

a .X /Ab ^ Ac F a D dAa C 12 Cbc

(8.7)

are the generalizations of the YM-curvatures, which we prefer to call field strengths in the general case. We use the notation A˛ D a .q ˛ /, where q ˛ are graded coordinates z 2 and ƒ˛ D a .p˛ /, the corresponding momenta (or anti-fields) on M2 D on M z 2 . Note that the map a W T Œ1† ! M2 is degree-preserving, so that X i W T Œ1M i A D a .x i / are 0-forms on † or scalar fields, Aa D a . a / are 1-forms, and ƒi and ƒa are .d 1/- and .d 2/-forms, respectively. We remark in parenthesis that although (8.5) is inherently coordinate and frame independent, the expression (8.7) is not. It is the splitting of the field strength(s) into two independent parts (8.6) and (8.7), which cannot be performed canonically. One way of curing this is by introducing an additional connection on the (target) Lie algebroid E, cf., e.g., [4], [39]. Better is to realize that the whole information is captured by the map (6.25), taking values in the tangent bundle over the target z 2 . We now turn to this perspective. M2 D T M The map f defined in (6.25) covers the map a W M1 ! M2 (it is also this fact that makes the difference in (6.25) well-defined, being the difference between two elements in the fiber, which is a vector space, over the same point)

T Œ1M2

> || f ||| || || | | / M2 M1 a


255

and measures the deviation of a to be a Q-morphism, in which case the following diagram would commute: a

T Œ1M O 1

/ T Œ1M2 O Q2

Q1

M1

a

/ M2 .

z 2 and by Denote by x A D .q ˛ ; p˛ / the graded coordinates on M2 D T Œd 1M A dx the induced fiber-linear coordinates on T Œ1M2 . To use the map f in practise, the following lemma is helpful: Lemma 8.1. Let .x A ; dx A / be local coordinates on T Œ1M2 , a 2 Mor.T Œ1†; M2 /, and f the map defined in (6.25). Then f .x A / D a .x A /;

f .dx A / D .da a Q2 /x A :

(8.8)

Proof. The first part is evident from f covering a as remarked above already. The second part is rather straightforward and we show only one first step in the calculation. The vector field Q2 of degree 1 is a degree-preserving section of the tangent bundle over M2 , if the latter is shifted by 1 in degree, i.e., as here, one considers T Œ1M2 . The section maps a point with coordinates x A to the point .x A ; Q2A .x//. Thus we find for the pull-back of dx A by this section: .Q2 / dx A D Q2A .x/, which, however, can also be rewritten as Q2 applied to x A . The remaining similar steps we leave as an exercise to the reader. We just remark that we had used Q1 D dd and that certainly pull-backs go the reverse order as the respective map, cf the second diagram above. We remark also that dx A is considered as a function on T Œ1M2 and is pulled back as such, not as a differential form on T Œ1M2 , while a function on T Œ1M2 is, on the other hand, equally well a differential form on M2 . In fact, the so-understood pull back will not be a chain map w.r.t. the de Rahm differential (a vector field on T Œ1M2 ), but only w.r.t. a modified vector field on T Œ1M2 , cf. Proposition 8.3 below. We see that the field strengths F i and F a are nothing but the pull back of dx i and a d by f and thus find [40] Corollary 8.2. The action functional (8.5) can be written more compactly as Z f .p˛ dq ˛ /; SLABF Œa D

(8.9)

†

where a W T Œ1† ! M2 D T Œd 1EŒ1 and p˛ dq ˛ is the canonical 1-form on M2 . Now we are in the position to generalize the Yang–Mills action functional (1.7) or better (8.2) to the Lie algebroid setting, i.e., to a situation where the structural Lie algebra g is replaced by a Lie algebroid E (cf. Definition 2.1). A Lie algebroid Yang– Mills theory is defined as a functional on a 2 Mor.T Œ1†; M2 D T Œd 1EŒ1/ and

256


is of the form [39] 0 SLAYM Œa

Z D SLABF Œa C

E 1

g

†

.ƒ.d 2/ ^; ƒ.d 2/ /

(8.10)

where ƒ.d 2/ 2 d 2 .†; X E/ are the Lagrange multiplier fields of the 2-form field strengths only. Equivalently, eliminating precisely those Lagrange multipliers like in the transition from (8.2) to (1.7), we can regard the somewhat more explicit action functional Z SLAYM ŒAa ; X i ; ƒi D ƒi ^ F i C 12 Eg.X /ab F a ^ F b ; (8.11) †

i

a

where F and F are the field strengths (8.6) and (8.7), respectively, detailed above. In a standard YM theory, the metric on the Lie algebra needs to be ad-invariant. Here we can ask that .E; Eg/ define a maximally symmetric E-Riemannian space (cf. Definition 2.10) or that the fiber metric is invariant w.r.t. a representation (2.13) induced by any auxiliary connection on E. (For still less restrictive conditions cf. [29]). This is now easy to generalize to higher form degrees of the gauge field. Let us consider a tower of gauge fields X i ; Aa ; : : : ; B B , which are 0-forms, 1-forms etc, respectively, up to a highest degree p, so B B being p-forms. The role of the structural z 2 ; Q2 /. The gauge fields Lie algebra will now be played by a degree-p Q-manifold .M z 2 . This is extended to are collected into a (degree-preserving) map a W T Œ1† ! M z a 2 Mor.T Œ1†; M2 / with M2 D T Œd1M2 the corresponding PQ-manifold with canonical 1-form p˛ dq ˛ . We then consider [40] Z Z 0 ˛ V 1 Shigher YM Œa D f .p˛ dq / C g .ƒ.d p1/ ^; ƒ.d p1/ / (8.12) †

†

where V g is a fiber metric on the vector bundle V ! M corresponding to the degree-p variables on M2 (in fact, there is a canonical quotient of M2 yielding V Œp). Similarly to before we can also eliminate the lowest form degree Lagrange multipliers ƒ.d p1/ / 0 in Shigher YM to obtain Shigher YM Œa in which the highest form degree field strength is squared by means of the fiber metric V g. There is always a condition on this fiber metric generalizing the ad-invariance in the standard YM case. Let us specify this condition in the case p D 2. We were mentioning in Section 5 that a degree-two Q-manifold corresponds to a Lie 2-algebroid, cf. Definition 2.13 and the ensuing text. In this case the condition to be placed on V g is its invariance w.r.t. the E-connection E r on V [40]: E

r V g D 0:

(8.13)

For p D 2 this is a possible definition of an action functional for nonabelian gerbes. In this contribution we never discussed the gauge transformations in any detail. In a gauge theory they are certainly of utmost importance and much further motivation for the theories discussed here come from looking at their gauge invariance also.


257

We make here only two general remarks on the gauge transformations: First, for a general algebroid type gauge theory the gauge symmetries have a “generic part” that comes from the structural algebroid (possibly including some particularly important geometrical ingredients from the target like a compatible symplectic form), but there can be also contributions to them that show all possible structural ingredients used in the definition of the action functional. An (impressive) example for the latter scenario is provided by the Dirac sigma model, cf. Section 7 and [25]. Second, for the higherYang–Mills action functionals proposed above, or also any other physical gauge theory proposed in this setting, the BV-formulation, which also captures elegantly the gauge symmetries of a theory, will not just consist in replacing the morphisms a by supermaps aQ as it was the case with the AKSZ type sigma models. In fact, the space Map.T Œ1†; M2 / is too big: It contains ghosts and ghosts for ghosts etc for gauge symmetries of the BF-part of the action that are broken by the fiber metric part. It is tempting, however, to try to obtain the BV-formulation of the full physical model by a kind of supersymplectic reduction induced by the symmetry breaking terms. In the present chapter we used the framework of the AKSZ sigma models to develop the field content of the higher gauge theories and, by the material at hand from the previous sections, it was natural to require the structural Lie algebra to be replaced by a general Q-manifold. This is in so far a physical approach as that the notion of Q-manifolds arose in a physical context, namely to describe the supergeometric framework underlying the BV-formulation [37]. In this way, the generalization of the connection 1-forms of a Yang–Mills theory to the field content considered sounds much less compulsory than it really is. As mentioned in the introduction, physical considerations often suggest geometrical notions and generalizations into very particular directions. This is also the case in the context of “higher gauge theories” – that is gauge theories where the field content is required to consist locally of a tower of differential forms up to some highest degree p. It is evident that then locally we can introduce (in a canonical way) a graded manifold M2 of degree p and interpret the field content as a morphism from T Œ1† to M2 . Even without yet considering gauge transformations acting on these fields, there are sensible requirements on the properties of the theory related to a generalization of the Bianchi identities in ordinary gauge theories that require that the target M2 has to be a Q-manifold. This approach would have fitted the logic of the present chapter very well, too, but it would have extended its length further as well so that we decided to not reproduce these considerations, which were presented in [40] and can be found in [17]. Up to now we only considered the generalization of gauge theories in the context of trivial bundles. In principle, it is not at all clear that the local construction generalizes in a straightforward way to a global one. But fortunately here this is indeed the case. We briefly sketch how this works, since, from the mathematical point of view, much of the interest in such gauge theories would precisely stem from the global picture. For further details we have to refer the reader to [26]. We again start from ordinary Yang–Mills theories. There the space of fields are not Lie algebra valued 1-forms on our spacetime †, but rather connections in a principal

258


G-bundle P over †. Associated to each principal bundle, however, there is the socalled Atiyah algebroid, which is a very particular Lie algebroid over †. As a total space it is equal to TP =G, which is canonically a Lie algebroid by use of the de Rahm differential on TP . The image of its anchor map is T †. Shifting the fiber degrees by 1, this becomes a bundle in the category of Q-manifolds T Œ1P =G ! T Œ1†, the typical fiber of which is the Q-manifold gŒ1 (as before, g is the Lie algebra of the structure group G). We call this a Q-bundle. Moreover, a connection on P is in bijection to a section of this bundle viewed as a bundle of graded manifolds (it is the flat connections that are the sections of the Q-bundle). The generalization is obvious now: As total spaces we consider Q-bundles

W .M; Q/ ! .M1 ; Q1 /, where as the base .M1 ; Q1 / we keep .T Œ1†; d/ for simplicity, but as typical fibers we permit any (non-negatively graded) Q-manifold .M2 ; Q2 /. Gauge fields are any degree-preserving maps a W M1 ! M such that

B a D idM1 . Many constructions that were done for gauge fields a W M1 ! M2 , like the definition of the field strength f , can now be repeated without modification by replacing merely M2 by the total space M. The (generic part of the) gauge symmetries receive the nice interpretation of (possibly a graded subgroup of) vertical inner automorphisms of this bundle. And the gauge invariance of action functionals like (8.12) permits to glue local expressions for functionals together to global ones. The global setting certainly also opens gates for studying characteristic classes for the bundles. In particular, there exists a generalization of the Chern–Weil formalism. In the classical setting, a connection on a principal bundle together with an ad-invariant polynomial on the Lie algebra gives a cohomology class on the base manifold, which does not depend on the connection and which vanishes when the bundle is trivial. In the context of the classical construction one considers the so-called Weil algebra W .g/ D S .g / ˝ ƒ.g / with a certain differential QW on it – the connection on P then induces a map from a subcomplex of so-called G-basic elements of W to the de Rahm complex on †. To generalize this construction, one may first observe that W can be identified with differential forms on .gŒ1/, which we can identify with functions on T Œ1gŒ1. This reminds one of the target of the map f , namely when M2 D gŒ1, cf. the first diagram above. Given a general Q-manifold M2 and a degree-preserving map a W M1 ! M2 , can we equip T Œ1M2 with a Q-structure QT M2 , such that f W .M1 ; Q1 / ! .T Œ1M2 ; QT M2 / is always a Q-morphism? The answer is affirmative [26]: Proposition 8.3. The map (6.25) is a Q-morphism, if we equip T Œ1M2 with the Qstructure QT M2 D d C LQ2 , where d is the de Rahm differential on M2 and Lv the Lie derivative w.r.t. a vector field v on M2 , both viewed as degree-1 vector fields on T Œ1M2 : Q1 f D f QT M2 :

(8.14)

Indeed, the canonical lift QT M2 of the differential Q2 to its tangent reproduces the Weil differential QW mentioned above for M2 D gŒ1. One now generalizes also


259

the notion of being basic, where in the general setting a graded gauge or holonomy group G plays an important role. In the end one obtains [26] a generalization of the Chern–Weil map in the form Theorem 8.4. Let W M ! T Œ1† be a Q-bundle with a typical fiber M2 , a holonomy group G , and a a “gauge field”, i.e., a section of (in the graded sense). Then there is a well-defined map in cohomology p .†/; H p ..M2 /G ; QT M2 / ! HdeRahm

(8.15)

which does not depend on homotopies of a. We want to end this section and thus this chapter by commenting on a rather intriguing link of this construction with the topological sigma models we were considering in a previous section, namely Section 6. There is a famous relation of a characteristic class on a principal bundle with a quadratic structure group, the second Chern class or the first Pontryagin class, with the integrand of the Chern–Simons gauge theory: locally “Second Chern form D d Chern–Simons form”. This class results from the Chern–Weil formalism by applying the map to the ad-invariant metric on g, viewed as quadratic polynomial. In Section 5 we learnt that can be viewed as a symplectic form ! of degree 2 on gŒ1. Locally, the above generalized Chern–Weil map is nothing but f . On the other hand, the AKSZ sigma model resulted from choosing a PQ-manifold .M2 ; Q2 ; !/ as a target – and for the Chern–Simons theory it is this ! that corresponds to . We are thus lead to ask for a relation of the AKSZ sigma models with f ! for a general PQ-manifold (that would serve as a typical fiber in a Q-bundle). In Corollary 8.2 we found that the integrand of the AKSZ sigma model can be written as the pullback z 2 with M z2 of the canonical 1-form, if M2 is the cotangent bundle M2 D T Œd 1M 29 ˛ a degree-1 Q-manifold. We thus compute df .p˛ dq / D f .d C LQ2 /p˛ dq ˛ , where we used the chain property (8.14). But the canonical 1-form is invariant w.r.t. the z 2 ; so only the first term remains z 2 on its base M Hamiltonian lift Q2 of a vector field Q on the r.h.s., which indeed has the form f !. In fact, this statement that we proved for M2 D T Œd 1EŒ1, is true for any PQ-manifold; one has [26] Theorem 8.5. Consider a PQ-manifold M2 with symplectic form ! of positive degree d 1 and a .d C 1/-dimensional manifold N with boundary @N D †, and a 2 Mor.T Œ1N; M2 /. Then Z f ! D SAKSZ Œa; (8.16) N

the AKSZ sigma model (6.22) on †. fact, this is true for a general Q-manifold, as one can prove directly in generalization of Corollary 8.2 or deduce from Theorem 8.5 below. 29 In

260


As mentioned, these observations generalize the relation of the Pontryagin class to the topological Chern–Simons theory to a much wider range of bundles and topological models. Certainly, in some cases Q-manifolds and Q-bundles may be “integrable” (already not every Lie algebroid is integrable to a Lie groupoid – cf. [11] for the necessary and sufficient conditions for the integrability of Lie algebroids). Similar to the relation of Atiyah algebroids to principal bundels, integrable Q-bundles will correspond to rather intricate other type of bundles – in the case of the Lie algebroid Yang–Mills theories, e.g., these are bundles with typical fiber being a groupoid, in the case of the fibers being degree-2 Q-manifolds over a point these will be the nonabelian bundle gerbes of [6]. What we called gauge fields and field strengths and others might call “connections” and “curvatures”, respectively, are often rather intricate to describe in the integrated pictures – in contrast to what happens in the Q-bundle setting. It is our impression that the Q-bundle picture has several advantages in the description of higher gauge theories. Also, they exist and give rise to potentially interesting action functionals even if the underlying Q-manifolds are not integrable; a topological example of this kind is the Poisson sigma model, which exists and has its merits even if the target Poisson manifold M (the Lie algebroid T M ) is not integrable. In any case, we hope to have convinced the reader who was not yet familiar with all the concepts used in this chapter, that algebroids of various kinds are interesting as mathematical objects on the one hand, interpolating in a novel way between various established notions like tensor fields on manifolds and Lie algebras or symplectic and complex geometry, and, on the other hand, that they appear in many different facets in the context of sigma models and higher gauge theories, in fact in such a way that they have to be considered indispensable for an elegant and technically and conceptually convincing discussion of such theories. Acknowledgement. We are deeply indebted to V. Cortés for his incredible patience with us in finishing this contribution.

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[25] A. Kotov, P. Schaller, and T. Strobl, Dirac sigma models. Comm. Math. Phys. 260 (2005), no. 2, 455–480. 212, 226, 248, 249, 250, 251, 257 [26] A. Kotov and T. Strobl, Characteristic classes associated to Q-bundles. Preprint arXiv:0711.4106v1 [math.DG] 257, 258, 259 [27] U. Lindström, R. Minasian, A. Tomasiello, and M. Zabzine, Generalized complex manifolds and supersymmetry. Comm. Math. Phys. 257 (2005), no. 1, 235–256. 230 [28] Z.-J. Liu, A. Weinstein, and P. Xu, Manin triples for Lie bialgebroids. J. Differential Geom. 45 (1997), 547–574. [29] C. Mayer and T. Strobl, Lie algebroid Yang-Mills with matter fields. J. Geom. Phys. 59 (2009), 1613–1623. 256 [30] J.-S. Park, Topological open p-branes. In Symplectic geometry and mirror symmetry (Seoul, 2000), World Scientific Publishing, River Edge, NJ, 2001, 311–384. 210 [31] D. Roytenberg, AKSZ-BV formalism and Courant algebroid-induced topological field theories. Lett. Math. Phys. 79 (2007), 143–159. 212, 242, 246, 247, 252 [32] D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids. In Quantization, Poisson brackets and beyond (Manchester, 2001), Contemp. Math. 315, Amer. Math. Soc., Providence, RI, 2002, 169–185. 216, 236, 237, 245 [33] P. Ševera, Letters to A. Weinstein. Unpublished, 1998. 223 [34] P. Ševera and A. Weinstein, Poisson geometry with a 3-form background. In Noncommutative geometry and string theory (Yokohama, 2001), Progr. Theoret. Phys. Suppl. No. 144 (2001), Progress of Theoretical Physics, Kyoto 2001, 145–154. 210, 219 [35] P. Schaller and T. Strobl, Quantization of field theories generalizing gravity Yang-Mills systems on the cylinder. In Integrable models and strings (Espoo, 1993), Lecture Notes in Phys. 436, Springer-Verlag, Berlin 1994, 98–122. 210 [36] P. Schaller and T. Strobl, Poisson structure induced (topological) field theories. Mod. Phys. Lett. A 9 (1994), 3129–3136. 210, 247 [37] A. Schwarz, Semiclassical approximation in Batalin–Vilkovisky formalism. Comm. Math. Phys. 158 (2) (1993), 373–396. 235, 257 [38] T. Strobl, Gravity from Lie algebroid morphisms Comm. Math. Phys. 246 (3) (2004), 475–502. 254 [39] T. Strobl, Algebroid Yang-Mills theories. Phys. Rev. Lett 93 (2004), 211601. 254, 256 [40] T. Strobl, Talks given e.g. at Vietri or the IHES in 2005. 255, 256, 257 [41] Edward Witten. Nonabelian bosonization in two dimensions. Comm. Math. Phys. 92 (1984), 455–472. 249 [42] M. Zabzine, Lectures on generalized complex geometry and supersymmetry. Arch. Math. (Brno) (supplement) 42 (2006), 119–146. 213

Chapter 8

A potential for generalized Kähler geometry Ulf Lindström, Martin Roˇcek, Rikard von Unge, and Maxim Zabzine

Contents 1 Introduction . . . . . . . . . . . . . . . . . . 2 Generalized complex geometry (GCG) . . . 3 Bihermitian geometry . . . . . . . . . . . . 4 Poisson-structures . . . . . . . . . . . . . . 5 The structure of cokerŒJC ; J . . . . . . . 6 The general case . . . . . . . . . . . . . . . 7 Linearization of generalized Kähler geometry References . . . . . . . . . . . . . . . . . . . . .

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263 264 265 266 267 269 271 273

1 Introduction Generalized Kähler geometry [1] is a special and particularly interesting example of generalized geometry [2], [1]. It succinctly encodes the bihermitian geometry of Gates, Hull and Roˇcek [3] in terms of a pair of commuting generalized complex structures .J1 ; J2 /. Here we use the equivalent bihermitian data to give a complete (local) description of generalized Kähler geometry in a neighbourhood of a regular point. We show that there exists a function K with the interpretation of a generating function for symplectomorphisms between two sets of coordinates in terms of which all geometric quantities may be described. In particular, the combination E D 12 .g C B/ of the metric and antisymmetric B-field is given as a nonlinear expression in terms of second derivatives of K. When further analyzed, this expression has the structure of a quotientquotient from some higher dimensional space. We find this auxiliary space, which we call an ALP space, and display the quotient structure. Finding a generalized Kähler geometry is thus reduced to a linear construction in terms of the corresponding K on the ALP. The bihermitian geometry was first derived as the target space geometry of N D .2; 2/ supersymmetric nonlinear sigma models [3]. Correspondingly, all our constructions have a sigma model realization, and many of the proofs were derived in that setting. In particular, the “generalized Kähler potential” K has an interpretation as the

264

Ulf Lindström, Martin Roˇcek, Rikard von Unge, and Maxim Zabzine

superspace sigma model Lagrangian1 . Detailed descriptions of this approach may be found in the two articles [5], [6] on which the presentation is based.

2 Generalized complex geometry (GCG) This section contains a brief recapitulation of the salient features of GCG. The references for the mathematical aspects are chiefly [2], [1], while a presentation more accessible to physicists (containing, e.g., coordinate and matrix descriptions) may be found in [7], [8]. A generalized almost complex structure is an algebraic structure on the sum of the tangent and cotangent bundles of a manifold M defined via J 2 End.T ˚ T /

(2.1)

J 2 D 1

(2.2)

J t J D ;

(2.3)

such that

and where is the metric corresponding to the natural pairing of elements X C ; Y C 2 T ˚ T : hX C ; Y C i D 12 .{X C {Y /:

(2.4)

The generalized almost complex structure is a (twisted) generalized complex structure iff it is integrable with respect to the (twisted) Courant bracket ŒX C ; Y C C WD ŒX; Y L C LX LY 12 d.{X {Y / C {X {Y H;

(2.5)

where ŒX; Y L is the Lie-bracket and “twisted” refers to inclusion of the last term involving the closed three-form H . Integrability is defined by the requirement that the distributions defined by the projection operators 1 2

.1 ˙ i J/

(2.6)

are involutive with respect to the Courant-bracket. Of great importance to physical applications is the fact that the automorphisms of the Courant-bracket, in addition to diffeomorphisms, also contain the b-transform: e b .X C / D X C C {X b

(2.7)

where b is a closed two-form: db D 0. When acting on a generalized complex structure J it produces an equivalent generalized complex structure Jb . In a matrix 1A

similar situation arises in the projective superspace description of hyper-Kähler sigma models [4].

Chapter 8. A potential for generalized Kähler geometry

representation this reads as follows: 1 0 1 0 Jb D J : b 1 b 1

265

(2.8)

We now restrict our attention to a subset of generalized complex geometries: generalized Kähler geometries (GKG). These are GCG’s with two commuting generalized complex structures J1;2 whose product defines a positive definite metric G on T ˚ T that squares to the identity [1]; ŒJ1 ; J2 D 0; G D J1 J2 ;

(2.9)

2

G D 1: This is the proper setting for the bihermitian geometry of [3], which we now describe.

3 Bihermitian geometry The data found in [3] to be necessary and sufficient to describe the target space geometry of a supersymmetric nonlinear sigma model with .2; 2/ supersymmetry consists of two (ordinary) complex structures JC and J , a metric g, hermitian with respect to both complex structures, and a closed three-form H with a (local) two-form potential B. The structures .J˙ ; g; B/ satisfy J˙2 D 1; J˙t gJ˙ D g; r .˙/ J˙ D 0; c H D dC !C D dc ! ;

(3.1)

c is d c with respect to J˙ respectively, and the where !˙ are the two-forms gJ˙ , d˙ covariant derivatives have (Bismut) torsion [9]

r .˙/ WD r ˙ g 1 H;

dH D 0;

(3.2)

where r is the Levi-Civita connection. When J˙ commute, it was found in [3] that there exist coordinates z, zN and z 0 , zN 0 that coordinatize ker.JC J /, respectively, and for which the metric and three-form are given as derivatives of a function K.z; zN ; z; zN 0 /. Explicitly: c c c d C d… d˙ /K () gz zN D @z @zN K; gz 0 zN 0 D @z 0 @zN 0 K; !˙ D .d c c H D d dC d K () Hz;z;z N 0 D @z @zN @z 0 K; Hz;z; N zN 0 D @z @zN @zN 0 K; (3.3) 0 0 0 0 Hz ;zN ;z D @z @zN @z K; Hz 0 ;zN 0 ;zN D @z 0 @zN 0 @zN K

266


c where d… is the d c operator defined with respect to the product … WD JC J and the c c ; dc ; d… anticommute. differentials dC A complete description for geometries with a nonvanishing cokerŒJC ; J was an open problem until our paper [5]. Before describing the solution to the problem in sections 5 and 6, we recall Gualtieri’s map of the bihermitian data .J˙ ; g; B/ into generalized Kähler geometry [1]: 1 1 0 1 0 JC ˙ J .!C !1 / J1;2 D ; (3.4) .JCt ˙ Jt / B 1 B 1 !C ! 1 0 1 0 0 g 1 : (3.5) G D g 0 B 1 B 1

Note that what looks like a b-transform is not, since dB D H ¤ 0. It is an interesting fact that if we introduce the usual GCS’s corresponding to J˙ , J˙ 0 J˙ D ; (3.6) 0 J˙t then the map (3.4) may be summarized as2 J1;2 D …C JCB ˙ … JB ;

(3.7)

with …˙ WD

1 2

.1 ˙ G / :

(3.8)

4 Poisson-structures There are three Poisson-structures relevant to our discussion. We first consider two real ones: ˙ WD .JC ˙ J /g 1 D g 1 .JC ˙ J /t : (4.1) These were introduced in [10], where they were used essentially as described below, simplifying an earlier derivation in [11]. They ensure the existence of coordinates adapted to ker.JC J / ˚ ker.JC C J /. In a neighborhood of a regular point x0 of we may choose coordinates x A whose tangents lie in the kernel of : A A D J : A D 0 H) JC

(4.2) 0

Similarly, in a neighborhood of a regular point of C we have coordinates x A such that A0 A0 A0 C D 0 H) JC D J : (4.3) 2 This

structure is related to how the map is derived in chapter 6 of [1]


267

It then follows from the nondegeneracy of C ˙ that the Poisson brackets defined by C and by cannot have common Casimir functions. In other words, the directions A and A0 cannot coincide. The result is that we may write 0 1 B C C; J˙ D B (4.4) @ 0 0 Ic 0 A 0 0 0 ˙I t in coordinates adapted to ker.JC J / ˚ ker.JC C J / ˚ cokerŒJC ; J ; where we write a canonical complex structure as i 0 I D 0 i

(4.5)

(4.6)

and c labels the A and t labels the A0 directions. A third Poisson structure , related to the real Poisson structures, was introduced in [12]: (4.7) WD ŒJC ; J g 1 D ˙.JC J /˙ D .JC ˙ J / : The relation to ˙ implies that ker D ker C ˚ ker :

(4.8)

Tangents to the symplectic leaf for lie in cokerŒJC ; J . Using (4.8), we can use to investigate the remaining directions in (4.4). We need that J˙ J˙t D H) D .2;0/ C N .0;2/ ; with the decomposition with respect to both JC and J . Further [12]: N .2;0/ D 0; @

(4.9) .2;0/

is holomorphic (4.10)

We now investigate the structure of the cokernel.

5 The structure of cokerŒJC ; J It is convenient to first treat the case kerŒJC ; J D ;. Then we have at our disposal a symplectic form : WD 1 ;

d D 0;

J˙t J˙ D :

Choosing complex coordinates with respect to JC , I 0 JC D s ; 0 Is

(5.1)

(5.2)

268


with I as in (4.6), we have the decomposition N .0;2/ ; C D .2;0/ C C

(5.3)

and D 0; @.2;0/ C

N .2;0/ D 0; @ C

(5.4)

which means in particular that .2;0/ is holomorphic. We may then choose Darboux C coordinates such that D dq a ^ dp a ; .2;0/ C

N .0;2/ D d qN aN ^ d pN aN : C

(5.5)

Alternatively, we can choose complex coordinates with respect to J and similarly derive that 0 0 0 0 N .0;2/ D dQa ^ dP a ; D d QN aN ^ d PN aN : (5.6) .2;0/ The transformation fq; pg ! fQ; P g is a canonical transformations (symplectomorphism) and can thus be specified by a generating function3 K.q; P /. Thus in a neighborhood, the canonical transformation is given by the generating function K.q; P / @K @K pD ; QD : (5.7) @q @P We calculate all our geometric structures JC , J , and g in the “mixed” coordinates fq; P g, using the transformation matrices ! ! 1 0 1 0 @.q; p/ 1 0 D @p @p D @2 K @2 K WD ; (5.8) KLL KLR @.q; P / @q @P @q@q @P @q and @.Q; P / D @.q; P /

@Q @q

@Q @P

0

1

! D

@2 K @q@P

@2 K @P @P

0

1

!

WD

KRL KRR ; 0 1

(5.9)

0 0 where the labels L and R are shorthand for the coordinates fq a ; qN aN g and fP a ; PN aN g in (5.5) and (5.6) above. The expressions for J˙ are nonlinear and can be read off from the formulae given below for the general case with kerŒJC ; J ¤ ;. The symplectic structure is the only linear function of K 0

D KAA0 dq A ^ dP A ;

(5.10)

where we use the collective notation A D fa; ag, N A0 D fa0 ; aN 0 g. We write as a matrix 0 KLR D : (5.11) KRL 0 3 There

always exists at least one polarization such that any symplectomorphism, can be written in terms of such a generating function [13].


269

and find the metric and B-field using this matrix as follows [14] (cf. 4.7) g D ŒJC ; J ;

B D fJC ; J g:

(5.12)

6 The general case In the general case when both kerŒJC ; J and cokerŒJC ; J are nonempty, we combine the discussion leading to (4.4) in section 4 with that of the previous section. The formulas that we compute are rather complicated, and we have not found a suitable coordinate free way to express them. We assume that in a neighborhood of x0 , the ranks of ˙ are constant, and as result, the rank of is constant. We work in coordinates fq; p; z; z 0 g adapted to the symplectic foliation of as well as to the description of kerŒJC ; J given in section 4. In such coordinates JC may be taken to have the canonical form4 0 1 Is 0 0 0 B 0 Is 0 0 C C JC D B (6.1) @ 0 0 Ic 0 A ; 0 0 0 It where fq; pg are Darboux coordinates for a symplectic leaf of and fz; z 0 g parametrize the kernels of . Similarly, there are coordinates fQ; P; z; z 0 g where J takes a diagonal form (identical to that of JC in (6.1) except for a change of sign in the last entry; I t ! I t ). For every leaf separately we may now apply the arguments of section 5. There thus exists a generating function K.q; P; z; z 0 / for the symplectomorphisms between the two sets of coordinates. The transformation matrices to the coordinates fq; P; z; z 0 g are given by the obvious extensions of (5.8) and (5.9) to include z; z 0 . The expression for J˙ in the “mixed” coordinates are then5 : 0 1 Is 0 0 0 BK 1 CLL K 1 Is KLR K 1 CLc K 1 CLt C RL RL RL RL C; (6.2) JC D B @ A 0 0 Ic 0 0 0 0 It and

0

1 1 KLR Is KRL KLR CRR B 0 Is J D B @ 0 0 0 0

1 KLR CRc 0 Ic 0

1 1 KLR ARt C 0 C; A 0 I t

(6.3)

4 For historical reasons related to the origin in sigma models involving chiral and twisted chiral superfields, we again use the labels c and t for the z and z 0 directions. 5 The rows and columns of this matrix correspond to the directions along fq; P; z; z 0 g.

270


where 1 D .KRL /1 ; KLR

0 2iK ; 2iK 0 2iK 0 A D IK C KI D ; 0 2iK

C D IK KI D

(6.4)

where we suppress the indices in the last two entries6 . The relations (5.12) no-longer hold when kerŒJC ; J ¤ ;. The definition (4.7) of the Poisson structure still determines the metric, except along the kernel. However, the additional relation c !˙ from (3.1) provides us with an equation for the remaining components H D ˙d˙ of g and allows us to find the B-field. From the sigma model we already know the solution; the sum E D 12 .g C B/ of the metric g and B-field takes on the explicit form: 1 ELL D CLL KLR Is KRL 1 ELR D Is KLR Is C CLL KLR CRR 1 ELc D KLc C Is KLc Ic C CLL KLR CRc 1 ELt D KLt Is KLt I t C CLL KLR ARt 1 ERL D KRL Is KLR Is KRL 1 ERR D KRL Is KLR CRR 1 ERc D KRc KRL Is KLR CRc 1 ERt D KRt KRL Is KLR ARt 1 Is KRL EcL D CcL KLR 1 EcR D Ic KcR Is C CcL KLR CRR 1 Ecc D Kcc C Ic Kcc Ic C CcL KLR CRc 1 Ect D Kct Ic Kct I t C CcL KLR ARt 1 E tL D C tL KLR Is KRL 1 E tR D I t K tR Is C C tL KLR CRR 1 E tc D K tc C I t K tc Ic C C tL KLR CRc 1 E t t D K t t I t K t t I t C C tL KLR ARt

(6.5)

In view of this last relation (6.5) as well as (6.2), (6.3) and (5.11) it seems appropriate to call K the generalized Kähler potential. 6 In (6.4), the rows and columns correspond to fA; Ag, N where fAg are any one of the fq; P; z; z 0 g directions

in (6.2), (6.3), (6.5).


271

7 Linearization of generalized Kähler geometry As can be seen in (6.5), the expressions for the metric and the B-field in terms of the generalized Kähler potential are highly nonlinear in contrast to what is the case in ordinary Kähler geometry. The nonlinearities all stem from the cokernel of ŒJC ; J : when ŒJC ; J D 0, the metric and B-field can be expressed as linear functions of the Hessian of K. We note that the structure of the nonlinearities are such as one might expect from a quotient construction. One is therefore faced with a natural question: Is there a space in which all the geometric data is encoded linearly with respect to the generalized Kähler potential and which gives the nonlinear generalized Kähler geometry through a quotient construction? Inspired by superspace sigma-models, we have found the following local prescription for generating such a space: In local coordinates, we simply make the substitution 0 q ! zL C zL0 ; P ! zR C zN R (7.1) in the generalized Kähler potential K and use the naive commuting complex structures J˙ in which the z-coordinates are biholomorphic and the z 0 -coordinates are JC holomorphic and J antiholomorphic. This introduces isometries of the metric (3.3) corresponding to shifts of zL ; zL0 that preserve the sum, and similarly for zR ; zNR ; taking a quotient with respect to these isometries leads to the original nonlinear model. We now attempt to make mathematical sense of this prescription. We begin with some definitions: Definition 1 (BiLP). A generalized Kähler geometry with commuting complex structures will be called a bihermitian local product space (BiLP). We also give a special name to a BiLP with the isometries sketched above: Definition 2 (ALP). A BiLP with these additional isometries (further characterized below) we will call an auxiliary product space (ALP). The isometries are generated by 2n complex Abelian Killing vectors that preserve all the BiLP geometric data. They can be separated into two groups of n Killing vectors fkLA g and fkRA g, A D 1; : : : ; n, satisfying the further requirement {kLA H D d˛LA ;

{kRA0 H D d˛RA0 ;

(7.2)

where the one-form ˛ is the dual of the killing vector k: ˛ D g.k/; this is equivalent to saying that the Killing vectors are covariantly constant with respect to the covariant derivative with torsion (3.2). Such Killing vectors are called Kac–Moody Killing vectors. We require that each group of Killing vectors kL and kR by themselves span maximally isotropic subspaces, i.e., that g.kLA ; kLB / D 0;

g.kRA0 ; kRB 0 / D 0;

(7.3)

272


but that the inner product hA0 B D g.kRA0 ; kLB /

(7.4)

is nondegenerate. Theorem. Locally, any generalized Kähler manifold M is a quotient of an ALP by its Kac–Moody isometries. The quotient is performed by going to the orbits of the action of the Killing vector and choosing a horizontal subspace by specifying a connection. In the particular case with left and right Kac–Moody Killing vectors there is a corresponding left and right connection given by 0

0

BA ; .R /A D g kLB h

0

AB .L /A kRB 0 g ; Dh

(7.5)

0

where hAB is the inverse of the nondegenerate matrix (7.4). The connections (7.5) satisfy the following properties L .kL / D 1;

R .kR / D 1;

L .kR / D 0;

R .kL / D 0:

(7.6)

Vectors in the horizontal subspaces are defined as the vectors lying in the kernel of both L and R . All geometric structures defined on the ALP can now be defined on the quotient space. As usual, forms are projected using the connection so that the contraction with any vertical vector is zero, whereas vectors are pushed forward to the quotient using the bundle projection p. When we project the complex structures JQ˙ .X / WD p J˙ .X/ L .X/ J˙ .kL / R .X / J˙ .kR / for all X , (7.7) it is clear that the projected complex structures JQ˙ do not commute even though J˙ do. Projecting the metric and B-field, we find gQ D g h.L ; R / h.R ; L /; BQ D B h.L ; R / C h.R ; L /:

(7.8) (7.9)

It is straightforward to see that gQ gives zero when contracted with any vertical vector Q This (apparent) mystery is resolved by showing that HQ D d BQ while this is not so for B. can be written as HQ D H L ^ {kL H R ^ {kR H L ^ R ^ {kL {kR H

(7.10)

so that the geometrically meaningful object HQ defined by d BQ is indeed well defined on the quotient. The quotients gQ and HQ agree with the expressions for the original generalized Kähler manifold. Acknowledgement. We are grateful to the 2006 Simons Workshop for providing the stimulating atmosphere where this work was initiated. The work of UL was


273

supported by EU grant (Superstring theory) MRTN-2004-512194 and VR grant 6212003-3454. The work of MR was supported in part by NSF grant no. PHY-0354776. The research of R.v.U. was supported by Czech ministry of education contract No. MSM0021622409. The research of M.Z. was supported by VR-grant 621-2004-3177

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[10] S. Lyakhovich and M. Zabzine, Poisson geometry of sigma models with extended supersymmetry. Phys. Lett. B 548 (2002), 243–251. 266 [11] I. T. Ivanov, B. B. Kim, and M. Roˇcek, Complex structures, duality and WZW models in extended superspace. Phys. Lett. B 343 (1995), 133–143. 266 [12] N. J. Hitchin, Instantons, Poisson structures and generalized Kähler geometry. Comm. Math. Phys. 265 (2006), 131–317. 267 [13] V. I. Arnold, Mathematical methods of classical mechanics. Second edition, Grad. Texts in Math. 60. Springer-Verlag, New York 1989. 268 [14] J. Bogaerts, A. Sevrin, S. van der Loo, and S. Van Gils, Properties of semichiral superfields. Nucl. Phys. B 562 (1999), 277–290. 269 [15] T. Buscher, U. Lindström, and M. Roˇcek, New supersymmetric -models with WessZumino terms. Phys. Lett. B 202 (1988), 94–98.

Part C

Geometries with torsion

Chapter 9

Non-integrable geometries, torsion, and holonomy Ilka Agricola

Contents 1

Background and motivation . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Mathematical motivation . . . . . . . . . . . . . . . . . . . . . . 1.3 Physical motivation – torsion in gravity . . . . . . . . . . . . . . 1.4 Physical motivation – torsion in superstring theory . . . . . . . . 1.5 First developments since 1980 . . . . . . . . . . . . . . . . . . . 2 Metric connections with torsion . . . . . . . . . . . . . . . . . . . . . 2.1 Types of connections and their lift into the spinor bundle . . . . . 2.2 Naturally reductive spaces . . . . . . . . . . . . . . . . . . . . . 2.3 Almost Hermitian manifolds . . . . . . . . . . . . . . . . . . . . 2.4 Hyper-Kähler manifolds with torsion (HKT-manifolds) . . . . . . 2.5 Almost contact metric structures . . . . . . . . . . . . . . . . . . 2.6 3-Sasaki manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Holonomy theory . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Geometric stabilizers . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 U.n/ and SU.n/ in dimension 2n . . . . . . . . . . . . . . . . . . 3.2 U.n/ and SU.n/ in dimension 2n C 1 . . . . . . . . . . . . . . . 3.3 G2 in dimension 7 . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Spin.7/ in dimension 8 . . . . . . . . . . . . . . . . . . . . . . . 4 A unified approach to non-integrable geometries . . . . . . . . . . . . 4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 G-structures on Riemannian manifolds . . . . . . . . . . . . . . . 4.3 Almost contact metric structures . . . . . . . . . . . . . . . . . . 4.4 SO.3/-structures in dimension 5 . . . . . . . . . . . . . . . . . . 4.5 Almost Hermitian manifolds in dimension 6 . . . . . . . . . . . . 4.6 G2 -structures in dimension 7 . . . . . . . . . . . . . . . . . . . . 4.7 Spin.7/-structures in dimension 8 . . . . . . . . . . . . . . . . . 5 Weitzenböck formulas for Dirac operators with torsion . . . . . . . . . 5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The square of the Dirac operator and parallel spinors . . . . . . . 5.3 Naturally reductive spaces and Kostant’s cubic Dirac operator . . 5.4 Some remarks on the common sector of type II superstring theory Appendix. Compilation of remarkable identities for connections with skew-symmetric torsion . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Background and motivation 1.1 Introduction Since the fifties, the French school founded by M. Berger had developed the idea that Riemannian manifolds should be subdivided into different classes according to their holonomy group, and the name special (integrable) geometries has become customary for those which are not of general type. At the beginning of the seventies, A. Gray generalized the classical holonomy concept by introducing a classification principle for non-integrable special Riemannian geometries discovered in this context nearly Kähler manifolds in dimension six and nearly parallel G2 -manifolds in dimension seven. Non-integrable geometries then entered mathematical physics when it became clear in the eighties that they are strongly related to twistor theory and the study of small eigenvalues of the Dirac operator. The interest in non-integrable geometries was revived in the past years through developments of superstring theory. Firstly, integrable geometries (Calabi–Yau manifolds, Joyce manifolds etc.) are exact solutions of the Strominger model (1986), though with vanishing B-field. If one deforms these vacuum equations and looks for models with non-trivial B-field, a new mathematical approach going back a decade implies that solutions can be constructed geometrically from non integrable geometries with torsion. In this way, manifolds not belonging to the field of algebraic geometry (integrable geometries) become candidates for interesting models in theoretical physics. Before discussing the deep mathematical and physical backgrounds, let us give a – very intuitive – explanation of why the traditional Yang-Mills approach needs modification in string theory and how torsion enters the scene. Point particles move along world-lines, and physical quantities are typically computed as line integrals of some potential that is, mathematically speaking, just a 1-form. The associated field strength is then its differential – a 2-form – and interpreted as the curvature of some connection. In contrast, excitations of extended 1-dimensional objects (the ‘strings’) are ‘world-surfaces’, and physical quantities have to be surface integrals of certain potential 2-forms. Their field strengths are thus 3-forms and cannot be interpreted as curvatures anymore. The key idea is to supply the (pseudo)-Riemannian manifold underlying the physical model with a non-integrable G-structure admitting a ‘good’ metric G-connection r with torsion, which in turn will play the role of a B-field strength; and the art is to choose the G-structure so that the connection r admits the desired parallel objects, in particular spinors, interpreted as supersymmetry transformations. In this chapter, we shall establish connections with skew-symmetric torsion as one of the main tools to understand non-integrable geometries. To this aim a series of key examples is presented and successively dealt with using the notions of intrinsic torsion and characteristic connection of a G-structure as unifying principles. The General Holonomy Principle bridges over to parallel objects, thus motivating the discussion

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of geometric stabilizers, with emphasis on spinors and differential forms. Several Weitzenböck formulas for Dirac operators associated with torsion connections enable us to discuss spinorial field equations, such as those governing the common sector of type II superstring theory. They also provide the link to Kostant’s cubic Dirac operator. My warmest thanks go to all colleagues who, by their countless remarks and corrections, helped improving the quality of this text, in particular to Simon Chiossi, Richard Cleyton, Thomas Friedrich, Mario Kassuba, Nils Schoemann (Humboldt University Berlin) as well as Paweł Nurowski and Andrzej Trautman (Warsaw University). For more details than presented in this survey, I refer the reader to the original literature as well as my lecture notes [5].

1.2 Mathematical motivation While the classical symmetry approach in differential geometry was based on the isometry group of a manifold, it turned out by the mid-fifties that a second intrinsic group associated to a Riemannian manifold was deeply related to fundamental features like curvature and parallel objects. The so-called holonomy group determines how a vector can change under parallel transport along a closed loop inside the manifold. Berger’s theorem (1955) classifies all possible restricted holonomy groups of a simply connected, irreducible and non-symmetric Riemannian manifold .M; g/ (see [37], [217] for corrections and simplifications in the proof and [56] for a status report). The holonomy group can be either SO.n/ in the generic case or one of the groups listed in Table 1 (here and in the sequel, r g denotes the Levi-Civita connection). Table 1. Possible Riemannian holonomy groups (‘Berger’s list’).

dim M

Hol.M /

name

parallel object

curvature

4n

Sp.n/Sp.1/

quaternionic Kähler manifold

—

Ric D g

2n

U.n/

Kähler manifold

rg J D 0

—

2n

SU.n/

Calabi–Yau manifold

rg J D 0

Ric D 0

4n

Sp.n/

hyper-Kähler manifold

rg J D 0

Ric D 0

7

G2

parallel G2 -manifold

rg !3 D 0

Ric D 0

8

Spin.7/

parallel Spin.7/-manifold

r g ˇ4 D 0

Ric D 0

Œ16

ŒSpin.9/

[parallel Spin.9/-manifold

—

—

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Manifolds having one of these holonomy groups are called manifolds with special (integrable) holonomy, or special (integrable) geometries for short. We put the case n D 16 and Hol.M / D Spin.9/ into brackets, because Alekseevski and Brown/Gray showed independently that such a manifold is necessarily symmetric ([16], [54]). The point is that Berger proved that the groups on this list were the only possibilities, but he was not able to show whether they actually occurred as holonomy groups of compact manifolds. It took another thirty years to find out that – with the exception of Spin.9/ – this is indeed the case: The existence of metrics with holonomy SU.m/ or Sp.m/ on compact manifolds followed from Yau’s solution of the Calabi Conjecture posed in 1954 [247]. Explicit non-compact metrics with holonomy G2 or Spin.7/ are due to R. Bryant [55] and R. Bryant and S. Salamon [58], while compact manifolds with holonomy G2 or Spin.7/ were constructed by D. Joyce only in 1996 (see [156], [157] [158] and the book [159], which also contains a proof of the Calabi Conjecture). Later, compact exceptional holonomy manifolds have also been constructed by other methods by Kovalev ([175]). As we will explain later, the General Holonomy Principle relates manifolds with Hol.M / D SU.n/; Sp.n/; G2 or Spin.7/ with r g -parallel spinors (see Section 3). Already in the sixties it had been observed that the existence of such a spinor implies in turn the vanishing of the Ricci curvature ([48] and Proposition 2.2) and restricts the holonomy group of the manifold ([139], [241]), but the difficulties in constructing explicit compact manifolds with special integrable Ricci-flat metrics inhibited further research on the deeper meaning of this result. There was progress in this direction only in the homogeneous case. Symmetric spaces are the “integrable” geometries inside the much larger class of homogeneous reductive spaces. Given a non-compact semisimple Lie group G and a maximal compact subgroup K such that rank G D rank K, consider the associated symmetric space G=K. The Dirac operator can be twisted by a finite-dimensional irreducible unitary representation of K, and it was shown by Parthasarathy, Wolf, Atiyah and Schmid that for suitable most of the discrete series representations of G can be realized on the L2 -kernel of this twisted Dirac operator ([201], [246], [29]). The crucial step is to relate the square of the Dirac operator with the Casimir operator G of G; for trivial , the corresponding formula reads D 2 D G C

1 Scal: 8

(1)

Meanwhile many people began looking for suitable generalizations of the classical holonomy concept. One motivation for this was that the notion of Riemannian holonomy is too restrictive for vast classes of interesting Riemannian manifolds; for example, contact or almost Hermitian manifolds cannot be distinguished merely by their holonomy properties (they have generic holonomy SO.n/), and the Levi-Civita connection is not adapted to the underlying geometric structure (meaning that the defining objects are not parallel).


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In 1971 A. Gray introduced the notion of weak holonomy ([128]), “one of his most original concepts” and “an idea much ahead of its time” in the words of N. Hitchin [141]. This concept turned out to yield interesting non-integrable geometries in dimensions n 8 and n D 16. In particular, manifolds with weak holonomy U.n/ and G2 became known as nearly Kähler and nearly parallel G2 -manifolds, respectively. But whereas metrics of compact Ricci-flat integrable geometries have not been realized explicitly (so far), there are many well-known homogeneous reductive examples of non-integrable geometries ([127], [86], [33], [107], [49], [92] and many others). The relation to Dirac operators emerged shortly after Th. Friedrich proved in 1980 a seminal inequality for the first eigenvalue 1 of the Dirac operator on a compact Riemannian manifold M n of non-negative curvature [95]: n min.scal/; (2) .1 /2 4.n 1/ M n In this estimate, equality occurs precisely if the corresponding eigenspinor the Killing equation s 1 min.scal/ g X μ X : rX D ˙ 2 n.n 1/

satisfies

The first non-trivial compact examples of Riemannian manifolds with Killing spinors were found in dimensions 5 and 6 in 1980 and 1985, respectively ([95], [101]). The link to non-integrable geometry was established shortly after; for instance, a compact, connected and simply connected 6-dimensional Hermitian manifold is nearly Kähler if and only if it admits a Killing spinor with real Killing number [133]. Similar results hold for Einstein–Sasaki structures in dimension 5 and nearly parallel G2 manifolds in dimension 7 ([105], [106]). Remarkably, the proof of inequality (2) relies on introducing a suitable spin connection – an idea much in line with recent developments. A. Lichnerowicz established the link to twistor theory by showing that on a compact manifold the space of twistor spinors coincides – up to a conformal change of the metric – with the space of Killing spinors [184].

1.3 Physical motivation – torsion in gravity The first attempts to introduce torsion as an additional ‘datum’for describing physics in general relativity goes back to Cartan himself [64]. Viewing torsion as some intrinsic angular momentum, he derived a set of gravitational field equations from a variational principle, but postulated that the energy-momentum tensor should still be divergencefree, a condition too restrictive for making this approach useful. The idea was taken up again in broader context in the late fifties. The variation of the scalar curvature (and an additional Lagrangian generating the energy-momentum tensor) on a space-time endowed with a metric connection with torsion yielded the two fundamental equations of Einstein–Cartan theory, first formulated by Kibble [163] and Sciama (see his article in [148]). The first equation is (formally) Einstein’s classical field equation of general

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relativity with an effective energy momentum tensor Teff depending on torsion, the second one can be written in index-free notation as Q.X; Y / C

n X

Q.Y; ei / ³ ei X Q.X; ei / ³ ei Y D 8S.X; Y /:

iD1

Here Q denotes the torsion of the new connection r, S the spin density and e1 ; : : : ; en any orthonormal frame. A. Trautman provided an elegant formulation of Einstein– Cartan theory in the language of principal fibre bundles [229]. The most striking predictions of Einstein–Cartan theory are in cosmology. In the presence of very dense spinning matter, nonsingular cosmological models may be constructed because the effective energy momentum tensor Teff does not fulfill the conditions of the Penrose– Hawking singularity theorems anymore [230]. The first example of such a model was provided by W. Kopczyński [171], while J. Tafel found a large class of such models with homogeneous spacial sections [226]. For a general review of gravity with spin and torsion including extensive references, we refer to the article [136]. In the absence of spin, the torsion vanishes and the whole theory reduces to Einstein’s original formulation of general relativity. In practice, torsion turned out to be hard to detect experimentally, since all tests of general relativity are based on experiments in empty space. Einstein–Cartan theory is pursued no longer, although some concepts that it inspired are still of relevance (see [137] for a generalization with additional currents and shear, [231] for optical aspects, [208] for the link to the classical theory of defects in elastic media). Yet, it may be possible that Einstein–Cartan theory will prove to be a better classical limit of a future quantum theory of gravitation than the theory without torsion.

1.4 Physical motivation – torsion in superstring theory Superstring theory (see for example [131], [185]) is a physical theory aiming at describing nature at small distances (' 1025 m). The concept of point-like elementary particles is replaced by one-dimensional objects as building blocks of matter – the so-called strings. Particles are then understood as resonance states of strings and can be described together with their interactions up to very high energies (small distances) without internal contradictions. Besides gravitation, string theory incorporates many other gauge interactions and hence is an excellent candidate for a more profound description of matter than the standard model of elementary particles. Quantization of superstrings is only possible in the critical dimension 10, while M -theory is a non-perturbative description of superstrings with “geometrized” coupling, and lives in dimension 11. Mathematically speaking, a 10- or 11-dimensional configuration space Y (a priori not necessarily smooth) is assumed to be the product Y 10;11 D V 3-5 M 5-8


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of a low-dimensional spacetime V describing the ‘external’part of the theory (typically, Minkowski space or a space motivated from general relativity like anti-de-Sitter space), and a higher-dimensional ‘internal space’ M with some special geometric structure. The metric is typically a direct or a warped product. On M , internal symmetries of particles are described by parallel spinor fields, the most important of which being the existing supersymmetries: a spinor field has spin 1=2, so tensoring with it swaps bosons and fermions. By the General Holonomy Principle (see Theorem 2.3), the holonomy group has to be a subgroup of the stabilizer of the set of parallel spinors inside Spin.9; 1/. These are well known and summarized in Table 2. We shall explain how to derive this result and how to understand the occurring semidirect products in Section 3.4. Table 2. Possible stabilizers of invariant spinors inside Spin.9; 1/.

# of invariant spinors

stabilizer groups

1

Spin.7/ Ë R8

2

G2 , SU.4/ Ë R8

3

Sp.2/ Ë R8

4

SU.3/, .SU.2/ SU.2// Ë R8

8

SU.2/, R8

16

feg

Since its early days, string theory has been intricately related with some branches of algebraic geometry. This is due to the fact that the integrable, Ricci-flat geometries with a parallel spinor field with respect to the Levi-Civita connection are exact solutions of the Strominger model for a string vacuum with vanishing B-field and constant dilaton. This rich and active area of mathematical research lead to interesting developments such as the discovery of mirror symmetry.

1.5 First developments since 1980 In the early eighties, several physicists independently tried to incorporate torsion into superstring and supergravity theories in order to get a more physically flexible model, possibly inspired by the developments in classical gravity ([238], [113], [142], [80], [207]). In fact, simple supergravity is equivalent to Einstein–Cartan theory with a massless, anticommuting Rarita–Schwinger field as source. But contrary to general relativity, one difficulty stems from the fact that there are several models in superstring

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theory (type I, II, heterotic,…) that vary in the excitation spectrum and the possible interactions. In his article “Superstrings with torsion” [222], A. Strominger describes the basic model in the common sector of type II superstring theory in form of a 6-tuple .M n ; g; r; T; ˆ; ‰/ consisting of a Riemannian spin manifold .M n ; g/, a 3-form T , a dilaton function ˆ and a spinor field ‰. The field equations can be written in the following form (recall that r g denotes the Levi-Civita connection): 1 Ricij Timn Tj mn C 2rig @j ˆ D 0; 4

rXg

1 C X ³T 4

ı.e 2ˆ T / D 0;

D 0;

.2dˆ T /

D 0:

If one introduces a new metric connection r whose torsion is given by the 3-form T , 1 rX Y ´ rXg Y C T .X; Y; /; 2 one sees that the third equation is equivalent to r‰ D 0. The remaining equations can similarly be rewritten in terms of r. For constant dilaton ˆ, they take the particularly simple form [151] Ricr D 0;

ı g .T / D 0;

r‰ D 0;

T ‰ D 0;

(3)

and the second equation (ı g .T / D 0) now follows from the first equation (Ricr D 0). For M compact, it was shown in [4, Theorem 4.1] that a solution of all equations necessarily forces T D 0, i.e., an integrable Ricci-flat geometry with classical holonomy given by Berger’s list. By a careful analysis of the integrability conditions, this result could later be extended to the non-compact case ([14], see also Section 5.4). Together with the well-understood Calabi–Yau manifolds, Joyce manifolds with Riemannian holonomy G2 or Spin.7/ thus became of interest in recent times (see [30], [75]). From a mathematical point of view, this result stresses the importance of tackling easier problems first, for example partial solutions. As first step in the investigation of metric connections with totally skew-symmetric torsion, Dirac operators, parallel spinors etc., Th. Friedrich and S. Ivanov proved that many non-integrable geometric structures (almost contact metric structures, almost Hermitian and weak G2 -structures) admit a unique invariant connection r with totally skew-symmetric torsion [102], thus being a natural replacement for the Levi-Civita connection. Non-integrable geometries could then be studied by their holonomy properties. In fact, in mathematics the times had been ripe for a new look at the intricate relationship between holonomy, special geometries, spinors and differential forms: in 1987, R. Bryant found the first explicit local examples of metrics with exceptional Riemannian holonomy (see [55] and [58]), Ch. Bär described their relation to Killing spinors via the cone construction [31]. Building on the insightful vision of Gray, S. Salamon realized the centrality of the concept of intrinsic torsion ([209] and, for recent results, [91], [69], [74]). Swann successfully tried weakening holonomy [225],


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and N. Hitchin characterized non-integrable geometries as critical points of some linear functionals on differential forms [141]. In particular, he motivated a generalization of Calabi–Yau-manifolds [140] and of G2 -manifolds [244], and discovered a new, previously unknown special geometry in dimension 8 (“weak PSU.3/-structures”, see also [245]). Friedrich reformulated the concepts of non-integrable geometries in terms of principle fiber bundles [99] and discussed the exceptional dimension 16 suggested by A. Gray years before ([97], [98]). Analytic problems – in particular, the investigation of the Dirac operator – on non-integrable Riemannian manifolds contributed to a further understanding of the underlying geometry ([42], [23], [115]). Finally, the Italian school and collaborators devoted over the past years a lot of effort to the explicit construction of homogeneous examples of non-integrable geometries with special properties in small dimensions (see for example [2], [93], [94], [210] and the literature cited therein), making it possible to test the different concepts on explicit examples. The first non-integrable geometry that raised the interest of string theorists was the squashed 7-sphere with its weak G2 -structure, although the first steps in this direction were still marked by confusion about the different holonomy concepts. A good overview about G2 in string theory is the survey article by M. Duff ([84]). It includes speculations about possible applications of weak Spin.9/-structures in dimension 16, which a priori are of too high dimension to be considered in physics. In dimension three, it is well known (see for example [220]) that the Strominger equation r‰ D 0 can be solved only on a compact Lie group with bi-invariant metric, and that the torsion of the invariant connection r coincides with the Lie bracket. In dimension four, the Strominger model leads to a HKT structure (see Section 2.4 for more references), i.e., a hyper-Hermitian structure that is parallel with respect to r, and – in the compact case – the manifold is either a Calabi–Yau manifold or a Hopf surface [151]. Hence, the first interesting dimension for further mathematical investigations is five. Obviously, besides the basic correspondence outlined here, there is still much more going on between special geometries and detailed properties of physical models constructed from them. Some weak geometries have been rederived by physicists looking for partial solutions by numerical analysis of ODE’s and heavy special function machinery [117].

2 Metric connections with torsion 2.1 Types of connections and their lift into the spinor bundle The notion of torsion of a connection was invented by Elie Cartan, and appeared for the first time in a short note at the Académie des Sciences de Paris in 1922 [63]. Although the article contains no formulas, Cartan observed that such a connection may or may not preserve geodesics, and initially turns his attention to those who do so. In this

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sense, Cartan was the first to investigate this class of connections. He then goes on to explain in very general terms how the connection should be adapted to the geometry under consideration. We believe that this point of view should be taken into account in Riemannian geometry, too. We give a short review of the eight classes of geometric torsion tensors. Consider a Riemannian manifold .M n ; g/. The difference between its Levi-Civita connection r g and any linear connection r is a .2; 1/-tensor field A, rX Y D rXg Y C A.X; Y /;

X; Y 2 TM n :

The vanishing of the symmetric or the antisymmetric part of A has immediate geometric interpretations. The connection r is torsion-free if and only if A is symmetric. The connection r has the same geodesics as the Levi-Civita connection r g if and only if A is skew-symmetric. Following Cartan, we study the algebraic types of the torsion tensor for a metric connection. Denote by the same symbol the .3; 0/-tensor derived from a .2; 1/-tensor by contraction with the metric. We identify TM n with .TM n / using g from now on. Let T be the n2 .n 1/=2-dimensional space of all possible torsion tensors, T D fT 2 ˝3 TM n j T .X; Y; Z/ D T .Y; X; Z/g Š ƒ2 TM n ˝ TM n : A connection r is metric if and only if A belongs to the space Ag ´ TM n ˝ .ƒ2 TM n / D fA 2 ˝3 TM n j A.X; V; W / C A.X; W; V / D 0g: In particular, dim Ag D dim T , reflecting the fact that metric connections can be uniquely characterized by their torsion. Proposition 2.1 ([65, p. 51], [232], [209]). The spaces T and Ag are isomorphic as O.n/ representations, an equivariant bijection being T .X; Y; Z/ D A.X; Y; Z/ A.Y; X; Z/; 2 A.X; Y; Z/ D T .X; Y; Z/ T .Y; Z; X / C T .Z; X; Y /: For n 3, they split under the action of O.n/ into the sum of three irreducible representations, T Š TM n ˚ ƒ3 .M n / ˚ T 0 : The last module will also be denoted A0 if viewed as a subspace of Ag and is equivalent to the Cartan product of representations TM n ˝ ƒ2 TM n , X;Y;Z ˚ P T 0 D T 2 T j S T .X; Y; Z/ D 0; niD1 T .ei ; ei ; X/ D 0 for all X; Y; Z

for any orthonormal frame e1 ; : : : ; en . For n D 2, T Š Ag Š R2 is O.2/-irreducible. The eight classes of linear connections are now defined by the possible components of their torsions T in these spaces. The nice lecture notes by Tricerri and Vanhecke [232] use a similar approach in order to classify homogeneous spaces by the algebraic


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properties of the torsion of the canonical connection. They construct homogeneous examples of all classes, study their “richness” and give explicit formulas for the projections on every irreducible component of T in terms of O.n/-invariants. Definition 2.1 (Connection with vectorial torsion). The connection r is said to have vectorial torsion if its torsion tensor lies in the first space of the decomposition in Proposition 2.1, i.e., if it is essentially defined by some vector field V on M . The tensors A and T can then be directly expressed through V as A.X; Y / D g.X; Y /V g.V; Y /X; T .X; Y; Z/ D g g.V; X /Y g.V; Y /X; Z : These connections are particularly interesting on surfaces, in as much that every metric connection on a surface is of this type. In [232], F. Tricerri and L. Vanhecke showed that if M is connected, complete, simply-connected and V is r-parallel, then .M; g/ has to be isometric to the hyperbolic space. V. Miquel studied in [192] and [193] the growth of geodesic balls of such connections, but did not investigate the detailed shape of geodesics. The study of the latter was outlined in [15], whereas [12] and [152] are devoted to holonomy aspects and a possible role in superstring theory. Notice that there is some similarity to Weyl geometry. In both cases, we consider a Riemannian manifold with a fixed vector field V on it ([62], [114]). A Weyl structure is a pair consisting of a conformal class of metrics and a torsion-free non-metric connection preserving the conformal structure. This connection is constructed by choosing a metric g in the conformal class and is then defined by the formula rXw Y ´ rXg Y C g.X; V / Y C g.Y; V /X g.X; Y /V: Weyl geometry deals with the geometric properties of these connections, but in spite of the resemblance, it turns out to be a rather different topic. Yet in special geometric situations it may happen that ideas from Weyl geometry can be useful. Definition 2.2 (Connection with skew-symmetric torsion). The connection r is said to have (totally) skew-symmetric torsion if its torsion tensor lies in the second component of the decomposition in Proposition 2.1, i.e., it is given by a 3-form. They are by now – for reasons to be detailed later – a well-established tool in superstring theory and weak holonomy theories (see for example [222], [185], [117], [75], [88], [84], [10] etc.). In Examples 2.2 to 2.5, we describe large classes of interesting manifolds that carry natural connections with skew-symmetric torsion. Geometrically, these connections can be characterized as those which are metric and geodesic-preserving. In contrast to the case of vectorial torsion, manifolds admitting invariant metric connections r with r-parallel skew-symmetric torsion form a vast class that is worth a separate investigation ([20], [74], [212]). Suppose now that we are given a metric connection r with torsion on a Riemannian spin manifold .M n ; g/ with spin bundle †M n . We slightly modify our notation and

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write r as

rX Y ´ rXg Y C AX Y;

where AX defines an endomorphism TM n ! TM n for every X. The condition for r to be metric g.AX Y; Z/ C g.Y; AX Z/ D 0 means that AX preserves the scalar product g, which can be expressed as AX 2 so.n/. After identifying so.n/ with ƒ2 .Rn /, AX can be written relative to some orthonormal frame X ˛ij ei ^ ej : AX D i <j

Since the lift into spin.n/ of ei ^ ej is Ei Ej =2, AX defines an element in spin.n/ and hence an endomorphism of the spinor bundle. In fact, we need not introduce a different notation for the lift of AX . Rather, observe that if AX is written as a 2-form, .1/ its action on a vector Y as an element of so.n/ is just AX Y D Y ³ AX , so our connection takes on vectors the form rX Y D rXg Y C Y ³ AX ; .2/ the action of AX on a spinor as an element of spin.n/ is just AX D .1=2/ AX , where denotes the Clifford product of a k-form by a spinor. The lift of the connection r to the spinor bundle †M n (again denoted by r) is thus given by rX

D rXg

C

1 AX : 2

We denote by .; / the Hermitian product on the spinor bundle †M n induced by g. When lifted to the spinor bundle, r satisfies the following properties that are well known for the lift of the Levi-Civita connection. In fact, the proof easily follows from the corresponding properties for the Levi-Civita connection [96, p. 59] and the Hermitian product [96, p. 24]. Lemma 2.1. The lift of any metric connection r on TM n into the spinor bundle †M n satisfies rX .Y

/ D .rX Y /

C Y .rX /;

X.

1;

2/

D .rX

1;

2/

C.

1 ; rX

2 /:

Any spinorial connection with the second property is again called metric. The first property (chain rule for Clifford products) makes only sense for spinorial connections that are lifts from the tangent bundle, not for arbitrary spin connections. Example 2.1 (Connection with vectorial torsion). For a metric connection with vectorial torsion given by V 2 TM , AX D 2 X ^ V , since Y ³ .2 X ^ V / D 2.X ^ V /.Y; / D .X ˝ V /.Y; / .V ˝ X /.Y; / D g.X; Y /V g.Y; V /X:


289

Example 2.2 (Connection with skew-symmetric torsion). For a metric connection with skew-symmetric torsion defined by some T 2 ƒ3 .M /, AX D X ³ T . Examples of manifolds with a geometrically defined torsion 3-form are given in the next section. Example 2.3 (Connection defined by higher order differential forms). As example of a metric spinorial connection not induced from the tangent bundle, consider rX

´ rXg

C .X ³ ! k /

C .X ^ l /

for some forms ! k 2 ƒk .M /, l 2 ƒl .M / (k; l 4). These are of particular interest in string theory as they are used for the description of higher dimensional membranes ([9], [205]). Example 2.4 (General case). The class A0 of Proposition 2.1 cannot be directly interpreted as vectors or forms of a given degree, but it is not complicated to construct elements in A0 either. Connections of this type have not yet been investigated as a class of their own, but they are used as an interesting tool in several contexts – for example, in closed G2 -geometry ([57], [70]). The canonical connection of an almost Kähler manifold is also of this type. What makes metric connections with torsion so interesting is the huge variety of geometric situations that they unify in a mathematically useful way. Let us illustrate this fact by some examples.

2.2 Naturally reductive spaces Naturally reductive spaces are a key example of manifolds with a metric connection with skew-symmetric torsion. Consider a Riemannian homogeneous space M D G=H . We suppose that M is reductive, i.e., the Lie algebra g of G splits as vector space direct sum of the Lie algebra h of H and an Ad.H /-invariant subspace m: g D h ˚ m and Ad.H /m m, where Ad W H ! SO.m/ is the isotropy representation of M . We identify m with T0 M and we pull back the Riemannian metric h ; i0 on T0 M to an inner product h ; i on m. By a theorem of Wang ([170, Chapter X, Theorem 2.1]), there is a one-to-one correspondence between the set of G-invariant metric affine connections and the set of linear mappings ƒm W m ! so.m/ such that ƒm .hXh1 / D Ad.h/ƒm .X/Ad.h/1

for X 2 m and h 2 H:

A homogeneous Riemannian metric g on M is said to be naturally reductive (with respect to G) if the map ŒX; m W m ! m is skew-symmetric, g.ŒX; Y m ; Z/ C g.Y; ŒX; Zm / D 0 t

for all X; Y; Z 2 m:

The family of metric connections r defined by ƒtm .X /Y WD .1 t /=2 ŒX; Y m has then skew-symmetric torsion T t .X; Y / D t ŒX; Y m . The connection r 1 is

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of particular interest and is called the canonical connection. Naturally reductive homogeneous spaces equipped with their canonical connection are a well studied (see for example [78]) generalization of symmetric spaces since they satisfy r 1 T 1 D r 1 R1 D 0, where R1 denotes the curvature tensor of r 1 (Ambrose–Singer, [25]; see [234, Theorem 2.3] for a converse). Naturally reductive spaces have been classified in small dimensions by Kowalski, Tricerri and Vanhecke, partially in the larger context of commutative spaces (in the sense of Gel’fand): the 3-dimensional naturally reductive homogeneous spaces are SU.2/, the universal covering group of SL.2; R/ and the Heisenberg group H3 , all with special families of left-invariant metrics ([232]). A simply connected fourdimensional naturally reductive space is either symmetric or decomposable as direct product ([176]). In dimension 5, it is either symmetric, decomposable or locally isometric to SO.3/ SO.3/=SO.2/, SO.3/ H3 =SO.2/ (or any of these with SO.3/ replaced by SL.2; R/), to the five-dimensional Heisenberg group H5 or to the Berger sphere SU.3/=SU.2/ (or SU.2; 1/=SU.2/), all endowed with special families of metrics ([178]). Other standard examples of naturally reductive spaces are • Geodesic spheres in two-point homogeneous spaces,with the exception of the complex and quaternionic Cayley planes [248], [233] • Geodesic hyperspheres, horospheres and tubes around totally geodesic non-flat complex space forms, described and classified in detail by S. Nagai [194], [195], [196] • Simply connected '-symmetric spaces [45]. They are Sasaki manifolds with complete characteristic field for which reflections with respect to the integral curves of that field are global isometries. • All known left-invariant Einstein metrics on compact Lie groups [78]. In fact, every simple Lie group apart from SO.3/ and SU.2/ carries at least one naturally reductive Einstein metric other than the bi-invariant metric. Similarly, large families of naturally reductive Einstein metrics on compact homogeneous spaces were constructed in [242]. In contrast, non-compact naturally reductive Einstein manifolds are necessarily symmetric [125].

2.3 Almost Hermitian manifolds An almost Hermitian manifold .M 2n ; g; J / is a manifold with a Riemannian metric g and a g-compatible almost complex structure J W TM 2n ! TM 2n . We denote by .X; Y / WD g.JX; Y / its Kähler form and by N the Nijenhuis tensor of J , defined by N.X; Y / ´ ŒJX; J Y J ŒX; J Y J ŒJX; Y ŒX; Y g J /Y .rJg Y J /X; D .rXg J /J Y .rYg J /JX C .rJX


291

where the second expression follows directly from the vanishing of the torsion of r g and the identity (4) .rXg J /.Y / D rXg .J Y / J.rXg Y /: The reader is probably acquainted with the first canonical Hermitian connection1 (see the nice article [115] by Paul Gauduchon, which we strongly recommend for further reading on Hermitian connections) xX Y ´ r g Y C 1 .r g J /J Y: r X 2 X x D 0 is equivalent to the identity (4), and the antisymmetry Indeed, the condition rJ of the difference tensor g..rXg J /J Y; Z/ in Y and Z can for example be seen from the standard identity2 2 g..rXg J /Y; Z/ D d .X; Y; Z/ d .X; J Y; J Z/ C g.N.Y; Z/; JX/:

(5)

Sometimes, one finds the alternative formula 1=2J.rXg J /Y for the difference tensor x but this is the same, since J 2 D 1 implies r g J 2 D 0 D .r g J /J C J.rX J /, of r; X X i.e., r g J 2 u.n/? so.2n/. Let us now express the difference tensor of the x using the Nijenhuis tensor and the Kähler form. Since r g .Y; Z/ D connection r X g g .rX J /Y; Z , the differential d is just g J /X; Y : (6) d .X; Y; Z/ D g .rXg J /Y; Z g .rYg J /X; Z C g .rZ Together with the expression for N in terms of covariant derivatives of J , this yields g..rXg J /J Y; Z/ D g N.X; Y /; Z C d .JX; J Y; J Z/ g..rYg J /J Z; X / g..rJgZ J /Y; X /: A priori, .rXg J /Y has no particular symmetry properties in X and Y , hence the last two terms cannot be simplified any further (in general, they are a mixture of the two other Cartan types). An exceptional situation occurs if M is nearly Kähler (.rXg J /X D 0), for then .rXg J /Y D .rYg J /X and the last two terms cancel each other. Furthermore, this antisymmetry property implies that the difference tensor is totally skew-symmetric, hence we can conclude: Lemma 2.2. On a nearly Kähler manifold .M 2n ; g; J /, the formula xX Y ´ r g Y C 1 .r g J /J Y D r g Y C 1 ŒN.X; Y / C d .JX; J Y; J / r X X 2 X 2 defines a Hermitian connection with totally skew-symmetric torsion. This connection – the so-called characteristic connection – was first defined and studied by Alfred Gray (see [127, p. 304] and [129, p. 237]). It is a non-trivial result 1 By definition, a connection r is called Hermitian if it is metric and has r-parallel almost complex structure J .

2 For a proof, see [170, Proposition 4.2]. Be aware of the different conventions in this book: is defined with J in the second argument, the Nijenhuis tensor is twice our N and derivatives of k-forms differ by a multiple of 1=k, see [169, Proposition 3.11].

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x of Kirichenko that it has r-parallel torsion ([167], see also [22] for a modern indexfree proof). Furthermore, it is shown in [129] that any 6-dimensional nearly Kähler manifold is Einstein and of constant type, i.e., it satisfies scalg kXk2 kY k2 g.X; Y /2 g.X; J Y /2 : 30 Together with Lemma 2.2, this identity yields by a direct calculation that any 6x x dimensional nearly Kähler manifold is also r-Einstein with Ricr D 2.scalg =15/ g (see Theorem A.1 for the relation between Ricci tensors). Now let us look for a Hermitian connection with totally skew-symmetric torsion on a larger class of Hermitian manifolds generalizing nearly Kähler manifolds. k.rXg J /.Y /k2 D

Lemma 2.3 ([102]). Let .M 2n ; g; J / be an almost Hermitian manifold with skewsymmetric Nijenhuis tensor N.X; Y; Z/ WD g.N.X; Y /; Z/. Then the formula 1 ŒN.X; Y; Z/ C d .JX; J Y; J Z/ 2 defines a Hermitian connection with skew-symmetric torsion. g.rX Y; Z/ ´ g.rXg Y; Z/ C

Besides nearly Kähler manifolds, Hermitian manifolds (N D 0) trivially fulfill the condition of the preceding lemma and r coincides then with the Bismut connection; however, in the non-Hermitian situation, r is not in the standard family of canonical Hermitian connections that is usually considered (see [115, 2.5.4]). Proposition 2 in this same reference gives the decomposition of the torsion of any Hermitian connection in its .p; q/-components and gives another justification for this precise form for the torsion. Later, we shall see that r is the only possible Hermitian connection with skew-symmetric torsion and that the class of almost Hermitian manifolds with skewsymmetric Nijenhuis tensor is the largest possible where it is defined. We will put major emphasis on almost Hermitian manifolds of dimension 6, although one will find some general results formulated independently of the dimension. Two reasons for this choice are that nearly Kähler manifolds are of interest only in dimension 6, and that 6 is also the relevant dimension in superstring theory.

2.4 Hyper-Kähler manifolds with torsion (HKT-manifolds) We recall that a manifold M is called hypercomplex if it is endowed with three (integrable) complex structures I , J , K satisfying the quaternionic identities IJ D JI D K. A metric g compatible with these three complex structures (a so-called hyper-Hermitian metric) is said to be hyper-Kähler with torsion or just a HKT-metric if the Kähler forms satisfy the identity I d I D J d J D K d K :

(7)

Despite the misleading name, these manifolds are not Kähler (and hence even less hyper-Kähler). HKT-metrics were introduced by Howe and Papadopoulos as target


293

spaces of some two-dimensional sigma models with .4; 0/ supersymmetry with Wess– Zumino term [144]. Their mathematical description was given by Grantcharov and Poon in [126] and further investigated by several authors since then (see for example [239], [82], [204], [93], [94], [150]). From the previous example, we can conclude immediately that 1 d I .IX; I Y; IZ/ 2 defines a metric connection with skew-symmetric torsion such that rI D rJ D rK D 0; one easily checks that r is again the only connection fulfilling these conditions. Equation (7) implies that we could equally well have chosen J or K in the last term. In general, a hyper-Hermitian manifold will not carry an HKT-structure, except in dimension 4 where this is proved in [116]. Examples of homogeneous HKTmetrics can be constructed using a family of homogeneous hypercomplex structures associated with compact semisimple Lie groups constructed by Joyce [155]. Inhomogeneous HKT-structures exist for example on S 1 S 4n3 [126]. The question of existence of suitable potential functions for HKT-manifolds was first raised and discussed in the context of super-conformal quantum mechanics by the physicists Michelson and Strominger [191] (a maximum principle argument shows that compact HKT-manifolds do not admit global potentials); Poon and Swann discussed potentials for some symmetric HKT-manifolds [203], while Banos and Swann were able to show local existence [32]. g.rX Y; Z/ ´ g.rXg Y; Z/ C

2.5 Almost contact metric structures An odd-dimensional manifold M 2nC1 is said to carry an almost contact structure if it admits a .1; 1/-tensor field ' and a vector field (sometimes called the characteristic or Reeb vector field) with dual 1-form (. / D 1) such that ' 2 D Id C ˝ . Geometrically, this means that M has a preferred direction (defined by ) on which ' 2 vanishes, while ' behaves like an almost complex structure on any linear complement of . An easy argument shows that '. / D 0 [44, Theorem 4.1]. If there exists in addition a '-compatible Riemannian metric g on M 2nC1 , i.e., satisfying g.'X; 'Y / D g.X; Y / .X/.Y /; then we say that .M 2nC1 ; g; ; ; '/ carries an almost contact metric structure or that it is an almost contact metric manifold. The condition says that is a vector field of unit length with respect to g and that g is '-compatible in the sense of Hermitian geometry on the orthogonal complement ? . Unfortunately, this relatively intuitive structural concept splits into a myriad of subtypes and leads to complicated equations in the defining data . ; ; '/, making the investigation of almost contact metric structures look rather unattractive at first sight (see [19], [66], [67], and [90] for a classification). Yet, they constitute a rich and particularly interesting class of non-integrable geometries, as they have no integrable analogue on Berger’s list. An excellent general source

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on contact manifolds with extensive references are the books by David Blair, [43] and [44] (however, the classification is not treated in these). The fundamental form F of an almost contact metric structure is defined by F .X; Y / D g.X; '.Y //, its Nijenhuis tensor is given by a similar, but slightly more complicated formula as in the almost Hermitian case and can also be written in terms of covariant derivatives of ', N.X; Y / ´ Œ'.X/; '.Y / 'ŒX; '.Y / 'Œ'.X/; Y C ' 2 ŒX; Y C d.X; Y / g D .rXg '/'.Y / .rYg '/'.X/ C .r'.X/ '/Y g .r'.Y '/X C .X /rYg .Y /rXg : /

Let us emphasize some particularly interesting cases. A manifold with an almost contact metric structure .M 2nC1 ; g; ; ; '/ is called .1/ a normal almost contact metric manifold if N D 0, .2/ a contact metric manifold if 2F D d. Furthermore, a contact metric structure is said to be a K-contact metric structure if is in addition a Killing vector field, and a Sasaki structure if it is normal (it is then automatically K-contact, see [44, Corollary 6.3]). Einstein–Sasaki manifolds are just Sasaki manifolds whose '-compatible Riemannian metric g is Einstein. Without doubt, the forthcoming monograph by Ch. Boyer and Kr. Galicki on Sasakian geometry [51] is set to become the standard reference for this area of contact geometry in the future; in the meantime, the reader will have to be contented with the shorter reviews [49] and [50]. Much less is known about metric connections on almost contact metric manifolds than on almost Hermitian manifolds. In fact, only the so-called generalized Tanaka connection (introduced by S. Tanno) has been investigated. It is a metric connection defined on the class of contact metric manifolds by the formula rX Y ´ rXg Y C .X/'.Y / .Y /rXg C .rX /.Y / satisfying the additional conditions r D 0 (which is of course equivalent to r D 0), see [227] and [44, 10.4]. One easily checks that its torsion is not skewsymmetric, not even in the Sasaki case. In fact, from the point of view of non-integrable structures, it seems appropriate to require in addition r ' D 0 (compare with the almost Hermitian situation). Following a similar but more complicated line of arguments as in the almost Hermitian case, Th. Friedrich and S. Ivanov showed: Theorem 2.1 ([102, Theorem 8.2]). Let .M 2nC1 ; g; ; ; '/ be an almost contact metric manifold. It admits a metric connection r with totally skew-symmetric torsion T and r D r' D 0 if and only if the Nijenhuis tensor N is skew-symmetric and if is a Killing vector field. Furthermore, r D r g C .1=2/T is uniquely determined by T D ^ d C d ' F C N ^ . ³ N /;


295

where d ' F stands for the '-twisted derivative, d ' F .X; Y; Z/ WD dF .'.X /; '.Y /; '.Z//: For a Sasaki structure, N D 0 and 2F D d implies d ' F D 0, hence T is given by the much simpler formula T D ^ d. This connection had been noticed before, for example in [179]. In fact, one sees that rT D 0 holds, hence Sasaki manifolds endowed with this connection are examples of non-integrable geometries with parallel torsion. A. Fino studied naturally reductive almost contact metric structures such that ' is parallel with respect to the canonical connection in [89]. In general, potentials are hardly studied in contact geometry (compare with the situation for HKT-manifolds), but a suitable analogue of the Kähler potential was constructed on Sasaki manifolds by M. Godlinski, W. Kopczynski and P. Nurowski [121].

2.6 3-Sasaki manifolds Similarly to HKT-manifolds and quaternionic-Kähler manifolds, it makes sense to investigate configurations with three ‘compatible’ almost metric contact structures .'i ; i ; i /, i D 1; 2; 3 on .M 2nC1 ; g/ for some fixed metric g. The compatibility condition may be formulated as 'k D 'i 'j j ˝ i D 'j 'i C i ˝ j ;

k D 'i j D 'j i

for any even permutation .i; j; k/ of .1; 2; 3/, and such a structure is called an almost contact metric 3-structure. By defining on the cone M 2nC1 R three almost complex structures J1 , J2 , J3 , one sees that it carries an almost quaternionic structure and hence has dimension divisible by 4. Consequently, almost contact metric 3-structures exist only in dimensions 4n C 3; n 2 N, and it is no surprise that the structure group of its tangent bundle turns out to be contained in Sp.n/ f1g. What is surprising is the recent result by T. Kashiwada that if all three structures .'i ; i ; i / are contact metric structures, they automatically have to be Sasakian [161]. A manifold with such a structure will be called a 3-Sasaki(an) manifold. An earlier result by T. Kashiwada claims that any 3-Sasaki manifold is Einstein [160]. The canonical example of a 3Sasaki manifold is the sphere S 4nC3 realized as a hypersurface in HnC1 : each of the three almost complex structures forming the quaternionic structure of HnC1 applied to the exterior normal vector field of the sphere yields a vector field i (i D 1; 2; 3) on S 4nC3 , leading thus to three orthonormal vector fields on S 4nC3 . Th. Friedrich and I. Kath showed that every compact simply connected 7-dimensional spin manifold with regular 3-Sasaki structure is isometric to S 7 or the Aloff–Wallach space N.1; 1/ D SU.3/=S1 (see [106] or [33]). By now, many non-homogeneous examples have been constructed and the homogeneous 3-Sasaki manifolds have been classified [52]. The analogy between 3-Sasaki manifolds and HKT-manifolds breaks down when one starts looking at connections, however. For an arbitrary 3-Sasaki structure with 1-forms i (i D 1; 2; 3), each of the three underlying Sasaki structures yields one

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possible choice of a metric connection r i with r i i D 0 and torsion T i D i ^ di as detailed in Theorem 2.1. However, these three connections do not coincide; hence, the 3-Sasaki structure itself is not preserved by any metric connection with skewsymmetric torsion. A detailed discussion of these three connections and their spinorial properties in dimension 7 can be found in [10]. Nevertheless, 3-Sasaki manifolds have recently appeared and been investigated in the context of the AdS/CFT-correspondence by Martelli, Sparks and Yau [186].

2.7 Holonomy theory Let .M n ; g/ be a (connected) Riemannian manifold equipped with any connection r. For a curve .s/ from p to q, parallel transport along is the linear mapping P W Tp M ! Tq M such that the vector field V defined by V.q/ ´ P V .p/

along

is parallel along , rV.s/=ds D rP V D 0. P is always an invertible endomorphism, hence, for a closed loop through p 2 M , it can be viewed as an element of GL.n; R/ (after choice of some basis). Consider the loop space C.p/ of all closed, piecewise smooth curves through p, and therein the subset C0 .p/ of curves that are homotopic to the identity. The set of parallel translations along loops in C.p/ or C0 .p/ forms a group acting on Rn Š Tp M , called the holonomy group Hol.pI r/ of r or the restricted holonomy group Hol0 .pI r/ of r at the point p. Let us now change the point of view from p to q, a path joining them; then Hol.qI r/ D P Hol.pI r/P1 and similarly for Hol0 .pI r/. Hence, all holonomy groups are isomorphic, so we drop the base point from now on. Customary notation for them is Hol.M I r/ and Hol0 .M I r/. Their action on Rn Š Tp M shall be called the (restricted ) holonomy representation. In general, it is only known that ([169, Theorem IV.4.2]) .1/ Hol.M I r/ is a Lie subgroup of GL.n; R/, .2/ Hol0 .p/ is the connected component of the identity of Hol.M I r/. If one assumes in addition – as we will do through this text – that r be metric, parallel transport becomes an isometry: for any two parallel vector fields V .s/ and W .s/, being metric implies

d rW .s/ rV.s/ g V .s/; W .s/ D g ; W .s/ C V.s/; D 0: ds ds ds Hence, Hol.M I r/ O.n/ and Hol0 .M I r/ SO.n/. For convenience, we shall henceforth speak of the Riemannian (restricted ) holonomy group if r is the LeviCivita connection, to distinguish it from holonomy groups in our more general setting. Example 2.5. This is a good moment to discuss Cartan’s first example of a space with torsion (see [63, p. 595]). Consider R3 with its usual Euclidean metric and the


connection

297

rX Y D rXg Y X Y;

corresponding, of course, to the choice T D 2 e1 ^ e2 ^ e3 . Cartan observed correctly that this connection has same geodesics than r g , but induces a different parallel transport. Indeed, consider the z-axis .t / D .0; 0; t /, a geodesic, and the vector field V which, in every point .t /, consists of the vector .cos t; sin t; 0/. Then one checks immediately that rgP V D P V , that is, the vector V is parallel transported according to a helicoidal movement. If we now transport the vector along the edges of a closed triangle, it will be rotated around three linearly independent axes, hence the holonomy algebra is hol.r/ D so.3/. Example 2.6 (Holonomy of naturally reductive spaces). Consider a naturally reductive space M n D G=H as in Example 2.2 with its canonical connection r 1 , whose torsion is T 1 .X; Y / WD ŒX; Y m . Recall that ad W h ! so.m/ denotes its isotropy representation. The holonomy algebra hol.r 1 / is the Lie subalgebra of ad.h/ so.m/ generated by the images under ad of all projections of commutators ŒX; Y h on h for X; Y 2 m, hol.r 1 / D Lie ad.ŒX; Y h / ad.h/ so.m/: For all other connections r t in this family, the general expression for the holonomy is considerably more complicated [170, Theorem X.4.1]. Remark 2.1 (Holonomy and contact properties). As we observed earlier, all contact structures are non-integrable and therefore not covered by Berger’s holonomy theorem. Via the cone construction, it is nevertheless possible to characterize them by a Riemannian holonomy property (see [52], [49]). Consider a Riemannian manifold .M n ; g/ and its cone over the positive real numbers N WD RC M n with the warped product metric gN WD dr 2 C r 2 g. Then, .M n ; g/ is /, that is, its positive cone is Kähler, .1/ Sasakian if and only if Hol.N I r g / U. nC1 2 .2/ Einstein–Sasakian if and only if Hol.N I r g / SU. nC1 /, that is, its positive 2 cone is a (non-compact) Calabi–Yau manifold, .3/ 3-Sasakian if and only if Hol.N I r g / Sp. nC1 /, that is, its positive cone is 4 hyper-Kähler.

The Lie algebra of any holonomy group can be determined by computing all parallel transports of its curvature tensor (Ambrose–Singer, 1953 [24]). Yet, the practical use of this result is severely restricted by the fact that the properties of the curvature transformation of a metric connection with torsion are more complicated than the Riemannian ones. For example, R.U; V / is still skew-adjoint with respect to the metric g, g.R.U; V /W1 ; W2 / D g.R.U; V /W2 ; W1 /; but there is no relation between g.R.U1 ; U2 /W1 ; W2 / and g.R.W1 ; W2 /U1 ; U2 / in general (but see Remark 2.2 below); in consequence, the Bianchi identities are less

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tractable, the Ricci tensor is not necessarily symmetric etc. As an example of these complications, we cite (see [151] for the case of skew-symmetric torsion): Theorem 2.2 (First Bianchi identity). .1/ A metric connection r with vectorial torsion V 2 TM n satisfies X;Y;Z

X;Y;Z

S R.X; Y /Z D S d V .X; Y /Z:

.2/ A metric connection r with skew-symmetric torsion T 2 ƒ3 .M n / satisfies X;Y;Z

S R.X; Y; Z; V / D d T .X; Y; Z; V / T .X; Y; Z; V / C .rV T /.X; Y; Z/; P where T is a 4-form that is quadratic in T defined by 2 T D niD1 .ei ³ T / ^ .ei ³ T / for any orthonormal frame e1 ; : : : ; en . Remark 2.2. Consequently, if the torsion T 2 ƒ3 .M n / of a metric connection with X;Y;Z

skew-symmetric torsion happens to be r-parallel, S R.X; Y; Z; V / is a 4-form and thus antisymmetric. Since the cyclic sum over all four arguments of any 4-form vanishes, we obtain X;Y;Z;V X;Y;Z S S R.X; Y; Z; V / D 2 R.Z; X; Y; V / 2 R.Y; V; Z; X/ D 0; as for the Levi-Civita connection. Thus the r-curvature tensor is invariant under swaps of the first and second pairs of arguments. Extra care has to be taken when asking which properties of the Riemannian holonomy group are preserved. For example, there are many instances of irreducible manifolds with metric connections whose holonomy representation is not irreducible. This implies that no analogue of de Rham’s splitting theorem can hold. Similarly, there is no theoretical argument ensuring the closure of the restricted holonomy group of a metric connection with torsion, although the examples that I know of this effect are rather pathological. We are particularly interested in the vector bundle of .r; s/-tensors T r;s M over n M , that of differential k-forms ƒk M and its spinor bundle †M (assuming that M is spin, of course). At some point p 2 M , the fibers are just .Tp M /r ˝ .Tp M /s , ƒk Tp M or n , the n-dimensional spin representation (which has dimension 2Œn=2 ); an element of the fibre at some point will be called an algebraic tensor, form, spinor or just algebraic vector for short. Then, on any of these bundles, .1/ the holonomy representation induces a representation of Hol.M I r/ on each fibre (the “lifted holonomy representation”); .2/ the metric connection r induces a connection (again denoted by r) on these vector bundles (the “lifted connection”) compatible with the induced metric (for tensors) or the induced Hermitian scalar product (for spinors), it is thus again metric;


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.3/ in particular, there is a notion of “lifted parallel transport” consisting of isometries, and its abstract holonomy representation on the fibres coincides with the lifted holonomy representation. We now formulate the general principle underlying our study. Theorem 2.3 (General Holonomy Principle). Let M be a differentiable manifold and E a (real or complex) vector bundle over M endowed with (any!) connection r. The following three properties are equivalent: .1/ E has a global section ˛ invariant under parallel transport, i.e., ˛.q/ D P .˛.p// for any path from p to q; .2/ E has a parallel global section ˛, i.e., r˛ D 0; .3/ at some point p 2 M , there exists an algebraic vector ˛0 2 Ep which is invariant under the holonomy representation on the fibre. The following two consequences are immediate, but of the utmost importance. Corollary 2.1. (1) The number of parallel global sections of E coincides with the number of trivial representations occurring in the holonomy representation on the fibres. (2) The holonomy group Hol.r/ is a subgroup of the isotropy group G˛ ´ fg 2 O.n/ j g ˛ D ˛g of any parallel global section ˛ of E. This is a powerful tool for (dis-)proving existence of parallel objects. Example 2.7. The fact that the determinant is an SO.n/-invariant element in ƒn .Rn / which is not O.n/-invariant implies that a Riemannian manifold .M n ; g/ is orientable if and only if the holonomy Hol.M I r/ of any metric connection r is a subgroup of SO.n/. Furthermore, the volume form is then r-parallel. Remark 2.3. In fact, an arbitrary P connection r admits a r-parallel n-form (possibly with zeroes) if and only if g.R.U; V /ei ; ei / D 0 for any orthonormal frame e1 ; : : : ; en . This property is weaker than the skew-adjointness of R.U; V / that holds for all metric connections; the holonomy is a subgroup of SL.n; R/. In 1924, J. A. Schouten called such connections “inhaltstreue “Ubertragungen” (volumepreserving connections), see [213, p. 89]. This terminology seems not to have been used anymore afterwards3 . The existence of parallel objects imposes restrictions on the curvature of the connection. For example, if a connection r admits a parallel spinor , we obtain by contracting the identity 0 D rr

D

n X

Rr .ei ; ej /ei ej

i;j D1

D’Atri space is a Riemannian manifold whose local geodesic symmetries are volume-preserving. Although every naturally reductive space is a D’Atri space [77], the two notions are only loosely related. 3A

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the following integrability condition: Proposition 2.2. Let .M n ; g/ be a Riemannian spin manifold, r a metric connection with torsion T 2 ƒ3 .M n /. A r-parallel spinor satisfies

1 X ³ d T C rX T D Ricr .X / 2 In particular, the existence of a r g -parallel spinor .T D 0/ implies Ricci-flatness.

3 Geometric stabilizers By the General Holonomy Principle, geometric representations with invariant objects are a natural source for parallel objects. This leads to the systematic investigation of geometric stabilizers, which we shall now discuss.

3.1 U.n/ and SU.n/ in dimension 2n A Hermitian metric h.V; W / D g.V; W / ig.J V; W / is invariant under A 2 End.R2n / if and only if A preserves the Riemannian metric g and the Kähler form .V; W / WD g.J V; W /. Thus U.n/ is embedded in SO.2n/ as U.n/ D fA 2 SO.2n/ j A D g: To fix ideas, choose a skew-symmetric endomorphism J of R2n with square 1 in the normal form

0 1 J D diag.j; j; j; : : : / with j D : 1 0 Then a complex .n n/-matrix A D .aiij / 2 U.n/ is realized as a real .2n 2n/h Re aij Im aij matrix with .22/-blocks Im aij Re aij . An adapted orthonormal frame is one such that J has the given normal form; U.n/ consists then exactly of those endomorphisms transforming adapted orthonormal frames into adapted orthonormal frames. Allowing now complex coefficients, one obtains an .n; 0/-form ‰ by declaring ‰ ´ .e1 C ie2 / ^ ^ .en1 C i e2n / μ ‰ C C i‰ in the adapted frame above. An element A 2 U.n/ acts on ‰ by multiplication with det A. Lemma 3.1. Under the restricted action of U.n/, ƒ2k .R2n / contains the trivial representation once; it is generated by ; 2 ; : : : ; n . The action of U.n/ O.2n/ cannot be lifted to an action of U.n/ inside Spin.2n/ – reflecting the fact that not every Kähler manifold is spin. For the following arguments


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though, it is enough to consider u.n/ inside spin.2n/. It then appears that u.n/ has no invariant spinors, basically because u.n/ has a one-dimensional center, generated precisely by after identifying ƒ2 .R2n / Š so.2n/. Hence one-dimensional u.n/representations are usually not trivial. More precisely, the complex 2n-dimensional spin representation 2n splits into two irreducible components ˙ 2n described in terms of eigenspaces of 2 u.n/. Set (see [96] and [164] for details on this decomposition of spinors) ! n ; 0 r n: Sr D f 2 2n j D i.n 2r/ g; dim Sr D r Sr is isomorphic to the space of .0; r/-forms with values in S0 (which explains the dimension), Sr Š ƒ0;r ˝ S0 : Since the spin representations decompose as ˇ ˇ ˇ ˇ C 2n u.n/ Š Sn1 ˚ Sn3 ˚ 2n u.n/ Š Sn ˚ Sn2 ˚ ; we conclude immediately that they cannot contain a trivial u.n/-representation for n odd. For n D 2k even, has zero on Sk , but this space is an irreducible eigenvalue representation of dimension 2k ¤ 1, hence not trivial either. The representations S0 k and Sn are one-dimensional, but again not trivial under u.n/. If one restricts further to su.n/, they are indeed: Lemma 3.2. The spin representations ˙ 2n contain no u.n/-invariant spinor. If one restricts further to su.n/, there are exactly two invariant spinors (both in C 2n for n for n odd). even, one in each ˙ 2n All other spinors in ˙ 2n have geometric stabilizer groups that do not act irreducibly on the tangent representation R2n . They can be described explicitly in a similar way; By de Rham’s splitting theorem, they do not appear in the Riemannian setting. To finish, we observe that the almost complex structure J (and hence ) can be C recovered from the invariant spinor C 2 C WD iX C (X 2 TM ), a 2n by J.X / formula well known from the investigation of Killing spinors on 6-dimensional nearly Kähler manifolds (see [133] and [33, Section 5.2]). Remark 3.1. The group Sp.n/ SO.4n/ can be deduced from the previous discussion: Sp.n/ with quaternionic entries aCbj is embedded into SU.2n/ by .22/-blocks i h a b , and SU.2n/ sits in SO.4n/ as before. We shall not treat Sp.n/- and quaterbN aN nionic geometries in this exposition (but see [223], [224], [18], [126], [20], [204], [17], [189] for a first acquaintance).

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3.2 U.n/ and SU.n/ in dimension 2n C 1 These G-structures arise from contact structures and are remarkable inasmuch they manifest a genuinely non-integrable behaviour – they do not occur in Berger’s list because the action of U.n/ on R2nC1 is not irreducible, hence any manifold with this action as Riemannian holonomy representation splits by de Rham’s theorem. Given an almost contact metric manifold .M 2nC1 ; g; ; ; '/, we may construct an adapted local orthonormal frame by choosing any e1 2 ? and setting e2 D '.e1 / (as well as fixing e2nC1 D once and for all); now choose any e3 perpendicular to e1 ; e2 ; e2nC1 and set again e4 D '.e3 / etc. With respect to such a basis, ' is given by

0 1 ' D diag.j; j; : : : ; j; 0/ with j D 1 0 and we conclude that the structural group of M 2nC1 is reducible to U.n/ f1g. If we denote the fundamental form by F (see Example 2.5), then U.n/ f1g D fA 2 SO.2n C 1/ j A F D F g: The U.n/ f1g-action on R2nC1 inherits invariants from the U.n/-action on R2n in a canonical way; one then just needs to check that no new one appears. Hence, we can conclude: Lemma 3.3. Under the action of U.n/, ƒ2k .R2nC1 / contains the trivial representation once; it is generated by F; F 2 ; : : : ; F n . The action of U.n/ O.2n C 1/ cannot, in general, be lifted to an action of U.n/ inside Spin.2n C 1/. As in the almost Hermitian case, let us thus study the u.n/ action on 2nC1 . The irreducible Spin.2n C 1/-module 2nC1 splits into C 2n ˚ 2n under the restricted action of Spin.2n/, and it decomposes accordingly into 2nC1 D S0 ˚ ˚ Sn ; where Sr D f

2 2nC1 j F

D i.n 2r/ g

and

! n dim Sr D ; r

0 r n:

Hence, 2nC1 can be identified with C 2n ˚ 2n , yielding finally the following result:

Lemma 3.4. The spin representation 2nC1 contains no u.n/-invariant spinor. If one restricts further to su.n/, there are precisely two invariant spinors.

3.3 G2 in dimension 7 While invariant 2-forms exist in all even dimensions and lead to the rich variety of almost Hermitian structures, the geometry of 3-forms played a rather exotic role


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in classical Riemannian geometry until the nineties, as it occurs only in apparently random dimensions, most notably dimension seven. That G2 is the relevant simple Lie group is a classical, although unfortunately not so well known result from invariant theory. A mere dimension count shows already this effect (see Table 3): the stabilizer of a generic 3-form ! 3 G!n 3 ´ fA 2 GL.n; R/ j ! 3 D A ! 3 g cannot be contained in the orthogonal group for n 6, it must lie in some group between SO.n/ and SL.n; R/ (for n D 3, we even have G!3 3 D SL.3; R/). The Table 3. Dimension count for possible geometries defined by 3-forms.

n

dim GL.n; R/ dim ƒ3 Rn

dim SO.n/

3

91D8

3

4

16 4 D 12

6

5

25 10 D 15

10

6

36 20 D 16

15

7

49 35 D 14

21

8

64 56 D 8

28

case n D 7 is the first dimension where G!n 3 can sit in SO.n/. That this is indeed the case was shown as early as 1907 in the doctoral dissertation of Walter Reichel in Greifswald, supervised by F. Engel ([206]). More precisely, he computed a system of invariants for a 3-form in seven variables and showed that there are exactly two open GL.7; R/-orbits of 3-forms. The stabilizers of any representatives ! 3 and !Q 3 of these orbits are 14-dimensional simple Lie groups of rank two, one compact and the other non-compact: G!7 3 Š G2 SO.7/;

G!7Q 3 Š G2 SO.3; 4/:

Reichel also showed the corresponding embeddings of Lie algebras by explicitly writing down seven equations for the coefficients of so.7/ resp. so.3; 4/. As in the case of almost Hermitian geometry, every author has his or her favourite normal 3-form with isotropy group G2 , for example, ! 3 ´ e127 C e347 e567 C e135 e245 C e146 C e236 : An element of the second orbit with stabilizer the split form G2 of G2 may be obtained by reversing any of the signs in ! 3 .

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Lemma 3.5. Under G2 , one has the decomposition ƒ3 .R7 / Š R ˚ R7 ˚ S0 .R7 /, where R7 denotes the 7-dimensional standard representation given by the embedding G2 SO.7/ and S0 .R7 / denotes the traceless symmetric endomorphisms of R7 (of dimension 27). Now let us consider the spinorial picture, as G2 can indeed be lifted to a subgroup of Spin.7/. From a purely representation theoretic point of view, this case is trivial: dim 7 D 8 and the only irreducible representations of G2 of dimension 8 are the trivial and the 7-dimensional representations. Hence 8 D 1 C 7 yields: Lemma 3.6. Under the restricted action of G2 , the 7-dimensional spin representation 7 decomposes as 7 Š R ˚ R7 . This lemma has an important consequence: the ‘spinorial’ characterization of G2 manifolds. Corollary 3.1. Let .M 7 ; g/ be a Riemannian manifold, r a metric connection on its spin bundle. Then there exists a r-parallel spinor if and only if Hol.r/ G2 . One direction follows from the fact that G2 is the stabilizer of an algebraic spinor, the converse from Lemma 3.6. In fact, the invariant 3-form and the invariant algebraic spinor are equivalent data. They are related (modulo an irrelevant constant) by ! 3 .X; Y; Z/ D hX Y Z ; i:

(8)

We now want to ask which subgroups G G2 admit other invariant algebraic spinors. Such a subgroup has to appear on Berger’s list and its induced action on R7 (viewed as a subspace of 7 ) has to contain one or more copies of the trivial representation. Thus, the only possibilities are SU.3/ with R7 Š R ˚ C 3 (standard SU.3/-action on C 3 ) and SU.2/ with R7 Š 3 R ˚ C 2 (standard SU.2/-action on C 2 ). Both indeed occur, with a total of 2 resp. 4 invariant spinors. Remark 3.2. A modern account of Reichel’s results can be found in the article [243] by R. Westwick; it is interesting (although it seems not to have had any further influence) that J. A. Schouten also rediscovered these results in 1931 [214]. A classification of 3-forms is still possible in dimensions 8 ([134], [134], [81]) and 9 ([240]), although the latter one is already of inexorable complexity. Based on these results, J. Bureš and J. Vanžura started recently the investigation of so-called multisymplectic structures ([237], [60], [59]). For a historically account on the discovery of G2 and its impact on differential geometry we refer to [6].


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3.4 Spin.7/ in dimension 8 As we just learned from G2 geometry, Spin.7/ has an irreducible 8-dimensional representation isomorphic to 7 , hence it can be viewed as a subgroup of SO.8/, and it does lift to Spin.8/. By restricting to SO.7/, Spin.7/ certainly also has a 7-dimensional representation. What is so special in this dimension is that Spin.7/ has two conjugacy classes in SO.8/ that are interchanged by means of the triality automorphism; hence the decomposition of the spin representation depends on the (arbitrary) choice of one of these classes. Lemma 3.7. Under the restricted action of Spin.7/, the 8-dimensional spin repre 8 7 sentations decompose as C 8 Š R Š 7 and 8 Š R ˚ R for one choice of C Spin.7/ SO.8/; the other choice swaps 8 and 8 . In particular, there is exactly one invariant spinor in C 8 . Again, it corresponds one-to-one to an invariant form, of degree 4 in this case: ˇ 4 .X; Y; Z; V / D hX Y Z V ; i: Yet, Spin.7/-geometry in dimension eight is not just an enhanced version of G2 geometry in dimension seven. Because dim GL.8; R/ D 64 < 70 D dim ƒ4 .R8 /, there are no dense open orbits under the action of GL.8; R/. Thus, there is no result in invariant theory similar to that of Reichel for G2 in the background. Let us fix the first choice for embedding Spin.7/ in SO.8/ made in Lemma 3.7. A C second invariant spinor can either be in C 8 or in 8 . If it is in 8 , we are asking for 7 a subgroup G Spin.7/ whose action on R contains the trivial representation once – obviously, G2 is such a group. Under G2 , C 8 and 8 are isomorphic, ˇ ˇ ˇ 7 ˇ C 8 G Š 8 G Š R ˚ R : 2

2

Thus, SU.3/ G2 Spin.7/ and SU.2/ G2 Spin.7/ are two further admissible groups with 2C2 and 4C4 invariant spinors. On the other hand, if we impose a second invariant spinor to live in 8 , we need a subgroup G Spin.7/ that has partially 8 trivial action on R7 , but not on C 8 Š R . A straightforward candidate is G D Spin.6/ with its standard embedding and R7 D R6 ˚ R; the classical isomorphism 4 Spin.6/ D SU.4/ shows that G acts irreducibly on C 8 Š C . The group SU.4/ in turn has subgroups Sp.2/ D Spin.5/ and SU.2/ SU.2/ D Spin.4/ that still act irreducibly on C 8 , and act on 8 by ˇ ˇ 5 4 ˇ ˇ 8 Sp.2/ Š 3 R ˚ R ; 8 SU.2/SU.2/ Š 4 R ˚ R : The results are summarized in Table 4. The resemblance between Tables 4 and 2 in Section 1.4 is no coincidence. A convenient way to describe Spin.9; 1/ is to start with Spin.10/ generated by elements e1 ; : : : ; e10 and acting irreducibly on C 10 . The vector C C spaces 10 and 9;1 can be identified, and Spin.9; 1/ can be generated by ei WD ei

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for i D 1; : : : ; 9 and e10 WD i e10 . Elements ! 2 spin.9; 1/ can thus be written X X X !ij !ij ei ^ ej D !ij ei ^ ej C i !k;10 ek ^ e10 !D 1i<j 10

i<j 9

k9

and we conclude that spin.9; 1/ can be identified with spin.9/ Ë R9 . A spinor 2 C 9;1 is stabilized by an element ! 2 spin.9; 1/ if and only if X X X 0D !ij ei ej C i !k;10 ek e10 : !ij ei ej D 1i <j 10

i<j 9

k9

In this last expression, both real and imaginary part have to vanish simultaneously, leading to 16 equations. A careful look reveals that they define spin.7/ Ë R8 , and the C statements from Table 4 imply those from Table 2 since C 9;1 Š 8 ˚ 8 . Table 4. Possible stabilizers of invariant spinors in dimension 8.

group

invariant spinors in C 8

invariant spinors in 8

Spin.7/

0

1

SU.4/ Š Spin.6/

0

2

Sp.2/ Š Spin.5/

0

3

SU.2/ SU.2/ Š Spin.4/

0

4

G2

1

1

SU.3/

2

2

SU.2/

4

4

feg

8

8

Remark 3.3 (Weak PSU.3/-structures). Recently, Hitchin observed that 3-forms can be of interest in 8-dimensional geometry as well ([141]). The canonical 3-form on the Lie algebra su.3/ spans an open orbit under GL.8; R/, and the corresponding 3-form on SU.3/ is parallel with respect to the Levi-Civita connection of the biinvariant metric. The Riemannian holonomy reduces to SU.3/=Z2 DW PSU.3/. More generally, manifolds modelled on this group lead to the investigation of closed and coclosed 3-forms that are not parallel, see also [245]. Sorting out the technicalities that we purposely avoided, one obtains Wang’s classification of Riemannian parallel spinors. By de Rham’s theorem, only irreducible


307

holonomy representations occur for the Levi-Civita connection. From Proposition 2.2, we already know that these manifolds are Ricci-flat. Theorem 3.1 (Wang’s Theorem, [241]). Let .M n ; g/ be a complete, simply connected, irreducible Riemannian manifold of dimension n. Let N denote the dimension of the space of parallel spinors with respect to the Levi-Civita connection. If .M n ; g/ is non-flat and N > 0, then one of the following holds: .1/ n D 2m (m 2), the holonomy representation is the vector representation of SU.m/ on C m , and N D 2 (“Calabi–Yau case”). .2/ n D 4m (m 2), the holonomy representation is the vector representation of Sp.m/ on C 2m , and N D m C 1 (“hyper-Kähler case”). .3/ n D 7, the holonomy representation is the unique 7-dimensional representation of G2 , and N D 1 (“parallel G2 - or Joyce case”). .4/ n D 8, the holonomy representation is the spin representation of Spin.7/, and N D 1 (“parallel Spin.7/- or Joyce case”).

4 A unified approach to non-integrable geometries 4.1 Motivation For G-structures defined by some tensor T , it has been for a long time customary to classify the possible types of structures by the isotypic decomposition under G of the covariant derivative r g T . The integrable case is described by r g T D 0, all other classes of non-integrable G-structures correspond to combinations of non-vanishing contributions in the isotypic decomposition and are described by some differential equation in T . This was carried out in detail for example for almost Hermitian manifolds (Gray/Hervella [130]), for G2 -structures in dimension 7 (Fernández/Gray [87]), for Spin.7/-structures in dimension 8 (Fernández [85]) and for almost contact metric structures (Chinea/Gonzales [66]). In this section, we shall present a simpler and unified approach to non-integrable geometries. The theory of principal fibre bundles suggests that the difference between the Levi-Civita connection and the canonical G-connection induced on the G-structure is a good measure for how much the given G-structure fails to be integrable. By now, is widely known as the intrinsic torsion of the G-structure (see Section 1.5 for references). Although this is a “folklore” approach, it is still not as popular as it could be. Our presentation will follow the main lines of [99]. We will see that it easily reproduces the classical results cited above with much less computational work whilst having the advantage of being applicable to geometries not defined by tensors. Furthermore, it allows a uniform and clean description of those classes of geometries admitting G-connections with totally skew-symmetric torsion, and led to the discovery of new interesting geometries.

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4.2 G -structures on Riemannian manifolds Let G SO.n/ be a closed subgroup of the orthogonal group and decompose the Lie algebra so.n/ into the Lie algebra g of G and its orthogonal complement m, i.e., so.n/ D g ˚ m. Denote by prg and prm the projections onto g and m, respectively. Consider an oriented Riemannian manifold .M n ; g/ and denote its frame bundle by F .M n /; it is a principal SO.n/-bundle over M n . By definition, a G-structure on M n is a reduction R F .M n / of the frame bundle to the subgroup G. The Levi-Civita connection is a 1-form Z on F .M n / with values in the Lie algebra so.n/. We restrict the Levi-Civita connection to R and decompose it with respect to the splitting g ˚ m: ˇ Z ˇT .R/ WD Z ˚ : Then, Z is a connection in the principal G-bundle R and is a tensorial 1-form of type Ad, i.e., a 1-form on M n with values in the associated bundle R G m. By now, it has become standard to call the intrinsic torsion of the G-structure (see Section 1.5 for references). The G-structure R on .M n ; g/ is called integrable if vanishes, for this means that it is preserved by the Levi-Civita connection and that Hol.r g / is a subgroup of G. All G-structures with ¤ 0 are called non-integrable; the basic classes of non-integrable G-structures are defined – via the decomposition of – as the irreducible G-components of the representation Rn ˝ m. For an orthonormal frame e1 ; : : : ; en adapted to the reduction R, the connection forms !ij WD g.r g ei ; ej / of the Levi-Civita connection define a 1-form WD .!ij / with values in the Lie algebra so.n/ of all skew-symmetric matrices. The form can then be computed as the m-projection of ,

D prm ./ D prm .!ij /: Interesting is the case in which G happens to be the isotropy group of some tensor T . Suppose that there is a faithful representation % W SO.n/ ! SO.V / and a tensor T 2 V such that ˚ G D g 2 SO.n/ j %.g/T D T : The Riemannian covariant derivative of T is then given by r g T D % . /.T /, where % W so.n/ ! so.V / is the differential of the representation. As a tensor, r g T is an element of Rn ˝ V . The algebraic G-types of r g T define the algebraic G-types of

and vice versa. Indeed, we have Proposition 4.1 ([99, Proposition 2.1]). The G-map Rn ˝ m ! Rn ˝ End.V / ! Rn ˝ V given by ! . /.T / is injective. An easy argument in representation theory shows that for G ¤ SO.n/, the Grepresentation Rn does always appear as summand in the G-decomposition of Rn ˝m. Geometrically, this module accounts precisely for conformal transformations of Gstructures. Let .M n ; g; R/ be a Riemannian manifold with a fixed geometric structure and denote by gO WD e 2f g a conformal transformation of the metric. There is a natural


309

identification of the frame bundles y n ; g/ F .M n ; g/ Š F .M O y At the infinitesimal level, the conformal change and a corresponding G-structure R. is defined by the 1-form df , corresponding to an Rn -part in . We shall now answer the question under which conditions a given G-structure admits a metric connection r with skew-symmetric torsion preserving the structure. For this, consider for any orthonormal basis ei of m the G-equivariant map X .ei ³ T / ˝ ei : ‚ W ƒ3 .Rn / ! Rn ˝ m; ‚.T / WD i

Theorem 4.1 ([102, Proposition 4.1]). A G-structure R F .M n / of a Riemannian manifold admits a connection r with skew-symmetric torsion if and only if the 1-form

belongs to the image of ‚, 2 D ‚.T / for some T 2 ƒ3 .Rn /: In this case the 3-form T is the torsion form of the connection. Definition 4.1. A metric G-connection r with torsion T as in Theorem 4.1 will be called a characteristic connection and denoted by r c , T DW T c is called the characteristic torsion. By construction, the holonomy Hol.r c / is a subgroup of G. Thus, not every G-structure admits a characteristic connection. If that is the case, T c is unique for all geometries we have investigated so far, and it can easily be expressed in terms of the geometric data (almost complex structure etc.). Henceforth, we shall just speak of the characteristic connection. Due to its properties, it is an excellent substitute for the Levi-Civita connection, which in these situations is not adapted to the underlying geometric structure. Remark 4.1. The canonical connection r c of a naturally reductive homogeneous space is an example of a characteristic connection that satisfies in addition r c T c D r c Rc D 0; in this sense, geometric structures admitting a characteristic connection such that r c T c D 0 constitute a natural generalization of naturally reductive homogeneous spaces. As a consequence of the General Holonomy Principle (Corollary 2.1), r c T c D 0 implies that the holonomy group Hol.r c / lies in the stabilizer GT c of T c . With this technique, we shall now describe special classes of non-integrable geometries, some new and others previously encountered. We order them by increasing dimension.

4.3 Almost contact metric structures At this stage, almost contact metric structures challenge any expository article because of the large number of classes. Qualitatively, the situation is as follows. The first

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classifications of these structures proceeded in analogy to the Gray–Hervella set-up for almost Hermitian manifolds (see Section 4.5) by examining the space of tensors with the same symmetry properties as the covariant derivative of the fundamental form F (see Section 2.5) and decomposing it under the action of the structure group G D U.n/ f1g using invariant theory. Because the G-action on R2nC1 is already not irreducible, this space decomposes into four G-irreducibles for n D 1, into ten summands for n D 2 and into twelve for n 3, leading eventually to 24 , 210 and 212 possible classes of almost contact metric structures ([19], [66], [67]). Obviously, most of these classes do not carry names and are not studied, and the result being what it is, the investigation of such structures is burdened by technical details and assumptions. From the inner logic of non-integrable geometries, it makes not so much sense to base their investigation on the covariant derivative r g F of F or some other fundamental tensor, as the Levi-Civita connection does not preserve the geometric structure. This accounts for the technical complications that one faces when following this approach. For this section, we decided to restrict our attention to dimension five, this being the most relevant for the investigation of non-integrable geometries (in dimension seven, it is reasonable to study contact structures simultaneously with G2 -structures). Besides, this case will illustrate the power of the intrinsic torsion concept outlined above. We shall use throughout that R5 D R4 ˚ R with standard U.2/-action on the first term and trivial action on the second term. Let us look at the decompositions of the orthogonal Lie algebras in dimension 4 and 5. First, we have so.4/ D ƒ2 .R4 / D u.2/ ˚ n2 : Here n2 is U.2/-irreducible, while u.2/ splits further into su.2/ and the span of . Combining this remark with the characterization of these subspaces via the complex structure J defining u.2/, we obtain u.2/ D f! 2 ƒ2 .R4 / j J ! D !g D su.2/ ˚ R ; n2 D f! 2 ƒ2 .R4 / j J ! D !g: In particular, ƒ2 .R4 / is the sum of three U.2/-representations of dimensions 1, 2 and 3. For so.5/, we deduce immediately that so.5/ D ƒ2 .R4 ˚ R/ D ƒ2 .R4 / ˚ R4 D u.2/ ˚ .R4 ˚ n2 / μ u.2/ ˚ m6 : Thus, the intrinsic torsion of a 5-dimensional almost metric contact structure is an element of the representation space R5 ˝ m6 D .R4 ˚ R/ ˝ .R4 ˚ n2 / D n2 ˚ R4 ˚ .R4 ˝ n2 / ˚ .R4 ˝ R4 /: The last term splits further into trace-free symmetric, trace and antisymmetric part, written for short as R4 ˝ R4 D S02 .R4 / ˚ R ˚ ƒ2 .R4 /: The 9-dimensional representation S02 .R4 / is again a sum of two irreducible ones of dimensions 3 and 6, but we do not need this here. To decompose the representation


311

R4 ˝ n2 , we observe that the U.2/-equivariant map ‚ W ƒ3 .R4 / ! R4 ˝ n2 (see Section 4.2) has 4-dimensional irreducible image isomorphic to ƒ3 R4 (which is again of dimension 4); its complement is an inequivalent irreducible U.2/-representation of dimension 4 which we call V4 . Consequently, R5 ˝ m6 D R ˚ n2 ˚ R4 ˚ S02 .R4 / ˚ ƒ2 .R4 / ˚ ƒ3 .R4 / ˚ V4 : Taking into account the further splitting of S02 .R4 / ˚ ƒ2 .R4 /, this is the sum of 10 irreducible U.2/-representations as claimed. On the other side, ƒ3 .R5 / D ƒ3 .R4 ˚ R/ D ƒ2 .R4 / ˚ ƒ3 .R4 /: We found a unique copy of this 10-dimensional space in the 30-dimensional space R5 ˝ m6 . Thus, we conclude from Theorem 4.1: Proposition 4.2. A 5-dimensional almost metric contact structure .M 5 ; g ; ; '/ admits a unique characteristic connection if and only if its intrinsic torsion is of class ƒ2 .R4 / ˚ ƒ3 .R4 /. In dimension 5, skew-symmetry of the Nijenhuis tensor N implies that it has to be zero, hence in the light of the more general Theorem 2.5, the almost metric contact manifolds of class ƒ2 .R4 / ˚ ƒ3 .R4 / should coincide with the almost metric contact structures with N D 0 and a Killing vector field. That this is indeed the case follows from the classifications cited above. This class includes for example all quasi-Sasakian manifolds (N D 0 and dF D 0), see [168]. Example 4.1. Consider R5 with 1-forms 2e1 D dx1 ; 2e2 D dy1 ; 2e3 D dx2 ; 2e4 D dy2 ; 4e5 D 4 D dz y1 dx1 y2 dx2 ; P metric g D i ei ˝ ei , and almost complex structure ' defined in h i? by '.e1 / D e2 ;

'.e2 / D e1 ;

'.e3 / D e4 ;

'.e4 / D e3 ;

'.e5 / D 0:

Then .R5 ; g; ; '/ is a Sasakian manifold, and the torsion of its characteristic connection is of type ƒ2 .R4 / ([89, Example 3.D]) and explicitly given by T c D ^ d D 2.e1 ^ e2 C e3 ^ e4 / ^ e5 : This example is in fact a left-invariant metric on a 5-dimensional Heisenberg group c with scalg D 4 and scalr D scalg 3kT k2 =2 D 16 (see Theorem A.1). In a left-invariant frame, spinors are simply functions W R5 ! 5 with values in the 5dimensional spin representation. In [104], it is shown that there exist two r c -parallel spinors i with the additional property F i D 0 .i D 1; 2/. This implies T i D 0, an equation of interest in superstring theory (see Section 5.4). It turns out that these spinors are constant, hence the same result holds for all compact quotients R5 = ( a discrete subgroup).

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We recommend the articles [89] and [90] for a detailed investigation of the representation theory of almost metric contact structures (very much in the style of the book [210]) – in particular, the decomposition of the space of possible torsion tensors T of metric connections (see Proposition 2.1) under U.n/ is being related to the possible classes for the intrinsic torsion.

4.4 SO.3/-structures in dimension 5 These structures were discovered by Th. Friedrich in a systematic investigation of possible G-structures for interesting non-integrable geometries (see [99]); until that moment, it was generally believed that contact structures were the only remarkable G-structures in dimension 5. The group SO.3/ has a unique, real, irreducible representation in dimension 5. We consider the corresponding non-standard embedding SO.3/ SO.5/ as well as the decomposition so.5/ D so.3/ ˚ m7 : It is well known that the SO.3/-representation m7 is the unique, real, irreducible representation of SO.3/ in dimension 7. We decompose the tensor product into irreducible components R5 ˝ m7 D R3 ˚ R5 ˚ m7 ˚ E9 ˚ E11 : There are five basic types of SO.3/-structures on 5-dimensional Riemannian manifolds. The symmetric spaces SU.3/=SO.3/ and SL.3; R/=SO.3/ are examples of 5-dimensional Riemannian manifolds with an integrable SO.3/-structure ( D 0). On the other hand, 3-forms on R5 decompose into ƒ3 .R5 / D R3 ˚ m7 : In particular, a conformal change of an SO.3/-structure does not preserve the property that the structure admits a connection with totally skew-symmetric torsion. M. Bobienski and P. Nurowski investigated SO.3/-structures in their articles [47] and [46]. In particular, they found a ternary symmetric form describing the reduction to SO.3/ and constructed many examples of non-integrable SO.3/-structures with non-vanishing intrinsic torsion. Recently, P. Nurowski suggested a link to Cartan’s work on isoparametric surfaces in spheres, and predicted the existence of similar geometries in dimensions 8, 14 and 26; we refer the reader to [199] for details. The case of SO.3/-structures illustrates that new classes of non-integrable geometries are still to be discovered beyond the well-established ones, and that their study reveals deeper connections between areas which used to be far from each other.


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4.5 Almost Hermitian manifolds in dimension 6 We begin with the Gray–Hervella classification of almost Hermitian manifolds and the consequences for the characteristic connection. Although most of these results hold in all even dimensions, we shall henceforth restrict our attention to the most interesting case, namely dimension 6. Let us consider a 6-dimensional almost Hermitian manifold .M 6 ; g; J /, corresponding to a U.3/-structure inside SO.6/. We decompose the Lie algebra into so.6/ D u.3/ ˚ m and remark that the U.3/-representation in R6 is the real representation underlying ƒ1;0 . Similarly, m is the real representation underlying ƒ2;0 . We decompose the complexification under the action of U.3/: C 6 C R ˝ m D ƒ1;0 ˝ ƒ2;0 ˚ ƒ1;0 ˝ ƒ0;2 R : The symbol .: : : /C R means that we understand the complex representation as a real representation and complexify it. Next we split the complex U.3/-representations ƒ1;0 ˝ ƒ2;0 D C 3 ˝ ƒ2 .C 3 / D ƒ3;0 ˚ E8 ; x 3 / D C 3 ˝ ƒ2 .C 3 / D .C 3 / ˚ E6 : ƒ1;0 ˝ ƒ0;2 D C 3 ˝ ƒ2 .C E6 and E8 are irreducible U.3/-representations of complex dimensions 6 and 8, respectively. Finally we obtain R6 ˝ m D ƒ3;0 ˚ E8 ˚ E6 ˚ .C 3 / μ W1.2/ ˚ W2.16/ ˚ W3.12/ ˚ W4.6/ : Consequently, R6 ˝m splits into four irreducible representations of real dimensions 2, 16, 12 and 6, that is, there are four basic classes and a total of sixteen classes of U.3/structures on 6-dimensional Riemannian manifolds, a result known as Gray/Hervellaclassification ([130]). Recently, F. Martín Cabrera established the defining differential equations for these classes solely in terms of the intrinsic torsion (see [187]), as we shall state them for G2 -manifolds in the next section. In case we restrict the structure group to SU.3/, the orthogonal complement su.3/? is now 7- instead of 6-dimensional, and we obtain R6 ˝ su.3/? D W1 ˚ W2 ˚ W3 ˚ W4 ˚ W5 ; where W5 is isomorphic to W4 Š .C 3 / . Furthermore, W1 and W2 are not irreducible anymore, but they split into W1 D W1C ˚ W1 D R ˚ R and W2 D W2C ˚ W2 D su.3/ ˚ su.3/ (see [69], [187]). Table 5 summarizes some remarkable classes of U.3/-structures in dimension 6. Most of these have by now well-established names, while there is still some confusion for others; these can be recognized by the parentheses indicating the different names to be found in the literature. In the last column, we collected characterizations of these classes (where several are listed, these are to be understood as equivalent characterizations, not as simultaneous requirements). Observe that we included in the last line a remarkable class of SU.3/-structures, the so called half-flat SU.3/-structures (‰ C is the real part of the .3; 0/-form defined by J , see Section 3.1). The name is chosen in order to suggest that half of all W -components vanish

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Ilka Agricola Table 5. Some types of U.3/- and SU.3/-structures in dimension six. name

class

characterization

nearly Kähler manifold

W1

almost Kähler manifold

W2

d D 0

balanced (almost Hermitian) or (Hermitian) semi-Kähler manifold

W3

N D 0 and ı D 0

locally conformally Kähler manifold

W4

g a) .rX J /.X/ D 0

b) 9 real Killing spinor

N D 0 and d D ^ (: Lee form) g rX .Y; Z/

g C rJX .J Y; Z/ D 0

quasi-Kähler manifold

W1 ˚ W 2

Hermitian manifold

W3 ˚ W 4

N D0

(almost-)semi-Kähler or (almost) cosymplectic manifold

W1 ˚ W 2 ˚ W 3

a) ı D 0, b) ^ d D 0

KT- or G1 -manifold

W1 ˚ W 3 ˚ W 4

a) N is skew-symmetric b) 9 char. connection r c

half-flat SU.3/-manifold

W1 ˚ W2 ˚ W3

^ d D 0 and d ‰ C D 0

for these structures. Relying on results of [141], S. Chiossi and S. Salamon described in [69] explicit metrics with Riemannian holonomy G2 on the product of any half-flat SU.3/-manifold with a suitable interval. A construction of half-flat SU.3/-manifolds as T 2 -principal fibre bundles over Kählerian 4-manifolds goes back to Goldstein and Prokushkin [124] and was generalized by Li, Fu and Yau [181], [110]. We refer to Section 5.2 for examples of half-flat SU.3/-structures on nilmanifolds. The following result may easily be deduced from Theorem 4.1 and the decomposition of 3-forms into isotypic U.3/-representations, ƒ3 .R6 / D ƒ3;0 ˚ E6 ˚ .C 3 / D W1 ˚ W3 ˚ W4 : Theorem 4.2 ([102]). An almost Hermitian 6-manifold .M 6 ; g; J / admits a characteristic connection r c if and only if it is of class W1 ˚ W3 ˚ W4 , i.e., if its Nijenhuis


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tensor N is skew-symmetric. Furthermore, r c is unique and given by the expression g.rXc Y; Z/ ´ g.rXg Y; Z/ C

1 ŒN.X; Y; Z/ C d .JX; J Y; J Z/ 2

Remark 4.2 (Parallel torsion). In Example 2.3, it had been observed that the characteristic torsion of nearly Kähler manifolds is always parallel (Kirichenko’s Theorem). Another interesting class of almost Hermitian G1 -manifolds with this property are the so-called generalized Hopf structures, that is, locally conformally Kähler manifolds (class W4 , sometimes abbreviated lcK-manifolds) with parallel Lee form WD ı B J ¤ 0 (in fact, r g D 0 and r c T c D 0 are equivalent conditions for W4 -manifolds). Besides the classical Hopf manifolds, they include for example total spaces of flat principal S 1 -bundles over compact 5-dimensional Sasaki manifolds (see [235], [236] for details); generalized Hopf structures are never Einstein. We recommend the book by S. Dragomir and L. Ornea [83] as a general reference and the articles [35], [111] for complex lcK-surfaces. In his thesis, N. Schoemann investigates almost Hermitian structures with parallel skew-symmetric torsion in dimension 6. A full classification of the possible algebraic types of the torsion form is worked out, and based on this a systematic description of the possible geometries is given. In addition numerous new examples are constructed (and, partially, classified) on naturally reductive spaces (including compact spaces with closed torsion form) and on nilmanifolds (see [22] and [212]). Remark 4.3 (Almost Kähler manifolds). The geometry of almost Kähler manifolds is strongly related to famous problems in differential geometry. W. Thurston was the first to construct an explicit compact symplectic manifold with b1 D 3, hence that does not admit a Kähler structure [228]. E. Abbena generalized this example and gave a natural associated metric which makes it into an almost Kähler non-Kähler manifold [1]; many more examples of this type have been constructed since then. In 1969, S. I. Goldberg conjectured that a compact almost Kähler–Einstein manifold is Kähler [123]. In this generality, the conjecture is still open. K. Sekigawa proved it under the assumption of non-negative scalar curvature [216], and it is known that the conjecture is false for non-compact manifolds: P. Nurowski and M. Przanowski gave the first example of a 4-dimensional Ricci-flat almost-Kähler non-Kähler manifold [200], J. Armstrong showed some non-existence results [28], while V. Apostolov, T. Dr˘aghici and A. Moroianu constructed non-compact counterexamples to the conjecture in dimensions 6 [27]. Different partial results with various additional curvature assumptions are now available. The integrability conditions for almost Kähler manifolds were studied in full generality in [26] and [166]. Remark 4.4 (Nearly Kähler manifolds). We close this section with some additional remarks on nearly Kähler manifolds. Kirichenko’s Theorem (r c T c D 0) implies that Hol.r c / SU.3/, that the first Chern class of the tangent bundle c1 .TM 6 ; J / vanishes, M 6 is spin and that the metric is Einstein. The only known examples are homogeneous metrics on S 6 , CP 3 , S 3 S 3 and on the flag manifold F .1; 2/ D

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U.3/=U.1/U.1/U.1/, although (many?) more are expected to exist. It was shown that these exhaust all nearly Kähler manifolds that are locally homogeneous (see [61]) or satisfying Hol.r c / ¤ SU.3/ (see [36]). A by now classical result asserts that a 6dimensional spin manifold admits a real Killing spinor if and only if it is nearly Kähler (see [101], [133] and [33]). Finally, more recent structure theorems justify why nearly Kähler manifolds are only interesting in dimension 6: any complete simply connected nearly Kähler manifold is locally a Riemannian product of Kähler manifolds, twistor spaces over Kähler manifolds and 6-dimensional nearly Kähler manifolds (see [197], [198]).

4.6 G2 -structures in dimension 7 We consider 7-dimensional Riemannian manifolds equipped with a G2 -structure. Since G2 is the isotropy group of a 3-form ! of general type, a G2 -structure is a triple .M 7 ; g; !/ consisting of a 7-dimensional Riemannian manifold and a 3-form ! of general type at any point. We decompose the G2 -representation (see [102]) R7 ˝ m D R ˚ ƒ214 ˚ ƒ327 ˚ R7 μ X1.1/ ˚ X2.14/ ˚ X3.27/ ˚ W4.7/ ; and, consequently, there are again four basic classes and a total of sixteen classes G2 structures (namely, parallel G2 -manifolds and 15 non-integrable G2 -structures). This result is known as the Fernández/Gray-classification of G2 -structures (see [87]); some important classes are again summarized in tabular form, see Table 6. The different classes of G2 -structures can be characterized by differential equations. They can be written in a unified way as 3 d! D ! C ^ ! C 3 ; 4

ı! D d ! D . ^ !/ C .2 ^ !/;

where is a scalar function corresponding to the X1 -part of the intrinsic torsion , 2 , 3 are 2- resp. 3-forms corresponding to its X2 resp. X3 -part and is a 1-form describing its X4 -part, which one sometimes calls the Lee form of the G2 -structure. This accounts for some of the characterizations listed in Table 6. For example, a G2 -structure is of type X1 (nearly parallel G2 -structure) if and only if there exists a number (it has to be constant in this case) such that d! D ! holds. Again, this condition is equivalent to the existence of a real Killing spinor and the metric has to be Einstein [107]; more recently, the Riemannian curvature properties of arbitrary G2 -manifolds have been discussed in detail by R. Cleyton and S. Ivanov [72]. G2 structures of type X1 ˚ X3 (cocalibrated G2 -structures) are characterized by the condition that the 3-form is coclosed, ı! 3 D 0. Under the restricted action of G2 , one obtains the following isotypic decomposition of 3-forms on R7 : ƒ3 .R7 / D R ˚ ƒ327 ˚ R7 D X1 ˚ X3 ˚ X4 :

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Chapter 9. Non-integrable geometries, torsion, and holonomy Table 6. Some types of G2 -structures in dimension seven. name

class

characterization

nearly parallel G2 -manifold

X1

a) d! D ! for some 2 R b) 9 real Killing spinor

almost parallel or closed (or calibrated symplectic) G2 -manifold

X2

d! D 0

balanced G2 -manifold

X3

ı! D 0 and d! ^ ! D 0

locally conformally parallel G2 -manifold

X4

d! D 34 ^ ! and d ! D ^ ! for some 1-form

cocalibrated (or semi-parallel or cosymplectic ) G2 -manifold

X1 ˚ X 3

ı! D 0

locally conformally (almost) parallel G2 -manifold

X2 ˚ X 4

d! D 34 ^ !

G2 T -manifold

X1 ˚ X 3 ˚ X 4

a) d ! D ^ ! for some 1-form b) 9 char. connection r c

This explains the first part of the following theorem and the acronym ‘G2 T -manifolds’ for this class: it stands for ‘G2 with (skew) torsion’. The explicit formula for the characteristic torsion may be derived directly from the properties of r c . Theorem 4.3 ([102], Theorem 4.8). A 7-dimensional manifold .M 7 ; g; !/ with a fixed G2 -structure ! 2 ƒ3 .M 7 / admits a characteristic connection r c if and only if it is of class X1 ˚ X3 ˚ X4 , i.e., if there exists a 1-form such that d ! D ^ !. Furthermore, r c is unique, admits (at least) one parallel spinor and is given by the expression rXc Y

´

rXg Y

1 1 C d! g.d!; !/! C . ^ !/ : 2 6

This last remarkable property is a direct consequence of our investigation of geometric stabilizers, as explained in Corollary 3.1. For a nearly parallel G2 -manifold,

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the r c -parallel spinor coincides with the Riemannian Killing spinor and the manifold turns out to be r c -Einstein [102]. But there is no general argument identifying Killing spinors with parallel spinors. The characteristic connection of a Sasaki-Einstein manifold does not necessarily admit parallel spinors ([102], [104]), though there always exist Killing spinors [105]. A 3-Sasaki manifold admits three real Killing spinors, but it does not admit a characteristic connection in any reasonable sense (each Sasaki structure has a characteristic connection, but it does not preserve the other two Sasaki structures), see Section 2.6 and [10]. This reflects the fact that Sasaki-Einstein and 3-Sasaki manifolds do not fit too well into the general framework of G-structures. Remark 4.5 (Parallel torsion). For a nearly parallel G2 -manifold, the explicit formula from Theorem 4.3 implies that the characteristic torsion T c is proportional to !, hence it is trivially r c -parallel. For the larger class of cocalibrated G2 -manifolds (class X1 ˚ X3 ), the case of parallel characteristic torsion has been investigated systematically by Th. Friedrich (see [100]). Again, many formerly unknown examples have been constructed, for example, from deformations of -Einstein Sasaki manifolds, from S 1 -principal fibre bundles over 6-dimensional Kähler manifolds or from naturally reductive spaces.

4.7 Spin.7/-structures in dimension 8 Let us consider Spin.7/-structures on 8-dimensional Riemannian manifolds. The subgroup Spin.7/ SO.8/ is the real Spin.7/-representation 7 D R8 , the complement m D R7 is the standard 7-dimensional representation and the Spin.7/-structures on an 8-dimensional Riemannian manifold M 8 correspond to the irreducible components of the tensor product R8 ˝ m D R8 ˝ R7 D 7 ˝ R7 D 7 ˚ K D R8 ˚ K; where K denotes the kernel of the Clifford multiplication 7 ˝ R7 ! 7 . It is well known that K is an irreducible Spin.7/-representation, i.e., there are two basic classes of Spin.7/-structures (the Fernández classification of Spin.7/-structures, see [85]). For 3-forms, we find the isotypic decomposition ƒ3 .R8 / D 7 ˚ K; showing that ƒ3 .R8 / and R8 ˝ m are isomorphic. Theorem 4.1 yields immediately that any Spin.7/-structure on an 8-dimensional Riemannian manifold admits a unique connection with totally skew-symmetric torsion. The explicit formula for its characteristic torsion may be found in [149].

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5 Weitzenböck formulas for Dirac operators with torsion 5.1 Motivation The question whether or not the characteristic connection of a G-structure admits parallel tensor fields differs radically from the corresponding problem for the LeviCivita connection. In particular, one is interested in the existence of parallel spinor fields, interpreted in superstring theory as supersymmetries of the model. The main analytical tool for the investigation of parallel spinors is the Dirac operator and several remarkable identities for it. We discuss two identities for the square of the Dirac operator. While the first one is straightforward and merely of computational difficulty, the second relies on comparing the Dirac operator corresponding to the connection with torsion T with the spinorial Laplace operator corresponding to the connection with torsion 3T . Such an argument has been used in the literature at several places. The first was probably S. Slebarski ([218], [219]) who noticed that on a naturally reductive space, the connection with torsion one-third that of the canonical connection behaves well under fibrations; S. Goette applied this property to the computation of the -invariant on homogeneous spaces [122]. J.-P. Bismut used such a rescaling for proving an index theorem for Hermitian manifolds [42]. It is implicit in Kostant’s work on a ‘cubic Dirac operator’, which can be understood as an identity in the Clifford algebra for the symbol of the Dirac operator of the rescaled canonical connection on a naturally reductive space ([173], [4]).

5.2 The square of the Dirac operator and parallel spinors Consider a Riemannian spin manifold .M n ; g; T / with a 3-form T 2 ƒ3 .M n / as well as the one-parameter family of linear metric connections with skew-symmetric torsion (s 2 R), rXs Y ´ rXg Y C 2s T .X; Y; /: In particular, the superscript s D 0 corresponds to the Levi-Civita connection and s D 1=4 to the connection with torsion T considered before. As before, we shall also sometimes use the superscript g to denote the Riemannian quantities corresponding to s D 0. These connections can all be lifted to connections on the spinor bundle †M n , where they take the expression rXs

´ rXg

C s.X ³ T / :

Two important elliptic operators may be defined on †M n , namely, the Dirac operator and the spinor Laplacian associated with the connection r s : Ds ´

n X kD1

ek resk D D 0 C3sT;

s . / D .r s / r s D

n X kD1

resk resk Crrs g e

ei i

:

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By a result of Th. Friedrich and S. Sulanke [109], the Dirac operator D r associated with any metric connection r is formally self-adjoint if and only if the rP g.r X; ei / of any vector field X coincides with its divergence divr .X/ WD e i i g Riemannian r -divergence. Writing r D r g C A, this is manifestly equivalent P to i g.A.ei ; X/; ei / D 0 and trivially satisfied for metric connections with totally skew-symmetric torsion4 . Shortly after P. Dirac introduced the Dirac operator, E. Schrödinger noticed the existence of a remarkable formula for its square [215]. Of course, since the concept of spin manifold had not yet been established, all arguments of that time were of local nature, but contained already all important ingredients that would be established in a more mathematical way later. By the sixties and the seminal work of Atiyah and Singer on index theory for elliptic differential operators, Schrödinger’s article was almost forgotten and the formula rediscovered by A. Lichnerowicz [182]. In our notation, the Schrödinger–Lichnerowicz formula states that 1 .D 0 /2 D 0 C scal0 : 4 Our goal is to derive useful relations for the square of D s . In order to state the first formula, let us introduce the first order differential operator Ds

´

n X kD1

.ek ³ T/ resk

D D0

Cs

n X

.ek ³ T/ .ek ³ T/ :

(9)

kD1

Theorem 5.1 ([102, Theorems 3.1, 3.3]). Let .M n ; g; r s / be an n-dimensional Riemannian spin manifold with a metric connection r s of skew-symmetric torsion 4s T . Then, the square of the Dirac operator D s associated with r s acts on an arbitrary spinor field as 1 D s . /C3s d T 8s 2 T C2s ıT 4s D s C scals : (10) 4 s Furthermore, the anticommutator of D and T is .D s /2

D s B T C T B D s D d T C ıT 8s T 2 D s :

(11)

scals denotes the scalar curvature of the connection r s . Remark that scal0 D scalg is the usual scalar curvature of the underlying Riemannian manifold .M n ; g/ and that the relation scals D scal0 24s 2 kT k2 holds. Moreover, the divergence ıT can be taken with respect to any connection r s from the family, hence we do not make a notational difference between them (see Proposition A.2). This formula for .D s /2 has the disadvantage of still containing a first order differential operator with uncontrollable spectrum as well as several 4-forms that are difficult to treat algebraically, hence it is not suitable for deriving vanishing theorems. 4 One checks that it also holds for metric connections with vectorial torsion, but not for connections of Cartan

type A0 .


321

It has however a nice application in the study of r s -parallel spinors for different values of s. As a motivation, let us consider the following example: Example 5.1. Let G be a simply connected Lie group, g a bi-invariant metric and consider the torsion form T .X; Y; Z/ WD g.ŒX; Y ; Z/. The connections r ˙1=4 are flat [170], hence they both admit non-trivial parallel spinor fields. Such a property for the connections with torsions ˙T is required in some superstring models. Theorem 5.1 now implies that there cannot be many values s admitting r s -parallel spinors. Theorem 5.2 ([10, Theorem 7.1]). Let .M n ; g; T / be a compact spin manifold, r s the family of metric connections defined by T as above. For any r s -parallel spinor , the following formula holds: Z Z 2 hT ; i C scals k k2 D 0: 64 s Mn

Mn

If the mean value of hT ; i does not vanish, the parameter s is given by Z Z ı 1 hd T ; i hT ; i: sD 8 Mn Mn If the mean value of hT ; i vanishes, the parameter s depends only on the Riemannian scalar curvature and on the length of the torsion form, Z Z Z s g 2 Scal D scal 24s kT k2 : 0D Mn

Mn

Mn

Finally, if the 4-forms d T and T are proportional (for example, if r 1=4 T D 0), there are at most three parameters with r s -parallel spinors. Remark 5.1. The property that d T and T are proportional is more general than requiring parallel torsion. For example, it holds for the whole family of connections r t on naturally reductive spaces discussed in Sections 2.2 and 5.3, but its torsion is r t -parallel only for t D 1. Example 5.2. On the 7-dimensional Aloff–Wallach space N.1; 1/ D SU.3/=S 1 , one can construct a non-flat connection such that r s0 and r s0 admit parallel spinors for suitable s0 , hence showing that both cases from Theorem 5.2 can actually occur in non-trivial situations. On the other hand, it can be shown that on a 5-dimensional Sasaki manifold, only the characteristic connection r c can have parallel spinors [10]. Inspired by the homogeneous case (see Section 5.3), we were looking for an alter0 native comparison of .D s /2 with the Laplace operator of some other connection r s s 2 from the same family. Since .D / is a symmetric second order differential operator with metric principal symbol, a very general result by P. B. Gilkey claims that there

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exists a connection r and an endomorphism E such that .D s /2 D r r C E [120]. Based on the results of Theorem 5.1, one shows: Theorem 5.3 (Generalized Schrödinger–Lichnerowicz formula, [10, Theorem 6.2]). The spinor Laplacian s and the square of the Dirac operator D s=3 are related by .D s=3 /2 D s C s d T C

1 scalg 2s 2 kT k2 : 4

We observe that D s=3 appears basically by quadratic completion. Under additional assumptions on T , this result can be used as a basis for eigenvalue estimates for D s=3 – see [13]. Theorem 5.4 ([10, Theorem 6.3]). Let .M n ; g; T / be a compact, Riemannian spin manifold of non positive scalar curvature, scalg 0. If there exists a solution ¤ 0 of the equations 1 C .X ³ T / D 0; hd T ; i 0; 2 the 3-form and the scalar curvature vanish, T D 0 D scalg , and is parallel with respect to the Levi-Civita connection. rX

D rXg

Theorem 5.4 applies, in particular, to Calabi–Yau or Joyce manifolds, where we know that r g -parallel spinors exist by Wang’s Theorem (Theorem 3.1). Let us perturb the connection r g by a suitable 3-form (for example, a closed one). Then the new connection r does not admit r-parallel spinor fields: the Levi-Civita connection and its parallel spinors are thus, in some sense, rigid. Nilmanifolds are a second family of examples where the theorem applies. A further family of examples arises from certain naturally reductive spaces with torsion form T proportional to the torsion form of the canonical connection, see [4]. From the high energy physics point of view, a parallel spinor is interpreted as a supersymmetry transformation. Hence the physical problem behind the above question (which in fact motivated our investigations) is really whether a free “vacuum solution” can also carry a non-vacuum supersymmetry, and how the two are related. Naturally, Theorem 5.4 raises the question to which extent compactness is really necessary. We shall now show that it is by using the equivalence between the inclusion Hol.r/ G2 and the existence of a r-parallel spinor for a metric connection with skew-symmetric torsion known from Corollary 3.1. For this, it is sufficient to find a 7-dimensional Riemannian manifold .M 7 ; g/ whose Levi-Civita connection has a parallel spinor (hence is Ricci-flat, in particular), but also admits a r-parallel spinor for some other metric connection with skew-symmetric torsion. Gibbons et al. produced non-complete metrics with Riemannian holonomy G2 in [119]. Those metrics have among others the interesting feature of admitting a 2-step nilpotent isometry group N acting on orbits of codimension one. By [68] such metrics are locally conformal to homogeneous metrics on rank-one solvable


323

extension of N , and the induced SU.3/-structure on N is half-flat. In the same article all half-flat SU.3/ structures on 6-dimensional nilpotent Lie groups whose rank-one solvable extension is endowed with a conformally parallel G2 structure were classified. Besides the torus, there are exactly six instances, which we considered in relation to the problem posed. It turns out that four metrics of the six only carry integrable G2 structures, thus reproducing the pattern of the compact situation, whilst one admits complex solutions, a physical interpretation for which is still lacking. The remaining solvmanifold .Sol; g/ – which has exact Riemannian holonomy G2 – provides a positive answer to both questions posed above, hence becoming the most interesting. The Lie algebra associated to this solvmanifold has structure equations Œei ; e7 D 35 mei ; i D 1; 2; 5; Œe1 ; e5 D 25 me3 ;

Œej ; e7 D 65 mej ; j D 3; 4; 6;

Œe2 ; e5 D 25 me4 ;

Œe1 ; e2 D 25 me6 :

The homogeneous metric it bears can be also seen as a G2 metric on the product R T , where T is the total space of a T 3 -bundle over another 3-torus. For the sake of an easier formulation of the result, we denote by r T the metric connection with torsion T . Theorem 5.5 ([7, Theorem 4.1]). The equation r T ‰ D 0 admits 7 solutions for some 3-form, namely: a) A two-parameter family of pairs .Tr;s ; ‰r;s / 2 ƒ3 .Sol/ †.Sol/ such that r Tr;s ‰r;s D 0; for r D s the torsion Tr;r D 0 and ‰r;r is a multiple of the r g -parallel spinor. b) Six ‘isolated’ solutions occurring in pairs, .Ti" ; ‰i" / 2 ƒ3 .Sol/ †.Sol/ for i D 1; 2; 3 and " D ˙. All these G2 structures admit exactly one parallel spinor, and for jrj ¤ jsj: !r;s is of general type R ˚ S02 R7 ˚ R7 , r D s: !r;r is r g -parallel, r D s: the G2 class has no R-part. Here !r;s denotes the defining 3-form of the G2 -structure, see equation (8). Remark 5.2. A routine computation establishes that hd T ‰; ‰i < 0 for all solutions found in Theorem 5.5, except for the integrable case r D s of solution a) where it vanishes trivially since T D 0. Remark 5.3. The interaction between explicit Riemannian metrics with holonomy G2 on non-compact manifolds and the non-integrable G2 -geometries as investigated with the help of connections with torsion was up-to-now limited to “cone-type arguments”, i.e., a non-integrable structure on some manifold was used to construct an integrable structure on a higher dimensional manifold (like its cone, an so on). It is thus a natural

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Ilka Agricola

question whether the same Riemannian manifold .M; g/ can carry structures of both type simultaneously. This appears to be a remarkable property, of which the above example is the only known instance. To emphasize this, consider that the projective space CP 3 with the well-known Kähler–Einstein structure and the nearly Kähler one inherited from triality does not fit the picture, as they refer to different metrics.

5.3 Naturally reductive spaces and Kostant’s cubic Dirac operator On arbitrary manifolds, only Weitzenböck formulas that express D 2 through the Laplacian are available. On homogeneous spaces, it makes sense to look for expressions for D 2 of Parthasarathy type, that is, in terms of Casimir operators. Naturally reductive spaces M n D G=H with their family of metric connections (X; Y 2 m) t rXt Y ´ rXg Y ŒX; Y m 2 were in fact investigated prior to the more general case described in the previous section. As symmetric spaces are good toy models for integrable geometries, homogeneous non-symmetric spaces are a very useful field for ‘experiments’ in non-integrable geometry. Furthermore, many examples of such geometries are in fact homogeneous. We will show that the main achievement in [173] was to realize that, for the parameter value t D 1=3, the square of D t may be expressed in a very simple way in terms of Casimir operators and scalars only ([173, Theorem 2.13], [221, 10.18]). It is a remarkable generalization of the classical Parthasarathy formula for D 2 on symmetric spaces (formula (1) in this chapter, see [201]). We shall speak of the generalized Kostant–Parthasarathy formula in the sequel. S. Slebarski used the connection r 1=3 to prove a “vanishing theorem” for the kernel of the twisted Dirac operator, which can be easily recovered from Kostant’s formula (see [180, Theorem 4]). His articles [218] and [219] contain several formulas of Weitzenböck type for D 2 , but none of them is of Parthasarathy type. In fact, Kostant needed his operator not for the investigation of non-integrable geometries, but as a tool for explaining a distinguished property of the representations of Spin.9/ discovered by the physicists Ramond and Pengpan [132]. In representation theory this opened the possibility to realize infinite-dimensional representations in kernels of twisted Dirac operators on homogeneous spaces ([145], [190]), as it had been carried out on symmetric spaces in the seventies. In order to exploit the full power of harmonic analysis, it is necessary to extend the naturally reductive metric h ; i on m to the whole Lie algebra g of G. By a classical theorem of B. Kostant, there exists a unique Ad.G/ invariant, symmetric, non degenerate, bilinear form Q on g such that Q.h \ g; m/ D 0 n

and

Qjm D h ; i

if G acts effectively on M and g D m C Œm; m, which we will tacitly assume from now on [172]. In general, Q does not have to be positive definite; if it is, the metric


325

is called normal homogeneous. Assume furthermore that there exists a homogeneous W H ! Spin.m/ of the isotropy representation such spin structure on M , i.e., a lift Ad that the diagram Spin.m/ x< O xx xx Ad xx xx x x x Ad / SO.m/ H the correspondcommutes, where denotes the spin covering. Moreover, denote by ad ing lift into spin.m/ of the differential ad W h ! so.m/ of Ad. Let W Spin.m/ ! GL. m / be the spin representation, and identify sections of the spinor bundle †M n D G Ad W G ! m satisfying m with functions 1 // .g/: .gh/ D .Ad.h The Dirac operator takes for Dt

2 †M n the form D

n X

ei . / C

iD1

1t H ; 2

where H is the third degree element in the Clifford algebra Cl.m/ of m induced from the torsion, ˛ 3 X ˝ Œei ; ej m ; ek ei ej ek : H ´ 2 i<j 0 such that r V D k1H and moreover the eigenvalues together with the corresponding eigenspaces for the tensors r, C , Ric are given in the following table: Eigenvalue

r

Ric

C

Eigenspace

1

nd k d

nC7d k 4d

4.n3d / k d

V

2

2k

nC2d k 2d

/ 2.n3d k d

H

(ii) If C has exactly 3-eigenvalues we have a h-invariant, orthogonal and J -invariant splitting V D V ˚ H1 ˚ H2 such that

C

.H1 ; H2 / D

C

.H2 ; H2 / D 0 and

C

.H1 ; H2 / D V .

(iii) In all cases, including when C D 0, we have RicH D gjH for some > 0, where RicH denotes the Ricci contraction of RH . We refer the reader to [49] for details of the proof and also mention that in case (ii) above it is also possible to display the relations between the eigenvalues of all relevant symmetric endomorphisms.

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5.5 The final classification By Theorem 5.2 it is enough to consider an SNK-holonomy system .V 2m ; g; J; C ; R/ with a special invariant subspace V and such that the Riemannian holonomy system .hH ; H / is irreducible, in the notations of the previous section. We shall combine here some well-known results for irreducible Riemannian holonomy systems and the information which is available in our present context. To progress in this direction we will first make clear the relationship between the Lie algebra r and the Riemannian holonomy algebra hH . Proposition 5.9. The following hold: (i) r is an ideal in hH ; (ii) RH D 2 1r on r for some > 0 such that rjV D 1V . Proof. (i) That p hH follows from (5.3) hence q D Œp; p is contained in hH as well. To see that r is an ideal we use (5.1) to observe that ŒR.x; y/;

C v

belongs to p for all x, y in H and for all v in V . The definition of RH and that of r now yields ŒRH .x; y/; vC in r and it easy to conclude by using the Jacobi identity and Proposition 5.5. (ii) First of all let us observe that r preserves V , since the latter is special. The h-invariance of r, together with the irreducibility of .h; V/ implies the existence of a constant > 0 such that rjV D 1V . For the action of RH on p our claim follows now from (5.3). To prove it on q we observe we proceed as follows. Given that R belongs to S 2 .ƒ2 / we can alternatively rewrite (5.1) under the form R.

C x1 x2 ; x3 /

C R.x2 ;

C x1 x3 /

D R.x1 ;

C x 2 x3 /

whenever xk are in V , k D 1; 2; 3. Using this for x1 D v; x2 D wC ei ; x3 D ei where v, w are in V and fei g is some orthonormal basis in H yields after summation and rearrangement of terms X R.ei ; Œ vC ; wC ei / D R.v; rw/ i C v ;

C w /

Œ 2

C C D that is R.Œ v ; w on H , after also using (5.2). After a straightforward invariance argument this leads to RH .Œ vC ; wC / D 2 Œ vC ; wC and the claim is proved.

Now we recall that for a given metric representation .l; W / of some Lie algebra l we have the Berger list of irreducible Riemannian holonomies, and proceed to the

Chapter 10. Totally skew-symmetric torsion and nearly-Kähler geometry

393

final classification result, hence proving Theorem 1.1 in the introduction. l

W

so.n/

Rn

u.m/

R2m

su.m/

R2m

sp.m/ ˚ sp.1/

R4m

sp.m/

R4m

spin.7/

R8

g2

R7

Theorem 5.3. Let .M 2m ; g; J / be a complete SNK-manifold. Then M is, up to finite cover, a Riemannian product whose factors belong to the following classes: (i) homogeneous SNK-manifolds; (ii) twistor spaces over positive quaternionic Kähler manifolds; (iii) 6-dimensional SNK-manifolds. Proof. First of all, because M must be compact with finite fundamental group we may assume up to a finite cover that it is simply connected. Pick now a point x in x Tx M /. If this is irreducible M and consider the holonomy representation .Holx .r/; over R or complex irreducible but real reducible we conclude by Theorem 4.4. x is reducible over C at x we consider the SNKIf the holonomy representation of r C x holonomy system .Tx M; gx ; Jx ; x ; Rx /. By the theorem of Ambrose–Singer we x Also note that when starting the whole reduction procedure have that h holx .r/. x complex invariant sub-space we end up leading to Theorem 5.2 from some holx .r/ with a splitting Tx M D V1 ˚ ˚ Vp x with the properties in Theorem 5.2 and which is furthermore holx .r/-invariant together with the special sub-spaces contained in each factor. Using parallel transport x this extends to a r-parallel x w.r.t. to r decomposition, which is actually r-parallel x is split along the decomposition. The splitting theorem of de since the torsion of r Rham applies and our study is reduced to that of SNK-manifolds such that at each point, we have an invariant special subspace having the property that the associated holonomy system is irreducible in the sense of Definition 5.5. Then we show (see [49]) that M is the total space of a Riemannian submersion W .M; g/ ! .N 2n ; h/

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whose fibres are totally geodesic and w.r.t. the induced metric and almost complex structure are simply connected, irreducible and compact Hermitian symmetric spaces. The tensor RH defined in Section 5.3 projects onto the Riemann curvature tensor of h making of hH a subalgebra of hol.N; h/ at each point. Using this, one shows that the Riemannian manifold .N 2n ; h/ is irreducible and Proposition 5.8 (iii) gives that h is Einstein of positive scalar curvature. If the tensor C has three eigenvalues a short manipulation of the second Bianchi x combined with the algebraic structure of the splitting of TM given in identity for r x is an Ambrose–Singer connection. (ii) of Proposition 5.8 gives that r Let us discuss now the case when C has two eigenvalues or it vanishes. If .N 2n ; h/ is a symmetric space the relations of O’Neill (see [6]) for the Riex is an mannian submersion W M ! N essentially say that the curvature tensor R C H ; R and the curvature of the fibre (which is a symmetric explicit expression of x space) along the r-parallel splitting TM D V ˚ H . This information results easily x an Ambrose–Singer connection again. in having r Suppose now that .N 2n ; h/ is not a symmetric space. Then the holonomy representation of .N; h/, at the Lie algebra level, corresponds to the representation .hH ; H / and as it is well known (see [52]), must be one of the entries in the Berger list above. We shall mainly use now Proposition 5.9. The Lie algebras su.m/, sp.m/, spin.7/, g2 are excluded because their curvature tensors are Ricci-flat. Supposing that hH D u.n/, given that the ideal r is at least two-dimensional we can only have r D su.n/; u.n/. But using Proposition 5.9 combined with the fact that a curvature tensor of KählerEinstein type is completely determined by its restriction to su.m/ we find that .N 2n ; h/ is symmetric a contradiction. Similar arguments together with results from [5] enable us to conclude in the case when hH D so.2n/ or when r D sp.m/. The only case remaining is when hH D sp.m/ ˚ sp.1/ and r D sp.1/ which implies that V is of x is contained in U.1/ U.2m/ and the fact rank 2. Hence the holonomy group of r 2m that .M ; g; J / is a twistor space over a positive quaternionic-Kähler manifold has been proved in [48].

6 Concluding remarks Due to considerations of space and time we have omitted to present some important facts related to nearly-Kähler geometry. The first aspect is related to the construction of examples, which is implicit in our present treatment. Given a quaternion-Kähler manifold .M 4m ; g; Q/ of positive scalar curvature the Salamon twistor construction [51] yields a Kähler manifold .Z; h; J / such that there is a Riemannian submersion with totally geodesic, complex fibres S 2 ,! .Z; h/ ! .M; g/:


395

Using this structure and a canonical variation of the metric h it has been shown [19] that Z admits an SNK-metric. Enlarging the context to that of Riemannian submersions with complex, totally geodesic fibres from Kähler manifolds one draws a similar conclusion [48] (see also [50] for some related facts). This observation can also be used to describe the homogeneous SNK-manifolds which appear in Theorem 5.3 as twistor spaces in the sense of [11] over symmetric spaces of compact type. We have also omitted to discuss NK-structures in dimension 6, which have a rich geometry, although not yet fully understood. If .M 6 ; g; J / is a strict NK manifold the structure group automatically reduces to SU.3/, hence c1 .M; J / D 0 [33]. Moreover the metric g must be Einstein [33], of positive scalar curvature. There is also a 1-1 correspondence between 6-dimensional manifolds admitting real Killing spinors and SNK-structures in dimension 6 [35]. In the compact case the only known examples are homogeneous, and various characterisations of these instances are available [12], [45]. However, compact examples with two conical singularities have been very recently constructed in [23] building on the fact that one can suitably rotate the SU.2/-structure of a Sasakian–Einstein 5-manifold and then extend it to a SU.3/-structure of NK-type through a generalised cone construction. Acknowledgments. Part of this material was elaborated during the “77ème rencontre entre physiciens théoriciens et mathématiciens” held in Strasbourg in 2005. Warm thanks to the organisers and especially to V. Cortés for a beautiful and inspiring conference. I am also grateful to V. Cortés and A. Moroianu for useful comments.

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[49] P.-A. Nagy, Nearly Kähler geometry and Riemannian foliations. Asian J. Math. 6 (2002), 481–504. 349, 382, 385, 386, 390, 391, 393 [50] P.-A. Nagy, Rigidity of Riemannian foliations with complex leaves on Kähler manifolds. J. Geom. Anal. 13 (2003), 659–667. 395 [51] S. Salamon, Quaternionic Kähler manifolds. Invent. Math. 67 (1982), 143–171. 394 [52] S. Salamon, Riemannian geometry and holonomy groups. Pitman Research Notes in Math. Ser. 201, Longman Scientific & Technical, Harlow 1989. 347, 394 [53] U. Semmelmann, Conformal Killing forms on Riemannian manifolds. Math. Z. 245 (2003), 503–527. 363, 377 [54] A. Strominger, Superstrings with torsion. Nuclear Phys. B 274 (1986), 253–284. 348 [55] A. Swann, Weakening holonomy. In Quaternionic structures in mathematics and physics (Rome, 1999), Università degli Studi di Roma “La Sapienza”, Rome, 1999, 405–415 (http://www.emis.de/proceedings/QSMP99/index.html). 348 [56] F. Tricerri and L. Vanhecke, Curvature tensors on almost Hermitian manifolds. Trans. Amer. Math. Soc. 267 (1981), 365–397. 354 [57] F. Tricerri and L. Vanhecke, Homogeneous structures on Riemannian manifolds. London Math. Soc. Lecture Note Ser. 83, Cambridge University Press, Cambridge 1983. 381 [58] F. Witt, Special metrics and triality. Adv. Math. 219 (2008), 1972–2005. 360

Chapter 11

Homogeneous nearly Kähler manifolds Jean-Baptiste Butruille

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Preliminaries: nearly Kähler manifolds and 3-symmetric spaces . . . . . . . 2 The case of S 3 S 3 : the natural reduction to SU.3/ . . . . . . . . . . . . . 3 Twistor spaces: the complex projective space CP 3 and the flag manifold F 3 4 Weak holonomy and special holonomy: the case of the sphere S 6 . . . . . . 5 Classification of 3-symmetric spaces and proof of the theorems . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

399 402 406 411 414 417 421

Introduction Probably the first known example of a nearly Kähler manifold is the round sphere in dimension 6, equipped with its well-known non integrable almost complex structure, introduced in [21]. The resulting almost Hermitian structure is invariant for the action of G2 on S 6 coming from the octonions (we look at S 6 as the unit sphere in the imaginary set = O). Thus S 6 ' G2 =SU.3/ is an example of a 6-dimensional homogeneous nearly Kähler manifold as we consider in the present exposition. Notice also that the representation of SU.3/ on the tangent spaces – the isotropy representation – is irreducible. Then Joseph A. Wolf in his book Spaces of constant curvature [46] discovered a class of isotropy irreducible homogeneous spaces G=H that generalizes S 6 , where G is a compact Lie group and H , a maximal connected subgroup, centralizing an element of order 3. Later on, Wolf and Gray [46] investigated the homogeneous spaces defined by Lie group automorphisms i.e. such that H is the fixed point set of some automorphism s W G ! G. They asked the following question: which of these spaces admit an invariant almost Hermitian structure with good properties? In particular, nearly Kähler manifolds are associated, through their work, to a type of homogeneous spaces – the 3-symmetric spaces – corresponding to s of order 3. Since their study was quite general, they felt legitimate to make a conjecture, which I reformulate using the terminology of the later article [24] by Gray alone: Conjecture 1 (Gray and Wolf). Every nearly Kähler homogeneous manifold is a 3-symmetric space equipped with its canonical almost complex structure.

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Another way to construct examples is by twistor theory. The twistor space Z of a self-dual 4-manifold has a natural complex structure by [3], but also a non integrable almost complex structure (see [16]). The latter may be completed into a nearly Kähler structure when the base is furthermore Einstein, with positive scalar curvature. The same construction holds for the twistor spaces of the positive quaternion-Kähler manifolds (see [2], [36]) or of certain symmetric spaces (as explained for instance in [40]). In this case Z is a 3-symmetric space again. Next, in the ’70s [22], [25] and more recently (we mention, as a very incomplete list of references on the topic, [38], [6], [36] and [35]), nearly Kähler manifolds have been studied for themselves. Some very interesting properties were discovered, especially in dimension 6, that give them a central role in the study of special geometries with torsion. These properties can all be interpreted in the setting of weak holonomy. Nearly Kähler 6-dimensional manifolds are otherwise called weak holonomy SU.3/. Many definitions of this notion have been proposed. One is by spinors [4]; others by differential forms [41], [10]; the original one by Gray [23] was found incorrect (see [1]). Finally, Cleyton and Swann [42], [14], [15] have worked on a definition based on the torsion of the canonical connection of a G-structure. They obtain a theorem which can be applied to our problem to show that nearly Kähler manifolds whose canonical connection has irreducible holonomy are either 3-symmetric or 6-dimensional. This constitutes an advance towards and gives a new reason to believe Gray and Wolf’s conjecture. However, the theorem of Cleyton and Swann does not say anything about dimension 6. Moreover, when the holonomy is reducible, we do not have a de Rhamlike theorem, like in the torsion-free situation. It was Nagy’s main contribution to this issue to show that we can always lead back, in this case, to the twistor situation. Using this, he was able to reduce Conjecture 1 to dimension 6 [35]. This is where we resume his work. We classify 6-dimensional nearly Kähler homogeneous spaces and show that they are all 3-symmetric. Theorem 1. Nearly Kähler, 6-dimensional, Riemannian homogeneous manifolds are isometric to a finite quotient of G=H equipped with the naturally reductive metric induced by the Killing form of G, where the groups G, H are given in the following list: • G D SU.2/ SU.2/ SU.2/ and H D SU.2/ (diagonal). • G D G2 and H D SU.3/. In this case G=H is the round 6-sphere. • G D Sp.2/ and H D SU.2/U.1/. Then, G=H ' CP 3 , the 3-dimensional complex projective space. • G D SU.3/, H D U.1/ U.1/ and G=H ' F 3 is the space of flags of C 3 . Each of the spaces S 3 S 3 , CP 3 and F 3 carries a unique invariant nearly Kähler structure, up to homothety. The invariant nearly Kähler almost complex structures on the sphere S 6 are all conjugated to each other by an isometry. They form a space isomorphic to RP 7 .

Chapter 11. Homogeneous nearly Kähler manifolds

401

As a corollary we obtain Theorem 2. Conjecture 1 is true in dimension 6 and thus, by the work of Nagy, in all dimensions. The proof is a case by case study. We start (Lemma 5.1) by making a list of the pairs .G; H / such that G=H is likely, for topological reasons, to admit an invariant nearly Kähler structure. The homogeneous spaces that appear in this list are the four compact 6-dimensional examples of 3-symmetric spaces found in [46]. Then we show, for each space, that there exists no other homogeneous nearly Kähler structure than the canonical almost complex structure on it. As a consequence, this chapter is mainly focused on examples. However, the four spaces in question are quite representative of a number of features in nearly Kähler geometry. Thus, it might be read as a sort of survey on the topic. In Section 1, we introduce the notions and speak of 3-symmetric spaces in general. Section 2 contains the most difficult point in the proof of Theorem 1. We look for left-invariant nearly Kähler structures on S 3 S 3 . For this we had to use the algebra of differential forms on the manifold, nearly Kähler manifolds in dimension 6 being characterized, by the work of Reyes Carrión [38] or Hitchin [31], by a differential system on the canonical SU.3/-structure. Section 3 is devoted to two homogeneous spaces CP 3 and F 3 that are the twistor spaces of two 4-dimensional Riemannian manifolds: respectively S 4 and CP 2 . We take the opportunity to specify the relation between nearly Kähler geometry and twistor theory. Weak holonomy stands in the background of Section 4 on the 6-sphere. Indeed, we may derive this notion from that of special holonomy trough the construction of the Riemannian cone as in [4]. Six-dimensional nearly Kähler – or weak holonomy SU.3/ – structures on S 6 are in one-to-one correspondence with constant 3-forms, inducing a reduction of the holonomy to G2 on R7 . Finally, in Section 5, we provide the missing elements for the proof of Theorems 1 and 2. We should mention here the remaining conjecture on nearly Kähler manifolds: Conjecture 2. Every compact nearly Kähler manifold is a 3-symmetric space. That this conjecture is still open means in particular that the homogeneous spaces presented along this chapter are the only known compact (or equivalently, complete) examples in dimension 6. Again by the work of Nagy [35], it may be separated in two restricted conjectures. The first one relates to a similar conjecture on quaternionKähler manifolds and symmetric spaces, for which there are many reasons to believe that it is true (to begin with, it was solved by Poon and Salamon [37] in dimension 8 and recently by Haydeé and Rafael Herrera [28] in dimension 12). The second may be formulated: the only compact, simply connected, irreducible (with respect to the holonomy of the intrinsic connection), 6-dimensional, nearly Kähler manifold is the sphere S 6 – and concerns the core of the nearly Kähler geometry: the fundamental explanation of the rareness of such manifolds or the difficulty to produce non-homogeneous examples.

402


1 Preliminaries: nearly Kähler manifolds and 3-symmetric spaces Nearly Kähler manifolds are a type of almost Hermitian manifolds i.e. 2n-dimensional real manifolds with a U.n/-structure (a U.n/-reduction of the frame bundle) or equivalently, with a pair of tensors .g; J / or .g; !/, where g is a Riemannian metric, J an almost complex structure compatible with g in the sense that g.JX; J Y / D g.X; Y /

for all X; Y 2 TM

(J is orthogonal with respect to g pointwise) and ! is a differential 2-form, called the Kähler form, related to g, J by !.X; Y / D g.JX; Y / for all X; Y 2 TM: Associated with g there is the well-known Levi-Civita connection, r, metric preserving and torsion-free. But nearly Kähler manifolds, as every almost Hermitian x called the intrinsic connection or the manifolds, have another natural connection r, canonical Hermitian connection, which shall be of considerable importance in the sequel. Let so.M / be the bundle of skew-symmetric endomorphisms of the tangent spaces (the adjoint bundle of the metric structure). The set of metric connections of .M; g/ is an affine space O modelled on the space of sections of T M ˝ so.M /. Then the set U of Hermitian connections (i.e. connections which preserve both the metric and the almost complex structure or the Kähler form) is an affine subspace of O with vector space .T M ˝ u.M //, where u.M / is the subbundle of so.M / formed by the endomorphisms which commute with J (or in other words, the adjoint bundle of the U.n/-structure). Finally, we denote by u.M /? the orthogonal complement of u.M / in so.M /, identified with the bundle of skew-symmetric endomorphisms of TM , anti-commuting with J . x is the orthogonal projection of Definition 1.1. The canonical Hermitian connection r x r 2 O on U. Equivalently, it is the unique Hermitian connection such that r r ? is a 1-form with values in u.M / . x is known explicitly: The difference D r r 1 J B .rX J / for all X 2 TM: 2 It measures the failure of the U.n/-structure to admit a torsion-free connection, in other words its torsion or its 1-jet (see [8]). Thus, it can be used to classify almost Hermitian manifolds as in [26]. For example, Kähler manifolds are defined by r itself x Equivalently, since determines both d! and being a Hermitian connection: r D r. the Nijenhuis tensor N , ! is closed and J is integrable. X D

Definition 1.2. Let M be an almost Hermitian manifold. The following conditions are equivalent and define a nearly Kähler manifold:


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x is totally skew-symmetric, (i) the torsion of r (ii) .rX J /X D 0 for all X 2 TM , (iii) rX ! D 13 X d! for all X 2 TM , (iv) d! is of type .3; 0/ C .0; 3/ and N is totally skew-symmetric. The following result, due to Kirichenko [32], is the base of the partial classification by Nagy in [35] of nearly Kähler manifolds. Proposition 1.3 (Kirichenko). For a nearly Kähler manifold, the torsion of the intrinsic connection is totally skew-symmetric (by definition) and parallel: x D 0: r x x x is Hermitian. Moreover, this is equivalent to rr! D 0 or rd! D 0 because r x is also parallel: r xR x D 0. Then M is locally Now, suppose that the curvature of r homogeneous or an Ambrose–Singer manifold. Besides, the associated infinitesimal model is always regular (for a definition of these notions, see [15], citing [43]) and thus, if it is simply connected, M is an homogeneous space. Examples obtained in this way belong to a particular class of homogeneous manifolds: the 3-symmetric spaces, defined by Gray [24], which shall interest us in the rest of this section. As expected, the 3-symmetric spaces are a generalization of symmetric spaces: Definition 1.4. A 3-symmetric space is a homogeneous space M D G=H , where G has an automorphism s of order 3 (instead of an involution, for a symmetric space) such that (1) G0s H G s ; where G s D fg 2 G j s.g/ D gg is the fixed points set of s and G0s is the identity component of G s . Let g, h be the Lie algebras of G, H , respectively. For a symmetric space, the eigenspace m for the eigenvalue 1 of ds W g ! g (the derivative of s) is an Ad.H /invariant complement of h in g, so that symmetric spaces are always reductive. Conversely, for a reductive homogeneous space, we define an endomorphism f of g by f jh D Idjh and f jm D Idjm , which integrates into a Lie group automorphism of G if and only if Œm; m h: (2) For a 3-symmetric space, things p are slightly more complicated because ds has now p 3 3 1 1 2 N three eigenvalues, 1, j D 2 C i 2 and j D j D 2 i 2 , two of which are complex. The corresponding eigenspaces decomposition is g C D hC ˚ mj ˚ mj 2 :

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Setting m D .mj ˚ mj 2 / \ g we get g D h ˚ m;

Ad.H /m m;

(3)

so 3-symmetric spaces are also reductive. As a consequence, invariant tensors, for the left action of G on M , are represented by constant, Ad.H /-invariant tensors on m. For example, an invariant almost complex structure is identified, first with the subbundle T C M T C M of .1; 0/-vectors, then with a decomposition mC D mC ˚ m

where Ad.H /mC mC and m D mC :

(4)

Definition 1.5. The canonical almost complex structure of a 3-symmetric space is the invariant almost complex structure associated to the decomposition mj ˚ mj 2 . In other words, the restriction ds W m ! m represents an invariant tensor S of M satisfying (i) S 3 D Id, (ii) for all x 2 M , 1 is not an eigenvalue of Sx . One can then write S as a (non trivial) third root of unity: p 1 3 S D Id C J; (5) 2 2 where J is the canonical almost complex structure of M . Similarly, an Ad.H /-invariant scalar product g on m defines an invariant metric on M , also denoted by g, and the pair .M; g/ is called a Riemannian 3-symmetric space if and only if g, J are compatible. Conversely a decomposition like (4) comes from an automorphism of order 3 if and only if h, mC , m satisfy ŒmC ; mC m ;

Œm ; m mC

and

ŒmC ; m hC

(6)

instead of (2). Now, conditions involving the Lie bracket might be interpreted, on a reductive homogeneous space, as conditions on the torsion and the curvature of the y The latter is defined as the H -connection on G whose horizontal normal connection r. distribution is G m T G ' G g. y viewed y of the normal connection r, Lemma 1.6. The torsion Ty and the curvature R as constant tensors, are respectively the m-valued 2-form and the h-valued 2-form on m given by yX;Y D ŒX; Y h for all X; Y 2 m: Ty .X; Y / D ŒX; Y m and R Proposition 1.7. A reductive almost Hermitian homogeneous space M D G=H is x a 3-symmetric space if and only if it is quasi-Kähler and the intrinsic connection r y coincides with r.


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An almost Hermitian manifold is said quasi-Kähler or .2; 1/-symplectic iff d! has type .3; 0/C.0; 3/ or equivalently (or rJ ) is a section of ƒ1 ˝u.M /? \ŒŒ2;0 ˝TM where ŒŒ2;0 ƒ2 is the bundle of real 2-forms of type .2; 0/ C .0; 2/. Proof. By Lemma 1.6, (6) is equivalent to Ty .mC ; mC / m ;

Ty .m ; m / mC ;

Ty .mC ; m / D f0g

y C ; mC / D R.m y ; m / D f0g: R.m y 2 ƒ1 ˝ u.M /? (for a metric connection, The first line implies that O D r r the torsion and the difference with the Levi-Civita connection are in one-to-one cory is the canonical Hermitian connection. It also implies that respondence), i.e. r 2;0 D O 2 ŒŒ ˝ TM (note that the bundles u.M /? ' ŒŒ2;0 are isomorphic through the operation of raising, or lowering, indices). The second line is automatically satisfied for a quasi-Kähler manifold, see [17]. There is also a local version of Proposition 1.7, as we announced after Proposition 1.3. Theorem 1.8. An almost Hermitian manifold M is locally 3-symmetric if and only x if it is quasi-Kähler and the torsion and the curvature of the intrinsic connection r satisfy x D 0 and r xR x D 0: rT The definition of a locally 3-symmetric space given in [24] relates to the existence of a family of local cubic isometries .sx /x2M such that, for all x 2 M , x is an isolated fixed point of sx (for a 3-symmetric space, the automorphism s provides such a family, moreover the isometries are globally defined). Then the requirement that M , the Riemannian 3-symmetric space, is a nearly Kähler manifold (which is more restrictive than quasi-Kähler) translates to a structural condition on the homogeneous space. Definition 1.9. A reductive Riemannian homogeneous space is called naturally reductive iff the scalar product g on m representing the metric satisfies g.ŒX; Y ; Z/ D g.ŒX; Z; Y / for all X; Y; Z 2 m: Equivalently, the torsion Ty of the normal connection is totally skew-symmetric. Now, for a 3-symmetric space, the intrinsic connection coincides with the normal connection by Proposition 1.7. Thus, we get the following result from [24]: Proposition 1.10 (Gray). A Riemannian 3-symmetric space equipped with its canonical almost complex structure is nearly Kähler if and only if it is naturally reductive. Remark. Let M D G=H be a compact inner 3-symmetric space such that G is compact, simple. (NB: A 3-symmetric space is called inner if the automorphism s of

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order 3 coincides with an inner automorphism.) The Killing form B of G is negative definite so it induces a scalar product q D B on g. Then the summand m, associated to the eigendecomposition of ds D Ad.h/, is orthogonal to h and the restriction of q to m defines a naturally reductive metric that makes M a nearly Kähler manifold.

2 The case of S 3 S 3 : the natural reduction to SU.3/ In [33], Ledger and Obata gave a procedure to construct a nearly Kähler 3-symmetric space for each compact Lie group G. The Riemannian product G G G has an obvious automorphism of order 3, given by the cyclic permutation, whose fixed point set is the diagonal subgroup G D f.x; x; x/ j x 2 Gg ' G. The resulting homogeneous space is naturally isomorphic to G G: to fix things, we shall identify .x; y/ with Œx; y; 1. In other words, we get a 3-symmetric structure on G G, invariant for the action: 1 ..h1 ; h2 ; h3 /; .x; y// 7! .h1 xh1 3 ; h2 yh3 /:

Now, let q be an Ad.G/-invariant scalar product on g, representing a bi-invariant metric on G. We choose, for the Ad.G/-invariant complement of ıg (the Lie algebra of G) in g˚g˚g, m D f0g˚g˚g, the sum of the last two factors, so that the restriction of q ˚ q ˚ q to m defines a naturally reductive metric g which is not the bi-invariant metric of G G. Indeed, the vector .X; Y / 2 g ˚ g ' Te .G G/ is identified with .0; Y X; X/ 2 m so we have the explicit formula ge ..X; Y /; .X 0 ; Y 0 // D q.Y X; Y 0 X 0 / C q.X; X 0 /. Following this procedure and setting G D SU.2/ ' S 3 , we obtain a 6-dimensional example: S 3 S 3 . In this section we will look for nearly Kähler structures on S 3 S 3 invariant for the smaller group SU.2/ SU.2/ ,! SU.2/ SU.2/ SU.2/; .h1 ; h2 / 7! .h1 ; h2 ; 1/; or for the left action of SU.2/ SU.2/ on itself. Such a structure is then simply given by constant tensors on the Lie algebra su.2/ ˚ su.2/. The latter is a 6-dimensional vector space so, for example, the candidates for the metric belong to a 21-dimensional space S 2 .su.2/˚su.2//. Thus, the calculations involving the Levi-Civita connection as in Definition 1.2 are too hard and we shall look for another strategy. Nearly Kähler manifolds in dimension 6 are special. In particular they are always Einstein [25], spin, and possess a Killing spinor [27]. But the most important feature, for us, is a natural reduction to SU.3/. Indeed, an SU.3/-structure in dimension 6, unlike an almost Hermitian structure, may be defined, without the metric, only by means of differential forms. This should be compared to the fact that a G2 -structure is determined by a special differential 3-form on a 7-manifold.


407

In first instance, an SU.3/-manifold is an almost Hermitian manifold with a complex volume form (a complex 3-form of type .3; 0/ and constant norm) ‰. This complex 3-form can be decomposed into real and imaginary parts: ‰ D C i where X D JX for all X 2 TM: (7) As a consequence, one of the real 3-forms or , together with J , determines the reduction of the manifold to SU.3/. Now, for a nearly Kähler manifold, such a form is naturally given by the differential of the Kähler form. Indeed, because of our preliminaries, d! has type .3; 0/ C .0; 3/ x (see Definition 1.2) and constant norm, since it is parallel for the metric connection r. Definition 2.1. The natural SU.3/-structure of a 6-dimensional nearly Kähler manifold is defined by 1 WD d!; 3 where ! is the Kähler form. On the other hand, Hitchin has observed in [31] that a differential 2-form ! and a differential 3-form , satisfying certain algebraic conditions, are enough to define a reduction of the manifold to SU.3/. In particular they determine the metric g and the almost complex structure J . Indeed, SU.3/ may be seen as the intersection of two groups, Sp.3; R/ and SL.3; C/, which are themselves the stabilizers of two exteriors forms on R6 . The symplectic group Sp.3; R/ stabilizes a non-degenerate 2-form of course. The second group, GL.6; R/, has two open orbits O1 and O2 on ƒ3 R6 . The stabilizer of the forms in the first orbit is SL.3; C/. To define the action of the latter, we must see R6 as the complex vector space C 3 . Consequently, if a differential 3-form belongs to O1 at each point, it determines an almost complex structure J on M . Then !, J determine g under certain conditions. We need to write this explicitly (after [30], [31]). For 2 ƒ3 , we define K 2 6 End.TM / ˝ ƒ by K.X/ D A.X ^ /; where A W ƒ5 ! TM ˝ ƒ6 is the isomorphism induced by the exterior product. Then . / D 16 tr K 2 is a section of .ƒ6 /2 and it can be shown that K 2 D Id ˝ . /: The 3-form Then, for D

belongs to O1 at each point if and only if p

. / < 0: . /, J D

1 K

(8)

408


is an almost complex structure on M . Moreover, the 2-form ! is of type .1; 1/ with respect to J if and only if !^

D 0:

(9)

Finally, ! has to be non degenerate ! ^ ! ^ ! ¤ 0;

(10)

.X; Y / 7! g.X; Y / D !.X; J Y / > 0:

(11)

and g has to be positive:

This is at the algebraic level. At the geometric level, Salamon and Chiossi [13] have shown that the 1-jet of the SU.3/-structure (or the intrinsic torsion) is completely determined by the differentials of !, , . In particular, nearly Kähler manifolds are viewed, in this section, as SU.3/-manifolds satisfying a first order differential system involving these three forms: ´ d! D 3 ; (12) d D 2 ! ^ !; where 2 RC . This differential system was first written by Reyes Carrión in [38]. As a consequence, looking for a nearly Kähler structure on a manifold is the same as looking for a pair of forms .!; / satisfying (8)–(11) together with (12) or, considering the particular form of (12), for a 2-form ! only, satisfying a highly non linear second order differential equation. We shall resolve this system on the space of invariant 2-forms of S 3 S 3 . We work with a class of co-frames .e1 ; e2 ; e3 ; f1 ; f2 ; f3 /, called cyclic co-frames, satisfying (i) .e1 ; e2 ; e3 ; f1 ; f2 ; f3 / is invariant for the action of SU.2/ SU.2/ on itself, (ii) the 1-forms ei (resp. fi ), i D 1; 2; 3 vanish on the tangent space of the first (resp. the second) factor, (iii) dei D eiC1 êiC2 where the subscripts are viewed as elements of Z3 . Similarly, dfi D fiC1 ^ fiC2 . The group of isometries of the sphere S 3 ' SU.2/, equipped with its round bi-invariant metric is SO.4/, with isotropy subgroup SO.3/. Moreover the isotropy representation lifts to the adjoint representation of SU.2/ ' Spin.3/. We denote .u; l/ 7! u:l the action of SO.3/ on the dual su.2/ of the Lie algebra. Then SO.3/ SO.3/ acts transitively on the set of cyclic co-frames by .u; v/:.e1 ; e2 ; e3 ; f1 ; f2 ; f3 / 7! .u:e1 ; u:e2 ; u:e3 ; v:f1 ; v:f2 ; v:f3 /:

(13) 3

In other words, two such co-frames are exchanged by a diffeomorphism of S S 3 and more precisely by an isometry of the canonical metric.


409

Now, any invariant 2-form may be written !D

3 X

ai eiC1 ^ eiC2 C

iD1

3 X

bi fiC1 ^ fiC2 C

iD1

3 X

ci;j ei ^ fj :

(14)

i;j D1

Let A be the column vector of the ai , B the column vector of the bi and C the square matrix .ci;j /i;j D1;2;3 . The latter is subject to the following transformation rule in a change of cyclic co-frame (13): C 7! M C tN;

(15)

where the 3 3 matrices M (resp. N ) represent u (resp. v) in the old base. We have the first essential simplification of (14): Lemma 2.2. Let ! be a non degenerate invariant 2-form on S 3 S 3 . Then !, D 13 d! satisfy (9) if and only if there exists a cyclic co-frame .e1 ; e2 ; e3 ; f1 ; f2 ; f3 / such that (16) ! D 1 e1 ^ f1 C 2 e2 ^ f2 C 3 e3 ^ f3 ; where i 2 R for all i D 1; 2; 3. Proof. Starting from (14), we calculate ! ^ ! ^ !. The 2-form ! is non degenerate if and only if t ACB C det C ¤ 0: (17) Then we calculate D 13 d! using the relations (iii) in the definition of a cyclic co-frame. We find that ! ^ D 0 is equivalent to tAC D CB D 0. Reintroducing these equations in (17), we get det C ¤ 0, i.e. C is nonsingular, and so A D B D 0. Secondly, we can always suppose that C is diagonal. Indeed, we write C as the product of a symmetric matrix S and an orthogonal matrix O 2 SO.3/. We diagonalize S: there exists an orthogonal matrix P such that S D tPDP where D is diagonal. Thus C D tPD.PO/ and by (15), ! can always be written in the form (16) where D D diag.1 ; 2 ; 3 /. This is a key lemma that will constitute the base of our next calculations. Lemma 2.3. Let ! be the invariant 2-form given by (16) in a cyclic co-frame. Then !, D 13 d! define an SU.3/-structure on S 3 S 3 if and only if (i) .1 2 3 /.1 C 2 3 /.1 2 C 3 /.1 C 2 C 3 / < 0, (ii) 1 2 3 > 0. Proof. The first condition is simply (8). Indeed, 3

D 1 .e23 ^ f1 e1 ^ f23 / C 2 .e31 ^ f2 e2 ^ f31 / C 3 .e12 ^ f3 e3 ^ f12 /

and we calculate 81. / D .41 C 42 C 43 221 22 222 23 221 23 / ˝ vol2

(18)

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where vol D e123 ^ f123 . Now, the polynomial of degree 4 in the i ’s in (18) factors into (i) of Lemma 2.3. The second condition comes from the positivity of the metric. Note that the product 1 2 3 D det C is independent on the choice of the co-frame .e1 ; e2 ; e3 ; f1 ; f2 ; f3 /, the determinant of the matrices M , N , in (15), being equal to 1. First, we compute the almost complex structure in the dual frame .X1 ; X2 ; X3 ; Y1 ; Y2 ; Y3 /: JXi D ˛i Xi C ˇi Yi ; where ˛i D

J Yi D ˇi Xi ˛i Yi ;

(19)

1 2 2 .i 2iC1 2iC2 / and ˇi D iC1 iC2 ; k k 9 D k vol:

Now, X 7! !.X; JX/ is the sum of three quadratic forms of degree 2: 2i iC1 iC2 xi2 .2i 2iC1 2iC2 /xi yi C iC1 iC2 yi2 : qi W .xi ; yi / 7! k The discriminant of these forms is k 2 > 0 so they are definite and their sign is given by 1 2 3 . Using (19) it is easy to compute , by (7), and translate the second line of (12) into a system of equations in the i ’s. We refer to [11] for a detailed proof. Lemma 2.4. Let ! be the invariant 2-form given by (16) in a cyclic co-frame. Then !, D 13 d! induce a nearly Kähler structure on S 3 S 3 if and only if there exists

2 R such that, for all i D 1; 2; 3, c D 2i .2i 2iC1 2iC2 /;

(20)

where c D 2 k det C: Finally we can conclude: Proposition 2.5. There exists a unique (up to homothety, up to a sign) left-invariant nearly Kähler structure on S 3 S 3 , corresponding to Ledger and Obata’s construction of a 3-symmetric space. Proof. Thanks to the preparatory work, we only need to solve the system (20) of equations of degree 4 in the i ’s. Let ƒ D 21 C 22 C 23 . For all i D 1; 2; 3, 2i is a solution of the unique equation of degree 2: 2x 2 ƒx c D 0:

(21)

Suppose that 21 , 22 are two distinct solutions of (21) and 23 D 22 , for example. The sum of the roots 21 C22 equals ƒ . But then we also have ƒ D 21 C222 , by definition 2 of ƒ. We immediately get 1 D 0, i.e. ! is degenerate, a contradiction. Thus, the i ’s


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must be equal, up to a sign. The positivity of the metric, (11) or Lemma 2.3 (ii), implies that either the three signs are positive, or only one of them. These two solutions are in fact the same: one is obtained from the other by a rotation of angle in the first factor. Finally, one can always write !, for a left-invariant nearly Kähler structure, ! D .e1 ^ f1 C e2 ^ f2 C e3 ^ f3 / C where D 1 D 2 D p . 3 2R 2 We also have k D 3, 1 1 (22) JXi D p .Xi C 2Yi /; J Yi D p .2Xi C Yi /: 2 2 3 3 Now, (22) coincides with the canonical almost complex structure of SU.2/ SU.2/ SU.2/=SU.2/. Indeed, the automorphism of order 3, s W .h1 ; h2 ; h3 / 7! .h2 ; h3 ; h1 /, induces the endomorphism S W .X; Y / 7! .Y X; X / on m ' su.2/˚su.2/. Then, by (5), J is identified with

1 J W .X; Y / 7! p .2Y X; 2X C Y /: 3 This is nothing else than (22) with D 1 for an appropriate choice of the base. p NB: c D 2 5 3 so by (20), D 1p is inversely proportional to the norm of !. 2 3

3 Twistor spaces: the complex projective space CP 3 and the flag manifold F 3 The twistor space Z of a 4-dimensional, Riemannian, oriented manifold .N; h/ is equipped with two natural almost complex structures. The first, JC , studied by Atiyah, Hitchin and Singer [3], is integrable as soon as N is self-dual, i.e one half of the Weyl tensor of h vanishes, while the second J , which was first considered by Salamon and Eels in [16], is never integrable. Now, on Z, varying the scalar curvature of the fibre, there are also a 1-parameter family of metrics .g t / t 20;C1Œ such that the twistor fibration over .N; h/ is a Riemannian submersion and for all t , .g t ; J˙ / is an almost Hermitian structure on Z. A natural problem is then to look at the type of this almost Hermitian structure. We are particularly interested in the case where N is compact, self-dual and Einstein. Theorem 3.1 (Hitchin, Eels and Salamon). Let .N; h/ be a compact, Riemannian, oriented, self-dual, Einstein 4-manifold, Z its twistor space and .g t / t 2Œ0;C1Œ , the twistor metrics. There exists a choice of the parameter t such that the scalar curvature of the fibre of W Z ! N is proportional to t and (i) .Z; g2 ; JC / is Kähler,

412


(ii) .Z; g1 ; J / is nearly Kähler. This provides us with two compact, homogeneous, nearly Kähler structures in dimension 6 on the complex projective space CP 3 and the flag manifold F 3 , the twistor spaces of S 4 and CP 2 , respectively. Moreover, we shall see that they correspond to a 3-symmetric structure. The goal of this section is to prove that these are the only invariant nearly Kähler structures on the above mentioned spaces. This is quite easy for CP 3 . The complex projective space is seen, in this context, as Sp.2/=U.1/Sp.1/. More generally, CP 2qC1 is isomorphic to Sp.q C 1/=U.1/Sp.q/. Indeed, Sp.q C 1/ acts transitively on C 2qC2 ' HqC1 , preserving the complex lines, and the isotropy subgroup at x 2 CP 2qC1 fixes also jx and acts on the orthogonal of fx; jxg, identified with Hq , as Sp.q/. Representing sp.q/ as usual in the set of q q matrices, the embedding of h D u.1/ ˚ sp.1/, the Lie algebra of H D U.1/Sp.1/, in g D sp.2/ is given by the composition of the natural maps u.1/ ,! sp.1/ and the identity of sp.1/, followed by the diagonal map sp.1/ ˚ sp.1/ ,! sp.2/. Thus, a natural choice of complement of h in g is m D p ˚ v where b 0 ˇˇ 0 a ˇˇ a 2 H and v D b D jx C ky; x; y 2 R : pD 0 0 a 0 These two subsets are Ad.H /-invariant so their sum too and m may be identified with the isotropy representation of Sp.2/=U.1/Sp.1/. The restriction of Ad.H / to p is irreducible because the induced representation of Sp.1/ is isomorphic to the standard one on H. Similarly, the restriction of Ad.H / to v induces the standard representation of U.1/ on C. As a consequence, m has exactly two irreducible summands and the set of invariant metrics on CP 3 has dimension 2. Moreover, we can gain one degree of freedom by working "up to homothety". Finally, we get a 1-parameter family of metrics which may be identified with .g t / t 2Œ0;C1Œ . On the other hand, CP 3 has 22 D 4 invariant almost complex structures according to [46], Theorem 4.3: ˙JC and ˙J . Thus, we are in the hypothesis of Muskarov’s work [34]. The conclusion is, as in Theorem 3.1, Proposition 3.2. The homogeneous space CP 3 ' Sp.2/=U.1/Sp.1/ has a unique – up to homothety, up to a sign – invariant nearly Kähler structure .g1 ; J /, associated to the twistor fibration of S 4 . Things are more complicated for F 3 because the isotropy representation has three irreducible summands. We could adapt Muskarov’s proof [34] for this case, as was done in [11], calculating rJ for every g with Levi-Civita connection r in the 3dimensional space of invariant metrics, and every invariant almost complex structure J , or we can look more carefully at the structure of F 3 , as we will do now. The flag manifold is the space of pairs .l; p/ where p C 3 is a complex plane and l p is a complex line. It is isomorphic to G=H where G D U.3/ and H D S 1 S 1 S 1 . Indeed, we see U.3/ as the space of unitary bases of C 3 . Then the

413


map .u1 ; u2 ; u3 / 7! .l; p/, where l D Cu1 and p D Cu1 ˚ Cu2 , is an H -principal bundle over F 3 with total space G. Now, the maps a W .l; p/ 7! Cua , a D 1; 2; 3 are well-defined (because Cu1 D l; Cu2 D l ? is the orthogonal line of l in p and finally Cu3 D p ? is the orthogonal of p) and give three different CP 1 -fibrations from F 3 to CP 2 (or, if we identify the base spaces, three different realizations of F 3 as an almost complex submanifold of Z, the twistor space of CP 2 ). This has the following geometrical interpretation: on the fibre of 3 , l varies inside p; on the fibre of 1 , it is the plane p that varies around the line l; finally, on the fibre of 2 , both l and p vary while l ? is fixed. These fibrations are all twistor fibrations over an Einstein self-dual 4-manifold. Let IC , JC , KC and I , J , K be the associated almost complex structures, as in Proposition 3.1. What is remarkable is that the Kählerian structures are all distinct but the non-integrable nearly Kähler structures coincide: I D J D K . This observation was already made by Salamon in [40], Section 6. We shall prove this at the infinitesimal level. As a consequence, the four almost complex structures described above, and their opposites, exhaust all the 8 D 23 invariant almost complex structures on F 3 ' U.3/=.S 1 /3 . Let g D u.3/ be the set of the trace-free, anti-Hermitian, 3 3 matrices. Then h D 3u.1/ is identified with the subgroup of the diagonal matrices and the set m of the matrices with zeros on the diagonal is an obvious Ad.H /-invariant complement of h in g. Denote 0 1 0 aN b 0 cN A ha; b; ci D @ a N b c 0 for all a; b; c 2 C. We have

0

1 0 0A2H Adh ha; b; ci D he i.st/ a; e i.t r/ b; e i.rs/ ci e it (23) It is easily seen from (23) that the isotropy representation decomposes into e ir for all h D @ 0 0

0 e is 0

mDp˚q˚r

(24)

where p D fha; 0; 0i j a 2 Cg; q D fh0; b; 0i j b 2 Cg; r D fh0; 0; ci j c 2 Cg: Each of these 2-dimensional subspaces, a, has a natural scalar product ga and a natural complex structure Ja . For example, on p, gp .ha; 0; 0i; ha0 ; 0; 0i/ D Re.aaN 0 / C

and

Jp ha; 0; 0i D hi a; 0; 0i:

Moreover, we denote by p ˚ p the decomposition of pC associated to Jp (and similarly for q, r). The relations between the three subspaces p, q and r are given by

414


the Lie brackets: Œha; 0; 0i; h0; b; 0i D h0; 0; abi; etc.

(25)

and Œha; 0; 0i; ha0 ; 0; 0i D diag.iy; iy; 0/ 2 h; where y D 2 Im.aaN 0 /; etc.

(26)

From (25) and (26), it is easy to calculate ŒpC ; qC r ;

ŒpC ; r qC

and

ŒpC ; pC D f0g; etc.

Thus, pC ˚ qC ˚ r is a subalgebra of mC , corresponding to an invariant complex structure on F 3 : KC . In the same way, IC , JC are represented by p ˚ qC ˚ rC , pC ˚q ˚rC , respectively. On the contrary, mC D pC ˚qC ˚rC is not a subalgebra. However, it satisfies (6) with m D mC D p ˚ q ˚ r . Thus, F 3 is a 3-symmetric space with mC as canonical almost complex structure. By Proposition 1.7, each pair .g; ˙J /, where g is a generic invariant metric, g D rgp C sgq C tgr ; is a .2; 1/-symplectic homogeneous structure on F 3 . Moreover, since IC , JC , KC are integrable, every invariant (strictly) nearly Kähler structure on F 3 has that form, where g is naturally reductive (see Proposition 1.10). It is not hard to see that this corresponds to r D s D t . Proposition 3.3. The nearly Kähler structures associated to the three natural twistor fibrations over CP 2 on F 3 ' U.3/=.S 1 /3 coincide. Moreover, every invariant nearly Kähler structure on F 3 is proportional to this one (or to its opposite). Remark. The decomposition (24) is still the irreducible decomposition for SU.3/ U.3/, so the results remain valid for this smaller group of isometries. This observation will be useful in Section 5.

4 Weak holonomy and special holonomy: the case of the sphere S 6 While S 3 S 3 is considered the hardest case because the isotropy is reduced to f0g, the case of S 6 is apparently the easiest because the isotropy is maximal, H D SU.3/. The isotropy representation is the standard representation of SU.3/ in dimension 6, in particular it is irreducible so there exists only one metric up to homothety and also one almost complex structure up to a sign preserved by H . However, a difficulty occurs since on S 6 unlike on the other spaces considered in sections 2 or 3, the metric doesn’t determine the almost complex structure: Proposition 4.1. Let .M; g/ be a complete Riemannian manifold of dimension 6, not isomorphic to the round sphere. If there exists an almost complex structure J on M


415

such that .M; g; J / is nearly Kähler (non Kählerian) then it is unique. Moreover, in this case, J is invariant by the isometry group of g. This can be proved using the spinors (by [27], a 6-dimensional Riemannian manifold admits a nearly Kähler structure if and only if it carries a real Killing spinor): see [11], Proposition 2.4, and the reference therein, [5], Proposition 1, p. 126, or by a “cone argument” (see below) as in [44], Proposition 4.7. On the contrary, on the sphere S 6 equipped with its round metric g0 , there exist infinitely many compatible nearly Kähler structures: Proposition 4.2. The set J of almost complex structures J such that .S 6 ; g0 ; J / is nearly Kähler, is isomorphic to SO.7/=G2 ' RP 7 . Corollary 4.3. Nearly Kähler structures compatible with the canonical metric on S 6 , or almost complex structures J 2 J, are all conjugated by the isometry group SO.7/ of g0 . To show this, we shall use a theorem of Bär [4]: the Riemannian cone of a nearly Kähler manifold has holonomy contained in G2 . However, in order to remain faithful to the point of view of differential forms adopted in this chapter we prefer the presentation by Hitchin [31] of this fact. According to Section 2, a nearly Kähler structure is determined by a pair of differential forms .!; / satisfying (12) as well as algebraic conditions (8)–(11). Moreover there exists, around each point, an orthonormal coframe .e1 ; : : : ; e6 / such that ! D e12 C e34 C e56 and D e135 e146 e236 e245 ; where e12 D e1 ^ e2 , e135 D e1 ^ e3 ^ e5 , etc. Now, the cone of .M; g/ is the x ; g/, x D M RC and gN D r 2 g C dr 2 in the Riemannian manifold .M N where M x by coordinates .x; r/. We define a section of ƒ3 M

D r 2 dr ^ ! C r 3

(27)

Let u0 D dr and for all i D 1; : : : ; 6, ui D 1r ei .

D u012 C u034 C u056 C u135 u146 u236 u245 x such that Thus, is a generic 3-form, inducing a G2 -structure on the 7-manifold M .u0 ; : : : ; u6 / is an orthonormal co-frame for the underlying Riemannian structure i.e. gN is the metric determined by , given the inclusion of G2 in SO.7/. Moreover, if we denote by the Hodge dual of g, N

D r 3 dr ^ C r 4 ! ^ !: 2

416 Then (12) implies


´

d D 0 d D0

By Gray–Fernandez [18], the last couple of equations is equivalent to r gN D 0, where x ; g/ N In other words, the holonomy of .M N is r gN is the Levi-Civita connection of g. contained in G2 . x can always be written as in (27) where Conversely, a parallel, generic 3-form on M .!; / define a nearly Kähler SU.3/-structure on M . We are now ready to prove Proposition 4.2. Proof. The Riemannian cone of the 6-sphere is the Euclidean space R7 . According to the discussion above, a nearly Kähler structure on S 6 compatible with g0 , defines a parallel or equivalently, a constant 3-form on R7 . This form must have the appropriate algebraic type, i.e. be an element of the open orbit O ' GL.7; R/=G2 ƒ3 R7 . But it must also induce the good metric (the cone metric) on R7 . Finally the 3-forms that parametrize J belong to a subset of O, isomorphic to SO.7/=G2 . The homogeneous nearly Kähler structure on S 6 is defined using the octonions. The octonian product .x; y/ 7! x:y may be described in the following way. First, the 8-dimensional real vector space O decomposes into R ˚ =. The subspace = ' R7 is called the space of imaginary octonions and equipped, for our purposes, with an inner product .x; y/ 7! hx; yi. Secondly, rules (i) to (iv) below are satisfied with respect to this decomposition: (i) 1:1 D 1, (ii) for all x 2 =, 1:x D x, (iii) for all imaginary quaternion x of norm 1, x:x D 1, (iv) finally, for all orthogonal x; y 2 =, x:y D P .x; y/ where P W R7 R7 ! R7 is the 2-fold vector cross product. The latter satisfies itself .x; y; z/ 7! hP .x; y/; zi is a 3-form. In particular P .x; y/ is orthogonal to x and y. Now, let S 6 be the unit sphere in = ' R7 . The tangent space at x 2 S 6 is identified with the subspace of R7 orthogonal to x. Then J is defined by Jx W Tx M ! Tx M; y 7! x:y: This is a well defined almost complex structure because x:y is orthogonal to x by rule (iv) and J 2 D Id by rules (ii), (iii). Moreover J is compatible with the metric induced by h ; i on S 6 (the round metric g0 ). Conversely, starting from J 2 J, we may rebuild an octonian product on R7 by bilinearity. Then J is invariant for the associated group of automorphisms, isomorphic to G2 . Consequently, all nearly Kähler structures on the round sphere are of the same kind (homogeneous). They correspond to different choices of embeddings of


417

G2 into the group of isometries of .S 6 ; g0 /. These embeddings are parametrized by SO.7/=G2 ' RP .7/ thus the above discussion gives an alternative proof of Proposition 4.2. The same question was raised by Friedrich in [19]. His proof is very similar to our first one though it uses the Hodge Laplacian instead of the differential system (12). Moreover, another proof is mentioned, that uses the Killing spinors. Finally, a third or fifth way of looking at this is the following: each SU.3/-structure .!x ; x / on a tangent space Tx S 6 , x 2 S 6 , may be extended in a unique way to a nearly Kähler structure on the whole manifold. Let be the constant 3-form on R7 whose value at .x; 1/ is

.x;1/ D dr ^ !x C x : Then is parallel for the Levi-Civita connection of the flat metric and in return, D 13 d! determine a nearly Kähler structure on S 6 whose values at ! D @r , x coincide with !x , x , consistent with our notations. Now, SU.3/-structures on a 6-dimensional vector space are parametrized by SO.6/=SU.3/ so this constitutes a (differential) geometric proof of the isomorphism SO.7/ SO.6/ ' ' RP .7/ ' J: SU.3/ G2

5 Classification of 3-symmetric spaces and proof of the theorems In this section, we draw all the useful conclusions of the facts gathered in the previous sections about nearly Kähler manifolds and synthesize all the results to achieve the proof of Theorems 1 and 2. As such, the conjecture of Gray and Wolf is not easy to settle because the notion of a nearly Kähler manifold, or even of a 3-symmetric space, are too rich. Indeed, the classification of 3-symmetric spaces [46] discriminates between three types A, B and C that correspond to quite different geometries: A. The first type consists of twistor spaces of symmetric spaces. Indeed, the situation described in the beginning of Section 3 has a wider application than the 6-dimensional twistor spaces of Einstein self-dual 4-manifolds. First, the study of quaternion-Kähler manifolds, i.e. Riemannian manifolds whose holonomy is contained in Sp.q/Sp.1/, provides an analog of that situation in dimension 4q, q 2. Such manifolds still admit a twistor space Z ! M with fibre CP 1 and two almost complex structures JC and J related by JC jV D J jV ;

JC jH D J jH ;

where V is the vertical distribution, tangent to the fibres, and H is the horizontal distribution of the Levi-Civita connection of the base, such that the first one, JC , is

418


always integrable (see for example [39]) while the second, J , is non-integrable. Moreover, for a positive quaternion-Kähler manifold, there exist two natural metrics g1 and g2 such that .g2 ; JC / is a Kählerian structure and .g1 ; J / is a nearly Kähler structure on Z (compare with Proposition 3.1). This includes the twistor spaces of the Wolf spaces: the compact, symmetric, quaternion-Kähler manifolds. Now, this construction can be extended to a larger class of inner symmetric spaces G=K (see [40]). The total space G=H is a generalized flag manifold (i.e. H is the centralizer of a torus in G) and a 3-symmetric space. In particular H contains a maximal torus of G (or has maximal rank) but it is not a maximal subgroup since the inclusion H K is strict. B. The 3-symmetric spaces of the second type are those studied by Wolf [45], Theorem 8.10.9, p. 280. They are characterized by H being the connected centralizer of an element of order 3. Thus, H is not the centralizer of a torus anymore. However it still has maximal rank, i.e. the 3-symmetric space is inner, and is furthermore maximal (for an explicit description, using the extended Dynkin diagram of h, see [45], [46] or [12]). C. Finally, the 3-symmetric spaces of the third type have rank H < rank G. Equivalently M is an outer 3-symmetric space. This includes two exceptional spaces – Spin.8/=G2 and Spin.8/=SU.3/ – and the infinite family G G G=G of Section 2. This division, which was obtained in [46] by algebraic means (or group theory), has a profound geometrical interpretation. Indeed, the three classes can be characterized by the type of the isotropy representation: A. In the first case, the vertical distribution V and the horizontal distribution H of G=H ! G=K are invariant by the left action of G. Moreover, by definition of the natural almost complex structures, they are stable by J˙ . Thus, the isotropy representation is complex reducible (we identify J with the multiplication by i on the tangent spaces). B. The exceptional spaces that constitute the second class of 3-symmetric spaces are known to be isotropy irreducible. Moreover, by [45], Corollary 8.13.5, these are the only non-symmetric isotropy irreducible homogeneous spaces G=H such that H has maximal rank. C . Finally, for the spaces of type C, the isotropy representation is reducible but not complex reducible. Let J be the canonical almost complex structure of G=H , viewed as a constant tensor on m. There exists an invariant subset n such that m decomposes into n ˚ J n. This situation is called real reducible by Nagy [35]. Remark. The dimension 6 is already representative of Gray and Wolf’s classification. Indeed, we have seen that CP 3 and F 3 are the twistor spaces of S 4 and CP 2 , respectively. Secondly, the sphere S 6 ' G2 =SU.3/ is isotropy irreducible (see Section 4). And thirdly, S 3 S 3 belongs to the infinite family of class C.


419

Now, for a general nearly Kähler manifold, we cannot look at the isotropy representation anymore. However we must remember Proposition 1.7 that the normal connection coincides with the intrinsic connection for a 3-symmetric space and so the x The quesisotropy representation is equivalent to the holonomy representation of r. tion then becomes, what can we say about the geometry of a nearly Kähler manifold whose holonomy is respectively: complex reducible, irreducible or real reducible? a. Belgun and Moroianu, carrying out a program of Reyes Carrión [38], p. 57 (especially Proposition 4.24), have shown, in [6], that the holonomy of a 6dimensional nearly Kähler manifold M is complex reducible (or equivalently x is contained in U.2/ SU.3/) if and only if M is the holonomy group of r the twistor space of a positive self-dual Einstein 4-manifold. Thus, by the result of Hitchin [30], the only compact, simply connected, complex reducible, nearly Kähler manifolds in dimension 6 are CP 3 and F 3 . P. A. Nagy generalized this result in higher dimensions. Let M be a complete, irreducible (in the Riemannian sense) nearly Kähler manifold of dimension 2n, n 4, such that the holonomy x is complex reducible. Then M is the twistor space of a representation of r quaternion-Kähler manifold or of a locally symmetric space. b. The irreducible case relates to weak holonomy. Cleyton and Swann [14], [15] have shown an analog of Berger’s theorem on special holonomies [7] for the special geometries with torsion. By this we mean G-manifolds, or real manifolds of dimension m, equipped with a G-structure, G SO.m/, such that the LeviCivita connection of the underlying metric structure is not a G-connection, or equivalently the torsion of the intrinsic connection is not identically zero. More x precisely they made the hypothesis that M , or the holonomy representation of r, are irreducible and that is totally skew-symmetric and parallel (the last condition is automatically satisfied for a nearly Kähler manifold by Proposition 1.3). Then M is (i) a homogeneous space or (ii) a manifold with weak holonomy SU.3/ or G2 . The first case in (ii) corresponds exactly to the 6-dimensional irreducible nearly Kähler manifolds while the second is otherwise called nearly parallel G2 . Moreover, the geometry of the homogeneous spaces in (i) may be specified. Indeed, the proof consists in showing that the curvature of the intrinsic connection x is an Ambrose–Singer connection. This reminds us of is also parallel. Then r Theorem 1.8. Finally, irreducible nearly Kähler manifolds are (i) 3-symmetric of type B or (ii) 6-dimensional. c. Eventually, Nagy has proved in [35], Corollary 3.1, that the complete, simply connected, real reducible, nearly Kähler manifolds are 3-symmetric of type C. He summarized his results in a partial classification theorem ([35], Theorem 1.1): let M be a complete, simply connected, (strictly) nearly Kähler manifold. Then M is an almost Hermitian product of the following spaces: • 3-symmetric spaces of type A, B or C; • twistor spaces of non locally symmetric, quaternion-Kähler manifolds;

420


• 6-dimensional irreducible nearly Kähler manifolds. If we suppose furthermore that the manifold is homogeneous, then there remains only 3-symmetric spaces and 6-dimensional, nearly Kähler, homogeneous manifolds. As a consequence, Conjecture 1 is reduced to dimension 6. Now, the proof of the conjecture in dimension 6 has two parts. First, we must show that the only homogeneous spaces G=H admitting an invariant nearly Kähler structure are those considered in Sections 2, 3, 4. Secondly, for each of these spaces, we must look for all the invariant nearly Kähler structures on it and prove they are all 3-symmetric, as was already done. Lemma 5.1. Let .G=H; g; J / be a simply connected 6-dimensional almost Hermitian homogeneous space, such that the almost Hermitian structure .g; J / is nearly Kähler. Then the Lie algebras of G, H and the manifold are given in following table. dim h

h

g

0

f0g

su.2/ ˚ su.2/

S3 S3

1

iR

i R ˚ su.2/ ˚ su.2/

S3 S3

2

iR ˚ i R

i R ˚ i R ˚ su.2/ ˚ su.2/

S3 S3

iR ˚ i R

su.3/

F3

su.2/

su.2/ ˚ su.2/ ˚ su.2/

S3 S3

3 4

8

(28)

iR ˚ su.2/ i R ˚ su.2/ ˚ su.2/ ˚ su.2/ S 3 S 3 iR ˚ su.2/

sp.2/

CP 3

su.3/

g2

S6

Proof. By a result of Nagy [36], the Ricci tensor of a nearly Kähler manifold M is positive (in dimension 6, this a consequence of Gray’s theorem in [25] that M is Einstein, with positive scalar curvature). Then, by Myer’s theorem, M is compact z of M is a nearly Kähler with finite fundamental group and the universal cover M z is manifold of the same dimension. Moreover, if M ' G=H is homogeneous, M z H z where the groups G, G z and H , H z have the same Lie algebras. isomorphic to G= z instead of M . Consequently, we shall work with M For an homogeneous space, we have the following homotopy sequence: ! 2 .G=H / ! 1 .H / ! 1 .G/ ! 1 .G=H / ! H=H 0 ! 0:

(29)

If the manifold is simply connected, this provides us with a compact surjective morphism ' from the fundamental group of H to the fundamental group of G. We shall


421

look only at the consequences at the Lie algebra level (i.e. at the S 1 or i R factors and not at the finite quotients). The second feature we use is the natural reduction to SU.3/ defined in Section 2. If g and J or ! are invariant for the left action of G, then, so is D d!. As a consequence, the isotropy group H is a subgroup of SU.3/. This leaves the following possibilities for h: f0g, u.1/, u.1/ ˚ u.1/, su.2/, u.2/ D su.2/ ˚ u.1/ and su.3/. Next, to find g, we use the fact that the difference between the dimensions of the two algebras is 6, the dimension of the manifold, and the existence of ' above. The latter allows us to eliminate the following: g D su.2/ ˚ 3u.1/ or 6u.1/ for h D f0g and g D su.3/ ˚ u.1/ for h D su.2/. Table (28) is a list of the remaining cases. Now, h acts as a subgroup of su.3/ on the 6-dimensional space m. Using this, we determine the isotropy representation and the embedding of H into G. In particular we show, when g D h ˚ su.2/ ˚ su.2/, that G=H is isomorphic to S 3 S 3 and G contains SU.2/SU.2/ acting on the left. So the nearly Kähler structures arising from these cases will always induce a left-invariant nearly Kähler structure on S 3 S 3 ' SU.2/ SU.2/. Thus, we need only to consider this more general situation. This was done in Section 2. Now, Theorem 1 is a consequence of Lemma 5.1 and Propositions 2.5, 3.2, 3.3 and Corollary 4.3.

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[33] A. J. Ledger and M. Obata, Affine and Riemannian s-manifolds. J. Differential Geometry 2 (1968), 451–459. 406 [34] O. Muškarov, Structures presque hermitiennes sur des espaces twistoriels et leurs types. C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), 307–309. 412 [35] P.-A. Nagy, Nearly Kähler geometry and Riemannian foliations. Asian J. Math. 6 (2002), 481–504. 400, 401, 403, 418, 419 [36] P.-A. Nagy, On nearly-Kähler geometry. Ann. Global Anal. Geom. 22 (2002), 167–178. 400, 420 [37] Y. S. Poon and S. M. Salamon, Quaternionic Kähler 8-manifolds with positive scalar curvature. J. Differential Geom. 33 (1991), 363–378. 401 [38] R. Reyes Carrión, Some special geometries defined by Lie groups. Ph.D. thesis, Oxford 1993. 400, 401, 408, 419 [39] S. Salamon, Quaternionic Kähler manifolds. Invent. Math. 67 (1982), 143–171. 418 [40] S. Salamon, Harmonic and holomorphic maps. In Geometry seminar “Luigi Bianchi” II—1984, Lecture Notes in Math. 1164, Springer-Verlag, Berlin 1985, 161–224. 400, 413, 418 [41] U. Semmelmann, Conformal Killing forms on Riemannian manifolds. Math. Z. 245 (2003), 503–527. 400 [42] A. Swann, Weakening holonomy. In Quaternionic structures in mathematics and physics (Rome, 1999), Università degli Studi di Roma “La Sapienza”, Rome 1999, 405–415. 400 [43] F. Tricerri, Locally homogeneous Riemannian manifolds. Rend. Sem. Mat. Univ. Politec. Torino 50 (1992), 411–426. 403 [44] M. Verbitsky, An intrinsic volume functional on almost complex 6-manifolds and nearly Kähler geometry. Pacific J. Math. 235 (2008), no. 2, 323–344. 415 [45] J. A. Wolf, Spaces of constant curvature. McGraw-Hill Book Co., New York 1967. 418 [46] J. A. Wolf and A. Gray, Homogeneous spaces defined by Lie group automorphisms. I. J. Differential Geometry 2 (1968), 77–114. 399, 401, 412, 417, 418

Chapter 12

Nearly pseudo-Kähler and nearly para-Kähler six-manifolds Lars Schäfer and Fabian Schulte-Hengesbach

Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Almost pseudo-Hermitian and almost para-Hermitian geometry . . 2.2 Stable three-forms in dimension six and "-complex structures . . . 2.3 Structure reduction of almost "-Hermitian six-manifolds . . . . . 3 Nearly pseudo-Kähler and nearly para-Kähler manifolds . . . . . . . . 3.1 General properties . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Characterisations by exterior differential systems in dimension six 3.3 Consequences for automorphism groups . . . . . . . . . . . . . . 4 Left-invariant nearly "-Kähler structures on SL.2; R/ SL.2; R/ . . . 4.1 An algebraic prerequisite . . . . . . . . . . . . . . . . . . . . . . 4.2 Proof of the uniqueness result . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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425 428 428 430 431 433 433 437 441 442 442 445 451

1 Introduction The notion of a nearly Kähler manifold was introduced and studied in a series of papers by A. Gray in the seventies in the context of weak holonomy. In the last two decades, six-dimensional nearly Kähler manifolds turned out to be of interest in a multitude of different areas including SU.3/-geometries, stable forms, geometries with torsion, existence of Killing spinors, (weak) holonomy, supersymmetric models and compactifications of string theories. For a survey explaining the relations between most of these areas we refer to Chapter 9 of this handbook ([1]). One observes that most of the literature on nearly Kähler geometry deals with Riemannian signature. To our best knowledge the paper on 3-symmetric spaces [16] is the only article by Gray considering also indefinite nearly Kähler metrics. Killing spinors on pseudo-Riemannian manifolds were studied in [23] where nearly pseudoKähler and nearly para-Kähler manifolds appear in a natural way. The subject of nearly para-Kähler manifolds was further developed in [21]. The prefix “para” roughly means

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that the anti-involutive complex structure is replaced by an involutive para-complex structure. We refer to Section 2.1 for details on para-complex geometry. Motivated by a class of solutions of the topological-antitopological fusion equations on the tangent bundle [31], [30] and the similarity to special Kähler geometry, we became interested in Levi-Civita flat nearly Kähler and Levi-Civita flat nearly paraKähler manifolds. A classification of these manifolds in a constructive manner has been established in [10], [11]. From these results it follows that non-Kählerian examples only exist in pseudo-Riemannian geometry. In other words, nearly Kähler geometry in the pseudo-Riemannian world can be very different from the better-understood Riemannian world. There is a left-invariant nearly Kähler structure on S 3 S 3 which arises from a classical construction of 3-symmetric spaces by Ledger and Obata [25]. It is shown in [5] (and also in the extended English version [6] contained in this handbook) that this nearly Kähler structure is the only one on S 3 S 3 up to homothety. In fact, the proof of this uniqueness result has been the most difficult step in the classification of homogeneous nearly Kähler structures in dimension six. The main tool is the well-known characterisation [27] of a nearly Kähler structure on a six-manifold as an SU.3/-structure .!; C ; / satisfying the exterior system d! D 3 C ; d D!^!

(1.1) (1.2)

for a real constant which depends on sign and normalisation conventions. The starting point of this article is the following observation. The construction of a 3-symmetric space from G D SL.2; R/ instead of SU.2/ defines a left-invariant nearly pseudo-Kähler structure on SL.2; R/ SL.2; R/. We shortly recall this construction explicitly. The group G G G admits a symmetry of order three given by .g1 ; g2 ; g3 / 7! .g2 ; g3 ; g1 / which stabilises the diagonal : The tangent space of M 6 D G G G= is identified with p D f.X; Y; Z/ 2 g ˚ g ˚ g j X C Y C Z D 0g: Denote by Kg the Killing form of g and define an invariant scalar product on g˚g˚g by g D Kg ˚ Kg ˚ Kg : This yields a naturally reductive metric on M 6 : Using Proposition 5.6 of [16] this metric is nearly pseudo-Kähler. For completeness sake we recall that the complex structure is given by 2 1 J.X; Y; Z/ D p .Z; X; Y / C p .X; Y; Z/: 3 3 Considering Butruille’s results, it is natural to ask how many left-invariant nearly pseudo-Kähler structures there are on SL.2; R/ SL.2; R/. Comparing with the results mentioned in the last paragraph, the answer seems a priori hard to guess. The main result of this article is the proof that there is a unique left-invariant nearly pseudo-Kähler structure on all Lie groups with Lie algebra sl.2; R/ ˚ sl.2; R/. A byproduct of the proof is the result that there are no nearly para-Kähler structures

Chapter 12. Nearly pseudo-Kähler and nearly para-Kähler six-manifolds

427

on these Lie groups. We add the remark that there exist co-compact lattices for these Lie groups. Indeed, the article [28] contains a complete list of the compact quotients of Lie groups with Lie algebras sl.2; R/, which also give rise to compact quotients on a direct product of such groups. When dealing with nearly pseudo-Kähler structures, the problem arises that many facts which are well known for the Riemannian signature have never been shown for indefinite metrics. For instance, a hyperbolically nearly Kähler structure is defined in [4] as a SU.2; 1/-structure satisfying the same exterior system (1.1), (1.2) as a nearly Kähler structure. It is not obvious whether this definition is equivalent to Gray’s classical definition of an (indefinite) nearly pseudo-Kähler manifold which is used in [10], [31]. However, the proof of our main result in Section 3 essentially relies on this exterior system and we have to prove the equivalence. The close analogy between the pseudo-Hermitian and the para-Hermitian case makes it desirable to give a unified proof dealing with all possible cases at the same time. Therefore, we seize the opportunity and introduce a language that allows us to treat analogous aspects of almost pseudo-Hermitian geometry and almost paraHermitian geometry simultaneously. This language is consistent with [29] and similar to [24]. In the preliminary section, the necessary basic notions are recalled in this unified language. In particular, we recall some facts about stable forms in dimension six which turn out to be very useful in characterising special almost Hermitian structures and special almost para-Hermitian structures. Section 3 is devoted to proving the mentioned characterisation of six-dimensional nearly pseudo-Kähler and nearly para-Kähler manifolds by the exterior system. Since we have to generalise many facts from the Riemannian setting, we give a self-contained proof. Although we follow the ideas of the proof in [27], we clarify the structure of the proof by elaborating the role of the Nijenhuis tensor. In particular, we prove that a half-flat structure is additionally nearly half-flat if and only if the Nijenhuis tensor is skew-symmetric. As a first application, we prove some results on the automorphism group of a nearly (para-)Kähler six-manifold in Section 3.3. In Section 4, we finally obtain the aforementioned structure results on SL.2; R/ SL.2; R/. It turns out that the proof is considerably more technical than in the compact case S 3 S 3 , cf. [5] or [6]. We also extend the results on S 3 S 3 by proving the non-existence of nearly (para-)Kähler structures of indefinite signature. The authors wish to thank V. Cortés and P.-A. Nagy for useful discussions. In particular, Section 3.3 has been inspired by P.-A. Nagy.

428


2 Preliminaries 2.1 Almost pseudo-Hermitian and almost para-Hermitian geometry We recall that an almost para-complex structure on a 2m-dimensional manifold M is an endomorphism field squaring to the identity such that both eigendistributions (for the eigenvalues ˙1) are m-dimensional. An almost para-Hermitian structure consists of a neutral metric and an antiorthogonal almost para-complex structure. For a survey on para-complex geometry we refer to [2] or [12]. In the following, we introduce the unified language describing almost pseudoHermitian and almost para-Hermitian geometry simultaneously. The philosophy is to put an “"” in front of all notions which is to be replaced by “para” for " D 1 and is to be replaced by “pseudo” or to be omitted for " D 1. From now on, we always suppose " 2 f˙1g. To begin with, we consider the "-complex numbers C" D fx C i " y; x; y 2 Rg with i2" D ". For the para-complex numbers, " D 1, there are obvious analogues of conjugation, real and imaginary parts and the square of the (not necessarily positive) N absolute value given by jzj2 D z z: Moreover, let V be a real vector space of even dimension n D 2m. We call an endomorphism J an "-complex structure if J 2 D " idV and if additionally, for " D 1, the ˙1-eigenspaces V ˙ are m-dimensional. An "-Hermitian structure is an "-complex structure J together with a pseudo-Euclidean scalar-product g which is "-Hermitian in the sense that it holds g.J ; J / D "g.; /: We denote the stabiliser in GL.V / of an "-Hermitian structure as the "-unitary group U" .p; q/ D fL 2 GL.V / j ŒL; J D 0; L g D gg ´ U.p; q/; p C q D m; for " D 1, Š GL.m; R/; for " D 1. Here, the pair .2p; 2q/ is the signature1 of the metric for " D 1. For " D 1, the group GL.m; R/ acts reducibly such that V D V C ˚ V and the signature is always .m; m/. An almost "-Hermitian manifold is a manifold M of dimension n D 2m endowed with a U" .p; q/-structure or, equivalently, with an almost "-Hermitian structure which consists of an almost "-Hermitian structure J and an "-Hermitian metric g. The nondegenerate two-form ! WD g.; J / is called fundamental two-form. Given an almost "-Hermitian structure .g; J; !/, there exist pseudo-orthonormal local fe1 ; : : : ; e2m g such that Jei D eiCm for i D 1; : : : ; m and ! D P frames i.iCm/ e , where i WD g.ei ; ei / for i D 1; : : : ; m. Upper indices will always " m i i D1 denote dual (not metric dual) one-forms and e ij stands for e i ^ e j . We call such a 1 Please

note that in our convention 2p refers to the negative directions.


429

frame "-unitary. If m 3, we can always achieve 1 D 2 by reordering the basis vectors. For both values of ", the "-complexification TM ˝ C" of the tangent bundle decomposes into the ˙i " -eigenbundles TM 1;0 and TM 0;1 . This induces the wellknown bi-grading of C" -valued exterior forms r;s D .ƒr;s / D .ƒr .TM 1;0 / ˝ ƒs .TM 0;1 / /: If X is a vector field on M , we use the notation X 1;0 D

1 .X C i " "JX/ 2 .TM 1;0 /; 2

X 0;1 D

1 .X i " "JX / 2 .TM 0;1 /; 2

for the real isomorphisms from TM to TM 1;0 respectively TM 0;1 . As usual in almost Hermitian geometry, we define the bundles ƒr;s for r ¤ s and Œƒr;r by the property

ƒr;s ˝ C" D ƒr;s ˚ i " ƒr;s D ƒr;s ˚ ƒs;r ; Œƒr;r ˝ C" D Œƒr;r ˚ i " Œƒr;r D ƒr;r : The sections in these bundles are denoted as real forms of type .r; s/C.s; r/ respectively of type .r; r/ and the spaces of sections by r;s respectively by Œr;r . For instance, it holds that Œ1;1 D f˛ 2 2 M j ˛.X; Y / D "˛.JX; J Y /g; such that the fundamental form is of type .1; 1/ and similarly

3;0 D f˛ 2 3 M j ˛.X; Y; Z/ D "˛.X; J Y; J Z/g:

(2.1)

Only in the para-complex case, " D 1, there is a decomposition of the real tangent bundle TM D V ˚ H into the ˙1-eigenbundles of J which also induces a bi-grading of real forms. It is also straightforward to show that

ƒ3;0 Š ƒ3 V ˚ ƒ3 H ;

(2.2)

when considering the characterisation (2.1). Returning to analogies, we recall that the Nijenhuis tensor of the almost "-complex structure J satisfies N.X; Y / D "ŒX; Y ŒJX; J Y C J ŒJX; Y C J ŒX; J Y D .rJX J / Y C .rJ Y J / X C J.rX J / Y J.rY J / X

(2.3)

for real vector fields X, Y , Z and for any torsion-free connection r on M . For both values of ", it is well known that the Nijenhuis tensor is the obstruction to the integrability of the almost "-complex structure. In the following, let r always denote the Levi-Civita connection of the metric g of an almost "-Hermitian manifold. Differentiating the almost "-complex structure,

430


its square and the fundamental two-form yields for both values of " the formulas .rX J / J Y D J.rX J / Y; g..rX J /Y; Z/ D .rX !/.Y; Z/;

(2.4)

for all vector fields X; Y; Z. Using these formulas, it is easy to show that for any almost "-Hermitian manifold, the tensor A defined by A.X; Y; Z/ D g..rX J /Y; Z/ D .rX !/.Y; Z/ has the symmetries A.X; Y; Z/ D A.X; Z; Y /; A.X; Y; Z/ D "A.X; J Y; J Z/

(2.5) (2.6)

for all vector fields X; Y; Z. The decomposition of the U" .p; q/-representation space of tensors with the same symmetries as A into irreducible components leads to a classification of almost "Hermitian manifolds which is classical for U.m/ [18]. The para-complex case for the group GL.m; R/ is completely worked out in [15]. In [24], the Gray–Hervella classes are generalised to almost "-Hermitian structures, which are denoted by generalised almost Hermitian or GAH structures there. Analogues of all sixteen Gray–Hervella classes are established. These are invariant under the respective group action, but obviously not irreducible for the para-Hermitian case when compared to the decomposition in [15]. Finally, we mention the useful formula 2.rX !/.Y; Z/ D d!.X; Y; Z/ C "d!.X; J Y; J Z/ C "g.N.Y; Z/; JX /

(2.7)

holding true for all vector fields X , Y , Z on any almost "-Hermitian manifold. A short direct proof for " D 1, g Riemannian, is given in the handbook article [26], which also holds literally for pseudo-Riemannian metrics and with sign modifications for " D 1. Alternatively, we refer to [22] for " D 1.

2.2 Stable three-forms in dimension six and "-complex structures We review a construction given in [19] which associates to a stable three-form on a six-dimensional oriented real vector space V an "-complex structure on the same vector space. Therefore we recall that a k-form 2 ƒk V is said to be stable if its orbit U under GL.V / is open. Denote by the canonical isomorphism ƒk V Š ƒ6k V ˝ ƒ6 V : For any three-form , one considers K W V ! V ˝ ƒ6 V defined by K .v/ WD ..v ³ / ^ /


431

and the quartic invariant

./ WD

1 tr .K2 / 2 .ƒ6 V /˝2 : 6

This invariant is different from zero if and only if is stable. Since L D ƒ6 V is of dimension one, there exists a well-defined notion of positivity and norm in L ˝ L: Therefore we can, by means of the orientation, associate a volume form ./ to a stable three-form by p ./ WD j ./j: Using this volume we define an endomorphism J .v/ WD

1 K .v/; ./

which can be proven (cf. [19], [9]) to be an "-complex structure, where " is the sign of

./: For both values of ", a stable three-form is of type .3; 0/ C .0; 3/ with respect to its induced "-complex structure J or, in other words, ‰ D C i " J is a .3; 0/-form (where J .X; Y; Z/ D .J X; J Y; J Z/). Moreover, a stable three-form is non-degenerate in the sense that for v 2 V , v ³ D 0 H) v D 0

(2.8)

and the induced volume form satisfies the formula 1 (2.9) ./ D J ^ : 2 Almost all assertions are straightforward to verify when choosing a basis such that the stable three-form is in the normal form D e 123 C ".e 156 C e 426 C e 453 /

(2.10)

which satisfies ./ D 4".e 1:::6 /˝2 , J2 D " idV and J.ei / D ˙eiC3 for i D 1; 2; 3 where the sign ˙ depends on the orientation. It is worth mentioning that for every stable three-form in the orbit with " D 1, there is also a basis such that D e 123 C e 456

(2.11)

where fe1 ; e2 ; e3 g and fe4 ; e5 ; e6 g span the ˙1-eigenspaces V ˙ of J .

2.3 Structure reduction of almost "-Hermitian six-manifolds Let .V; g; J; !/ be a 2m-dimensional "-Hermitian vector space and ‰ D C C i " be an .m; 0/-form of non-zero length. We define the special "-unitary group SU" .p; q/

432


as the stabiliser of ‰ in the "-unitary group U" .p; q/ such that ´ SU.p; q/ ; p C q D m; for " D 1; " SU .p; q/ D StabGL.V / .g; J; ‰/ Š SL.m; R/; for " D 1, where SL.m; R/ acts reducibly such that V D V C ˚ V . With this notation, an SU" .p; q/-structure on a manifold M 2m is an almost "Hermitian structure .g; J; !/ together with a global .m; 0/-form ‰ of non-zero constant length. Locally, there exists an "-unitary frame fe1 ; : : : ; em ; emC1 D Je1 ; : : : ; e2m D Jem g which is adapted to the SU" .p; q/-reduction ‰ in the sense that ‰ D a.e 1 C i " e .mC1/ / ^ ^ .e m C i " e 2m /

(2.12)

for a constant a 2 R . In dimension six, there is a characterisation of SU" .p; q/-structures in terms of stable forms. Given a six-dimensional real vector space V; we call a pair .!; / of a stable ! 2 ƒ2 V and a stable 2 ƒ3 V compatible if it holds ! ^ D 0:

(2.13)

We claim that the stabiliser in GL.V / of a compatible pair is StabGL.V / .!; / D SU" .p; q/;

p C q D 3;

where " 2 f˙1g is the sign of ./, that is, J2 D " idV . This can be seen as follows. For the two-form !, stability is equivalent to non-degeneracy and we choose the orientation on V such that ! 3 is positive. By the previous section, we can associate an "-complex structure J to the stable three-form . For instance in an adequate basis, it is easy to verify that ! ^ D 0 is equivalent to the skew-symmetry of J with respect to !. Equivalently, the pseudo-Euclidean metric g D " !.; J /;

(2.14)

induced by ! and is "-Hermitian with respect to J . Since ‰ D C i " J is a .3; 0/-form and the stabiliser of ! and also stabilises the tensors induced by them, the claim follows. We conclude that an SU" .p; q/-structure, p C q D 3, on a six-manifold is characterised by a pair .!; C / 2 2 M 3 M of everywhere stable and compatible forms such that the induced .3; 0/-form ‰ D C C i " J C C D C C i " has constant non-zero length with respect to the induced metric (2.14). In an "-unitary frame which is adapted to ‰ in the sense of (2.12), the formula

^

C

Dk

C 21

k

6

!3

(2.15)


433

is easily verified. Thus, given a compatible pair .!; C / of stable forms, it can be checked that the induced .3; 0/-form ‰ has constant non-zero length without explicitly computing the induced metric. We remark that in the almost Hermitian case, the literature often requires ‰ to be normalised such that k C k2 D 4, for instance in [8]. In the more general almost "-Hermitian case, we have in an adapted local "-unitary frame (2.12) with 1 D 2 , C

D a.e 123 C ".e 156 C e 426 C e 453 //

and

k

C 2

k D 4a2 3

(2.16)

for a real constant a. Therefore we have to consider two different normalisations k C k D ˙4 or we multiply the metric by 1 if necessary such that k C k is always positive. Finally, we remark that SU.3/-structures are classified in [8] and it is shown that the intrinsic torsion is completely determined by the exterior derivatives d!, d C and d .

3 Nearly pseudo-Kähler and nearly para-Kähler manifolds The main objective of this section is to generalise the characterisation of six-dimensional nearly Kähler manifolds by an exterior differential system to nearly pseudoKähler and nearly para-Kähler manifolds. We remark that Riemannian nearly Kähler manifolds are discussed intensively in the articles [26] and [6] of this handbook.

3.1 General properties Definition 3.1. An almost "-Hermitian manifold .M 2m ; g; J; !/ is called nearly "-Kähler manifold if its Levi-Civita connection r satisfies the nearly "-Kähler condition .rX J / X D 0 for all X 2 .TM /: A nearly "-Kähler manifold is called strict if rX J ¤ 0 for all non-trivial vector fields X. A tensor field B 2 ..TM /˝2 ˝ TM / is called totally skew-symmetric if the tensor g.B.X; Y /; Z/ is a three-form. The following characterisation of a nearly "-Kähler manifold is well known in the Riemannian context. Proposition 3.2. An almost "-Hermitian manifold .M 2m ; g; J; !/ satisfies the nearly "-Kähler condition if and only if d! is of real type .3; 0/ C .0; 3/ and the Nijenhuis tensor is totally skew-symmetric. Proof. The nearly "-Kähler condition is satisfied if and only if the tensor A D r! is a three-form because of the antisymmetry (2.5).

434


Assume first that .g; J; !/ is a nearly "-Kähler structure. Comparing the identities (2.1) and (2.6), we see that the real three-form A is of type .3; 0/ C .0; 3/. Since d! is the alternation of r!, we have d! D 3r! D 3A 2 3;0 :

(3.1)

Furthermore, if we apply the nearly "-Kähler condition to the expression (2.3), the Nijenhuis tensor of a nearly "-Kähler structure simplifies to N.X; Y / D 4 J.rX J / Y:

(3.2)

We conclude that the Nijenhuis tensor is skew-symmetric since (2.6)

g.N.X; Y /; Z/ D 4A.X; Y; J Z/ D 4"J A.X; Y; Z/:

(3.3)

The converse follows immediately from the identity (2.7) when considering (2.1). For self-containedness we give a direct proof. Assume that d! 2 3;0 and the Nijenhuis tensor is skew-symmetric. To begin with, we observe that .rY !/ .X; X/ D 0 D .rJ Y !/ .X; JX / by (2.5) and (2.6). With this identity, we have on the one hand 0 D "g.N.JX; J Y /; JX / D g.N.X; Y /; JX / (2.3)

D g..rJX J / Y; JX / C g..rJ Y J / X; JX / C g.J.rX J / Y; JX / g.J.rY J / X; JX /

(2.4)

D .rJX !/ .Y; JX / C ".rX !/ .Y; X /

(2.5)

D .rJX !/ .Y; JX / ".rX !/ .X; Y /;

and on the other hand (2.1)

0 D "d!.X; X; Y / D d!.X; JX; J Y / D .rX !/.JX; J Y / C .rJX !/.J Y; X / C .rJ Y !/.X; JX / (2.6)

D ".rX !/.X; Y / C .rJX !/.Y; JX /:

It follows that .rX !/ .X; Y / D 0 which is equivalent to the nearly "-Kähler condition. Remark 3.3. The notion of nearly "-Kähler manifold corresponds to the generalised class W1 in [24]. However, in the para-Hermitian case, there are two subclasses, see [15]. Indeed, we already observed that (2.2)

A D r! 2 3;0 D .ƒ3 V ˚ ƒ3 H / for a nearly para-Kähler manifold. x on an almost "-Hermitian manifold .M 2m ; g; J; !/ "-HerWe call a connection r x D 0 and rJ x D 0. mitian if rg


435

Proposition 3.4. An almost "-Hermitian manifold .M 2m ; g; J; !/ admits an "-Hermitian connection with totally skew-symmetric torsion if and only if the Nijenhuis x and its torsion tensor is totally skew-symmetric. If this is the case, the connection r T are uniquely defined by xX Y; Z/ D g.rX Y; Z/ C 1 g.T .X; Y /; Z/; g.r 2 g.T .X; Y /; Z/ D "g.N.X; Y /; Z/ d!.JX; J Y; J Z/; x the canonical "-Hermitian connection (with skew-symmetric torsion). and we call r Proof. The Riemannian case is proved in [14], the para-complex case in [21]. In fact, the sketched proof in [14] holds literally for the almost pseudo-Hermitian case with indefinite signature as well. For completeness, we give a direct proof for all cases simultaneously. x Y X ŒX; Y D SX Y SY X be the totally skewxX Y r Let T .X; Y / D r x where SX Y D r xX Y rX Y is symmetric torsion of an "-Hermitian connection r the difference tensor with respect to the Levi-Civita connection r of g. Then, the Nijenhuis tensor is totally skew-symmetric as well, since we have g.N.X; Y /; Z/ D "g.T .X; Y /; Z/ C g.T .JX; J Y /; Z/ C g.T .JX; Y /; J Z/ C g.T .X; J Y /; J Z/;

(3.4)

x D 0. Moreover, the difference tensor SX is skew-symmetric with using only rJ x D 0. Combining this fact with the total skew-symmetry of the respect to g, for rg torsion, we find that SX Y D SY X and consequently xX Y; Z/ D g.rX Y; Z/ C 1 g.T .X; Y /; Z/: g.r 2 x With this identity and r! D 0, the equation 2rJX !.Y; Z/ D g.T .JX; Y /; J Z/ C g.T .JX; J Y /; Z/

(3.5)

follows. Finally, we verify the claimed formula for the torsion: (2.6)

d!.JX; J Y; J Z/ D ".rJX !.Y; Z/ C rJ Y !.Z; X / C rJ Z !.X; Y // (3.5)

D ".g.T .JX; J Y /; Z/ C g.T .JX; Y /; J Z/ C g.T .X; J Y /; J Z//

(3.4)

D "g.N.X; Y /; Z/ g.T .X; Y /; Z/:

Conversely, if the Nijenhuis tensor is skew-symmetric, it is straightforward to verify that the defined connection is "-Hermitian with skew-symmetric torsion. Remark 3.5. An almost Hermitian manifold is said to be of type G1 if it admits a Hermitian connection with skew-symmetric torsion. Such manifolds are studied in [26] in this handbook. More generally, the proposition justifies to say that an almost

436


"-Hermitian manifold is of type G1 if it admits an "-Hermitian connection with skewsymmetric torsion. In particular, the proposition applies to nearly "-Kähler manifolds. In this case, the skew-symmetric torsion T of the canonical "-Hermitian connection simplifies to T .X; Y / D "J.rX J /Y D

1 "N.X; Y / 4

due to the identities (3.1), (3.2) and (3.3). x of a nearly "-Kähler manProposition 3.6. The canonical "-Hermitian connection r 2m ifold .M ; J; g; !/ satisfies x x / D 0: r.rJ / D 0 and r.T x D 0. A short proof of the first Proof. The two assertions are equivalent since rJ assertion for the Hermitian case is given in [3]. This proof generalises without changes to the pseudo-Hermitian case since it essentially uses the identity 2 J /Y; Z/ D X;Y;Z g..rW J /X; .rY J /J Z/; 2g..rW;X

which was proved in [17] for Riemannian metrics and also holds true in the pseudoRiemannian setting (see [23], Proposition 7.1). The para-Hermitian version is proved in [21], Theorem 5.3. Corollary 3.7. On a nearly "-Kähler manifold .M 2m ; J; g; !/, the tensors rJ and N D 4"T have constant length. Proof. This is obvious since both tensors are parallel with respect to the connection x which preserves in particular the metric. r Remark 3.8. In dimension six, the fact that rJ has constant length is usually expressed by the equivalent assertion that a nearly "-Kähler six-manifold is of constant type, i.e. there is a constant 2 R such that g..rX J / Y; .rX J / Y / D f g.X; X/g.Y; Y / g.X; Y /2 C "g.JX; Y /2 g: In fact, the constant is D 14 krJ k2 . Furthermore, it is well known in the Riemannian case that strict nearly Kähler six-manifolds are Einstein manifolds with Einstein constant 5 [17]. The same is true in the para-Hermitian case [21] and in the pseudoHermitian case [32]. The case krJ k2 D 0 for a strict nearly "-Kähler six-manifold can only occur in the para-complex world. We give different characterisations of such structures which provide an obvious break in the analogy of nearly para-Kähler and nearly pseudoKähler manifolds.


437

Proposition 3.9. Let .M 6 ; g; J; !/ be a six-dimensional strict nearly para-Kähler manifold. Then the following properties are equivalent: (i) krJ k2 D kAk2 D 0: (ii) The three-form A D r! 2 3;0 is either in .ƒ3 V / or in .ƒ3 H /. (iii) The three-form A D r! 2 3;0 is not stable. (iv) The metric g is Ricci-flat. Proof. We choose a local frame fe1 ; : : : ; e6 g such that fe 1 ; e 2 ; e 3 g spans the C1-eigenspace V of J , fe 4 ; e 5 ; e 6 g spans the 1-eigenspace H of J and g.ei ; eiC3 / D 1 for i D 1; 2; 3. According to (2.2), there are local functions a and b such that A D ae 123 C be 456 . Thus, it holds 1 (2.15) kAk2 ! 3 D J A ^ A D .ae 123 be 456 / ^ .ae 123 C be 456 / D 2abe 123456 : 6 With ! D e 14 e 25 e 36 and ! 3 D 6e 123456 , we have kAk2 D 2ab. Since A is nowhere zero due to the strictness and considering also (2.9), the first three assertions are equivalent to a D 0 or b D 0. Finally, assertions (i) and (iv) are equivalent by Theorem 5.5 in [21]. Flat strict nearly para-Kähler manifolds .M; g; J; !/ are classified in [11]. It turns out that they always satisfy krJ k2 D 0. In [15], almost para-Hermitian structures on tangent bundles T N of real three-dimensional manifolds N 3 are discussed. It is shown that the existence of nearly para-Kähler manifolds satisfying the second condition of Proposition 3.9 is equivalent to the existence of a certain connection on N 3 : However, to the authors best knowledge, there exists no reference for an example of a Ricci-flat nearly para-Kähler structure which is not flat.

3.2 Characterisations by exterior differential systems in dimension six The following lemma explicitly relates the Nijenhuis tensor to the exterior differential. For " D 1, it gives a characterisation of Bryant’s notion of a quasi-integrable U.p; q/structure, p C q D 3, in dimension six [4]. Let .M 6 ; g; J; !/ be a six-dimensional almost "-Hermitian manifold. Given a local "-unitary frame fe1 ; : : : ; e6 D Je3 g, we define a local frame fE 1 ; E 2 ; E 3 g of .TM 1;0 / by E i WD .e i C i " "Je i / D .e i C i " e iCm / for i D 1; 2; 3 and denote it as a local "-unitary frame of .1; 0/-forms. The dual vector fields of the .1; 0/-forms are Ei D ei1;0 D

1 1 .ei C i " "Jei / D .ei C i " "eiCm / 2 2

438


such that the C" -bilinearly extended metric satisfies 1 g.Ei ; Exj / D i ıij 2

and

g.Ei ; Ej / D 0

in such a frame. Lemma 3.10. The Nijenhuis tensor of an almost "-Hermitian six-manifold .M 6 ; g; J; !/ is totally skew-symmetric if and only if for every local "-unitary frame of .1; 0/-forms, there exists a local C" -valued function such that .dE .1/ /0;2 D .1/ E .2/ .3/

(3.6)

for all even permutations of f1; 2; 3g. Proof. First of all, the identities S / D 4"ŒVx ; W S 1;0 N.Vx ; W

and

S/ D 0 N.V; W

for any vector fields V D V 1;0 , W D W 1;0 in TM 1;0 follow immediately from the definition of N . Using the first identity, we compute in an arbitrary local "-unitary frame dE i .Exj ; Exk / D E i .ŒExj ; Exk / D 2i g.ŒExj ; Exk ; Exi / 1 D 2i g.ŒExj ; Exk 1;0 ; Exi / D " i g.N.Exj ; Exk /; Exi / 2 for all possible indices 1 i; j; k 3. If the Nijenhuis tensor is totally skewsymmetric, equation (3.6) follows by setting 1 " g.N.Ex1 ; Ex2 /; Ex3 /: (3.7) 2 Conversely, the assumption (3.6) for every local "-unitary frame implies that the Nijenhuis tensor is everywhere a three-form when considering the same computation S / D 0. and N.V; W

D

If there is an SU" .p; q/-reduction with closed real part, this characterisation can be reformulated globally in the following sense. Proposition 3.11. Let .!; C / be an SU" .p; q/-structure on a six-manifold M such that C is closed. Then the Nijenhuis tensor is totally skew-symmetric if and only if d

D!^!

(3.8)

for a global real function . Proof. It suffices to proof this locally. Let fE i g be an "-unitary frame of .1; 0/forms with 1 D 2 which is adapted to the SU" .p; q/-reduction such that ‰ D

Chapter 12. Nearly pseudo-Kähler and nearly para-Kähler six-manifolds C

C i"

439

D aE 123 for a real constant a as in (2.12). The fundamental two-form is m

1 X N ! D i" k E k k 2 kD1

x in such a frame. Furthermore, as is closed, we have d ‰ D i " d D d ‰; 2;2 which implies that d ‰ 2 ƒ . Considering this, we compute the real 4-form d D "i " d ‰ D "i " a .dE 1 /0;2 ^ E 23 C .dE 2 /0;2 ^ E 31 C .dE 3 /0;2 ^ E 12 C

and compare this expression with 1 N N N N N N ".2 3 E 2233 C 1 3 E 1133 C 1 2 E 1122 / 2 1 NN NN NN D "3 .1 E 2323 C 2 E 3131 C 3 E 1212 /: 2 Hence, by Lemma 3.10, the Nijenhuis tensor is totally skew-symmetric if and only if d D ! ^ ! holds true for a real function . More precisely, the two functions and are related by the formula !^! D

D 23 i " a :

(3.9)

An SU" .p; q/-structure .!; / is called half-flat if d

D 0;

d! 2 D 0;

and nearly half-flat if d

D!^!

for a real constant . These notions are defined for the Riemannian signature in [8] respectively [13] and extended to all signatures in [9]. Corollary 3.12. Let .!; C / be a half-flat SU" .p; q/-structure on a six-manifold M . Then, the Nijenhuis tensor is totally skew-symmetric if and only if .!; / is nearly half-flat. Proof. If .!; / is nearly half-flat, the equation (3.8) is satisfied by definition and the Nijenhuis tensor is skew-symmetric by the previous proposition. In particular one has d! 2 D 0: Conversely, if the Nijenhuis tensor is skew, we know that (3.8) holds true for a real function , since we have d C D 0. Differentiating this equation and using d! 2 D 0, we obtain d ^ ! 2 D 0. The assertion follows as wedging by ! 2 is injective on one-forms. Remark 3.13. An interesting property of SU.p; q/" -structures which are both half-flat and nearly half-flat in the sense of the corollary is the fact that, given that the manifold and the SU.p; q/" -structure are analytic, the structure can be evolved to both a parallel

440


G2 -structure and a nearly parallel G2 -structure via the Hitchin flow. For details, we refer to [20] and [33] for the compact Riemannian case and [9] for the non-compact case and indefinite signatures. In [7], six-dimensional nilmanifolds N admitting an invariant half-flat SU.3/structure .!; C / such that .!; / is nearly half-flat are classified. As six nilmanifolds admit such a structure, we conclude that these structures are not as scarce as nearly Kähler manifolds. It is also shown in this reference that these structures induce invariant G2 -structures with torsion on N S 1 . We give another example of a (normalised) left-invariant SU.3/-structure on S 3 3 S which satisfies d C D 0; d D ! ^ ! such that d! neither vanishes nor is of type (3,0) + (0,3). We choose a global frame of left-invariant vector fields fe1 ; e2 ; e3 ; f1 ; f2 ; f3 g on S 3 S 3 such that de 1 D e 23 ;

de 2 D e 31 ;

df 1 D f 23 ; p and set with x D 2 C 3,

de 3 D e 12 ;

df 2 D f 31 ; df 3 D f 12 ;

! D e1f 1 C e2f 2 C e3f 3; C

1 D x 2 e 123 C 2xe 12 f 3 2xe 13 f 2 2xe 1 f 23 C 2xe 23 f 1 2 C 2xe 2 f 13 2xe 3 f 12 C .4x 8/f 123 ;

D

1 123 2e 1 f 23 C 2e 2 f 13 2e 3 f 12 C 4f 123 ; xe 2

g D x.e 1 /2 C x .e 2 /2 C x .e 3 /2 C 4 .f 1 /2 C 4 .f 2 /2 C 4 .f 3 /2 2x e 1 f 1 2x e 2 f 2 2xe 3 f 3 : Finally, we come to the characterisation of six-dimensional nearly "-Kähler manifolds by an exterior differential system generalising the classical result of [27] which holds for " D 1 and Riemannian metrics. Theorem 3.14. Let .M; g; J; !/ be an almost "-Hermitian six-manifold. Then M is a strict nearly "-Kähler manifold with krJ k2 ¤ 0 if and only if there is a reduction ‰ D C C i " to SU.p; q/" which satisfies d! D 3 C ; d D 2 ! ^ !;

(3.10) (3.11)

where D 14 krJ k2 is constant and non-zero. Remark 3.15. Due to our sign convention ! D g.:; J:/, the constant is positive in the Riemannian case and the second equation differs from that of other authors.


441

Furthermore, we will sometimes use the term nearly "-Kähler manifold of non-zero type if krJ k2 ¤ 0. Proof. By Proposition 3.2, the manifold M is nearly "-Kähler if and only if d! is of type .3; 0/ C .0; 3/ and the Nijenhuis tensor is totally skew-symmetric. Therefore, when .g; J; !/ is a strict nearly "-Kähler structure such that kAk2 D krJ k2 is constant (by Corollary 3.7) and not zero (by assumption), we can define the reduction ‰ D C Ci " by C D 13 d! D A and D J C such that the first equation is satisfied. Since ! is of type .1; 1/ and therefore d.! ^!/ D 2d! ^! D 0, this reduction is half-flat. Thus, Corollary 3.12 and the skew-symmetry of N imply that there is a constant 2 R such that d D ! ^ !. According to (2.12), we can choose an "-unitary local frame with 1 D 2 such that ‰ D A i " J A D aE 123 ; where a is constant and satisfies 4 D krJ k2 D k C k2 D 4a2 3 by (2.16). Now, the functions defined in Lemma 3.10 and Proposition 3.11 evaluate as (3.7)

D

1 (3.3) "g.N.Ex1 ; Ex2 /; Ex3 / D 2J A.Ex1 ; Ex2 ; Ex3 / D " i " a 2

and (3.9)

D 23 i " a D 23 a2 D 2 : Conversely, if a given SU.p; q/" -structure satisfies the exterior system, the real three-form C is obviously closed and the Nijenhuis tensor is totally skew-symmetric by Corollary 3.12. Considering that d! D 3r! is of type .3; 0/ C .0; 3/ by the first equation, the structure is nearly "-Kähler. Since A D C is stable, the structure is strict nearly "-Kähler by (2.8) and krJ k D kAk ¤ 0 by Proposition 3.9. Now, the computation of the constants in the adapted "-unitary frame shows that in fact krJ k D 4 .

3.3 Consequences for automorphism groups An automorphism of an SU" .p; q/-structure on a six-manifold M is an automorphism of principal fibre bundles or equivalently, a diffeomorphism of M preserving all tensors defining the SU" .p; q/-structure. By our discussion on stable forms in Section 2.3, an SU" .p; q/-structure is characterised by a pair of compatible stable forms .!; / 2 2 M 3 M . Since the construction of the remaining tensors J; and g is invariant, an diffeomorphism preserving the two stable forms is already an automorphism of the SU" .p; q/-structure and in particular an isometry. This easy observation has the following consequences when combined with the exterior systems of the previous section and the naturality of the exterior derivative.

442


Proposition 3.16. Let .!;

C

/ be an SU" .p; q/-structure on a six-manifold M .

(i) If the exterior differential equation d! D

C

is satisfied for a constant ¤ 0, then a diffeomorphism ˆ of M preserving ! is an automorphism of the SU" .p; q/-structure and in particular an isometry. (ii) If the exterior differential equation

d

D !^!

is satisfied for a constant ¤ 0, then a diffeomorphism ˆ of M preserving (a) the real volume form and

C

(b) or the real volume form and

;

;

(c) or the "-complex volume form ‰ D

C

C i"

;

is an automorphism of the SU" .p; q/-structure and in particular an isometry. We like to emphasise that both parts of the proposition apply to strict nearly "Kähler structures of non-zero type. Conversely, it is known for complete Riemannian nearly Kähler manifolds that orientation-preserving isometries are automorphism of the almost Hermitian structure except for the round sphere S 6 , see for instance Proposition 4.1 in [6] in this handbook. However, this is not true if the metric is incomplete. In [13], Theorem 3.6, a nearly Kähler structure is constructed on the incomplete sine-cone over a Sasaki–Einstein five-manifold .N 5 ; ; !1 ; !2 ; !3 /. In fact, the Reeb vector field dual to the one-form is a Killing vector field which does not preserve !2 and !3 . Thus, by the formulae given in [13], its lift to the nearly Kähler six-manifold is a Killing field for the sine-cone metric which does neither preserve ‰ nor ! nor J .

4 Left-invariant nearly "-Kähler structures on SL.2 ; R/ SL.2 ; R/ 4.1 An algebraic prerequisite The following lemma is the key to proving the forthcoming structure result, since it considerably reduces the number of algebraic equations on the nearly "-Kähler candidates. Lemma 4.1. Denote by .R1;2 ; h; i/ the vector space R3 endowed with its standard Minkowskian scalar-product and denote by SO0 .1; 2/ the connected component of the


443

identity of its group of isometries. Consider the action of SO0 .1; 2/ SO0 .1; 2/ on the space of real 3 3 matrices Mat.3; R/ given by ˆ W SO0 .1; 2/ Mat.3; R/ SO0 .1; 2/ ! Mat.3; R/; .A; C; B/ 7! At CB: Then any invertible element C 0 ˛ @0 0

2 Mat.3; R/ lies in the orbit of an element of the form 1 0 1 x y 0 ˇ z ˇ z A or @˛ x y A 0 0 0

with ˛; ˇ; ; x; y; z 2 R and ˛ˇ ¤ 0: Proof. Let an arbitrary invertible element C 2 Mat.3; R/ be given. Denote by fe1 ; e2 ; e3 g the standard basis of R1;2 : There are three different cases: 1.) Suppose that the first column c of C has negative p length. We extend c to a Lorentzian basis fl1 D c=˛; l2 ; l3 g with ˛ WD jhc; cij. The linear map L defined by extension of L.li / D ei is by definition a Lorentz transformation. The transformation L can be chosen time-oriented (by replacing l1 by ˙ l1 ) and oriented (by replacing l3 by ˙ l3 ). With this definition we obtain ˛ t with an element C 0 2 Mat.2; R/: ˆ.L ; C; 1/ D 0 C0 Using the polar decomposition we can express C 0 D O1 S as a product of O1 2 SO.2/ and a symmetric matrix S in Mat.2; R/ and diagonalise S by O2 2 SO.2/: If we put 1 0 1 0 and L D L1 D 2 0 O21 O11 0 O2 we obtain

0

1 ˛ x y ˆ.Lt1 ; ˆ.Lt ; C; 1/; L2 / D @ 0 ˇ 0 A : 0 0

2.) Next suppose that the first column c of C has positiveplength. Again, we extend c to a Lorentzian basis fl1 ; l2 D c=˛; l3 g with ˛ WD jhc; cij. The linear map L defined by extension of L.li / D ei is by definition a Lorentz transformation. The transformation L can be chosen time-oriented (by replacing l1 by ˙ l1 ) and oriented (by replacing l3 by ˙ l3 ). We get 1 0 0 ˆ.Lt ; C; 1/ D @˛ C 0 A with an element C 0 2 Mat.2; R/: 0

444


The first column of this matrix is stable under the right-operation of 1 0 with O1 2 SO.2/ L1 D 0 O1 and there exists an element O1 2 SO.2/ such that it holds 0 1 0 ˇ z ˆ.1; ˆ.Lt ; C; 1/; L1 / D @˛ x y A : 0 0 3.) Finally suppose that it holds hc; ci D 0: Then there exists an oriented and timeoriented Lorentz transformation L such that L.c/ D .e1 C e2 / with ¤ 0: Afterwards one finds as in point 2.) an element O 2 SO.2/; such that it holds 1 0 c1 C 0 WD ˆ.Lt ; C; O/ D @ c2 A : 0 0 Let

0

1 cosh.q/ sinh.q/ 0 B.q/ WD @ sinh.q/ cosh.q/ 0A : 0 0 1

Claim. There exist q1 ; q2 2 R such that

1 ˛ x y ˆ B.q1 /t ; C 0 ; B.q2 / D @ 0 ˇ z A : 0 0

0

To prove this claim let us first consider the right-action of B.q/ on C 00 WD t 0 ˆ B.q1 / ; C ; 1 1 0 00 00 c11 cosh.q/ C c12 sinh.q/ 00 00 cosh.q/ C c22 sinh.q/ A for q 2 R: ˆ 1; C 00 ; B.q/ D @c21 0 0 00 00 We choose q2 such that c21 cosh.q2 / C c22 sinh.q2 / vanishes. This is only pos00 00 00 00 =c21 j > 1: sible if c22 =c21 is in the range of coth; i.e. jc22 In the sequel we show that this can always be achieved by the left-action of an 00 ¤ 0. In fact, it is element B.q1 / on C 0 and that c21 00 c22 D c1 sinh.q1 / C c2 cosh.q1 /; 00 c21 D .sinh.q1 / C cosh.q1 // D e q1 ; 00 c22 c2 c1 2q1 c1 C c2 C e : 00 D 2 2 c21


445

We observe that c1 ¤ c2 ; since the matrix C is invertible. Therefore we can 00 00 =c21 j > 1: This proves the claim and finishes the proof of always achieve jc22 the lemma.

4.2 Proof of the uniqueness result Finally, we prove our main result which is the following theorem. By a homothety, we define the rescaling of the metric by a real number which we do not demand to be positive since we are working with all possible signatures. Theorem 4.2. Let G be a Lie group with Lie algebra sl.2; R/. Up to homothety, there is a unique left-invariant nearly "-Kähler structure with krJ k2 ¤ 0 on G G. This is the nearly pseudo-Kähler structure of signature .4; 2/ constructed as 3-symmetric space in the introduction. In particular, there is no left-invariant nearly para-Kähler structure. Remark 4.3. The proof also shows that there is a left-invariant nearly "-Kähler structure of non-zero type on G H with Lie.G/ D Lie.H / D sl.2; R/ if G ¤ H which is unique up to homothety and exchanging the orientation. Proof. More precisely, we will prove uniqueness up to equivalence of left-invariant almost "-Hermitian structures and homothety. We will consider the algebraic exterior system d! D 3 C ; d D 2! ^ !

(4.1) (4.2)

on the Lie algebra sl.2; R/ ˚ sl.2; R/. By Theorem 3.14, solutions of this system are in one-to-one correspondence to left-invariant nearly "-Kähler structure on G G with krJ k2 D 4. This normalisation can always be achieved by applying a homothety. Furthermore, two solutions which are isomorphic under an inner Lie algebra automorphism from Inn.sl.2; R/ ˚ sl.2; R// D SO0 .1; 2/ SO0 .1; 2/ are equivalent under the corresponding Lie group isomorphism. Since both factors are equal, we can also lift the outer Lie algebra automorphism exchanging the two summands to the group level. In summary, it suffices to show the existence of a solution of the algebraic exterior system (4.1), (4.2) on the Lie algebra which is unique up to inner Lie algebra automorphisms and exchanging the summands. A further significant simplification is the observation that all tensors defining a nearly "-Kähler structure of non-zero type can be constructed out of the fundamental two-form ! with the help of the first nearly Kähler equation (4.1) and the stable form formalism described in Section 2.3. We break the main part of the proof into three

446


lemmas, step by step simplifying ! under Lie algebra automorphisms in a fixed Lie bracket. We call fe1 ; e2 ; e3 g a standard basis of so.1; 2/ if the Lie bracket satisfies de 1 D e 23 ;

de 2 D e 31 ;

de 3 D e 12 :

In this basis, an inner automorphism in SO0 .1; 2/ acts by usual matrix multiplication on so.1; 2/. Lemma 4.4. Let g D h D so.1; 2/ and let ! be a non-degenerate two-form in ƒ2 .g ˚ h/ D ƒ2 g ˚ .g ˝ h/ ˚ ƒ2 h : Then we have d! 2 D 0 () ! 2 g ˝ h: Proof. By inspecting the standard basis, we observe that all two-forms on so.1; 2/ are closed whereas no non-trivial 1-form is closed. Thus, when separately taking the exterior derivative of the components of ! 2 in ƒ4 D .ƒ3 g ˝h /˚.ƒ2 g ˝ƒ2 h /˚ .g ˝ ƒ3 h ), the equivalence is easily deduced. Lemma 4.5. Let g D h D so.1; 2/ and let fe 1 ; e 2 ; e 3 g be a basis of g and fe 4 ; e 5 ; e 6 g a basis of h such that the Lie brackets are given by de 1 D e 23 ; 4

de 2 D e 31 ; 56

de D e ;

5

de 3 D e 12 ; 64

de D e ;

6

de D e

and 45

(4.3)

for some 2 f˙1g. Then, every non-degenerate two-form ! on g ˚ h satisfying d! 2 D 0 can be written !

D ˛ e 14 C ˇ e 25 C e 36 C x e 15 C y e 16 C z e 26

(4.4)

for ˛; ˇ; 2 R f0g and x; y; z 2 R modulo an automorphism in SO0 .1; 2/ SO0 .1; 2/. Proof. We choose standard bases fe 1 ; e 2 ; e 3 g for g and fe 4 ; e 5 ; e 6P g for h. Using the previous lemma and the assumption d! 2 D 0, we may write ! D 3i;j D1 cij e i.j C3/ for an invertible matrix C D .cij / 2 Mat.3; R/. When a pair .A; B/ 2 SO0 .1; 2/ SO0 .1; 2/ acts on the two-form !, the matrix C is transformed to At CB. Applying Lemma 4.1, we can achieve by an inner automorphism that C is in one of the normal forms given in that lemma. However, an exchange of the base vectors e1 and e2 corresponds exactly to exchanging the first and the second row of C . Therefore, we can always write ! in the claimed normal form by adding the sign in the Lie bracket of the first summand g.

447


Lemma 4.6. Let fe 1 ; : : : ; e 6 g be a basis of so.1; 2/ so.1; 2/ such that de 1 D e 23 ;

de 2 D e 31 ;

de 4 D e 56 ;

de 3 D e 12 and

de 5 D e 64 ;

(4.5)

de 6 D e 45 :

Then the only SU" .p; q/-structure .!; C / modulo inner automorphisms and modulo exchanging the summands, which solves the two nearly "-Kähler equations (4.1) and (4.2), is determined by p 3 14 .e C e 25 C e 36 /: (4.6) !D 18 Proof. Since d! 2 D 0 by the second equation (4.2), we can choose a basis satisfying (4.3) such that ! is in the normal form (4.4). In order to satisfy the first equation (4.1), we have to set 3

C

D d! D ˛ e 234 C ˛ e 156 x e 235 C x e 146 y e 236 y e 145 ˇ e 135 C ˇ e 246 z e 136 z e 245 C e 126 e 345 :

The compatibility ! ^ C D 0 is equivalent to d.! 2 / D 0: It remains to determine all solutions of the second nearly "-Kähler equation (4.2) modulo automorphisms. For the sake of readability, we identify ƒ6 .g ˚ h/ with R by means of e 123456 . Supported by Maple, we compute K

C

.e1 / D .x 2 C y 2 C z 2 ˛ 2 C ˇ 2 C 2 /e1 .2xˇ C 2yz/e2 2 ye3 C 2 ˇe4 ;

K

C

.e2 / D .2xˇ C 2yz/e1 C .x 2 y 2 z 2 C ˛ 2 ˇ 2 C 2 /e2 2 ze3 C 2 xe4 2 ˛e5 ;

K

C

.e3 / D 2ye1 2ze2 C .x 2 y 2 C z 2 C ˛ 2 C ˇ 2 2 /e3 C .2yˇ 2xz/e4 C 2˛ze5 2˛ˇe6 ;

K

C

.e4 / D 2ˇe1 C 2xe2 C .2yˇ 2xz/e3 C .x 2 C y 2 z 2 C ˛ 2 ˇ 2 2 /e4 2˛xe5 2˛ye6 ;

K

C

.e5 / D 2˛e2 2˛ze3 C 2˛xe4 C .x 2 C y 2 z 2 ˛ 2 C ˇ 2 2 /e5 C .2ˇz 2xy/e6 ;

K

C

.e6 / D 2˛ˇe3 C 2˛ye4 C .2ˇz 2xy/e5 C .x 2 y 2 C z 2 ˛ 2 ˇ 2 C 2 /e6 :

We assume that . C / ¤ 0 and check this a posteriori for the solutions we find. Hence, we can set k WD p 1 C and J C D kK C . Since C C i " J C C is a ˙

j.

.3; 0/-form with respect to J

C

/j

, we have

D J C

C

D"

C

.J

C

:; :; :/ which

448


turns out to be "

27 k

D 2 ˛ˇ e 123 C 2 y˛ e 124 C 2 .xy ˇz/ e 125 2.xˇ C yz/˛ e 134 C .x 2 C y 2 z 2 C ˛ 2 C ˇ 2 2 / e 126 fˇ.x 2 y 2 z 2 C ˛ 2 ˇ 2 C 2 / C 2xyzg e 135 C fz.x 2 y 2 C z 2 ˛ 2 C ˇ 2 C 2 / 2xyˇg e 136 fy.x 2 y 2 C z 2 C ˛ 2 ˇ 2 C 2 / C 2xzˇg e 145 fx.x 2 C y 2 C z 2 ˛ 2 ˇ 2 C 2 / 2yzˇg e 146 ˛.x 2 C y 2 C z 2 ˛ 2 C ˇ 2 C 2 / e 156 ˛.x 2 C y 2 C z 2 ˛ 2 C ˇ 2 C 2 / e 234 fx.x 2 C y 2 C z 2 ˛ 2 ˇ 2 C 2 / 2yzˇg e 235 C fy.x 2 y 2 C z 2 C ˛ 2 ˇ 2 C 2 / C 2xzˇg e 236 fz.x 2 y 2 C z 2 ˛ 2 C ˇ 2 C 2 / 2xyˇg e 245 fˇ.x 2 y 2 z 2 C ˛ 2 ˇ 2 C 2 / C 2xyzg e 246 C .x 2 C y 2 z 2 C ˛ 2 C ˇ 2 2 / e 345 2.xˇ C yz/˛ e 256 2.xy ˇz/ e 346 2y˛ e 356 C 2˛ˇ e 456 :

Furthermore, we compute the exterior derivative "

27 d k

D 4 ˛y e 1256 4 .xy ˇz/ e 1246 C 4˛.xˇ C yz/ e 1356 C 2 .x 2 C y 2 z 2 C ˛ 2 C ˇ 2 2 / e 1245 C 2fˇ.x 2 y 2 z 2 C ˛ 2 ˇ 2 C 2 / C 2xyzg e 1346 C 2fz.x 2 C y 2 C z 2 ˛ 2 C ˇ 2 C 2 / 2xyˇg e 1345 C 2fy.x 2 y 2 C z 2 C ˛ 2 ˇ 2 C 2 / C 2xzˇg e 2345 C 2fx.x 2 C y 2 C z 2 ˛ 2 ˇ 2 C 2 / 2yzˇg e 2346 C 2˛.x 2 C y 2 C z 2 ˛ 2 C ˇ 2 C 2 / e 2356

and ! 2 D 2..yˇ xz/ e 1256 ˛z e 1246 x e 1356 ˛ˇ e 1245 ˛ e 1346 ˇ e 2356 /: The second nearly Kähler equation (4.2) is therefore equivalent to the following nine coefficient equations: .˛ˇ 27"k 1 / x D ˛ yz;

(e 1356 )

. ˛ 27"k 1 ˇ/ y D 27"k 1 xz;

(e 1256 )

. ˇ 27"k 1 ˛/ z D xy;

(e 1246 )


ˇ ; ˛ z.x 2 C y 2 C z 2 ˛ 2 C ˇ 2 C 2 / D 2ˇyx; ˛ xyz 2 ; x 2 y 2 z 2 C ˛ 2 ˇ 2 C 2 D 54"k 1 ˇ ˇ y.x 2 y 2 C z 2 C ˛ 2 ˇ 2 C 2 / D 2ˇzx; ˛ˇ x 2 C y 2 z 2 C ˛ 2 C ˇ 2 2 D 54 "k 1 ; x.x 2 y 2 z 2 C ˛ 2 C ˇ 2 2 / D 2ˇyz: x 2 C y 2 C z 2 ˛ 2 C ˇ 2 C 2 D 54"k 1

449

(e 2356 ) (e 1345 ) (e 1346 ) (e 2345 ) (e 1245 ) (e 2346 )

Recall that ˛; ˇ; ¤ 0 because ! is non-degenerate. We claim that there is no solution if any of x, y or z is different from zero. On the one hand, assume that one of them is zero. Using one of the first three equations respectively, we find that at least one of the other two has to be zero as well. However, in all three cases, we may easily deduce that the third one has to be zero as well by comparing equations 4 and 5 respectively 6 and 7 respectively 8 and 9. On the other hand, if we assume that all three of them are different from zero, the bracket in the first equation is necessarily different from zero and we may express x by a multiple of yz. Substituting this expression into equations 2 and 3, yields expressions for y 2 and z 2 in terms of ˛,ˇ, and k. But if we insert all this into equation 4 (or 6 or 8 alternatively), we end up with a contradiction after a slightly tedious calculation. To conclude, we can set x D y D z D 0 without losing any solutions of the second nearly Kähler equation which simplifies to the equations ˛ 3 ˛ˇ 2 ˛ 2 54"k 1 ˇ D 0; ˇ 3 ˇ 2 ˇ˛ 2 54"k 1 ˛ D 0; 3 ˛ 2 ˇ 2 54"k 1 ˛ˇ D 0: Setting c1 D ˛ 2 C ˇ 2 C 2 and c2 D 54"k 1 ˛ˇ , these are equivalent to 2˛ 4 c1 ˛ 2 c2 D 0; 2ˇ 4 c1 ˇ 2 c2 D 0; 4

(4.7)

2

2 c1 c2 D 0: To finish the proof, we have to show that all real solutions of the system (4.7) are isomorphic under SO0 .1; 2/ SO0 .1; 2/ to p 3 ˛DˇD D ; D 1: 18 Since ˛ 2 , ˇ 2 and 2 satisfy the same quadratic equation, at least two of them have to be identical, say ˛ 2 D ˇ 2 . However, if 2 was the other root of the quadratic equation, we would have ˛ 2 C 2 D 12 c1 and by definition of c1 at the same time

450


2˛ 2 C 2 D c1 . This would only be possible if was zero, a contradiction to the non-degeneracy of !. Therefore has to be C1 and ˛,ˇ and have to be identical up to sign. By applying one of the proper and orthochronous Lorentz transformations 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 @0 1 0 A ; @0 0 1A ; @0 0 1A 0 0 1 0 1 0 0 1 0 on, say, the second summand, it is always possible to achieve that the signs of ˛, ˇ and are identical. So far, we found a basis satisfying (4.5) such that ! D ˛.e 14 C e 25 C e 36 /: It is straightforward to check that the quartic invariant in this basis is 1 1 (4.8)

. d!/ D ˛ 4 : 3 27 Therefore, there cannot existpa nearly para-Kähler structure and we can set " D 1. Inserting k D ˙ p1 D ˙3 3˛ 2 into equations (4.7) yields

54 1p 2˛ 4 3˛ 4 ˙ p ˛ 5 D 0 () ˛ D ˙ 3: 18 3 3 Finally, we can achieve that ˛ is positive by applying the Lie algebra automorphism exchanging the two summands, i.e. ei 7! eiC3 mod 6 and the lemma is proven. In fact, the uniqueness, existence and non-existence statements claimed in the theorem follow directly from this lemma and formula (4.8). As explained in the introduction, we know that there is a left-invariant nearly pseudo-Kähler structure of indefinite signature on all the groups in question. After applying a homothety, we can achieve krJ k2 D 4 and this structure has to coincide with the unique structure we just constructed. Therefore, the indefinite metric has to be of signature (4,2) by our sign conventions. We summarise the data of the unique nearly pseudo-Kähler structure in the basis (4.5) and can easily double-check the signature of the metric explicitly: 1 p 14 !D 3.e C e 25 C e 36 /; 18 1 p 126 C D 3.e e 135 C e 156 e 234 C e 246 e 345 /; 54 1 D .2 e 123 C e 126 e 135 e 156 e 234 e 246 C e 345 C 2e 456 /; 54 1p 2p 2p 1p J.e1 / D 3e1 3e4 ; J.e4 / D 3e1 C 3e4 ; 3 3 3 3 1p 2p 2p 1p J.e2 / D 3e2 C 3e5 ; J.e5 / D 3e2 C 3e5 ; 3 3 3 3


J.e3 / D gD

1p 2p 3e3 C 3e6 ; 3 3

J.e6 / D

451

2p 1p 3e3 C 3e6 ; 3 3

1 1 2 ..e / .e 2 /2 .e 3 /2 C .e 4 /2 .e 5 /2 .e 6 /2 e 1 e 4 e 2 e 5 e 3 e 6 /: 9

Observing that in [5] very similar arguments have been applied to the Lie group S 3 S 3 , we find the following non-existence result. Proposition 4.7. On the Lie groups G H with Lie.G/ D Lie.H / D so.3/, there is neither a left-invariant nearly para-Kähler structure of non-zero type nor a leftinvariant nearly pseudo-Kähler structure with an indefinite metric. Proof. The unicity of the left-invariant nearly Kähler structure S 3 S 3 is proved in [5], Section 3, with a strategy analogous to the proof of Theorem 4.2. In the following, we will refer to the English version [6]. There, it is shown in the proof of Proposition 2.5 that for any solution of the exterior system d! D 3 d

C

C

D 2 ! 2

there is a basis of the Lie algebra of S 3 S 3 and a real constant ˛ such that de 1 D e 23 ;

de 2 D e 31 ;

de 3 D e 12

de 4 D e 56 ;

de 5 D e 64 ;

de 6 D e 45 ;

and ! D ˛.e 14 C e 25 C e 36 /: In this basis, a direct computation or formula (18) in [6] show that the quartic invariant that we denote by is

D

1 4 ˛ 27

with respect to the volume form e 123456 . Therefore, a nearly para-Kähler structure cannot exist on all the Lie groups with the same Lie algebra as S 3 S 3 by Theorem 3.14. A nearly pseudo-Kähler structure with an indefinite metric cannot exist either, since the induced metric is always definite as computed in the second part of Lemma 2.3 in [6].

References [1]

I. Agricola, Non-integrable geometries, torsion, and holonomy. In Handbook of pseudoRiemannian geometry and supersymmetry, ed. by V. Cortés, IRMA Lect. Math. Theor. Phys. 16, European Math. Soc. Publishing House, Zürich 2010, 277–346. 425

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[2]

D. V. Alekseevsky, C. Medori, and A. Tomassini, Homogeneous para-Kähler Einstein manifolds. Uspekhi Mat. Nauk 64 (2009), no. 1, 3–50; English transl. Russ. Math. Surveys 64 (2009), no. 1, 1–43. 428

[3]

F. Belgun and A. Moroianu, Nearly Kähler 6-manifolds with reduced holonomy. Ann. Global Anal. Geom. 19 (2001), 307–319. 436

[4]

R. L. Bryant, On the geometry of almost complex 6-manifolds. Asian J. Math. 10 (2006), 561–605. 427, 437

[5]

J.-B. Butruille, Classification des variétés approximativement kähleriennes homogènes. Ann. Global Anal. Geom. 27 (2005), 201–225. 426, 427, 451

[6]

J.-B. Butruille, Homogeneous nearly Kähler manifolds. In Handbook of pseudoRiemannian geometry and supersymmetry, ed. by V. Cortés, IRMA Lect. Math. Theor. Phys. 16, European Math. Soc. Publishing House, Zürich 2010, 399–423. 426, 427, 433, 442, 451

[7]

S. G. Chiossi and A. Swann, G2 -structures with torsion from half-integrable nilmanifolds. J. Geom. Phys. 54 (2005), 262–285. 440

[8]

S. Chiossi and S. Salamon, The intrinsic torsion of SU.3/ and G2 structures. In Differential geometry, Valencia, 2001, World Scientific Publishing, River Edge, NJ 2002, 115–133. 433, 439

[9]

V. Cortés, Th. Leistner, L. Schäfer, and F. Schulte-Hengesbach, Half-flat structures and special holonomy. Preprint 2009, arXiv:0907.1222v1 [math.DG]. 431, 439, 440

[10] V. Cortés and L. Schäfer, Flat nearly Kähler manifolds. Ann. Global Anal. Geom. 32 (2007), 379–389. 426, 427 [11] V. Cortés and L. Schäfer, Geometric structures on Lie groups with flat bi-invariant metric. J. Lie Theory 19 (2009), no. 2, 423–437. 426, 437 [12] V. Cruceanu, P. Fortuny, P. M. Gadea, A survey on paracomplex geometry. Rocky Mountain J. Math. 26 (1996), no. 1, 83–115 428 [13] M. Fernández, S. Ivanov, V. Muñoz, and L. Ugarte, Nearly hypo structures and compact nearly Kähler 6-manifolds with conical singularities. J. Lond. Math. Soc. (2) 78 (2008), 580–604. 439, 442 [14] T. Friedrich and S. Ivanov, Parallel spinors and connections with skew-symmetric torsion in string theory. Asian J. Math. 6 (2002), 303–335. 435 [15] P. M. Gadea and J. M. Masque, Classification of almost para-Hermitian manifolds. Rend. Mat. Appl. (7) 11 (1991), 377–396. 430, 434, 437 [16] A. Gray, Riemannian manifolds with geodesic symmetries of order 3. J. Differential Geom. 7 (1972), 343–369. 425, 426 [17] A. Gray, The structure of nearly Kähler manifolds. Math. Ann. 223 (1976), 233–248. 436 [18] A. Gray and L. M. Hervella, The sixteen classes of almost Hermitian manifolds and their linear invariants. Ann. Mat. Pura Appl. (4) 123 (1980), 35–58. 430 [19] N. Hitchin, The geometry of three-forms in six dimensions. J. Differential Geom. 55 (2000), 547–576. 430, 431 [20] N. Hitchin, Stable forms and special metrics. In Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000), Contemp. Math. 288, Amer. Math. Soc., Providence, RI, 2001, 70–89. 440


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[21] S. Ivanov and S. Zamkovoy, Parahermitian and paraquaternionic manifolds. Differential Geom. Appl. 23 (2005), 205–234. 425, 435, 436, 437 [22] S. Kaneyuki and M. Kozai, Paracomplex structures and affine symmetric spaces. Tokyo J. Math. 8 (1985), 81–98. 430 [23] I. Kath, Killing spinors on pseudo-Riemannian manifolds. Habilitationsschrift, HumboldtUniversität, Berlin 1999. 425, 436 [24] V. F. Kirichenko, Generalized Gray-Hervella classes and holomorphically projective transformations of generalized almost Hermitian structures. Izv. Ross. Akad. Nauk Ser. Mat. 69 (2005), 107–132. 427, 430, 434 [25] A. J. Ledger and M. Obata, Affine and Riemannian s-manifolds. J. Differential Geometry 2 (1968), 451–459. 426 [26] P.-A. Nagy, Connections with totally skew-symmetric torsion and nearly-Kähler geometry. In Handbook of pseudo-Riemannian geometry and supersymmetry, ed. by V. Cortés, IRMA Lect. Math. Theor. Phys. 16, European Math. Soc. Publishing House, Zürich 2010, 347–398. 430, 433, 435 [27] R. Reyes Carrión, Some special geometries defined by Lie groups. Ph.D. thesis, Oxford 1993. 426, 427, 440 [28] F. Raymond and A. T. Vasquez, 3-manifolds whose universal coverings are Lie groups. Topology Appl. 12 (1981), 161–179. 427 [29] L. Schäfer, t t -geometry and pluriharmonic maps. Ann. Global Anal. Geom. 28 (2005), 285–300. 427 [30] L. Schäfer, Para-t t -bundles on the tangent bundle of an almost para-complex manifold. Ann. Global Anal. Geom. 32 (2007), 125–145. 426 [31] L. Schäfer, t t -geometry on the tangent bundle of an almost complex manifold. J. Geom. Phys. 57 (2007), 999–1014. 426, 427 [32] L. Schäfer, On the structure of nearly pseudo-Kähler manifolds. Monatsh. Math., to appear; DOI 10.1007/s00605-009-0184-1. 436 [33] S. Stock, Lifting SU.3/-structures to nearly parallel G2 -structures. J. Geom. Phys. 59 (2009), 1–7. 440

Chapter 13

Quaternionic geometries from superconformal symmetry Andrew Swann

Contents 1 2 3

Introduction . . . . . . . . . . . . . . . . . . . . . HKT geometry . . . . . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . 3.1 Four dimensions . . . . . . . . . . . . . . . 3.2 Higher dimensions . . . . . . . . . . . . . . 4 Potentials . . . . . . . . . . . . . . . . . . . . . . 4.1 Change of potential and metric . . . . . . . . 5 Some physical motivation . . . . . . . . . . . . . 6 A family of simple superalgebras . . . . . . . . . 7 Superconformal symmetry and special homotheties 8 Relation to QKT geometry . . . . . . . . . . . . . 9 Examples . . . . . . . . . . . . . . . . . . . . . . 9.1 Quaternionic Kähler bases . . . . . . . . . . 9.2 Group manifolds . . . . . . . . . . . . . . . 9.3 Dimension four and connected sums . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

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455 456 458 458 459 460 462 462 464 465 467 469 469 470 471 471

1 Introduction Although often presented as abstract algebraic objects, quaternions have a long association with physics. They were introduced by Hamilton [29] who wrote extensively on applications, and the unit quaternions i, j and k appear as standard notation in elementary discussion of three-dimensional geometry and mechanics. Quaternions reappear in at least two guises in modern theoretical physics. One is through the geometry of four-dimensions, where the close links between the orthogonal group, groups of unit quaternions and SU.2/ arise in the conformal geometry and selfduality conditions. This is particularly relevant for theYang–Mills and Seiberg–Witten theories, and the quaternions often occur again in descriptions of the geometries of related moduli spaces. Another situation where quaternions arise is in the discussion of supersymmetry. Bagger and Witten [3] demonstrated how hyper-Kähler and quaternionic Kähler

456

Andrew Swann

geometry are necessary in certain models. More recently, extensive discussion of one-dimensional quantum mechanics with supersymmetry has shown the relevance of geometries with torsion. This article will discuss certain aspects of these geometries mostly from a mathematical viewpoint. It will concentrate on the so-called hyper-Kähler manifolds with torsion, taking in discussion of both weak and strong geometries. Following the work of Michelson and Strominger [47], I will describe the specialisation to the geometries arising from models with superconformal symmetry. Although quantum mechanically the main interest is in positive definite metrics one finds that there are direct relations with corresponding geometries in mixed signature and so we will discuss indefinite geometries from the start. It turns out that there are intimate links with another type of quaternion-based geometry, namely quaternionic geometry with torsion, that arises in certain two-dimensional field theories. Much of the discussion is based on my paper [56] with Yat Sun Poon, but with inclusion of newer developments. I will not attempt to give a comprehensive guide to the literature, which particularly in the physics directions, is vast. In particular, while allowing indefinite metrics, we will not discuss geometries based on the split quaternions, cf. work of Stefan Ivanov in this volume, nor non-linear realisations of the supersymmetry algebra as discussed in [34]. I wish to thank the organisers of the workshop in Strasbourg for a stimulating meeting. This work is supported by grants from the Danish and Spanish research councils.

2

HKT

geometry

Let M be a manifold and suppose I , J and K are endomorphisms of the tangent bundle satisfying I 2 D 1 D J 2 D K 2 ;

IJ D K D JI:

(2.1)

We say that .I; J; K/ is an almost hypercomplex structure on M . A metric g, possibly indefinite, is said to be compatible if g.AX; AY / D g.X; Y / for all X; Y 2 TM and each A D I; J; K. In this situation the tensors FA .; / D g.A; /, A D I; J; K, are two-forms and .M; g; I; J; K/ is called an almost hyper-Hermitian manifold. Definition 2.1. A manifold M is (weak) hyper-Kähler with torsion hkt if it is almost hyper-Hermitian and dI FI D dJ FJ D dK FK : Here dA is given on r-forms by dA D .1/r AdA and A acts on an r-tensor ˛ by A˛.X1 ; : : : ; Xr / D .1/r ˛.AX1 ; : : : ; AXr /; On an hkt manifold M , consider the three-form c D IdFI I

A D I; J; K:

Chapter 13. Quaternionic geometries from superconformal symmetry

457

this is independent of I . Since c is totally skew, the connection r D r LC C 12 c is metric rg D 0 with torsion c 2 ƒ3 T M . The hkt condition ensures that r also preserves the almost complex structures: rI D 0 D rJ D rK: An almost complex structure I is said to be integrable if there are local complex coordinates z 1 ; : : : ; z 2n such that I @=@z k D i @=@z k . The obstruction to integrability is expressed by the Nijenhuis tensor NI .X; Y / D ŒIX; I Y I ŒIX; Y I ŒX; I Y ŒX; Y : Proposition 2.2 ([45]). On an almost hyper-Hermitian manifold the Nijenhuis tensors satisfy 2NI D J.23/ I.23/ I.12/ I.13/ 1 .dJ FJ dK FK /; etc. In particular, on an hkt manifold, the almost complex structures I , J and K are integrable. Here I.abc::: / ˇ is I applied in slots a; b; c; : : : of ˇ. A manifold with integrable complex structures I , J and K satisfying (2.1) is called hypercomplex. It is important to note that such manifolds do not in general admit local quaternionic coordinates. Indeed a hypercomplex manifold carries a unique torsionfree connection r Ob , the Obata connection [49], such that r Ob I D 0 D r Ob J D r Ob K: The manifold admits local quaternionic coordinates if and only if the curvature tensor ROb of r Ob is zero. In the hkt case, as I , J and K are integrable we have a hyper-Hermitian structure. The pair .g; I / is a Hermitian structure on M . Gauduchon [23] showed that a Hermitian manifold .M; g; I / admits a unique connection with skew-symmetric torsion satisfying rIB g D 0 and rIB I D 0: Due to its applications in [7], rIB is known as the Bismut connection. The hkt B condition therefore says that the Bismut connections rIB , rJB and rK coincide and so equal r. Definition 2.3. An hkt structure is strong if dc D 0, i.e., the torsion three-form is closed. When c D 0 we have a hyper-Kähler structure. Strong, as opposed to weak, hkt structures are important in some physical applications, but not all. Hyper-Kähler metrics are historically much better understood

458

Andrew Swann

than general hkt structrues: hyper-Kähler metrics are Ricci-flat, so Calabi–Yau, and appear in the classification of Riemannian holonomies.

3 Examples 3.1 Four dimensions In four dimensions any hyper-Hermitian manifold is hkt and the metric is necessarily definite. Indeed a hypercomplex structure in dimension four determines a unique conformal class Œg of compatible metrics: if X is a non-zero tangent vector then fX; IX; JX; KXg is an oriented conformal basis for TM . The Obata connection r Ob preserves the conformal structure, so r Ob g D ! ˝ g for a representative metric g 2 Œg. One now checks that .M; g; I; J; K/ is hkt with torsion-form c D .I! ^ J ! ^ K!/= k!k2 D !. The first example here is M D R4 Š H, the quaternions, with standard basis f1; i; j; kg. As M is a vector space we have Tp M D H canonically, and one takes I , J and K to be given by right multiplication by i, j and k respectively; the signs being necessary to ensure IJ D K. The Obata connection is flat differentiation and the N gives the metric g0 D standard inner product on R4 Š H given by hp; qi D Re.p q/ N that is hyper-Hermitian. Indeed .dx 1 /2 C .dx 2 /2 C .dx 3 /2 C .dx 4 /2 D Re.dp dp/ r Ob g0 D 0, so g0 is hyper-Kähler. If g D fg0 , we have ! D d log f and c D 0 df . Thus g is a strong hkt metric if and only if f is harmonic. The important case here is if f D f .r 2 /, where r 2 D p pN is the radius squared. The resulting non-flat strong metrics are g D Re..a C b=p p/dp N dp/, N with a, b constant. When a D 0, this descends to the standard product metric on S 3 S 1 . Four-dimensional compact hypercomplex manifolds have been classified by Boyer [9]. They are the four-torus T 4 D R4 =Z4 D H=Z4 , K3 surfaces (simply-connected complex surfaces with c1 D 0) and S 3 S 1 . The first two admit hyper-Kähler metrics, and we have seen that the last manifold has a strong hkt structure locally conformal to the flat hyper-Kähler metric. We thus have in dimension four that a compact hkt manifold is (i) conformal to a strong hkt structure, (ii) locally conformal to a hyperKähler metric, (iii) if not conformally flat, the space is a K3 surface. Passing to the non-compact realm, there are many examples of four-dimensional hyper-Kähler structures known. Two important series are the multi-instanton metrics of Gibbons and Hawking [24], the first of which is the Eguchi–Hanson metric [19] on the cotangent bundle T CP.1/ (see equation (4.2) below with a D 0), and the multi-Taub-nut metrics [30]. Both series live on the same topological manifolds, which have the homotopy type of a chain of two-spheres, and each metric admits a tri-holomorphic circle symmetry. As such one may apply a version of t-duality to construct hkt structures in a new conformal class, as we will now explain.


459

For a little more generality consider a four-dimensional hkt manifold .M; g/ admitting a circle symmetry X D @=@ preserving both the metric and the complex structures, and such that the one-form X [ D d C ! metric dual to X satisfies AdX [ D dX [ for A D I; J and K. (If g is hyper-Kähler, the last condition is automatically satisfied.) Let V D 1= kXk ²;

' D V X [:

Then LX ' D 0, '.X/ D 1 and gD

1 1 2 ' C V h D .d C !/2 C V hij dy i dy j ; V V

where h is a metric on the three-dimensional manifold Y D M=hXi and ! is the connection one-form on the circle bundle W M ! Y . The function V descends to Y and the hkt condition forces V and h to satisfy a generalised monopole equation. (In the hyper-Kähler case d V D h d!.) Write c D ' ^ 2 C 3 ; where X ³ i D 0 and i 2 i .Y /. Then one can show that the four-dimensional manifold W D Y R, or Y S 1 , carries an hkt structure with metric and torsion gW D V .dt 2 C h/;

cW D 3 C dt ^ .V 2 d'/:

As Callan, Harvey and Strominger [12] found in particular cases, starting with a hyperKähler metric on M , c D 0, we always obtain a strong hkt structure on W , cf. [25]. One can check that in general the metrics gW are not conformal to hyper-Kähler [8], [52], unlike the compact case.

3.2 Higher dimensions In dimensions greater than four, an hkt metric will in general have signature .4p; 4q/ and these structures will be the focus of much of the rest of this article. First examples may be constructed from products of four-dimensional examples and from totally geodesic submanifolds (this concept is the same for r LC and r) as successfully exploited in [25]. Beyond that there is a wide range of constructions of hypercomplex manifolds, and it may be shown that many of these admit compatible positive-definite hkt metrics, see [18], [20], [26], [28], [32], [61]. One particularly interesting class here is given as follows. Let G be any compact Lie group. Then Joyce [38] showed that there is a torus T r , 0 6 r 6 rank G, such that G T r admits a left-invariant hypercomplex structure. One may adapt Joyce’s construction so that it is compatible with a biinvariant metric g and in this way obtain strong hkt structures on G T r with c.X; Y; Z/ D B.ŒX; Y ; Z/, where B is the Killing form. Note that some extra factor such as T r is necessary in general in order that the dimension of the manifold be

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divisible by 4. However, in some cases, for example when G D SU.3/, we can take r D 0 and obtain a strong hkt structure on a compact simply-connected manifold. Locally any hypercomplex structure admits a compatible positive definite hkt metric, as we will see below. However, Grantcharov and Fino [21] showed that there exist compact 8-dimensional hypercomplex manifolds for which this is impossible globally. On the other hand it is interesting to note that their concrete example admits an hkt structure of signature .4; 4/.

4 Potentials A function on M is an hkt potential if FI D 12 .ddI C dJ dK / D 12 .1 J /dId:

(4.1)

In complex coordinates for I , this says 2 @2 qN p @ J J : i |N @z i @zN j @z p @Nz q Equation (4.1) implies similar equations for FJ and FK and that

2i .FI /i |N D

g D 14 .1 C I C J C K/.r Ob /2 : Given a hypercomplex manifold we can always find a local so that FI D 12 .d dI C dJ dK / is non-degenerate and hence a local hkt metric; we may even ensure that this is positive definite. Example 4.1. Let g be a hyper-Kähler metric and choose a local Kähler potential for FI , i.e., FI D 12 i @I @N I . Then is a local hkt potential for g. Many hyper-Kähler metrics can be presented in this way globally by specifying a single Kähler potential. These include Biquard and Gauduchon’s descriptions [5], [6] of hyper-Kähler metrics on T .G=K/, where G=K is a Hermitian symmetric space, local versions in [48], and families of G-invariant hyper-Kähler metrics on nilpotent orbits O D G C :X gC , where G is compact and adX D ŒX; W gC ! gC is nilpotent [40], [42], [43], [41], [62], [63]. Indeed Calabi’s original construction [11] of a non-flat hyper-Kähler metric is essentially obtained this way. The prototype for most of these constructions is the family of Eguchi–Hanson metrics gD

r2 1 dr 2 C .12 C 22 C W 32 / W 4

(4.2)

with W D .1a2 r 4 /, where i is a basis of left-invariant one-forms on S 3 D SU.2/ with d1 D 2 ^ 3 , etc. This has potential 2 1 2 1 r r a tan : D 2 a


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In general, if is an hkt potential, then the torsion tensor is given by c D 12 dI dJ dK D 12 IdKdId so the structure is strong if and only if d dI dJ dK D 0. For the standard hypercomplex structure on R4 , we have that is the potential for a non-degenerate hkt structure if ¤ 0, and that the structure is strong if 2 D 0, i.e., if is bi-harmonic. Taking 4 3 2 D .r 2 /, the bi-harmonic condition is R2 ddR4 C 6R ddR3 C 6 ddR2 D 0, for R D r 2 , which has elementary solutions R, log R, R1 and 1; the first gives the flat metric, the second the strong structure of S 3 S 1 , whereas the last two are harmonic and so do not give non-degenerate metrics. The question of which hkt structures are determined by a potential is answered by Theorem 4.2 (Banos–Swann [4]). Every hkt structure locally admits a potential. Note that in the positive definite case on a compact manifold this potential can not be global due to the maximum principle [28]. The local existence is obtained as follows. Salamon [57] introduced a differential complex for hypercomplex manifolds. There is a two-dimensional family of complex structures on M given by D aI C bJ C cK, where a2 C b 2 C c 2 D 1. For each 0;1 we may decompose the complex forms in to types: TC M D ƒ1;0 ˚ ƒ is the k1;1 eigenspace decomposition under ; and ƒk TC M D ƒk;0 ˚ ˚ƒ0;k ˚ƒ , with fp;kpg p;q p;kp 1;0 0;1 p q k ƒ D ƒ .ƒ / ˝ ƒ .ƒ /. Let ƒ D ƒ T M \ .ƒ ˚ ƒkp;p / and define [ fk;0g \ X fp;kpg ƒ ; B k D ƒ ; Ak D

k

k

16p6k=2

k

so ƒ T M D A ˚ B . The Salamon operator D Sal W Ak ! AkC1 is then exterior differentiation d W Ak ! ƒkC1 T M followed by projection. That I; J; K form a hypercomplex structure implies that .A ; D Sal / is a complex, i.e., D Sal B D Sal D 0. Using results of Verbitsky [60], one finds that the hkt condition on a hypercomplex manifold is equivalent to D Sal FI D 0 for a non-degenerate FI 2 ƒI1;1 with JFI D FI . This is equivalent to the relations .1 J.12/ J.13/ J.23/ /dFI D 0 where I; J anti-commute. There is a local potential if and only if FI D D Sal D 14 .3 I J K/d

for some one-form : when there is a potential one can take D Id, conversely N when FI D D Sal one finds locally via d D d dI , i.e., the usual @@-lemma.

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On a hypercomplex manifold the cohomology in degree k 6 12 dim M is equal to N the @-cohomology of the twistor space Z by results of Mamone Capria and Salamon [44]. In particular, H 2 .U; D Sal / D 0 locally, so the hkt condition D Sal FI D 0 implies FI is D Sal -exact. Alternatively, Michelson and Strominger [47] showed that any hkt structure on the flat hypercomplex manifold Hn D R4n admits a potential via considerations in quaternionic coordinates. However, to first order any hkt structure may be identified with the flat model, and the equation FI D D Sal only uses first order information, so is solved by the Michelson and Strominger proof. This argument, avoiding the twistor space, was used in [36] for analogous split signature structures.

4.1 Change of potential and metric As we noted above, the hkt property is not conformally invariant in dimensions greater than 4. However, using potentials there is a related type of transformation as described by Grantcharov and Poon [28]. Suppose f W R ! R is smooth, then f ./ is an hkt potential for the metric gf D f 0 ./g C 12 f 00 ./.d H /2 ;

(4.3)

where .d H /² D .1 C I C J C K/.d/², whenever gf is non-degenerate. Note these transformations may change the signature.

5 Some physical motivation In the work of Michelson and Strominger [47], [10], [46] the principal motivation is quantum mechanics of certain one-dimensional systems. One considers N particles moving on a line with the dynamics governed by a potential function V .x/ D V .x 1 ; : : : ; x N / defined on some open subset of RN . The Hamiltonian takes the form H D 12 g ab Pa Pb C V .x/; for some symmetric tensor g ab , which we may take to be non-degenerate, where Pa are the generalised momenta. One quantises this system by replacing the coordinates x a by the multiplication operator x a and the generalised momenta Pa by i @x@a , for a D 1; : : : ; N . The Hamiltonian now becomes the operator H D 12 Pa g ab Pb C V .x/;

(5.1) p 1 and the operators are applied in the given order. Acting where Pa D det g Pa pdet g

on functions one has H D 12 C V , where is the Laplacian of g. The system is said to have conformal symmetry if it scales homogeneously under dilations. One looks for mechanical systems that reflect this symmetry, meaning that


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˚ b W a; b; c 2 R . the quantisation contains a copy of the Lie algebra sl.2; R/ D ac a In the physics literature a standard basis for this algebra is 1 0 0 1 0 0 iD D ; iH D ; iK D ; (5.2) 0 1 0 0 1 0 which satisfies ŒD; H D 2iH;

ŒD; K D 2iK;

ŒH; K D iD:

Michelson and Strominger find Proposition 5.1 ([47]). Let H be the operator given in equation (5.1). (1) There exists D with ŒD; H D 2iH if and only if there exists a vector field X with LX g D 2g and LX V D 2V: (2) There exists K completing the sl2 -triple fH; D; Kg if and only if X [ D g.X; / is closed. (3) In this situation, K D 12 g.X; X/. On the other hand introducing supersymmetry, one puts V D 0 and adds supercommuting odd-operators Qi , i D 1; : : : ; N , such that Qi2 D H . The actions of Qi and Qj are related via some endomorphism Aij of the tangent bundle Qi

D Aij Qj :

The algebra relations imply immediately that for i ¤ j , Aij are anti-commuting almost complex structures and a result of Coles and Papadopoulos [13] is that each Aij is integrable with the mixed Nijenhuis tensors N.Aij ; Ak` / also zero. Thus when N D 2 we have a complex manifold, and hypercomplex geometry can arise in the case N D 4 if the algebra of Aij ’s is closed. In addition, the operators Qi are required to preserve a superspace action determined not only by g but also by an additional three-form c on M . In the N D 1 case the action is usually given as Z i . 12 gij D i @ t j C 16 cij k D i D j D k / dt d

where .t; / are coordinates on R1j1 , is the superfield thought of as map from R1j1 to the target space M , and D D @=@ C i @=@t . The geometric consequences of this were determined in [22], [13], [25] who found in the N D 4 case (strictly N D 4B) that g is Hermitian with respect to each complex structure and hkt manifolds are amongst the possible target spaces. Examples that are not hkt may be found in [25] and further discussion of this case is given by Hull [33]. However, Michelson and Strominger [47] showed that if the endomorphisms Aij satisfy the quaternion algebra, then hkt is the only possible geometry.

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Superconformal symmetry is present when there is an action of a superalgebra containing both an even subalgebra isomorphic to sl.2; R/ and supercharges Qi , as above. In the next section, we will look at the essentially the smallest possible such superalgebra.

6 A family of simple superalgebras A Lie superalgebra is a Z=2-graded algebra g D g0 C g1 with Lie bracket satisfying ŒX; Y D .1/pqC1 ŒY; X for X 2 gp , Y 2 gq , and a super Jacobi identity. The simple Lie superalgebras were classified by Kac [39]. One notable difference with the classification of complex simple Lie algebras is the occurrence of one continuous family D.2; 1I ˛/ with a real parameter ˛. For g in this family one has g0 D sl.2; C/ ˚ sl.2; C/C ˚ sl.2; C/ ; 2 2 g1 D C 2 ˝ CC ˝ C :

In particular, the even part g0 contains the ideal sl.2; C/ which is the complexification of the sl.2; R/ of conformal symmetry. The algebra r D sl.2; C/C ˚ sl.2; C/ Š so.4; C/ is often referred to as the algebra of R-symmetries. The action of g0 on the odd part g1 is as the tensor product of the standard two-dimensional representations. To specify the Lie superalgebra it remains to give the map Œ; W g1 g1 ! g0 . This map is symmetric and g0 equivariant, so by Schur’s Lemma there is one free-parameter for each sl.2; C/-factor of g0 . However, the Jacobi identity on elements of g1 imposes a homogeneous linear constraint on these parameters and an overall scaling leads to isomorphic algebras. There is thus one degree of freedom. The customary description of this via the parameter ˛ is as follows. 4 C CS4 Let D; H; K be the basis of sl.2; C/ given in (5.2). Write g1 D CQ corresponding to the i and Ci eigenspaces of D. Let Q1 ; : : : ; Q4 be a standard 4 basis of CQ under the action of r D so.4; C/, and put S a D iŒK; Qa , which gives a corresponding standard basis for CS4 . The Jacobi identity then gives ŒQa ; S a D 1 D for some 1 . One also finds ŒQ1 ; S 2 D 2 DC C 3 D 2 so.4; C/ D sl.2; C/C ˚ sl.2; C/ . The Jacobi identity implies 1 C 2 C 3 D 0. When 1 ¤ 0 one rescales to 1 D 1 and writes 2 D 4˛=.1 C ˛/ and 3 D 4=.1 C ˛/. Note that ˛ D 1 corresponds to 1 D 0, in that case one may take 2 D 1, 3 D 1. The algebras D.2; 1I ˛/C are simple for ˛ ¤ 1; 0; 1. Over C, there are isomorphisms between the cases ˛ ˙1 , .1 C ˛/˙1 , .˛=.1 C ˛//˙1 , corresponding to permuting the labelling of the sl.2; C/-factors in g0 . To obtain real forms of this algebra one chooses real structures on each sl.2; C/factor. These induce either a real or quaternionic structure on the corresponding two-dimensional representation C 2 depending on whether one has a non-compact or 2 ˝ compact form of sl.2; C/. Thus in order to have a real structure on g1 D C 2 ˝ CC 2 C one needs an odd number of non-compact real forms in the factors of g0 .


465

For superconformal symmetry related to quaternionic geometry the appropriate real form is g0 D sl.2; R/ C su.2/C C su.2/ . These are the algebras we will refer to as D.2; 1I ˛/. Note that only the algebras corresponding to ˛ ˙1 are isomorphic.

7 Superconformal symmetry and special homotheties The conditions on the target space of N D 4B models with superconformal symmetry D.2; 1I ˛/ were determined by Michelson and Strominger [47] in the hkt case, see also Papadopoulos [53]. The extra conformal ingredient comes from the generator D of the sl.2; R/-factor which gives rise to a vector field X on M . Let .M; g; I; J; K/ be an hkt manifold. A vector field X on M is called a special homothety of type .a; b/ if (i) LX g D ag, (ii) LX A D 0, for A D I; J; K, and (iii) LAX A D 0 and LAX B D bC , whenever .A; B; C / is a cyclic permutation of .I; J; K/. Note that scaling X by a constant does the same for the pair .a; b/, so we make speak of the type Œa; b 2 RP.1/ if IX is not tri-holomorphic. In the standard flat model M D Hn , one has that X D @=@r is a special homothety of type .2; 2/. For a ¤ 0 one may wish to describe the geometry as an hkt cone. When a D 0, we will sometimes talk of a special isometry since X now preserves the metric g; we now have an hkt cylinder. This is the case in the S 1 S 3 -model thought of as H =Z where the vector field X D @=@r above gives a special homothety of type .0; 2/. Theorem 7.1 ([47], [53]). D.2; 1I ˛/-symmetry corresponds to the existence of a special homothety of type .˛ C 1; 1/. The existence of a special homothety usually leads directly to an hkt potential. Proposition 7.2 ([56], [55]). Suppose X is a special homothety of type .˛ C 1; 1/. (1) For ˛ ¤ 1, we have dX [ D 0 and X ³ c D 0. (2) For ˛ ¤ 1; 0, the function D

2 g.X; X / ˛.˛ C 1/

is an hkt potential. In the cases ˛ ¤ 1; 0, we may now consider the related hkt metrics gf with potential f ./ of (4.3). To do this it is necessary to work away from the (possibly empty) hypersurface where X is null. Then the cases for which gf still has X as a special homothety are (1) for f ./ D jjk , gf has X of type .k.˛ C 1/; 1/ ¤ .1; 1/, and (2) for f ./ D log jj, gf has X of type .0; 1/, i.e., ˛ D 1

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The isomorphic algebras D.2; 1I ˛/ and D.2; 1I 1=˛/ are obtained by taking f ./ D jj1=˛ . The signature of gf may differ from that of g. First note that by replacing g by g we may assume > 0 in the neighbourhood of a given point. Suppose g has signature .4p; 4q/. Then gk has signature .4p; 4q/ precisely when the parameters ˛ for the two structures have the same sign; if the signs are different then the new signature is .4p 4; 4q C 4/. The metric glog has signature .4p 4; 4q C 4/ if ˛ > 0, otherwise it is signature .4p; 4q/. .4p; 4q/ 7! .4p; 4q/ jjk

jjk

log jj 1

0 ˛ log jj jjk

.4p; 4q/ 7! .4p 4; 4q C 4/ Figure 7.1. Signature changes for D.2; 1I ˛/ symmetry.

Locally, structures with ˛ D 1 may be converted to other values provided X is not null and X ³ c D 0. This is done by solving d log D X [ and considering gk , with k ¤ 1. In conclusion, working in the category of indefinite hkt structures we have: Proposition 7.3 ([56]). The existence of an hkt structure with a nowhere null special homothety X of type .˛ C 1; 1/, ˛ ¤ 1; 0, is equivalent to the existence of such a structure for any other such value ˛. Remark 7.4. This equivalence extends to the case ˛ D 1 locally when X ³ c D 0, cf. [51]. But other cases can arise for ˛ D 1 via t-duality. This requires the existence of a closed invariant 2-form F that is of type .1; 1/ for each I and has X ³ F D 0, where X is a special homothety of type .0; 1/. The cohomology class ŒF , when integral, then determines a circle bundle P over M to which one may lift the action of X. The quotient W D P =hX i carries an hkt structure with D.2; 1I 1/-symmetry generated by XW such that XW ³ cW is a constant times F when X ³ c D 0. Locally, every special isometry with X ³ c of type .1; 1/ for all I may be obtained in this way [59].


467

8 Relation to QKT geometry Let us continue to consider an hkt manifold M with a special homothety X of type .˛ C 1; 1/, ˛ ¤ 0, and assume that X is nowhere null. In this situation the vector fields IX, JX and KX generate a local action of the compact group SU.2/, that is isometric. When ˛ D 1, the action of X is also isometric and one may consider the manifold N D M=hX; IX; JX; KXi as a Riemannian submersion. For ˛ ¤ 1, X scales the metric so this procedure is no longer possible. However, the level sets of the potential are transverse to the homothety X and we consider the quotient N D

1 .1/ : hIX; JX; KXi

(8.1)

We will concentrate on the case when N is a manifold, but the discussion carries over easily to the general case, when N is an orbifold, and generates a wide range of examples. For all ˛ ¤ 0, the space N inherits a metric g N and local almost complex structures ; J; K, induced by I; J; K, defining an integrable quaternionic structure, cf. [54]; when dim N D 4 this means that N has a self-dual conformal structure. For ˛ ¤ 1 there is also a torsion three-form c N , induced by restriction of c to 1 .1/, which is of type ƒf1;2g for each . For ˛ D 1, we get a torsion three-form when X ³ c D 0, cf. Remark 7.4. Using c N , we may define first a connection r with skew-symmetric torsion via r D r LC C 12 c N that preserves the metric g N and the quaternionic structure. This gives N the structure of a quaternionic Kähler manifold with torsion. Definition 8.1. A manifold N is quaternionic Kähler with torsion (qkt) if there is a metric g, a connection r N and a subbundle G of End T N satisfying the following: (i) there are local bases fIN ; JN ; KN g of G satisfying the quaternion relations (2.1) and compatible with g; (ii) the connection r N is metric, r N g D 0, and quaternionic, meaning r N IN is a linear combination of JN and KN , etc.; (iii) the torsion tensor c N of r N is totally skew-symmetric and of type .2; 1/ C .1; 2/ with respect to each IN . Ivanov [35] showed that c N is uniquely determined by g N and the local almost complex structures when dim N > 4. In [56] it was shown that there is a another connection r Q on N , that is torsion-free and preserves the quaternionic structure, given via the formula rQ D r C

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with 12 Y Z D 6c N .Y; Z/ C 2

X

Ac N .Y; AZ/ Ac N .AY; Z/

AD;J;K

S Ac N .BY; C Z/ Ac N .BY; C Z/:

.A;B;C / D.;J;K/

This connection is characterised by the property that r Q is torsion-free and agrees with r on G D Spanf; J; Kg. It turns out that N arising in (8.1) is not a general qkt manifold, but rather that it satisfies two stronger conditions. The first is that of being of instanton type, meaning that the curvature of the quaternionic connection r Q on R D ƒtop T N is of type .1; 1/ for each A D ; J; K. Equivalently, \ 1;1 ƒA ; d 2 A

where is the torsion 1-form given by X 2.X/ D c N .AX; ei ; Aei /: AD;J;K iD1;:::;4n

Note that conformally rescaling g N and using Ivanov’s construction [35] of a qkt connection preserves the condition of being of instanton type. The second condition satisfied by N in (8.1) is a compatibility between the curvature Rr of r and the metric g N . The bundle G D Spanf; J; Kg End T N is preserved by r and has curvature Rr D RQ D ˇK J C ˇJ K;

etc.

When N is of instanton type, one may show that D 12 ˇI .; / C ˇI .J; K/ D 12 .1 J/ˇI .; / is symmetric, independent of , and Hermitian with respect to each . We call the curvature metric of N . Note that conformally rescaling we can locally ensure that is non-degenerate on any given qkt manifold of instanton type. Theorem 8.2. Suppose M 4nC4 is an hkt manifold with a nowhere null special homothety of type .˛ C 1; 1/, ˛ ¤ 1; 0. Then N 4n D

1 .1/ hIX; JX; KXi

is qkt of instanton type, with g N D the curvature metric.

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This result extends to the case ˛ D 0 by replacing with g.X; X /. It also applies locally to the case ˛ D 1 if X ³ c D 0. For a converse, consider the following two bundles over a quaternionic manifold N : .N / WD f.; J; K/ 2 Gx satisfying the quaternion relations (2.1) W x 2 N g; U.N / WD .N / R>0 : The bundle .N / has fibres SO.3/ D RP.3/; the second bundle U.N / was called the associated bundle in [58], but it is also known as the Swann bundle. U.N /

hkt

4n C 4

RP.3/

.N / 4n C 3

N

qkt

4n

Figure 8.1. Bundles over the qkt base N .

Theorem 8.3. Suppose N 4n is qkt of instanton type, then using the curvature metric horizontally, the Swann bundle M 4nC4 D U.N / admits hkt structures with special homotheties of type .˛ C 1; 1/ for all ˛ ¤ 0. The structures with ˛ D 1 descend to hkt structures with special isometry on the discrete quotient M=Z D .N / S 1 . This result is proved by constructing a potential for a special homothety of type .2; 2/ and then applying the results on type change in §7.

9 Examples 9.1 Quaternionic Kähler bases A quaternionic Kähler manifold of dimension 4n is a (pseudo-) Riemannian manifold .N 4n ; g N / such that either (1) n > 1 and for some 0 6 p 6 n the Levi-Civita connection has holonomy contained in Sp.p; n p/ Sp.1/, or (2) n D 1 and g N is definite, self-dual and Einstein. In both cases, the Levi-Civita connection is also a qkt and a quaternionic connection and the metric g N is Einstein with constant scalar curvature s N . The curvature metric is a positive constant times s N g N , and so is non-degenerate precisely when s N ¤ 0.

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The constructions of Swann [58] are a special case of Theorem 8.3 and provide U.N / with a pseudo-hyper-Kähler metric g and a special homothety of type .2; 2/, ˛ D 2. This metric g has signature .4p C 4; 4n 4p/ if s N > 0, or .4n 4p C 4; 4p/ if s N < 0. In particular, it is definite if g N is positive definite with s N > 0. Compact examples of such N are provided by the Wolf spaces U.n C 2/ Sp.n C 1/ f4 .RnC4 / D SO.n C 4/ ; ; Gr 2 .C nC2 / D ; Gr Sp.n/ Sp.1/ U.n/ U.2/ SO.n/ SO.4/ F4 E6 E7 E8 G2 : ; ; ; ; E7 Sp.1/ SO.4/ Sp.3/ Sp.1/ SU.6/ Sp.1/ Spin.12/ Sp.1/

HP.n/ D

These are exactly the compact symmetric spaces of the form G=K Sp.1/. A full list of the quaternionic Kähler symmetric spaces in all signatures may be found in [2]. Other homogeneous examples have been found in [15]. When these metrics are definite they are the metrics found in [1], [16], [14] and have s N < 0. The hyper-Kähler metric on U.N / is then of signature .4; 4n/, but using the type change 7! 2 of §7 one obtains positive definite hkt structures with special homotheties of type .4; 2/, ˛ D 1. Note that the question of when U.N / might be hyper-Kähler was addressed in [31]. The authors gave conditions that are equivalent to the requirement that the curvature metric on N be quaternionic Kähler.

9.2 Group manifolds As discussed in §3.2, for a compact group G we can always find a torus T r such that M D G T r admits a strong hkt structure. Opfermann and Papadopoulos have described in [50] when such manifolds admit a qkt quotient N . In practice, their quotients arise by considering a special isometry on G T r leading to an action of U.2/, so N D G T r = U.2/ and these are instances of the ˛ D 1 variation of Theorem 8.2. For G simple, the corresponding M are SU.2n C 1/; SO.4n/ T 2n ;

SU.2n/ T 1 ;

SO.2n C 1/ T n ;

SO.4n C 2/ T 2n1 ; E7 T 7 ;

G2 T 2 ;

Sp.n/ T n ; F4 ;

E6 T 2 ;

E8 T 8 :

The examples where M D G has no torus factor can not arise from the inverse construction of Theorem 8.3. For such M the homotopy exact sequence shows that b2 .N / D 1. Pulling a 2-form on N representing a non-trivial integral cohomology class back to .N /S 1 , we may now recover the hkt structure on M via t-duality as in Remark 7.4 up to finite quotients. A particular case is M D G D SU.3/, N D CP.2/, where one obtains SU.3/=.Z=2/; see [54] for the hypercomplex geometry.


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9.3 Dimension four and connected sums If the base manifold N has dimension four, then the condition to be qkt of instanton type is no more than g N being self-dual. Choosing any one-form t , one may construct a qkt connection with c N D t , this is of instanton type if and only if dt is selfdual. For N 4 compact this forces t to be closed. Theorem 8.3 now implies that .N 4 / S 1 carries an hkt structure .g; c/ provided the curvature metric is nondegenerate. Although not strong in general, one has that dc is the pull-back of a four-form on N . If b2 .N /C ¤ 0 we may now find a t-dual M 8 of .N 4 / S 1 that is hkt and has finite fundamental group. Examples of this are obtained by taking N 4 D n CP.2/, the connected sum of n-copies of CP.2/, which is self-dual by [17]. In this case, an alternative to checking the non-degeneracy of the curvature metric is provided by Joyce’s construction [37] of n CP.2/ as a quaternionic quotient of HP.n/ by U.1/n1 . At the level of Swann bundles this may be viewed as a construction of .n CP.2// S 1 as a hypercomplex quotient of HnC1 , which in turn may be regarded as an hkt quotient using the results of [27]. Thus there are compact simply-connected hkt manifolds Mn8 fibering over n CP.2/.

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D. V. Alekseevsky, Classification of quaternionic spaces with a transitive solvable group of motions. Math. USSR Izv. 9 (1975), 297–339. 470

[2]

D. V. Alekseevsky and V. Cortés, Classification of pseudo-Riemannian symmetric spaces of quaternionic Kähler type. In Lie groups and invariant theory, Amer. Math. Soc. Transl. (2) 213, Amer. Math. Soc., Providence, RI 2005, 33–62. 470

[3]


[4]

B. Banos and A. Swann, Potentials for hyper-Kähler metrics with torsion. Classical Quantum Gravity 21 (2004), 3127–3135. 461

[5]

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Part D

Para-geometries

Chapter 14

Twistor and reflector spaces of almost para-quaternionic manifolds Stefan Ivanov, Ivan Minchev, and Simeon Zamkovoy

Contents 1 2 3

Introduction and statement of the results . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Twistor and reflector spaces of almost para-quaternionic manifolds 3.1 Dependence on the para-quaternionic connection . . . . . . . 3.2 Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Para-quaternionic Kähler manifolds with torsion . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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477 481 482 485 489 493 494

1 Introduction and statement of the results We study the geometry of structures on a differentiable manifold related to the algebra of paraquaternions. This structure leads to the notion of (almost) hyper-paracomplex and almost paraquaternionic manifolds in dimensions divisible by four. These structures are also attractive in theoretical physic since they play a role in string theory [36], [24], [8], [25], [15], [16] and integrable systems [19]. For example, hyperparacomplex geometry arises in connection with different versions of the c-map [16]. New versions of the c-map are constructed in [16] which allow the authors to obtain the target manifolds of hypermultiplets in Euclidean theories with rigid N D 2 supersymmetry. The authors show that the resulting hypermultiplet target spaces are para-hyper-Kähler manifolds. One might also consult the contribution of Th. Mohaupt [35] in this volume where the role of paraquaternions in theories with Euclidean supersymmetry is discussed as well as the exposition written by M. Rocek, C. Vafa and S. Vandoren [37] and the work of A. Swann [40] (booth in this book) concerning the implement of quaternions in string theories. z are real Clifford algebras, H D Both quaternions H and paraquaternions H z z of paraquaternions is C.2; 0/, H D C.1; 1/ Š C.0; 2/. In other words, the algebra H 0 0 0 generated by the unity 1 and the generators J1 ; J2 ; J3 satisfying the paraquaternionic identities, .J10 /2 D .J20 /2 D .J30 /2 D 1;

J10 J20 D J20 J10 D J30 :

(1.1)

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Stefan Ivanov, Ivan Minchev, and Simeon Zamkovoy

We recall the notion of almost hyper-paracomplex manifold introduced by Libermann [34]. An almost quaternionic structure of the second kind on a smooth manifold consists of two almost product structures J1 , J2 and an almost complex structure J3 which mutually anti-commute, i.e., these structures satisfy the paraquaternionic identities (1.1). Such a structure is also called complex product structure [5], [4]. We recall that an almost product structure P on M is a .1; 1/ tensor field such that P 2 D idM . If moreover the two eigenbundles T C M and T M associated to the two eigenvalues C1 and 1 of P , respectively, have the same rank then the structure P is said to be an almost paracomplex structure (see [17] for a survey of paracomplex geometry). Sometimes these structures are also called almost bilagrangian [30]. An almost hyper-paracomplex structure on a 4n-dimensional manifold M is a z D .J˛ /, ˛ D 1; 2; 3, where J1 and J2 are almost paracomplex structures triple H J1 ; J2 W TM ! TM , and J3 W TM ! TM is an almost complex structure, satisfying the para-quaternionic identities (1.1). We note that on an almost hyper-paracomplex manifold there is actually a 2-sheeted hyperboloid worth of almost complex structures: S12 .1/ D fc1 J1 C c2 J2 C c3 J3 j c12 C c22 c32 D 1g and a 1-sheeted hyperboloid worth of almost paracomplex structures: S12 .1/ D fb1 J1 C b2 J2 C b3 J3 j b12 C b22 b32 D 1g: z is said to be a hyper-paraWhen each J˛ , ˛ D 1; 2; 3 is an integrable structure, H complex structure on M . Such a structure is also called sometimes pseudo-hypercomplex [19]. It is well known that the structure J˛ is integrable if and only if the corresponding Nijenhuis tensor N˛ D ŒJ˛ ; J˛ C J˛2 Œ ; J˛ ŒJ˛ ; J˛ Œ ; J˛ vanishes, N˛ D 0. In fact an almost hyper-paracomplex structure is hyper-paracomplex if and only if any two of the three structures J˛ , ˛ D 1; 2; 3 are integrable due to the existence of a linear identity between the three Nijenhuis tensors [29], [14]. In this case all almost complex structures of the two-sheeted hyperboloid S12 .1/ as well as all para-complex structures of the one-sheeted hyperboloid S12 .1/ are integrable. Examples of hyperparacomplex structures on the simple Lie groups SL.2n C 1; R/, SU.n; n C 1/ are constructed in [27]. A hyper-paraHermitian metric is a pseudo Riemannian metric which is compatible z D .J˛ /, ˛ D 1; 2; 3 in the sense with the (almost) hyper-paracomplex structure H that the metric is skew-symmetric with respect to each J˛ , ˛ D 1; 2; 3. Such a metric is necessarily of neutral signature .2n; 2n/. Such a structure is called (almost) hyper-paraHermitian structure. An almost para-quaternionic structure on M is a rank-3 subbundle P End.TM / z D .J˛ /; such which is locally spanned by an almost hyper-paracomplex structure H z will be called admissible basis of P . A linear connection a locally defined triple H r on TM is called para-quaternionic connection if r preserves P . We denote the space all para-quaternionic connections on an almost para-quaternionic manifold by .P /.

Chapter 14. Twistor and reflector spaces of almost para-quaternionic manifolds

479

An almost para-quaternionic structure is said to be a para-quaternionic if there is a torsion-free para-quaternionic connection. An almost para-quaternionic (resp. para-quaternionic) manifold with hyper-paraHermitian metric is called an almost para-quaternionic Hermitian (resp. para-quaternionic Hermitian) manifold. If the Levi-Civita connection of a para-quaternionic Hermitian manifold is para-quaternionic connection, then the manifold is said to be para-quaternionic Kähler manifold. This condition is equivalent to the statement that the holonomy group of g is contained in Sp.n; R/Sp.1; R/ for n 2 [22]. A typical example is the para-quaternionic projective space endowed with the standard para-quaternionic Kähler structure [13]. The para-quaternionic analogue of the quaternionic geometry is developed in [22] where it is also shown that any para-quaternionic Kähler manifold of dimension 4n 8 is Einstein. If on a para-quaternionic Kähler manifold z / such that each J˛ , ˛ D 1; 2; 3 is parallel with there exists an admissible basis .H respect to the Levi-Civita connection, then the manifold is said to be (locally) hyperparaKähler. Such manifolds are also called hypersymplectic [23], neutral hyperKähler [31], [21]. The equivalent characterization is that the holonomy group of g is contained in Sp.n; R/ if n 2 [41]. For n D 1 an almost para-quaternionic structure is the same as an oriented neutral conformal structure [19], [22], [41], [14] and turns out to be always quaternionic. The existence of a (local) hyper-paracomplex structure is a strong condition since the integrability of the (local) almost hyper-paracomplex structure implies that the corresponding neutral conformal structure is anti-self-dual [1], [24], [29]. When n 2, the para-quaternionic condition, i.e., the existence of torsionfree para-quaternionic connection is a strong condition which is equivalent to the z / Sp.1; R/ Š GL.2n; R/ Sp.1; R/- structure 1-integrability of the associated GL.n; H [4], [5]. The paraquaternionic condition controls the Nijenhuis tensors in the sense that NJ˛ WD N˛ preserves the subbundle P . An invariant first order differential operator D is defined on any almost paraquaternionic manifolds which is two-step nilpotent i.e., D 2 D 0 exactly when the structure is paraquaternionic [28]. Para-quaternionic structure is a type of a para-conformal structure [7] as well as a type of generalized hypercomplex structure [10]. Let .M; P / be an almost para-quaternionic manifold. The vector bundle P carries a Lorentz structure of signature .C; C; / (note that this is the opposite to the naturally induced signature .; ; C/ as discussed in [12]) such that .J1 ; J2 ; J3 / forms an orthonormal local basis of P . There are two kinds of “unit sphere” bundles according to the existence of the 1-sheeted hyperboloid S12 .1/ and the 2-sheeted hyperboloid S12 .1/. The twistor space Z .M / is the unit pseudo-sphere bundle with fibre (the one sheet of) S12 .1/. The reflector space Z C .M / is the unit pseudo-sphere bundle with fibre S12 .1/. In other words, the fibre of Z .M / consists of all almost complex structures compatible with the given paraquaternionic structure while the fibre of Z C .M / consists of all almost paracomplex structures compatible with the given paraquaternionic structure. Reflector spaces where introduced in [30] in the study of neutral surfaces in 4-dimensional neutral pseudo-

480


Riemannian manifolds. The construction of the reflector space in [30] is based on the splitting of the non-semi-simple structure group SO.2; 2/, (on the Lie algebra level so.2; 2/ Š sl.2; R/ ˚ sl.2; R/ Š su.1; 1/ ˚ su.1; 1/ Š sp.1; R/ ˚ sp.1; R// and the reflector spaces are the neutral space analogs of the twistor spaces of Riemannian geometry [30]. Keeping in mind the formal similarity with the quaternionic geometry where there are two natural almost complex structures on the corresponding twistor space [6], [20], one observes the existence of two naturally arising almost complex structures I1r , I2r on Z .M / and two almost paracomplex structures P1r , P2r on Z C .M / defined with the help of the horizontal spaces of an arbitrary para-quaternionic connection r 2 .P /. The almost paracomplex structures on the reflector space of a 4-dimensional manifold with neutral signature metric are defined using the horizontal spaces of the Levi-Civita connection in [30]. The authors show that one of the almost paracomplex structure is never integrable while the other almost paracomplex structure is integrable if and only if the neutral metric is anti-self-dual. The almost complex structures on the twistor space of a para-quaternionic Kähler manifold are defined and investigated in [12] using the horizontal spaces of the Levi-Civita connection (see also [2]). The authors show that one of the almost complex structure is never integrable while the other almost complex structure is always integrable. These construction are generalized in the case of twistor and reflector space of a para-quaternionic manifold [11] (see also [29]). In the present chapter we investigate the dependence on the para-quaternionic connection of these structures on the twistor and reflector spaces over an almost paraquaternionic manifold. We obtain conditions on the torsion of the para-quaternionic connection which imply the coincidence of the corresponding structures (Theorem 3.4, Corollary 3.3). We show that the existence of an integrable almost complex structure on the twistor space (resp. the existence of an integrable almost paracomplex structure on the reflector space) does not depend on the para-quaternionic connection and it is equivalent to the condition that the almost para-quaternionic manifold is quaternionic provided the dimension is bigger than four (Theorem 3.9, Theorem 3.12). In dimension four we find new relations between the Riemannian Ricci forms,i.e., the 2-forms which determine the Sp.1; R/-component of the Riemannian curvature, which are equivalent to the ant-self-duality of the oriented neutral conformal structure corresponding to a given para-quaternionic structure (Theorem 3.7). In the last section we apply our considerations to the paraquaternionic Kähler manifold with torsion recently described by the third author in [42].


481

2 Preliminaries z be the para-quaternions and identify H z n D R4n . To fix notation we assume Let H n z z that H acts on H by right multiplication. This defines an antihomomorphism W funit para-quaternionsg D fx C j1 y C j2 z C j3 w j x 2 y 2 z 2 C w 2 D ˙1g ! SO.2n; 2n/ GL.4n; R/; z n on the left. Denote the image where our convention is that SO.2n; 2n/ acts on H 0 0 by Sp.1; R/ and let J1 D .j1 /, J2 D .j2 /, J30 D .j3 /. The Lie algebra of Sp.1; R/ is sp.1; R/ D spanfJ10 ; J20 ; J30 g and we have 2

2

2

J10 D J20 D J30 D 1;

J10 J20 D J20 J10 D J30 :

z / D fA 2 GL.4n; R/ j A.sp.1; R//A1 D sp.1; R/g. The Lie Define GL.n; H z / is gl.n; H z / D fA 2 gl.4n; R/ j AB D BA for all B 2 algebra of GL.n; H sp.1; R/g. z D .J˛ /, ˛ D 1; 2; 3 Let .M; P / be an almost paraquaternionic manifold and H be an admissible local basis. We shall use the notation 1 D 2 D 3 D 1. Let B 2 .ƒ2 .TM / ˝ TM /. We say that B is of type .0; 2/J˛ with respect to J˛ if B.J˛ X; Y / D J˛ B.X; Y / 0;2 and denote this space by ƒ0;2 J˛ . The projection BJ˛ is given by

.X; Y / D BJ0;2 ˛

1 .B.X; Y / C ˛ B.J˛ X; J˛ Y / 4 ˛ J˛ B.J˛ X; Y / ˛ J˛ B.X; J˛ Y //:

For example, the Nijenhuis tensor N˛ is an element of ƒ0;2 J˛ . Let r 2 .P / be a para-quaternionic connection on an almost paraquaternionic manifold .M; P /. This means that there exist locally defined 1-forms !˛ , ˛ D 1; 2; 3 such that rJ1 D !3 ˝ J2 C !2 ˝ J3 ; rJ2 D !3 ˝ J1 C !1 ˝ J3 ;

(2.1)

rJ3 D !2 ˝ J1 C !1 ˝ J2 : An easy consequence of (2.1) is that the curvature Rr of any para-quaternionic connection r 2 .P / satisfies the relations ŒRr ; J1 D A3 ˝ J2 C A2 ˝ J3 ; A1 D d!1 C !2 ^ !3 ; ŒRr ; J2 D A3 ˝ J1 C A1 ˝ J3 ;

A2 D d!2 C !3 ^ !1 ;

ŒRr ; J3 D A2 ˝ J1 C A ˝ J2 ;

A3 D d!3 !1 ^ !2 :

(2.2)

482


The Ricci 2-forms of a para-quaternionic connection are defined by 1 tr.Z ! J˛ Rr .X; Y /Z/; ˛ D 1; 2; 2 1 3r .X; Y / D tr.Z ! J3 Rr .X; Y /Z/: 2 It is easy to see using (2.2) that the Ricci forms are given by ˛r .X; Y / D

1r D d!1 C !2 ^ !3 ;

2r D d!2 !3 ^ !1 ;

3r D d!3 !1 ^ !2 : (2.3)

z /-valued part .Rr /0 and sp.1; R/-valued part We split the curvature of r into gl.n; H r 00 .R / following the classical scheme (see e.g. [3], [26], [9]) Proposition 2.1. The curvature of an almost para-quaternionic connection on M splits as follows 1 r . .X; Y /J1 C 2r .X; Y /J2 C 3r .X; Y /J3 /; 2n 1 Œ.Rr /0 .X; Y /; J˛ D 0; ˛ D 1; 2; 3;

Rr .X; Y / D .Rr /0 .X; Y / C

Let , ‚ be the curvature 2-form and the torsion 2-form of r on the principle z /Sp.1; R/-bundle P .M /, respectively (see e.g. [32]). We denote the splitGL.n; H z / ˚ sp.1; R/-valued curvature 2-form on P .M / according to ting of the gl.n; H z /-valued 2-form and 00 is Proposition 2.1, by D 0 C 00 , where 0 is a gl.n; H a sp.1; R/-valued form. Explicitly, 00 D 001 J10 C 002 J20 C 003 J30 ; where 00˛ , ˛ D 1; 2; 3, are 2-forms. If ; ; 2 R4n , then the 2-forms 00˛ , ˛ D 1; 2; 3, are given by 00˛ .B. /; B. // D

1 ˛ .X; Y / 2n

and

X D u. /; Y D u. /:

3 Twistor and reflector spaces of almost para-quaternionic manifolds z1 of imaginary para-quaternions. It is isomorphic to the Lorentz Consider the space H 2 space R1 with a Lorentz metric of signature .C; C; / defined by hq; q 0 i D Re.qq 0 /, where qN D q is the conjugate imaginary para-quaternion. In R21 there are two kinds of ’unit spheres’, namely the pseudo-sphere S12 .1/ of radius 1 (the 1-sheeted hyperboloid) which consists of all imaginary para-quaternions of norm 1 and the pseudo-sphere S12 .1/ of radius .1/ (the 2-sheeted hyperboloid) which contains all imaginary para-quaternions of norm .1/. The 1-sheeted hyperboloid S12 .1/ carries a natural paracomplex structure while the 2-sheeted hyperboloid S12 .1/ carries a


483

z1 Š R2 defined by natural complex structure, both induced by the cross-product on H 1 X Y D

X

x i y k Ji Jk

i6Dk

for vectors X D x i Ji , Y D y k Jk . Namely, for a tangent vector X D x i Ji to the k 1-sheeted hyperboloid S12 .1/ at a point qC D qC Jk (resp. tangent vector Y D yk Jk k to the 2-sheeted hyperboloid S12 .1/ at a point q D q Jk ) we define PX WD qC X (resp. J Y D q Y ). It is easy to check that PX is again tangent vector to S12 .1/ and P 2 X D X (resp. J Y is tangent vector to S12 .1/ and J 2 Y D Y ). Let M be a 4n-dimensional manifold endowed with an almost para-quaternionic structure P . Let J1 , J2 , J3 be an admissible basis of P defined in some neighborhood of a given point p 2 M . Any linear frame u of Tp M can be considered as an isomorphism u W R4n ! Tp M . If we pick such a frame u we can define a subspace of the space of the all endomorphisms of Tp M by u.sp.1; R//u1 . Clearly, this subset is a para-quaternionic structure at the point p and in general this para-quaternionic structure is different from Pp . We define P .M / to be the set of all linear frames u which satisfy u.sp.1; R//u1 D P . It is easy to see that P .M / is a z /Sp.1; R/, it is also called principal frame bundle of M with structure group GL.n; H z a GL.n; H /Sp.1; R/-structure on M . Let W P .M / ! M be the natural projection. For each u 2 P .M / we consider two linear isomorphisms j C .u/ and j .u/ on T.u/ M defined by j C .u/ D uJ10 u1 and j .u/ D uJ30 u1 . It is easy to see that .j C .u//2 D id and .j .u//2 D id. For each point p 2 M we define ZpC .M / D fj C .u/ j u 2 P .M /; .u/ D pg and Zp .M / D fj .u/ j u 2 P .M /; .u/ D pg. In other words, Zp .M / is the connected component of J3 of the space of all complex structures (resp. ZpC .M / is the space of all paracomplex structures) in the tangent space Tp M which are compatible with the almost para-quaternionic structure on M . S We define the twistor space Z of M , by setting Z D p2M Zp .M /. Let z /Sp.1; R/. There is a bijective corH3 be the stabilizer of J30 in the group GL.n; H z /Sp.1; R/=H3 Š S 2 .1/C D respondence between the symmetric space GL.n; H 1 f.x; y; z/ 2 R3 j x 2 C y 2 z 2 D 1; z > 0g and Zp .M / for each p 2 M . So we can consider Z as the associated fibre bundle of P .M / with stanz /Sp.1; R/=H3 . Hence, P .M / is a principal fibre bundle over dard fibre GL.n; H Z with structure group H3 and projection j . We consider the symmetric spaces z /Sp.1; R/=H3 . We have the following Cartan decomposition gl.n; H z/ ˚ GL.n; H sp.1; R/ D h3 ˚ m3 where z / ˚ sp.1; R/ j AJ30 D J30 Ag h3 D fA 2 gl.n; H z / ˚ sp.1; R/ j AJ 0 D J 0 Ag. It is the Lie algebra of H3 and m3 D fA 2 gl.n; H 3 3 0 0 is clear that m3 is generated by J1 , J2 , i.e., m3 D spanfJ10 ; J20 g. Hence, if A 2 m3 then J30 A 2 m3 .

484


S We proceed with defining the reflector space Z C of M . Put Z C D p2M ZpC .M /. z /Sp.1; R/. There is a bijective Let H1 be the stabilizer of J10 in the group GL.n; H z / Sp.1; R/=H1 Š S 2 .1/ D correspondence between the symmetric space GL.n; H 1 2 C 3 2 2 f.x; y; z/ 2 R j x Cy z D 1g and Zp .M / for each p 2 M . So we can consider z /Sp.1; R/=H1 . Z C as the associated fibre bundle of P .M / with standard fibre GL.n; H C Hence, P .M / is a principal fibre bundle over Z with structure group H1 and proz /Sp.1; R/=H1 . We have the jection j C . We consider the symmetric spaces GL.n; H z / ˚ sl.1; R/ D h1 ˚ m1 where following Cartan decomposition gl.n; H z / ˚ sl.1; R/ j AJ10 D J10 Ag h1 D fA 2 gl.n; H z / ˚ sl.1; R/ j AJ 0 D J 0 Ag. It is the Lie algebra of H1 and m1 D fA 2 gl.n; H 1 1 0 0 is clear that m1 is generated by J2 , J3 , i.e., m1 D spanfJ20 ; J30 g. Hence, if A 2 m1 then J10 A 2 m1 . Let r be a para-quaternionic connection on M , i.e., r is a linear connection in the principal bundle P .M / (see e.g. [32]). Note that we make no assumptions on the torsion or on the curvature of r. Keeping in mind the formal similarity with the quaternionic geometry where one uses a quaternionic connection to define two almost complex structures on the corresponding twistor space [6], [20], [38], [39], we use r to define two almost complex structures I1r and I2r on the twistor space Z and two almost paracomplex structures P1r and P2r on the reflector space Z C . We denote by A (resp. B. /) the fundamental vector field (resp. the standard z / ˚ sl.1; R/ (resp. horizontal vector field) on P .M / corresponding to A 2 gl.n; H 4n 2 R ). Let u 2 P .M / and Qu be the horizontal subspace of the tangent space Tu P .M / induced by r (see e.g. [32]). The vertical space i.e., the vector space tangent to a fibre is isomorphic to z / ˚ sl.1; R//u D .h3 /u ˚ .m3 /u D .h1 /u ˚ .m1 /u ; .gl.n; H where .hi /u D fAu j A 2 hi g and .mi /u D fAu j A 2 mi g; i D 1; 3. Hence, Tu P .M / D .hi /u ˚ .mi /u ˚ Qu . For each u 2 P .M / we put Vj .u/ D ju ..m3 /u /; Hj .u/ D ju Qu ;

C C C VjCC .u/ D ju ..m1 /u /; Hj.u/ D ju Qu :

Thus we obtain vertical and horizontal distributions V and H on Z (resp. V C and H C on Z C ). Since P .M / is a principal fibre bundle over Z (resp. Z C ) with C structure group H3 (resp. H1 ) we have Ker ju D .h3 /u (resp. Ker ju D .h1 /u ). Hence Vj .u/ D ju .m3 /u and juj.m / ˚Q W .m3 /u ˚Qu ! Tj .u/ Z is an iso3 u

u

C C .m1 /u and juj.m W .m1 /u ˚ Qu ! Tj C .u/ Z C morphism (resp. VjCC .u/ D ju 1 /u ˚Qu is an isomorphism).


485

We define two almost complex structures I1r and I2r on Z by A / D ju .J30 A/ ; I1r .ju

I2r .ju A / D ju .J30 A/

Œ4ptIir .ju B. // D ju B.J30 /;

i D 1; 2;

(3.1)

for A 2 m3 , 2 R4n . Similarly, we define two almost paracomplex structures P1r and P2r on Z C by C C A / D ju .J10 A/ ; P1r .ju

C C P2r .ju A / D ju .J10 A/

C C Pir .ju B. // D ju B.J10 /;

i D 1; 2;

(3.2)

for A 2 m1 ; 2 R4n . Clearly, the construction of these structures depends on the choice of the para-quaternionic connection r. The almost paracomplex structures (3.2) on the reflector space of a 4-dimensional manifold with neutral signature metric are defined using the horizontal spaces of the Levi-Civita connection r g in [30]. The authors show that the almost paracomplex g g structure P2r is never integrable while the almost paracomplex structure P1r is integrable if and only if the neutral metric is anti-self-dual. The almost complex structures (3.1) on the twistor space of a para-quaternionic Kähler manifold are defined and investigated in [12] with the help of the horizontal spaces of the Levi-Civita connection. g The authors show that the almost complex structure I2r is never integrable while the g almost complex structure I1r is always integrable. Both construction are generalized in the case of twistor and reflector space of a para-quaternionic manifold in [11], [29]. Twistor space of para-quaternionic Kähler manifold is investigated also in [18] where the LeBrun’s inverse twistor construction for quaternionic Kähler manifolds [33] has been adapted to the case of para-quaternionic Kähler manifolds. We finish this section with the next useful Lemma 3.1. Let J 2 Z be an almost complex structure or JC 2 Z C be an almost paracomplex structure and B 2 .ƒ2 .TM / ˝ TM /. If BJ0;2 D 0 for all J 2 Z then BJ0;2 D 0 for all JC 2 Z C and vice versa. C Proof. Let J t D sinh tJ1 C cosh tJ3 ; t 2 R be an almost complex structure in Z . D 0 D BJ0;2 , we calculate Using the conditions BJ0;2 t 3 1 1 C sinh 2t ŒB D 0: .1 C cosh 2t /BJ0;2 1 2 2 D 0. Similarly, BJ0;2 D 0 and the lemma follows. The latter leads to BJ0;2 1 2

3.1 Dependence on the para-quaternionic connection In this section we investigate when different almost para-quaternionic connections induce the same structure on the twistor or reflector space over an almost para-quaternionic manifold.

486


Let r and r 0 be two different almost para-quaternionic connections on an almost para-quaternionic manifold .M; P /. Then we have rX0 D rX C SX ;

X 2 .TM /;

z / ˚ sp.1; R/ for where SX is a .1; 1/ tensor on M and u1 .SX /u belongs to gl.n; H any u 2 P .M /. Thus we have the splitting SX .Y / D SX0 .Y / C s 1 .X /J1 Y C s 2 .X /J2 Y C s 3 .X /J3 Y;

(3.3)

where X; Y 2 .TM /, s i are 1-forms and ŒSX0 ; Ji D 0, i D 1; 2; 3. Proposition 3.2. Let r and r 0 be two different para-quaternionic connections on an almost para-quaternionic manifold .M; P /. The following conditions are equivalent. 0

(i) The two almost complex structures I1r and I1r on the twistor space Z coincide. (ii) The 1-forms s 1 , s 2 , s 3 are related as follows: s 1 .J1 X/ D s 2 .J2 X / D s 3 .J3 X /;

X 2 .TM /: 0

(iii) The two almost paracomplex structures P1r and P1r on the reflector space Z C coincide. Proof. We fix a point J of the twistor space Z . We have J D a1 J1 C a2 J2 C a3 J3 with a12 C a22 a32 D 1. Let W Z ! M be the natural projection and x D .J /. The connection r induces a splitting of the tangent space of Z into vertical and horizontal components: TJ Z D VJ ˚HJ . Let v and h be the vertical and horizontal 0 projections corresponding to this splitting. Let TJ Z D V 0 J ˚ H J be the splitting 0 0 0 induced by r with the projections v and h , respectively. It is easy to observe the following identities: v C h D 1; v 0 C h0 D 1; vv 0 D v 0 ; v 0 C vh0 D v:

(3.4)

In fact, VJ D V 0 J and we may regard this space as a subspace of Px . We have that VJ D fW 2 Px j W J C J W D 0g D fw1 J1 C w2 J2 C w3 J3 j w1 a1 C w2 a2 w3 a3 D 0g; where J D a1 J1 C a2 J2 C a3 J3 . It follows that for any W 2 VJ , I1r .W / D 0 I1r .W / D J W . In general, for any W 2 TJ Z , we have I1r .W / D J.vW / C .J .W //h ; 0

0

I1r .W / D J.v 0 W / C .J .W //h ;


487

0

where ./h (resp. ./h ) denotes the horizontal lift on Z of the corresponding vector field on M with respect to r (resp. r 0 ). Using (3.4), we calculate that 0

0

0

v.I1r W / D J.v 0 W / C v.J .W //h D J..v vh0 /W / C v.J .W //h 0

D v.I1r W / J.vh0 W / C v.J .W //h :

(3.5)

We investigate the equality 0

J.vh0 W / D v.J .W //h ;

W 2 TJ Z :

(3.6)

h0

Take W D Y , Y 2 .TM / in (3.6) to get 0

0

J.vY h / D v.J Y /h ;

Y 2 Tx M:

(3.7)

0 I1r

I1r

D because of (3.5). Hence, (3.7) is equivalent to P i @ Let .U; x1 ; : : : ; x4n / be a local coordinate system on M and let Y D Y @x i . The horizontal lift of Y with respect to r 0 at the point J 2 Z is given by 0

YJh D

4n 3 X X @ .Y i B / i as r 0 Y Js : @x sD1 iD1

We calculate 0

0

0

0

v.J Y /h D .J Y /h h.J Y /h D .J Y /h .J Y /h D

3 X

as .rJ0 Y Js C rJ Y Js / D ŒSJ Y ; J :

(3.8)

sD1

On the other hand, we have 0

0

J.vY h / D J.Y h Y h / D J

3 X

as .rY0 Js C rY Js / D J ŒSY ; J :

(3.9)

sD1 0

Substitute (3.8) and (3.9) into (3.7) to get that I1r D I1r is equivalent to the condition J ŒSY ; J D ŒSJ Y ; J ; Y 2 .TM /; J 2 Z : (3.10) Now, (3.10) and (3.3) easily lead to the equivalence of (i) and (ii). 0 Similarly, we obtain that P1r D P1r if and only if P ŒSY ; J D ŒSP Y ; J

(3.11)

for any choice of P 2 Z C and Y 2 TM . The equality (3.11) together with (3.3) implies the equivalence of (ii) and (iii). Corollary 3.3. Let r and r 0 be two different para-quaternionic connections on an almost para-quaternionic manifold .M; P /. The following conditions are equivalent: 0

(i) The two almost complex structures I2r and I2r on the twistor space Z coincide.

488


(ii) The 1-forms s 1 , s 2 , s 3 vanish, s 1 D s 2 D s 3 D 0. 0

(iii) The two almost paracomplex structures P2r and P2r on the reflector space Z C coincide. 0

Proof. It is sufficient to observe from the proof of Proposition 3.2 that I2r D I2r is 0 equivalent to J ŒSY ; J D ŒSJ Y ; J , Y 2 .TM /, J 2 Z while P1r D P1r if and C only if P ŒSY ; J D ŒSP Y ; J for any choice of P 2 Z and Y 2 TM . Each one of the last two conditions imply s 1 D s 2 D s 3 D 0. Theorem 3.4. Let r and r 0 be two different para-quaternionic connections with 0 torsion tensors T r and T r , respectively, on an almost para-quaternionic manifold .M; P /. The following conditions are equivalent: 0

(i) The two almost complex structures I1r and I1r on the twistor space Z coincide. 0

(ii) The .0; 2/J parts with respect to all J 2 P of the torsion T r and T r coincide, r 0 0;2 .T r /0;2 /J . J D .T 0

(iii) The two almost paracomplex structures P1r and P1r on the reflector space Z C coincide. Proof. The equivalence of (i) and (iii) has been proved in Proposition 3.2. Let S D r 0 r. Then we have 0

T r .X; Y / D T r .X; Y / C SX .Y / SY .X /:

(3.12)

The .0; 2/J -part with respect to J of (3.12) gives 0

r 0;2 .T r /0;2 J .T /J D ŒSJX ; J Y J ŒSX ; J Y ŒSJ Y ; J X C J ŒSY ; J X: (3.13)

Suppose (iii) holds. Substitute (3.11) into the right hand side of (3.13) and use 0 r 0;2 Lemma 3.1 to get .T r /0;2 J D .T /J , i.e., (ii) is true. For the converse, put J D J2 in (3.13) and use the splitting (3.3) to obtain 1 r 0 0;2 .T /J2 .T r /0;2 J2 2 1 D s .X/ C s 3 .J2 X / J1 Y C s 1 .J2 X / C s 3 .X / J3 Y s 1 .Y / C s 3 .J2 Y / J1 X s 1 .J2 Y / C s 3 .Y / J3 X:

(3.14)

r 0;2 Hence, s 1 .J1 X/ D s 3 .J3 X / is equivalent to .T r /0;2 J2 D .T /J2 Substitute J D J1 in (3.13) and use the splitting (3.3) to get s 1 .J2 X / D s 3 .J3 X / is equivalent to 0 r 0;2 .T r /0;2 J1 D .T /J1 . Now, Lemma 3.1 together with Proposition 3.2 completes the proof. 0

489


3.2 Integrability In this section we investigate conditions on the para- quaternionic connection r which imply the integrability of the almost complex structure I1r on Z and almost paracomplex structure P1r on Z C . We also show that I2r and P2r are never integrable, i.e., for any choice of the para-quaternionic connection r each of these two structures has non-vanishing Nijenhuis tensor. We denote by I Ni , PNi , i D 1; 2 the Nijenhuis tensors of Ii and Pi , respectively and recall that I Ni .U; W / D ŒIi U; Ii W ŒU; W Ii ŒIi U; W Ii ŒU; Ii W ; PNi .U; W / D ŒPi U; Pi W C ŒU; W

U; W 2 .T Z /;

Pi ŒPi U; W Pi ŒU; Pi W ; U; W 2 .T Z C /: Proposition 3.5. Let r be a para-quaternionic connection on an almost para-quaternionic manifold .M; P / with torsion tensor T r . The following conditions are equivalent: (i) The almost complex structure I1r on the twistor space Z of .M; P / is integrable. (ii) The .0; 2/J -parts .T r /0;2 J of the torsion with respect to all J 2 P vanish, .T r /0;2 J D 0; J 2 P

(3.15)

and the .2; 0/ C .0; 2/ parts of the Ricci 2-forms with respect to an admissible basis J1 , J2 , J3 of P coincide in the sense that the following identities hold ˇ ˛ .Jˇ X; Jˇ Y / C ˛ .X; Y / .Jˇ X; Y / .X; Jˇ Y / D 0; (3.16) where f˛; ˇ; g is a cyclic permutation of f1; 2; 3g and 1 D 2 D 3 D 1. (iii) The almost paracomplex structure P1r on the reflector space Z C of .M; P / is integrable. Proof. Let J1 , J2 , J3 be an admissible basis of the almost para-quaternionic structure P . Let hor be the natural projection Tu P ! .m3 /u ˚ Qu , with ker.hor/ D .h3 /u . We define a tensor field I10 on P .M / by I10 .U / 2 .m3 /u ˚ Qu ; .j /u .I10 .U // D I1 ..j /u U /;

U 2 Tu P:

For any U; W 2 .TP .M // we define I N10 .U; W / D horŒI10 U; I10 W horŒhor U; hor W I10 ŒI10 U; hor W I10 Œhor U; I10 W : It is easy to check that I N10 is a tensor field on P .M /. We also observe that ju .I N10 .U; W // D I N1 .ju U; ju W /;

U; W 2 Tu P .M /:

(3.17)

490


Let A; B 2 m3 and ; 2 R4n . Using the well-known general commutation relations among the fundamental vector fields and standard horizontal vector fields on the principal bundle P .M / (see e.g. [32]), we calculate taking into account (3.17) that .Au /; ju .Bu // D 0; I N1 .ju I N1 .ju .Au /; ju .B. /u // D 0; .B. /u //; ju .B. /u //H ŒI N1 .ju D ju .B.‚.B.J30 /; B.J30 // C ‚.B. /; B. //

(3.18)

C J30 ‚.B.J30 /; B. // C J30 ‚.B. /; B.J30 ///u /; .B. /u //; ju .B. /u //V ŒI N1 .ju

D f1 .B.J30 /; B.J30 // C 1 .B. /; B. // C 2 .B.J30 /; B. // C 2 .B. /; B.J30 //gju .J10 /

(3.19)

C f2 .B.J30 /; B.J30 // C 2 .B. /; B. // .J20 /; 1 .B.J30 /; B. // 1 .B. /; B.J30 ///gju .Au /; ju .B. /u // D 4ju .B.A /u / 6D 0: I N2 .ju

(3.20)

Concerning the reflector space, let horr be the natural projection Tu P ! .m1 /u ˚Qu , with ker.horr/ D .h1 /u . In a very similar way as above, we calculate .Au /; ju .Bu // D 0; PN1 .ju .Au /; ju .B. /u // D 0; PN1 .ju .B. /u //; ju .B. /u //H C ŒPN1 .ju D ju .B.‚.B.J10 /; B.J10 // ‚.B. /; B. //

(3.21)

C J10 ‚.B.J10 /; B. // C J10 ‚.B. /; B.J10 ///u /; .B. /u //; ju .B. /u //V C ŒPN1 .ju

D f2 .B.J10 /; B.J10 // 2 .B. /; B. // .J20 / C 3 .B.J10 /; B. // C 3 .B. /; B.J10 //gju

C C

(3.22)

f3 .B.J10 /; B.J10 // 3 .B. /; B. // 2 .B.J10 /; B. // C 2 .B. /; B.J10 //gju .J30 /;

C C C PN2 .ju .Au /; ju .B. /u // D 4ju .B.A /u / 6D 0:

(3.23)

Take X D u. /; Y D u. / we see that (3.18) and (3.19) are equivalent to r r r r .T r /0;2 J3 D T .X; Y / T .J3 X; J3 Y / C J3 T .J3 X; Y / C J3 T .X; J3 Y / D 0; (3.24)

1r .J3 X; J3 Y / 1r .X; Y / 2r .J3 X; Y / 2r .X; J3 Y / D 0;

(3.25)

491


for any admissible basis J1 , J2 , J3 of P which proves the equivalence of (i) and (ii). Similarly, (3.21) and (3.22) are equivalent to r r r r .T r /0;2 J1 D T .J1 X; J1 Y / C T .X; Y / J1 T .J1 X; Y / J1 T .X; J1 Y / D 0; (3.26)

3r .J1 X; J1 Y / C 3r .X; Y / 2r .J1 X; Y / 2r .X; J1 Y / D 0;

(3.27)

for any admissible basis J1 ; J2 ; J3 of P which proves the equivalence of (iii) and (ii). The equations (3.20) and (3.23) in the proof of Proposition 3.5 yield Corollary 3.6. Let r be a para-quaternionic connection on an almost para-quaternionic manifold .M; P / with torsion tensor T r . (1) The almost complex structure I2r on the twistor space Z of .M; P / is never integrable. (2) The almost paracomplex structure P2r on the reflector space Z C of .M; P / is never integrable. In the 4-dimensional case we derive Theorem 3.7. Let .M 4 ; g/ be a 4-dimensional pseudo-Riemannian manifold with neutral metric g and let P be the para-quaternionic structure corresponding to the conformal class generated by g with a local basis J1 , J2 , J3 . Then the following conditions are equivalent (i) The neutral metric g is anti-self-dual. (ii) The Ricci forms ag of the Levi-Civita connection r g satisfy (3.16), i.e., ˇ ˛g .Jˇ X; Jˇ Y / C ˛g .X; Y / g .Jˇ X; Y / g .X; Jˇ Y / D 0: (iii) The torsion condition (3.15) for a linear connection r always implies the curvature condition (3.16). Proof. The proof is a direct consequence of Proposition 3.5, Theorem 3.4 and the g result in [30] (resp. [12]) which states that the almost paracomplex structure P1r (resp. g the almost complex structure I1r ) is integrable exactly when the neutral conformal structure generated by g is anti-self-dual. Remark 3.8. We note that for any compatible with the orientation integrable almost (para) complex structure, say J D J3 , the condition (ii) of Theorem 3.7 holds for ˛ D 1, ˇ D 2, D 3 but the oriented neutral conformal structure may not be anti-self-dual. Indeed, if J˛ is a (para)complex structure then the connection 1-forms !ˇg ; !g of r g satisfy the equality [42] !ˇg .X/ D ˛ !g .J˛ X /:

(3.28)

492


An easy calculations, using (2.1) and (3.28), give d!ˇg .X; Y / D ˛ .r gX !g /J˛ Y C ˛ .r g Y !g /J˛ X

˛ !ˇg .X /!g .J Y / C ˇ !g .X /!g .Jˇ Y / C

˛ !ˇg .Y /!g .J X /

(3.29)

ˇ !g .Y /!g .Jˇ X /:

g

The Ricci 2-forms ˛r of r g with respect to .J1 ; J2 ; J3 D J / are given by (2.3). Insert (2.3) and (3.29) into (3.16) taken for ˛ D 1; ˇ D 2; D 3 to check that (3.16) holds. In higher dimensions, the curvature condition (3.16) is always a consequence of the torsion condition (3.15) in the sense of the next Theorem 3.9. Let r be a para-quaternionic connection on an almost para-quaternionic 4n-dimensional n 2 manifold .M; P / with torsion tensor T r . Then the following conditions are equivalent: (i) The almost complex structure I1r on the twistor space Z of .M; P / is integrable. (ii) The .0; 2/J -part .T r /0;2 J of the torsion with respect to all J 2 P vanishes, .T r /0;2 J D 0;

J 2 P:

(iii) The almost paracomplex structure P1r on the reflector space Z C of .M; P / is integrable. Proof. We use Proposition 3.5. Since the connection r is a para-quaternionic connection, r 2 .P /, the condition (3.15) yields the next expression of the Nijenhuis tensor NJ of any local J 2 P , NJ .X; Y / 2 spanfJ1 X; J1 Y; J2 X; J2 Y; J3 X; J3 Y g;

(3.30)

where J1 , J2 , J3 is an admissible local basis of P . To prove that (ii) implies the integrability of I1r and P1r , we apply the result of Zamkovoy [42] which states that an almost para-quaternionic 4n-manifold .n 2/ is para-quaternionic if and only if the three Nijenhuis tensors N1 , N2 , N3 satisfy the condition .N1 .X; Y / C N2 .X; Y / N3 .X; Y // 2 spanfJ1 X; J1 Y; J2 X; J2 Y; J3 X; J3 Y g: (3.31) Clearly, (3.31) follows from (3.30) which shows that the almost para-quaternionic 4n-manifold (n 2) .M; P / is a para-quaternionic manifold. Let r 0 be a torsion-free 0 para-quaternionic connection on .M; P /. Then the almost complex structure I1r on 0 the twistor space Z as well as the almost paracomplex structure P1r on the reflector 0 0 space Z C are integrable [29] and I1r D I1r , P1r D P1r due to Theorem 3.4. Hence, the equivalence between (i), (ii) and (iii) is established, which completes the proof.


493

From the proof of Proposition 3.5 and Theorem 3.9, we easily derive Corollary 3.10. Let r be a para-quaternionic connection on a 4n-dimensional (n 2) almost para-quaternionic manifold .M; P / with torsion tensor T r . Then the torsion condition (3.15) implies the curvature condition (3.16). We note that Corollary 3.10 generalizes the same statement proved in the case of PQKT-connection (see below) in [42] where the first Bianchi identity is used. Theorem 3.9 and Theorem 3.4 imply Corollary 3.11. Let .M; P / be an almost para-quaternionic manifold. (i) Among the all almost complex structures I1r , r 2 .P / on the twistor space Z at most one is integrable. (ii) Among the all almost paracomplex structures P1r , r 2 .P / on the reflector space Z C at most one is integrable. A direct consequence of the proof of Theorem 3.9, Theorem 3.7 and Corollary 3.11 is the following Theorem 3.12. Let .M; P / be an almost para-quaternionic 4n-manifold. The next three conditions are equivalent: (1) Either .M; P / is a para-quaternionic manifold (if n 2) or .M; P D Œg/ is anti-self dual for n D 1. (2) There exists an integrable almost complex structure I1r on the twistor space Z which does not depend on the para-quaternionic connection r. (3) There exists an integrable almost paracomplex structure P1r on the reflector space Z C which does not depend on the para-quaternionic connection r.

4 Para-quaternionic Kähler manifolds with torsion An almost para-quaternionic Hermitian manifold .M; P ; g/ is called para-quaternionic Kähler with torsion (PQKT) if there exists an almost para-quaternionic Hermitian connection r T 2 .P / whose torsion tensor T is a 3-form which is .1; 2/C.2; 1/ with respect to each Ja , i.e., the tensor T .X; Y; Z/ WD g.T .X; Y /; Z/ is totally skewsymmetric and satisfies the conditions T .X; Y; Z/ D T .J˛ X; J˛ Y; Z/ T .J˛ X; Y; J˛ Z/ T .X; J˛ Y; J˛ Z/; ˛ D 1; 2; T .X; Y; Z/ D T .J3 X; J3 Y; Z/ C T .J3 X; Y; J3 Z/ C T .X; J3 Y; J3 Z/: We recall that each PQKT is a para-quaternionic manifold [42]. The condition on the torsion implies that the .0; 2/-part of the torsion of a PQKT connection vanishes. Applying Theorem 3.9, we obtain

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Theorem 4.1. Let .M; P ; r T / be a PQKT and r 0 2 .P / be a torsion-free paraquaternionic connection. Then T (i) The almost complex structure I1r on the twistor space Z is integrable and 0 therefore it coincides with I1r . T

(ii) The almost paracomplex structure P1r on the reflector space Z C is integrable 0 and therefore it coincides with P1r .

References [1]

M. Akivis and V. Goldberg, Conformal differential geometry and its generalizations. Pure Appl. Math. (N. Y.), Wiley, New York 1996. 479

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D. V. Alekseevsky and V. Cortés, The twistor spaces of a para-quaternionic Kähler manifold. Osaka J. Math. 45 (2008), no. 1, 215–251. 480

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D. V. Alekseevsky and S. Marchiafava, Quaternionic structures on a manifold and subordinated structures. Ann. Mat. Pura Appl. (4) 171 (1996), 205–273. 482

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A. Andrada, Complex product structures on differentiable manifolds. Preprint. 478, 479

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A. Andrada and S. Salamon, Complex product structures on Lie algebras. Forum Math. 17 (2005), no. 2, 261–295. 478, 479

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M. F. Atiayah, N. Hitchin, and I. M. Singer, Self-duality in four-dimensional Riemannian geometry. Proc. Roy. Soc. London Ser. A 362 (1978), 425–461. 480, 484

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T. Bailey and M. Eastwood, Complex paraconformal manifolds-their differential geometry and twistor theory. Forum Math. 3 (1991), 61–103. 479

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J. Barret, G. W. Gibbons, M. J. Perry, C. N. Pope, and P. Ruback, Kleinian geometry and the N D 2 superstring. Internat. J. Modern Phys. A 9 (1994), no. 9, 1457–1493. 477

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[10] R. Bielawski, Manifolds with an SU.2/-action on the tangent bundle. Trans. Amer. Math. Soc. 358 (2006), no. 9, 3997–4019. 479 [11] D. Blair, A product twistor space. Serdica Math. J. 28 (2002), no. 2, 163–174. 480, 485 [12] D. Blair, J. Davidov and O. Muskarov, Hyperbolic twistor spaces. Rocky Mountain J. Math. 35 (2005), no. 5, 1437–1465. 479, 480, 485, 491 [13] N. Blazic, Paraquaternionic projective space and pseudo-Riemannian geometry. Publ. Inst. Math. (Beograd) (N.S.) 60 (74) (1996), 101–107. 479 [14] N. Blazic and S. Vukmirovic, Para-hypercomplex structures on a four-dimensional Lie group. In Contemporary geometry and related topics, World Sci. Publ., River Edge, NJ, 2004, 41–56. 478, 479 [15] V. Cortés, Ch. Mayer, Th. Mohaupt, and F. Saueressig, Special geometry of Euclidean supersymmetry I: Vector multiplets. J. High Energy Phys. 0403 (2004), 028. 477 [16] V. Cortés, Ch. Mayer, Th. Mohaupt, and F. Saueressig, Special geometry of Euclidean supersymmetry II: Hypermultiplets and the c-map. J. High Energy Phys. 0506 (2005), 025. 477


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[17] V. Cruceanu, P. Fortuny, and P. M. Gadea, A survey on paracomplex geometry. Rocky Mountain J. Math. 26 (1996), no. 1, 83–115. 478 [18] A. S. Dancer, H. R. Jorgensen, and A. F. Swann, Metric geometries over the split quaternions. Rend. Sem. Mat. Univ. Politec. Torino 63 (2005), no. 2, 119–139. 485 [19] M. Dunajski, Hyper-complex four-manifolds from the Tzitzéica equation. J. Math. Phys. 43 (2002), 651–658. 477, 478, 479 [20] J. Eells and S. Salamon, Constructions twistorielles des applications harmoniques. C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), 685–687. 480, 484 [21] A. Fino, H. Pedersen, Y.-S. Poon, and M. W. Sorensen, Neutral Calabi-Yau structures on Kodaira manifolds. Comm. Math. Phys. 248 (2004), no. 2, 255–268. 479 [22] E. Garcia-Rio, Y. Matsushita, and R. Vasquez-Lorentzo, Paraquaternionic Kähler manifold. Rocky Mountain J. Math. 31 (2001), 237-260. 479 [23] N. Hitchin, Hypersymplectic quotients. Acta Acad. Sci. Tauriensis 124 (1990), supl., 169– 180. 479 [24] C. M. Hull, Actions for .2; 1/ sigma models and strings. Nuclear Phys. B 509 (1988), no. 1, 252–272. 477, 479 [25] C. M. Hull, A geometry for non-geometric string backgrounds. J. High Energy Phys. 065 (2005), 10. 477 [26] S. Ishihara, Quaternion Kählerian manifolds. J. Differential Geom. 9 (1974), 483–500. 482 [27] S. Ivanov and V. Tsanov, Complex product structures on some simple Lie groups. Preprint 2004; arXiv:math/0405584v2 [math.DG]. 478 [28] S. Ivanov, V. Tsanov, and S. Zamkovoy, Hyper-ParaHermitian manifolds with torsion. J. Geom. Phys. 56 (2006), no. 4, 670–690. 479 [29] S. Ivanov and S. Zamkovoy, Para-Hermitian and Para-Quaternionic manifolds. Differential Geom. Appl. 23 (2005), 205–234. 478, 479, 480, 485, 492 [30] G. Jensens and M. Rigoli, Neutral surfaces in neutral four spaces. Mathematiche (Catania) 45 (1990), 407–443. 478, 479, 480, 485, 491 [31] H. Kamada, Neutral hyper-Kähler structures on primary Kodaira surfaces. Tsukuba J. Math. 23 (1999), 321–332. 479 [32] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol I, II, Interscience Publ., New York 1963, 1969. 482, 484, 490 [33] C. R. LeBrun, Quaternionic Kähler manifolds and conformal geometry. Math. Ann. 284 (1989), 353–376. 485 [34] P. Libermann, Sur le problème d’équivalence de certains structures infinitésimales. Ann. Mat. Pura Appl. 36 (1954), 27–120. 478 [35] Th. Mohaupt, Special geometry, black holes and Euclidean supersymmetry. In Handbook of pseudo-Riemannian geometry and supersymmetry, ed. by V. Cortés, IRMA Lect. Math. Theor. Phys. 16, European Math. Soc. Publishing House, Zürich 2010, 149–181. 477 [36] H. Ooguri and C. Vafa, Geometry of N D 2 strings. Nuclear Phys. B 361 (1991), 469–518. 477 [37] M. Rocek, C. Vafa and S. Vandoren, Quaternion-Kähler spaces, hyper-Kähler cones, and the c-map. In Handbook of pseudo-Riemannian geometry and supersymmetry, ed. by

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[38] S. Salamon, Quaternionic Kähler manifolds. Invent. Math. 67 (1982), 143–171. 484 [39] S. Salamon, Differential geometry on quaternionic manifolds. Ann. Sci. École Norm. Sup. Ser. (4) 19 (1986), 31–55. 484 [40] A. Swann, Quaternionic geometries from superconformal symmetry. In Handbook of pseudo-Riemannian geometry and supersymmetry, ed. by V. Cortés, IRMA Lect. Math. Theor. Phys. 16, European Math. Soc. Publishing House, Zürich 2010, 475–494. 477 [41] S. Vukmirovic, Paraquaternionic reduction. arXiv:math.DG/0304424. 479 [42] S. Zamkovoy, Geometry of paraquaternionic Kähler manifolds with torsion. J. Geom. Phys. 57 (2006), no. 1, 69–87, arXiv:math/0511595v1 [math.DG]. 480, 491, 492, 493

Chapter 15

Para-pluriharmonic maps and twistor spaces Matthias Krahe

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Para-complex vector spaces . . . . . . . . . . . . . . . . . . 1.2 Para-complex manifolds . . . . . . . . . . . . . . . . . . . 1.3 Para-hermitian metrics and para-Kähler manifolds . . . . . . 1.4 Para-quaternionic Kähler manifolds . . . . . . . . . . . . . 2 Para-Kähler submanifolds of para-quaternionic Kähler manifolds 2.1 Para-complex symplectic and contact structures . . . . . . . 2.2 Para-Kähler submanifolds and twistor lift . . . . . . . . . . 3 Para-pluriharmonic maps into symmetric spaces . . . . . . . . . 3.1 Para-pluriharmonic maps . . . . . . . . . . . . . . . . . . . 3.2 Isotropic para-pluriharmonic maps . . . . . . . . . . . . . . 3.3 Canonical elements and twistor spaces . . . . . . . . . . . . 3.4 Isotropic para-pluriharmonic maps into Grassmannians . . . 3.5 Para-quaternionic Kähler symmetric spaces . . . . . . . . . 3.6 The classical para-quaternionic Kähler symmetric spaces . . 4 tt*-bundles and para-pluriharmonic maps . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Introduction Twistor theory goes back to the 1960s, where twistor methods were introduced by R. Penrose to provide an approach to quantum gravity. The general idea is to translate a problem on a differentiable manifold M endowed with a certain geometric structure to a problem on a complex manifold Z, the twistor space. Usually, Z is a fibre bundle over M , with the fibres over p 2 M being complex structures on Tp M which are compatible with the geometric structure on M . Z carries a canonical almost complex structure, whose integrability can be derived from the properties of M . An important application of twistor theory is the construction of minimal and pluriminimal submanifolds: In 1982, Bryant used the fibration CP3 ! HP1 to construct minimal surfaces in HP1 D S4 [Br]. This has been generalized to the construction of minimal surfaces in various Grassmannians and symmetric spaces by considering horizontal maps into certain flag manifolds and flag domains [BR], [BS].

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The reinterpretation of Bryant’s work by Lawson [L] using a birational equivalence CP3 F1;1;1 .C 3 / WD SU.3/=S.U.1/ U.1/ U.1// led to the remarkable observation that the twistor spaces of any two symmetric quaternionic Kähler spaces are birationally equivalent [Bu]. The purpose of this work is to transfer some of these methods to para-complex geometry: A para-complex (or hyperbolic complex) manifold is a smooth manifold with a field of involutions Jp W Tp M ! Tp M , J 2 D 1, such that the ˙1-eigendistributions are integrable and of the same dimension. The obvious example is a product of two manifolds MC M of the same dimension; in fact, every para-complex manifold is locally of this form. Para-complex and para-quaternionic manifolds have recently been studied in mathematics as well as in physics, where they are related to supersymmetric field theories with Euclidean (rather than Lorentzian) space-times [CMMS1], [CMMS2]. The first section of this chapter contains the definitions of para-complex manifolds, para-Kähler manifolds and para-quaternionic Kähler manifolds. We give a proof of the para-complex Dolbeault Lemma, which states that in a para-complex manifold, N the @-operator is locally exact (Theorem 1.2.8 and Corollary 1.2.11). Extending earlier results by Blair et al. [BDM], it has been shown by Alekseevsky and Cortés [AC2], that for any para-quaternionic Kähler manifold M there is a twistor bundle W Z ! M of para-complex structures (actually there is also a twistor bundle Z ! M of complex structures, which we neglect here). Z has a canonical paraKähler structure, and the horizontal distribution is a para-holomorphic contact structure. A submanifold N M such that there exists a parallel section of ZjN is called para-complex submanifold. If the pseudo-Riemannian metric gM is nondegenerate on TN , then N is a para-Kähler submanifold, and it follows that N is a minimal submanifold (i.e. the trace of the shape operator is zero). Any para-complex submanifold can be lifted to a para-holomorphic horizontal submanifold of Z, in other words, paracomplex immersions f W N ! M are precisely the projections of para-holomorphic horizontal immersions F W N ! Z. F

N

|

| f

|

Z |>

/M

It follows that N has at most half the dimension of M , and the para-complex submanifolds with maximal dimension correspond to the Legendrian submanifolds of Z. Our aim in Section 2 is to show that these Legendrian submanifolds and therefore the maximal para-complex submanifolds of M are locally given by a para-holomorphic function. The main step will be the proof of the para-complex equivalent of the Darboux theorem, which states that on each para-complex contact manifold, there exist local co-

Chapter 15. Para-pluriharmonic maps and twistor spaces

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ordinate charts which identify the contact structure with the standard contact structure on C 2nC1 , where C denotes the algebra of para-complex numbers (Theorem 2.1.11). The notion of twistors can be generalized to the setting where M a pseudoRiemannian symmetric space (not necessarily para-quaternionic Kähler). There is a twistor construction analogous to the one in Section 2, where the role of paracomplex immersions is played by isotropic para-pluriharmonic maps (in fact, in the case of para-quaternionic Kähler symmetric spaces, these constructions coincide). A map f W N ! M is called para-pluriharmonic, if it is harmonic along every paracomplex curve in N . Under certain conditions, these maps have a one-parameter family .f /2R of deformations, the associated family. A special case occurs when this family is constant in ; such maps f are called isotropic. In Section 3 we prove the following: Each full isotropic para-pluriharmonic map f W N ! M from a Kähler manifold N into a symmetric space M D G=G defines a twistor fibration Z ! M and a holomorphic superhorizontal lift F W N ! Z; vice versa, given such a map F , its projection f D B F is isotropic para-pluriharmonic (Theorem 3.3.6). x D .G=P / The twistor space is an open subset of the para-complex coset space Z .G=P /, where the para-complex structure is just multiplication by C1 on the first factor and by 1 on the second. If G is a real linear algebraic group, then P is a parabolic subgroup, and thus G=P a real projective variety. This variety has a cell decomposition (Bruhat decomposition) with exactly one cell N in the largest dimension, which is an open (even Zariski-open) and dense subset. Now, if Z is the twistor space associated to a symmetric para-quaternionic Kähler space, then it turns out that there is a contact isomorphism between the product x of the big cells and the para-complex Heisenberg group. It follows that, N N Z given the twistor spaces of two symmetric para-quaternionic Kähler spaces of the same dimension, then there is a biholomorphism defined on Zariski-open subsets preserving the contact distribution; moreover, this biholomorphism extends to a birational equivalence. This can be used to transfer the construction of para-pluriharmonic maps from one para-quaternionic Kähler symmetric space to another. Among these spaces is the para-complex Grassmannian Gr 2 .C nC2 / with corresponding twistor fibration F1;n;1 .C nC2 / ! Gr 2 .C nC2 /, where this construction is fairly easy. In the last section we consider tt*-bundles. (Para-)pluriharmonic maps are closely related to a geometric structure in topological field theories called topological-antitopological fusion, or tt*-geometry: There is a bijection between tt*-bundles and (para-)pluriharmonic maps into the symmetric space GL.p Cq/=O.p; q/ which admit an associated family (called admissible in [S2]). Here we relate this to the construction of associated families and give a criterion for the (para-)pluriharmonic map to be isotropic.

Differences to the complex theory For those who are familiar with the twistor theory in the complex case, we point out a few differences to the para-complex case:

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• Para-complex manifolds: A para-complex structure defines a splitting of the tangent bundle into eigenbundles – there is no need to consider the para-complexified tangent bundle (at times we do so for convenience, but this yields no new information). • Regularity: Para-holomorphic and para-pluriharmonic maps are not necessarily analytic (any pair of differentiable functions f˙ defines a para-holomorphic map fC f W R R ! R R). • Associated families: All pluriharmonic maps into Riemannian symmetric spaces have associated families, provided that the symmetric space has nonpositive or nonnegative curvature. In the para-complex context, this is different: Unless the symmetric space is flat, there exist para-pluriharmonic maps which do not admit an associated family. Moreover, the parameter range of the associated family is R rather than S1 . Consequently, in our context the loop group of G will be defined as the set of smooth curves R ! G. • Topology: In contrast to hermitian symmetric spaces, para-hermitian spaces are not necessarily simply connected, an example being the para-complex projective space C Pn D SL.n C 1; R/=S.GL.1; R/ GL.n; R//: This example also shows that the twistor fibration G=H ! G=K is not defined globally, if G0 K ¨ G is “too small” (where H is the centralizer of a canonical element and G denotes the fixed-point set of the Cartan involution). Acknowledgements. This chapter essentially contains my thesis. I am grateful for the support and motivation I received from a number of people. Specifically I would like to thank my advisors Prof. Dr. Vicente Cortés and Prof. Dr. Jost-Hinrich Eschenburg, Christian Boltner, Walter Freyn, Marie-Amélie Lawn, Peter Quast, Lars Schäfer and Kerstin Weinl, and above all, my family and my parents.

1 Preliminaries 1.1 Para-complex vector spaces Definition 1.1.1. Let V be a finite-dimensional real vector space. A para-complex structure on V is an involution J W V ! V , J 2 D idV , such that the eigenspaces V ˙ WD ker.id J / have the same dimension. The pair .V; J / is then called a paracomplex vector space. A homomorphism between para-complex vector spaces .V; J / and .V 0 ; J 0 / is a linear map W V ! V 0 satisfying B J D J 0 B .


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Just like complex vector spaces (i.e. .V; I / with I 2 D 1) are vector spaces over the field C, para-complex vector spaces are free modules over the algebra of paracomplex numbers: Definition 1.1.2. The algebra C D R ˚ j R of para-complex numbers (also called split complex or hyperbolic complex numbers) is the real algebra generated by 1 and j , with the relation j 2 D 1 (in fact, C is the Clifford algebra Cl.0; 1/). The conjugation map N W C ! C is defined by x C jy D x jy for x; y 2 R. The real numbers x D Re.z/ WD .z C zN /=2 and y D Im.z/ WD .z z/=2j N are called the real and imaginary part of z D x C jy, respectively. Note that zz N D x 2 y 2 2 R. Thus, the group of units is C D fx C jy W x ¤ ˙yg D fz 2 C W z … .1 ˙ j /Rg. It has four connected components; in fact, each z 2 C can be written uniquely as ˙re j or ˙jre j , where r 2 RC , 2 R, and e j D cosh. / C j sinh. /. This is the para-complex analogue of the polar decomposition; the unit circle is replaced by the four hyperbolas f.x; y/ 2 R2 W x 2 y 2 D ˙1g. The free module C n is a para-complex vector space, with the multiplication by j as para-complex structure. Conversely, any para-complex vector space .V; J / can be regarded as a C -module via .x C jy/v WD xv C yJ v. In order to see that this module is free, let fv1C ; : : : ; vnC ; v1 ; : : : ; vn g be a basis of V , such that va˙ 2 V ˙ . Then the P P a P a equation .x a C jy a /.vaC C va / D 1Cj .x C y a /vaC C 1j .x y a /va for 2 2 a C a coefficients x C jy 2 C shows that fw1 ; : : : ; wn g, wa WD va C va is a free basis of the C -module V . Homomorphisms between para-complex vector spaces correspond to C -linear maps. Remark 1.1.3. Since C is not a field, some basic facts from linear algebra must be treated with caution: (1) Let W V ! V 0 be C -linear. Then the image and the kernel of are J -invariant real subspaces of V 0 and V , respectively. They are not necessarily para-complex subspaces, because the dimensions of .V ˙ /, resp. ker./˙ may not be the same. – Of course, it still holds that injective () ker./ D f0g: (2) Let W V ! V be a C -linear endomorphism and let z D x C jy 2 C . Then the eigenspace Vz WD ker. z id/ is J -invariant, and therefore we have Vz D VzC ˚ Vz . The Vz˙ are eigenspaces of with real eigenvalue .x ˙ y/. In particular, eigenspaces to different para-complex eigenvalues need not be disjoint. (3) The C -span of a single vector v 2 V is not necessarily a para-complex vector space; this motivates the following definition: Definition 1.1.4. Let .V; J / be a para-complex vector space. A vector v 2 V is called regular, if it fulfills the following equivalent conditions:

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v … V C and v … V . C v is a one-dimensional (over C ) para-complex vector space. v and J v are linearly independent over R. For z 2 C it holds that .zv D 0 () z D 0/.

Proof of the equivalence. .1/ () .2/: We write v D v C C v 2 V C ˚ V . Then .C v/˙ D R v ˙ . So C v is a one-dimensional para-complex vector space, if and only if v ¤ 0 and v C ¤ 0. .1/ H) .3/: We may assume that v ¤ 0. If J v D rv, r 2 R, then v D J 2 v D r 2 v, so r D ˙1. .3/ H) .1/ is trivial. .3/ () .4/ follows immediately from .x C jy/v D xv C yJ v. The following example is fundamental since it illustrates the principle that paracomplex objects can be described either by using para-complex numbers or by using real numbers: Example 1.1.5. The map Š ! .Rn Rn ; .id/.id//; .C n ; j / .x a C jy a /aD1;:::;n 7! .x a C y a ; x a y a /aD1;:::;n ; 1j a 1Cj a a a zC C z .zC ; z /aD1;:::;n ; 2 2 aD1;:::;n

is an isomorphism of para-complex vector spaces. It induces an isomorphism of groups Š GL.n; C / ! GL.n; R/ GL.n; R/; B CC 0 B C jC 7! ; 0 B C 1j 1Cj AC 0 AC C A : 0 A 2 2

Lemma 1.1.6. Let AC ; A 2 GL.n; R/. Then 1j 1j 1Cj 1Cj AC C A D det.AC / C det.A /: det C 2 2 2 2 2 Proof. This follows immediately from the Leibniz formula and from 1Cj D 2 1j 2 1j 1Cj 1j D 2 , 2 2 D 0. 2

1Cj 2

,

Corollary 1.1.7. 1Cj 1j AC C A 2 C () det.AC / ¤ 0 ^ det.A / ¤ 0: detC 2 2 Corollary 1.1.8. SL.n; C / Š SL.n; R/ SL.n; R/:


503

1.2 Para-complex manifolds Definition 1.2.1. An almost para-complex structure on a manifold M is a smooth endomorphism field J on TM , such that for every p 2 M , Jp is a para-complex structure on Tp M . An almost para-complex structure is said to be integrable, if the eigendistributions T ˙ M WD ker.id J / are integrable. In this case, .M; J / is called a para-complex manifold. As a consequence of the Frobenius theorem – see for example [CMMS1, Chapter 2.1] – we have Proposition 1.2.2. Let .M; J / be an almost para-complex manifold. Then the following conditions are equivalent: (1) J is integrable. (2) ŒT C M; T C M T C M and ŒT M; T M T M . (3) NJ D 0, where NJ is the Nijenhuis tensor NJ .X; Y / WD ŒX; Y C ŒJX; J Y J ŒX; J Y J ŒJX; Y : 1 n 1 n (4) For any point p 2 M , there exist local real coordinates .zC ; : : : ; zC ; z ; : : : ; z /W n n z z U ! UC U R R defined on an open neighbourhood U of p, such a a B J D ˙ dz˙ . that dz˙ 0 (4 ) For any point p 2 M , there exist local para-complex coordinates .z 1 ; : : : ; z n / W z U ! U C n defined on an open neighbourhood U of p, such that dz a B J D j dz a .

These coordinates are para-holomorphic in the following sense: Definition 1.2.3. A smooth map W .M; JM / ! .M 0 ; JM 0 / between para-complex manifolds is called para-holomorphic (or just holomorphic, if no confusion is possible), if d B JM D JM 0 B d: Para-holomorphic maps .M; J / ! .C ; j / or .M; J / ! .R R; .id/ .id// are called para-holomorphic functions. a a zC C 1j z , Definition 1.2.4. Let M be a para-complex manifold and let z a D 1Cj 2 2 a a a D 1 : : : n, be a system of local holomorphic coordinates. Then .dz ; d zN /aD1:::n is local frame of the complexified cotangent bundle T M ˝ C . The dual frame of TM ˝ C is given by

1Cj @ 1j @ WD a C @z a 2 @zC 2 1j @ 1Cj @ WD a C a @zN 2 @zC 2

@ ; a @z @ : a @z

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a a Proposition 1.2.5. Let M be a para-complex manifold and let z a D 1Cj zC C 1j z , 2 2 a D 1 : : : n, be a system of local holomorphic coordinates. Then a smooth function f W M ! C is para-holomorphic, if and only if it fulfills the Cauchy–Riemann equations @f D 0: @zN a

fC C 1j f , where f˙ WD Re.f / ˙ Im.f /. Then it is Proof. We write f D 1Cj 2 2 easy to see that the Cauchy–Riemann equations are equivalent to @fC @f D a D 0: a @z @zC But this in turn is just a rewriting of df B JM D JRR B df .

Lemma 1.2.6. Let U C be a simply connected domain and let f W U ! C be a para-holomorphic function. Then f has an antiderivative, i.e. a para-holomorphic Df. function F , such that @F @z Proof. Since d.f dz/ D exact, i.e. f dz D dF D

@f d zN ^ dz D 0, the C -valued 1-form f @zN @F dz C @F d z. N It follows that @F Df @zN @z @z

dz is closed, hence and @F D 0. @zN

1.2.1 Bigrading of differential forms, Dolbeault sequences Real version. Let .M; J / be an almost para-complex manifold. The decomposition T M D .T M /C ˚ .T M / into the ˙1-eigenspaces of J induces a bigrading on exterior and differential forms: M V.Cp;q/ Vk T M D T M; pCqDk k

.M / D

M

.Cp;q/ .M /:

pCqDk

Definition 1.2.7. The elements of .Cp;q/ .M / are called differential forms of degree .Cp; q/. In the case that J is integrable, then we can use holomorphic coordinates to show that d .Cp;q/ .M / .C.pC1/;q/ .M / ˚ .Cp;.qC1// .M /. Hence the exterior derivative splits as d D @C C @ , where @C W .Cp;q/ .M / ! .C.pC1/;q/ .M /; @ W .Cp;q/ .M / ! .Cp;.qC1// .M /:


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From d 2 D 0 we get @2C D @2 D @C @ C @ @C D 0, so for any para-complex manifold the operators @˙ can be used to define real versions of the Dolbeault sequence. This sequence is locally exact: Theorem 1.2.8. Let U D UC U be a para-complex manifold (with the paracomplex structure on T UC T U given by id id), such that both UC and U are contractible. Then any @C -closed form ! 2 .Cp;q/ .U /, p 1, is @C -exact. Likewise, any @ -closed form ! 2 .Cp;q/ .U /, q 1, is @ -exact. Remark. (1) From Proposition 1.2.2 (4) it follows that each point in an arbitrary para-complex manifold has a neighbourhood U of this form. (2) The condition that U be contractible is not sufficient, as shown by the following counterexample: Let U WD R2 R2 n .f0g f0g R0 R/ and !.zC ; wC ; z ; w / WD f .z /

zC dwC wC dzC , 2 Cw 2 zC C

where f .z / is a smooth nonzero

function that vanishes on R>0 . Then ! is a smooth .1; 0/-form, which is @C -closed, but not @C -exact. Proof of the theorem. For the proof of the first statement, let ! 2 .Cp;q/ .U /, p 1, and @C ! D 0. Let the contraction maps on UC and U be denoted by hC and h respectively. We can define a para-holomorphic homotopy map h W Œ0; 12 U ! U by h.sC ;s / .uC ; u / D .hsC .uC /; hs .u //. Then h.1;1/ D idU and dh.0;1/ T C U D 0 and therefore, since p 1, ! 0 D h.1;1/ ! h.0;1/ ! Z 1 d D .h.sC ;1/ !/dsC : 0 dsC

(1)

In order to compute the integrand, we write h.sC ;1/ D h B ‰sC B {, where { is the embedding U ! f0g f1g U Œ0; 12 U and ‰ is the flow of the vector field @ on Œ0; 12 U . Then by Cartan’s formula we have @sC d h.sC ;1/ ! D { ‰sC LsC .h !/ dsC @ @ D { ‰sC ³ d.h !/ C d ³h ! @sC @sC @ @ D { ‰sC ³ @C .h !/ C @C ³ h ! @sC @sC @ @ C ³ @ .h !/ C @ ³ h ! : @sC @sC

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Because the maps {, ‰sC , h are para-holomorphic, the respective pull-back maps commute with @˙ and preserve the types of differential forms. Hence the sum of the terms containing @ must be zero, since these terms are of degree .C.p 1/; .q C 1// and the rest is of degree .Cp; q/. Further, by assumption we have @C .h !/ D h @C ! D 0 and thus the expression reduces to

@ d h ! D @C { ‰sC ³ h ! dsC .sC ;1/ @sC

:

Inserting this in (1) yields Z

! D @C

0

1

{ ‰sC

@ ³ h ! dsC : @sC

The proof of the second statement is completely analogous.

Para-complex version. There is also a para-complex version of the Dolbeault sequence defined on the para-complexified cotangent bundle T M c WD T M ˝R C : The C -linear extension of J W T M ! T M to an endomorphism field on T M c V V yields a decomposition T M c D .1;0/ T M ˚ .0;1/ T M into the ˙j -eigenbundles V.1;0/ T M D f˛ C jJ˛ W ˛ 2 T M g; V.0;1/ T M D f˛ jJ˛ W ˛ 2 T M g: This decomposition extends to a bigrading of C -valued exterior and differential forms: M V.p;q/ Vk c T M D T M; pCqDk k

c

.M / D

M

.p;q/ .M /:

pCqDk

If J is integrable, then the C -linear extension of the exterior derivative splits as N where d c D @ C @, @ W .p;q/ .M / ! .pC1;q/ .M /; @N W .p;q/ .M / ! .p;qC1/ .M /: Definition 1.2.9. The elements of .p;q/ .M / are called para-complex differential N forms of degree .p; q/. A @-closed .p; 0/-form is called para-holomorphic. N @@ N D 0, so for any para-complex manifold the From d 2 D 0 we get @2 D @N 2 D @@C N @ can be used to define para-complex versions of the Dolbeault sequence. operators @, The real and para-complex versions are related in the following way:


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Proposition 1.2.10. For a para-complex manifold M , there is an (R-linear) isomorphism V V V.Cp;q/ ' T M .Cq;p/ T M ! .p;q/ T M; 1j 0 1Cj C ; .; 0 / 7! 2 2 such that the following diagram commutes: '

.Cp;q/ .M / .Cq;p/ .M / @ @C

.Cp;.qC1// .M / .C.qC1/;p/ .M /

/ .p;q/ .M /

'

@N

/ .p;qC1/ .M /.

V Proof. The fact that the image of ' is contained in .p;q/ T M is an easy consequence 2 1j of 1˙j D 1˙j , 1Cj D 0 and of the equations 2 2 2 2 1Cj a dz ; 2 1Cj a 1Cj a dz D d zN ; 2 2

1Cj 2

a dzC D

1j a dz ; 2 1j a 1j a dzC D d zN 2 2 1j 2

a dz D

a for local holomorphic coordinates .z˙ /kD1:::n and z a D ' is clearly a bijection, with the inverse given by

1Cj a zC 2

C

1j a z . 2

The map

! 7! .Re.!/ C Im.!/; Re.!/ Im.!// : For the commutativity of the diagram, it is sufficient to consider functions, i.e. the f D 1jJ d 1Cj f D case that f; f 0 2 .C0;0/ . Then we calculate @N 1Cj 2 2 2 1Cj 1J df D 1Cj @ f and in the same way @N 1j f 0 D 1j @C f 0 . 2 2 2 2 2 N Corollary 1.2.11. For U D UC U with UC and U contractible, any @-closed .p;q/ N .U /, q 1, is @-exact. form ! 2

1.3 Para-hermitian metrics and para-Kähler manifolds We recall that a pseudo-Euclidean vector space is a real vector space V equipped with a nondegenerate symmetric bilinear form g W V V ! R. If W V is a subspace with orthogonal complement W ? WD fv 2 V W g.w; v/ D 0 for all w 2 W g, then W is called • nondegenerate, if W ? \ W D f0g, • degenerate, if W ? \ W ¤ f0g, • totally degenerate, if W ? W , i.e. if gjW W D 0.

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Fact 1.3.1. A subspace W is totally degenerate if and only if it consists only of degenerate vectors. Proof. This follows from the polarization identity 4g.w; w 0 / D g.w C w 0 ; w C w 0 / g.w w 0 ; w w 0 /. Definition 1.3.2. Let .V; J / be a para-complex vector space. A pseudo-Euclidean scalar product g on V is said to be compatible with J or para-hermitian, if J is g-skew, i.e. if g.J ; / C g.; J / D 0. Since J 2 D 1, this is equivalent to J being an anti-isometry, i.e. g.J ; J / D g. In particular, g has real signature .n; n/ (where n D 12 dimR V ), and the subspaces V ˙ are totally degenerate. On V D C n the standard para-hermitian scalar product is given by X g .z 1 ; : : : ; z n /; .w 1 ; : : : ; w n / D Re za wa : a

The group of C -linear automorphisms of C n preserving g is called the para-unitary group U .n/ WD fA 2 GL.n; C / W A A D 1n g (where A WD ANt ): It is easy to see that ˚ U .n/ D 1Cj BC 2

B t W B 2 GL.n; R/ (where B t WD .B 1 /t ): B C 1j B t D 1Cj det.B/ C 1j det.B/1 ; By Lemma 1.1.6 we have det C 1Cj 2 2 2 2 this proves 1j 2

U .n/ Š GL.n; R/; SU .n/ Š SL.n; R/: Under the identification C n Š Rn Rn , the standard para-hermitian scalar product corresponds to the symmetric bilinear form given by the matrix 0n 1n : E WD 1n 0n ˚ The isometry group is X 2 GL.2n; R/ W X t EX D E Š O.n; n/, and U .n/ is identified with the subgroup ³ ² B 0n W B 2 GL.n; R/ : 0 B t n

Proposition 1.3.3. Every para-hermitian vector space has a unitary basis, i.e. a free C -basis .ea /aD1:::n in which the metric is given by g.ea ; eb / D ıab . Proof. Let V be a para-hermitian vector space, dimC .V / D n. V is nondegenerate with respect to g, so there is a vector e1 , such that g.e1 ; e1 / ¤ 0. After multiplying


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p p e1 by g.e1 ; e1 / (if g.e1 ; e1 / > 0) or by j g.e1 ; e1 / (if g.e1 ; e1 / < 0), we may assume that g.e1 ; e1 / D 1. It follows that e1 … V ˙ , so e1 is regular and C e1 is a 1-dimensional para-complex subspace. The orthogonal complement .C e1 /? is an .n 1/-dimensional para-complex subspace, which is nondegenerate, so we can proceed by induction. Definition 1.3.4. A para-Kähler manifold is an almost para-complex manifold .M; J / equipped with a compatible pseudo-Riemannian metric g, such that DJ D 0, where D denotes the Levi-Civita connection. Equivalently, a para-Kähler manifold is a pseudo-Riemannian manifold with holonomy group GL.n; R/ O.n; n/. From the parallelity of J it follows that NJ D 0 (in particular, .M; J / is a paracomplex manifold) and that the .C1; 1/-form ! WD g.; J / is also parallel, and therefore closed. The converse is also true: If NJ D 0 and d! D 0, then DJ D 0 (see [CMMS1, Theorem 1]).

1.4 Para-quaternionic Kähler manifolds Definition 1.4.1. A para-quaternionic structure on a finite-dimensional vector space V is a three-dimensional subalgebra Q gl.V / with a basis I; J; K that satisfies the para-quaternionic relations I 2 D 1;

J 2 D K 2 D C1;

IJ D JI D K:

Remark. From IJ D JI we see that I maps the C1-eigenspace of J isomorphically to the 1-eigenspace, so these eigenspaces have the same dimension. In other words, J (as well as K) is a para-complex structure in the sense of Definition 1.1.1. Para-quaternionic vector spaces are identified with free right-modules over H , where H D R ˚ i R ˚ j R ˚ kR, i 2 D j 2 D k 2 D 1; ij D j i D k, is the algebra of para-quaternions (also called split quaternions) and I; J; K correspond to the multiplication by i; j; k from the right. Accordingly, H n is the space of paraquaternionic column vectors with scalar multiplication from the right. Multiplication by para-quaternionic matrices from the left is H -linear. The identification H D .R ˚ i R/ ˚ j.R ˚ i R/ D C ˚ j C shows that any paraquaternionic vector space is a complex vector space of the double dimension. A map is H -linear if and only if it is C-linear and commutes with the right multiplication by j (which is a C-antilinear involution). In the same way, H D .R ˚ j R/ ˚ i.R ˚ j R/ D C ˚ i C shows that any paraquaternionic vector space is a para-complex vector space of the double dimension. A map is H -linear if and only if it is C -linear and commutes with the right multiplication by i (which is a C -antilinear complex structure).

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Definition 1.4.2. Let .V; Q/ be a para-quaternionic vector space. A pseudo-Euclidean metric g on V is said to be compatible with Q or para-quaternion hermitian, if the elements of Q are g-skew. The standard para-quaternion hermitian metric on H n is given by X ha h0a : g..h1 ; : : : ; hn /; .h01 ; : : : ; h0n // WD Re The isometry group Sp .n/ WD fA 2 GL.n; H / W A A D idn g is called the para-symplectic group. By the same argument as in Proposition 1.3.3, we see that any two para-quaternion hermitian vector spaces of the same dimension are isometric. Proposition 1.4.3. Sp .n/ Š Sp.2n; R/. Proof. Let 2n D 10n 10n be the standard symplectic structure on R2n , then H n Š .R4n ; I; J; K; g/, where 1 0 0 12n ; J D 2n ; K D JI; I D 0 0 12n 12n 2n 0 : g D !I .; I /; !I D 0 2n Thus, Sp .n/ is identified with the endomorphism of R4n that commute with I and J and preserve !I . This shows that ³ ² A 0 W A 2 Sp.2n; R/ : Sp .n/ Š 0 A Definition 1.4.4. A para-quaternionic Kähler manifold.M; g; Q/ is a pseudo-Riemannian manifold .M; g/ endowed with a g-compatible para-quaternionic structure Q End.TM / which is parallel with respect to the Levi-Civita connection. It is not difficult to see (cf. [AC1, Proposition 1], [AC2, Proposition 9]) that the normalizer of QR4n D RI C RJ C RK in O.2n; 2n/ is G0 [ J G0 , where G0 D Sp .1/ Sp .n/ is the subgroup of GL.4n; R/ D GLR .H n / generated by left multiplication by elements of Sp .n/ and right multiplication by elements of Sp .1/ (which is just the group of unit para-quaternions), and where J denotes the right N D 1). So we have: multiplication by j (which is an anti-unit, i.e. jj Proposition 1.4.5. A pseudo-Riemannian manifold is para-quaternionic Kähler if and only if Hol.M / .Sp .1/ Sp .n// [ J .Sp .1/ Sp .n//:


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Note. In the definition in [AC1], it is also required that M be strongly oriented, i.e. that Hol.M / SOC .2n; 2n/. Of course, this is always fulfilled, if M is simply connected. It can be shown [AC1, Theorem 3] that any para-quaternionic Kähler manifold is Einstein and that its curvature tensor admits a decomposition RD

scal R0 C W; 4n.n C 2/

where R0 is the curvature tensor of H Pn and W is an algebraic curvature tensor of type sp .n/. It follows that, if scal ¤ 0, then sp .1/ hol (see [AC1, Corollary 1]).

2 Para-Kähler submanifolds of para-quaternionic Kähler manifolds 2.1 Para-complex symplectic and contact structures The classical Darboux theorem states that all 2n-dimensional symplectic manifolds are locally isomorphic to T Rn equipped with the canonical symplectic form dpa ^ dq a . As shown in this paragraph, the analogous statement in the para-holomorphic setting also holds. 2.1.1 Para-complex symplectic vector spaces Definition 2.1.1. Let .V; J / be a para-complex vector space. A para-complex symplectic form is an anti-symmetric C -bilinear map ! W V V ! C which is nondegenerate, i.e. .!.X; / D 0 () X D 0/. V Equivalently, a para-complex symplectic form is an element ! 2 .2;0/ V , such V that the map ! W V .1;0/ ! .1;0/ V is an isomorphism. Proposition 2.1.2. Let .V; J / be a para-complex vector space and ! a para-complex symplectic form. Then there is a basis fZ a ; Wb ga;bD1;:::;n of the C -module V such that !.Z a ; Wb / D ıba , !.Z a ; Z b / D 0 D !.Wa ; Wb / for a; b 2 f1; : : : ; ng. Proof. Choose a regular vector Z 1 2 V . We claim that there exists W1 2 V such that !.Z 1 ; W1 / D 1: Otherwise the image of W 7! !.Z 1 ; W / would be contained in one of the maximal ideals .1 ˙ j /R C . But then !..1 j /Z 1 ; / 0, and therefore .1 j /Z 1 D 0 by the nondegeneracy of !, in contradiction to the regularity of Z 1 . From !.Z 1 ; W1 / D 1 it now follows that W1 is regular and that Z 1 and W1 are C -linearly independent. Therefore, spanC .Z 1 ; W1 / is a paracomplex subspace, as well as the symplectic complement .spanC .Z 1 ; W1 //?! WD

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fX 2 V W !.Z 1 ; X/ D !.W1 ; X / D 0g. The latter is again nondegenerate with respect to !, and the proposition follows by induction. Corollary 2.1.3. An antisymmetric C -bilinear 2-form ! is nondegenerate if and only V if ! n 2 .2n;0/ V is regular (where n D dimC .V /=2). Proof. If the form ! is nondegenerate and fZ a ; Wb ga;bD1;:::;n a basis as above, then ! n .Z 1 ; W1 ; : : : ; Z n ; Wn / D nŠ, in particular .1 ˙ j /! n ¤ 0. n Let fB1 ; : : : ; B2n g be an arbitrary C -basis Conversely, suppose P a that ! is regular. of V and X D a x Ba 2 ker.!/, x a 2 C . We have to show that X D 0. By the skew-symmetry of ! n we have 0 D ! n .B1 ; : : : ; Ba1 ; X; BaC1 ; : : : ; B2n / D x a ! n .B1 ; : : : ; B2n /: But by assumption we have ! n .B1 ; : : : ; B2n / 2 C , so x a D 0.

2.1.2 Para-holomorphic symplectic structures on manifolds Definition 2.1.4. Let .M; J / be a para-complex manifold. A para-holomorphic symplectic structure is a nondegenerate closed form ! 2 .2;0/ M . Lemma 2.1.5. Let .M; J / be a para-complex manifold and !0 ; !1 two para-holomorphic symplectic forms on M , which coincide in a point p 2 M , !0 .p/ D !1 .p/. Then there is a para-holomorphic diffeomorphism ˆ defined on a neighbourhood of p, such that ˆ.p/ D p and ˆ !1 D !0 . We want to apply Moser’s method of constructing ˆ as time-1-map of a paraholomorphic flow, i.e. a flow ˆX W .; / U ! U of a (time-dependent) vector field Xs , such that .ˆX s / J D J for any s 2 .; /. The vector field Xs then needs to be an infinitesimal automorphism of the para-complex structure, i.e. LXs J D 0. Remark (on notation). If Xs is a time-dependent vector field, then ˆX s; denotes the d X X X flow of X starting at time s, i.e. dt ˆs;t .p/ D X.ˆs;t .p//, ˆs;s .p/ D p. Instead of X ˆX 0;s we will also write ˆs . The Lie-derivative of a (possibly also time-dependent) tensor Ts is defined as d j tD0 ˆs;sCt TsCt : dt For functions, differential forms and vector fields this yields LXs Ts WD

LXs fs D Xs .fs / C

@ fs ; @s

LXs ˛s D d.Xs ³ ˛s / C Xs ³ d˛s C LXs Ys D ŒXs ; Ys C

@ Ys : @s

@ ˛s ; @s


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Lemma 2.1.6. Let .M; J / be a para-complex manifold. Then LXs J D 0 if and only if the .1; 0/-vector field 12 .Xs C jJXs / is para-holomorphic. a a a Proof. Using P para-holomorphic coordinates /aD1:::n we write Xs D P.z a D x aC jy P 1 @ @ a @ a @ C , .X C jJX/ D . C j / . s s @z a The equations J @x a D a s @x a a s @y a 2 @ @y a

and J @x@a D

@ @y a

yield

@ .LXs J / @x a

@ @ D ŒXs ; J a J Xs ; a @x @x b X @b @ @ @ sb @ s @bs s D C : a a a a b @x @y @x @x @y @y b b

The vanishing of this expression is equivalent to the para-Cauchy–Riemann equations for the function . sb C jbs /. The calculation for .LX J /.@yk / is analogous. Remark 2.1.7. LX J D 0 implies LJX J D 0 and ŒX; JX D 0. If X is timeindependent, the flows ˆX and ˆJX commute, so ˆX can be extended to a flow of Z D 12 .X C jJX/, depending on a para-complex parameter s C jt 2 C , by Z d X JX Z ˆZ sCjt .p/ WD ˆs .ˆ t .p//, fulfilling dz ˆz .p/ D Z ˆz .p/ . Proof of Lemma 2.1.5. Let ! t D !0 C t .!1 !0 /; 0 t 1. We have ! t .p/ D !0 .p/ D !1 .p/ for all t , so by the compactness of Œ0; 1 there is a neighbourhood of p, such that in this neighbourhood ! t is nondegenerate for all t 2 Œ0; 1. We will now construct a 1-parameter family ˆ t of diffeomorphisms (defined on a neighbourhood of p), such that ˆ t .p/ D p; ˆt ! t D !0 :

(2)

d ! t D !1 !0 is closed by assumption, it locally is the differential of Since ! 0 WD dt 0 a 1-form, ! D d˛. Further, we may assume that ˛.p/ D 0 (by adding the differential N .0;1/ / D .! 0 /.0;2/ D 0, we can apply Corollary 1.2.11 and of a function). Since @.˛ N Replacing ˛ with ˛ dˇ, find a (locally defined) function ˇ, such that ˛ .0;1/ D @ˇ. .1;0/ N we obtain ˛ 2 M . Further, by construction, @˛ D ! 0.1;1/ D 0, i.e. ˛ is paraholomorphic. Thus the equation Z t ³ ! t D ˛ defines a time-dependent vector field Z t , which is para-holomorphic since ˛ and ! t1 are, the latter by Cramer’s x t is a 1rule. According to Remark 2.1.7, the flow of X t D 2Re.Z t / D Z t C Z parameter family ˆ t of para-holomorphic diffeomorphisms. For all t 2 Œ0; 1 we have X t .p/ D 0, thus ˆ t .p/ D p, and ˆ t is defined on a neighbourhood of p. By writing LX .U C jV / D ŒX; U C jV and extending the definition of the Lie bracket and Lie derivative C -linearly to the tensor algebra on T C M , we calculate (using

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Matthias Krahe

x t ³ ! t D 0): Cartan’s formula and Z

d d ˆ ! t D ˆt d.X t ³ ! t / C X t ³ d! t C ! t dt t dt d D ˆt d.Z t ³ ! t / C 0 C ! t dt D ˆt .d˛ C ! 0 / D 0:

Integration then yields (2).

Theorem 2.1.8. Let .M; J / be a 2n-dimensional para-complex manifold and let ! be a para-holomorphic symplectic form. Then in the neighbourhood of any p 2PM , there is a para-holomorphic coordinate system .z a ; wb /a;bD1;:::;n such that ! D a dz a ^ dwa . Proof. In a sufficiently small neighbourhood of p 2 M , choose an arbitrary paraholomorphic coordinate system .zQ a ; wQ b /. From Proposition 2.1.2 it follows that by n n changing P a the coordinates by a C -linear map C ! C , we can arrange thatP!.p/a D d zQ .p/ ^ d wQ a .p/. Now apply the above lemma to !0 D ! and !1 D d zQ ^ d wQ a . Then .z a WD zQ a B ˆ; wb WD wQ b B ˆ/ is the desired coordinate system. 2.1.3 Para-holomorphic contact forms Definition. Let M be a .2n C 1/-dimensional para-complex manifold. A paraholomorphic contact form is a para-holomorphic 1-form ˛ 2 .1;0/ .M /, such that .˛ ^ .d˛/n / 2 .2nC1;0/ .M / is regular. Remark. This condition is equivalent to ker.˛/ being a 2n-dimensional para-holomorphic distribution in the sense of [AC2], Chapter 2.2 (see Proposition 3) and d˛jker.˛/ being nondegenerate (cf. Corollary 2.1.3). Lemma 2.1.9. There is a unique vector field R˛ , called Reeb vector field, such that R˛ ³ d˛ D 0; R˛ ³ ˛ D 1: Proof. At each p 2 M , d˛.p/ has rank 2n, so ker.d˛.p// is 1-dimensional. With similar reasoning as in the proof of Proposition 2.1.2, there is a (unique) vector R.p/ 2 ker.d˛.p// which normalizes ˛.p/: Otherwise, either .1 C j /˛.p/ or .1 j /˛.p/ would be zero on ker.d˛.p//, and therefore .1 ˙ j /˛ ^ .d˛/n D 0. Lemma 2.1.10. Let M be a para-complex manifold and let ˛0 ; ˛1 be two para-holomorphic contact forms on M . Let p 2 M and ˛0 .p/ D ˛1 .p/, d˛0 .p/ D d˛1 .p/. Then there is a para-holomorphic diffeomorphism ˆ defined on a neighbourhood of p, such that ˆ.p/ D p and ˆ ˛1 D ˛0 .

515


Proof. As in the proof of Lemma 2.1.5, let ˛ t D ˛0 C t .˛1 ˛0 /; 0 t 1. ˛ t is a contact form for all t 2 Œ0; 1 in a neighbourhood of p. We will construct a 1-parameter family ˆ t of diffeomorphisms, such that ˆ t .p/ D p; ˆt ˛ t D ˛0 :

(3)

As before, if ˆ t is defined as the flow of a time-dependent vector field X t D 2Re.Z t /, with Z t a para-holomorphic vectorfield to be specified below, then (3) follows from d d ˆ t ˛ t D Z t ³ d˛ t C d.Z t ³ ˛ t / C ˛ t : dt dt We decompose Z t D r t R˛ t C W t with W t tangential to ker.˛ t / and obtain 0D

d ˛ t D 0: (4) dt Now d˛ t is nondegenerate on ker.˛ t / and zero on span.R˛ t /, so for given r t there is a vector field W t which solves (4), if and only if r t satisfies W t ³ d˛ t C dr t C

d (5) ˛ t D 0: dt This equation can be solved as follows: Let ‰ t;z denote the flow of R˛ t (see Remark 2.1.7) and let N be a 2n-dimensional para-complex submanifold of M such that p 2 N and such that N is transversal to R˛ t in a neighbourhood of p (note that R˛ t .p/ D R˛0 .p/, so we can take the same N for all t ). Then .z; q/ 7! ‰ t;z .q/ is an isomorphism from a neighbourhood of .0; p/ in C N onto a neighbourhood of p in M , and the equation (5) reads: R˛ t ³ dr t C

d d r t .‰ t;z .q// D R˛ t .‰ t;z .q// ³ ˛t dz dt The right-hand side is para-holomorphic in z, so a solution for r t exists by Lemma 1.2.6. d Moreover, by construction and because of dt ˛ t .p/ D 0, we have dr t .p/ D 0 D r t .p/, so that Z t .p/ D 0, hence ˆ t .p/ D p, and ˆ t is well defined for all t 2 Œ0; 1 in a neighbourhood of p. Theorem 2.1.11. Let M be a .2n C 1/-dimensional para-complex manifold and let ˛ be a para-holomorphic contact form. Then in the neighbourhood of any p 2 M , there isP a para-holomorphic coordinate system .w0 ; z a ; wb /a;bD1;:::;n such that ˛ D dw0 wa dz a . Proof. Let .wQ 0 ; zQ a ; wQ b /a;bD1;:::;n be a coordinate system such that . @Qz@a .p/; @w@Q b .p//

@ .p/ @w Q0

D R˛ .p/

is a symplectic basis for d˛.p/jker.˛.p// . By adding and such that a constant, we can assume that p isP mapped to zero. Then it follows that ˛.p/ D P a .d wQ 0 wQ a d zQ /.p/ and d˛.p/ D d zQ a .p/^d wQ a .p/. Now apply Lemma 2.1.10

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to ˛0 D ˛, ˛1 D d wQ 0

P

wQ a d zQ a and set z D zQ B ˆ, v a D vQ a B ˆ, wb D wQ b B ˆ.

2.2 Para-Kähler submanifolds and twistor lift Let .M; Q; g/ be a para-quaternionic Kähler manifold. Definition 2.2.1. The twistor bundle W Z ! M is the bundle of para-complex structures in Q End.TM /: Z WD fq 2 Q W q 2 D idTM g: The fibres of Z are one-sheeted hyperboloids: If Ip , Jp , Kp is a basis for Qp as in Definition 1.4.1, then Zp D frIp C sJp C tKp W r 2 C s 2 C t 2 D 1g. This can be identified with the homogenous space SO.2; 1/=SO.1; 1/ D Sp .1/=U .1/. The Levi-Civita connection on TM induces a splitting of T Z into a vertical and a horizontal distribution T Z D V ˚ H D ker.d / ˚ H D . With respect to this splitting, we can define an almost complex structure JZ WD JV C JH , where at each point z 2 Z, JH .z/ is the para-complex structure given by z 2 Q.z/ End.T.z/ M / Š End.Hz /, and where JV .z/ is a SO.2; 1/-invariant para-complex structure on the homogenous space Z.z/ (see [AC2, Proposition 6]). t Likewise, we can define a one-parameter family gZ D tgV C gH , t 2 R n f0g, of para-hermitian metrics, such that the map W Z ! M is a pseudo-Riemannian submersion (see [AC2, Proposition 7]), and where 2gV is the Killing form on so.2; 1/. Theorem 2.2.2 ([AC2, Theorem 2 and 3]). Let Z be the twistor space of a paraquaternionic Kähler manifold with nonzero scalar curvature. Then (1) The almost para-complex structure JZ is integrable. 4n.nC2/=scal (2) gZ is a para-Kähler-Einstein metric. (3) The horizontal distribution H T Z is a para-holomorphic contact structure.

Definition 2.2.3. Let .M; Q; g/ be a para-quaternionic Kähler manifold. A paracomplex submanifold of M is a pair .N; J /, where N M is a smooth submanifold and J a parallel section of ZjN D fq 2 QjN W q 2 D 1g, such that J.TN / D TN , and such that the ˙1-eigenspaces TN ˙ have the same dimension. A para-complex submanifold is called para-Kähler submanifold, if the induced pseudo-Riemannian metric gN WD gjTN is nondegenerate. Remark. It follows that the almost complex structure JN WD J jTN is integrable, since it is parallel with respect to a torsion-free connection. Remark. Let N M be a para-Kähler submanifold. Then the map f W .N; gN / ! .M; g/ is an isometric immersion which fulfills df B JN D J B df;

(6)


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where J is a parallel section of f Q f End.TM /. Thus, in the language of Section 3.2, the map f is isotropic para-pluriharmonic, see equation (10) (with ˆ WD e J ). Note that in Section 3.2, M is a symmetric space, but even without that assumption, equation (6) implies that f is para-pluriharmonic: Because JN and J are parallel, we have .Ddf /JN D D.df JN / D D.J df / D J.Ddf /, and therefore .Ddf /.X; JN Y / .Ddf /.Y; JN X/ D J ..Ddf /.X; Y / .Ddf /.Y; X // D 0: It follows that .Ddf /.C1;1/ D 0, and that f is para-pluriharmonic by Proposition 3.1.3. The following theorem by Alekseevsky and Cortés relates the para-complex submanifolds of a para-quaternionic Kähler manifold to the horizontal submanifolds of its twistor space: Theorem ([AC2, Theorem 4]). Let .M; Q; g/ be a para-quaternionic Kähler manifold with nonzero scalar curvature and let W Z ! M be its twistor bundle. (1) If .N; J / is a para-complex submanifold of M , then F W N ! Z, F .p/ WD Jp defines an embedding of N into Z such that BF D idN . The image Nz D F .N / is a horizontal para-complex submanifold of Z (called the canonical lift of N ). (2) Conversely, if Nz Z is a para-complex horizontal submanifold such that W Nz ! .Nz / M is a diffeomorphism, then its projection .N WD .Nz /; J WD d B JZ B d 1 / is a para-complex submanifold of M . (3) Moreover, N is a para-Kähler submanifold, if and only if the one-parameter t family of metrics gZ is nondegenerate on Nz Z. Because the horizontal distribution H T Z is a contact distribution, it follows that the dimension of N cannot be greater than half the dimension of M . The maximal (i.e. of maximal dimension) para-complex submanifolds of M correspond to Legendrian submanifolds (i.e. submanifolds tangent to the contact distribution with maximal possible dimension) of Z. Combining this with the results of the preceding section, we can show that the maximal para-complex submanifolds are locally given by para-holomorphic functions: Theorem 2.2.4. Let M be a para-quaternionic Kähler manifold of real dimension 4n, and let W Z ! M be its twistor bundle. Then at any point q 2 Z there exist local coordinates ! Uz Uw C 2nC1 .z 1 ; : : : z n ; w0 ; : : : ; wn / W U

defined on aPneighbourhood U of q, such that the horizontal distribution is H D ker.dw0 wa dz a /.

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For any para-holomorphic function h W Uz ! C , the equations w0 D h.z 1 ; : : : ; z n /; wa D @h=@z a define a horizontal submanifold Nz Z (provided that .w0 ; : : : ; wn / 2 Uw ), and the projection .Nz / is a para-complex submanifold of M . Any maximal para-complex submanifold N M is locally of this form with respect to a locally finite covering of Z by such coordinate domains. Proof. Since H is a para-holomorphic contact distribution, the existence of local coordinates as above follows from Theorem 2.1.11. Since Z is paracompact, it is locallyP finitely covered by such coordinates. Clearly, with respect to the contact form dw0 wa dz a , a section Uz ! Uz Uw is horizontal, if and only if wa D @w0 =@z a . It remains to show that, given a horizontal n-dimensional submanifold Nz U , we can write it as such a section. Let .z 1 ; : : : ; z n ; w0 ; : : : ; wn / be a system of local coordinates as above. We may assume that Uz Uw is a sufficiently small neighbourhood of 0 2 C 2nC1 , so that the 2n differential forms dz 1 ; : : : dz n ; dw1 ; : : : ; dwn are pointwise C -linearly independent on H jU . In other words, the 2 2n real-valued differential 1 n forms dz˙ ; : : : ; dz˙ , dw1˙ ; : : : ; dwn˙ , are R-linearly independent on H ˙ T Z ˙ . Therefore a subset of 2n of these forms (n for each sign) is R-linearly independent on T Nz ˙ . After exchanging some of the z˙ - and w ˙ -coordinates (note that dw0˙ 1 n w ˙ dz˙ D d.w0˙ w ˙ z˙ /Cz˙ dw ˙ ), we can assume that dz˙ ; : : : ; dz˙ are linearly ˙ independent on T Nz , so by the inverse function theorem, the z-coordinates form a local coordinate system on Nz . 2.2.1 Example: The twistor space C P2 nC1 ! H Pn . Let C Pn be the space of nondegenerate para-complex lines in C nC1 , ˚ P C Pn D .zo ; : : : ; zn / 2 C nC1 W naD0 zN a za ¤ 0 =C : Accordingly, let

˚ P H Pn D .h0 ; : : : ; hn / 2 H nC1 W naD0 hN a ha ¤ 0 =H :

In each para-quaternionic line there is a unit element, which can be completed to a unitary basis of H nC1 (see the proof of Proposition 1.3.3), so Sp .n C 1/ acts transitively on H Pn ; therefore we can identify the latter with the coset space Sp .n C 1/=Sp .n/ Sp .1/. Consider the identification H nC1 D C nC1 ˚ i C nC1 Š C 2nC2 and note that .z C iw/.z C iw/ D zN z C ww N for z; w 2 C , so the scalar products (and the notions of nondegeneracy) on H nC1 and C 2nC2 agree. In particular, there is a fibration C P2nC1 ! H Pn , which is the fibration of homogenous spaces Sp .n C 1/ Sp .n C 1/ ! : Sp .n/ U .1/ Sp .n/ Sp .1/


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P We want to give an explicit expression for the contact form: Let ! D naD0 dz a ^ P n @ dwa be the standard symplectic form on C 2nC2 and let Xrad D aD0 .za @z a C P n wa @w@ a / be the radial vector field. The 1-form ˛ 0 WD Xrad ³ ! D aD0 .z a dwa wa dz a / is homogenous of degree one and vanishes in the radial direction, so it yields a holomorphic form ˛ on C P2nC1 which is well-defined up to multiplication by a non-vanishing function. The kernel of ˛ 0 in C 2nC2 P is the orthogonal complement .Xrad i/? of the vertical vector field Xrad i D i naD0 .zN a @w@ a wN a @z@a /, and thus ker.˛/ is indeed the horizontal distribution in C P2nC1 . On the subset U0 WD fŒz 0 ; w0 ; : : : z n ; wn W z0 2 C g C P2nC1 we introduce affine coordinates Œz 0 ; : : : ; wn 7! .z 0 /1 .w0 ; z 1 ; : : : ; wn /, thereby identifying U0 with the hyperplane fz 0 1g C 2nC2 . In these coordinates we have ˛jz 0 1 D dw0 C

n X

.z a dwa wa dz a /:

aD1 a

Coordinates .Qz ; wQ b / as in Theorem 2.1.11 are given by X z a wa ; wQ 0 D w0 C zQ a D z a ; wQ a D 2wa : Using these coordinates, we can construct the following horizontal immersions C n U ! C P2nC1 :

X @h=2 1 n a 1 @h=2 n @h=2 z Wz W W W z W ; .z ; : : : ; z / 7! 1 W h @z a @z 1 @z n where h W U ! C is a holomorphic function. Accordingly, the para-complex immersions U ! H Pn are

X @h=2 @h=2 @h=2 n za a / W z1 C i W W z C i : .z 1 ; : : : ; z n / 7! 1 C i.h @z @z 1 @z n

3 Para-pluriharmonic maps into symmetric spaces 3.1 Para-pluriharmonic maps 3.1.1 Definitions Definition 3.1.1. Let .N; gN /; .M; gM / be pseudo-Riemannian manifolds. A map f W N ! M is called harmonic if it is a critical point of the energy functional under variations with compact support Z 1 tr g .f gM / d volN : E.f / WD 2 N N

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The first variation of the energy is given by Z @f t d E.f t / D ; trgN .Ddf // C divX d volN ; gM . dt @t N where D is the connection on End.TN; f TM / induced by the Levi-Civita connections on TN and TM and where X is the vector field on N defined by gN .X; / D t gM . @f ; df /. Therefore, f is harmonic if and only if @t trgN .Ddf / D 0: If N is 2-dimensional, then the energy functional is invariant under conformal changes of the metric gN , so the definition depends only on the conformal class of gN . But in two dimensions, the conformal classes of definite metrics correspond to the complex structures (up to sign), and the conformal classes of indefinite metrics correspond to the para-complex structures (any indefinite scalar product in two dimensions has two null directions, which we take as the ˙1-eigenspaces of the para-complex structure). The former case leads to the definition of pluriharmonic maps, which have already been studied (see [BR], [ET2]), the latter case to para-pluriharmonic maps: Definition 3.1.2. Let .N; J; g/ be a para-Kähler manifold and let .M; gM / be a pseudo-Riemannian manifold. A smooth map f W N ! M is called para-pluriharmonic if it is harmonic along every 1-dimensional para-complex submanifold. Proposition 3.1.3. For a map f W N ! M with N , M as above the following conditions are equivalent: (1) f is para-pluriharmonic. (2) .Ddf /.C1;1/ D 0, i.e. Ddf vanishes on T C N ˝ T N . (2’) .Ddf /.1;1/ D 0, i.e. .Ddf /c vanishes on T .1;0/ N ˝ T .0;1/ N . Proof. .2/ H) .1/: Let L N be a para-complex curve and z˙ D x ˙ y a pair of holomorphic coordinates on L. The Levi Civita connection on L differs from the one on N only by the (traceless) second fundamental form, and at any point p 2 L the pseudo-Riemannian metric gjTp L is proportional to dzC _dz D dx ˝dx dy ˝dy. @ @ @ @ ; @x /.Ddf /. @y ; @y / D .Ddf /. @z@C ; @z@ / D 0. So we have tr g .Ddf / .Ddf /. @x

.1/ H) .2/: Any pair of vectors Z˙ 2 Tp˙ N is tangent to a para-complex curve through p. The same calculation as before shows that .Ddf /.ZC ; Z / tr g .Ddf / D 0. .2/ () .20 / is immediate from _.1;1/ .TN / D _.C1;1/ .TN / ˝ C (cf. Proposition 1.2.10).

3.1.2 A remark on complexification. We observe that if I is a complex structure on V , then I C is a C-linear complex structure on V ˝ C, and J C D i I C is a para-complex structure. This leads to the idea to construct para-pluriharmonic maps


521

in the following way: Start with a pluriharmonic map f W N ! M , extend this to map f C W N C ! M C and then restrict it to an alternate real form Nz of N C , such that iINC is a para-complex structure of T Nz . – The problem with this is that one cannot z of M C . expect the image f .Nz / to be contained in a totally real submanifold M Toillustrate this, we consider N D R2 with the standard complex structure I D 0 1 and M D R: A function f W R2 ! R is harmonic, if and only if 1 0 d.df B I / D 0: Then the complex analytic continuation f C W C 2 ! C of f is harmonic and paraharmonic, d.df C B I C / D d.df C B iI C / D 0: Restricting this to .C ; j / Š .Re1 ˚ i Re2 ; iI /, we obtain a para-harmonic function f C jRe1 ˚iRe2 W Re1 ˚ i Re2 ! C: However, the image of f C .Re1 ˚ i Re2 / is usually not contained in a onedimensional real subspace, e.g. if f .x; y/ D x C y, then f .x; iy/ D C. 3.1.3 Pseudo-Riemannian symmetric spaces. Recall that a pseudo-Riemannian symmetric space is a pseudo-Riemannian manifold M , such that for each point p 2 M ! M (the geodesic reflection) whose differential there exists an isometry sp W M at p is dsp .p/ D idTp M . We will always assume that M is connected. It follows that the identity component of the isometry group G WD Iso0 .M / acts transitively on M . Therefore M can be identified with the coset space G=K, where K D fg 2 G W g o D og is the stabilizer subgroup of a chosen base point o 2 M . The conjugation by so is an involutive automorphism W G ! G, the Cartan involution. It holds that G0 K G , where G denotes the fixed-point set of . The differential d e W g ! g defines the AdK -invariant Cartan decomposition g D ker.d e id/ ˚ ker.d e C id/ DW k ˚ m. In fact, k is the Lie algebra of K, so the differential of G ! M annihilates k and maps m isomorphically to To M . Conversely, let G be a Lie group, an involution of G, K G a closed subgroup such that G0 K G and suppose that there exists an AdK -invariant pseudo-Euclidean scalar product on m WD ker.d e C idg / (e.g. the Killing form, if it is nondegenerate) for which d is an isometry. Then M D G=K has the structure of a pseudo-Riemannian symmetric space with the geodesic reflection given by sgK .hK/ D g .g 1 hK/. The group G acts by isometries. By saying that M D G=K is a symmetric space, we will implicitly assume that the image of G ! Iso.M / contains Iso0 .M /. (This is automatically the case if G is semisimple.) Usually, we will also demand that the kernel of G ! Iso.M / be discrete, in other words, that the G-action on M be almost effective. The following facts hold (see e.g. [H], [O’N], [E]) Fact 3.1.4. Let G=K be a pseudo-Riemannian symmetric space with almost effective G-action. Then

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(1) Œk; k k; Œk; m m; Œm; m k. (2) If g is semisimple then Œm; m D k. (3) ŒŒm; m; m m. The isomorphism m !To M identifies the Lie triple product on m with the Riemannian curvature tensor (possibly up to sign, depending on the definition of the latter).

(4) The group gKg 1 acts on TgK M by linear isometries preserving R. Moreover, if G is semisimple then the map gK0 g 1 ! Aut0 .TgK M; g; R/ is surjective. The same holds if M is simply connected and the image of G in Iso.M / contains Iso0 .M / ([O’N, Theorem 8.14]). 3.1.4 Existence of associated families. Pluriharmonic maps into Riemannian symmetric spaces of non-positive or non-negative curvature are characterized by the fact that they have associated families. In our setting this is slightly different: Theorem 3.1.5. Let N be a connected and simply connected para-complex manifold and M a pseudo-Riemannian symmetric space. The following conditions for a smooth map f W N ! M are equivalent: (1) f has an associated family, i.e. there exists a 1-parameter family .f ; ˆ /2R of maps f W N ! M and parallel bundle isomorphisms ˆ W f TM ! f TM preserving the pseudo-Riemannian metric g M and the curvature RM , such that df B e J D ˆ B df :

(7)

(2) f is para-pluriharmonic and RM .df .T C N /; df .T C N // D 0; RM .df .T N /; df .T N // D 0: (2’) f is para-pluriharmonic and RM .df .T .1;0/ N /; df .T .1;0/ N // D 0: Proof. The equivalence .1/ () .2/ follows from the answer to a more general question, which is, roughly speaking: Given a vector bundle E ! N , such that the fibres are isomorphic to those of TM , when is a bundle homomorphism F W TN ! E the differential of a function f W N ! M? The answer is given in [ET1, Theorem 1]: Theorem (Eschenburg, Tribuzy). Let N be a simply connected smooth manifold and let M be a pseudo-Riemannian symmetric space with Levi-Civita connection D M and curvature tensor RM . Let E be a vector bundle over N , equipped with V2a connection D, E E ! End.E/, a D-parallel fibre metric g and a D-parallel triple product R W


523

such that there exists a linear isometry ˆ0 W To M ! Ep for some fixed points E M o 2 M , p 2 N , such that ˆ0 Rp D Ro . Let F W TN ! E be a vector bundle homomorphism. Then there exists a smooth map f W N ! M and a parallel bundle ! E preserving R and g such that isomorphism ˆ W f TM

F D ˆ B df if and only if .DX F /.Y / .DY F /.X / D 0; DX DY DY DX DŒX;Y D R.FX; F Y /

(8) (9)

for all vector fields X; Y 2 X.N / and sections of E, where in the first equation D denotes the connection on Hom.TN; E/ induced by D and a torsion-free connection on TN . Applying this theorem to the bundle E D f TM with the pull-back connection, metric and curvature tensor and to the map F D df B e J , we see that a solution .ˆ ; f / of the equation F D ˆ B df exists if and only if the conditions (8) and (9) are fulfilled. We already know that they are fulfilled for F0 D df . We consider the first equation: Since DJ D 0, we have DF D DF e J . Now, if X and Y are of the same type, say X; Y 2 T C N , then e J is just multiplication by e , so equation (8) remains valid for all 2 R. Thus, it has only be checked for vectors of different types; in this case it holds for all 2 R, if and only if both terms are zero – but this is precisely the pluriharmonicity. A similar consideration can be done for the second equation: This time, if X; Y are of different types, the factors e and e cancel, so we have to test equation (9) with vectors X; Y of the same type: Then the right-hand side picks up the factor e ˙2 , so equation (9) can only be fulfilled for all 2 R if both sides are zero. But for F0 D df , this is precisely the condition for the images of T ˙ N given in (2). The equivalence .2/ () .20 / follows immediately from T .1;0/ N D .1Cj /T C N ˚ .1 j /T N . Remark 3.1.6. If M is simply connected, then the solution .ˆ ; f / is unique up to isometries of M ([ET1, Theorem 1]). Moreover, if f is full (i.e. if the image of f is not contained in a totally geodesic subspace of M ), then ˆ is determined by f ; this follows from [ET1, Theorem 2]. Remark 3.1.7. In the Riemannian case, the condition RM .df .T .1;0/ N /; df .T .1;0/ N // D 0 actually follows from the pluriharmonicity, provided that the curvature tensor of M V (viewed as a symmetric bilinear form on .1;1/ TM ) is semi-definite; therefore, if M is a Riemannian symmetric space of nonpositive or nonnegative curvature, all pluriharmonic maps have associated families. In our case this is not true, as shown by the following counterexample:

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Let M be any symmetric space with nonzero curvature and let U M be an open subset diffeomorphic to Rn . Let N WD U U , JN D idT U .idT U / and 1 let g be a compatible metric. The projection on the first factor f W N ! U ,! M is obviously para-pluriharmonic since it depends only on the +-coordinates. But since RM .df .T C N /; df .T C N // ¤ 0 by assumption, f does not have an associated family. Remark 3.1.8. If M D G=K is a para-Kähler symmetric space, then the conditions for the existence of an associated family (Theorem 3.1.5 (2)) can be expressed in a very nice way; this has been discovered recently by Eschenburg and Quast [EQ]: Suppose that G is semisimple. We consider the standard immersion J W M ! g, which assigns to each point p 2 M the para-complex structure Jp , which is a derivation of the Lie triple algebra .Tp M; Rp / and therefore given by an element of g acting by the adjoint representation. From the parallelity of the endomorphism field J we conclude that the image J.M / g is the adjoint orbit AdG . /, ad WD JeK (see the proof of Proposition 3.2.2). Let g D k ˚ m be the Cartan decomposition of g. By definition, ad is a para-complex structure on m and zero on k. In particular, k coincides with the stabilizer subalgebra of . It follows that the map M ! AdG . / g is a Gequivariant covering. The tangent and normal bundles of J.M / g are AdG .m/ and AdG .k/, respectively. At the point 2 AdG . /, the para-complex structure on T J.M / is ad . Now let f W N ! M be a map from a para-Kähler manifold N into M . The differential of fQ W N ! M ! g can be considered as a g-valued differential form on N . Let WD JM d fQ JN D adfQ .d fQ JN / 2 1 .N I g/: We claim that f has an associated family if and only if is closed [EQ, Theorem 5.1]. Proof. A short calculation shows that d.X; JN Y / D Œf;Q rd fQ.X; Y / rd fQ.JN X; JN Y / C Œd fQ.X/; d fQ.Y / C Œd fQ.JN X /; d fQ.JN Y /; where r denotes the connection on T N ˝ g defined by the Levi-Civita connection on TN . Since adfQ vanishes on the normal bundle, the expression in the first line takes values in the tangent bundle. It is zero for all X; Y , if and only if 0 D .rd fQ.X; Y / rd fQ.JN X; JN Y //T D Dd fQ.X; Y / Dd fQ.JN X; JN Y /; which is precisely the definition of fQ being para-pluriharmonic.


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The expression in the second line takes values in the normal bundle. It is zero for all X; Y , if and only if 0 D Œd fQ.X/; d fQ.Y / C Œd fQ.JN X /; d fQ.JN Y / D RJ.M / .d fQ.X /; d fQ.Y // C RJ.M / .d fQ.JN X /; d fQ.JN Y //; which is equivalent the condition RM .df .T ˙ N /; df .T ˙ N // D 0.

In [EQ] this is used to generalize the Sym–Bobenko construction of surfaces in R3 with prescribed constant mean curvature from their (harmonic) Gauß map. 3.1.5 The Maurer–Cartan form. If G is a matrix group, i.e. a closed subgroup of GL.; R/, then there is an alternative approach to associated families using the Maurer–Cartan form. The following is completely analogous to the Riemannian case, see [DE] (note that our ˆ is ˆ1 there). Let N0 be a contractible open subset of N . Then any map f W N0 ! M D G=K can be lifted to a map F W N0 ! G such that f D B F ; this is called a framing of f . This framing is unique up to right multiplication by K-valued maps. Without loss of generality, we may assume that po 2 N0 is mapped to o D eK 2 M , and that F .po / D e. To the framing F we assign the g-valued differential form ˛ WD F 1 dF 2 1 .N0 I g/; the Maurer–Cartan form. It holds that d˛ C ˛ ^ ˛ D 0. Suppose that f W N ! M D G=K fulfills the conditions of Theorem 3.1.5. Let .ˆ ; f / be the associated family of f . For any p 2 N , ˆ .p/ is a linear isometry Tf .p/ M ! Tf .p/ M which preserves R, therefore under the requirements of Fact 3.1.4 (4), ˆ .p/ is given by an element of G. We can easily check that F WD ˆ1 F is a framing of f with corresponding Maurer–Cartan form ˛ D F1 dF D ˛ C F 1 ˆ dˆ1 F: From the parallelity of ˆ it follows (cf. [DE, Lemma 1]) that the last term on the right-hand side takes values in m g and that .˛ /k D ˛k for any 2 R (where ˛ D ˛k C ˛m due to the splitting g D k ˚ m). By definition, the component .˛ /m is the horizontal lift of df , up to left multiplication by elements of G, therefore from equation (7) we obtain .C1;0/ .0;1/ C e ˛m : .˛ /m D ˛m B e J D e ˛m

For fixed p 2 N0 , the map 7! F .p/ can be considered as an element of the loop group ƒ WD f W R ! GI smoothg:

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Thus, we have defined a map F W N0 ! ƒ, F .p/. / WD F .p/ which satisfies .C1;0/ .0;1/ F 1 d F D ˛k C e ˛m C e ˛m I

this is called an extended framing. By construction, F is unique up to constant loops with values in K, hence the map Fz WD B F W N ! ƒ=K is uniquely determined by f and globally defined on N . Conversely we can reconstruct the associated family from F : Let F D F . / and f D B F . Then a short calculation shows that df B e J D .F0 F1 / B df0 and that ˆ WD F0 F1 is parallel.

3.2 Isotropic para-pluriharmonic maps We now want to focus our attention to the case where the family .f / is constant in . Such maps are called isotropic para-pluriharmonic maps. Equation 7 then reads df B e J D ˆ B df;

(10)

where ˆ is a parallel bundle automorphism of f TM preserving R and g. We now assume that f is full, i.e. that the image is not contained in a totally geodesic subspace of M . Proposition 3.2.1. Let f W N ! M be a full isotropic para-pluriharmonic map. Then we have the following: (1) .ˆ/2R is a one-parameter group of automorphisms of f TM , i.e. ˆ B ˆ 0 D ˆC 0 ; L (2) f TM has a decomposition into parallel eigenbundles: f TM D odd E , where ˆ jE D e ; (3) The image of df is contained in E1 ˚ EC1 , and E1 ˚ EC1 generates f TM as Lie triple algebra; (4) g.E ; E / D 0 if C ¤ 0 and dim E D dim E . T Proof. Let E WD 2R ker.ˆ e /. From equation (10) we see immediately that 0 df .T ˙ N / E˙1 . Let E˙1 be the smallest parallel subbundle of f TM containing 0 E˙1 , because ˆ commutes with the parallel translation on df .T ˙ N /, then E˙1 0 0 f TM . Now let E f TM be the bundle generated by .E1 ˚ EC1 / asL a Lie triple algebra. By definition we have ŒŒE ; E ; E EC C and thus E odd E . We claim that E D f TM : In fact, because E is parallel and R-invariant and because f is full, this is the consequence of [ET1, Theorem 2]. This proves (2) and (3). (1) is an immediate consequence. Now (4) follows from the fact that ˆ is an isometry of f TM : We have gjE E D ˆ gjE E D e .C / gjE E , which obviously ! E , which proves implies the first statement. But then, g is an isomorphism E the equality of dimensions.


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Assume now that G is semisimple or that M is simply connected (and that the image of G in Iso.M / contains Iso0 .M /). For each p 2 N , ˆ .p/ is a one-parameter group of automorphisms of the Lie triple algebra .Tf .p/ M; R/, so by Fact 3.1.4(4), it can be considered as a one-parameter d subgroup of gKg 1 (where gK D f .p/). Its generator d ˆ .p/ is a derivation of the .TgK M; R/ and therefore given by the adjoint representation of an element of Adg .k/. This element is uniquely determined, if the action of G on M is almost effective. Let po 2 N and choose o D f .po / as the base point in M . Proposition 3.2.2. Under the identification ad W Adg .k/ ! Der.TgK M; g; R/, the map d F D jD0 ˆ d takes values in the adjoint orbit AdG . / g, where ad WD F .p0 /.

Proof. Let p 2 N and let W Œ0; 1 ! N be a smooth path from po to p. Recall that the left invariant distribution gm Tg G yields a connection on the K-principal bundle G ! M which agrees with the Levi-Civita connection on TM D G K m. Therefore the parallel translation along f B is given by a horizontal curve g.t / in G, such that gK D f B , g.0/ D e and g 0 2 gm. It follows that F ..t // D Adg.t / , because both sides are parallel endomorphism fields along f B , which coincide at t D 0. From Proposition 3.2.1 we see immediately that ad has odd integer eigenvalues on m Š To M . Therefore, if g is semisimple, ad has even integer eigenvalues on k D Œm; m, and the sum of the ˙1-eigenspaces generates all of g as Lie algebra. Elements 2 g with this property are called canonical:

3.3 Canonical elements and twistor spaces Definition 3.3.1. Let g be a real semisimple Lie algebra. An element 2 g is called canonical, if L (1) g is the sum g D 2Z g of ad -eigenspaces g WD ker.ad idg / with integer eigenvalues, (2) g1 ˚ g1 generates g as Lie algebra. Some authors replace the condition (2) by (20 ) the Lie algebra generated by g1 ˚ g1 contains

L ¤0

g .

To distinguish these notions, we call an element semi-canonical, if (1) and (20 ) hold. Lemma 3.3.2. If g is simple, then any semi-canonical element of g is either canonical or zero.

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More generally, let g be semisimple and let g D g0 ˚g00 ˚ be the decomposition into simple summands. Then a semi-canonical element D 0 C 00 C is canonical, if and only if all the 0 ; 00 ; : : : are nonzero. Proof. Let g be simple and semi-canonical. From (2’) and Œg0 ; g˙1 g˙1 and the Jacobi identity it follows that the Lie algebra generated by g1 ˚ g1 is an ideal in g and therefore either all of g or trivial. This proves the first statement. The rest is straightforward. Definition 3.3.3. Let 2 g be semi-canonical. The adjoint orbit Z. / WD AdG . / is called the twistor space associated to . Fact. Z is the homogenous space G=H , where H D fg 2 G W Adg . / D g is the centralizer of with Lie algebra h D g0 . Proposition 3.3.4. Let G be a connected real form of a complex semisimple Lie group. Let g be the Lie algebra of G and 2 g semi-canonical. (1) The map W g 7! e i ge i .i 2 D 1/ is an involutive automorphism of G with corresponding Cartan decomposition L L gDk˚mD even g ˚ odd g : (2) It holds that H G , and H G0 is a subgroup of G. Let K be a subgroup of G such that H G0 K G . Then there is a G-equivariant projection map W Z ! M onto the symmetric space M D G=K, called the twistor fibration. (3) The G-action on M is almost effective (i.e. the kernel of the map G ! Iso.M / is discrete) if and only if is canonical. Proof. (1) The stated Cartan decomposition follows from the fact that d D e iad is C1 on g if is even, and 1 if is odd. This also shows that d .g/ D g and d 2 D id, which implies .G/ D G and 2 D id, since G is connected. (2) If g 2 H , then e i D e i Adg D ge i g 1 , so g 2 G . This shows that H G . It now follows that hG0 h1 D G0 for any h 2 H . Therefore H G0 D G0 H , which implies that H G0 is a subgroup of G. The rest is straightforward. (3) Let be canonical. We have to show that the adjoint action k ! gl.m/ is effective. Suppose that X 2 k and adX jm D 0. Since m g1 ˚ g1 generates g, it follows by the Jacobi identity that adX jg D 0, hence X D 0. Vice versa, let be semi-canonical, but not canonical. Then the orthogonal complement (with respect to the Killing form) of the subalgebra generated by g1 ˚ g1 is an ideal contained in g0 k and therefore in the kernel of ad W k ! gl.m/. L At 2 Z, the tangent space of Z is T Z D adg . / D 2Znf0g g DW z. This eigenspace decomposition is AdH -invariant and thus defines a decomposition of T Z into AdG -invariant distributions AdG .g /, 2 Z n f0g.


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Proposition 3.3.5. The twistor space Z D AdG . / has a canonical para-complex structure which is given by multiplication by sgn. / on AdG .g / T Z. Proof. The proposition follows from the fact that Z can be written as a quotient of para-complex groups (see below). However, there is a direct proof: From the fact that e ad is an isometry with respect to the (nondegenerate) Killing form it follows that ˙ dim.gL / D dim.g /, cf. the proof of 3.2.1 (4). Moreover, the distributions T Z D integrable, because they are tangent to the leaves AdgN˙ . / Z, AdG . ?0 g / areL where N˙ WD exp. ?0 g /. Theorem 3.3.6. Let N be a connected para-Kähler manifold and let G be a connected real form of a semisimple complex Lie group. (1) Let M D G=K be a symmetric space with K D G and suppose that G acts almost effectively. Then any full isotropic para-pluriharmonic map f W N ! M defines a canonical element 2 g and a twistor fibration W Z ! M . The map f has a para-holomorphic lift F W N ! Z, f D B F , such that the differential dF takes values in the subbundle H1 WD AdG .g1 ˚ g1 / T Z (called superhorizontal bundle) (2) Let 2 g be canonical and let W Z ! M D G=K a twistor fibration as in Proposition 3.3.4. For any para-holomorphic superhorizontal map F W N ! Z the map f WD B F is isotropic para-pluriharmonic. d j D0 ˆ is paraProof. For the proof of (1), there is only left to show that F D d ˙ holomorphic and superhorizontal, i.e. that dF maps T N to AdG .g˙1 /: In the proof 1 0 of Proposition 3.2.2 we have already seen that LF B .t / D Adg.t / with g g 2 m 0 and therefore dF 2 Adg Œm; D Adg . odd g /. The map d W T Z ! TM takes AdG .g / to E , and from equation (10) it follows that d B dF .T ˙ N / D df .T ˙ N / E˙1 , hence dF .T ˙ N / AdG .g˙1 /. In order to prove (2), recall that the para-complex structure on the subspace Adg .g1 ˚ g1 / T 0 Z, 0 D Adg , is just ad. 0 /. The fact that F is holomorphic and the image of dF is contained in AdG .g1 ˚ g1 /, yields dF B e J D e adF B dF . Composing this equation with d W T Z ! TM (which is multiplication by .1/ on ! Adg .g /; odd), and recalling that the isomorphism Der.Adg .m/; ŒŒ ; ; / J F Der.TgK M; g; R/ identifies adF with F , we obtain df B e D e B df , where e F DW ˆ is an automorphism field of f TM , which preserves the curvature tensor and any AdK -invariant metric. F and ˆ are parallel, since dF has values in the horizontal bundle AdG .m/ (see the proof of Proposition 3.2.2). Thus, f D B F is isotropic para-pluriharmonic by the definition.

Corollary 3.3.7. A necessary condition for the existence of full isotropic para-pluriharmonic maps N ! M D G=G is that is of the form .g/ D e i ge i for 2 g.

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Example 3.3.8 (Para-holomorphic maps). The easiest example of all this is as follows: Assume that M D G=G is a para-Kähler symmetric space with para-complex structure JM . Any para-holomorphic map f W N ! M is isotropic para-pluriharmonic, since df B e JN D e JM B df . The canonical element 2 g for this map is exactly the para-complex structure JM ; it defines a depth-1 grading g D g1 ˚ g0 ˚ g1 . By definition, is invariant under G , so G H . We have already seen that H G , therefore Z D M , and the twistor fibration is just the identity map. 3.3.1 Para-complex description of the twistor space. We want to give a description of Z as a quotient of para-complex groups: Definition 3.3.9. Let G be a (not necessarily semisimple) Lie group. The paracomplexification of G is the product G c WD G G. This is a para-complex manifold, with the para-complex structure given by J D idT G .idT G /. Its Lie algebra is gc D g g Š g ˝ C ; .X; Y / Š 1Cj X C 1j Y . (cf. Example 1.1.5). We can 2 2 identify G with the diagonal subgroup f.g; g/ W g 2 Gg, being the fixed-point set of the anti-holomorphic involution .g; h/ 7! .h; g/ of G c . Let G be semisimple and 2 g semi-canonical. There is a decomposition of gc into ad.;/ -eigenspaces with imaginary eigenvalues (see Remark 1.1.3): For 2 Z, -eigenspace of ad.;/ (j equals C1 on g f0g let gc WD g g . Then gc is the jL c and 1 on f0gg), and we have g D 2Z gc . We define the following subalgebras of g and gc : L n˙ WD ?0 g ; nc˙ WD n˙ n ; p˙ WD g0 ˚ n˙ ;

pc˙ WD p˙ p :

The p˙ are parabolic subalgebras (see below) with nilradical (maximal nilpotent ideal) n˙ . Let N˙ WD exp.n˙ /;

P˙ WD fg 2 G W Adg 2 C n˙ g:

These are closed subgroups with respective Lie algebras n˙ and p˙ (see [ET2]). Let P˙c WD P˙ P and N˙c WD NC N . x WD G c =Pc D .G=P / Proposition 3.3.10. Z D G=H is an open subset of Z .G=PC /, called the closed twistor space. The para-complex structure on Z inherited x coincides with the one given in Proposition 3.3.5. from Z

Proof. The map G ,! G c ! G c =Pc has kernel P \ PC D H , so we have a welldefined embedding G=H ! G c =Pc . The differential of this map is the isomorphism ! gc =pc Š nC n which sends g to g f0g or to f0g g , depending g=g0 on the sign of .


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3.3.2 Parabolic subgroups. In order to describe the coset spaces G=P , we need some facts concerning (real) parabolic subgroups. We start with some basic definitions from algebraic geometry. (All of the following can be found in [Bo].) An affine algebraic variety is a subset V C n which is the set of zeroes of a family of polynomials. It can be endowed with the Zariski topology, in which the closed sets are by definition the subvarieties. Let I.V / CŒC n be the ideal of polynomials vanishing on V . Then CŒV WD CŒC n =I.V / is the ring of regular functions. A morphism of varieties is a map W V ! V 0 , such that .CŒV 0 / CŒV . V is said to be irreducible, if V is not the proper union of two nonempty varieties, which is equivalent to I.V / being a prime ideal. If V is irreducible, then the quotient field of CŒV is called the field of rational functions, denoted by C.V /. A variety V is said to be defined over R, or an R-variety, if I.V / is generated by polynomials with real coefficients. In this case, the set V .R/ of real points of V (i.e. the points of V with real coordinates) is a real variety. A morphism W V ! V 0 of R-varieties is defined over R (or an R-morphism), if .RŒV 0 / RŒV . A group is called linear algebraic, if it is both a subgroup and a subvariety of 2 GLn .C/ (where GLn .C/ is the variety f.g; det.g/1 / W g 2 GLn .C/g C n C1 ). It follows that any linear algebraic group over C is a complex Lie group. If G is an Rgroup (i.e. a linear algebraic group defined over R), then G.R/ is a real Lie group and a real form of G. Important examples of linear algebraic groups are the classical (real and complex) matrix groups. In fact, any connected semisimple complex Lie group is linear algebraic, as well as the adjoint form of any connected semisimple real Lie group. (In the complex case, the notions of connectivity in the ordinary (Hausdorff) topology and the Zariski topology coincide). Let G be a connected semisimple R-group. A subgroup P G is called parabolic, if it contains a Borel subgroup, i.e. a maximal solvable connected (in fact, the connectivity follows) subgroup. In this case, G=P is compact and a projective algebraic variety (i.e. a subset of CPn defined by homogenous polynomials). We assume that G.R/ is non-compact (otherwise the following would be trivial). Let S be a maximal R-split torus, i.e. a maximal algebraic torus S , such that the adjoint action of S.R/ on g has only real eigenvalues. The dimension r of S is called R-rank of G (this is well-defined, since all maximalL R-split tori are conjugate, [Bo, 20.9]). Then g is the sum of relative root spaces g D ˛2ˆR g˛ (in general not one-dimensional), where ˆR s is the relative root system. The group generated by the orthogonal reflections through the hyperplanes f˛ ? W ˛ 2 ˆR g is the relative Weyl group WR D N .S/=Z.S/, which can beSrepresented by elements of G.R/ [Bo, 21.2]. The connected components of s n ˛2ˆR ˛ ? are called relative Weyl chambers. The choice of a relative Weyl chamber determines a partition ˆR D C ˆC R [ ˆR of the set of roots into positive and negative ones. Let R ˆR denote the set of simple roots, i.e. the positive roots that are not the sum of two other positive

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roots. Clearly, the simple roots .˛1 ; : : : ; ˛r /, r D rank R .G/ form a basis of s . Let . 1 ; : : : ; r / be the basis of s dual to this, i.e. ˛a . b / D ıab . The following proposition relates subsets of R to semi-canonical elements and to parabolic R-subgroups (cf. [Bo, 21.12]): Proposition 3.3.11. There is a bijection between • subsets of R , • conjugacy classes of semi-canonical elements, and • conjugacy classes of parabolic R-subgroups. More explicitly: (1) For each subset I R , the element I WD

P ˛a 2R nI

a is semi-canonical.

(2) Each semi-canonical element is G.R/-conjugate to exactly one of the I . (3) If is R-subgroup P D fg 2 G W Adg 2 Lsemi-canonical, then the connectedL C ˛./>0 g˛ g with Lie algebra p D ˛./ 0 g˛ is parabolic. (4) For each parabolic R-subgroup P there exists a unique semi-canonical element , such that P D P . L Proof. (1) By definition, the ad -eigenspaces are g D ˛./D g˛ . Let be as described. Then we have ˛. / 2 f0; 1g for each simple root ˛. Since each positive root is the sum of simple ones, it follows that ˛. / 2 N0 for each positive root ˛ (and of course ˛. / 2 N0 for each negative rootL˛). Because g0 ˚ g1 contains all simple root spaces, it generates L the subalgebra 0 g . From Œg0 ; g1 g1 it then follows that g1 generates >0 g . Analogously for g1 . (3) By definition, is semisimple, so it is contained in a maximal torus t g0 . Choose a Weyl chamber in t whose closure contains . This defines a positive direction in t, such that p contains the sum of t and all positive root spaces. The latter is easily seen to be a Borel subalgebra, therefore p is parabolic. Now (2) and (4) follow from the fact that each parabolic R-group is G.R/conjugate to exactly one of the PI [Bo, 21.12], and that P D P 0 only if D 0 : This follows from the observation that the ad -eigenspaces g (and therefore , by semisimplicity) are L determined by p and p : Let n˙ be the nilradical of p˙ . Then Da n˙ D aC1 g˙ (where D0 n WD n, DaC1 n WD Œn; Da n). Because g ? g for ¤ with respect to the Killing form, it follows that g0 D pC \ n? and g˙a D Da1 n˙ \ .Da n /? for a 2 N. There are two special cases of this construction: For I D R we obtain D 0 and PR D G. – For I D ; we get the minimal parabolic R-subgroups, which play the same role in the real Bruhat decomposition as the Borel subgroups in the complex case:


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Theorem ([Bo, 21.15, 21.16]). Let G be a connected semisimple R-group, S a maximal R-split torus and P; a minimal parabolic R-subgroup. Then [ P; .R/wP; .R/ (disjoint union): G.R/ D w2WR

More generally, let PI P; be a parabolic R-subgroup of G and let WI WR be the subgroup generated by the reflections through the f˛ ? W ˛ 2 I g, then [ P; .R/wPI .R/: G.R/ D w2WR =WI

Therefore, G.R/=PI .R/ is a union of the Bruhat cells [ P; .R/wPI .R/=PI .R/: G.R/=PI .R/ D w2WR =WI

Let w be the element of shortest length in its class w WI ; then the cell P; wPI =PI is R-isomorphic to the affine space Nw WD N; \ wN; w 1 , where N; , N; are the unipotent radicals of P; and of its opposite parabolic subgroup P; WD P; , respectively [Bo, 21.20, 21.29]. Note that exp W nw 7! Nw is an R-isomorphism, because nw is nilpotent and the exponential series and its inverse are finite. There is one big cell in this decomposition, belonging to the element w0 which maps R n I to ˆ R : The image of Nw0 Š NI is R-Zariski-open and -dense in G=PI . Moreover, the map NI ! G=PI is a birational equivalence (meaning that ! C.NI /) it induces an isomorphism of the ring of rational functions C.G=PI / [Bo, 21.20]. Summarizing this we conclude (back to our original notation, where the real groups are denoted G; P˙ ; : : : rather than G.R/; P˙ .R/; : : : ): Theorem 3.3.12. Let G be a noncompact real form of a connected semisimple linear algebraic group. Let 2 g be semi-canonical and G c =Pc the closed twistor space associated to . Then the following facts hold: (1) The quotient G c =Pc is a product .G=P / .G=PC / of real projective varieties. (2) .G=P / .G=PC / is a disjoint union of affine spaces, namely the products of the Bruhat cells [ .P; wC P =P / .P; w PC =PC /: .wC ;w /2W W exp

(3) The map nC n ! NC N ! .G=P / .G=PC / is an embedding onto a Zariski-open and -dense subset (namely the product of the respective big Bruhat cells) and induces a birational equivalence nC n .G=P / .G=PC /. 3.3.3 Twistor fibrations of R-split simple Lie groups. By Proposition 3.3.4, in order to find the twistor fibrations of a given Lie group G and therefore the isotropic

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para-pluriharmonic maps into G-symmetric spaces (Theorem 3.3.6), we have to find the canonical elements in g. A direct consequence of Proposition 3.3.11 is the following Corollary 3.3.13. Let G be a simple Lie group of R-rank r. Then there exist, up to covering or isomorphism, 2r 1 nontrivial G-twistor fibrations, each of which corresponds to a proper subset of R . This construction is fairly straightforward, if G has an R-split maximal torus (i.e. one that is maximal among all tori). These groups are called R-split. As an example, we will carry this out for Sp.2n; R/: The group Sp.2n; R/ is defined as the group of automorphisms of R2n which 0 1n preserve the standard symplectic form given by 2n D 1n 0 . Its Lie algebra is ³ ² A C W A; B; C 2 gl.n; R/; B; C symmetric : sp.2n; R/ D B At The group Sp.2n; R/ has R-rank n, and a system of simple coroots spanning the maximal torus is given by Da 0a ; Da WD for 1 a n 1 a WD Da 1na and n WD

1

1 2 n

12 1n

d d: : : d (˛n is the long root of the Dynkin diagram ˛1 There are two types of canonical elements:

d< d

˛n ).

Case 1: ˛n 2 I , i.e. does not contain the summand n . In this case we have 1 0 0 1m1 C B 1 1m2 D 0 C B D ; DDB C: 2 1 m3 0 D A @ :: : The centralizer of is H D Sp.2m1 ; R/ GL.m2 ; R/ GL.m3 ; R/ o n 0 W A 2 GL.n; R/ Sp.2n; R//, (where GL.n; R/ D A0 .A1 /t and the centralizer of e i is

where modd

G D Sp.2modd ; R/ Sp.2mev ; R/; P P WD a odd ma ; mev WD a even ma .


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The resulting twistor fibration G=H ! G=K (with K WD H G0 D G ) is Sp.2n; R/ = Sp.2m1 ; R/ GL.m2 ; R/ GL.m3 ; R/ # Sp.2n; R/ = Sp.2mev ; R/ Sp.2modd ; R/: Case 2: ˛n … I , i.e. contains the summand n . In this case we have 1 01 1 2 m1 3 C B 1 D 0 C B 2 m2 5 D ; DDB C: 1 m 0 D 3 A @ 2 :: : In order to compute the twistor fibration, it is practical to switch to a different Weyl chamber and change into 1 01 1 2 m1 0 C B 32 1m2 D 0 C B 0 0 5 D ; D D C: B 1 0 D 0 A @ 2 m3 :: : Then the corresponding twistor fibration G=H ! G=K (with K D H G0 D G ) is Sp.2n; R/=GL.m1 ; R/ GL.m2 ; R/ # Sp.2n; R/=GL.n; R/: (The original would lead to a subgroup conjugate to this.) Note that Sp.2n; R/=GL.n; R/ is the adjoint orbit AdG . n / and therefore has a para-complex structure. However, the map AdSp.2n;R/ . 0 / ! AdSp.2n;R/ . n / is not holomorphic, except in the trivial case 0 D n . In the same way we can compute the twistor fibrations G=H ! G=.H G0 / of all classical R-split simple Lie groups (see Table 1). Remark. The reader may have observed that SO.n; n/ and SO.n C 1; n/ are not connected. However, the only place where this assumption was needed, was the proof of .G/ D G (Proposition 3.3.4). Since this holds nonetheless, we stick with these groups rather than use their (non-algebraic) identity components.

3.4 Isotropic para-pluriharmonic maps into Grassmannians The fibrations in the first row of this table are in fact the fibrations of para-complex flag manifolds over para-complex Grassmannians, which we will now describe more in detail, giving an explicit construction of para-pluriharmonic maps.

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Matthias Krahe Table 1. Twistor fibrations of classical R-split simple groups. SL.n; R/

An

Bn

d

˛1

d: : : d

˛n

SO.n C 1; n/

d: : : d

˛1

d> d

˛n1 ˛n

Sp.2n; R/

Cn

d: : : d

˛1

d< d

˛n1 ˛n

SO.n; n/

Dn

d: : : d

˛1

SL.n; R/ = S .GL.m1 ; R/ GL.m2 ; R/ / # SL.n; R/ = S .GL.modd ; R/ GL.mev ; R//

d

˛n … I

SO.n C 1; n/ = GL.m1 ; R/ GL.m2 ; R/ GL.m3 ; R/ # SO.n C 1; n/ = SO.modd ; modd / SO.mev C 1; mev /

˛n 2 I

SO.n C 1; n/ = SO.m1 C 1; m1 / GL.m2 ; R/ GL.m3 ; R/ # SO.n C 1; n/ = SO.modd C 1; modd / SO.mev ; mev /

˛n … I

Sp.2n; R/ = GL.m1 ; R/ GL.m2 ; R/ # Sp.2n; R/ = GL.n; R/

˛n 2 I

Sp.2n; R/ = Sp.2m1 ; R/ GL.m2 ; R/ GL.m3 ; R/ # Sp.2n; R/ = Sp.2modd ; R/ Sp.2mev ; R/

either ˛n 2 I or ˛n1 2 I

SO.n; n/ = GL.m1 ; R/ GL.m2 ; R/ GL.m3 ; R/ # SO.n; n/ = GL.n; R/

both or neither

SO.n; n/ = SO.m1 ; m1 / GL.m2 ; R/ GL.m3 ; R/ # SO.n; n/ = SO.modd ; modd / SO.mev ; mev /

˛n1

d d P Pd

˛n

P Definition 3.4.1. Let m1 ; : : : ; ml 2 N and n WD ma . The para-complex flag manifold Fm1 ;:::;ml .C n / is the space of nondegenerate .m1 ; : : : ; ml /-flags in C n ; such a flag is an orthogonal decomposition of C n into nondegenerate (with respect to the standard para-hermitian scalar product) subspaces C n D E1 ˚ ˚El , dimC .Ea / D ma . Equivalently, a flag is a sequence of nondegenerate subspaces f0g D W0 ¨ W1 ¨ ¨ Wl D C n , dim.Wa / D m1 C C ma , where Wa WD E1 ˚ ˚ Ea . (To see the equivalence, we use the fact that the orthogonal complement of a nondegenerate subspace is again nondegenerate.) The flag manifolds Fm;nm .C n / DW Gr m .C n / are called para-complex Grassmannians. The standard action of U .n/ on C n induces a transitive (by Proposition 1.3.3) U .n/-action on Fm1 ;:::;ml .C n /. There is an equivariant projection map U .n/ ! Fm1 ;:::;ml .C n / which sends a matrix g 2 U .n/ to the flag .E1 ; : : : ; El /, where E1 is the span of the first m1 columns of g, E2 the span of the next m2 columns, and so on. In particular, the identity matrix is mapped to the standard flag C n D C m1 ˚ ˚ C ml .


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The restriction of this map to SU .n/ is still surjective, since we can multiply the N D 1g without first column of a given matrix g 2 U .n/ by det.g/1 2 fz 2 C W zz changing its span. This yields the identification F.m1 ;:::;ml / .C n / Š SU .n/ = S .U .m1 / U .ml // : Now recall that SU .n/ Š SL.n; R/. So the fibration SL.n; R/ = S .GL.m1 ; R/ GL.ml ; R// # SL.n; R/ = S .GL.modd ; R/ GL.mev ; R// is the mapL F.m1 ;:::;ml / .C n / ! Gr modd .C n /, which sends the flag .E1 ; : : : ; El / to the subspace a odd Ea C n (cf. Table 1). The corresponding canonical element is 1 0 0 1m1 C B :: D@ A C 1n 2 sl.n; R/ : .l 1/1ml (where is the unique real number, such that is trace free). By the identification X 1j X t , this corresponds to sl.n; R/ ! su .n/, X 7! 1Cj 2 2 1 0 0 j 1m1 C B :: D@ A C j 1n 2 u .n/: : .l 1/j 1ml Lemma.

˚ L D0 ; T.E1 ;:::;El / Z D .Xab / 2 a¤b Hom.Ea ; Eb / W Xab C Xba M c T.E Z D Hom.Ea ; Eb /: 1 ;:::;El / a¤b

Proof. The second equation follows immediately from the first. So let a denote the orthogonal projection on Ea . From a D a , a2 D a and a b D 0, a ¤ b it follows that a tangent vector to Z at E D .Ea / is given by a collection of maps 10 ; : : : l0 W C n ! C n satisfying a0 D a0 ;

a0 a C a a0 D a0 ;

a0 b C a b0 D 0

for a ¤ b:

Now it is easy to see that Xab WD b a0 D b0 a 2 Hom.Ea ; Eb / for a ¤ b satisfies D 0 and that any such collection .Xab / defines maps a0 with the above Xab C Xba properties. By definition, ad has eigenvalueL .b a/j on Hom.C ma ; C mb /, therefore the suc c perhorizontal bundle is Adg .g1 / D a Hom.Ea ; EaC1 / T.E Z. A holomorphic a/ map f W N ! Z from a para-Kähler manifold N to Z D F.m1 ;:::;ml / .C n / is superhor-

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L izontal, if and only if @f D df .1;0/ takes values in a Hom.Ea ; EaC1 /, or, equivalently (as follows by induction), if @f maps Wa D E1 ˚ ˚Ea to WaC1 . Thus we can construct para-pluriharmonic maps in the following way: Let f1 ; : : : ; fm1 W N ! C n be holomorphic maps such that for n 2 N the images fa .n/ are C -linearly independent vectors with nondegenerate span. Let W1 be the span of the fa , W2 the span of the fa and their derivatives and possibly further g1 ; g2 ; : : : W N ! C n and so on, as long as all these maps are (pointwise) linearly independent and span nondegenerate subspaces. This defines a superhorizontal map F W N ! F.m1 ;:::;ml / .C n /, and the composition f D B F W N ! Gr modd .C n / is para-pluriharmonic. A special case of this is the fibration F.1;n;1/ .C nC2 / ! Gr 2 .C nC2 /, which is the twistor fibration associated to the para-quaternionic Kähler structure on Gr 2 .C nC2 / (cf. Section 3.6.1). Para-pluriharmonic maps f W N ! Gr 2 .C nC2 /, dimC N D n, are obtained by (para-)holomorphic functions h W N ! C nC2 via f D W1 ˚ W2? , where W1 .n/ D C h.n/ and W2 .n/ D spanC .h.n/; grad.h.n///. In the next section we will prove that, given two para-quaternionic Kähler symmetric spaces of the same dimension, there is a biholomorphism between open and dense subsets (the big Bruhat cells, see above) of their twistor spaces preserving the horizontal structure. This can be used to transfer the preceding construction to any para-quaternionic Kähler symmetric space.

3.5 Para-quaternionic Kähler symmetric spaces In this section, we examine the twistor fibrations Z ! M associated to a para-quaternionic Kähler structure on M . We will see that these are the twistor spaces which belong to a depth-two grading g D g2 ˚ g1 ˚ g0 ˚ g1 ˚ g2 with dim.g˙2 / D 1. The main result is the birational equivalence between all these twistor spaces of the same dimension. Proposition 3.5.1. Let M D G=K be a pseudo-Riemannian symmetric space such that G acts effectively on M . (1) If M is quaternionic Kähler with nonzero scalar curvature (see Definition 1.4.4), then there exists a semi-canonical element 2 k which defines a grading g D g2 ˚ g1 ˚ g0 ˚ g1 ˚ g2 such that 2 Œg2 ; g2 , g2 ˚ g0 ˚ g2 D k, g1 ˚ g1 D m, dim.g2 / D dim.g2 / D 1 and Œ ; W g˙1 g˙1 ! g˙2 is nondegenerate. (2) Vice versa, if there exist a semi-canonical 2 k which defines a grading g D g2 ˚g1 ˚g0 ˚g1 ˚g2 as above and if K H G0 (where H is the centralizer of in G), then M has a G-invariant para-quaternionic Kähler structure. (3) If K D H G0 , the twistor space AdG . / D G=H is the twistor space of para-complex structures (Definition 2.2.1). Remark. Of course, there may be other twistor bundles over the same base space M . A simple example is the trivial fibration Gr 2 .C nC2 / ! Gr 2 .C nC2 /.


539

Proof. (1) Suppose M is a para-quaternionic Kähler manifold. Then k D hol sp .1/ ˚ sp .n/. If scal.M / ¤ 0 then the curvature decomposition shows that k contains the sp .1/-factor (see [AC1, Lemma 1]), so k D sp .1/ ˚ k 0 . The isomorphism To M Š m identifies the quaternionic structure Qo End.To M / with adsp .1/ jm . Now choose a para complex structure Jo 2 Qo . This corresponds to an element 2 sp .1/ k. By [AC1, Lemma 3 and 4], g is simple and corresponds to a long root with respect to a Cartan subalgebra a gC in the sense h˛; i that ˛. / D 2 h ; i for all ˛ 2 a . In particular, ˛. / 2 f0; ˙1; ˙2g, so defines a depth two grading of g, and g˙2 are the one-dimensional root spaces g˙ . To see that Œ ; W g˙1 g˙1 ! g˙2 is nondegenerate, we remark that for each root ˛ 2 a with ˛. / D ˙1, we have Œg˛ ; gw .˛/ D g˙ ¤ 0, where w 2 W denotes the element of the Weyl group that is the reflection throughLthe hyperplane f D 0g. Of course, the nondegeneracy implies that g˙1 generates ?0 g , so is semi-canonical. (2) Let g D g2 ˚ g1 ˚ g0 ˚ g1 ˚ g2 . Then g2 ˚ R ˚ g2 is a subalgebra isomorphic to sp .1/, and we have g0 D R ˚ Zg .sp .1//. It follows that k D sp .1/ ˚ k 0 with k 0 D Zg .sp .1//. This implies that Q0 WD adsp .1/ jm is a paraquaternionic structure on m Š To M , which is invariant under the adjoint action of k and therefore under K0 D G0 . By definition, it is also invariant under H , hence under K H G0 . Therefore Q0 extends to a G-invariant para-quaternionic structure on TM . (3) The twistor space Z Q of para-complex structures is the G-fibre bundle G K ZoQ , where ZoQ D fq 2 Qo W q 2 D 1g is the Sp .1/-orbit of in Qo . Since K contains the Sp .1/ factor, it acts transitively on ZoQ . It follows that G acts transitively on Z Q , so Z Q is the homogenous space G=H 0 , where H 0 D K \ H is the centralizer of in K. But since H K by assumption, we have H 0 D H and therefore Z Q D AdG . /. Birational equivalence of twistor spaces. Let G be a linear algebraic group. We x D G c =Pc , which is a real have seen that the twistor space Z is an open subset of Z c c c c variety, and that exp.nC / D NC ! G =P is an embedding onto a Zariski-open and -dense subset, which induces a birational equivalence NCc G c =Pc . If Z is the twistor space of a para-quaternionic Kähler symmetric space, then NCc is isomorphic to the (generalized) para-complex Heisenberg group: Definition 3.5.2. The (real, complex, para-complex) Heisenberg group H.n; K/ (K D R; C; C ) is the group of .n C 2/ .n C 2/-matrices of the form 1 0 1 C B z1 1 C B C B :: :: C 2 GL.n C 2; K/: B : : C B n A @z 1 w0 w1 wn 1

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Matthias Krahe

Its Lie algebra is the Heisenberg algebra h.n; K/ and consists of the matrices 0 1 0 Bu1 0 C B C B :: C : :: B : C 2 gl.n C 2; K/: B n C @u A 0 v0 v1 vn 0 Remark. Note that H.n; C / Š H.n; R/ H.n; R/. Lemma 3.5.3. A Lie algebra is isomorphic to the Heisenberg algebra if and only if it is two-step nilpotent with one-dimensional kernel. (If K D C this condition includes that the kernel is a para-complex subspace.) Proof. Let .u1 ; : : : ; un ; v 0 ; : : : ; v n / be the obvious basis of h.n; K/ (i.e. the one which is dual to the above coefficients). Then Œv a ; ub D ıba v 0 for 1 a; b n, and all other brackets are zero, so obviously h.n; K/ is two-step nilpotent with kernel K v 0 . Conversely, let g be a two-step nilpotent Lie algebra and let v 0 be a nonzero (resp. regular, if K D C ) element of its one-dimensional kernel. Let g0 a complementary subspace, i.e. g D K v 0 ˚ g0 . By assumption we have Œg0 ; g0 D K v 0 , so the Lie bracket defines a symplectic form ! on g0 via Œ; D !.; /v 0 . Let .v a ; ub /a;bD1:::n be a symplectic basis of .g0 ; !/ (see Proposition 2.1.2) , then Œv a ; ub D ıba v 0 , which defines an isomorphism g Š h.n; K/. Proposition 3.5.4. H.n; K/ has a left-invariant contact form given by dw0

n X

wa dz a :

aD1

x D G c =Pc be the closed twistor space associated to a para-quaternionic Let Z Kähler structure. Let NCc be the unipotent radical of PCc . Then NCc Š H.n; C / (as real linear algebraic groups). This isomorphism is biholomorphic and identifies the contact distribution on H.n; C / with the horizontal distribution on the orbit of NCc in x Moreover, it extends to a birational equivalence N c H.n; C /. Z. C Proof. By Proposition 3.5.1 we have ncC D gc1 ˚ gc2 , and that the Lie bracket Œ ; W gc1 gc1 ! gc2 is nondegenerate. So ncC is two-step nilpotent with one-dimensional kernel gc2 . This shows that ncC Š h.n; C /, or equivalently nC n Š h.n; R/ h.n; R/. Exponentiating this, we get an isomorphism of the groups NC N Š H.n; R/ H.n; R/. This isomorphism is algebraic (i.e. has polynomial components) because both the exponential and logarithm series are finite. Obviously, it is a biholomorphism. To show that the contact distribution on NCc obtained through this isomorphism is mapped to the horizontal distribution by the map W NCc ! G c =Pc ,


541

it suffices to remark that both distributions are NCc -invariant and coincide at the point e 2 NCc , where they are given by the linear subspace gc1 ncC . The statement about the birational equivalence has already been made in Theorem 3.3.12. Thus we have shown a refinement of Theorem 2.1.11: The Darboux coordinates can be constructed explicitly by writing down the isomorphism NCc Š H.n; C /. They are defined on the open dense subset .NCc / \ Z of Z. Theorem 3.5.5. Let Z be the twistor space of a para-quaternionic Kähler symmetric space. Then there is an open dense subset U Z and para-holomorphic coordinates ; : : : ; wn / W U ! C 2nC1 such that the horizontal distribution is given .z 1 ; : : : ; z n ; w0P by ker.dw0 naD1 wa dz a /. Now let Z 0 be the twistor space of another para-quaternionic Kähler symmetric space of the same dimension, then we can put the birational maps together: G 0c =P0c

NC0c Š H.n; C / Š NCc ! G c =Pc :

This proves: x D G c =Pc and Z x 0 D G 0c =P0c be two closed twistor spaces Theorem 3.5.6. Let Z associated to para-quaternionic Kähler symmetric spaces of the same dimension. x and Then there is a biholomorphic map between Zariski-open and -dense subsets of Z 0 x Z , which preserves the horizontal distribution. This morphism induces a birational x Zx0 . equivalence Z This allows us to construct para-pluriharmonic maps into any para-quaternionic Kähler symmetric space by transferring the construction from Gr 2 .C nC2 / (see Section 3.4).

3.6 The classical para-quaternionic Kähler symmetric spaces The simply connected para-quaternionic Kähler symmetric spaces have been classified by Alekseevsky and Cortés [AC1], see also [DJS]: Theorem (Alekseevsky, Cortés). Let M D G=K be a simply connected para-quaternionic Kähler symmetric space. Then M is one of the following spaces: A)

SL.nC2;R/ , SU.pC1;qC1/ , S.GLC .2;R/GLC .n;R// S.U.1;1/U.p;q//

BD)

SOC .pC2;qC2/ , SO .2nC4/ , SOC .2;2/SOC .p;q/ SO .4/SO .2n/

C)

Sp.2nC2;R/ , Sp.2;R/Sp.2n;R/

E6)

E6.6/ E6.2/ E6.14/ , , , SL.2;R/SL.6;R/ SU.3;3/SU.1;1/ SU.5;1/SU.1;1/ E7.7/ E7.5/ E7.25/ , , , SL.2;R/Spin0 .6;6/ SL.2;R/SO .12/ SL.2;R/Spin0 .10;2/

E7)

542

Matthias Krahe E8.8/ E8.24/ , , SL.2;R/E7.7/ SL.2;R/E7.25/

E8)

F4.4/ , SL.2;R/Sp.6;R/ G2.2/ . SO.2;2/

F4) G2)

We want to describe the spaces belonging to the classical groups more in detail. However, to apply the results of this section, it is useful to consider spaces of the form G=.H G0 / rather than their universal covers. We will find that these are the following Grassmannians: (Kp;q (K D R; C) denotes the vector space K pCq , endowed with an indefinite pseudo-Euclidean or -hermitian scalar product of signature .p; q/. A subspace of signature .p 0 ; q 0 / is called .p 0 ; q 0 /-subspace.) (1)

SL.nC2;R/ S.GL.2;R/GL.n;R//

Š Gr 2 .C nC2 /, the set of nondegenerate 2-dimensional para-

complex subspaces of C nC2 . (2)

SU.pC1;qC1/ S.U.1;1/U.p;q// pC1;qC1

Š Gr 1;1 .C pC1;qC1 /, the set of complex (1,1)-subspaces of

SO.pC2;qC2/ SO.2;2/SO.p;q/ pC2;qC2

pC2;qC2 Š Gr or /, the set of oriented real (2,2)-subspaces of 2;2 .R

SO .2nC4/ SO .4/SO .2n/ nC2;nC2

nC2;nC2 Š Gr H /, the set of those complex (2,2)-subspaces of 2;2 .C

C (3)

R (4)

C (5)

.

.

which are invariant under a certain quaternionic structure.

Sp.2nC2;R/ Sp.2;R/Sp.2n;R/

Š H Pn , the para-quaternionic projective space.

Following the procedure described in the preceding section, we will construct Darboux coordinates on their twistor spaces, which in turn will give us a description of the isotropic para-pluriharmonic maps. (In the cases (1) and (5) it is natural to use C valued coordinates, whereas we will use .R R/-valued coordinates in the cases (2)–(4).) 3.6.1 SL.n C 2 ; R/ = S .GL.2 ; R/ GL.n; R//. Let G D SU .n C 2/ Š SL.n C 2; R/. The Lie algebra of G is g D fX 2 gl.n C 2; C / W X C X D 0; tr.X / D 0g : In Section 3.4 we have seen that the canonical element 1 0 j A 0n WD @ j leads to the twistor fibration G=H ! G=K (with K D G D H G0 ) SU .n C 2/ SU .n C 2/ ! : S .U .1/ U .n/ U .1// S .U .2/ U .n//


543

From Section 3.4 we also know that the fibration G=H ! G=K can be identified with F1;n;1 .C nC2 / ! Gr 2 .C nC2 /; where Gr 2 .C nC2 / is a para-complex Grassmannian and Z D F1;n;1 .C nC2 / the space of nondegenerate .1; n; 1/-flags in C nC2 . x D G c =Pc . In the present case, We have also proven that Z is an open subset of Z c c G D SL.n C 2; C /, and P is the group of upper block diagonal matrices of the form 1 0 @ 0 A 2 SL.1 C n C 1; C /: 0 0 x is the space of all .1; n; 1)-flags in C nC2 : Such a flag is a sequence 0 ¨ W1 ¨ Thus, Z W2 ¨ C nC2 , where W1 and W2 are para-complex subspaces (possibly with degenerate x is scalar product) of dimension 1 and 1 C n, respectively. The map G c =Pc !Z c defined as follows: Given an element of G , let W1 be the C -span of its first column and W2 the C -span of the first 1 C n columns. This map is easily seen to be surjective and invariant under multiplication by elements of Pc from the right. The unipotent radical NCc of PCc is the para-complex Heisenberg group H.n; C /. We want to determine the orbit of NCc in G c =Pc : By definition it consists of those g Pc , such that g 2 NCc Pc , i.e. 0 1 0 1 0 1 g1;nC2 g1;1 1 0 0 B :: C @ :: :: 1n 0 A @ 0 A : @ : AD : : 1 0 0 gnC2;1 gnC2;nC2 A necessary condition is that the square blocks on the diagonal of the right factor are invertible. From this it follows after short calculation that g1;1 2 C ; 0

g1;1 B :: :: det @ : : gnC1;1

(11) 1

g1;nC1 C :: A2C : : gnC1;nC1

(12)

The Gaussian elimination algorithm shows that this condition is also sufficient: Since g1;1 is invertible, we can annihilate the rest of the first column of g by adding multiples of the first row to the other rows. This is achieved by left multiplication of a suitable matrix, so we obtain 1 0 1 0 1 0 0 @ 1n 0 A g D @ 0 A 0 1 0

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This does not change the determinant of the upper left .n C 1/ .n C 1/ submatrix, which is invertible by equation (12). It follows that the middle n n-block on the right-hand side is invertible, so we can eliminate the middle block in the last row: 1 0 1 0 1 0 1 0 0 @ 0 1n 0 A @ 0 A D @ 0 A : 0 1 0 0 0 This yields the desired decomposition of g. From Corollary 1.1.7 it follows that the equations (11) and (12) define an R-Zarikix as claimed in Theorem 3.3.12. open subset of Z, It remains to give a description of the isotropic para-pluriharmonic maps: The coordinates on NCc , 1 0 1 A; @ z 1n t w0 w 1 z D .z 1 ; : : : ; z n /t , w D .w1 ; : : : ; wn /t , yield holomorphic Darboux coordinates on .NCc / G c =Pc . With respect to these coordinates, a holomorphic map F W C n Uz ! G c =Pc ; .z 1 ; : : : ; z n / 7! .z 1 ; : : : ; z n ; w0 ; : : : ; wn /; is horizontal if there is a holomorphic function h W Uz ! C , such that w0 D h.z 1 ; : : : ; z n /; wa D @h=@z a ; a D 1 : : : n: This holomorphic horizontal map projects onto the isotropic para-pluriharmonic map f D B F W Uz ! Gr 2 .C nC2 /; .z 1 ; : : : ; z n / 7! W1 ˚ W2? ; where

0

1 1 Bz 1 C B C B C W1 D C B ::: C ; B nC @z A h

80 1 0 1 0 19 0 0 > 1 ˆ ˆ > ˆ ˆBz 1 C B 1 C B 0 C> > ˆ = B :: C B :: C B :: C W2 D spanC B : C ; B : C ; : : : ; B : C : ˆ B C> C B C ˆB ˆ > @z n A @ 0 A @ 1 A> ˆ > ˆ > : ; @h @h h @z n @z 1

This is what we have seen in Section 3.4 (neglecting that we can change this map by z a biholomorphism C n Uz ! Uz C n ). 3.6.2 SU.p C 1; q C 1/ = S .U.1; 1/ U.p; q//. Let G D SU.p C 1; q C 1/ be the group of endomorphisms of C nC2 , n D p C q, with unit determinant which preserve


the hermitian scalar product given by the matrix 0 B E WD B @

1p

1

1q

545

1 C C: A

1 Its Lie algebra is g D su.p C 1; q C 1/ D fX 2 sl.n C 2; C/ W X E C EX D 0g and consists of traceless matrices of the form 1 0 C D a c B A C C C 2 sl.1 C n C 1; C/; b; c 2 i R: B u.p; q/ @ B D A A B b aN The canonical element

0 WD @

1

1

A

0pCq 1

leads to the twistor fibration SU.p C 1; q C 1/ SU.p C 1; q C 1/ ! S .U.p; q/ C / S .U.p; q/ U.1; 1// ˚ (where C D a0 1=0aN W a 2 C U.1; 1/). The coset space SU.p C 1; q C 1/ = S .U.p; q/ U.1; 1// can be identified with the Grassmannian Gr 1;1 .C pC1;qC1 / of 2-dimensional complex subspaces V C nC2 which have signature .1; 1/ with respect to the above pseudo-hermitian metric; an element of G is sent to the subspace V that is spanned by the first and the last column. The twistor fibre over V C nC2 is the set of compatible para-complex structures on V , i.e. the set of decompositions V D V C ˚ V into complementary complex (totally) degenerate spaces. The subgroups P˙ and N˙ are the groups of lower or upper block diagonal matrices, where the groups N˙ have unit matrices on the diagonal. The homogenous x of all pairs .V C ; V / of one-dimensional space .G=P / .G=PC / is the space Z x sends a pair .gC ; g / to degenerate spaces V ˙ C pC1;qC1 . The map G G ! Z C C .V ; V /, where V is the span of the first column of gC and V the span of the last column of g . As before, the algebra nC consists of the strictly lower block diagonal matrices. A Lie algebra isomorphism h.p C q; R/ ! nC is given by 0 1 0 1 0 0 B u1 C 0 B C C B u C iv 0p B :: C :: C; B : C 7! B : A @ Q Q 0 u i v B pCq C q @u A t t t t 0 2iv0 u C i v uQ C i vQ 0 v1 vpCq 0 v0

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Matthias Krahe

where u D .u1 ; : : : ; up /t ; v D .v1 ; : : : ; vp /t ;

uQ D .upC1 ; : : : upCq /t ; vQ D .vpC1 ; : : : vpCq /t :

Exponentiating this, we obtain an isomorphism H.p C q; R/ ! NC , and Darboux coordinates on NC : 1 0 1

@

zC C i w C QC zQ C i w t t t 1 t t Q Q Q Q Q C/ .z z w w C z C w / C i.2w0C ztC wC zQ C w z w C C C C C C C C 2

p t 1 ; : : : ; zC / , zQ C where zC D .zC C C ; : : : ; wpCq /t . .wpC1

0 @

D

1p 0 ztC Ci wtC

pC1 pCq t .zC ; : : : ; zC / , wC

D

0 1q t Q tC zQ C i w

A; 1

.w1C ; : : : ; wpC /t , wQ C

In the same way for N : 1

zt Ci wt 1p 0

t Q t zQ i w 0 1q

t 1 Q t w Q / .zt z wt w C zQ zQ C w 2

z C i w Q zQ i w 1

t

Q / C i.2w0 zt w zQ w

D

1 A:

Therefore, the isotropic para-pluriharmonic maps are of the form C n Š Rn Rn Uz ! Gr .1;1/ .C pC1;qC1 /; 1 n .z˙ ; : : : ; z˙ / 7! V C ˚ V ;

where VC is the span of the first column of NC and V the span of the last column of N (note that these vectors are degenerate): 0 1 1 B C zC C i wC C; VC D C B @ A zQ C i wQ C t t t C 1 t t t .zC zC wC wC C zQ C zQ C C wQ C wQ C / C i.2w0 zC wC zQ C wQ C / 2 01 1 t t t t t t Q Q Q Q Q Q .z z w w C z C w / C i.2w z w z / z w w 0 2 B C z C i w C; V D C B @ A zQ i wQ 1 1 n a ; : : : ; z˙ /, wa˙ D @h˙ =@z˙ for a para-holomorphic function and where w0˙ D h˙ .z˙ n n h D hC h W R R U ! R R.

3.6.3 SO.p C 2 ; q C 2/ = SO.p; q/ SO.2 ; 2/. Let G D SO.p C 2; q C 2/ be the group of endomorphism of RpCqC4 which preserve the pseudo-Riemannian metric given by the matrix 0 1 12 B C 1p C: EDB @ A 1q 12

547


The Lie algebra g Š so.p C 2; q C 2/ consists of the E-skew matrices; these are of the form 0 1 C t D t a c B A C C B C 2 gl.2 C n C 2/; b; c 2 o.2/: so.p; q/ @ B D A At B t b at 1 2 0pCq The canonical element WD leads to the twistor fibration G=H ! 12 G=.H G0 / SO.p C 2; q C 2/ SO.p C 2; q C 2/ ! : SO.p; q/ GL.2; R/ SO.p; q/ SO.2; 2/ The space M D SO.p C 2; q C 2/=SO.p; q/ SO.2; 2/ can be identified with the pC2;qC2 Grassmannian Gr or / of oriented 4-dimensional subspaces of RpC2;qC2 2;2 .R which have signature (2,2) with respect to the above pseudo-Riemannian metric. The map G ! M is defined as follows: Given an element of G, its first two and last two columns form an oriented basis of this (2,2)-subspace. Note that M has a two-fold quotient, namely the non-oriented Grassmannian Gr 2;2 .RpC2;qC2 / D SO.p C 2; q C 2/ = S.O.p; q/ O.2; 2//, which is not paraquaternionic Kähler (the para-quaternionic structure is not S.O.p; q/ O.2; 2//invariant). M also has a two-fold cover, the strongly oriented Grassmannian pC2;qC2 / D SO.p C2; q C2/ = SO.p; q/SOC .2; 2/, which is para-quaterGr stror 2;2 .R nionic Kähler, but has no twistor fibration in the sense of Proposition 3.3.4 (in other words, the twistor space defined in Section 2 is no adjoint orbit). The twistor space Z D G=H is the set of pairs .V C ; V / of 2-dimensional totally degenerate subspaces V ˙ of RpC2;qC2 , such that the pseudo-Euclidean metric is nondegenerate on V C ˚ V (which is therefore a subspace of signature (2,2)). The map Z ! M maps the pair .V C ; V / to the sum V C ˚ V ; note that a canonical ! .V / . orientation on V C ˚ V is obtained through the isomorphism V C x D .G=P /.G=PC / is the space of all pairs .V C ; V / The closed twistor space Z x of 2-dimensional totally degenerate subspaces of RpC2;qC2 . The map G G ! Z C C sends the pair .gC ; g / to .V ; V /, where V is the span of the first two columns of gC and V the span of the last two columns of g . This map is P PC -invariant x and defines an isomorphism .G=P / .G=PC / ! Z. An isomorphism h.n; R/ ! nC is given by 0

0 u1 :: :

B B B B B pCq @u v0

1 0 ::

: 0

v1

vpCq

0

1

0

B C B C B u C C 7! B B uQ C B A @ 0 0 v0

0 v 0p vQ v0 ut 0 vt

0q uQ t vQ t

C C C C; C C A

0 0

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where u D .u1 ; : : : ; up /t ; v D .v1 ; : : : ; vp /t ;

uQ D .upC1 ; : : : upCq /t ; vQ D .vpC1 ; : : : vpCq /t :

Thus we obtain Darboux coordinates on NC : 0 1 0 B 0 1 B B z w 1p 0 C C B B Q Q 0 1 z w C C q B 1 t @ .zt zC C zQ t zQ C / w C C zt wC C zQ t wC zt Q 1 0 z C C C C C C 0 2 t t 1 t t Q Q Q w0C .w w C w / w w w C C C C 0 1 C C 2 and similarly on N : 0 1 0 zt zQ t B 0 1 wt wQ t B B 1p 0 B B 0 1 q B @

C zQ t zQ / w0 C zt w C zQ t w 1 w0 .wt w C wQ t wQ / 2 z w zQ wQ 1 0 0 1

1 .zt z 2

1 C C C CI C C A

1 C C C C: C C A

Therefore, the isotropic para-pluriharmonic maps are given by pC2;qC2 Rn Rn U ! Gr or /; 2;2 .R 1 n 1 n 1 n .z˙ ; : : : ; z˙ / 7! VC .zC ; : : : zC / ˚ V .z ; : : : z /;

where

19 0 > > C> > B 1 C> > B = C B w C C ; B C> B wQ C C> B C > @w0 C ztC wC C zQ tC wC A> > > ; t 1 t Q Q .w w C w / w C C C C 2

80 1 1 ˆ ˆ ˆ ˆ B C 0 ˆ ˆ C > B 1 .wt w C wQ t wQ / C> C> > B 2 = C B w C ; B C> B wQ C> B > A> @ 0 > > ; 1

1 n a and where w0˙ D h˙ .z˙ ; : : : ; z˙ /, wa˙ D @h˙ =@z˙ for a para-holomorphic function n n h D hC h W R R U ! R R.

549


3.6.4 SO .2n C 4/ = SO .4/ SO .2 n/. Let H D R ˚ i R ˚ j R ˚ kR, i 2 D j 2 D 1, k D ij D j i, be the algebra of quaternions. We take quaternionic vector spaces to be right H-modules. Accordingly, Hn is the space of column vectors with scalar multiplication on the right, and left multiplication by quaternionic matrices is H-linear. Regarding Hn as the complex vector space Hn D C n ˚ j C n Š C 2n , right multiplication by j becomes a C-antilinear map rj W C 2n ! C 2n . A C-linear map C 2n ! C 2n is H-linear, if and only if it commutes with rj . The group SO .2n/ – sometimes also denoted U .n; H/ – is defined as the group of endomorphisms of Hn preserving the skew-hermitian form !.h; k/ WD h j k. Writing ! DW ˛ C jˇ, we obtain maps ˛; ˇ W C 2n C 2n ! C, where ˇ is C-bilinear symmetric and ˛ is skew-hermitian (whence i˛ is hermitian). These two are related by ˛.rj ; / D ˇ. It follows that an endomorphism of C 2n is an element of SO .2n/, if and only if it commutes with rj and preserves the hermitian form i˛. Moreover, we have lj ˛ D rj ˛ D ˛. It follows that i˛ DW h has signature .n; n/. Thus we have SO .2n/ D GL.n; H/ \ U.n; n/: Let G Š SO .2n C 4/ be the group of endomorphisms of C 2n which commute with 1 0 1 C B 1 C B C B 1n C rj D B B C B 1n C B @ 1 A 1 and preserve the indefinite hermitian scalar product h given by the matrix 1 0 1 B 1 C C B C B 1n C: B E WD B C 1 n C B A @ 1 1 The Lie algebra g Š so .2n C 4/ consists of matrices of the form 0 B B B B B B @

a b A B c 0

bN aN Bx AN 0 cN

C D D t C t so .2n/ A B B t At

d 0 0 dN x C D D Cx aN bN b a

1 C C C C 2 gl.2n C 4; C/; C C A

c; d 2 i R:

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The canonical element 0 WD @

12

1 02n

A 12

leads to the twistor fibration SO .2n C 4/ SO .2n C 4/ ! : GL.1; H/ SO .2n/ SO .4/ SO .2n/ The coset space SO .2n C 4/=SO .4/ SO .2n/ can be identified with the set of complex subspaces which are rj - invariant and have signature (2,2) with respect to the above pseudo-hermitian product. (In order to see that the SO .2n C 4/-action on this set is transitive, we have to verify that the orthogonal complement of such a subspace N The projection G ! G=K sends is again rj -invariant: This follows from rj h D h.) an element of G to the span of its first two and last two columns. The isomorphism h.2n; R/ ! nC is given by the coordinates 1 0 0 C B 0 C B C B u C iv uQ i vQ 0n C B C B uQ i vQ u i v 0n C B A @ 2iv0 0 ut C i vt uQ t C i vQ t 0 t t t t 0 2iv0 uQ C i vQ u i v 0 Thus, the isotropic para-pluriharmonic maps nC2;nC2 /; R R U ! Gr H 2;2 .C 1 2n 1 2n 1 2n .z˙ ; : : : ; z˙ / 7! VC .zC ; : : : ; zC / ˚ V .z ; : : : ; z /;

are given by 0

1 1 B C 0 B C B C zC C i wC B C VC D C B C Q Q i w z C C B1 C @ .zt zC wt wC C zQ tC zQ C C wQ tC wQ C / C i.2w C zt wC zQ tC wQ C /A C C C 0 2 2.zt zQ wt wQ C i.zt wQ wt zQ // 1 0 0 C B 1 C B C B QzC i wQ C C; B ˚CB C zC i wC C B A @ 2.zt zQ C wt wQ C i.zt wQ wt zQ // t t t C 1 t t t .zC zC wC wC C zQ C zQ C C wQ C wQ C / i.2w0 zC wC zQ C wQ C / 2


551

01

1 .zt z wt w C zQ t zQ C wQ t wQ / C i.2w0 zt w zQ t wQ / B C 2.zt zQ wt wQ C i.zt wQ wt zQ // B C B C z C i w B C V D C B C zQ i wQ B C @ A 1 0 1 0 2.zt zQ C wt wQ C i.zt wQ wt zQ // B 1 .zt z wt w C zQ t zQ C wQ t wQ / i.2w zt w zQ t wQ /C 0 C B2 C B Qz i wQ C; B ˚CB C z i w C B A @ 0 1 2

nC1 1 n 2n where z˙ D .z˙ ; : : : ; z˙ /, zQ ˙ D .z˙ ; : : : ; z˙ /, w˙ D .w1˙ ; : : : wn˙ /, wQ ˙ D ˙ ˙ 1 n a ˙ ˙ for a para-holomorphic .wnC1 ; : : : w2n / with w0 D h˙ .z˙ ; : : : ; z˙ /, wa D @h˙ =@z˙ n n function h D hC h W R R U ! R R.

3.6.5 Sp.2 n C 2 ; R/ = Sp.2 ; R/ Sp.2 n; R/. Recall that Sp.2n; R/ Š Sp .n/, therefore Sp.2n C 2; R/ = Sp.2; R/ Sp.2n; R/ can be identified P with the paraquaternionic projective space H Pn D f.h0 ; : : : ; hn / 2 H nC1 W a hN a ha ¤ 0g=H , cf. Example 2.2.1. It is useful to regard H n as a vector space over C and write Sp .n/ as a subgroup of GL.2n; C /: In fact, H n D C n ˚ i C n can be identified with the free C module C 2n . A C -linear map C 2n ! C 2n is H -linear, if and only if it commutes with the right multiplication by i , which is a C -antilinear involution. It is easy to see that ² ³ A Bx GL.n; H / D N 2 GL.2n; C / : B

A

Let h be a para-quaternionic hermitian metric on H n . The identification H D C ˚ i C yields a decomposition of h D g ˚ i ! into a para-hermitian scalar product g and a C -bilinear symplectic form ! on C 2n , which are related by ! D g.i; /. Therefore an endomorphism of C 2n preserves h, if and only if it preserves g and !, or equivalently, if it preserves ! and is H -linear. If h is standard, so is !. This shows that Sp .n/ D Sp.2n; C / \ GL.n; H /: Let G D Sp .nC1/. The Lie algebra of G is g D sp .nC1/ D sp.2nC2; C /\ gl.n C 1; H / and consists of matrices 1 0 A B bN a B A Bx C C 2 gl.2n C 2; C /; a 2 j R: B sp .n/ @ B AN A B t At b aN

552

Matthias Krahe

The canonical element 0 D@

1

j 02n

A j

leads to the twistor fibration Sp .n C 1/ Sp .n C 1/ ! Sp .n/ U .1/ Sp .n/ Sp .1/ (cf. Table 1). This is, of course, the fibration C P2nC1 ! H Pn of Example 2.2.1. The projection map Sp .n C 1/ ! C P2nC1 sends each matrix to the C -span of its first column. We x D G c =Pc is the space of have G c D Sp.2n C 2; C /. The closed twistor space Z all (possibly degenerate) one-dimensional para-complex subspaces in C P2nC1 (by the proof Proposition 2.1.2, any vector that spans such a subspace can be extended to x maps a matrix to the span of its first a symplectic basis). The projection G c ! Z column. The isomorphism of the Lie algebras 0

0 B u1 0 B B :: :: B : : B n @u v0 v1

h.n; C / ! ncC ; 1 0 0 C C B u 0n C C 7! B @ v 0n C A 0 2v0 vt ut vn 0

1 C C; A 0

(where u D .u1 ; : : : ; un /t , v D .v1 ; : : : ; vn /t ) yields the isomorphism 0

1 B z1 1 B B :: :: B : : B n @z w 0 w1

H.n; C / ! NCc ; 1 0 1 C C B z 1n C C 7! B @ w 0 C A 1 2w0 zt w wt wn 1

1 0 1n zt

C C: A 1

The holomorphic horizontal maps C n U ! NCc are given by the equations w0 D h.z 1 ; : : : ; z n /, wa D @h=@z a for a holomorphic function h W C n U ! C . The projection NC C P2nC1 ! H Pn maps this matrix to the H -span of its first column; in conclusion, the isotropic para-pluriharmonic maps f W C n U ! H Pn


553

are given by 0

1 P 1 C i.2h z a @h=@z a / B C z 1 C i @h=@z 1 B C 1 n .z ; : : : ; z / 7 ! H B C: :: @ A : z n C i @h=@z n

4 tt*-bundles and para-pluriharmonic maps Let .N; J / be a simply connected para-complex manifold. As shown in [S1] (see also [CS] for the complex case), any para-pluriharmonic map fQ W N ! GL.p C q; R/=O.p; q/ DW Sp;q which fulfills R.d fQ.T ˙ N /; d fQ.T ˙ N // D 0 defines a metric para-tt*-bundle over N and vice versa. In this paragraph we relate this to the construction of the associated family (cf. Section 3.1.4) and give a criterion for the map fQ to be isotropic. Definition 4.1.1. Let .N; J / be a simply connected para-complex manifold. A metric para-tt*-bundle .E; D; S; g/ over N is a real vector bundle E ! N endowed with a connection D, a (possibly indefinite) D-parallel fibre metric g and a section S of T N End.E/ which is g-symmetric (i.e. g.SX ; / D g.; SX /), such that the tt*-equation R D 0

for all 2 R

holds, where R is the curvature tensor of the connection rX WD DX C SeJ X . We denote by Symp;q .R/ the space of symmetric .p C q/ .p C q/ matrices with signature .p; q/. This space is identified with Sp;q via the map Sp;q 3 g 1

0

O.p; q/ 7! .g 1 /t 1p;q g 1 2 Symp;q .R/, where 1p;q D 0p 1q is the standard diagonal matrix of signature .p; q/. The geodesic reflection at eO.p; q/ 2 Sp;q yields the symmetric decomposition gl.p C q; R/ D o.p; q/ ˚ m, where m D fX 2 gl.p Cq; R/ W X t 1p;q D 1p;q Xg. The tangent space TgK Sp;q is canonically identified with Adg m D fX 2 gl.p Cq; R/ W X t .g t 1p;q g 1 / D .g t 1p;q g 1 /X g via the map .g/

Adg m ! .Adg m/g D gm ! TgK Sp;q (which is easily seen to be independent of the choice of g 2 gK). Theorem (Theorem 3 and 4 in [S1]). Let .N; J / be a simply connected para-complex manifold and let .E; D; S; g/ be a metric para-tt*-bundle of rank p Cq (and signature

554

Matthias Krahe

.p; q/) over N . Choose a family s .x/ W RpCq ! E of r -flat frames and set f W N ! Symp;q .R/; x 7! s .x/ gx D .g .s .x/a ; s .x/b //a;b ; f ! Sp;q ; fQ W N ! Symp;q .R/

fQ W N

f

/ Symp;q .R/

4/ Sp;q .

fQ

Then fQ is a 1-parameter family of para-pluriharmonic maps and d fQ .X/ D s1 B SeJ X B s :

(13)

Moreover, for each x 2 N , d fQ maps TxC N and Tx N into abelian subalgebras of TfQ .x/ Sp;q gl.p C q; R/. Conversely, any para-pluriharmonic map from a simply-connected para-complex manifold N into Sp;q such that the images of T ˙ N are abelian can be obtained in this way. Remark. The condition that the images of d fQ.T ˙ N / are abelian is equivalent to R.d fQ.T ˙ N /; d fQ.T ˙ N // D 0: Since R is preserved by parallel translation, it suffices to check this assertion at To Sp;q m, where R is given by the Lie triple bracket, so we have to show that for X; Y 2 m, ŒŒX; Y ; Z D 0 for all Z 2 m () ŒX; Y D 0 The diagonal matrices form a maximal abelian subalgebra of gl.p C q; R/, which is contained in m, so there is no nontrivial element of k D o.p; q/ that commutes with all Z 2 m. Since ŒX; Y 2 Œm; m k, the assertion follows. This 1-parameter family fQ of para-pluriharmonic maps is in fact an associated family in the sense of Theorem 3.1.5: Proposition 4.1.2. The map ˆ W fQ T Sp;q ! fQ0 T Sp;q ;

X 7! .s1 s0 /1 X.s1 s0 /;

is a parallel bundle homomorphism preserving the curvature of Sp;q , and ˆ B d fQ D d fQ0 B e J : Proof. The equation ˆ B d fQ D d fQ0 B e J is immediate from (13). ˆ .x/ is the differential of the isometry of the space Sp;q , given by left multiplication by s1 .x/s0 .x/ 2 GL.p C q; R/, hence it preserves the curvature tensor. In order to verify the parallelity of ˆ , let c W Œ0; 1 ! N be a curve and let W Œ0; 1 ! fQ T Sp;q


555

be a parallel section of fQ T Sp;q along c. Any such section is of the form t 7! .t /X , where X 2 m and is the horizontal lift of fQ B c, i.e. B D fQ B c and 1 P 2 m. We have to check that .ˆ B c/ is parallel, i.e. that Q WD ..s0 B c/1 .s B c// is horizontal: By the definition of s we have DX s D SeJ X s and DX s0 D SX s0 , (here D denotes the connection on E˝Rn induced by the given connection D on E and d .s01 s / D DcP .s01 /s C s0 DcP s D s01 Sce the trivial connection on Rn ) so dt P J cP s 1 P 1 1 1 and Q Q D P C s Sce s 2 m. J P „ ƒ‚ cP … 2Ad m

Proposition 4.1.3. fQ is isotropic if and only if there exists a family ˛ of D-parallel, g-orthogonal sections of End.E/, such that SeJ D ˛ B S B ˛1 : Proof. Suppose that fQ is isotropic, i.e. fQ D fQ0 and therefore d fQ D d fQ0 for all 2 R. Define ˛ WD s s01 . Then we see that SeJ D ˛ S˛1 , .D˛/s0 D D.˛s0 /˛D.s0 / D r s SeJ s ˛r 0 s0 C˛Ss0 D 0 and g.s0 ; s0 / D g.s ; s / D .˛ g/.s0 ; s0 /. Conversely, suppose that there exists a family ˛ as above. Choose a r 0 -flat frame s0 and set s WD ˛ s0 . Then by a similar calculation it follows that s is r -flat and that f D f0 . By Corollary 3.3.7, there are no full isotropic para-pluriharmonic maps into the symmetric space GL.pCq; R/=O.p; q/, since the Cartan involution .g/ D 1p;q g t 1p;q is not an inner automorphism of GL.p C q; C/. Among the totally geodesic submanifolds is the symmetric space Sp.2n; R/=GL.n; R/ GL.2n; R/=O.n; n/. Isotropic para-pluriharmonic maps into this space arise from special para-Kähler manifolds: Example: special Kähler manifolds. Let .N; J; g/ be a para-Kähler manifold and let r be a torsion-free flat connection on TN such that d r J D 0 and r! D 0, where ! WD g.; J / denotes the Kähler form of N . The quadruple .N; J; g; r/ is then called an affine special para-Kähler manifold. In [S1] it is shown that .E WD TN; D WD r S; S WD 12 J.rJ /; g/ is a metric para-tt*-bundle. From the definition of S , one sees after short calculation that SJX Y D JSX Y D SX J Y and SeJ D e .=2/J B S B e .=2/J ; .=2/J

(14)

is D-parallel and g-orthogonal, and the para-pluriharfurther DJ D 0, thus e monic map fQ W N ! Sn;n is isotropic by the above proposition. Indeed, fQ is a para-holomorphic map into the para-hermitian symmetric space Sp.2n; R/=U .n/ SL.2n; R/=SO.n; n/. In [S1] this is proved by relating fQ to the dual Gauß map of the canonical immersion N ,! C 2n . A direct argument is as follows: It is a direct consequence of r! D 0 and rs D 0 that the image of fQ is contained in Sp.2n; R/=U .n/ D Sp.2n; R/=.O.n; n/ \ Sp.2n; R//: The matrix-valued map

556

Matthias Krahe

x 7! s .x/!x is constant, and by adjusting the frame s we can assume that its image is the standard skew-symmetric form 2n . Now observe that 2 GL.2n; R/ is a representative for fQ.x/ if and only if s.x/ B is an orthonormal basis for gx . In particular, this is the case if s.x/ B is an orthonormal basis for the para-hermitian metric gx C j!x (such bases always exist by Proposition 1.3.3). For any such we have 2n D 2n , i.e. 2 Sp2n R. For the holomorphicity it is sufficient to consider one point x0 2 M . Again, by adjusting the frame s, we may assume fQ.x0 / D e U .n/. If gx0 and !x0 have standard form in the basis s.x0 / then so has Jx0 , i.e. Jx0 D s.x0 / B J2n B s.x0 /1 . From equation (14) we get ˆ D Ad.s1 s0 / D Ad.e .=2/J2n /. Differentiating the equation ˆ B d fQ D d fQ B e JN yields ad.J2n =2/ B d fQ D d fQ B JN ; which shows that fQ is holomorphic, because ad.J2n =2/ is the standard para-complex structure of Sp.2n; R/=U .n/: ad.J2n =2/ is zero on k D un D fX 2 sp.2n; R/ W XJ JX D 0g and has eigenvalues ˙1 on m D fX 2 sp.2n; R/ W XJ C JX D 0g. The twistor bundle is thus trivial (cf. Example 3.3.8).

References [AC1]

D. V. Alekseevsky and V. Cortés, Classification of pseudo-Riemannian symmetric spaces of quaternionic Kähler type. Amer. Math. Soc. Transl. Ser.(2) 213 (2005), 33–62. 510, 511, 539, 541

[AC2]

D. V. Alekseevsky and V. Cortés, The twistor spaces of a para-quaternionic Kähler manifold. Osaka J. Math. 45 (2008), no. 1, 215–251. 498, 510, 514, 516, 517

[AM]

R. Abraham and J. E. Marsden, Foundations of mechanics. Benjamin / Cummings Publishing, Reading, MA, 1978

[Be]

A. L. Besse, Einstein manifolds. Ergeb. Math. Grenzgeb. (3) 10, Springer-Verlag, Berlin 1987.

[BDM]

D. E. Blair, J. Davidov, and O. Mu˘skarov, Hyperbolic twistor spaces. Balkan J. Geom. Appl. 5 (2000), no. 2, 9–16. 498

[Bo]

A. Borel, Linear algebraic groups. 2nd enl. ed., Grad. Texts in Math. 126, SpringerVerlag, New York 1991. 531, 532, 533

[BT]

A. Borel and J. Tits, Groupes réductifs. Inst. Hautes Études Sci. Publ. Math. 27 (1965), 55–151.

[Br]

R. Bryant, Conformal and minimal immersions of compact surfaces into the 4sphere. J. Differential Geom. 17 (1982), 455–473. 497

[Bu]

F. E. Burstall, Minimal surfaces in quaternionic symmetric spaces. In Geometry of low-dimensional manifolds, 1, London Math. Soc. Lecture Note Ser. 150, Cambrigde University Press, Cambrigde 1990, 231–235. 498

[BEFT]

F. E. Burstall, J. H. Eschenburg, M. J. Ferreira, and R. Tribuzy, Kähler submanifolds with parallel pluri-mean curvature. Differential Geom. Appl. 20 (2004), 47–66.

Chapter 15. Para-pluriharmonic maps and twistor spaces [BR] [BS] [CLS]

[CMMS1]

[CMMS2]

[CS]

[DE] [DJS] [E] [EQ] [ET1]

[ET2] [H] [L] [MS] [O’N] [S1]

[S2] [W]

557

F. E. Burstall and J. H. Rawnsley, Twistor theory for Riemannian symmetric spaces. Lecture Notes in Math. 1424, Springer-Verlag, Berlin 1990. 497, 520 F. E. Burstall and S. Salomon, Tounaments, flags and harmonic maps. Math. Ann. 277 (1987), 249–265. 497 V. Cortés, M.-A. Lawn, and L. Schäfer, Affine hyperspheres associated to special para-Kähler manifolds. Int. J. Geom. Methods Mod. Phys. 3 (2006), no. 5–6, 995–1009. V. Cortés, Ch. Mayer, Th. Mohaupt, and F. Saueressig, Special geometry of Euclidean supersymmetry I: Vector multiplets. J. High Energy Phys. 0403 (2004), 028. 498, 503, 509 V. Cortés, Ch. Mayer, Th. Mohaupt, and F. Saueressig, Special geometry of Euclidean supersymmetry II: Hypermultiplets and the c-map. J. High Energy Phys. 0506 (2005), 025. 498 V. Cortés and L. Schäfer, Topological-antitopological fusion equations, pluriharmonic maps and special Kähler manifolds. In Complex, contact and symmetric manifolds (O. Kowalski, E. Musso, D. Perrone, eds.), Progr. Math. 234, Birkhäuser, Boston, MA, 2005, 59–74. 553 J. Dorfmeister and J.-H. Eschenburg, Pluriharmonic maps, loop groups and twistor theory. Ann. Global Anal. Geom. 24 (2003), 301–321. 525 A. S. Dancer, H. R. Jørgensen, and A. F. Swann, Metric geometries over the split quaternions. Rend. Semin. Mat. Torino 63 (2005), no. 2, 119–139. 541 J. H. Eschenburg, Lecture notes on symmetric spaces. Preprint, Augsburg 1997; http://www.math.uni-augsburg.de/ eschenbu/. 521 J. H. Eschenburg and P. Quast, Pluriharmonic maps into Kähler symmetric spaces and Sym’s formula. Preprint. 524, 525 J. H. Eschenburg and R. Tribuzy, Existence and uniqueness of maps into affine homogenous spaces. Rend. Sem. Mat. Univ. Padova 89 (1993), 11–18. 522, 523, 526 J. H. Eschenburg and R. Tribuzy, Isotropic pluriminimal submanifolds. Mat. Contemp. 17 (1999), 171–191. 520, 530 S. Helgason, Differential geometry, Lie groups and symmetric spaces, Pure Appl. Math. 80, Academic Press, New York 1978. 521 H. B. Lawson, Surfaces minimales et la construction de Calabi-Penrose. Sém. Bourbaki 624 (1983/84); Astérisque 121–122 (1985), 197–211. 498 D. McDuff and D. Salamon, Introduction to symplectic topology. Oxford University Press, Oxford, New York, 1998. B. O’Neill, Semi-Riemannian geometry. Academic Press, NewYork, London 1983. 521, 522 L. Schäfer, tt*-bundles in para-complex geometry, special para-Kähler manifolds and para-pluriharmonic maps. Differential Geom. Appl. 24 (2006), 60–89. 553, 555 L. Schäfer, Géométrie tt* et applications pluriharmoniques. PhD Thesis, Nancy/ Bonn 2006. 499 N. M. J. Woodhouse, Geometric quantization. Oxford University Press, Oxford 1994.

Chapter 16

Maximally homogeneous para-CR manifolds of semisimple type Dmitri V. Alekseevsky, Costantino Medori, and Adriano Tomassini

Contents 1 2

Introduction and notation . . . . . . . . . . . . . . . . . . . . Graded Lie algebras associated with para-CR structures . . . 2.1 Gradations of a Lie algebra . . . . . . . . . . . . . . . . 2.2 Fundamental algebra associated with a distribution . . . 2.3 Para-CR algebras and regular para-CR structures . . . . 3 Prolongations of graded Lie algebras . . . . . . . . . . . . . 3.1 Prolongations of negatively graded Lie algebras . . . . . 3.2 Prolongations of non-positively graded Lie algebras . . . 4 Standard almost para-CR manifolds . . . . . . . . . . . . . . 4.1 Maximally homogeneous Tanaka structures . . . . . . . 4.2 Tanaka structures of semisimple type . . . . . . . . . . . 4.3 Models of almost para-CR manifolds . . . . . . . . . . . 5 Fundamental gradations of a complex semisimple Lie algebra 6 Fundamental gradations of a real semisimple Lie algebra . . . 6.1 Real forms of a complex semisimple Lie algebra . . . . 6.2 Gradations of a real semisimple Lie algebra . . . . . . . 7 Classification of maximally homogeneous para-CR manifolds References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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559 561 561 562 562 563 563 563 564 564 565 566 567 569 569 570 572 576

1 Introduction and notation Let M be a 2n-dimensional manifold. An almost paracomplex structure on M is a field of endomorphisms K 2 End.TM / of the tangent bundle TM of M such that K 2 D id. It is called an (almost) paracomplex structure in the strong sense if its ˙1-eigenspace distributions T ˙ M D fX ˙ KX j X 2 .M; TM /g have the same rank (see e.g. [13], [9]). An almost paracomplex structure K is called a paracomplex structure, if it is integrable, i.e., S.X; Y / D ŒX; Y C ŒKX; K Y KŒX; K Y KŒKX; Y D 0

560

Dmitri V. Alekseevsky, Costantino Medori, and Adriano Tomassini

for any vector fields X; Y 2 .TM /. This is equivalent to say that the distributions T ˙ M are involutive. Recall that an almost CR-structure of codimension k on a 2n C k-dimensional manifold M is a distribution HM TM of rank 2n together with a field of endomorphisms J 2 End.HM / such that J 2 D id. An almost CR-structure is called CR-structure, if the ˙i -eigenspace subdistributions H˙ M of the complexified tangent bundle T CM are involutive. We define an almost para-CR structure in a similar way. Definition 1.1. An almost para-CR structure of codimension k on a 2n C k-dimensional manifold M (in the weak sense) is a pair .HM; K/, where HM TM is a rank 2n distribution and K 2 End.HM / is a field of endomorphisms such that K 2 D id and K ¤ ˙id. An almost para-CR structure is said to be a para-CR structure, if the eigenspace subdistributions H˙ M HM are integrable or equivalently if the following integrability conditions hold: ŒKX; K Y C ŒX; Y 2 .HM /;

(1)

S.X; Y / WD ŒX; Y C ŒKX; K Y K.ŒX; K Y C ŒKX; Y / D 0

(2)

for all X; Y 2 .HM /. If the eigenspace distributions H˙ M D fX ˙ KX j X 2 .M; HM /g of an almost para-CR structure have the same rank, then .HM; K/ is called an almost para-CR structure in the strong sense. A straightforward computation shows that the integrability condition is equivalent to the involutivness of the distributions HC M and H M . A manifold M , endowed with an (almost) para-CR structure, is called an (almost) para-CR manifold . Note that a direct product of (almost) para-CR manifolds is an (almost) para-CR manifold. One can associate with a point x 2 M of a para-CR manifold .M; HM; K/ a fundamental graded Lie algebra m. A para-CR structure is said to be regular if these Lie algebras mx do not depend on x. In this case, a para-CR structure can be considered as a Tanaka structure (see [3] and Section 4). A regular para-CR structure is called a structure of semisimple type if the full prolongation g D m1 D md C C m1 C g0 C g1 C of the associated non-positively graded Lie algebra gd C C g1 C g0 (which is an analogue of the generalized Levi form of a CR structure) is a semisimple Lie algebra. Such a para-CR structure defines a parabolic geometry and its group of automorphisms Aut.M; HM; K/ is a Lie group of dimension dim g.

Chapter 16. Maximally homogeneous para-CR manifolds of semisimple type

561

Recently in [16], P. Nurowski and G. A. J. Sparling consider the natural paraCR structure which arises on the 3-dimensional space M of solutions of a second order ordinary differential equation y 00 D Q.x; y; y 0 /. Using the Cartan method of prolongation, they construct the full prolongation G ! M of M with a sl.3; R/valued Cartan connection and a quotient line bundle over M with a conformal metric of signature .2; 2/. This is a para-analogue of the Feffermann bundle of a CR-structure. They apply these bundles to the initial ODE and get interesting applications. In [2] we proved that a para-CR structures of semisimple type on a simply connected manifold M with the automorphism group of maximal dimension dim g can be identified with a (real) flag manifold M D G=P where G is the simply connected Lie group with the Lie algebra g and P the parabolic subgroup generated by the parabolic subalgebra p D g0 C g1 C C gd : We gave a classification of maximally homogeneous para-CR structures of semisimple type such that the associated graded semisimple Lie algebra g has depth d D 2. In the present chapter we classify all maximally homogeneous para-CR structures of semisimple type in terms of graded real semisimple Lie algebras.

2 Graded Lie algebras associated with para-CR structures 2.1 Gradations of a Lie algebra Recall that a gradation (more precisely a Z-gradation) of depth k of a Lie algebra g is a direct sum decomposition X gi D gk C gkC1 C C g0 C C gj C (3) gD i2Z i

j

such that Œg ; g giCj , for any i; j 2 Z, and gk ¤ f0g. Note that g0 is a subalgebra of g and each gi is a g0 -module. We say that an element x 2 gj has degree j and we write d.x/ D j . The endomorphism ı of g defined by ıjgj D j id is a semisimple derivation of g (with integer eigenvalues), whose eigenspaces determine the gradation. Conversely, any semisimple derivation ı of g with integer eigenvalues defines a gradation where the grading space gj is the eigenspace of ı with eigenvalue j . If g is a semisimple Lie algebra, then any derivation ı is inner, i.e., there exists d 2 g such that ı D add . The element d 2 g is called the grading element. Definition 2.1. A gradation g D called

P

gi of a Lie algebra (or a graded Lie algebra g) is

(1) fundamental, if the negative part m D

P i 1.

Then evaluating vector fields at a point x 2 M , we get a flag Tx M Hd 1 .x/ D Hd .x/ © Hd C1 .x/ H2 .x/ H1 .x/ D Hx in Tx M , where

Hi .x/ D fXjx j X 2 .H /i g:

Let us assume that Hd .x/ D Tx M . The commutators of vector fields induce a structure of fundamental negatively graded Lie algebra on the associated graded space m.x/ D gr.Tx M / D md .x/ C md C1 .x/ C C m1 .x/; where mj .x/ D Hj .x/=Hj C1 .x/. Note that m1 .x/ D Hx . A distribution H is called a regular distribution of depth d and type m if all graded Lie algebras m.x/ are isomorphic to a given graded fundamental Lie algebra m D md C md C1 C C m1 : In this case m is called the Lie algebra associated with the distribution H . A regular distribution H is called non-degenerate if the associated Lie algebra is non-degenerate.

2.3 Para-CR algebras and regular para-CR structures We recall the following Definition 2.2. A pair .m; Ko /, where m D md C C m1 is a negatively graded fundamental Lie algebra and Ko is an involutive endomorphism of m1 , is called a 1 are para-CR algebra of depth d . If, moreover, the ˙1-eigenspaces m1 ˙ of Ko on m commutative subalgebras of m, then .m; Ko / is called an integrable para-CR algebra.

Chapter 16. Maximally homogeneous para-CR manifolds of semisimple type

563

Definition 2.3. Let .m; Ko / be a para-CR algebra of depth d . An almost para-CR structure .HM; K/ on M is called regular of type .m; Ko / and depth d if, for any x 2 M , the pair .m.x/; Kx / is isomorphic to .m; Ko /. We say that the regular almost para-CR structure is non-degenerate if the graded algebra m is non-degenerate. Note that a regular almost para-CR structure of type .m; K0 / is integrable if and only if the Lie algebra .m; K0 / is integrable.

3 Prolongations of graded Lie algebras 3.1 Prolongations of negatively graded Lie algebras The full prolongation of a negatively graded fundamental Lie algebra m D md C C m1 is defined as a maximal graded Lie algebra g.m/ D gd .m/ C C g1 .m/ C g0 .m/ C g1 .m/ C with the negative part

gd .m/ C C g1 .m/ D m

such that the following transitivity condition holds: if X 2 gk .m/; k 0; ŒX; g1 .m/ D f0g; then X D 0: In [17], N. Tanaka proved that the full prolongation g.m/ always exists and that it is unique up to isomorphisms. Moreover, it can be defined inductively by 8 i ˆ i < 0; <m ; i j j g .m/ D fA 2 Der.m; m/ j A.m / m for all j < 0g; i D 0; ˆ P : h j iCj for all j < 0g; i > 0; fA 2 Der.m; h B > ˆ t : ˆ > ˆ 0 t ; : 0 0 ˇ B c2R 0 0 0

Chapter 17. Recent developments in pseudo-Riemannian holonomy theory

609

In [56] A. Ikemakhen classified indecomposable, non-irreducible subalgebras of so.2; n C 2/ that contain the ideal 0 1 0 0 1 0

B A WD RB @ 0 0 0 0

0 0

0 0 0 0

1 0

0 0

C C so.2; n C 2/ : A

0

As in the Lorentzian case, to each subalgebra h so.2; n C 2/ one can associate its projection onto so.n/ so.2; n C 2/ . In [56] it was noted that such projection g of the holonomy algebra may not be a holonomy algebra of a Riemannian manifold. Moreover, in the next section we will see that there is no additional condition on the subalgebra g so.n/ induced by the Bianchi-identity and replacing the weak-Berger property.

4.1 The orthogonal part of indecomposable, non-irreducible subalgebras of so.2 ; n C 2/ Let us now consider one type of indecomposable, non-irreducible subalgebras of so.2; n C 2/. For any subalgebra g so.n/ define the Lie algebra 80 9 1 ˇ 0 0 Xt 0 c ˆ > ˇ ˆ > t ˇ ˆ > 0 0 : 0 0 0 ; ˇ 0

0

0

0

0

We identify an element of the Lie algebra hg with the 4-tuple .A; X; Y; Z/. It is easy to see that the subalgebra hg so.2; n C 2/ is indecomposable. The following theorem was proved in [44]. Theorem 4.1. For any subalgebra g so.n/, the Lie algebra hg can be realised as the holonomy algebra of a pseudo-Riemannian manifold of signature .2; n C 2/. Let g so.n/ be a subalgebra. To prove Theorem 4.1, in [44] a polynomial metric on RnC4 is constructed, the holonomy algebra of which at the point 0 is exactly hg . This was done in the following way. First we will explain why for any subalgebra g so.n/, the Lie algebra hg is a Berger algebra. For u; v 2 E WD spanfE1 ; : : : ; En g, any symmetric linear maps T1 ; T2 W E ! E and any linear map S W E ! E such that S S 2 g let RT1 .Z1 ; u/ D .0; T1 .u/; 0; 0/;

RT2 .Z2 ; u/ D .0; 0; T2 .u/; 0/;

RS .Z1 ; Z2 / D .S S ; 0; 0; 0/;

RS .Z1 ; u/ D .0; 0; S.u/; 0/;

RS .Z2 ; u/ D .0; S .u/; 0; 0/;

RS .u; v/ D .0; 0; 0; hS.u/; vi hu; S.v/i/

610

Anton Galaev and Thomas Leistner

and extend these linear maps to R2;nC2 ˝ R2;nC2 in the trivial way. It easy to check that RT1 ; RT2 ; RS 2 K.hg /. Obviously, curvature endomorphisms of these forms generate hg . Now let dim g D N and let A1 ; : : : ; AN be a basis of the vector space g. Denote by i the elements of the matrices A˛ . Let .x1 ; x2 ; y1 : : : ; yn ; z1 ; z2 / be the canonical Aj˛ coordinates on RnC4 . Consider the following metric on RnC4 : g D 2dx1 dz1 C 2dx2 dz2 C

n n X X .dyi /2 C 2 ui dyi dz2 C f .dz1 /2 ; iD1

iD1

where ui D

n N X X 1 i Aj˛ yj z1˛ ˛Š ˛D1 j D1

and

f D

n X .yi /2 : iD1

This metric is constructed in such a way that

pr so.n/ r@z1 : : : r@z1 R @z1 ; @z2 0 D Sr Sr D Ar ; „ ƒ‚ …

(1)

.r1/times

for r D 1; : : : ; N , where Sr D 12 Ar , and the images of other covariant derivatives of R are contained in hg .

4.2 Holonomy groups of pseudo-Kählerian manifolds of index 2 A pseudo-Riemannian manifold .M; g/ is called pseudo-Kählerian if there exists a parallel smooth field of endomorphisms J of the tangent bundle of M that satisfies J 2 D id and g.JX; J Y / D g.X; Y / for all vector fields X and Y on M . The holonomy algebra of a pseudo-Kählerian manifold of signature .2; 2nC2/ is contained in u.1; n C 1/. From Theorem 2.3 it follows that the holonomy algebra of a pseudo-Kählerian manifold of .2; 2n C 2/ is a direct sum of irreducible holonomy algebras of Kählerian manifolds and of the indecomposable holonomy algebra of a pseudo-Kählerian manifold of signature .2; 2k C 2/. If the last algebra is irreducible, The Berger list in Theorem 2.4 implies that it is either u.1; k C 1/ or su.1; k C 1/, or that it is the holonomy algebra of an irreducible hermitian symmetric space and listed in [15]. If the last algebra is not irreducible, again we are left with the problem of classifying indecomposable, non-irreducible holonomy algebras in u.1; k C 1/. In order to describe this classification we denote by R2;2nC2 the vector space 2nC4 endowed with a complex structure J and with a J -invariant metric h ; i of R signature .2; 2n C 2/, i.e., hJ x; Jyi D hx; yi for all x; y 2 R2;2nC2 . We fix a basis X1 , X2 , E1 ; : : : ; En , F1 ; : : : ; Fn , Z1 , Z2 of R2;2nC2 such that the Gram matrix of


the metric h ; i and the complex structure J 0 0 1 0 0 0 0 1 01 1 0 B B 0 0 0 0 1C B B0 0 I2n 0 0C and B0 0 B0 0 @ A @ 1 0 0 0 0 0

1

0

0

0 0

0

0 0

611

have the form 0 0 0 In 0 0

0 0 In 0 0 0

0 0 0 0 0 1

1

0 0C 0C C; 0C A 1 0

respectively:

We denote by u.1; n C 1/ the subalgebra of u.1; n C 1/ that preserves the J invariant 2-dimensional isotropic subspace D RX1 ˚ RX2 R2;2nC2 . The Lie algebra u.1; n C 1/ can be identified with the following matrix algebra 80 9 1 a1 a2 z1t z2t 0 c ˇ ˆ > ˆ ˇ t t ˆBa2 a1 > z2 z1 c 0 C ˇ a ; a ; c 2 R; > ˆ > ˆ ˇ > @ A 2 u.n/ ˆ 0 > 0 0 0 a1 a2 ˆ ˇ C B > : ; 0

Recall that u.n/ D

0

˚ B

C C B

0

0

a2

a1

ˇ ˇ B 2 so.n/; C 2 gl.n; R/; C t D C

and su.n/ D

˚ B

C C B

ˇ 2 u.n/ ˇ tr C D 0 :

We identify an element of u.1; n C 1/ with the 4-tuple .a1 C ia2 ; B C iC; z1 C iz2 ; c/: The non-vanishing components of the Lie brackets in u.1; n C 1/ are the following: Œ.0; B C iC; 0; 0/; .0; B1 C iC1 ; 0; 0/ D .0; ŒB C iC; B1 C iC1 u.n/ ; 0; 0/; Œ.a1 ; 0; 0; 0/; .0; 0; z1 C iz2 ; c/ D .0; 0; a1 .z1 C iz2 /; 2a1 c/; Œ.i a2 ; 0; 0; 0/; .0; 0; z1 C iz2 ; 0/ D .0; 0; a2 .z2 iz1 /; 0/; Œ.0; B C iC; 0; 0/; .0; 0; z1 C iz2 ; 0/ D .0; 0; Bz1 C z2 C i.C z1 C Bz2 /; 0/; Œ.0; 0; z1 C iz2 ; 0/; .0; 0; w1 C iw2 ; 0/ D .0; 0; 0; 2.z1 w2t C z2 w1t //: Hence we obtain the decomposition u.1; n C 1/ D .C ˚ u.n// Ë .C n Ë R/: Denote by su.1; nC1/ the subalgebra of su.1; nC1/ that preserves the subspace RX1 ˚ RX2 R2;2nC2 . Then su.1; n C 1/ D f.a1 C ia2 ; B C iC; z1 C iz2 ; c/ 2 u.1; n C 1/ j 2a2 C trR C D 0g and u.1; n C 1/ D su.1; n C 1/ ˚ RJ:

612


Note that

9 80 1 0 c ˇ > ˆ = < a1 a2 ˇ c 0 Cˇ Ba2 a1 ; a ; c 2 R a u.1; 1/ D @ 0 ˇ A 1 2 0 a1 a2 ˇ > ˆ ; : 0

0

a2

a1

If an indecomposable subalgebra h u.1; n C 1/ preserves a degenerate proper subspace W R2;2nC2 , then h preserves the J -invariant 2-dimensional isotropic subspace W1 R2;2nC2 , where W1 D .W \ J W / \ .W \ J W /? if W \ J W ¤ f0g and W1 D .W ˚ J W / \ .W ˚ J W /? if W \ J W D f0g. Therefore h is conjugated to an indecomposable subalgebra of u.1; n C 1/ . The classification of indecomposable, non-irreducible holonomy algebras in u.2; n C 2/ is given by the following theorem, for the second part of which we fix some notation: 0 m n is an integer, u u.m/ is a subalgebra (as above, u D z ˚ u0 ), we fix the decomposition C n D C m ˚ C nm , Rnm C nm is a real form and Jnm u.n m/ u.n/ u.1; n C 1/ is the complex structure on C nm . Let '; W u ! R be linear maps with 'ju0 D ju0 D 0. Theorem 4.2. 1) A subalgebra h u.1; 1/ is the indecomposable, non-irreducible holonomy algebra of a pseudo-Kählerian manifold of signature .2; 2/ if and only if h is conjugated to u.1; 1/ or one of the following subalgebras of it: 80 9 1 ˇ 0 0 ˆ > < a1 a2 = ˇ a1 0 0 Cˇ Ba h2nD0 D @ 02 ; a 2 R I a ˇ 0 a1 a2 A ˇ 1 2 ˆ > : ; 0

1 ;2 hnD0

0

a2

a1

80 9 1 ˇ 0 c ˆ > < a1 a2 ˇ = a1 c 0 Cˇ Ba D @ 02 a; c 2 R ; ˇ 0 a1 a2 A ˇ ˆ > : ; 0

0

a2

where 1 ; 2 2 R;

a1

2) Let n 1. Then a subalgebra h u.1; n C 1/ is the indecomposable, nonirreducible holonomy algebra of a pseudo-Kählerian manifold of signature .2; 2nC2/ if and only if h is conjugated to one of the following subalgebras of u.1; n C 1/ : hm;u D .R ˚ R.i C Jnm / ˚ u/ Ë ..C m ˚ Rnm / Ë R/I hm;u; D .R ˚ f .A/.i C Jnm / C A j A 2 ug/ Ë ..C m ˚ Rnm / Ë R/I hm;u;'; D f'.A/ C .A/.i C Jnm / C A j A 2 ug Ë ..C m ˚ Rnm / Ë R/I hm;u;' D .R.i C Jnm / ˚ f'.A/ C A j A 2 ug/ Ë ..C m ˚ Rnm / Ë R/I hm;u; D .R.1 C .i C Jnm // ˚ u/ Ë ..C m ˚ Rnm / Ë R/; h

n;u; ;k;l

D fA C

k

.A/ j A 2 ug Ë ..C ˚ R

nl

where 2 RI

/ Ë R/;

where k and l are integers such that 0 < k l n, we have the decomposition C n D C k ˚ C lk ˚ C nl , u u.k/ is a subalgebra with dim z.u/ n C l 2k


and

W u ! C lk ˚ iRnl is a surjective linear map with hm;u;

;k;l;r

D fA C

613

ju0 D 0;

.A/ j A 2 ug Ë ..C k ˚ Rml ˚ Rrm / Ë R/;

where k, l, r and m are integers such that 0 < k l m r n and m < n, we have the decomposition C n D C k ˚ C lk ˚ C ml ˚ C rm ˚ C nr , u u.k/ is a subalgebra with dim z.u/ n C m C l 2k r and W u ! C lk ˚ i Rml ˚ Rnr is a surjective linear map with ju0 D 0. 1 D1;2 D0 Note that hnD0 D su.1; 1/ . As an example for the Lie algebras in the theorem, an element in the Lie algebra hm;u;'; is given by 1 0 t 0t t

'.A/

B .A/ B B 0 B B 0 B B 0 B B 0 B @ 0 0

.A/ '.A/ 0 0 0 0 0 0

z1 z2t B 0 C 0 0 0

z1 0 0 0 0 .A/Inm 0 0

z2 z1t C 0 B 0 0 0

with c 2 R, z1 ; z2 2 Rm , z10 2 Rnm , A D

B

0 z10t 0 .A/Inm 0 0 0 0

C C B

0 c z1 z10 z2 0 '.A/ .A/

c 0 C C z2 C C 0 C C z1 C C z10 C C .A/A '.A/

2 u.

As a corollary, we get the classification of indecomposable, non-irreducible holonomy algebras contained in su.1; n C 1/, i.e., of the holonomy algebras of special pseudo-Kählerian manifolds (these manifolds are pseudo-Kählerian and Ricci-flat). Corollary 4.3. Let .M; h; J / be a indecomposable, non-irreducible pseudo-Kählerian manifold of signature .2; nC2/ which is Ricci-flat. Then its holonomy algebra is given by 1 D0;2 D0 1 D1;2 D0 or su.1; 1/ D hnD0 hnD0 in the case n D 0, and by hm;u; ; hm;u;'; with u su.n/ and .A/ D

1 trC A; nmC2

and hn;u;

;k;l

; hm;u;

;k;l;r

with u su.k/

for n > 0. As in the first case, we consider all subalgebras of u.1; 1/ and show which of these subalgebras are indecomposable Berger subalgebras. We realise the Lie algebras of Part 1) of Theorem 4.2 as the holonomy algebras of pseudo-Kählerian metrics on R2;2 , see Section 5.2. By analogy to the case of Lorentzian manifolds, the proof of Part 2) of Theorem 4.2 consists of the following 3 steps:

614


Step 1. Classification of indecomposable, non-irreducible subalgebras of su.1; nC1/. Step 2. Classification of indecomposable, non-irreducible Berger subalgebras of u.1; n C 1/. Step 3. Construction of a pseudo-Riemannian manifold with the holonomy algebra h for each indecomposable, non-irreducible Berger subalgebra h u.1; n C 1/. Step 1. First we classify all connected subgroups of SU.1; n C 1/ that act indecomposably and non-irreducibly on R2;2nC2 , that is equivalent to the classification of indecomposable, non-irreducible subalgebras of su.1; n C 1/. Any such subgroup preserves a 2-dimensional isotropic J -invariant subspace of R2;2nC2 . We use a generalization of the method from [47] (see Section 3.2). We denote by C 1;nC1 the .n C 2/-dimensional complex vector space given by 2;2nC2 .R ; J; /. Let g be the pseudo-Hermitian metric on C 1;nC1 of signature .1; n C 1/ corresponding to . If a subgroup G U.1; n C 1/ acts indecomposably on R2;2nC2 , then G acts indecomposably on C 1;nC1 , i.e., does not preserve any proper g-non-degenerate complex vector subspace. nC1 nC1 of the complex hyperbolic space HC . The We consider the boundary @HC nC1 nC1 boundary @HC consists of complex isotropic lines of C 1;nC1 . We identify @HC 2nC1 with the .2n C 1/-dimensional sphere S . Consider the complex isotropic line D CX1 D RX1 ˚ RX2 C 1;nC1 and denote by U.1; n C 1/ U.1; n C 1/ the connected Lie subgroup that preserves the line . The Lie algebra of the Lie group U.1; n C 1/ is u.1; n C 1/ . Any connected subgroup G U.1; n C 1/ that acts on C 1;nC1 indecomposably and non-irreducibly is conjugated to a subgroup of U.1; n C 1/ . From the above decomposition of the Lie algebra u.1; n C 1/ we obtain the decomposition U.1; n C 1/ D .C U.n// Ë .C n Ë R/: nC1 nfg D S 2nC1 nfpointg with the Heisenberg space We identify the set @HC n Hn D C ˚R. Any element f 2 U.1; nC1/ induces a transformation .f / of Hn , moreover, .f / 2 Sim Hn , where Sim Hn is the group of the Heisenberg similarity transformations of Hn . For the Lie group Sim Hn we have the decomposition

Sim Hn D .RC U.n// Ë .C n Ë R/: The elements 2 RC , A 2 U.n/ and .z; u/ 2 C n Ë R act on Hn D C n ˚ R in the following way: W .z; u/ 7! .z; 2 u/ (real Heisenberg dilation about the origin), A W .z; u/ 7! .Az; u/ (Heisenberg rotation about the vertical axis), .w; v/ W .z; u/ ! 7 .w C z; v C u C 2 Im g.w; z// (Heisenberg translations ). We show that W U.1; nC1/ ! Sim Hn is a surjective Lie group homomorphism with the kernel T , where T is the 1-dimensional subgroup generated by the complex structure J 2 U.1; n C 1/ . In particular, T is the center of U.1; n C 1/ . Let SU.1; nC1/ D U.1; nC1/ \SU.1; nC1/. Then U.1; nC1/ D SU.1; nC1/ T


615

and the restriction jSU.1;nC1/ W SU.1; n C 1/ ! Sim Hn is a Lie group isomorphism. We consider the natural projection W Sim Hn ! Sim C n , where Sim C n D .RC U.n// Ë C n is the group of similarity transformations of C n . The homomorphism is surjective and its kernel is 1-dimensional. We prove that if a subgroup G U.1; n C 1/ acts indecomposably on C 1;nC1 , then (1) the subgroup ..G// Sim C n does not preserve any proper complex affine subspace of C n ; (2) if ..G// Sim C n preserves a proper non-complex affine subspace L C n , then the minimal complex affine subspace of C n containing L is C n . This is the key statement for our classification. Since we are interested in connected Lie groups, it is enough to classify the corresponding Lie algebras. The classification is done in the following way: First we describe non-complex vector subspaces L C n with spanC L D C n (it is enough to consider only vector subspaces, since we do the classification up to conjugacy). Any such non-complex vector subspace has the form L D C m ˚ Rnm , where 0 m n. Here we have 3 types of subspaces: 1) m D 0 (L is a real form of C n ); 2) 0 < m < n; 3) m D n (L D C n ). We describe the Lie algebras f of the connected Lie subgroups F Sim C n preserving L. Without loss of generality, we can assume that each Lie group F does not preserve any proper affine subspace of L. This means that F acts irreducibly on L. By a theorem of D. V. Alekseevsky [2], [3], F acts transitively on L. In Theorem 3.4 we divided transitive similarity transformation groups of Euclidean spaces into 4 types. Here we unify type 2 and 3. The group F is contained in .RC SO.L/ SO.L? // Ë L, where RC is the group of real dilations of C n about the origin and L is the group of all translations in C n by vectors of L. In general situation we know only the projection of F on Sim L D .RC SO.L// Ë L, but in our case the projection of F on SO.L/ SO.L? / is also contained in U.n/ and we know the full information about F . On this step we obtain 9 types of Lie algebras. Then we describe subalgebras a LA.Sim Hn / with .a/ D f. For each f we have 2 possibilities: a D f C ker or a D fx C .x/ j x 2 fg, where

W f ! ker is a linear map. Using the isomorphism .jsu.1;nC1/ /1 we obtain a list of subalgebras h su.1; n C 1/ . This gives us 12 types of Lie algebras.

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Finally we check which of the so obtained subalgebras of su.1; n C 1/ so.2; 2n C 2/ are indecomposable. It turns out that some of the types contain Lie algebras that are not indecomposable. Giving new definitions to these types we obtain 12 types of indecomposable Lie algebras. Unifying some of the types we obtain 8 types of indecomposable subalgebras of su.1; n C 1/ so.2; 2n C 2/. Example 4.4. Let g so.n/ be a subalgebra with non-trivial center and W g ! R be a non-zero linear map with jg0 D 0. Then the subalgebra f.0; B; z1 ; .B// j B 2 g; z1 2 Rn g su.1; n C 1/ so.2; 2n C 2/ is indecomposable. Note that the Lie algebra of the above example was not considered in [56]. Step 2. In this step we classify indecomposable Berger subalgebras of u.1; n C 1/ . First we obtain a list of candidates for the indecomposable subalgebras of u.1; n C 1/ so.2; 2n C 2/. For each f LA.Sim Hn / as above and for each h su.1; n C 1/ with ..h// D f we consider the Lie algebras hJ D h ˚ RJ and h D fx C .x/ j x 2 hg, where W h ! R is a non-zero linear map. As we claimed above, any indecomposable subalgebra of u.1; n C 1/ so.2; 2n C 2/ is of the form h, hJ or h . These subalgebras are candidates for the indecomposable subalgebras of u.1; n C 1/ so.2; 2n C 2/. We associate with each of these subalgebras an integer 0 m n. If m > 0, then the subalgebras of the form h, hJ and h u.1; n C 1/ are indecomposable. We have inclusions u.m/ u.n/ u.1; nC1/ and projection maps pr u.m/ W u.1; n C 1/ ! u.m/, pr u.n/ W u.1; n C 1/ ! u.n/. For any integer 0 m n and any subalgebra u u.m/ ˚ ..so.n m/ ˚ u.1; n C 1/ and describe so.n m// \ u.n m// we consider a subalgebra hm;u 0 m;u the space K.hm;u /. The Lie algebras of the form h contain all candidates for the 0 0 the space indecomposable subalgebras of u.1; n C 1/ . For any subalgebra h hm;u 0 K.h/ can be found from the following condition: 2;2nC2 R 2 K.h/ if and only if R 2 K.hm;u ; R2;2nC2 / h: 0 / and R.R

Using this, we easily find all indecomposable, non-irreducible Berger subalgebras of u.1; n C 1/ . Step 3. As the last step of the classification, we construct metrics on R2nC4 that realise all Berger algebras obtained above as holonomy algebras. Idea of constructions of the metrics is similar to the one in Section 4.1. The coefficients of the metrics are polynomial functions, hence the corresponding Levi-Civita connections are analytic and in each case the holonomy algebra at the point 0 2 R2nC4 is generated by the operators images of the curvature tensor and of all its derivatives. We explicitly compute for each metric the components of the curvature tensor and its derivatives. Then using the induction, we find the holonomy algebra for each of the metrics.


617

4.3 Examples of 4-dimensional Lie groups with left-invariant pseudo-Kählerian metrics Let G be a Lie group endowed with a left-invariant metric g and let g be the Lie algebra of G. We will consider g as the Lie algebra of left-invariant vector fields on G and as the tangent space at the identity e 2 G. Let X; Y; Z 2 g. Since X; Y; Z and g are left-invariant, from the Koszul formula it follows that the Levi-Civita connection on .G; g/ is given by 2g.rX Y; Z/ D g.ŒX; Y ; Z/ C g.ŒZ; X ; Y / C g.X; ŒZ; Y /;

(2)

where X; Y; Z 2 g. In particular, we see that the vector field rX Y is also leftinvariant. Hence rX can be considered as the linear operator rX W g ! g. Obviously, rX 2 so.g; g/. For the curvature tensor R of .G; g/ at the point e 2 G we have R.X; Y / D ŒrX ; rY rŒX;Y ;

(3)

where X; Y 2 g. The holonomy algebra he at the point e 2 G is given by he D m0 C Œm1 ; m0 C Œm1 ; Œm1 ; m0 C ;

(4)

where m0 D spanfR.X; Y / j X; Y 2 gg

m1 D spanfrX j X 2 gg:

and

Now we consider 4-dimensional Lie algebras with the basis X1 , X2 , Z1 , Z2 and with the metric that has the Gram matrix

0 0 1 0

0 0 0 1

1 0 0 0

0 1 0 0

with respect to this basis. Define

the following Lie algebras by giving their non-zero brackets: g1 :

ŒX1 ; Z1 D X1 C Z2 ; ŒX2 ; Z1 D X2 C Z1 ;

g2 :

ŒX1 ; Z2 D X1 ;

ŒX1 ; Z2 D X2 Z1 ; ŒX2 ; Z2 D X1 C Z2 I

ŒX2 ; Z1 D X1 ;

ŒZ1 ; Z2 D X1 C Z1 I

Example 4.5. The holonomy algebras of the Levi-Civita connections on the simply connected Lie groups corresponding to the Lie algebras g1 and g2 are h2nD0 and 1 D0;2 D1 hnD0 , respectively.

5 Holonomy in neutral signature The Berger list in Theorem 2.4 shows that neutral signature .n; n/ has the largest variety of irreducible, non-symmetric holonomy groups of all signatures. Apart from SO.n; n/ we have the unitary groups U.p; p/ and SU.p; p/ and the symplectic groups Sp.q; q/ and Sp.1/ Sp.q; q/, but also SO.r; C/;

Sp.p; R/ SL.2; R/;

and

Sp.p; C/ SL.2; C/;

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and finally the exceptional groups G2C SO.7; 7/;

Spin.4; 3/ SO.4; 4/;

and

Spin.7/C SO.8; 8/:

Nevertheless, for a complete classification of holonomy groups of neutral signature one also has to consider indecomposable, non-irreducible ones. In [12] some partial results for the holonomy algebras of pseudo-Riemannian manifolds of signature .n; n/ were obtained. In particular, a complete classification for n D 2 can be given. For a point p in a pseudo-Riemannian manifold of signature .n; n/ we always fix a basis in Tp M such that the metric h at p is of the form 0 In ; (1) In 0 where In is the identity on Rn . If V is a degenerate subspace which is invariant under the holonomy, again we can form the totally isotropic subspace V \V ? , but in contrary to the previous cases this subspace can have any dimension from 1 up to n. In the case in which this dimension is n we have that V ? D V , i.e., the orthogonal complement gives no further information, but this case contains a very special situation which will be described first.

5.1 Para-Kähler structures Recall that we have defined a holonomy representation to be indecomposable if any proper invariant subspace is degenerate. Neutral signature .n; n/ is the only case where this property does not prevent the holonomy representation from decomposing, i.e., to split into two invariant subspaces which are complementary. This is the case if we have two invariant totally isotropic and complementary subspaces V C and V of dimension n, i.e., Tp M D V C ˚ V ; (2) with V C and V totally isotropic and holonomy invariant. This property is equivalent to the existence of a para Kähler structure on .M; h/ (see [35] and [34]). Definition 5.1. A para-Kähler manifold .M; J; h/ is a pseudo-Riemannian manifold .M; h/ equipped with a non-trivial section J in the endomorphism bundle such that: (1) rJ D 0 where r is the Levi-Civita connection of h, (2) J h D h, and (3) J 2 D Id and the eigendistributions V ˙ WD ker .Id J / have the same rank. The second condition forces the metric to be of neutral signature .n; n/, the third condition ensures that the eigendistributions are totally isotropic of dimension n. That J is parallel is equivalent to the fact that V ˙ are holonomy invariant. It can be shown that the local coefficients of the metric of a para-Kähler metric can be expressed as


619

second derivatives of a para-Kähler potential (for a proof see [81], [12], or [35] in terms of para-Kähler manifolds). Regarding the holonomy groups one gets the following result: Theorem 5.2 (Bérard-Bergery, Ikemakhen, [12]). Let .M; h/ be a pseudo-Riemannian manifold of signature .n; n/ and let H WD Holp .M; h/ be its holonomy group at a point p 2 M . If H leaves invariant two totally isotropic complementary subspaces of dimension n, then ˇ ³ ² U 0 ˇ U 2 GL.n; R/ : (3) H G WD ˇ t 1 0

.U /

Moreover, there exists coordinates .x1 ; : : : xn ; y1 ; : : : ; yn / around p sending p to 0 2 R2n , and a smooth function on a neighborhood of 0 2 R2n such that: n P 2

Dij dxi dyj with Dij D @x@i @y . (1) h D j i;j D1

(2) The Taylor series of in 0 starts with x1 y1 C C xn yn and continues with terms which are at least quadratic in the xi ’s and quadratic in the yj ’s. (3) H is the smallest connected subgroup of G which contains the element D 0 2 G: D WD 0 .D t /1 Here is the para-Kähler potential. In [5] it is shown that cones over para-Sasakian manifolds are examples of manifolds with the property (2).

5.2 Neutral metrics in dimension four Now we consider the case where the dimension of the manifold M is four, i.e., the signature is .2; 2/. Let H be the indecomposable, non-irreducible holonomy group of .M; h/. In this case the invariant totally isotropic subspace is a null-line or a totally isotropic plane. The first case is contained in the second: If .e1 ; : : : ; e4 / is a basis of Tp M such that the metric has the form (1) with e1 spanning the invariant null-line, then the plane spanned by e1 and e2 is invariant as well because Ae2 is orthogonal to e1 for A in H . Hence, in both case H leaves invariant a totally isotropic plane , i.e., the Lie algebra h of H is contained in ˇ ³ ² 0 1 U aJ ˇ U 2 gl.2; R/; a 2 R ; with J D so.2; 2/ D 0 U t ˇ 1 0 so.2; 2/ is a semi-direct sum, so.2; 2/ D gl.2; R/ËA, where A is the ideal spanned by 00 J0 , with the commutator Œ.U; a/; .V; b/ D .ŒU; V ; b trace.U / a trace.V // : Then one can prove:

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Theorem 5.3 (Bérard-Bergery, Ikemakhen, [12]). Let h be an indecomposable subalgebra of so.2; 2/ so.2; 2/. Then either h contains an ideal which is conjugated in SO0 .2; 2/ to A, or it is conjugated to one of the three exceptions ² ˇ ³ rI2 C sJ 0 ˇ h1 D so.2/ ˚ R D r; s 2 R ; 0 rI C sJ ˇ h2 D R

L J 0 L

, or h3 D R

J

2

J

, with J as above and L D

0 J

1

0 0 1

.

Note that the other cases of subalgebras contained in the Lie algebra g of G defined in Theorem 5.2 contain an ideal which is conjugated to A, namely the ideals spanned by matrices U 2 GL.2; R/ which are nonzero only at one entry. Note also that h1 D h2nD0 defined in Theorem 4.2. One can show that the algebras h2 and h3 are not Berger algebras and therefore cannot be holonomy algebras. Corollary 5.4. Let h be an indecomposable Berger algebra in so.2; 2/ so.2; 2/. Then h is conjugated to a subalgebra which is either contained in g or contains the ideal A. Based on this in [12] a complete classification of indecomposable, non-irreducible holonomy groups of pseudo-Riemannian manifolds of signature .2; 2/ is obtained. Theorem 5.5. Let H SO.2; 2/ be the indecomposable, non-irreducible acting holonomy group of a 4-dimensional pseudo-Riemannian manifold of signature .2; 2/ and h its Lie algebra. Then H leaves invariant a totally isotropic plane and it holds one of the following: (A) H leaves invariant another totally isotropic plane complementary to , in which case h g and h is conjugated in SO0 .2; 2/ to one of the following: gl.2; R/, sl.2; R/, the strictly upper triangular, the upper triangular matrices, ² ˇ ³ a b ˇ k D a; b 2 R for 2 R, 0 a ˇ or

²

a b

h1 D

b a

ˇ ³ ˇ ˇ a; b 2 R ;

under use of the identification g ' gl.2; R/. (B) H does not leave invariant a complementary totally isotropic plane. In this case the Lie algebra h of H is conjugated to a semi-direct sum h0 Ë A; where h0 is conjugated to one of the subalgebras of g listed in (A), to R

or to u WD R

1 1

.

1 1 01

,


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Remark 5.6. From this Theorem we can also recover the classification of unitary holonomy algebras in signature .2; 2/ given in Theorem 4.2. First we get the obvious subalgebras of u.1; 1/: 1 D0;2 D0 2 ;2 ¤0 ; h1 D h2nD0 ; h1 Ë A D u.1; 1/ ; and u Ë A D hnD0 : A D hnD0 1 D1;2 D0 D su.1; 1/ one For the remaining subalgebra R I02 I0 2 ˚ A D hnD0 has to pay attention to the fact that in order to classify subalgebras of u.1; 1/ one 1 D1 needs up to conjugation in U.1; 1/. In fact, hnD0 is conjugated ˚ to classify them to a0 ab j a; b 2 R D k1 of (A), which is obviously not a subalgebra of u.1; 1/ because the conjugation lies in O.2; 2/ but not in U.1; 1/.

From Theorem 5.5 we get a conclusion about the closedness of holonomy groups. Corollary 5.7. The holonomy group of a 4-dimensional pseudo-Riemannian manifold of neutral signature .2; 2/ is closed. Finally we want to address the question, which of the algebras obtained can be realised as holonomy algebras. First we recall results in [12], where it is shown that all the possible holonomy groups that leave invariant a pair of complementary totally isotropic planes listed in (A) of Theorem 5.5 can be realised as holonomy groups of metrics which are not locally symmetric. As seen in Theorem 5.2, for these manifolds the metric can be written in terms of the para-Kähler potential , which is given by D x1 y1 C x2 y2 C f .x1 ; x2 ; y1 ; y2 /; where f is a smooth function. In the generic case h D gl.2; R/, the Taylor series of f in 0 is at least quadratic in the xi ’s and quadratic in the yi ’s, f D x1 x2 y1 y2 is an example. In [12] the other algebras are realised by specifying f :

² a 0

b c

h ³

f

a;b;c2R

k6D0

k0 0 1 R 0 0

f D y12 f1 .x1 ; x2 ; y1 / C x22 f2 .x2 ; y1 ; y2 /; where the Taylor series of f1 and f2 in 0 start with polynomials of degree 2 in .x1 ; x2 / and .y1 ; y2 / respectively, e.g., f x2 y1 .x1 y1 C x2 y2 /. f D x1

y R1 0

x2 y1 fO.x2 ; t/dt Cy2

Rx2 ..1 C x2 y1 fO..t; y1 //2 1/dt; 0

with fO non-zero f D x1 x2 y12 fO.x1 ; x2 ; y1 /; with fO non-zero, f D x22 y12 fO.x2 ; y1 /; with fO non-zero.

622


A metric with holonomy h1 u.1; 1/ is given in terms of complex coordinates. For the existence of a metric with holonomy sl.2; R/ as listed in (A) of Theorem 5.5, in [12] is argued on general grounds referring to [23] and [24]. [12] leaves open all the cases in (B) of Theorem 5.5. We will solve this problem partially, first by giving metrics with holonomy sl.2; R/ Ë A. This metric has the form g D 2dx1 dy1 C 2dx2 dy2 C f1 dy12 C f2 dy22 C 2f3 dy1 dy2

(4)

with the functions f1 D x12 , f2 D x22 , f3 D 2x1 x2 . To all the subalgebras of u.1; 1/ obtained in Theorem 4.2 our method of constructing metrics as sketched in Section 4.2 applies and ensures that all of them can be realised as holonomy algebras. The metrics again have the form (4) with the following functions: h

f1 ; f2 ; f3

u.1; 1/

f1 D 2x2 y1 x1 y12 ,

f2 D f1 ,

f3 D 2x1 y1 x2 y12

f1 D x12 x22 ,

f2 D f1 ,

f3 D 2x1 x2

h2nD0 1 ;2 hnD0

f1 D 21 x2 y1 22 x1 y1 ,

.12 C 22 ¤ 0/ 1 D0;2 D0 hnD0

f2 D f1 ,

f3 D 21 x1 y1 22 x2 y1 f1 D y22 ,

f2 D f3 D 0

These metrics disprove a claim in [49], that the Lie algebras sl.2; R/;

sl.2; R/ Ë A;

u Ë A for 6D 0;

and

h1 Ë A D u.1; 1/

– in [49] denoted by A21 , A29 , A12 , and A24 – cannot be realised as holonomy algebras in so.2; 2/. The only case we have to leave undecided, and which is left undecided in [49] as A13 , is the one of 1 1 hDR Ë A: (5) 0 1 However, very recently L. Bérard-Bergery and T. Krantz developed a general construction to build connections on vector bundles which give new holonomies [13]. In particular, by a construction on the cotangent bundle of a surface they can show that all the algebras listed in Theorem 5.5, even the last one in Equation (5), are holonomy algebras of a metric of signature .2; 2/.


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Chapter 18

Geometric applications of irreducible representations of Lie groups Antonio J. Di Scala, Thomas Leistner, and Thomas Neukirchner

Contents 1 Background, results and applications . . . . . . . . . 2 The algebra of invariant endomorphisms . . . . . . . . 3 Irreducibly acting, connected subgroups of GL.n; R/ . 4 Irreducibly acting, connected subgroups of O.1; n/ . . 5 Invariant bilinear forms of irreducible representations . References . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Background, results and applications This chapter is motivated by three algebraic questions which are related to problems in holonomy theory of affine or semi-Riemannian manifolds and in the theory of homogeneous spaces. Two of these questions are known for most experts in differential geometry but, besides special cases, general proofs are not easy to find in the literature. Thus, one goal of this chapter is to supply such proofs. Indeed, we think that these results are not so well-known for experts in other areas of mathematics or physics. So, we hope this exposition will be useful for non experts in differential geometry. We will give some applications in order to illustrate the use of it. We also state some open problems in the form of conjectures. The three motivating problems are the following: Are holonomy groups closed? What are special holonomy groups of Lorentzian manifolds? And finally, how many G-invariant bilinear forms exist on a homogeneous space G=H ? Regarding the first question, the first thing to be observed is that the answer should not depend on the mere fact of being a holonomy group, because due to a result of [HO65] any linear Lie group can be realised as a holonomy group of a linear connection (usually with non trivial torsion). Hence, one may try to find algebraic conditions to ensure closedness. Secondly one finds that the answer is ‘yes’ if one restricts it to holonomy groups of Riemannian manifolds, because any irreducibly acting, connected subgroup of O.n/ is closed (see for example [KN63, Appendix 5]),

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and thus, by the de Rham decomposition theorem [dR52] the holonomy group is a direct product of closed ones. But for a general connection the answer is ‘no’, there are a lot of examples of non-closed holonomy groups, see [HO65] for affine, torsion-free connections, [Wu67] for pseudo-Riemannian manifolds and [BI93] for Lorentzian manifolds. In all of these examples the holonomy group does not act irreducibly. So it arises the question if all irreducibly acting subgroups of GL.n; R/ are closed? There is a note in [Bes87, Note 10.50, p. 290] that the following theorem was proved in [Wak71]. Theorem 1. Any irreducibly acting connected Lie subgroup of GL.n; R/ is closed in GL.n; R/. In this note we will prove this theorem (see Section 3) independently of the proof in [Wak71]. Regarding holonomy groups of linear connections, this has the following consequence. Corollary 1. If the restricted holonomy group of a linear connection acts irreducibly, then it is closed. Furthermore, the restricted holonomy group of a semi-Riemannian manifold is closed if it acts completely reducibly. This statement follows from the de Rham/Wu decomposition theorem [Wu64] and another theorem in [Wu67] (see below, last paragraph in Section 3). Our proof of Theorem 1 uses a theorem of Yosida [Yos37] and Malcev [Mal45] and a very explicit description of the centre of G. This description gives us two corollaries, the first of which will be useful in the proof of Theorem 2. Corollary 2. Let G O0 .p; q/ be a connected Lie subgroup of O0 .p; q/ which acts irreducibly. If G is not semisimple, then p and q are even and G is a subgroup of U.p=2; q=2/ with centre U.1/. Applying this to the spin representations of the orthogonal algebras we get: Corollary 3. Let G SO0 .p; q/ be a connected Lie subgroup which acts irreducibly z Spin.p; q/ the pre-image of the covering Spin.p; q/ ! SO0 .p; q/. If the and G z admits a trivial subrepresentation, then G is semisimple, or spin representation of G 0 0 G D U.1/ G with G ¤ SU.p=2; q=2/ and .p C q/=2 is even. Unfortunately, one cannot show that the existence of a trivial subrepresentation of the spin representation implies semisimplicity of G as the following example shows: Let G D U.1/ Sp.p; q/ with .p C q/ even, where Sp.p; q/ is the symplectic or quaternionic unitary group (for a definition see Section 5). Then the intersection of the spaces of spinors which are annihilated by U.1/ with the ones which are annihilated by Sp.p; q/ is a one-dimensional space (for details see [BK99]) . Nevertheless, this corollary has applications to geometric problems. The first application is a well known fact. If the holonomy group of a semi-Riemannian manifold

Chapter 18. Geometric applications of irreducible representations of Lie groups

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acts irreducibly and has a centre, the manifold cannot admit parallel spinors. It was obtained by the classification of irreducible holonomy groups of semi-Riemannian manifolds with parallel spinors ([Wan89] for Riemannian manifolds, and [BK99] for pseudo-Riemannian manifolds). But furthermore it gives results in any case where the holonomy group has a irreducibly acting component on which the existence of parallel spinors depends, as it is the case for indecomposable, non-irreducible Lorentzian manifolds, see [Lei02b]. Regarding the second question which special Lorentzian holonomy groups might exist, one distinguishes between the irreducible and the indecomposable, non-irreducible case. While in the latter case there are several possibilities (for a classification see [BI93], [Lei02a], [Lei03a], [Lei03b], [Lei07], and [Gal05]), for the irreducible case the situation is very limited. The irreducible holonomy groups of semi-Riemannian manifolds were determined by M. Berger in [Ber55] and [Ber57]. For definite metrics and many other signatures the list depends essentially on the property of being a holonomy group, whereas in the Lorentzian case it turns out that irreducibility is sufficient to determine the group. Theorem 2. SO0 .1; n/ is the only connected Lie subgroup of O.1; n/ which acts irreducibly. The consequence for irreducible Lorentzian holonomy groups follows immediately. Corollary 4. If the restricted holonomy group of a Lorentzian manifold acts irreducibly, then it is equal to SO0 .1; n/. Here is an application to isotropy irreducibly Lorentzian homogeneous spaces. Corollary 5. Isotropy irreducibly Lorentzian homogeneous spaces have constant sectional curvature, i.e., they are flat, de-Sitter or anti de-Sitter spaces. A direct and geometric proof of Theorem 2 was given in [DSO01]. An almost algebra-free proof which uses dynamical methods, can be found in [BZ04]. Finally, a purely algebraic proof was given in [BdlH04]. A nice and direct proof of Corollary 5, based on dynamical methods, can be found in [Zeg04]. In Section 4 we will give a short proof of Theorem 2 based on a theorem of Karpelevich [Kar53] and Mostow [Mos55] (i.e., Theorems 7 and 8). Theorem 2 also follows from Corollary 4.5.1 in [ChGr74] which, in turn, depends upon Karpelevich–Mostow’s theorem (notice that Lemma 4.4.3 in [ChGr74] is actually Theorem 8). Theorem 2 seems to be a well-known fact to experts in hyperbolic geometry but widely unknown to mathematicians working in representation theory and differential geometry. For this reason it would be nice to have a geometric and almost algebra-free proof of Karpelevich–Mostow’s Theorems 7 and 8. Unfortunately, the proofs in both papers [DSO01] and [BZ04] make use that the boundary at the infinity is homogenous (i.e., the isometry group acts transitively on it or equivalently the space has rank one).

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We would like to thank the referee to call our attention to the paper of S. S. Chen and L. Greenberg [ChGr74]. The result of the last Section 5 is motivated by the geometric problem of describing the space of metrics or symplectic forms on a homogeneous space G=H which are invariant under G. Any G-invariant metric or symplectic form corresponds to a non-degenerate bilinear form on g=h which is invariant under the linear isotropy representation AdG .H / GL.g=h/. In our context AdG .H / is assumed to act irreducibly. This is a special case of the following algebraic problem: Given an irreducibly acting Lie subgroup G GL.n; R/, what is the dimension of the space of G-invariant bilinear forms on Rn . We prove the following statement. Theorem 3. Let G be an irreducibly acting subgroup of GL.n; R/. The the space of G-invariant symmetric bilinear forms which are not of neutral signature .p; p/ is at most one-dimensional. Moreover, the space of invariant symmetric bilinear forms is at most three-dimensional. We will describe all possible cases for the dimension of the space of (skew-) symmetric bilinear forms and determine the maximal subgroup which fixes these bilinear forms. We should point out that many results in this chapter rely on the classification of G-invariant endomorphisms for G GL.n; R/. This classification follows from Schur’s lemma and the classification of associative division algebras by Frobenius, but we will give an elementary proof of it in Section 2.

2 The algebra of invariant endomorphisms If G is a Lie group and V and W two (real or complex) G-modules the algebra of invariant homomorphism is defined as HomG .V; W / WD fX 2 Hom.V; W / j A B X D X B A for all A 2 Gg: Now, Schur’s lemma says that HomG .V; W / Iso.V; W / [ f0g, and furthermore, if V D W is complex, then EndG .V / D C Id. In any case, it implies that HomG .V; W / is a real associative division algebra, and thus by their classification of Frobenius (1878), it is isomorphic to the algebra of real numbers R, complex numbers C or quaternions H (see e.g. [Pal68]). We are interested in the description of EndG .V / where V is a real vector space, and in this section we will recall some facts about real irreducible representations which provide an elementary proof of this result. Suppose that G is a real Lie group and V a real irreducible module. Then there are two cases which can occur for the complexified G-module V C . The first case is that W WD V C is still irreducible. In this case V or W is called of real type. One should remark that, if W is a complex irreducible G-module then one can consider W as a real vector space which we denote by WR . This is a reducible real G-module


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with invariant real subspace V if and only if W is the complexification of the real irreducible G-module V . In the other and more complicated case, regarding the application of Schur’s lemma, V C is a reducible G-module. In this case V C splits into two irreducible G-modules, S: VC DW ˚W S In fact, if W is an invariant complex subspace of V C then the conjugate module W defined by the conjugation with respect to V V C is invariant as well. Furthermore, S and W \ W S are invariant and equal to their conjugation. Hence they the spaces W C W S D V C. S D V C and W \ W are complexifications of real vector spaces, i.e., W C W 1 2 Of course V1 and V2 are invariant subspaces of V and thus V1 D V and V2 D f0g. The same argument ensures the irreducibility of W . S D f0g, the mapping W WR 3 v 7! 1 .v C v/ 2 V is an Now, since W \ W 2 isomorphism of real vector spaces yielding the identification SR WR ' V ' W

(1)

of real G-modules. In this case V , respectively W , are called of complex type, and again we have that a complex module W is irreducible as real module V D WR if and S is reducible. only if V C D W ˚ W Now we are able to describe the algebra of invariant endomorphisms of a real irreducible G-module V . Proposition 1. Let G be a Lie group and V a real irreducible G-module. Then EndG .V / is isomorphic to one of the real algebras R, C or H. Proof. As above, we consider two cases. First, assume that V C is irreducible which ensures that EndG .V C / D C Id by Schur’s lemma. Hence, if A 2 EndG .V /, its complexification AC 2 EndG .V C / is given by AC D Id with 2 C. Since AC leaves V invariant and V is invariant under conjugation we get for v D vN 2 V that N v D N vN D AC v D AC v D v; i.e., 2 R. A D AC jV gives that EndG .V / D R Id. For the second case we have to assume that V C is reducible, i.e., by the above V C D S , V ' WR and thus EndG .V / D EndG .WR /. Now any real endomorphism W ˚W on WR decomposes uniquely into a complex linear and complex anti-linear part: S /; End.WR / ' End.W / ˚ Hom.W; W 1 1 A D .A C iAi / C .A iAi /: 2 2 This decomposition descends to EndG .WR /: S /: EndG .WR / ' EndG .W / ˚ HomG .W; W S are Now Schur’s lemma implies that EndG .W / D C Id and, since both, W and W S S irreducible, that HomG .W; W / Iso.W; W / [ f0g.

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S / D f0g we get immediately If HomG .W; W C Id D EndG .WR / ' EndG .V / D spanR fId; I g; with the complex structure I WD B .i Id/ B 1 where is defined in (1). S / which is an isomorphism by Otherwise consider a non-zero j 2 HomG .W; W 2 2 Schur’s lemma. Then j 2 EndG .W /, hence j D Id with 0 6D 2 C. In fact, 2 R since N j.w/ D j. w/ D j j 2 .w/ D j 2 .j.w// D j.w/ for all w 2pW . Finally < 0 since otherwise WR would decompose into the Ginvariant ˙ -eigenspaces of jR . Thus, we may assume j 2 D 1. For another S / we get j B A 2 EndG .W / and therefore j B A D c Id for some A 2 HomG .W; W c 2 C. On the other hand j B .cj N / D c Id and thus A D cj N . Hence, we obtain S / D C Id ˚ C j; EndG .WR / ' EndG .W / ˚ HomG .W; W which gives finally EndG .V / D spanR fId; I; J; I B J g ' H; with I WD

Bi B

1

and J WD

Bj B

1

anti-commuting complex structures.

Corresponding to the structure of EndG .V /, the real irreducible G-module V is said to be of real, complex or quaternionic type. This corresponds to the convention to call a complex irreducible G-module W of real type if it is self-conjugated with respect to an anti-linear bijection J with J 2 D Id, of quaternionic type if it is selfconjugated with J 2 D Id and of complex type if it is not self-conjugated. Here is a useful consequence of the preceding proposition. Corollary 6. For a real irreducible G-module V any A 2 EndG .V / is of the form A D ˛ Id C ˇJ with ˛; ˇ 2 R and J a G-invariant complex structure (depending on A). Proof. Although this follows directly from Proposition 1 we will give another proof which will be useful later on. Applying the Schur-lemma we see that the minimal polynomial A .x/ of A is irreducible over R (cf. [KN63, Appendix 5, Lemma 1]). If A .x/ D x ˛ is of degree one 0 D A .A/ D A ˛ Id. Otherwise A .x/ D .x ˛/2 C ˇ 2 is a polynomial of degree 2 with strictly positive quadratic supplement, since A is irreducible. Thus J WD .A ˛ Id/=ˇ defines a complex structure on V . Finally, we describe the maximal representations of different types, i.e., any other irreducible representation occurs as a subrepresentation of them.


635

Proposition 2. Let G GL.n; R/ be an irreducibly acting subgroup. Then up to conjugation G is contained in one of the following subgroups L GL.n; R/. EndG .Rn / L GL.n; R/ R

GL.n; R/

C

GL.n=2; C/

H

GL.n=4; H/

Proof. We set V WD Rn and K WD EndG .V /. Thus V becomes a left K-vector space in a natural way. In order to make it a right K-vector space we choose an N Then anti-automorphism 7! N of K (i.e., C D N C N and D N ). N v WD .v/ defines a right-multiplication on V with respect to the scalar field K (This is essential only in case of the non-commutative field K D H). The group GL.V; K/ of K-linear invertible maps from V into itself is by definition the centralizer of the homothety group HK WD f fv 7! vg j 2 K g. By choosing a K-basis n=d fbi gn=d ' V . Under this identification GL.V; K/ i D1 , where d D dim R K, we get K corresponds to the group GL.n=d; K/ of invertible .n=d n=d /-matrices acting on Kn=d from the left. By definition HK is the centralizer of G and thus G is contained in L WD GL.V; K/. As explained, this yields an inclusion G GL.n=d; K/. Conversely it is known that the centralizer of GL.n=d; K/ equals HK , hence EndL .Rn / D HK . Finally, the embedding GL.n=d; K/ GL.n; R/ is obtained by associating to the K-basis fbi g the real basis fbi k g i D1;:::;n=d , where fk gkD1;:::;d is a basis of K. kD1;:::;d

Remark 1. In the proof of this proposition we see that if the action of a group G GL.n; R/ is defined by scalar multiplication from the right, the invariant endomorphism have to act from the left. Of course, this becomes only relevant in case of EndG .Rn / D H, and we can see this in the example of G WD GL.1; H/: It is GL.1; H/ D fRq W H ! H j q 2 H and Rq .p/ WD p qg D H ; whereas EndGL.1;H/ .R4 / D fA 2 GL.4; R/ j A.Rq .p// D Rq .A.p/g D fLq 2 GL.4; R/ j Lq .p/ WD q pg DH since Lq B Rp D Rp B Lq but Rp B Rq 6D Rq B Rp . This gives the seemingly paradoxical situation where both, the centraliser ZGL.4;R/ .G/ and the group G itself are equal to H , but its centre Z.G/ which is the intersection of G with its centraliser is commutative and thus equal to C .

636


3 Irreducibly acting, connected subgroups of GL.n; R/ In this section we shall give a proof of Theorem 1 by using the results of the first section and two general results from Lie theory. First we describe the identity component of the centre of an irreducibly acting Lie subgroup of GL.n; R/. We should remark that we use ‘Lie subgroup’ always in the weaker sense of being a submanifold but not necessarily an immersion in order to make the statement of Theorem 1 non-trivial. Proposition 3. Let G GL.n; R/ be an irreducibly acting, connected Lie subgroup, Z.G/ its centre and Z.G/0 the identity component of the centre. If Z.G/0 is nontrivial, then Z.G/0 is either (a) equal to RC Id, or (b) isomorphic to C D RC S 1 , or (c) isomorphic to a one-parameter subgroup of C . Cases .b/ and .c/ can only occur if n is even. Proof. Let g gl.n; R/ be the Lie algebra of G and suppose that the centre z of g is non trivial. Considering the three cases of Proposition 1 we first assume that the representation is of real type, i.e., that EndG .Rn / D R Id. Since z EndG .Rn / we obtain in this case that z D R Id and therefore Z.G/0 D exp.z/ D RC Id. Now suppose that Rn is a G-module of non-real type, i.e., EndG .R2n / isomorphic to C or H. Again z is an Abelian subalgebra of EndG .R2n /. In case EndG .R2n / ' H ' u.2/ any maximal Abelian subalgebra is isomorphic to C. Hence z is isomorphic to a subalgebra of C D spanR .Id; J / where J is a complex structure on R2n . But exp tJ D .cos t / Id C .sin t /J , i.e., exp.RJ / ' S 1 . But this implies that either isomorphic to C , i.e., Z.G/0 D RC Id f.cos t / Id C .sin t /J j t 2 Rg ' RC S 1 D C ; or to a one-parameter subgroup of it, i.e., Z.G/0 D exp .R .a Id C bJ // ˚ D eat Id B ..cos bt/ Id C .sin bt/J / j t 2 R ; for some real constants a and b. Of course if a or b are zero this is either RC or S 1 , if not this is a logarithmic spiral in C . Proposition 3 will be the main ingredient in our proof of Theorem 1 but it implies also Corollaries 2 and 3 given in the introduction. But before we can prove these we have to recall that for a completely reducibly acting Lie subgroup G GL.n; R/ the centre decides whether the Lie algebra is semisimple or not. This is due to the well known fact from the theory of Lie algebras, that a Lie algebra g which admits a completely reducible representation is reductive. Hence g admits a Lie algebra decomposition into its centre and its derived Lie algebra, g D z ˚ Œg; g ;

(1)


637

the derived Lie algebra being semisimple. A proof of this fact can be found in [Che47], see also [Bou71]. This means that the irreducibly acting, connected Lie subgroup in question is semisimple if the identity component of its centre is trivial. Remark 2. In this context we should remark that the centre of a semisimple subgroup G GL.n; R/ is finite (see e.g. [Got48]): if G is semisimple, due to Weyl’s theorem it acts completely reducibly, and furthermore its elements are of determinant 1, hence by Schur’s lemma the centre of G corresponds to the nk -th roots of 1 where nk are the dimensions of the irreducible subspaces. For verifying Corollary 2 we assume that G O.p; q/ is connected and acts irreducibly. If G is not semisimple, its Lie algebra g has a non trivial centre z, but the orthogonality of the representation implies that projection of the centre on R Id is trivial. Hence, the representation is not of real type, i.e., n D p C q is even, and z D RJ where J is the complex structure which commutes with g. But on the other hand, J 2 so.p; q/, i.e., J is compatible with the inner product, which gives that p and q are even as well. Thus, by Proposition 2, g so.p; q/ \ gl.n=2; C/ D u.p=2; q=2/: which is the statement of Corollary 2. This also implies Corollary 3: If g is not semisimple, then g D R J ˚ g0 where g0 D Œg; g is semisimple. Hence, g0 D Œg0 ; g0 su.r; s/, with su.r; s/ defined by the complex structure J , r D p=2 and s D q=2. The complex structure J is given by J D

rCs X

2k E2k1 2k ;

k

where i D ˙1 is defined by hei ; ej i D i ıij with ei an orthonormal basis of RpCq , and Eij is the corresponding basis of so.p; q/. Let u."rCs ; : : : ; "1 / be the basis of the spinor module .p;q/ as defined in [BK99]. This is an eigenbasis for the spin representation of J : J u."rCs ; : : : ; "1 / D i

rCs X

2k 2k1 2k "k u."rCs ; : : : ; "1 /

k

Di

rCs X

"k u."rCs ; : : : ; "1 /;

k

where i D i if i D 1 and 1 otherwise. If VRJ denotes the subspace in the spinor module which is annihilated by RJ under its spin representation, then ˚ P VRJ D span u."rCs ; : : : ; "1 / j rCs "i D 0 ; k which implies that J has zero eigenvectors only if r C s is even. This is the statement of Corollary 3. The example given in the introduction of a non-semisimple irreducibly

638


acting Lie algebra with trivial spin representation is given by g D RJ ˚ sp.r; s/ with r C s even. Again in the notation of [BK99] it follows that X ' rCs D u."rCs ; "rCs ; : : : ; "1 ; "1 / 2

"i D1 for

rCs 2

times

is annihilated by sp.r; s/. But it is also annihilated by J : X ."1 C : : : C "rCs / u."rCs ; "rCs ; : : : ; "1 ; "1 /: J' rCs D 2 2 ƒ‚ … „ " D1 for i

rCs 2

D0

many "i ’s

Hence, Vg D VRJ \ Vsp.r;s/ D R' rCs is one-dimensional. 2

Example. An example for an irreducible real representation of a Lie group with 2dimensional centre is the representation of S 1 CO.n; R/ on C n considered as real R/ denotes the conformal group of Rn . The Lie algebra vector space R2n . Here CO.n; A aIn consists of the matrices aI 2 gl.2n; R/ with a 2 R and A 2 co.n; R/, n A where In denotes the n-dimensional unit matrix. The centre of the identity component of this group is S 1 RC , the semisimple part is SO.n/. In the same manner we can built an example where the centre is a spiral in C by taking as Lie algebra ˚ ˇ In In g WD A0 A0 ˇA 2 so.n/ ˚ R I gl.2n; R/; n In and as group G the connected subgroup in GL.2n; R/ with this Lie algebra. Both groups do not act orthogonally. Now we can go ahead with the proof of Theorem 1. Let G be a connected, irreducibly acting Lie subgroup of GL.n; R/, and g be its Lie algebra. Our proof now relies on the following result of [Yos37] and [Mal45] (see also [Got48] where it is a corollary to a deeper result). Theorem 4 ([Yos37], [Mal45], [Got48]). A connected Lie subgroup of GL.n; R/ is closed in GL.n; R/ if and only if its radical is closed. In particular, if it is semisimple, it is closed. Recall that the radical of G is the connected Lie subgroup of G which corresponds to maximal solvable ideal in the Lie algebra g. Thus we have to show, that the radical of G is closed in GL.n; R/. But by the remarks above, the Lie algebra of G is reductive, and thus the radical of G is equal to the identity component of its centre, denoted by x of G is still connected, acts irreducibly and has a reductive Z.G/0 . Now the closure G x is closed in Lie algebra. By Theorem 4 the identity component of its centre Z G 0 x x it is GL.n; R/. But Z.G/ Z.G/ because for z 2 Z.G/ and g D lim gn 2 G z g D z lim gn D lim.z gn / D lim.gn z/ D lim gn z D g z:


639

If we now assume that G is not closed we get by Theorem 4 that Z.G/0 is not closed in GL.n; R/, i.e., x 0: Z.G/0 ¤ Z.G/0 Z.G/ x is irreducible and connected, Proposition 3 leaves us only with the Now, since G x 0 is isomorphic to C and Z.G/0 is a one-parameter subgroup possibility that Z.G/ of C . But these are closed in C . This is a contradiction which completes the proof of Theorem 1. Since holonomy groups are Lie subgroups of GL.n; R/, the first point of Corollary 1 is a direct consequence of the Theorem 1. The second can be obtained by a theorem in [Wu67] which contains several results with different algebraic conditions for subgroups of the pseudo-orthogonal group, having consequences for holonomy groups. Theorem 5 ([Wu67]). The following subgroups of GL.p C q/ are closed: (1) reductive, indecomposable subgroups of O.p; q/, (2) indecomposable subgroups of O.p; q/ if p C q < 6, (3) holonomy groups of affine symmetric spaces. Here ‘indecomposable’means ‘no proper non-degenerate invariant subspace’. One should remark that the restriction to the dimension in the second point is sharp: In [Wu67] is constructed a 6-dimensional Kähler manifold whose reduced holonomy group is non-closed in SO.4; 2/; also the Lorentzian examples in [BI93] are constructed in dimension 6. Also in [Wu67] is constructed an example of a symmetric space with solvable, non-Abelian holonomy group which shows that the third point does not follow from the first. Some of these examples are obtained by constructing subgroups containing a torus, which has non-closed 1-parameter subgroups. Our proof shows that such a situation can be excluded if the group acts irreducibly. In order to obtain the second statement of Corollary 1, note that the first point of Theorem 5 implies that semi-Riemannian holonomy groups which act completely reducibly are closed: by the de Rham/Wu decomposition theorem [Wu64] any semiRiemannian holonomy group is a product of indecomposably acting holonomy groups, but if the group is assumed to act completely reducibly it is reductive and hence closed by the first point of Theorem 5. Since the dense line on the Clifford torus provides an example of a completely reducibly acting group which is not closed in GL.2; C/, such a result cannot be true for holonomy groups of an arbitrary affine connection due to the result in [HO65], that any connected linear Lie group can be obtained as the holonomy group of an affine connection. It is not difficult to check that the connection of this example is not torsion-free. But such a result might be true for torsion-free connections. Conjecture. Let .M; r/ be an affine manifold where r is a torsion-free connection. Assume that the restricted holonomy group Holp .r/ acts completely reducible on Tp M . Then, Holp .r/ is closed inside GL.Tp M /.

640


4 Irreducibly acting, connected subgroups of O.1; n/ In this section we want to give a short proof of Theorem 2, that the only connected subgroup G of O.1; n/ which acts irreducibly on Minkowski space R1;n is the connected component of the identity of O.1; n/, i.e., G D SO0 .1; n/. This statement was proven in [DSO01] where the main goal was to generalize to real hyperbolic space the following result about minimal homogeneous submanifolds, i.e., orbits of isometry subgroups, in the Euclidean space. Theorem 6 ([DS02]). A (extrinsically) homogeneous minimal submanifold of the Euclidean space must be totally geodesic. It turns out that such result also holds in the real hyperbolic space (see [DSO01] for details). It is interesting to remark that further investigations of minimal homogeneous submanifolds were done in several directions [ADS03], [DS03]. In particular, the following conjecture was posed in [DS03]. Conjecture. Let M be a Riemannian manifold that is either locally homogeneous or Einstein. Then, any minimal isometric immersion f W M ! Rn must be totally geodesic. Now, in order to prove Theorem 2 we assume that G O.1; n/ acts irreducibly and is connected. By Corollary 2 it is semisimple and closed by Theorem 1. Our proof requires the following Karpelevich’s theorem. Theorem 7 ([Kar53], [Mos55]). Let M D Iso.M /=K be a Riemannian symmetric space of non-compact type. Then any connected and semisimple subgroup G of the full isometry group Iso.M / has a totally geodesic orbit G p M . The above theorem can also be stated in a purely algebraic way as follows. Theorem 8. Let g0 be a real semisimple Lie algebra of non compact type and let g g0 be a semisimple Lie subalgebra. Let g D k ˚ p be a Cartan decomposition for g. Then there exists a Cartan decomposition g0 D k 0 ˚ p0 for g0 such that k k 0 and p p0 . The original proof of Theorem 7 is as a corollary of Theorem 8. The proof of Theorem 8 is not trivial and very algebraic (see [Mos55, Theorem 6] or [Oni04]). Hence, it would be nice to have a proof of Karpelevich’s theorem by use of more geometric methods. Remark 3. Observe that if g0 D SL.n; R/ then Theorem 8 is a well-known fact related to the Cartan’s procedure to pass from a symmetric space to its dual one. Namely, if g D k ˚ p then the Lie algebra gc WD k ˚ i p is compact. So there exists an


641

Hermitian inner product .j/ on C n invariant by gc . Now it is not difficult to check that h; i WD Real.j/ is an inner product of Rn such that k consist of skew-symmetric matrices w.r.t h; i and p consist of symmetric matrices w.r.t. h; i. Remark 4. Note that in Theorems 7 and 8 the hypothesis on g0 of being of “non compact type” can not be deleted. Any irreducible representation of a compact simple Lie group G gives an example. Indeed, the group G SO.n C 1/ acts on a sphere S n D SO.n C 1/=SO.n/ without totally T geodesic orbits. Notice that if the orbit G x is totally geodesic then G x D S n V , where V is a linear subspace of RnC1 . So V is G-invariant giving a contradiction. Now let us continue the proof of Theorem 2. Since G O.1; n/ is semisimple, Karpelevich’s theorem applies to our situation. It implies that the action of G on the hyperbolic space H n D SO.1; n/=SO.n/ R1;n is transitive. Indeed, if the totally geodesic orbit G p is not the whole hyperbolic space H n , then G p is contained in a Lorentzian subspace L of R1;n . This is due to the fact that totally geodesic submanifolds of H n are intersections H n \ L where L is a Lorentzian subspaces of R1;n . Thus, G can not act irreducibly as we had assumed. Now, let K be a maximal connected compact subgroup of the semisimple group G. Then by Cartan’s fixed point theorem K has a fixed point p 2 H n . Since .Gp /0 is compact, we get K D .Gp /0 . Thus, .G; K/ is a symmetric pair such that H N D G=K. Then, from the uniqueness of such symmetric pairs (see [Hel78, pp. 243]) we get G D SO0 .1; n/ and K D SO.n/. This proves Theorem 2. Remark 5. Note that Theorem 8 implies the “uniqueness” of the presentation M D G=K with .G; K/ as a symmetric pair, where M is a symmetric space of non compact type. Namely, if g D k ˚ p and Lie.Iso.M // D k 0 ˚ p0 then the dimensions of p and p0 are the same since both are identified with the tangent space at some point of M . Thus, g D Lie.Iso.M //. In geometric words, the “uniqueness” of such pairs follows from the so-called Cartan’s construction of the Lie algebra of the group of isometries Iso.M / of M . Namely, the curvature tensor R of M belong to k. Since k is generated by R (i.e., k is the holonomy algebra) and R gives the Lie bracket on p it follows that Iso.M /0 D G. This is the geometric explanation of the “uniqueness” quoted above (see [Hel78, pp. 243]). A different, almost algebra-free proof of Theorem 2 which uses dynamical methods, can be found in [BZ04]. A purely algebraic proof was given in [BdlH04]. Theorem 2 also follows from Corollary 4.5.1 in [ChGr74] which in turn depends upon Karpelevich–Mostow’s theorem (cf. Lemma 4.4.3 in [ChGr74] with Theorem 8). Let M be a (locally) indecomposable Lorentzian manifold, i.e., the restricted holonomy group ˆp acts indecomposably on Tp M . In [DSO01] was proved that either ˆp acts transitively on hyperbolic spaces Hr WD fv 2 Tp M j hv; vi D r 2 g or transitively on horospheres Qz Hr centered at a point z 2 Hr .1/. Observe that in the Riemannian case the non-transitivity (on a sphere) of the holonomy group implies

642


the locally symmetry of the Riemannian manifold, i.e., the Berger–Simons Theorem (see [Olm05] for a direct and geometric proof). It is interesting to note that the holonomy group of a non irreducible indecomposable Lorentzian symmetric space is abelian [CaWa70]. Thus, such holonomy group must act transitively on horospheres Qz Hr centered at a point z 2 Hr .1/, i.e., z is associated to the (unique) light-like direction invariant by ˆp . There are also non symmetric global (i.e., geodesically complete) examples of (non irreducible and indecomposable) homogeneous Lorentzian spaces such that the restricted holonomy group is abelian, i.e., the so called homogeneous plane waves [BlLo]. Notice that such spaces had a rich family of totally geodesic Lorentzian surfaces through each point. Namely, the exponential of the normal space to each horosphere Qz Tp M . Anyway, it should be desirable to know if some type of Berger–Simons theorem holds in the Lorentzian setting under suitable assumptions. Finally, here is another application, i.e., Lemma 3.1 of [Sen06], see also [Hal91]. Proposition 4. Let .M; g/ be a simply connected Lorentzian manifold. Assume that there exist a non-zero covariantly constant symmetric tensor field h not proportional to the metric. Then, .M; g/ is reducible, and further it is indecomposable only if there exists a unique (up to multiples) light-like parallel vector field. Indeed, if .M; g/ is irreducible then the holonomy group ˆp is SO0 .1; n/. Thus, h being SO0 .1; n/-invariant should be proportional to the metric g (See the first line in the table of Proposition 6 of the next section). The second part follows from the fact that if ˆp leaves invariant two non proportional light-like vectors then ˆp must leave invariant the two dimensional Lorentzian space generated by these vectors. Thus, .M; g/ is decomposable. Notice that the “uniqueness” of a light-like parallel vector field does not imply indecomposability of the Lorentzian manifold, e.g. the product of a Riemannian manifold and an indecomposable Lorentzian manifold. A modern exposition of Karpelevich’s theory is [Oni04]. A step towards the generalization of Theorem 2 to arbitrary signatures has recently been made in [DSL10], where irreducible subalgebras of so.2; n/ are classified.

5 Invariant bilinear forms of irreducible representations As in the second section we consider an irreducibly acting subgroup G GL.V / of a real vector space V and denote by BG .V / the vector space of G-invariant bilinear forms. If BG .V / is non-trivial it is intimately related to EndG .V /. By Schur’s lemma a non-zero a 2 BG .V / is non-degenerate since its kernel is G-invariant and not equal to V . Thus Riesz’ theorem provides a one-to-one correspondence between BG .V /


643

and EndG .V / via ! BG .V / EndG .V / B 7! b D a.B. /; /:

(1)

In particular, this map endows BG .V / with the structure of an associative algebra and by Proposition 1, BG .V / is isomorphic to R, C, or H. The unique decomposition of a bilinear form into symmetric and skew-symmetric parts applies also to G-invariant bilinear forms, since the (skew-)symmetrization of a G-invariant form inherits this property. Thus, (2) BG .V / D SG .V / ˚ ƒG .V /: This induces a decomposition of EndG .V / into a-selfadjoint and a-skew-adjoint operators, (3) EndG .V / D SGa .V / ˚ ƒaG .V /: If a is symmetric, SG .V / corresponds to SGa .V / under (1) and if a is skew-symmetric, SG .V / corresponds to ƒaG .V /. The main question is what are the possible dimensions of SG .V / and what are the occurring signatures. A first step towards an answer is given by the following statement. Proposition 5. Let a; b 2 BG .V / be linearly independent, b D a.B. /; / by (1) and B D ˛ Id C ˇJ according to Corollary 6. (i) If a and b are both symmetric (or skew-symmetric), J 2 SGa .V / and thus J is an anti-isometry with respect to both a and b. In particular Sig.a/ D Sig.b/ D .n=2; n=2/ where n D dim V . (ii) If a is symmetric and b is skew-symmetric, B D ˇJ 2 ƒaG .V / and thus J is an isometry with respect to both a and b. Proof. (i) Since B and Id are a-selfadjoint, the same holds for J . Using ŒB; J D 0 we obtain a.J.x/; J.y// D a.J 2 .x/; y/ D a.x; y/; and b.J.x/; J.y// D a.B B J.x/; J.y// D a.B.x/; y/ D b.x; y/: (ii) Here B is a-skew-adjoint. This implies for its minimal polynomial, B .x/ D B .x/ D B .x/, hence B .x/ D x 2 C ˇ 2 (cf. Corollary 6), i.e., B D ˇJ . The remaining part is analogous to (i). Next we determine all possible pairs dim SG .V /; dim ƒG .V / by describing their maximal representations analogous to Proposition 2 of Section 2. Recall that a representation is self-dual if and only if the space of non-degenerate invariant bilinear forms is non-trivial. Proposition 6. Let W G ! GL.n; R/ be an irreducible self-dual representation on Rn . Then up to conjugation .G/ is contained in one of the following subgroups L GL.n; R/ with p C q D n:

644


End.G/ .Rn / dim S.G/ .Rn / dim ƒ.G/ .Rn / L GL.n; R/ R

C

H

1

0

O.p; q/

0

1

Sp.n=2; R/

2

0

O.n=2; C/; .n 4/

0

2

Sp.n=4; C/

1

1

U.p=2; q=2/

1

3

Sp.p=4; q=4/; .n 8/

3

1

O .n=4/; .n 8/

Moreover we have the following isomorphisms:1 O.n=2; C/ ' O.n=2; n=2/ \ GL.n=2; C/; Sp.n=2; C/ ' Sp.n; R/ \ GL.n=2; C/; U.p=2; q=2/ ' O.p; q/ \ Sp.n; R/; Sp.p=4; q=4/ ' U.p=2; q=2/ \ Sp.n=4; C/; O .n=4/ ' U.n=4; n=4/ \ O.n=2; C/:

(4)

Remark 6. This proposition raises the question if there are proper subrepresentations of the different groups L GL.n; R/. The answer depends very much on the group L in question. To illustrate this, note that any compact simple Lie group admits irreducible representations in arbitrary high dimensions. All these representations are contained in O.n/ due to Weyl’s trick. Considered the other way around, O.n/ has in general a lot of irreducible subrepresentations. In contrast, there are no proper subgroups of SO0 .1; n/ which act irreducibly, see Section 4. Proof. For general considerations set V WD Rn ; we will return at the end to Rn by choosing an appropriate basis. First note that .; V / is self-dual if and only if B.G/ .V / ¤ f0g. In particular End.G/ .V / ' B.G/ .V / according to (1) and we may distinguish between the various types of .; V /. We determine in each case the maximal subgroup L GL.V / fixing every element of B.G/ .V / and thus .G/ L. .; V / of real type: This is the simplest case, since B.G/ .V / is 1-dimensional and thus spanned either by a symmetric or skew-symmetric bilinear form a (cf. (2)). In 1 For

details how the groups are embedded into GL.n; R/ resp. GL.n; C/ we refer to the proof.


645

the symmetric case we can find a (pseudo-)orthonormal basis, i.e., Ip : .aij / D Ip;q WD Iq Its isometry group is the (pseudo-)orthogonal group O.p; q/ WD f A 2 GL.n; R/ j At Ip;q A D Ip;q g: In the skew-symmetric case we can find a symplectic basis, i.e., In=2 .aij / D Jn=2 WD : In=2 Its isometry group is the real symplectic group Sp.n=2; R/ WD f A 2 GL.n; R/ j At Jn=2 A D Jn=2 g: .; V / of complex type: First we fix a (skew-)symmetric a 2 BG .V / and consider its bilinear extension aC as well as its sesquilinear extension aN C to the complexification VC : aC .x C iy; u C iz/ WD a.x; u/ a.y; z/ C i a.x; z/ C a.y; u/ ; aN C .x C iy; u C iz/ WD a.x; u/ C a.y; z/ C i a.x; z/ a.y; u/ : aC is (skew-)symmetric, aN C is (skew-)Hermitian and they are linked by the formula aC .v; N w/ D aN C .v; w/. From this it follows aC .v; N w/ N D aC .v; w/

and

aN C .v; N w/ N D aN C .v; w/:

(5)

Exactly one of them has to vanish on the ˇC -irreducible subspace W . Indeed, if we ˇ ˇ ˇ suppose both to be non-zero and set aC W D aN C W .J./; / one shows that J 2 ˇ Endˇ .W /, hence J D 0.ˇ On the otherˇ hand aC ˇW W D 0 together with (5) implies aC ˇWS WS D 0. Thus, aC ˇWS W D aN C ˇW W has to be non-degenerate and vice versa. Lets denote by aQ the non-vanishing form on W . Since aQ is -invariant, it induces the -invariant C-valued R-bilinear form aQ on V via (1). In the following we will suppress the isomorphism . Real and imaginary part of aQ are related by Im .a/.x; Q y/ D Re .a/.x; Q I.y//: In particular they are linear independent and thus B.G/ .V / is spanned by these two forms. So the isometry group of aQ is isomorphic to the maximal subgroup of L GL.V / which fixes any element of B.G/ .V /. Note that any element of L commutes with I and thus it is complex linear. If aQ is symmetric, the same holds for its real and imaginary part and their signature has to be .n=2; n=2/ (cf. Proposition 5 (i)). We can find a complex orthonormal basis, i.e., .aQ ij / D In=2 and the isometry group is the complex orthogonal group O.n=2; C/ D f A 2 GL.n=2; C/ j At A D In=2 g:

646


If aQ is skew-symmetric, the same holds for its real and imaginary part. We can find a complex symplectic basis, i.e., .aQ ij / D Jn=4 and the isometry group is the complex symplectic group Sp.n=2; C/ D f A 2 GL.n=2; C/ j At Jn=4 A D Jn=4 g: Finally, aQ might be Hermitian (a complex skew-Hermitian form turns into a Hermitian one by multiplication with i ). We can find a complex (pseudo-)orthonormal basis, i.e., .aQ ij / D Ip=2;q=2 and the isometry group is the unitary group U.p=2; q=2/ WD f A 2 GL.n=2; C/ j ANt Ip=2;q=2 A D Ip=2;q=2 g: In this case the real part is symmetric and has signature .p; q/ and the imaginary part is skew-symmetric. As mentioned in Proposition 2, the complex basis fbi gn=2 iD1 of V (with respect to I ) n=2 induces the real basis fbi ; I.bi /giD1 . Thus, I D Jn=2 and we obtain the embedding GL.n=2; C/ ! f C 2 GL.n; R/ j C B Jn=2 D Jn=2 B C g; A B A C iB 7! : B A

Now the real part form aQ D In=2 is given in the associated real basis of the symmetric I

by Re.a/ Q D n=2 In=2 . Its isometry group is O.n=2; n=2/, thus we get the first identity of (4). Analogously, the real part of the symplectic form aQ D Jn=4 is given Jn=4 by Re.a/ Q D Jn=4 . Its isometry group is conjugated to O.n=2; n=2/ which yields the second identity of (4). The real part of the Hermitian form aQ D Ip=2;q=2 I

is given by Re.a/ Q D p=2;q=2 Ip=2;q=2 . Its isometry group is conjugate to O.p; q/. Instead of taking the intersection with the centralizer of Jn=2 as above we take here the isometry group of the imaginary part Im.a/ Q D Jn=2 which is Sp.n=2; R/, hence the third identity of (4). .; V / of quaternionic type: For representations of real or complex type all possible dimensions for the subspaces S.G/ .V / and ƒ.G/ .V / occurred. This is no longer true in the quaternionic case.

Lemma 1. If .; V / is self-dual and of quaternionic type then S.G/ .V / and ƒ.G/ .V / are odd-dimensional, i.e., the dimension of one space is 1 and of the other is 3. In particular, is both, orthogonal and symplectic. If the 1-dimensional subspace is spanned by fag then under the identification End.G/ .V / ' H the decomposition (3) is given by a Re.H/ D S.G/ .V / and Im.H/ D ƒa.G/ .V /: a .V /. On the other hand ƒa.G/ .V / Im.H/ Proof. Clearly Re.H/ D R Id S.G/ a by Proposition 5, hence Im.H/ D S.G/ .V / \ Im.H/ ˚ ƒa.G/ .V /. One of the


647

subspaces has dimension greater than or equal to two and is spanned by anti-commuting complex structures I; J . Irrespective of whether I; J are self- or skew-adjoint with respect to a, their product is skew-adjoint: .I BJ / D J BI D J BI D I BJ . N As in Proposition 2 we consider V as right H-vector space via x D .x/. Then an element a 2 B.G/ .V / as in the preceding lemma yields the following quaternionic sesquilinear form on V : aH .x; y/ WD a.x; y/ C i a.xi; y/ C j a.xj; y/ C k a.xk; y/: N H .x; y/ and aH .x; y/ D aH .x; y/. Recall that one has to check aH .x; y/ D a Since multiplication (from the right) with imaginary quaternions is an a-skew-adjoint operation according to Lemma 1, aH is Hermitian if a is symmetric and skewHermitian otherwise. By construction, B.G/ .V / is spanned by the real and imaginary parts of aH , hence the group L which fixes all elements of B.G/ .V / is the isometry group of aH (again H-linearity is ensured already by leaving aH invariant). Now, for any (skew-)Hermitian form one can find an orthogonal basis [Die71, Chapter I, §8]. In the Hermitian case the basis can be normed to the length ˙1. Thus the isometry group is the quaternionic unitary group Sp.p=4; q=4/ D f A 2 GL.n=4; H/ j ANt Ip=4;q=4 A D Ip=4;q=4 g: In the skew-Hermitian case the basis can be normed to the length i . Thus the isometry group is O .n=4/ D f A 2 GL.n=4; H/ j ANt i In=4 A D i In=4 g: The embedding L GL.n; R/ follows from the embedding GL.n=4; H/ GL.n; R/ as in Proposition 2. In order to obtain the remaining identities of (4), we fix i as complex structure and thus represent any quaternionic matrix by two complex matrices: A C iB C jC C kD D .A C iB/ C .C C iD/j D U C Vj . This yields the algebra isomorphism GL.n=4; H/ ! f C 2 GL.n=2; C/ j Cx B Jn=4 D Jn=4 B C g; U V U C Vj 7! x : V Ux

Since under this identification the operation C 7! Cx t is the same in GL.n=4; H/ and GL.n=2; C/ it is easily seen, that Sp.p=4; q=4/ is equal to the intersection of Ip=4;q=4 the isometry group U.p=2; q=2/ of the Hermitian form with the Ip=4;q=4 Ip=4;q=4 isometry group Sp.n=4; C/ of the symplectic form Ip=4;q=4 . Analogously, as intersection of the isometry group U.n=4; n=4/ of the skewwe obtain O .n=4/ iIn=4 Hermitian form iIn=4 with the isometry group O.n=2; C/ of the symmetric I form In=4 n=4 . This yields the remaining identities of (4).

648


We conclude the proof by showing that the maximal groups L GL.n; R/ are acting irreducibly on Rn . One knows even more: For any subgroup L GL.n=d; K/ GL.n; R/ occurring in the list of the proposition its centralizer coincides with the corresponding homothety group HK : EndL .Rn / D HK ;

L GL.K/:

For the symplectic groups this can be easily verified. For the unitary groups this is true beginning with n=d 2 and for the orthogonal groups it is true for n=d 3 (see [Die71, Chapter II, §3]). In this context the quaternionic unitary groups Sp.p=4; q=4/ and O .n=4/ are understood as unitary groups. Since any homothety is invertible, the above groups act irreducibly, otherwise the projection onto an invariant subspace would be an element of the centralizer which is certainly not invertible. It remains to discuss irreducibility in the excluded small dimensions. Remark 7. A quaternionic vector space does not admit any symmetric or skewsymmetric bilinear form. This is reflected in the fact that the space of symmetric or skew-symmetric bilinear forms is never 4-dimensional (cf. Lemma 1). i we obtain the emRemark 8. Changing the complex basis by the matrix p1 11 i 2 bedding O .n=4/ GL.n=2; C/ as given in [Hel01, Chapter X, §2,1]. ı There it is explained how O .n=4/ occurs as dual of the symmetric space O.n=2/ U.n=4/ which justifies the notation. The considerations above can be generalized to non-irreducible representations of Lie groups or Lie algebras. This has been done in [MR93]. Of course the structure of the algebra EndG .V / becomes more involved. On the other hand we may restrict our attention to special representations as such as the adjoint representation Ad.G/ GL.g/ of a Lie group G. To ask for an Ad.G/-invariant non-degenerate symmetric bilinear form on g becomes interesting from a geometrical point of view, since any such bilinear form induces a pseudo-Riemannian metric on G which is invariant under left- and right-multiplication. In particular G becomes a symmetric space. Hereafter we cite some results in this direction. As shown above there are representations which are symplectic but not orthogonal. This fails for adjoint representations: Proposition 7 ([MR93], Theorem 1.4). A Lie algebra g admits an ad.g/-invariant non-degenerate symmetric bilinear form if and only if it is self-dual. On the other hand it has been shown: Proposition 8 ([MR93], Corollary 1.7). A Lie algebra g admits an ad.g/-invariant skew-symmetric bilinear form if and only if codimg Œg; g 2.


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In particular, for simple Lie algebras the adjoint representation is irreducible and Œg; g D g. Thus, they cannot be symplectic which excludes many cases of Proposition 6.

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D. V. Alekseevsky and A. J. Di Scala, Minimal homogeneous submanifolds of symmetric spaces. In Lie groups and symmetric spaces, Amer. Math. Soc. Transl. Ser. (2) 210, Amer. Math. Soc., Providence, RI, 11–25. 2003. 640

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Y. Benoist and P. de la Harpe, Adhérence de Zariski des groupes de Coxeter. Compositio Math. 140 (5) (2004), 1357–1366. 631, 641

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M. M. Berger, Sur les groupes d’holonomie homogène des variétés a connexion affine et des variétés riemanniennes. Bull. Soc. Math. France 83 (1955), 279–330. 631

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M. M. Berger, Les espace symétriques non compacts. Ann. Sci. École Norm. Sup. 74 (1957), 85–177. 631

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A. J. Di Scala and T. Leistner, Connected subgroups of SO.2; n/ acting irreducibly on R2;n . Israel J. Math., in press; preprint 2008, arXiv:0806.2586v1 [math.DG]. 642

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A. J. Di Scala and C. Olmos, The geometry of homogeneous submanifolds in hyperbolic space. Math. Z. 237 (1) (2001), 199–209. 631, 640, 641

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A. S. Galaev, Metrics that realize all types of Lorentzian holonomy algebras. Int. J. Geom. Methods Mod. Phys. 3 (5–6) (2006), 1025–1045. 631

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M. Gotô, Faithful representations of Lie groups. I. Math. Japonicae 1 (1948) 107–119. 637, 638

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G. S. Hall, Covariantly constant tensors and holonomy structure in General Relativity. J. Math. Phys. 32 (1991), 181–187. 642

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S. Helgason, Differential geometry, Lie groups, and symmetric spaces. Pure Appl. Math. 80, Academic Press, New York 1978. 641

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S. Helgason, Differential geometry, Lie groups, and symmetric spaces. Grad. Stud. Math. 34, Amer. Math. Soc., Providence, RI, 2001. 648

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T. Leistner, Berger algebras, weak-Berger algebras and Lorentzian holonomy. SFB 288-Preprint no. 567, 2002, ftp://ftp-sfb288.math.tu-berlin.de/pub/Preprints/ preprint567.ps.gz. 631

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T. Leistner, Towards a classification of Lorentzian holonomy groups. Preprint 2003; arXiv:math.DG/0305139. 631

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T. Leistner, Towards a classification of Lorentzian holonomy groups. Part II: Semisimple, non-simple weak Berger algebras. Preprint 2003; arXiv:math.DG/0309274. 631

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T. Leistner, On the classification of Lorentzian holonomy groups. J. Differential Geom. 76 (3) (2007), 423–484. 631

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G. Mostow, Some new decomposition theorems for semi-simple groups. Mem. Amer. Math. Soc. 14 (1955), 31–54. 631, 640

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A. Médina and P. Revoy, Algèbres de Lie orthogonales. Modules orthogonaux. Commun. Algebra 21 (7) (1993), 2295–2315. 648


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R. S. Palais, The classification of real division algebras. Amer. Math. Monthly 75 (1968), 366–368. 632

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J. M. M. Senovilla, Second-order symmetric Lorentzian manifolds I: Characterization and general results. Classical Quantum Gravity 25 (2008), no. 24. 642

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H. Wakakuwa, Holonomy groups. Publications of the Study Group of Geometry 6, Department of Mathematics, Okayama University, Okayama 1971 630

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H. Wu, On the de Rham decomposition theorem. Illinois J. Math. 8 (1964), 291–311. 630, 639

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H. Wu, Holonomy groups of indefinite metrics. Pacific J. Math. 20 (2) (1967), 351–392. 630, 639

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Chapter 19

Surface holonomy Konrad Waldorf

Contents 1 Introduction . . . . . . . . . . . . . . . . . 2 Gerbes . . . . . . . . . . . . . . . . . . . 3 From line bundles to bundle gerbes . . . . 4 Morphisms of bundle gerbes . . . . . . . . 5 Holonomy around closed oriented surfaces 6 The line bundle over the loop space . . . . 7 D-branes and surfaces with boundary . . . 8 Unoriented closed surfaces . . . . . . . . . References . . . . . . . . . . . . . . . . . . . .

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1 Introduction From the theory of hermitian line bundles with connection we single out one interesting aspect: such a line bundle L ! M assigns to each loop W S 1 ! M a certain complex number holL ./ in U.1/ which is called the holonomy of L around . This assignment has many properties, and we will gradually encounter some of them during this chapter. It is worthwhile to specify three of them here, which will turn out to be analogous for surface holonomy. To start with, the holonomy of L measures the curvature of L in the following way: for an embedded two-dimensional submanifold D of M with parameterized boundary @D one finds Z exp curv.L/ D holL .@D/; (1) D

where we have normalized the exponential function by a factor of 2i such that it is 1-periodic. Equation (1) can be seen as an improvement of Stokes’Theorem in two dimensions, Z Z F D D

@D

which expresses the integral of the closed 2-form F by something on the boundary, just like the above formula for the integral of the curvature of the line bundle L. However, Stokes’ Theorem is restricted to exact 2-forms F D d whose cohomology class

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ŒF 2 H2 .M; R/ vanishes. Our improvement allows us at the price of exponentiation not only to take 2-forms with trivial cohomology class, but also – and more general – 2-forms whose class lies in the image of the homomorphism Hk .M; Z/ ! Hk .M; R/. Those 2-forms F – which we shall call forms with integral class – arise as the curvature of a hermitian line bundle L with connection. Then, Z F D ln .holL .@D// mod Z D

generalizes Stokes’ Theorem. A second reason why holonomy is interesting lies in understanding the assignment 7! holL ./ as a U.1/-valued function holL W LM ! U.1/ on the loop space LM that consists of all smooth loops S 1 ! M . The loop space can canonically be endowed with the structure of an infinite dimensional manifold, so that the map holL becomes smooth [Bry93]. One can consider going in the other direction, when one is concerned with a smooth U.1/-valued function on the loop space and is able to express it as the holonomy of a hermitian line bundle with connection over the – finite dimensional – manifold M . Also of importance is the application of line bundles in physics: a hermitian line bundle L ! M with connection offers a natural description of a U.1/-gauge theory. On the quantum level, such a theory can be defined by assigning a complex number to each particle moving (on a closed line) through the target space M ; this number acts as an amplitude in some path integral. If a particle moves on a circle W S 1 ! M , its amplitude is given by A./ D eSkin . / holL . /; where Skin ./ is a kinetic term, and the holonomy expresses the coupling to the gauge field. The line bundle L comes up with all features you would expect from a gauge field: its curvature is a 2-form F and may be called the field strength of the gauge field. The second Bianchi identity dF D 0 implies two of the four Maxwell equations, and – last but not least – the fact that F has an integral class is nothing but Dirac’s quantization condition for the electric charge. The question for an appropriate concept of surface holonomy has similar origins: • it could provide a generalization Stokes’ Theorem in three dimensions similar to that in two dimensions indicated above. • it could be used in string theory to couple strings to non-trivial background fluxes, analogous to the coupling of a point particle to a gauge field. • it could also provide actions for 3-form gauge fields in certain (classical) gauge theories. • it could provide a way to describe structure on the loop space by structure on a finite-dimensional space, very much in the same way like smooth U.1/-valued functions may be interpreted as the holonomy of a hermitian line bundle with connection over M .

Chapter 19. Surface holonomy

655

Let us pick out the first two points and describe how they are related to surface holonomy. The generalization of Stokes’ Theorem is obvious: if H is a 3-form on a three-dimensional manifold B, which may be not exact but with integral class, its integral over B could be expressed as (the logarithm of) the holonomy around the surface † D @B. In fact, this is exactly a question which arises in two-dimensional conformal field theory, when studying non-linear sigma models on a Lie group G. Such a model can be defined by amplitudes A./ for some path integral, where is a map from a closed complex surface † – the worldsheet – into the target space G of the model. In [Wit84], Witten gives the following definition for G D SU.2/. † is the boundary of a three dimensional manifold B, and because the homotopy groups i .SU.2// vanish for i D 1; 2, every map W † ! M can be extended into the interior B to a map ˆ W B ! G. From the theory of compact, simple, connected and simply connected Lie groups it is known that the Ad-invariant trilinear form kh; Œ; i on the Lie algebra su.2/ induces a closed, bi-invariant 3-form H on SU.2/ with integral class, provided k is an integer. Witten showed that – due to the integrality of H – Z ˆ H A./ WD exp Skin ./ C B

neither depends on the choice of B nor on the choice of the extension ˆ, so that he obtained a well-defined amplitude. Here Skin ./ is a kinetic term, and with a certain relative normalization this model is called the Wess–Zumino–Witten model on G at level k. Now, if we could express the integral of ˆ H over B by something on the boundary † – for instance via a generalization of Stokes’ Theorem – the definition of the amplitude A./ would be independent of conditions on the existence of the extension ˆ, in particular of the simply-connectedness of G. It would hence provide a proper definition of Wess–Zumino–Witten models on arbitrary simple and compact Lie groups. Acknowledgements. I thank U. Schreiber and C. Schweigert for the many discussions during the preparation of this chapter, and J. Fjelstad and again C. Schweigert for useful remarks and criticism.

2 Gerbes To define surface holonomy, we first need a mathematical object which plays the role of the hermitian line bundle with connection. Such an object is collectively called gerbe with connective structure. What makes gerbes a bit mysterious is that there are numerous definitions which look outmost different. To give an impression we show some examples of the different manifestations of gerbes and their connective structures.

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Gerbes as stacks. This is the original definition given in 1971 by J. Giraud with a view to non-abelian cohomology [Gir71]. A stack is a fibred category satisfying a gluing (or descent) axiom. According to Giraud, a gerbe is a stack, whose fibres are groupoids and satisfy a certain transitivity and non-emptiness condition, also see [Moe02]. As an important characterization of a gerbe, Giraud defines the band of a gerbe, a certain sheaf of groups. The definition of a connective structure on a gerbe in this sense was given thirty years later by L. Breen and W. Messing [BM05], who considered gerbes with arbitrary bands living over a scheme. Gerbes as cohomology classes. Around 1972, P. Deligne invented a cohomology theory which is now called Deligne cohomology [Del91], [Bry93]. It is build up on cochain complexes 0

/ D 0 .n/

D

/ D 1 .n/

D

/ :::

D

/ D k .n/ ,

one for each natural number n. Deligne was originally interested in algebraic geometry, and realized that the cohomology group H1 .M; D.1// classifies hermitian line bundles with connection [Del91]. A class in the cohomology group H2 .M; D.2// can be seen as a U.1/-banded gerbe with connective structure in the sense that it provides a definition of holonomy around surfaces. This was shown by K. Gaw¸edzki and applied to topological field theory [Gaw88]. Gaw¸edzki also showed that gerbes are related to a certain structure on the loop space, namely to a line bundle, which is obtained by a so-called transgression procedure. Deligne cohomology provides a natural way to see gerbes with connective structure in a hierarchy of objects, starting with U.1/-valued functions, hermitian line bundles with connection and gerbes with connective structure, and which is continued by n-gerbes which are classified by HnC1 .M; D.n C 1//. Gerbes as sheaves of groupoids. This concept is strongly related to the original one of a certain kind of stack, and defines a gerbe over a manifold M by an assignment U 7! G .U / of a groupoid G .U / to any open subset U of M . It is developed in great detail in J.-L. Brylinski’s book [Bry93]. Analogous to the gluing axiom of a sheaf, gluing axioms for such a gerbe a given. For C -banded sheaves of groupoids, Brylinski develops the definition and properties of a connective structure. He gives precise relations and classification result between sheaves of groupoids and Deligne cohomology. Furthermore he constructs a line bundle over the loop space and shows that it coincides with the one Gaw¸edzki constructed from Deligne cohomology. Gerbes as bundle gerbes. The concept of a sheaf of groupoids and even more the one of a connective structure on it is quite general but also quite complex. For some purposes, e.g. for the definition of holonomy and for applications in conformal field


657

theory, it is sufficient to use a simplified version – simplified in the sense that it just uses well-known geometric structure like line bundles and differential forms. For U.1/- or C -banded gerbes, M. K. Murray invented bundle gerbes [Mur96], which we will use in this chapter to develop surface holonomy. As we will learn, bundle gerbes admit a very simple and natural definition of a connective structure. To make contact to other concepts of gerbe, we will see, that every bundle gerbe induces a sheaf of groupoids, and that their isomorphism classes are in bijection to the Deligne cohomology group H2 .M; D.2//. Bundle gerbes have also been used in two-dimensional conformal field theory [GR02], [Gaw05], [SSW05]. Gerbes defined on open covers. Closely related to bundle gerbes are gerbes defined on open covers [Hit01], although bundle gerbes are a bit more general. Gerbes as 2-bundles. A somewhat different approach to surface holonomy is by categorification of a vector bundle with connection. This leads to the concept of a 2-bundle with certain additional structure [BS04]. This approach also covers gerbes with non-abelian bands. Furthermore it realizes consequently the 2-categorial nature of gerbes, which we will also discover during this chapter. For the purposes of this exposition we consider gerbes with band U.1/, also called abelian (hermitian) gerbes. Accordingly we drop the qualifier hermitian for gerbes and for line bundles to improve the readability.

3 From line bundles to bundle gerbes As indicated before, bundle gerbes are built up of line bundles and differential forms. One of basic features of a line bundle L ! M is that it is locally trivializable. This is usually stated with respect to an open cover, but here we state it with respect to a covering W Y ! M . From any open cover fVi gi2I of M one can produce such a covering by defining Y as the disjoint union of the sets, G Vi ; Y D i2I

and as patched together from the inclusions Vi ,! M . Locally trivializable means that there is a covering W Y ! M and a commutative diagram /L

Y C Y

/ M:

A local trivialization defines a transition function g W Y Œ2 ! U.1/, where Y Œk denotes the k-fold fibre product of Y with itself. The transition function satisfies the cocycle

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condition g 23 g D 13 g 12

(1)

over Y Œ3 , where 12 W Y Œ3 ! Y Œ2 are the projections on the respective components. If Y comes from on open cover, Y Œk is the disjoint union of all k-fold intersections of the open sets Vi . Accordingly, the transition function g decomposes in functions gij W Vi \ Vj ! U.1/, and the cocycle condition becomes gij gj k D gik as functions on Vi \ Vj \ Vk . Bundle gerbes do not have a total space as line bundles do. For the definition of a bundle gerbe we step in after having locally trivialized, i.e. after having chosen a covering W Y ! M , ‹

Y

/ M.

Now we define the bundle gerbe analogous to what remains of the locally trivialized line bundle. According to our remarks about Deligne cohomology, U.1/-valued functions, hermitian line bundles with connection and gerbes with connective structure form a certain series of objects. Now we move up one step: instead of a transition function on Y Œ2 , we take a line bundle L ! Y Œ2 . The next steps are predicted: because we cannot multiply line bundles like the pullbacks of the transition function g in (1), the cocycle condition has to be relaxed to an isomorphism W 12 L ˝ 23 L ! 13 L

of line bundles over Y Œ3 – called the groupoid multiplication. To capture an essential aspect of the multiplication of functions, we demand that this isomorphism is associative. It is straightforward to define a connective structure on a bundle gerbe. Consider first a connection on a line bundle L ! M . In a local trivialization W Y ! M , this connection defines a 1-form A 2 1 .Y /, which is related to the transition function g by g 1 dg D 2 A 1 A: For the bundle gerbe, we take a connection on the line bundle L ! Y Œ2 and impose that the isomorphism respects connections. Additionally, we take a 2-form C 2 2 .Y / – called the curving – which has to be related to the connection on L by curv.L/ D 2 C 1 C: The connection on L together with the curving C form the connective structure. It is shown in [Mur96] that every bundle gerbe admits a connective structure. In the rest of this chapter, we will only be concerned with bundle gerbes with connective structure, and hence drop the last suffix. We will also understand a line bundle as a (hermitian) line bundle with connection. Accordingly, all isomorphisms

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of line bundles will be isomorphisms of line bundles which preserve the connections. With these conventions, we are arrived at the following definition: Definition 1. A bundle gerbe G over M consists of a covering W Y ! M , a 2-form C 2 2 .Y /, a line bundle L ! Y Œ2 and an isomorphism W 12 L ˝ 23 L ! 13 L

of line bundles over Y Œ3 . Two axioms have to be satisfied: (G1) The curvature of L is related to the curving by curv.L/ D 2 C 1 C: (G2) The groupoid multiplication is associative in the sense that the diagram L ˝ 23 L ˝ 34 L 12

˝1 123

1˝234

12 L ˝ 24 L

/ L ˝ L 13 34 134

124

/ L 14

of isomorphisms of line bundles over Y Œ4 is commutative. In the following we will specify some properties of this definition and show several constructions what one can do with it. Similar to line bundles (with connection), each bundle gerbe G determines a closed 3-form curv.G / on M , called the curvature of G : the derivative of axiom (G1) gives 1 dC D 2 dC , since the curvature of the line bundle L is a closed form. This means that dC descends along W Y ! M to a 3-form on M – the curvature of the bundle gerbe G . It is obviously a closed form, and it will turn out later that it has an integral class. To give an example of a bundle gerbe, we introduce trivial bundle gerbes. Just as for every 1-form A 2 1 .M / there is a trivial line bundle over M having this 1-form as its connection, we find a trivial bundle gerbe for every 2-form 2 2 .M /. The construction of this bundle gerbe is quite easy: for the covering we take the identity id W M ! M , and the curving is the given 2-form . The line bundle L is the trivial line bundle with the trivial connection, and the groupoid multiplication is the identity isomorphism between trivial line bundles. Now, axiom (G1) is satisfied since curv.L/ D 0 and 1 D 2 D idM . The associativity axiom (G2) is surely satisfied by the identity isomorphism. Thus we have defined a bundle gerbe, which we denote by . The curvature of a trivial gerbe is curv. / D d. Less elementary examples of bundle gerbes have been constructed in [GR02], [Mei02], [GR03], namely all (bi-invariant) bundle gerbes over all simple compact Lie groups. The availability of concrete examples in such non-trivial cases is an important advantage of bundle gerbes. Their constructions were possible by an explicit use of the geometric nature of bundle gerbes. For example, to construct a bundle gerbe

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over SU.N /, the line bundle L occurs as the canonical line bundle on the coadjoint orbit through a simple weight of some representation of SU.N /. Another remarkable aspect is that the Lie-theoretic construction of bundle gerbes over Lie groups apart from SU.N / or Sp.4n/ makes use of the fact that a covering W Y ! M is more general that having an open cover of M : the dimension of Y may be greater than the dimension of M . Let us again assume that the covering W Y ! M of a bundle gerbe G comes from an open cover fVi gi2I of M (in fact this is the kind of gerbe which was called before a gerbe defined on open covers [Hit01]). Remember that we introduced the gerbe data – namely the line bundle L, the groupoid multiplication and the curving C – as analogues of the data of a trivialized line bundle. Surely the curving restricts to a 2-form Bi on each open set Vi . To get similar local expressions for the line bundle and the groupoid multiplication, we can trivialize once more: if the open sets Vi are chosen such that every double intersection Vi \ Vj is contractible, we are able to choose sections ij W Vi \ Vj ! L of unit length. Then, the connection on L pulls back to 1-forms Aij on each double intersection Vi \ Vj . Furthermore, over a three-fold intersection Vi \ Vj \ Vk , we can multiply two sections using the groupoid multiplication, and compare the result to a third section, . ij ˝ j k / D gij k ik ; via a function gij k W Vi \ Vj \ Vk ! U.1/. Summarizing, we have extracted U.1/valued functions gij k on three-fold intersections, 1-forms Aij on two-fold intersections, and 2-forms Bi on each open set. One can deduce the following relations among this local data: gij k gikl D gj kl gij l ; gij1k dgij k D Aj k Aik C Aij ; dAij D Bj Bi : The first one is a consequence of the associativity of from axiom (G2), the second is the fact that preserves connections, and the third is the curvature condition (G1). These equations look like analogues of the two conditions for local data of a line bundle – the transition functions gij W Vi \ Vj ! U.1/ and the connection 1-forms Ai on Vi – namely gij gj k D gik ; gij1 dgij D Aj Ai : A precise meaning of this analogy is given by Deligne cohomology. We already have indicated that Deligne cohomology comes from a cochain complex with cochain groups D k .n/ and a coboundary operator D. Their definition is such that the collection of local data .g; A; B/ of the bundle gerbe G defines an element in D 2 .2/. The three


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relations among this local data give the cocycle condition D.g; A; B/ D 0, so that we see, that the bundle gerbe G defines a class in the cohomology group H2 .M; D.2//. Similarly, the collection of local data .g; A/ of a line bundle forms an element in D 1 .1/, and the two relations above give the cocycle condition D.g; A/ D 0. So, a line bundle defines a class in H1 .M; D.1//. In the next section we will see, that with the correct definition of isomorphisms between bundle gerbes, the correspondence of isomorphism classes of bundle gerbes with classes in Deligne cohomology is one-toone. To continue with specifying properties of a bundle gerbe, we come to the definition of characteristic classes. The transition function gij W Vi \ Vj ! U.1/ of a line bundle L ! M with respect to an open cover fVi gi2I defines a class Œg in the ˇ Cech cohomology of the sheaf U.1/ with respect to the chosen open cover. Via the exponential sequence, it hence gives rise to a class Œg 2 H2 .M; Z/; which is independent of the cover and of the sections chosen to get the transition function. It is thus an intrinsic quantity of the line bundle L, called the (first) Chern class and denoted by c1 .L/. The image of the Chern class in de Rham cohomology is equal to the class of the curvature of L, c1 .L/ D Œcurv.L/; which proves that the curvature of a line bundle with connection is a 2-form with integral class. In the same way, the function gij k W Vi \ Vj \ Vk ! U.1/ defined by the groupoid multiplication of a bundle gerbe G trivialized on an open cover, defines a class Œg 2 H3 .M; Z/ which is also independent of the open cover and of the sections ij which were chosen to extract the local data. This class is called the Dixmier–Douady class of the bundle gerbe G and denoted by dd.G /. The Dixmier–Douady class and the curvature of G obey the same relation as the Chern class and the curvature of a line bundle [Mur96]: dd.G / D Œcurv.G /: This shows that the curvature of a bundle gerbe is a 3-form with integral class. As a little example, we may compute the Dixmier–Douady class of a trivial bundle gerbe . It is easy to see that – for any open cover – the functions gij k can be chosen to be constantly 1, so that the local connection 1-forms vanish, Aij D 0. Solely the curving restricts to non-trivial 2-forms jVi on Vi . Now it is clear that the Deligne class of is represented by .1; 0; /, and that the Dixmier–Douady class vanishes. Indeed, the curvature is an exact form, whose class also vanishes. Finally, let us remark that there are three standard constructions one can do with bundle gerbes: tensor products, duals and pullbacks. Without giving the details of these constructions, one observes that they behave naturally under the correspondence

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with classes in Deligne cohomology: the tensor product of bundle gerbes corresponds to the sum of classes, taking the dual bundle gerbe corresponds to taking opposite sign, and the pullback of bundle gerbes corresponds to the pullback of classes.

4 Morphisms of bundle gerbes To find the appropriate definition of a morphism between two bundle gerbes G1 and G2 , assume for a moment that the coverings i W Yi ! M come from open covers for i D 1; 2. To compare the gerbe data, it would be natural to go to a common refinement of these covers. On the double intersections of this common refinement the line bundles L1 and L2 could be compared. For general coverings 1 and 2 the common refinement amounts to consider the fibre product Z WD Y1 M Y2 thought of as a new covering W Z ! M sending .y1 ; y2 / to the point 1 .y1 / D 2 .y2 /. The two-fold intersections amount to consider Z Œ2 . The restriction of the line bundle Li to Z Œ2 is implemented by the pullback along the canonical map yi W Z Œ2 ! YiŒ2 . A first idea is to require that the line bundles y1 L1 and y2 L2 are isomorphic. In fact, this was the original definition of a morphism between bundle gerbes [Mur96]. However, it turned out that this definition was too restrictive, in other words: the isomorphism classes were too small, and there were many non-isomorphic bundle gerbes having the same Deligne class and hence the same surface holonomy. A solution to this was presented in [MS00]: the line bundles should not be isomorphic but stably isomorphic in the sense that there is a line bundle A ! Z with an isomorphism y1 L1 ˝ 2 A Š 1 A ˝ y2 L2 of line bundles over Z Œ2 . Here 1 and 2 are two natural projections from Z Œ2 to Z. It is natural to demand that the data of a morphism of bundle gerbes – the line bundle A ! Z and an isomorphism ˛ as above – is compatible with the rest of the structure of the bundle gerbes, namely the curvings and the groupoid multiplications. Summarizing, with the additional generalization that A may be a vector bundle (of rank higher than 1), the correct definition of a morphism between bundle gerbes is Definition 2. A morphism A W G1 ! G2 of bundle gerbes is a vector bundle A ! Z together with an isomorphism ˛ W y1 L1 ˝ 2 A ! 1 A ˝ y2 L2 of vector bundles over Z Œ2 . Two axioms have to be satisfied: (M1) The curvature of A is a real 2-form and fixed by curv.A/ D y2 C2 y1 C1 :

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(M2) The isomorphism ˛ commutes with the groupoid multiplications 1 and 2 of the bundle gerbes in the sense that the diagram y1 1 ˝id

L1 ˝ y1 23 L1 ˝ 3 A y1 12

/ y L1 ˝ A 1 13 3

˛ 1˝23

12 y1 L1 ˝ 2 A ˝ 23 y2 L2

˛ 13

˛˝1 12

1 A ˝ y2 12 L2 ˝ y2 23 L2

1˝y2 2

/ A ˝ y L2 1 2 13

of isomorphisms of vector bundles over Z Œ3 is commutative. There is one important point to notice from this definition of a morphism. Given two such morphisms A W G1 ! G2

and

A0 W G1 ! G2

each providing a vector bundle A, A0 over Z, to compare both morphisms it does not make sense to state that they are equal or not: to compare vector bundles one needs isomorphisms between them. This leads us forthright to the fact that bundle gerbes form a 2-category [Ste00]: we have bundle gerbes as objects, morphisms between the objects as defined above, and 2-morphisms between the morphisms. Before we discuss the 2-categorial aspects of bundle gerbes, we give the precise definition of a 2-morphism. Definition 3. Let A W G1 ! G2 and A0 W G1 ! G2 be two morphisms. A 2-morphism ˇ W A ) A0 is an isomorphism ˇ W A ! A0 of vector bundles over Z, which is compatible with the isomorphisms ˛ and ˛ 0 in the sense that the diagram y1 L1 ˝ 2 A

˛

1˝2 ˇ

y1 L1 ˝ 2 A0

/ A ˝ y L2 1 2 1 ˇ ˝1

˛0

/ A0 ˝ y L2 2 1

of isomorphisms of vector bundles over Z Œ2 is commutative. Realizing that bundle gerbes form a 2-category is not a fault of the theory, it is a feature. To give an example, notice that – as in every 2-category – the set of all morphisms between two fixed bundle gerbes G1 and G2 , together with the set of all the 2-morphisms between such morphisms, forms a category. To be a bit more precise:

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Konrad Waldorf

for bundle gerbes they form a groupoid, since all 2-morphisms are invertible. In particular, we have a groupoid of endomorphisms from a bundle gerbe G to itself. This groupoid may be considered as the groupoid of gauge transformations of G . So we get a clear understanding what the gauge symmetry of a gauge theory for strings is: it is a groupoid rather than a group. Another feature of this 2-categorial point of view is the following. In any 2category, a morphism A W G1 ! G2 is called isomorphism, if it is invertible in the sense that there is another morphism B W G2 ! G1 in the opposite direction, such that there are 2-isomorphisms B B A ) idG1

and

A B B ) idG2 :

Now we ask, what this general definition means for morphisms of bundle gerbes from Definition 2. Of course one has to say two things: what are the identity morphisms idG1 and idG2 , and how the composition B is defined. The last point is quite involved, however it can be done [Ste00]. We only present the result here. Proposition 1. A morphism A W G1 ! G2 of bundle gerbes is an isomorphism, if and only if the vector bundle A ! Z is a line bundle, i.e. has rank 1. The standard literature about bundle gerbes, e.g. [CJM02], [GR02], takes the last proposition as the definition of morphisms between bundle gerbes (so-called stable isomorphisms), and neglects that there are morphisms with vector bundles of higher rank. As a consequence, in the standard literature bundle gerbes form a 2-groupoid rather than a 2-category. The advantages of our definition become apparent later on. For the rest of this section, we restrict ourselves to isomorphisms between bundle gerbes. We would like to get an overview over the set of possible isomorphisms between G1 and G2 . First we get rid of the 2-morphisms by going to equivalence classes: we call two such isomorphisms equivalent, if there is a 2-morphism between them (which is automatically a 2-isomorphism). In other words, we consider the skeleton of the groupoid of isomorphisms between G1 and G2 . Proposition 2 ([CJM02], [SSW05]). The set of equivalence classes of isomorphisms between two fixed bundle gerbes is a torsor over the group Pic0 .M / of isomorphism classes of flat line bundles over M . Supposed we are able to compose morphisms as indicated above, being isomorphic is an equivalence relation on the set of bundle gerbes. We present a result which classifies these isomorphism classes of bundle gerbes, in other words: we are looking for the skeleton of the 2-category of bundle gerbes. Recall that a bundle gerbe defines by its local data a cocycle in the Deligne cochain group D 2 .2/. Similarly it can be shown, that an isomorphism A W G1 ! G2 defines a cochain .t; W / in the cochain group D 1 .2/. It relates the cocycles of the two bundle gerbes by its coboundary, .g2 ; A2 ; B2 / D .g1 ; A1 ; B1 / C D.t; W /:


665

This equation means that two isomorphic bundle gerbes define the same class in Deligne cohomology. Even stronger is the following theorem. Theorem 1 ([MS00]). The set of isomorphism classes of bundle gerbes is in bijection to the Deligne cohomology group H2 .M; D.2//. This theorem makes contact to one of the other definitions of gerbes with connective structure and gives the precise relation between line bundles and bundle gerbes: line bundles are classified by the Deligne cohomology group H1 .M; D.1//, and bundle gerbes by H2 .M; D.2//. In the remainder of this section, we are going to sketch the relation between bundle gerbes and yet another realization of gerbes, namely sheaves of groupoids. To this purpose we need the trivial bundle gerbes defined in the last section as an example for bundle gerbes. In the same way as a trivialization of a line bundle is an isomorphism to a trivial line bundle, we say Definition 4. A trivialization of a bundle gerbe G is an isomorphism T W G ! : Let us briefly exhibit the details of a trivialization, which follow from the definitions of an isomorphism and of the trivial bundle gerbe . The isomorphism T consists of a line bundle T over the space Z D Y M M which we identify canonically with Y itself. Under this identification, the two projections to the coverings of the bundle gerbes become the identity id W Y ! Y and the covering W Y ! M itself, so that axiom (M1) becomes curv.T / D C: Further, the isomorphism T consists of an isomorphism W L ˝ 2 T ! 1 T of line bundles over Z Œ2 D Y Œ2 . Because the groupoid multiplication of the trivial bundle gerbe is the identity, axiom (M2) for reduces to the equation 13 B D 12 B 23

of isomorphisms of line bundles over Z Œ3 D Y Œ3 . Of course not every bundle gerbe admits a trivialization. In the same way as for line bundles the obstruction to the existence of a trivialization is given by the first Chern class, a bundle gerbe G admits a trivialization if and only if its Dixmier–Douady class vanishes [CJM02]. In this case, the curvature of the bundle gerbe G is an exact form, and curv.G / D d for any trivialization T W G ! . We define a sheaf of groupoids in the following way: for an open subset U of M consider the set of trivializations G jU ! for all 2-forms (the set may be empty). Since trivializations are nothing but isomorphisms of bundle gerbes, together with the

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2-morphisms they form naturally a groupoid G .U /. Furthermore, trivializations can clearly be restricted to smaller subsets, so that the assignment U 7! G .U / is a presheaf of groupoids. The gluing axiom can be shown by gluing together two trivializations over U1 and U2 , if there is a 2-morphisms between their restrictions to the intersection U1 \ U2 . This way, every bundle gerbe defines a sheaf of groupoids in Brylinski’s sense [MS00].

5 Holonomy around closed oriented surfaces The holonomy of a bundle gerbe around a closed oriented surface should be analogous to the holonomy of a line bundle around a loop W S 1 ! M , since S 1 is also closed and oriented. So, it is worthwhile to recall how the holonomy of a line bundle L over M around a loop W S 1 ! M can be defined. The pullback of L along gives a line bundle over the circle, whose first Chern class vanishes for dimensional reasons. Hence, it becomes isomorphic to a trivial line bundle with some connection 1-form A 2 1 .S 1 /. Then, Z holL . / WD exp A S1

is a number in U.1/ which is in fact independent of the choice of the trivialization. We also write out this definition in terms of local data .g; A/ of the line bundle L with respect to an open cover fVi gi2I . Choose a triangulation of S 1 that is subordinated to the open cover by a map i W ! I , such that .e/ Vi.e/ for any edge e and .v/ 2 Vi.v/ for any vertex v. Then, by splitting the integral over A with respect to the triangulation and using Stokes’ Theorem, one can derive the formula Z Y Y .e;v/ holN ./ WD exp Ai.e/ gi.e/i.v/ ..v// i 2I

e

v2@e

where .e; v/ 2 f1; 1g is positive, if v is the endpoint of e end negative otherwise. The meaning of this formula is that one has to integrate the local connection 1-forms along the edges, and to use the transition functions to intermediate at the vertices between two edges. For the definition of the holonomy of a bundle gerbe G we start with a configuration like in Figure 1 and mimic the same procedure as for line bundles. Definition 5 ([CJM02]). Let G be a bundle gerbe over M . For a closed oriented surface † and a smooth map W † ! M , let T W G !

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Chapter 19. Surface holonomy G

M

†

Figure 1. A surface is mapped into some space M with bundle gerbe G .

be a trivialization of the pullback of the bundle gerbe G along . Then we define Z holG ./ WD exp †

to be the holonomy of the bundle gerbe G around W † ! M . This sounds easy but we have to assure that the number holG ./ is independent of the choice of the trivialization, which exists for dimensional reasons. Different trivializations may have different 2-forms , however, in the next lemma we show that the difference 2 1 between to such 2-forms is the curvature of some line bundle over M , in particular: it is a closed form with integral class. Then, the calculation Z Z Z Z exp 2 D exp 2 1 exp 1 D exp 1 †

†

†

†

shows that the definition holG ./ is independent of the choice of the trivialization. Lemma 1. Two trivializations T1 W G ! 1 and T2 W G ! 2 of the same bundle gerbe G over M determine a line bundle over M with curvature 2 1 . Proof. Using the features of the 2-category of bundle gerbes, we can give a very relaxed proof: by taking the inverse and composition, we obtain an isomorphism T2 B T11 W 1 ! 2 of trivial bundle gerbes. From the definitions of isomorphisms and trivial bundle gerbes it follows immediately, that T2 B T11 is a line bundle with curvature 2 1 . But since we have not defined inverses and composition of isomorphisms, let us also give a more concrete proof. Recall that the two trivializations provide line bundles T1

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and T2 over Y . We show that the line bundle T2 ˝ T1 descends along W Y ! M . To do so we have to specify descent data, namely an isomorphism W 2 .T2 ˝ T1 / ! 1 .T1 ˝ T2 / of line bundles over Y Œ2 which satisfies the cocycle condition 13 D 12 B 23

over Y Œ3 . For the definition of recall that the two trivializations provide isomorphisms i W L ˝ 2 Ti ! 1 Ti for i D 1; 2. Then we declare to be the following isomorphism: W 2 .T2 ˝ T1 / Š .L ˝ 2 T2 / ˝ .2 T1 ˝ L /

1 ˝ 21

1 T2 ˝ 1 T1 Š 1 .T2 ˝ T1 / Using axiom (M2) for 1 and 2 one can now show that satisfies the cocycle condition. Hence, T2 ˝ T1 descends to a line bundle N over M with the property N Š T2 ˝ T1 : To finish the proof we have to compute the curvature of N . Notice that by axiom (M1) we find curv.T2 ˝ T1 / D 2 C . 1 C / D .2 1 /: This shows that the curvature of N is 2 1 .

Now that we have the definition of holonomy around a closed oriented surface we address the question if it provides the desired generalization of Stokes’ Theorem, which is important for the application in Wess–Zumino–Witten models. Proposition 3. Let G be a bundle gerbe over M with curvature H . For a threedimensional oriented manifold B with boundary and a map ˆ W B ! M , we find Z holG .ˆj@B / D exp ˆ H : B

Proof. Remember that for any trivialization T W ˆ G j@B ! we have ˆ H j@B D d. Then, by definition, Z Z holG .ˆj@B / D exp D exp ˆ H @B

B

It is important to recognize that the last step does not just follow from Stokes’Theorem (because ˆ H is not exact over B). In fact one can triangulate B, choose local trivializations with 2-forms i differing by closed 2-forms with integral class by Lemma 1, and then use Stokes’ Theorem on each 3-face. The remaining differences between the


2-forms disappear after exponentiation.

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This way we reproduce the amplitude of the coupling term of the Wess–Zumino– Witten model by A./ D exp .Skin .// holG ./: Notice that we did not impose any condition on the topology of the target space M . While in Witten’s definition the background field (apart from the metric) is just the three 3-form H , in the approach using bundle gerbes the background field is the bundle gerbe G . This is no contradiction since all bundle gerbes over SU.2/ with curvature curv.G / D H are isomorphic [GR02]. However, for general target spaces there may be bundle gerbes with same curvature, which are not isomorphic. This occurs for instance on the Lie group Spin.4n/=.Z2 Z2 /. Then, the bundle gerbe contains more information as just its curvature, and it becomes essential to recognize the bundle gerbe itself as the background field which defines the theory. Just as little as by its curvature, the bundle gerbe can be replaced by a 2-form, sometimes called the B-field, or Kalb–Ramond field. Such a 2-form exists in general only locally, namely when there is a trivialization G jU ! . Even then, it is not unique. This situation is analogous in every detail to what is called the Aharonov–Bohm effect in electrodynamics: in a (first) quantized theory of charged particles moving through a certain electric field of field strength F , not only the field strength, but also the gauge potential A is an observable quantity which is necessary to describe the theory – opposed to a classical theory of electrodynamics where everything is determined by the field strength and Maxwell’s equations. This Aharonov–Bohm effect (which in fact was predicted 10 years before by W. Ehrenberg and R. E. Siday) has been measured. Now one might think that the gauge potential A, a 1-form, is the object which describes the theory. But the gauge potential with dA D F is only defined locally, and if it is, it is not unique. This is exactly the behaviour of a line bundle and shows that a line bundle provides the correct description for this situation. To close this section, we reformulate the holonomy in terms of local data of the bundle gerbe, analogous the formula for the holonomy of a line bundle. Recall that a trivialization T W G ! chosen in Definition 5 implies the following relation between the local data of the bundle gerbe and the local data of the trivial gerbe, .1; 0; / D .g; A; B/ C D.t; W /: Now, following the strategy of the local expression for the line bundle, we choose a triangulation of the surface †, consisting of faces f , edges e and vertices v. It should be chosen subordinated to the same open cover fVi gi2I of M which was used to extract the local data .g; A; B/ of the bundle gerbe G . So there is a map i W ! I , assigning to each face, edge or vertex f an index i.f / so that .f / Vi.f / . Now the integral of the 2-form over † which defines the holonomy may be split up with

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respect to the triangulation. By a subsequent use of Stokes’ Theorem and the above formula for the local data [CJM02], one ends up with the following formula Z Y exp Bi.f / holG ./ D f 2

Y

e2@f

f

Z

exp e

Y

Ai.f /;i.e/

v2@e

.f;e;v/ gi.f ..v//: /;i.e/;i.v/

(1)

This formula shows explicitly what’s going on: the surface integral is expressed by local integrals over the local 2-forms Bi , the local Kalb–Ramond fields with dBi D H . But on the edges and vertices, the sum has to be corrected by the rest of the local data of the bundle gerbe. This way, the triangulated surface gets decorated like shown in Figure 2. Aij i

†

Bi

j

gij k Bj Aj k

k

Aki

Bk

Figure 2. The triangulation of the surface † is decorated by the local data: the faces with 2-forms Bi , the edges with 1-forms Aij and the vertices with the functions gij k .

Of course one can define the last expression without knowing bundle gerbes just by starting with a class in Deligne cohomology represented by local data .g; A; B/. In fact, surface holonomy appeared first in this form and was recognized to be useful for string theory [Alv85], [Gaw88].

6 The line bundle over the loop space As pointed out in the introduction, the holonomy of a line bundle L over a manifold M can be seen as a U.1/-valued function holL W LM ! U.1/ on its loop space. Recall that such a function is the first object in a series of mathematical objects and is followed by a line bundle. So one might guess that, when starting with a bundle gerbe instead of a line bundle, the function on the loop space is replaced by a line bundle over the loop space. This correspondence in question between gerbes


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over M and line bundles over LM was first discovered in terms of Deligne cohomology [Gaw88]. It was later redefined and extended to sheaves of groupoids [Bry93]. The associated homomorphism of Deligne cohomology groups Hk .M; D.k// ! Hk1 .LM; D.k 1// is called transgression, see also [GT01]. In particular, it reproduces for k D 1 the correspondence between line bundles over M and U.1/-valued functions on LM . Let us describe briefly how one can construct the line bundle over LM from a bundle gerbe G over M . The construction we present here is an adaption of Brylinski’s construction to bundle gerbes. Another construction starting from bundle gerbes is proposed in [GR02]. Let the fibre over a loop W S 1 ! M consist of all trivializations T W G ! 0 : Such trivializations exist, because the Dixmier–Douady class of the pullback bundle gerbe lives in H3 .S 1 ; Z/ D 0. We identify two trivializations T1 and T2 , if there is a 2-morphism T1 ) T2 : After that, the fibre over consists of equivalence classes of isomorphisms between G and 0 . This set is by Proposition 2 a torsor over the group Pic0 .S 1 / of isomorphism classes of flat line bundles over the circle. It is well-known that this group is canonically isomorphic to U.1/. Under this identification, the fibre over each loop is in a natural way a U.1/-torsor. One can further show that the union of the fibres carries a canonical smooth structure, which makes it into a principal U.1/-bundle over LM . On the associated line bundle L the bundle gerbe defines a connection. The Deligne cohomology class of this line bundle (with connection) is the image of the class of the bundle gerbe G we started with under the transgression homomorphism of Deligne cohomology groups. There is an interesting relation between the holonomy of the line bundle L over LM and the holonomy of the bundle gerbe G over M : if W S 1 ! LM is a loop in the loop space we can naturally identify it with a map W S 1 S 1 ! M . One can now consider both the holonomy of the line bundle L around the loop as well as the holonomy of the bundle gerbe G around . Both coincide [Bry93]: holL . / D holG ./: The line bundle L over the loop space plays an important role for Wess–Zumino– Witten models on Lie groups G. Its total space can be endowed with a group structure in a way that it becomes a central extension of the loop group LG. It can also be completed with respect to an appropriate scalar product, so that the space of holomorphic sections forms a Hilbert space, which acts as the space of states for the quantized Wess–Zumino–Witten model [Gaw99].

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7 D-branes and surfaces with boundary Some string theories involve not only loops (closed strings) but also open strings, whose worldsheets are surfaces with boundary. For those, we are not able to apply Definition 5 of the holonomy of a bundle gerbe G : the integral of the closed form 2 1 with integral class is not an integer anymore. More precisely, a boundary term emerges which has to be compensated to achieve a holonomy independent of the choice of the trivialization. Unfortunately, there is no analogous situation for the holonomy of line bundles. Therefore, we adopt the concept of D-branes from string theory [Pol96]. D-branes restrict the endpoints of open strings to submanifolds Q of the target space, and hence impose (generalized) Dirichlet boundary conditions to the motion of open strings. In the former definition of the Wess–Zumino–Witten model on a Lie group G by a 3-form H , a typical choice of the submanifold Q is a (twisted) conjugacy class of G. Here there is an additional condition, namely that the 3-form is fixed on the D-brane to H jQ D d! for some 2-form ! on Q. This 2-form can be interpreted as the field strength of a twisted U.1/-gauge field on Q. Let us for simplicity assume that the surface † has only one boundary component. Accordingly, there is one D-brane .Q; !/ chosen, and we consider a map W † ! G with .@†/ Q: Then, the following definition of the amplitudes is given [Gaw99]. Let D 2 be a disk acting as a cap for the surface †, so that there exists a three-dimensional manifold B whose boundary is @B D † [ D 2 . We again have to assume that there is an extension ˆ W B ! M of the map , which now has to send the cap D 2 into the D-brane Q. Then, the amplitude is defined by Z Z A./ WD exp Skin ./ C ˆ H ˆ ! : B

D2

There is a condition on the well-definedness of this amplitude; here it is not sufficient that H is a closed 3-form with integral class. The equations dH D 0 and H jQ D d! mean that the pair .H; !/ defines a class in the relative cohomology H3 .G; Q; R/, and the condition is, that this class is integral in the sense that it lies in the image of the relative cohomology with integer coefficients. There are explicit expressions for ! in the case that Q is a (twisted) conjugacy class, so that this integrality condition is satisfied. We have learned before that the theory of bundle gerbes extends the former 3-form approach to the Wess–Zumino–Witten-model. Accordingly, we have to adjust the definition of a D-brane. It is still build up on a submanifold Q. In the first attempt [GR02], the 2-form ! was replaced by a trivialization E W G jQ ! !

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of the bundle gerbe G restricted to Q. Notice that this reproduces in particular the old condition H jQ D d! for the curvature H of the bundle gerbe G . Later it was recognized [Gaw05] that having a trivialization, i.e. an isomorphism, was too strong. In fact a D-brane for a certain bundle gerbe over SO.4n/=Z2 was found which does not admit an isomorphism but a weaker structure – a morphism from G jQ to the trivial bundle gerbe ! . Such a morphism G ! ! is also called a G -module or bundle gerbe module. A slightly more general version was also considered in a purely mathematical context to obtain a geometric realization of twisted K-theory [BCMC 02]. Definition 6. Let G be a bundle gerbe over M . A D-brane for G is a submanifold Q of M together with a G jQ -module E W G jQ ! ! : The 2-form ! on Q is called the curvature of the D-brane. Recall that Lemma 1 implies that two trivializations G ! 1 and G ! 2 lead to a line bundle over M of curvature 2 1 . The same statement holds for a gerbe module G ! ! and an trivialization G ! : they define a vector bundle E ! M of curvature ! . To prove this, recall that a gerbe module is a trivialization with a vector bundle instead of a line bundle. To this situation, the proof of Lemma 1 extends without changes. The holonomy of this vector bundle will be the term which compensates the changes of on the boundary. Now consider a configuration like shown in Figure 3. We give the following definition of holonomy. G E M Q †

Figure 3. A surface is mapped into a target space with bundle gerbe G , so that its boundary is mapped into the submanifold Q with bundle gerbe module E.

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Definition 7 ([CJM02]). Let G be a bundle gerbe over M , and let .Q; E/ be a D-brane for G . For an oriented surface † and a map W † ! M which maps the boundary of † into Q, choose a trivialization T W G ! of the pullback of G along . Its restriction to @† determines in combination with the bundle gerbe module E W G j@† ! ! a vector bundle E ! @† of curvature ! . Then, we define Z holG ;E ./ D exp tr.holE .@†// †

to be the holonomy of the bundle gerbe G with D-brane .Q; E/ around . In fact it is easy to see that this definition does not depend on the choice of the trivialization T : for another trivialization T 0 recall that by Lemma 1 we obtain a line bundle N ! @† with curvature 0 . The following change in the first factor of holG ;E ./ emerges: Z Z Z 0 exp D exp exp curv.N / † † Z† D exp 0 .holN .@†//1 : †

This change has to be compensated by the second factor. Indeed, the second trivialization determines another vector bundle E 0 ! @† of curvature ! 0 . From the construction of these bundles in the proof of Lemma 1 it becomes clear that they satisfy E Š N ˝ E 0: This means for the second factor tr .holE .@†// D tr .holE 0 ˝N .@†// D holN .@†/ tr .holE 0 .@†// Thus we have shown that the holonomy defined in Definition 7 does not depend on the choice of the trivialization. With this definition of holonomy around a surface with boundary, we have to check that it reproduces the amplitude given above in terms of the 3-form H on M and the 2-form ! on Q. Proposition 4. Let G be a bundle gerbe over M with curvature H and let .Q; E/ be a D-brane with curvature !. For an oriented three-dimensional manifold B, whose boundary decomposes in two parts † and D 2 , and a map ˆ W B ! M with ˆ.D 2 / Q we find Z Z holG ;E .ˆj† / D exp

ˆ H

B

D2

ˆ ! :

675


This proposition can be proven similarly to Proposition 3. It shows that the theory of bundle gerbes and bundle gerbe modules is helpful for open string theories and allows for a proper definition of the amplitudes of worldsheets with boundary in the Wess–Zumino–Witten model – even in topologically non-trivial situations. As a last point, we have a look on the holonomy in terms of local data. Let .g; A; B/ be local data of the bundle gerbe G with respect to an open cover fVi gi2I of M . Similarly to the local data of an isomorphism it is possible to extract local data of a morphism, in particular of a bundle gerbe module [Gaw05]. It consists of functions Gij W Vi \ Vj ! U.n/ on double intersections, where n is the rank of the vector bundle, which is part of the structure of a morphism. It consists further of u.n/-valued 1-forms Pi on each open set Vi . Like an isomorphism, a morphism E W G ! ! relates the local data .g; A; B/ of the bundle gerbe G to that of the trivial bundle gerbe ! by the following equations: 1 gij k D 1; Gij Gj k Gik

Pj AdGij .Pi / Gij1 dGij C Aij D 0; dPi C Bi D !: In these equations, we identify U.1/ with the diagonal subgroup of U.n/, and correspondingly the Lie algebra of U.1/ – R – with a subalgebra of u.n/. Notice that if the local data of the bundle gerbe is trivial, i.e. .g; A; B/ D .1; 0; 0/, the three equations would be the usual cocycle conditions for a U.n/-vector bundle with connection of curvature !. With non-trivial local data, these cocycle conditions become twisted – for this reason, gerbe modules are also known as twisted vector bundles. We use a triangulation of † which is subordinated to the open cover of M just like we did to derive the formula (1) of the local expression of the holonomy around a closed surface. Splitting of the integral of the 2-form over †, which build the first factor of holG ;E ./, leads exactly to formula (1). It has to be amended by the local expression for the second factor, which is the holonomy of the vector bundle E around the boundary of †. The ladder is similar to the local expression for the holonomy of a line bundle around a loop, namely ² Y Z Y ³ .e;v/ tr .holE .@†// D tr P exp Pi.e/ Gi.e/;i.v/ : e2 \@†

e

v2@e

The only difference is that the terms now live in the non-abelian group U.n/ and have to be ordered with respect to the induced orientation on @†, which is indicated by the path-ordering operator P. The cyclic property of the trace assures that it does not depend on a specific point from where one starts multiplying terms. The complete picture where the local data is used is shown in Figure 4.

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Konrad Waldorf Aij i †

j

Bi

@†

k

Aki

gij k Bj

Bk

Pj Gj k Pk

Figure 4. The triangulation of the surface † is decorated by the local data as in Figure 2, completed by the local 1-forms Pi and the functions Gij coming from the bundle gerbe module, which are placed on the boundary @†.

8 Unoriented closed surfaces From the preceding section we learn two things. To incorporate boundaries we first had to choose structure – the D-branes – additional to the given bundle gerbe over M . Secondly, we restricted the possible maps W † ! M to those which respect this additional structure. To incorporate unoriented surfaces † we also have to do these two steps. The additional structure has been defined in [SSW05] and called a Jandl structure on the bundle gerbe G . It consists of an involution of M – i.e. a diffeomorphism k W M ! M with k B k D idM – and of a certain isomorphism between the pullback bundle gerbe k G and the dual bundle gerbe G . In the second step, we have to specify the space of maps we want to consider. As we will see, they have to be compatible in a certain sense with the involution k. For any (unoriented) closed surface † there is an oriented two-fold covering y ! †. It is unique up to orientation-preserving diffeomorphisms and it is pr W † connected if and only if † is not orientable. It has a canonical, orientation-reversing involution , which permutes the sheets and preserves the fibres. We call this two-fold covering the orientation covering of †. y ! M starting from the orienGiven a closed surface †, we consider maps O W † y which are equivariant with respect to the two involutions on † y and tation covering †, M , i.e. the diagram y †

y † has to be commutative.

O

/M

O

k

/M


677

Before we come to the details of a Jandl structure, let us briefly describe the idea O behind its definition. If we pullback the bundle gerbe G along an equivariant map , y we obtain a bundle gerbe on the orientation covering †, and could in principle compute y If we do so, we would get the square of the holonomy of this bundle gerbe around †. what we originally wanted, since each point of † is twice covered. To reveal this, we are going to establish a descent procedure – not for the bundle gerbe but rather for its holonomy. In a first attempt we assume that there is an isomorphism A W kG ! G : O it induces an isomorphism between O G and O G . Due to the equivariance of , This isomorphism says: changing the sheet by goes hand in hand with replacing the bundle gerbe with its dual. With the precise definition of the dual bundle gerbe G it becomes easy to see from Definition 7 that both processes – changing the orientation and taking the dual – give each a sign in the holonomy. So the isomorphism A implies that the holonomy of O G takes locally the same value on both sheets, which is the y to †. initial condition for a descent from † A detailed calculation shows that it is not enough to choose any isomorphism like above as additional structure. It shows that the isomorphism A itself has to be equivariant in a certain sense. To give a complete definition of the Jandl structure it is convenient to use the 2-categorial language. Definition 8. A Jandl structure J on a bundle gerbe G over M is an involution k of M together with an isomorphism A W kG ! G and a 2-morphism ' W k A ) A which satisfies the equivariance condition k' D ': Notice the remarkable symmetry of the three lines. Of course we have not developed the full theory of pullbacks and duals of morphisms and 2-morphisms in this chapter (although they turn out to be quite canonical constructions). For instance, k A as well as A are both isomorphisms from G to k G , so it makes sense to have a 2-morphism ' between them. Similarly, both k ' and ' are 2-morphisms from A to k A , so it makes sense to demand that they are equal. To give an impression of the details of a Jandl structure, recall that an isomorphism such as A consists of a line bundle A over the space Z which is build up from the two coverings of the bundle gerbes k G and G . In this particular situation, there is a canonical lift kQ of the involution k into the space Z, and it is in fact easy to Q work out that the 2-morphism ' defines a k-equivariant structure on the line bundle A.

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Summarizing, a Jandl structure J on G is an isomorphism A W kG ! G whose line bundle A is equivariant with respect to the involution kQ on Z. As always when there is an additional structure to choose, one would like to know how many inequivalent choices there are. To say what equivalent Jandl structures are amounts to define a morphism between two of them. Definition 9. A morphism ˇ W J ! J 0 between Jandl structures J D .k; A; '/ and J 0 D .k; A0 ; ' 0 / on the same bundle gerbe G over M with the same involution k is a 2-morphism ˇ W A ) A0 which commutes with ' and ' 0 in the sense that the diagram kA k ˇ

'

k A0

+3 A

'0

ˇ

+3 A0

is commutative. Recall that 2-morphisms are certain isomorphisms of vector bundles, so that the diagram is in fact a diagram of isomorphisms of line bundles over Z. The definition of a morphism between Jandl structures allows us to consider the set of equivalence classes of Jandl structures on G with involution k. Recall that by Proposition 2 the set of equivalence classes of isomorphisms is a torsor over the group of isomorphism classes of flat line bundles over M . The following theorem is a refinement for Jandl structures. Theorem 2 ([SSW05]). The set of equivalence classes of Jandl structures on a bundle gerbe G with involution k is a torsor over the group of isomorphism classes of flat k-equivariant line bundles over M . This theorem can be used to determine the number of inequivalent choices of a Jandl structure, for instance for manifolds which have been considered before in unoriented string theories – so called orientifolds, e.g. [BPS92], [BCW01], [HSS02]. The following known results are reproduced. • For M D SU.2/ with involution k.g/ WD zg 1 for any element z in the center of G, and for a bundle gerbe G over M whose curvature is an integral multiple of the 3-form H introduced in the introduction, there are two inequivalent Jandl structures on G with respect to k. • For M D SO.3/ with involution k.g/ D g 1 and a bundle gerbe G which is isomorphic to k G , there are four inequivalent Jandl structures on G with respect to k.

679


• For the 2-torus T D S 1 S 1 with involution k D idT and for a bundle gerbe G which admits Jandl structures, their equivalence classes are in bijection to the group Z2 U.1/ U.1/. Let us now use a Jandl structure on a bundle gerbe G . We are going to pursuit the idea of descent holonomy. The existence of the isomorphism A assures that the holonomy locally coincides on both sheets – now we have to make local selections y of one of them. This is exactly what a choice of a fundamental domain of † in † y does. It can be constructed locally as shown in Figure 5 as a submanifold F of † with y †

†

pr

Figure 5. On the left hand side we show a dual triangulation, i.e. every vertex has three edges. According to a choice of local orientations for every face of the triangulation – which is always y ! †. This possible – select one of the two preimages of this face under the covering pr W † defines a fundamental domain as the grey-shaded surface on the right hand side.

(piecewise smooth) boundary. A key observation, which can be heuristically seen from Figure 5, is that the involution restricts to an orientation-preserving involution y Accordingly, the quotient @F is an oriented closed submanifold of †. on @F †. Remember the following two situations: two trivializations of the same bundle gerbe define a line bundle over M with a certain curvature (Lemma 1). A trivialization together with a D-brane gives a vector bundle with a certain curvature. A similar situation appears for a trivialization together with a Jandl structure. Lemma 2 ([SSW05]). A trivialization T W G ! and a Jandl structure J on G with involution k determine a k-equivariant line bundle R ! M with curvature k C . Now we are ready to put the pieces together: Definition 10 ([SSW05]). Let G be a bundle gerbe over M with Jandl structure J y and let with involution k. Let † be a closed surface with orientation covering † O O O W † ! M be an equivariant map. The pullback along gives a bundle gerbe G y with Jandl structure O J with involution . A choice of a trivialization over † T W O G !

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y determines in combination with the Jandl structure a -equivariant line bundle R ! †. x In turn, this equivariant line bundle determines a quotient line bundle R ! †. Let F be a fundamental domain. Then we define Z O WD exp holG ;E;J ./ holRx .@F / F

O to be the holonomy of the bundle gerbe G with Jandl structure J around . In [SSW05] we show that this definition depends neither on the choice of the trivialization nor on the choice of the fundamental domain. To close, let us remark that also a Jandl structure can be understood in terms of local data. Recall that an isomorphism A relates local data of bundle gerbes, here .g; A; B/ D k .g; A; B/ C D.t; W /: It is also possible to extract a function ji W Vi ! U.1/ from the 2-morphism ', which relates the local data .t; W / of the isomorphism A to their pullback, .t; W / D k .t; W / C D.j /: Finally, the condition on the 2-morphism ' leads to j 1 D k j: O is derived in terms of the In [SSW05] an expression for the holonomy holG ;E;J ./ local data of the bundle gerbe and of the Jandl structure, analogous to (1). We do not give the full expression here, but indicate how the local data should be placed on a triangulated surface in Figure 6. y †

† Bi

ti k Ai k

i

Bk l

k

Wi

jk

Wk Bl

Wl

j Bj

Figure 6. The oriented triangulation of the closed surface † determines a fundamental domain, which is decorated by the local data of the bundle gerbe in its interior and by the local data of the Jandl structure along the orientation reversing edges.


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References O. Alvarez, Topological quantization and cohomology. Commun. Math. Phys. 100 (1985), 279–309. 670 [BCMC 02] P. Bouwknegt, A. L. Carey, V. Mathai, M. K. Murray, and D. Stevenson, Twisted K-theory and K-theory of bundle gerbes. Commun. Math. Phys. 228 (1) (2002), 17–49. 673 [BCW01] C. Bachas, N. Couchoud, and P. Windey, Orientifolds of the 3-sphere. J. High Energy Phys. 12 (2001), 003. 678 [BM05] L. Breen and W. Messing, Differential geometry of gerbes, Adv. Math. 198 (2) (2005), 732–846. 656 [BPS92] M. Bianchi, G. Pradisi, and A. Sagnotti, Toroidal compactification and symmetry breaking in open-string theories. Nucl. Phys. B 376 (1992)., 365–386 678 [Bry93] J.-L. Brylinski, Loop spaces, characteristic classes and geometric quantization. Progr. Math. 107, Birkhäuser, Boston, MA, 1993. 654, 656, 671 [BS04] J. Baez and U. Schreiber, Higher gauge theory: 2-connections on 2-bundles. Preprint 2004; arXiv:hep-th/0412325v1. 657 [CJM02] A. L. Carey, S. Johnson, and M. K. Murray, Holonomy on D-branes. J. Geom. Phys. 52 (2) (2002), 186–216. 664, 665, 666, 670, 674 [Del91] P. Deligne, Le symbole modéré. Inst. Hautes Études Sci. Publ. Math. 73 (1991), 147–181. 656 [Gaw88] K. Gaw¸edzki, Topological actions in two-dimensional quantum field theories. In Nonperturbative quantum field theory (Cargèse, 1987), NATO Adv. Sci. Inst. Ser. B Phys., 185, Plenum, New York 1988, 101–141. 656, 670, 671 [Gaw99] K. Gaw¸edzki, Conformal field theory: a case study. Preprint 1999; arXiv:hep-th/9904145v1 671, 672 [Gaw05] K. Gaw¸edzki, Abelian and non-abelian branes in WZW models and gerbes. Commun. Math. Phys. 258 (2005), 23–73. 657, 673, 675 [Gir71] J. Giraud, Cohomologie non-abélienne. Grundlehren Math. Wiss. 197, SpringerVerlag, Berlin 1971. 656 [GR02] K. Gaw¸edzki and N. Reis, WZW branes and gerbes. Rev. Math. Phys. 14 (12) (2002), 1281–1334. 657, 659, 664, 669, 671, 672 [GR03] K. Gaw¸edzki and N. Reis, Basic gerbe over non simply connected compact groups. J. Geom. Phys. 50 (1–4) (2003), 28–55. 659 [GT01] K. Gomi and Y. Terashima, Higher-dimensional parallel transports. Math. Res. Lett. 8 (2001), 25–33. 671 [Hit01] N. Hitchin, Lectures on special Lagrangian submanifolds. Adv. Math. 23 (2001), 151–182. 657, 660 [HSS02] L. R. Huiszoon, K. Schalm, and A. N. Schellekens, Geometry of WZW orientifolds. Nuclear Phys. B 624 (1–2) (2002), 219–252. 678 [Mei02] E. Meinrenken, The basic gerbe over a compact simple Lie group. Enseign. Math. (2) 49 (3–4) (2002), 307–333. 659 [Alv85]

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I. Moerdijk, Introduction to the language of stacks and gerbes. Preprint 2002; arXiv:math/0212266v1 [math.AT]. 656

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M. K. Murray, Bundle gerbes. J. London Math. Soc. 54 (1996), 403–416 . 657, 658, 661, 662

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J. Polchinski, TASI lectures on D-branes. Preprint 1996; arXiv:hep-th/9611050v2. 672

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U. Schreiber, C. Schweigert, and K. Waldorf, Unoriented WZW models and holonomy of bundle gerbes. Comm. Math. Phys. 274 (1) (2007), 31–64. 657, 664, 676, 678, 679, 680

[Ste00]

D. Stevenson, The geometry of bundle gerbes. PhD thesis, University of Adelaide, 2000; arXiv:math/0004117v1 [math.DG]. 663, 664

[Wit84]

E. Witten, Nonabelian bosonization in two dimensions. Commun. Math. Phys. 92 (1984), 455–472. 655

Part F

Symmetric spaces and spaces of constant curvature

Chapter 20

Classification results for pseudo-Riemannian symmetric spaces Ines Kath

Contents 1 2 3 4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Pseudo-Riemannian symmetric spaces and symmetric triples Quadratic cohomology and quadratic extensions . . . . . . Pseudo-Riemannian symmetric spaces with special geometric structures . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . 685 . . . . . . . . . . . . . 686 . . . . . . . . . . . . . 690 . . . . . . . . . . . . . 696 . . . . . . . . . . . . . 701

1 Introduction The aim of this chapter is to give an introduction to the classification problem for pseudo-Riemannian symmetric spaces. We will present the main ideas and an overview on several attempts to tackle the problem rather than the various classification results in full detail. Symmetric spaces constitute an important class of (pseudo-)Riemannian manifolds. On the one hand they are sufficiently complicated to serve as examples for many geometric phenomena and on the other hand they are simple enough to calculate various geometric quantities explicitly. The study of symmetric spaces began with the work of É. Cartan who succeeded to classify Riemannian symmetric spaces completely. Here we will be interested mainly in pseudo-Riemannian symmetric spaces, i.e., in the case where the metric is non-degenerate but not necessarily definite. In this case it is impossible to give a full classification in the sense of a list. Therefore the aim is to find at least a suitable description of the structure of pseudo-Riemannian symmetric spaces that allows a systematic study of these manifolds. The classification problem for symmetric spaces is closely related to the holonomy problem. Holonomy groups were also introduced by É. Cartan. Besides some low-dimensional manifolds the first large class of manifolds for which the holonomy group could be computed was that of symmetric spaces. By de Rham’s Theorem the study of holonomy groups and representations of pseudo-Riemannian manifolds is reduced to the case where the manifold is (locally) indecomposable, i.e., where it is not locally the non-trivial product of two pseudo-Riemannian manifolds. The

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holonomy group of a pseudo-Riemannian manifold .M; g/ has a natural orthogonal representation on the tangent space at a point of M called holonomy representation. .M; g/ is indecomposable if this representation does not have proper invariant subspaces which are non-degenerate with respect to the metric. The easiest case of an indecomposable pseudo-Riemannian manifold is that of an irreducible one. These are manifolds whose holonomy representation is irreducible. Lie groups which can appear as holonomy group of an irreducible pseudo-Riemannian manifold are classified. For locally symmetric manifolds this classification follows from Berger’s classification of irreducible symmetric spaces, for the non-locally symmetric case it is known as Berger’s list. In the Riemannian case each indecomposable manifold is irreducible. Of course, this is not true in the general pseudo-Riemannian case since here we can have degenerate invariant subspaces. While Berger’s list gives an answer to the question which groups can be holonomy groups of Riemannian manifolds this question is widely open for pseudo-Riemannian manifolds. Holonomy groups of Lorentzian manifolds were classified by Th. Leistner. However, except of some partial results there is no such classification known if the index of the metric is at least two. And even partial results are involved and hard to find. We will see that all this is reflected in the study of pseudo-Riemannian symmetric spaces. Irreducible symmetric spaces are easy to handle whereas the general case of indecomposable ones is very complicated.

2 Pseudo-Riemannian symmetric spaces and symmetric triples A connected pseudo-Riemannian manifold .M; g/ is called pseudo-Riemannian symmetric space if for all x 2 M there is an involutive isometry x of .M; g/ (called reflection) for which x is an isolated fixed point. This means that for any x 2 M the geodesic reflection at x extends to a globally defined isometry. The covariant derivative rR of the curvature tensor R of a pseudo-Riemannian symmetric space must vanish. In other words, R is parallel. Indeed, rR is invariant with respect to all isometries of .M; g/. In particular, we have x .rR/x D .rR/x for all x 2 M . On the other hand, x .rR/x D .rR/x , since rR depends on an odd number of tangent vectors and dx D Id W Tx M ! Tx M . Conversely one can show, that every complete and simply connected pseudo-Riemannian manifold with parallel curvature tensor is a symmetric space. Before we will start with the general pseudo-Riemannian case let us say a few words about the above mentioned classification of simply connected Riemannian symmetric spaces due to É. Cartan. Each of these symmetric spaces decomposes into so-called irreducible ones, i.e., into Riemannian symmetric spaces that have an irreducible holonomy representation. Hence it suffices to classify irreducible symmetric spaces. Obviously, a simply connected irreducible Riemannian symmetric space is flat if and only if it is isometric to .R; dt /. Now assume that .M; g/ is a non-flat irreducible Riemannian symmetric space. Then we have four cases:

Chapter 20. Classification results for pseudo-Riemannian symmetric spaces

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(I) M D G=H , where G is a compact simple simply connected Lie group and H is the unity component of the group of fixed points of some non-trivial involution of G, (II) M itself is a compact simple simply connected Lie group endowed with a biinvariant metric, i.e., a metric that is invariant with respect to all left and right translations, (III) M D G=H , where G is a non-compact simple Lie group with trivial centre, H is a maximal compact subgroup of G and, moreover, the complexification gC of the Lie algebra of G is simple as a complex Lie algebra, (IV) as in (III), but with non-simple gC (in which case G is a simple complex Lie group). Now the classification of Riemannian symmetric spaces essentially reduces to the classification of simple Lie groups, which is also due to Cartan. In each of the cases (I) and (III) one obtains seven infinite series and twelve exceptional spaces and in each of the cases (II) and (IV) four infinite series and five exceptional spaces. All these Riemannian symmetric spaces have pseudo-Riemannian analoga, e.g., instead of the Riemannian sphere S n D SO.n C 1/=SO.n/ we can consider pseudoRiemannian spheres S p;q D fx 2 Rp;qC1 j hx; xi D 1g D SO.p; q C 1/=SO.p; q/ and pseudo-Riemannian hyperbolic spaces H p;q D fx 2 RpC1;q j hx; xi D 1g D SO.p C 1; q/=SO.p; q/. Similarly, we have pseudo-Riemannian Grassmannians etc. However, there are much more pseudo-Riemannian symmetric spaces than these analoga of Riemannian ones. Let us consider for example the following series of symmetric spaces constructed by Cahen and Wallach [14]. Take M D RnC2 with co-ordinates .z; x1 ; : : : ; xn ; l/ and define a metric on M by g1 ;:::;n D

n X

dxi2 C 2dzd l C

iD1

n X

i xi2 d l 2 ;

iD1

, where 1 ; : : : ; n are real parameters. Then one can prove that the manifolds M1;nC1 1 ;:::;n 1 ; : : : ; n 2 R, are Lorentzian symmetric spaces. We will call them Cahen–Wallach spaces. We do not want to give the reflections explicitly here since later on we will learn about an easier way how to see that these manifolds are symmetric spaces. Note, that some of these spaces are isometric. If we assume that the parameters and M1;nC1 are ordered by magnitude, then M1;nC1 0 ;:::;0 are isometric if and only if 1 ;:::;n 1

n

.1 ; : : : ; n / D c.01 ; : : : ; 0n / for some real number c > 0. As another class of examples of pseudo-Riemannian symmetric spaces let us consider (finite-dimensional) Lie groups G with a bi-invariant metric. The reflection e at the unity e 2 G is just the inversion and we obtain any other reflection g by conjugation of e by the left translation Lg . Any simply connected Lie group G with a bi-invariant metric is uniquely determined by its Lie algebra g and a g-invariant (nondegenerate) inner product h ; i on g. We will call such a pair .g; h ; i/ metric Lie algebra. Of course, any finite-dimensional abelian Lie algebra with an arbitrary inner

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product is a metric Lie algebra. Moreover, each semi-simple Lie algebra g together with its Killing form is a metric Lie algebra. However, there are much more examples than these. E.g., take an arbitrary finite-dimensional Lie algebra l and define an inner product on l Ëad l by the dual pairing. More exactly, put hL1 C Z1 ; L2 C Z2 i WD Z1 .L2 / C Z2 .L2 / for L1 ; L2 2 l and Z1 ; Z2 2 l . Then .l Ëad l ; h ; i/ is a metric Lie algebra. In particular, if L is a Lie group, then we can consider its cotangent bundle T L D L ËAd l as a Lie group with bi-invariant metric. Generalising this example Medina and Revoy [31] considered the semi-direct sum g Ì l of a metric Lie algebra by an arbitrary Lie algebra l, where is a homomorphism from l into the Lie algebra of antisymmetric derivations of g. Then they constructed metric Lie algebras as suitable extensions of g Ì l by the abelian Lie algebra l . The inner product on these extensions is given by the inner product on g and the dual pairing between l and l . Such a metric Lie algebra is called double extension of g by l. If .p; q/ is the signature of (the inner product on) g, then the arising metric Lie algebra has signature .p C dim l; q C dim l/. Medina and Revoy [31] could prove that one can inductively produce all metric Lie algebras from simple and one-dimensional ones by taking direct sums and double extensions. More exactly, let g be an indecomposable metric Lie algebra, i.e., a metric Lie algebra that is not the direct sum of two non-trivial metric Lie algebras. Then either g is simple or g is one-dimensional or g is a double extension of a metric Lie algebra gQ by a one-dimensional or a simple Lie algebra l. Using this it is not hard to see that any indecomposable non-simple Lorentzian metric Lie algebra of dimension > 1 is the double extension of an abelian Euclidean metric Lie algebra by a one-dimensional Lie algebra. This allows the classification of Lorentzian metric Lie algebras up to isomorphisms [30]. However, for the classification of metric Lie algebras of higher index (of the metric) the method of double extensions is less suitable since, in general, a given metric Lie algebra can be obtained in many different ways by a series of subsequent double extensions. For another construction of metric Lie algebras, which avoids this problem see Section 3. Already the examples described above show that there is a huge variety of pseudoRiemannian symmetric spaces. Is it possible to get a grasp of the structure of these spaces? Is it possible to classify them? Here we will study only simply connected symmetric spaces. This restriction is not essential but it makes the presentation easier. For the non-simply connected case see e.g. [28]. A simply connected pseudo-Riemannian symmetric space is called indecomposable if it is not the direct product of two pseudoRiemannian symmetric spaces of dimension > 0. As far as the classification problem is concerned we may obviously concentrate on indecomposable pseudo-Riemannian symmetric spaces. Note, however, that in the pseudo-Riemannian case most of indecomposable symmetric spaces are not irreducible since the holonomy representation can have degenerate invariant subspaces. As in the case of Lie groups with bi-invariant metric it is much easier to study the infinitesimal objects associated with symmetric spaces than the symmetric spaces


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themselves. Because this is discussed also in the following chapter in this book ([1]) we will give only a very short description of the correspondence between pseudoRiemannian symmetric spaces and suitable algebraic objects, which we will call symmetric triples. Since involutions on Lie algebras will play a role in this description we will introduce the following notation. If is an involution on a Lie algebra g, we will denote by gC and g the eigenspaces with eigenvalues 1 and 1, respectively. A symmetric triple .g; ; h ; i/ consists of a metric Lie algebra .g; h ; i/ and an isometric involution on g whose eigenspaces satisfy Œg ; g D gC . Proposition 2.1. Isometry classes of simply-connected pseudo-Riemannian symmetric spaces correspond bijectively to isomorphism classes of symmetric triples. The correspondence is constructed as follows. Let .M; g/ be a simply-connected pseudo-Riemannian symmetric space. Then we can associate with .M; g/ a distinguished subgroup of the isometry group Iso.M; g/ of .M; g/, namely the so-called transvection group G D hx B y j x; y 2 M i: If G is semisimple, then its Lie algebra g equals the Lie algebra of Iso.M; g/. However, in general it is smaller. The transvection group G acts transitively on M . In particular, M is a homogeneous space and can be identified with G=GC , where GC is the stabiliser of a point x0 2 M . Then x0 acts by conjugation on G. Let us denote this action as well as the induced involution on the Lie algebra g of G by . Then it is easy to see, that gC is the Lie algebra of GC and g can be identified with Tx0 M . Moreover, one can prove that Œg ; g D gC holds. The metric g on M induces a scalar product h ; i on g Š Tx0 M , which is gC -invariant. Cahen and Parker [13] proved that h ; i can be uniquely extended to a scalar product on g that is invariant with respect to g and to . Hence we obtain a symmetric triple .g; ; h ; i/. The subgroup GC of G is the holonomy group of .M; g/. The holonomy representation equals the adjoint representation of GC on g . The pseudo-Riemannian symmetric space .M; g/ is indecomposable if and only if this representation does not contain a proper non-degenerate invariant subspace. Moreover, .M; g/ is indecomposable if and only if the associated symmetric triple is indecomposable, i.e., is not the direct sum of two non-trivial symmetric triples. By Proposition 2.1 the classification problem for pseudo-Riemannian symmetric spaces is equivalent to the classification problem for symmetric triples. We will call a symmetric triple .g; ; h ; i/ and the associated symmetric space semisimple, reductive etc., if g is semisimple, reductive etc. As explained above Riemannian symmetric spaces were classified by Cartan. We have seen that symmetric triples that are associated with these spaces are necessarily reductive and that their classification relies on the classification of semisimple Lie algebras. This result was generalised by Berger [9] to the case of arbitrary reductive symmetric triples. Similarly to the case of Riemannian symmetric spaces Berger obtained a complete list of reductive pseudo-Riemannian symmetric spaces.

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Contrary to that the non-reductive case appears to be much more involved. All of what we know of non-reductive pseudo-Riemannian symmetric spaces suggests that it is impossible to get a complete classification. However, there are results for metrics with small index. Lorentzian symmetric spaces were classified by Cahen and Wallach. In the language of double extensions introduced by Medina and Revoy (see above) we can formulate the result as follows. Any indecomposable symmetric triple .g; ; h ; i/ that is associated with such a space the following holds. Either g is one-dimensional or semisimple or .g; h ; i/ is the double extension of a pseudo-Euclidean abelian Lie algebra by R. In the semisimple case only symmetric spaces of constant curvature can have Lorentzian signature. In the third case we find the associated symmetric space by integration. This gives the following result. Theorem 2.2 (Cahen–Wallach [14]). Any indecomposable simply connected Lorentzian symmetric space is isometric to exactly one of the following manifolds (1) .R; dt 2 /, (2) a complete simply-connected Lorentzian manifold with non-vanishing constant curvature, (3) one of the above defined Cahen–Wallach spaces M1;nC1 with 1 ; : : : ; n 2 1 ;:::;n R n f0g, where 1 2 n and 1 D ˙1. Pseudo-Riemannian symmetric spaces with a metric of index 2 were studied by Cahen and Parker [12], [13]. Unfortunately, in their classification result an essential case is missing, namely, speaking in the language of [14], that of symmetric triples with maximal centre that are neither nilpotent nor ‘at least nilpotent’. This was noticed by Th. Neukirchner, who gave a revised classification in [32]. Because of the lack of a general theory he had to do a lot of calculations ‘by hand’, which had caused a few errors. We will discuss this further in the next section. All these attempts of getting lists for pseudo-Riemannian symmetric spaces show that even for small index of the metric this is a rather involved task. It seems to be a more realistic aim to find a ‘good description’ of the set of all isometry/isomorphism classes of symmetric spaces/triples. To reach this aim it is necessary to develop a structure theory for symmetric triples. The first result in this direction is due to M. Cahen and M. Parker. In [13] they study Levi decompositions of indecomposable symmetric triples. In particular, they describe the structure of the solvable radical as representation of the Levi factor.

3 Quadratic cohomology and quadratic extensions In this section we will present an approach to the classification problem for general pseudo-Riemannian symmetric spaces. We will give a description of the set of all isomorphism classes of symmetric triples by an algebraic object that is constructed


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from cohomology sets. We will call this description classification scheme. It will not be a classification in the true sense since, e.g., in general the appearing cohomology sets cannot be calculated explicitly. However, this classification scheme allows a systematic construction of symmetric triples and it gives a ‘recipe’ how to get an explicit classification under suitable additional conditions, e.g., for small index of the inner product. This approach uses the following construction due to Bérard-Bergery, which associates an isotropic ideal with each metric Lie algebra in a canonical way. This ideal will allow to consider any symmetric triple as the result of a certain extension procedure. Let .g; ; h ; i/ be a symmetric triple. There is a chain of -invariant ideals g D R0 .g/ R1 .g/ R2 .g/ Rk .g/ D 0 defined by the condition that Rj .g/ is the smallest ideal of g contained in Rj 1 .g/ such that the g-module Rj 1 .g/=Rj .g/ is semisimple. The ideal i.g/ WD

k1 X

Rj .g/ \ Rj .g/? g

j D1

is called canonical isotropic ideal of g. Proposition 3.1 ([6], [7]; [25], Lemma 3.4). If .g; ; h ; i/ is a symmetric triple, then i.g/ is a -invariant (totally) isotropic ideal and the g-module i.g/? =i.g/ is semisimple. If g does not contain simple ideals, then the Lie algebra i.g/? =i.g/ is abelian. Here we may assume that our symmetric triple .g; ; h ; i/ does not contain simple ideals. Indeed, any symmetric triple that contains simple ideals can be decomposed into the direct sum of a semisimple symmetric triple and one without simple ideals. Since semisimple symmetric Lie algebras are completely known we needn’t to consider them here. Hence, here a WD i.g/? =i.g/ is always abelian. Obviously, the inner product h ; i on g induces a (non-degenerate) inner product h ; ia on a and induces an involution a on a. Moreover, induces an involution l on l WD g=i.g/? . This means that the symmetric triple .g; ; h ; i/ is a quadratic extension of .l; l / by a in the following sense. Let .l; l / be a finite-dimensional Lie algebra with involution. An orthogonal .l; l /-module .; a; h ; ia ; a / (often abbreviated to a) consists of a (finite-dimensional) pseudo-Euclidean vector space .a; h ; ia /, an isometric involution a on a and an orthogonal representation of l on a satisfying .l .L// D a B .L/ B a . Definition 3.2. Let .; a; h ; ia ; a / be an orthogonal .l; l /-module. A quadratic extension of .l; l / by a is given by a quadruple .g; i; i; p/, where (i) g is a metric Lie algebra with involution (see below), (ii) i is an isotropic -invariant ideal of g,

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(iii) i and p are homomorphisms of Lie algebras with involution constituting an exact sequence i

p

0 ! a ! g=i ! l ! 0 that is consistent with the representation of l on a and has the property that i is an isometry from a to i? =i. Here a metric Lie algebra with involution is a triple .g; ; h ; i/, where .g; h ; i/ is a metric Lie algebra and is an involutive isometry on g. This is less than to be a symmetric triple, which has to satisfy in addition Œg ; g D gC . This kind of extension is called quadratic, since g is the result of two subsequent (ordinary) extensions. First we extend l by a and then we extend the resulting Lie algebra by i Š l . Similarly to ordinary extensions there is a natural notion of equivalence of quadratic extensions. And in the same way as one classifies (ordinary) extensions of a Lie algebra l by an l-module a by the second Lie algebra cohomology group H 2 .l; a/ we can describe quadratic extensions of .l; l / by an orthogonal .l; l /module by a suitable second cohomology set. This cohomology is a special case of non-linear cohomology sets introduced by Grishkov [21]. Let us describe this cohomology Lie algebra cochain complex C p .l; a/ WD Vp now. We start with the standard p l; a/ with differential d W C .l; a/ ! C pC1 .l; a/ given by Hom. d .L1 ; : : : ; LpC1 / D

pC1 X

.1/i1 .Li /.L1 ; : : : ; LO i ; : : : ; LpC1 /

iD1

C

X

.1/iCj .ŒLi ; Lj ; L1 ; : : : ; LO i ; : : : ; LO j ; : : : ; LpC1 /:

i <j

In the special case where a is the one-dimensional trivial representation of l we denote the standard cochain complex also by .C .l/; d /. The pair .l ; a / defines an involution on C p .l; a/ by 7! a B l . As usual, let us denote the space of invariant elements in C p .l; a/ by C p .l; a/C . We define a product C p .l; a/C C q .l; a/C ! C pCq .l/C by ^

h ;ia

h ^ i W C p .l; a/C C q .l; a/C ! C pCq .l; a ˝ a/C ! C pCq .l/C : Now we define the set of quadratic 1-cochains as 1 CQ .l; l ; a/ D C 1 .l; a/C ˚ C 2 .l/C :

This set is a group with group operation defined by .1 ; 1 / .2 ; 2 / D .1 C 2 ; 1 C 2 C 12 h1 ^ 2 i/: Let us consider the set Z2Q .l; l ; a/ D f.˛; / 2 C 2 .l; a/C ˚ C 3 .l/C j d˛ D 0; d D 12 h˛ ^ ˛ig


693

1 whose elements are called quadratic 2-cocycles. Then the group CQ .l; l ; a/ acts from 2 the right on ZQ .l; l ; a/ by

.˛; /.; / D .˛ C d ; C d C h.˛ C 12 d / ^ i/:

(1)

We define the second quadratic cohomology set as the orbit space 2 1 .l; l ; a/ WD Z2Q .l; l ; a/=CQ .l; l ; a/: HQ 2 .l; l ; a/ is denoted by Œ˛; . The equivalence class of .˛; / 2 Z2Q .l; l ; a/ in HQ H.-Ch. Herbig suggested to relate this cohomology set to another construction, which is well known to people who work with differential graded algebras. Namely, the 2 elements of HQ .l; l ; a/ can be regarded as equivalence classes of so-called twisting elements. This L can be seen in the following way. We define a differential graded algebra A D p2N Ap by

Ap D C pC1 .l; a/C ˚ C pC2 .l/C with differential d W Ap ! ApC1 , d.; / D .d; d / and product .1 ; 1 / .2 ; 2 / D .1/q .0; 12 h1 ^ 2 i/ for .1 ; 1 / 2 Ap and .2 ; 2 / 2 Aq . Now we adjoin to A an element 1 of degree 0 satisfying d1 D 0, 1 1 D 1 and 1 a D a 1 D a for all a 2 A. We denote the O Then Z2 .l; l ; a/ A1 D AO1 equals the resulting differential graded algebra by A. Q set fa 2 AO j da C a a D 0g AO1 O Now we consider the group .AO0 / of invertible of so-called twisting elements of A. 0 O elements in A . This group consists of all elements r 1 C c with r 2 R, r 6D 0 and 1 .l; l ; a/. It acts from the right on AO1 by c 2 A0 D CQ .a; g/ 7! g 1 ag C g 1 dg

(2)

and this action leaves invariant the set of twisting elements. Moreover, R D R 1 1 .l; l ; a/ on Z2Q .l; l ; a/ .AO0 / acts trivially. Comparing (2) with the action of CQ defined in (1) we see that H 2 .l; l ; a/ is just the set of twisting elements in AO modulo Q

the group of invertible elements of AO0 . Now we want to formulate a description of quadratic extensions by cohomology.

Theorem 3.3. Let .l; l / be a finite-dimensional Lie algebra with involution and let a be an orthogonal .l; l /-module. There is a bijection from the set of equivalence 2 .l; l ; a/. classes of quadratic extensions of .l; / by a to HQ We want to describe the inverse of this bijection explicitly since this description gives us a construction for all metric Lie algebras with involution, hence, in particular, 2 for all symmetric triples. Take Œ˛; 2 HQ .l; l ; a/. We consider the vector space

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d WD l ˚ a ˚ l and define an inner product h ; i and an isometry on d by hZ1 C A1 C L1 ; Z2 C A2 C L2 i WD hA1 ; A2 ia C Z1 .L2 / C Z2 .L1 /; .Z C A C L/ WD l .Z/ C a .A/ C l .L/ for Z; Z1 ; Z2 2 l , A; A1 ; A2 2 a and L; L1 ; L2 2 l. Moreover, we define an antisymmetric bilinear map Œ ; W d d ! d by Œl ; l ˚ a D 0 and ŒL1 ; L2 D .L1 ; L2 ; / C ˛.L1 ; L2 / C ŒL1 ; L2 l ; ŒL; A D hA; ˛.L; /i C L.A/; ŒA1 ; A2 D h./A1 ; A2 i; ŒL; Z D ad .L/.Z/ for L; L1 ; L2 2 l, A; A1 ; A2 2 a and Z 2 l . Because of the cocycle condition for .˛; / the bracket Œ ; satisfies the Jacobi identity. Moreover, is an involution and h ; i is d-invariant. Hence d˛; .l; l ; a/ WD .d; ; h ; i/ is a metric Lie algebra with involution. This construction is related to other known extension procedures. In the special case where ˛ D D 0 the metric Lie algebra .d; h ; i/ is a double extension of the abelian metric Lie algebra a by l in the sense of Medina and Revoy. For a D 0 it coincides with the T -extension introduced by Bordemann [11]. Now let us turn to the promised classification scheme for symmetric triples, more exactly for symmetric triples without simple ideals. Up to now we know that every such triple has the structure of a quadratic extension in a canonical way. Moreover, we can classify quadratic extensions of a Lie algebra with involution .l; l / by an orthogonal 2 .l; l ; a/. Now the following problem occurs. In general we .l; l /-module a by HQ can write a metric Lie algebra with involution in different ways as quadratic extension and only one of these possibilities is the canonical one. Hence we must find those quadratic extensions that come from our canonical construction. These are exactly those quadratic extensions .g; i; i; p/ for which i.g/ D i holds. We will call them 2 2 balanced. In [25] we managed to describe the subset HQ .l; l ; a/b HQ .l; l ; a/ of cohomology classes that correspond to balanced quadratic extensions. A further problem is caused by the fact that many quadratic extensions .g; i; i; p/ do not satisfy Œg ; g D gC , hence they are not symmetric triples. Therefore, in a second step, we 2 2 have to determine the subset HQ .l; l ; a/] HQ .l; l ; a/b of cohomology classes that correspond to balanced quadratic extensions .g; i; i; p/ satisfying Œg ; g D gC . This is an easy task, see [26]. We will call these cohomology classes and the corresponding extensions admissible. 2 .l1 ; l1 ; a1 / There is a natural notion of a direct sum of cohomology classes a1 2 HQ 2 2 and a2 2 HQ .l2 ; l2 ; a2 /. The direct sum a1 ˚ a2 is contained in HQ .l1 ˚ l2 ; l1 ˚ l2 ; a1 ˚ a2 /. A cohomology class is called indecomposable if it cannot be written as such a sum, where at least one of the pairs .l1 ; a1 /, .l2 ; a2 / is non-zero. We 2 .l; l ; a/] by will denote the subset of indecomposable cohomology classes in HQ


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2 HQ .l; l ; a/0 . It is not hard to prove that admissible quadratic extensions with indecomposable underlying symmetric triple correspond to indecomposable cohomology classes [26]. Finally we have to take into consideration that non-equivalent quadratic extensions of .l; l / by a can have isomorphic underlying symmetric spaces, hence we have to divide out automorphisms of .l; l / and a. We obtain:

Theorem 3.4. Let L be a complete set of representatives of isomorphism classes of finite-dimensional Lie algebras with involution, and let A be a complete set of representatives of isometry classes of pseudo-Euclidean vector spaces with involution. For .l; l / 2 L and a D .a; h ; ia ; a / 2 A we denote by Rl;l ;a the set of all semisimple representations of .l; l / on a such that a D .; a; h ; ia ; a / is an orthogonal .l; l /-module. Then there is a bijection from the set of isomorphism classes of indecomposable non-semisimple symmetric triples to ı a a a 2 HQ .l; l ; a /0 Aut.l; l / Aut.a; h ; ia ; a / .l;l /2L a2A

2Rl;l ;a

The inverse of this map assigns the isomorphism class of d˛; .l; l ; a/ to the orbit of 2 Œ˛; 2 HQ .l; l ; a /0 . Note that in the above formula one does not really have to consider all .l; l / 2 L. A contribution to the coproduct comes only from those .l; l / for which there exists 2 .l; l ; a/0 is not empty. This a rather an orthogonal .l; l /-module a such that HQ restrictive condition for .l; l /, see e.g. [22]. However, it does not appear to be strong enough to allow a classification for arbitrary dimension. Note that by definition of 2 2 .l; l ; a/0 HQ .l; l ; a/] can hold only if Œl ; l D lC . admissibility ; 6D HQ This formula gives us a classification scheme, which is especially suitable for the study of symmetric spaces with small index of the metric. Here the existence of additional geometric structures allows to increase the index of the metric. The general procedure is as follows. Suppose we are given a pseudo-Riemannian symmetric space .M; g/ such that the associated symmetric triple .g; ; h ; i/ does not have simple ideals. Then .g; ; h ; i/ can be written as an admissible quadratic extension of a Lie algebra with involution .l; l / by an orthogonal .l; l /-module a. By definition of quadratic extension the index of g is an upper bound for dim l , which gives an upper bound for the dimension of l. Lie algebras .l; l / of small dimension that 2 satisfy HQ .l; l ; a/0 6D ; for some semisimple orthogonal representation a can be classified. All such semisimple orthogonal .l; l /-modules a can be determined and 2 .l; l ; a/0 explicitly. This yields explicit classification results for we can calculate HQ pseudo-Riemannian symmetric spaces of small index. If, e.g., .M; g/ is an indecomposable, non-semisimple Lorentzian symmetric space, then .l; l / must be isomorphic to .R; Id/. Moreover, a is isometric to Rp;p ˚ R2q and the representation of l on a is given by non-zero weights 1 ; : : : ; pCq 2 l Š R.

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2 Obviously, in this case we have HQ .l; l ; a/0 D fŒ0; 0g. In this way it is easy to recover Cahen–Wallach’s classification of Lorentzian symmetric spaces. If .M; g/ is an indecomposable, non-semisimple symmetric space with metric of index 2, then l is isomorphic to one of the Lie algebras R, R2 , so.2/ Ë R2 , so.1; 1/ Ë R1;1 , su.2/, sl.2; R/ or the 3-dimensional Heisenberg algebra h.1/. All these Lie algebras have a unique (up to isomorphism) involution such that Œl ; l D lC except for sl.2; R/, which has two. For each of these .l; l / it is easy to determine all semisimple orthogonal .l; l /-modules and to calculate the quadratic cohomology sets. This gives a list of all pseudo-Riemannian symmetric spaces of index 2 (see [26]), which corrects the results by Cahen, Parker and Neukirchner discussed in Section 2. For index 3 up to now only Lie groups with biinvariant metric are classified [25]. More results on Lie groups with biinvariant metric or, equivalently, on metric Lie algebras can be found in [22], where in particular nilpotent metric Lie algebras of dimension 10 are classified.

4 Pseudo-Riemannian symmetric spaces with special geometric structures In this section we want to consider pseudo-Riemannian symmetric spaces that have additional structures coming from complex or quaternionic geometry. Let us recall some basic notions related to these geometries. Let .M; g/ be a pseudo-Riemannian manifold. Objects on .M; g/ that are covariant constant with respect to the Levi-Civita connection are called parallel. A Kähler structure on .M; g/ is a parallel section I of so.TM / that satisfies I 2 D Id. A hyper-Kähler structure on .M; g/ is a pair .I; J / of Kähler structures satisfying IJ D JI . If there is a hyper-Kähler structure on .M; g/, then .M; g/ is Ricci flat. Its signature equals .4p; 4q/, p; q 2 N. If we weaken the parallelity condition for I and J we are lead to the notion of a quaternionic Kähler structure. It consists of a 3-dimensional parallel subbundle of so.TM / that can be locally spanned by sections I; J and K satisfying I 2 D J 2 D Id and K D IJ D JI . Here we want to exclude the case, where this parallel subbundle is spanned by parallel sections, i.e., that it even defines a hyper-Kähler structure. If there exists a quaternionic Kähler structure on .M; g/, then g is an Einstein metric of non-vanishing scalar curvature. Moreover, .M; g/ is not locally the product of two non-trivial pseudo-Riemannian manifolds. It is easy to see, that .M; g/ admits a quaternionic Kähler structure if and only if the holonomy group of .M; g/ is contained in Sp.p; q/ Sp.1/ SO.4p; 4q/ but not in Sp.p; q/. Moreover, the holonomy group of a manifold admitting a quaternionic Kähler structure always contains the whole Sp.1/-factor of Sp.p; q/ Sp.1/. For these general facts we refer to [10] and note that all arguments given there for the Riemannian situation remain valid in the indefinite case.


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All these structures have ‘para’-analogs. A para-Kähler structure on .M; g/ is a parallel section I of so.TM / that satisfies I 2 D Id. It can exist only if g has neutral signature .q; q/. A para-hyper-Kähler structure is a pair .I; J /, where I is a Kähler structure and J is a para-Kähler structure such that IJ D JI . Para-quaternionic Kähler structures are defined analogously. A pseudo-Hermitian symmetric space .M; g; I / is a pseudo-Riemannian symmetric space .M; g/ together with a Kähler structure I on .M; g/. Analogously, hyperKähler symmetric spaces and quaternionic Kähler symmetric spaces are pseudoRiemannian symmetric spaces together with a fixed hyper-Kähler and quaternionic Kähler structure, resp. Similarly, we have para-Hermitian, para-hyper-Kähler (also called hypersymplectic) and para-quaternionic Kähler symmetric spaces. Let us start with the pseudo-/para-Hermitian case. The pseudo-/para-Kähler structure I on .M; g/ defines an additional structure on the symmetric triple .g; h ; i; / associated with .M; g/, namely a gC -invariant map IO 2 so.g / satisfying IO2 D Id. We will call .g; h ; i; ; IO/ a pseudo-Hermitian symmetric triple. For semisimple g such triples were classified by Berger [9]. If g does not contain simple ideals, then we can consider g again as a quadratic extension of a Lie algebra with involution .l; l / by an orthogonal .l; l /-module. It is not hard to see that IO induces maps IOl on l and IOa on a with properties analogous to those of IO. One can prove that every solvable pseudo- or para-Hermitian triple is nilpotent [28]. If we apply this to a pseudo-Hermitian symmetric space g of index 2 which has the structure of a quadratic extension, then either g has a non-trivial Levi factor and then l 2 fsu.2/; sl.2; R/g or g and therefore also l is nilpotent and then l 2 fR; h.1/g. Moreover, if l is nilpotent, then the representation of l on a must be trivial. In this way we easily find all pseudoHermitian symmetric spaces of index 2 without checking the whole (rather long) list of all symmetric spaces of index 2 [26], [28]. In [28] we also classify para-Hermitian symmetric spaces of index 2. For another approach to the classification of pseudoand para-Hermitian symmetric spaces see [1] in this book, where D. Alekseevsky describes such spaces that have a special kind of abelian holonomy. Now let us turn to quaternionic Kähler symmetric spaces. Riemannian quaternionic Kähler symmetric spaces were classified by J. Wolf [33]. Let us explain the main idea on which this classification is based. Since in the Riemannian situation compact spaces and non-compact ones are in duality we will restrict ourselves to the compact case. Let g be a compact simple Lie algebra and h g a Cartan subalgebra. We consider the corresponding roots and choose a set of positive ones. Let ˇ be the highest root and let sˇ WD gˇ ˚gˇ ˚Œgˇ ; gˇ gC be the 3-dimensional subalgebra generated by the root spaces of ˇ and ˇ. Then sˇ \ g is isomorphic to sp.1/. We define an involution on g by gC WD sˇ \ g ˚ Zg .sˇ \ g/;

g D .gC /? ;

where Zg denotes the centraliser in g. Then the symmetric triple which is constituted by g together with this involution and a non-zero multiple of the Killing form is always associated with a compact Riemannian quaternionic Kähler symmetric space. Con-

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versely, every symmetric triple associated with a compact Riemannian quaternionic Kähler symmetric space can be obtained in this way. Now we turn to the indefinite case. Recall that any quaternionic Kähler manifold is Einstein with non vanishing scalar curvature, hence it has a non-degenerate Ricci tensor. Since the Ricci tensor of a symmetric space is essentially given by the Killing form of the transvection group we see that the transvection group of a quaternionic Kähler symmetric space is semisimple. Since, moreover, such a symmetric space is (locally) irreducible it is either isometric to a simple group G or its transvection group is simple. The first case can be excluded since G would also be the holonomy group of the symmetric space and would therefore be of the form Sp.1/ K 0 , which contradicts the assumption that G is simple. Hence also in the indefinite case the transvection group of a quaternionic Kähler symmetric spaces is simple. Let us now consider the complex version of Wolf’s construction discussed above. We start with a complex simple Lie algebra g. We fix a Cartan subalgebra in g and choose ˇ and sˇ g as above. We define an involution on g by gC D sˇ ˚ Zg .sˇ /; g D .gC /? : Now suppose that is an anti-linear involution on g that preserves sˇ and for which sˇ is isomorphic to sp.1/. Then respects the decomposition g D gC ˚ g , thus we obtain an involution on g . The real Lie algebra g together with this involution and a non-zero multiple of the Killing form defines a symmetric triple such that the associated symmetric space is quaternionic Kählerian. As in the Riemannian case one proves that, conversely, every such symmetric triple can be obtained in this way. In [4] D.V. Alekseevsky and V. Cortés determine all these anti-linear involutions for every complex simple Lie algebra g. In this way they obtain a classification of indefinite simply connected quaternionic Kähler symmetric spaces avoiding to check Berger’s list of all semisimple pseudo-Riemannian symmetric spaces for quaternionic Kählerian ones. The para-quaternionic case was studied by D.V. Alekseevsky, N. Blažić, V. Cortés and S. Vukmirović in [2]. Next let us consider hyper-Kähler symmetric spaces. Since such spaces are Ricciflat the Killing form of the transvection group vanishes. In particular, there are no non-flat Riemannian hyper-Kähler symmetric spaces. In analogy to the Kählerian case hyper-Kähler symmetric spaces correspond to so-called hyper-Kähler symmetric O JO /, where .g; ; h ; i/ is a symmettriples. These are triples .g; ˆ; h ; i/, ˆ D .; I; O O ric triple and I and J are gC -invariant elements of so.g / such that IO2 D JO 2 D Id and KO WD IOJO D JO IO. First we want to explain a description of hyper-Kähler symmetric triples by certain quartic polynomials due to Alekseevsky and Cortés. This description will lead to a classification in the special case of abelian holonomy. Let .E; !/ be a complex symplectic vector space. Take S 2 S 4 E. For v; w 2 E let Sv;w be the contraction of


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S with v and w via the symplectic form !. Now let us consider the subspace hS D spanfSv;w 2 S 2 E j v; w 2 Eg of S 2 E Š sp.E; !/. If

S 2 .S 4 E/hS ;

(3)

then hS sp.E; !/ is a subalgebra. In this case there is a Lie bracket on gS D hS ˚ .H ˝C E/ such that hS is a subalgebra, hS sp.E; !/ acts on H ˝C E by the natural action on the second factor and the commutator of two elements of H ˝C E is in hS and defined by Œp ˝ v; q ˝ w D !H .p; q/Sv;w ;

p; q 2 H; v; w 2 E:

Here !H denotes the complex bilinear 2-form on H with !H .1; j / D 1. N Then J Now assume that E admits a quaternionic structure J such that J ! D !. 4 2 induces a real structure on S E and on S E Š sp.E; !/. Moreover, j ˝ J is a real structure on H ˝C E. Let us denote all these real structures by the same symbol . If S 2 .S 4 E/ satisfies (3), then gJ;S WD .hS / ˚ .H ˝C E/ is a real Lie algebra. We can define an involution on gJ;S by .gJ;S /C D .hS / and .gJ;S / D .H ˝C E/ . Then .gJ;S /C D Œ.gJ;S / ; .gJ;S / holds. In particular, the inner product !H ˝ ! on .gJ;S / D .H ˝C E/ extends to an invariant inner product on gJ;S . If we let, moreover, IO and JO be the left multiplication by i and j on the first factor of .H˝C E/ , then we obtain the structure of a hyper-Kähler symmetric triple on gJ;S . Theorem 4.1 (Alekseevsky–Cortés [3]). Every hyper-Kähler symmetric triple .g; ˆ; h ; i/ is isomorphic to some hyper-Kähler symmetric triple gJ;S for suitable data .E; !; J; S/, where S 2 S 4 E is a -invariant solution of (3). The quadruple .E; !; J; S / is uniquely determined by .g; ˆ; h ; i/ up to complex linear isomorphisms. Hence the classification of hyper-Kähler symmetric triples is equivalent to the classification of all -invariant solutions of (3). Up to now it is not known how to find all these solutions. However, there is a large family of special solutions that can easily be described. Indeed, if EC E is a Lagrangian subspace, then all S 2 S 4 EC S 4 E satisfy (3). We will call these solutions tame. If S is tame, then hS is abelian. Cortés proved that also the converse is true. This yields the following classification of hyper-Kähler symmetric triples with abelian holonomy. Proposition 4.2 (Cortés [15]). In the description of hyper-Kähler symmetric spaces given in Theorem 4.1 those with abelian holonomy group correspond exactly to tame solutions of (3). In [27] we prove that there exist indeed hyper-Kähler symmetric spaces that have non-abelian holonomy and we give several examples. All examples were constructed

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by our method of quadratic extensions. In particular this shows that there are solutions of .3/ that are not tame. Up to now it is not clear how to find such solutions in a more direct way. Let us sketch the ‘simplest’ example of a hyper-Kähler symmetric space with non-abelian holonomy. We start with the Lie algebra with involution .l; l / defined by l D l ˚ lC D H ˚ Im H, where lC D Im H is the centre of l and Œq1 ; q2 D Im qN 1 q2 2 lC D Im H for q1 ; q2 2 l D H. We consider a D a WD H2 with the standard scalar product as a trivial .l; l /-module. Let IOa , IOl , JOa , JOl be the left-multiplication by i and j on a D a D H2 and l D H, respectively. Then it is 2 not hard to find an element Œ˛; 2 HQ .l; l ; a/ which satisfies jlC ^lC ^lC 6D 0

(4)

and which is in a certain sense invariant with respect to the multiplication by i and j on a and l . Now we consider the quadratic extension d WD d˛; .l; l ; a/. By the mentioned invariance property the maps IO D IOl ˚ IOa ˚ IOl ;

JO D JOl ˚ JOa ˚ JOl

define dC -invariant elements of so.d / and d together with IO and JO constitutes a hyper-Kähler symmetric triple. The associated hyper-Kähler symmetric space has signature .4; 12/. The holonomy group of this symmetric space cannot be abelian. Indeed, its Lie algebra dC is not abelian because of (4). For more detailed explanations see [27]. In [27] one can also find a classification of indecomposable simply connected hyper-Kähler symmetric spaces of signature .4; 4q/. Besides the example of signature .4; 12/ discussed above the flat space H and a one-parameter family of spaces of signature .4; 4/ with abelian holonomy occur. Para-hyper-Kähler symmetric spaces are considered in [16]. Of course, the above discussed hyper-Kähler symmetric space with non-abelian holonomy has a para-hyperKählerian analogon. Finally let me draw your attention to manifolds that are symmetric from an extrinsic point of view. A non-degenerate connected submanifold M Rp;q is called extrinsic symmetric if it is invariant under the reflection at the (affine) normal space Tx? M for every x 2 M . Extrinsic symmetric spaces in Rp;q are exactly those complete submanifolds whose second fundamental form is parallel. Obviously, these spaces are pseudo-Riemannian symmetric spaces in the ordinary sense. Extrinsic symmetric spaces in the Euclidean space are well understood. Any such space is the product of a Euclidean space and a compact extrinsic symmetric space, which lies in a round sphere. A classification of compact Euclidean extrinsic symmetric spaces follows from a nice construction due to Ferus. This construction shows that compact extrinsic symmetric spaces in the Euclidean space are orbits of isotropy representations of semi-simple symmetric spaces, namely they are exactly the standard embedded symmetric R-spaces [18], [19], [20]. A classification of Euclidean symmetric spaces now follows from the classification of symmetric R-spaces due to Kobayashi and Nagano. J. R. Kim [29] showed that Ferus’ construction can be carried out also for


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pseudo-Riemannian extrinsic symmetric spaces that satisfy additional conditions. As in the Riemannian case we see that theses extrinsic symmetric spaces are orbits of the isotropy representation of symmetric spaces. But now these symmetric spaces are in general non-reductive and a full classification becomes impossible. However, it seems to be a realistic aim to classify Lorentzian extrinsic symmetric spaces by the method of quadratic extensions [23].

References [1]

D. V. Alekseevsky, Pseudo-Kähler and para-Kähler symmetric spaces. In Handbook of pseudo-Riemannian geometry and supersymmetry, ed. by V. Cortés, IRMA Lect. Math. Theor. Phys. 16, European Math. Soc. Publishing House, Zürich 2010, 701–727. 689, 697

[2]

D. V. Alekseevsky, N. Blažić, V. Cortés, and S. Vukmirović, A class of Osserman spaces. J. Geom. Phys. 53 (2005), no. 3, 345–353. 698

[3]

D. V. Alekseevsky and V. Cortés, Classification of indefinite hyper-Kähler symmetric spaces. Asian J. Math. 5 (2001), 663 – 684. 699

[4]

D. V. Alekseevsky and V. Cortés, Classification of pseudo-Riemannian symmetric spaces of quaternionic Kähler type. In Lie groups and invariant theory, Amer. Math. Soc. Transl. (2) 213, Amer. Math. Soc., Providence, RI, 2005, 33–62. 698

[5]

H. Baum and I. Kath, Doubly extended Lie groups – curvature, holonomy, and parallel spinors. J. Differential Geom. Appl. 19 (2003), 253–280.

[6]

L. Bérard-Bergery, Décomposition de Jordan-Hölder d’une représentation de dimension finie, adaptée à une forme réflexive. Handwritten notes. 691

[7]

L. Bérard-Bergery, Structure des espaces symétriques pseudo-riemanniens. Handwritten notes. 691

[8]

L. Bérard-Bergery, Semi-Riemannian symmetric spaces. In preparation.

[9]

M. Berger, Les espaces symétriques non compacts. Ann. Sci. École Norm. Sup. 74 (1957), 85–177. 689, 697

[10] A. L. Besse, Einstein manifolds. Ergeb. Math. Grenzgeb. (3) 10, Springer-Verlag, Berlin 1987. 696 [11] M. Bordemann, Nondegenerate invariant bilinear forms on nonassociative algebras. Acta Math. Univ. Comenian. 66 (1997), 151–201. 694 [12] M. Cahen and M. Parker, Sur des classes d’espaces pseudo-riemanniens symétriques. Bull. Soc. Math. Belg. 22 (1970), 339–354. 690 [13] M. Cahen and M. Parker, Pseudo-Riemannian symmetric spaces. Mem. Amer. Math. Soc. 24 (1980), no. 229. 689, 690 [14] M. Cahen and N. Wallach, Lorentzian symmetric spaces. Bull. Amer. Math. Soc. 76 (1970), 585–591. 687, 690 [15] V. Cortés, Odd Riemannian symmetric spaces associated to four-forms. Math. Scand. 98 (2006), 201-216. 699

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[16] A. S. Dancer, H. R. Jorgensen, and A. F. Swann, Metric geometries over the split quaternions. Rend. Sem. Mat. Univ. Politec. Torino 63 (2005), 119–139. 700 [17] J.-H. Eschenburg and E. Heintze, Extrinsic symmetric spaces and orbits of s-representations. Manuscripta Math. 88 (1995), 517–524. [18] D. Ferus, Produkt-Zerlegung von Immersionen mit paralleler zweiter Fundamentalform. Math. Ann. 211 (1974), 1–5. 700 [19] D. Ferus, Immersions with parallel second fundamental form. Math. Z. 140 (1974), 87–92. 700 [20] D. Ferus, Symmetric submanifolds of Euclidean space. Math. Ann. 247 (1980), 81–93. 700 [21] A. N. Grishkov, Orthogonal modules and nonlinear cohomologies, Algebra and Logic 37 (1998), 294–306. 692 [22] I. Kath, Nilpotent metric Lie algebras of small dimension. J. Lie Theory 17 (2007), no. 1, 41–61. 695, 696 [23] I. Kath, Indefinite extrinsic symmetric spaces I. J. Reine Angew. Math., to appear; preprint 2008, arXiv:0809.4713v2 [math.DG]. 701 [24] I. Kath and M. Olbrich, Metric Lie algebras with maximal isotropic centre. Math. Z. 246 (2004), 23–53. [25] I. Kath and M. Olbrich, Metric Lie algebras and quadratic extensions. Transform. Groups 11 (2006), no. 1, 87–131. 691, 694, 696 [26] I. Kath and M. Olbrich, On the structure of pseudo-Riemannian symmetric spaces. Transform. Groups 14 (2009), no. 4, 847–885. 694, 695, 696, 697 [27] I. Kath and M. Olbrich, New examples of indefinite hyper-Kähler symmetric spaces. J. Geom. Phys. 57 (2007), no. 8, 1697–1711. 699, 700 [28] I. Kath and M. Olbrich, The classification problem for pseudo-Riemannian symmetric spaces. In Recent developments in pseudo-Riemannian geometry, ESI Lect. Math. Phys., European Math. Soc. Publ. House, Zürich 2008, 1–52. 688, 697 [29] J. R. Kim, Extrinsic symmetric spaces. Dissertation, Augsburg 2005, Shaker Verlag. 700 [30] A. Medina, Groupes de Lie munis de métriques bi-invariantes. Tohoku Math. J. (2) 37 (1985), 405–421. 688 [31] A. Medina and Ph. Revoy, Algèbres de Lie et produit scalaire invariant. Ann. Sci. École Norm. Sup. (4) 18 (1985), 553–561. 688 [32] Th. Neukirchner, Solvable pseudo-Riemannian symmetric spaces. Preprint 2003, arXiv:math/0301326v1 [math.DG]. 690 [33] J. A. Wolf, Complex homogeneous contact manifolds and quaternionic symmetric spaces. J. Math. Mech. 14 (1965), 1033–1047. 697

Chapter 21

Pseudo-Kähler and para-Kähler symmetric spaces Dmitri V. Alekseevsky

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Pseudo-Riemannian symmetric spaces . . . . . . . . . . . . . . . . . . . . . 1.1 Basic properties of symmetric spaces . . . . . . . . . . . . . . . . . . . . 1.2 Pseudo-Kähler and para-Kähler symmetric spaces . . . . . . . . . . . . . 2 Structure of para-Kähler symmetric spaces . . . . . . . . . . . . . . . . . . . 2.1 Curvature bi-form of a para-Kähler symmetric space . . . . . . . . . . . 2.2 Ricci-flat para-Kähler symmetric spaces . . . . . . . . . . . . . . . . . . 2.3 Para-Kähler symmetric spaces with holonomy algebra h VC ^ V . . 2.4 Ricci-flat para-Kähler symmetric spaces of dimension 6 . . . . . . . . 3 Structure of pseudo-Kähler symmetric spaces . . . . . . . . . . . . . . . . . . 3.1 A construction of pseudo-Calabi–Yau symmetric spaces with commutative holonomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Pseudo-Calabi–Yau symmetric spaces of dimension n 6 . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

703 706 706 709 711 711 714 717 718 721

. . . 724 . . . 726 . . . 728

Introduction Pseudo-Riemannian symmetric spaces form a big class of pseudo-Riemannian manifolds. Riemannian symmetric spaces were classified by E. Cartan. The classification reduces to the description of involutive automorphisms of semisimple Lie algebras, since an irreducible Riemannian symmetric space has semisimple isometry group. This approach leads also to a classification of pseudo-Riemannian symmetric spaces with semisimple isometry group. However, the isometry group of a pseudoRiemannian (in particular Kähler and para-Kähler) symmetric space is non-semisimple in general. There is no hope to get a classification of all pseudo-Riemannian symmetric spaces, since the cotangent bundle TG of any Lie group G has a natural structure of pseudo-Riemannian symmetric space of neutral signature .n; n/. Nevertheless, many interesting special classes of pseudo-Riemannian symmetric spaces have been studied and classified, for example, Lorentzian symmetric spaces [CW] and symmetric spaces of signature .2; n/ [CP], [KO]. In [KO1], I. Kath and M. Olbrich developed a general approach for description and classification of pseudo-Riemannian symmetric spaces, based on a construction of quadratic extension of Lie algebras, and got many

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interesting results. In the paper [AC1], the classification of (pseudo-Riemannian) quaternionic Kähler symmetric space with non-zero scalar curvature is given. In the papers [AC] and [ABCV], pseudo-Riemannian hyper-Kähler and para-hyper-Kähler symmetric spaces were studied and their classification was reduced to the description of some homogenous polynomials of degree 4. In particular, a classification of such symmetric spaces with commutative holonomy group was given, see [Cor]. First examples of hyper-Kähler symmetric spaces with non-commutative holonomy were constructed in the paper [KO2], where also a classification of hyper-Kähler symmetric spaces of signature .4; n/ was obtained. The present chapter is devoted to the investigation of the structure of pseudo-Kähler and para-Kähler symmetric spaces, that is, pseudo-Riemannian symmetric spaces .M; g/ together with a parallel field J of skew-symmetric complex, or, respectively, para-complex structures (such that J 2 D Id or, respectively, J 2 D Id). A paraKähler structure .g; J / on a manifold M is also called a bi-Lagrangian structure since it is equivalent to a symplectic structure together with a decomposition of the tangent bundle into a direct sum of two integrable Lagrangian distributions. The classification of simply connected para-Kähler symmetric spaces reduces to the algebraic problem of classification of Z-graded Lie algebras of the form g D g1 C g0 C g1 (called 3-graded Lie algebras) such that the adjoint representations of g0 on g1 and g1 are exact and mutually contragredient (dual). Note that the structure of a 3-graded Lie algebra on a vector space g is equivalent to the structure of a Jordan pair, that is, a pair .V D g1 , V C D g1 / of vector spaces together with two 3-linear maps V V C V ! V ;

V V VC ! V C

which satisfies some identities, see [B]. Jordan pairs and 3-graded Lie algebras are closely related also with Jordan algebras. For example, if the grading derivation D of a 3-graded algebra g (defined by Djgi D i Id) is inner and the corresponding element d (such that add D D) can be imbedded into a 3-dimensional subalgebra sl2 with the standard basis .e 2 g1 ; h; f /, then the multiplication .x; y/ 7! x B y WD ŒŒx; e; y defines the structure of a Jordan algebra on V D g1 . Jordan pairs are related with many algebraic and differential-geometric objects and they have been intensively studied, see, for example, [B], [KO3], [N], [T], [Tan]. In particular, in [S] it is shown that one can associate with a Jordan pair a completely integrable equation of Schrödinger type. The classification of 3-graded semisimple Lie algebras is well known, see [KN], [Kac]. However, it seems that only a few constructions of nonsemisimple 3-graded Lie algebras have been proposed in the literature, see for example, [As], [deO], [Tan]. Note that Ricci-flat para-Kähler manifold has the holonomy group H SU.p; q/ and such a manifold is a pseudo-Riemannian analogue of a Calabi–Yau manifold. The aim of this chapter is to describe a class of Ricci-flat para-Kähler and pseudoKähler symmetric spaces. In particular, we classify all Ricci-flat para-Kähler and pseudo-Kähler symmetric spaces of dimension n 6.

Chapter 21. Pseudo-Kähler and para-Kähler symmetric spaces

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In Section 1, we recall some basic facts about pseudo-Riemannian symmetric spaces. We prove that a symmetric space has semisimple isometry group if and only if its Ricci form ric is non-degenerate and that ric D 0 if and only if the Lie algebra g generated by transvections has zero Killing form and, in particular, is solvable. In Section 2, we reduce the classification of simply connected para-Kähler symmetric spaces to the description of 3-graded Lie algebras g D g1 C g0 C g1 such that the g0 -modules V D g1 and V D g1 are dual to each other and the representation of h D g0 on V is exact. Then h is identified with a subalgebra h of the linear algebra V ^ V D gl.V / so.V C V /: We prove that the gradation of g is consistent with an appropriate Levi–Malcev decomposition of g. If a subalgebra h gl.V / is given, the structure of an admissible graded Lie algebra on g D V C h C V is determined by an h-invariant bi-form R 2 .S 2 V ˝ S 2 V /h such that the associated endomorphism R W V ^ V ! gl.V / D V ^ V ;

X ^ 7! R.X ^ /;

defined by the contraction take values in h: R.V ^ V / h. Using this, we describe all admissible graded Lie algebras of the form g D V C VC ^ V C V ; where V D VC C V is a direct sum decomposition of a vector space V and V D VC C V the dual decomposition. Such a gradation is determined by a bi-form R 2 S 2 VC ^ S 2 V : The associated para-Kähler symmetric space is Ricci flat and has commutative holonomy algebra h VC ^ V : Section 3 deals with pseudo-Kähler symmetric spaces .M; g; J /. We prove that a pseudo-Kähler symmetric space is determined by a complex admissible graded Lie algebra g D g1 C g0 C g1 D V C h C V ; and by its admissible anti-involution, that is, by an antilinear involutive automorphism of g (considered as a real Lie algebra) which satisfies certain conditions. We describe all admissible anti-involutions of a complex graded Lie algebra of the form g D V C VC ^ V C V , where V D VC C V is a decomposition of a complex vector space into the direct sum of two isomorphic subspaces and V D VC C V is the dual decomposition. This gives a big class of pseudo-Calabi–Yau symmetric spaces, i.e., Ricci-flat pseudo-Kähler symmetric spaces. We also classify all pseudo-Kähler and para-Kähler Ricci-flat symmetric spaces of dimension n 6. Acknowledgement. This work was supported by Leverhulme Trust, EM/9/2005/0069.

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1 Pseudo-Riemannian symmetric spaces 1.1 Basic properties of symmetric spaces A pseudo-Riemannian symmetric space is a pseudo-Riemannian manifold .M; g/ such that any point x is an isolated fixed point of an isometric involution Sx called the central symmetry with the center at x. The connected component G of the group generated by central symmetries is called the group of transvections. The group G acts transitively on M and M can be identified with the coset space G=H , where H D Go is the stabilizer of a point o 2 M . The central symmetry S0 defines an involutive automorphism W G 3 a 7! S0 B a B S0 of the Lie group G and its Lie algebra g. The -eigenspace decomposition gDhCm

(1.1)

of the Lie algebra g is a symmetric decomposition, that is, Œh; h h;

Œh; m m;

Œm; m h:

Moreover, the symmetric decomposition satisfies the following two properties: It is effective, that is, h does not contain a non-trivial ideal of the Lie algebra g and it is minimal, that is, Œm; m D h: We will identify m with the tangent space T0 M and we denote by h ; i D g0 a pseudo-Euclidean metric in m defined by the pseudo-Riemannian metric g. Then the isotropy representation of the stability subgroup H (respectively, stability subalgebra h D LieH ) is identified with the adjoint representation AdH jm (respectively, adh jm ). Moreover, the isotropy group AdH jm coincides with the holonomy group of .M D G=H; g/. Conversely, let g D h C m be a symmetric decomposition of a Lie algebra g and g0 D h ; i an adh -invariant pseudo-Euclidean metric in m. They define a simply connected pseudo-Riemannian symmetric space .M D G=H; g/, where G is the simply connected Lie group with the Lie algebra g, H connected subgroup of G, generated by the subalgebra h and g a G-invariant pseudo-Riemannian metric on M defined by the pseudo-Euclidean metric g0 in m D To .G=H /, o D eH 2 G=H . The central symmetry So with center at o is given by So .aH / D .a/H;

aH 2 G=H;

where is the involutive automorphism of G which corresponds to the involutive automorphism of the Lie algebra g defined by jh D Id, jm D Id. If the symmetric decomposition is effective, then the action of G on M D G=H is almost effective, that is, the kernel of effectivity N D fg 2 G; gx D x for all x 2 M g H is a discrete x D G=N acts on M effectively and transitively central subgroup of G and the group G x H x , where H x D H=N . Moreover, G x is the as a group of isometries and M D G=


707

group of transvections if and only if the symmetric decomposition is minimal. The following known theorem reduces the classification of simply connected Riemannian symmetric spaces to the classification of effective minimal symmetric decompositions (1.1) of Lie algebras with pseudo-orthogonal isotropy representation of h in m. Theorem 1. There exists a natural 1-1 correspondence between effective minimal symmetric decompositions (1.1) of a Lie algebra g together with an adh -invariant pseudo-Euclidean metric go on m and simply connected pseudo-Riemannian symmetric spaces .M D G=H; g/, where G is the simply connected Lie group with Lie.G/ D g and H the connected subgroup generated by h as described above. Moreover, any pseudo-Riemannian with the symmetric decomposition (1.1) has the y ; g/, where H y is a subgroup such that H H y G . Here G is the form .G=H fixed point set of the involutive automorphism of G associated with the symmetric decomposition. In the future, by a symmetric space .M; g/ we will understand a simply connected pseudo-Riemannian associated with a minimal effective symmetric decomposition of a Lie algebra g and a pseudo-Euclidean adh -invariant metric g0 on m and we will denote by G the simply connected Lie group with the Lie algebra g which acts on M D G=H almost effectively. Proposition 1. Let .M D G=H; g/ be a simply connected pseudo-Riemannian manifold associated with an effective minimal symmetric decomposition (1.1) and an adh invariant metric go D h ; i on m. Then the following hold: i) The curvature tensor R and the Ricci tensor ric of .M; g/ at the point o D eH is given by R.X; Y / D adŒX;Y jm ; X; Y 2 m D To M; 1 where B is the Killing form of g: ric D Bjm ; 2 ii) The holonomy algebra at o is spanned by the curvature operators R.X; Y /, X; Y 2 m. iii) The Ricci tensor ric is non-degenerate if and only if the Lie algebra g is semisimple. iv) The Ricci tensor vanishes if and only if the Killing form B of g is zero. This implies that g is solvable and h D Œm; m is nilpotent. v) Assume that .M; g/ has no flat factor. Then the following conditions are equivalent: (a) The holonomy algebra adh jm is totally reducible. (b) g is semisimple. (c) ric is non-degenerate.

708


Note that the Killing form B D 0 for a nilpotent Lie algebra. The converse statement is not true. For example, the solvable non-nilpotent Lie algebra g D RA C C 2 which is a semidirect sum of the commutative ideal C 2 and 1-dimensional subalgebra spanned by the derivation A D diag.i; 1/ of C 2 has B D 0. Proof. Claims i) and ii) are well known, see [Bess]. Since B.h; m/ D 0, the kernel of Bjm belongs to the kernel of B in g. So if g is semisimple, then B is non-degenerate by Cartan criterion and ric D 12 Bjm is non-degenerate. Conversely, if ric is nondegenerate, then ker B h is an ideal of g which is zero by effectivity. Hence, g is semisimple. This proves iii). If B D 0 then ric D 0 by i). Conversely, if ric D 0, the kernel of B is an ideal of g which contains m. Since g D Œm; m C m, we get ker B m? g and B D 0. This proves iv). v) Suppose that the holonomy algebra is totally reducible. We may also assume that .M; g/ is not a product of two pseudo-Riemannian manifolds. Then, by the De Rham–Wu decomposition theorem, either m is an irreducible non-trivial h-module or a direct sum of two totally isotropic irreducible modules m1 and m2 . Note that ric ¤ 0. Indeed, in the opposite case, g is solvable by iv) and adh D adŒm;m consists of nilpotent endomorphisms and cannot be totally reducible. Then the Schur Lemma shows that ric is non-degenerate in the first case. We prove that this is also true in the second case. First of all, we prove that Œm1 ; m1 D R.m1 ; m1 / D 0 and similar Œm2 ; m2 D 0. Since R.m; m/ 2 adh , the Bianchi identity implies that R.m1 ; m1 /m2 D 0. Using this and the fact that the curvature operators are skew-symmetric, we get hR.m1 ; m1 /m1 ; m2 i D hm1 ; R.m1 ; m1 /m2 i D 0. This implies that R.m1 ; m1 / D Œm1 ; m1 D 0. With respect to a basis which is consistent with the decomposition g D m1 C m2 C h, the matrix of operators adm1 has the form 0 1 0 0 g D @0 0 0 A : 0 0 Hence, B.X; Y / D tr adX adY D 0

for all X; Y 2 m1 :

(1.2)

Suppose now that ric is degenerate. Without loss of generality, we may assume that ker ric D ker Bjm m2 . This together with (1.2) implies that B.m; m/ D ric.m; m/ D 0 and we get a contradiction. Hence, the Ricci tensor is non-degenerate and g is semisimple. Conversely, if g is a semisimple Lie algebra, then it is well known that the fixed point set h D g of an involutive automorphism is a reductive subalgebra. Let .m; h ; i/ be a pseudo-Euclidean vector space and h a Lie subalgebra of the pseudo-orthogonal Lie algebra so.m/. We denote by ˚ V R.h/ D R 2 h ˝ 2 .m /; ˇ.R/ D 0


709

the space of curvature tensor of type h, that is, of h-valued 2-forms which satisfy the Bianchi identity ˇ.R/.X; Y; Z/ D R.X; Y /Z C R.Y; Z/X C R.Z; X/Y D 0

for all X; Y; Z 2 m:

For R 2 R.h/, we denote by h.R/ the subalgebra of h generated by the curvature operators R.X; Y /, X; Y 2 m. A tensor R 2 R.h/ is a curvature tensor of a pseudoRiemannian symmetric space if and only if it is invariant under the Lie algebra h.R/. Such a tensor defines a Lie algebra g with a symmetric decomposition g D h.R/ C m defined by ŒA; B D AB BA;

ŒA; X D AX; ŒX; Y D R.X; Y /; A; B 2 h; X; Y 2 m:

Note that the associated simply connected pseudo-Riemannian symmetric space .M D G=H; g/ has holonomy algebra h. We have Proposition 2. There is natural 1-1 correspondence between simply connected pseudoRiemannian symmetric spaces with holonomy algebra h so.m/ (considered up to an isometry) and elements R of the space R.h/h of h-invariants curvature tensors from R.h/ which satisfy the condition h D h.R/ (considered up to a transformation from the normalizer NO.m/ .h/.)

1.2 Pseudo-Kähler and para-Kähler symmetric spaces Recall that a pseudo-Kähler (respectively, para-Kähler ) manifold is a 2n-dimensional pseudo-Riemannian manifold .M; g/ together with a parallel filed J of skewsymmetric endomorphisms such that J 2 D Id (respectively, J 2 D Id ). A pseudoKähler manifold is called also a bi-Lagrangian manifold. The existence of parallel complex structure J (such that J 2 D Id) (respectively, parallel para-complex structure J (such that J 2 D Id)) on a pseudo-Riemannian manifold .M; g/ means that the holonomy group Hol .M; g/ is a subgroup of the pseudo-unitary group U.p; q/, p C q D n (respectively, the general linear group GL.n; R/ SO.n; n/ ). The curvature tensor R of a pseudo-Kähler manifold (respectively, para-Kähler manifold) .M; g/ belongs to the space R.u.p; q// (respectively, R.gl.n; R//). Lemma 1. The Ricci tensor ric.R/ of a para-Kähler or pseudo-Kähler manifold .M; g; J / with the curvature tensor R is given by 1 ric.X; J Y / D tr JR.X; J Y /: 2 In particular, the manifold .M; g/ is Ricci-flat if and only if the holonomy algebra is a subalgebra of sl.n; R/ or, respectively, of su.p; q/.

710


Proof. Let ei be an orthonormal basis of the tangent space Tx M such that hei ; eP ii D of "i 2 f˙1g. Then ric.X; Y / D "i hR.X; ei /ei ; Y i (where we omit the sign summation over repeated indices). Using the Bianchi identity and the fact that J is a skew-symmetric endomorphism with J 2 D " Id commuting with the curvature operator R.X; Y /, where " D 1 in Kähler case and " D 1 in para-Kähler case, we have tr JR.X; Y / D "i hJR.X; Y /ei ; ei i D "i hR.X; Y /ei ; Jei i D "i hR.ei ; Jei /X; Y i D "i hR.Jei ; X /ei ; Y i C "i hR.X; ei /Jei ; Y i D ""i hR.Jei ; X/Jei ; J Y i C "i hR.X; ei /Jei ; Y i D ""i hR.X; Jei /Jei ; J Y i "i hR.X; ei /ei ; J Y i D ric.X; J Y / ric.X; J Y / D 2 ric.X; J Y /: Here we use the identity hJei ; Jei i D ""i .

Lemma 2. The space R.u.p; q// (respectively, R.gl.n; R//) of curvature tensors of type u.p; q/ (respectively, of type gl.n; R/) consists of tensors which are invariant under the complex (respectively, the para-complex) structure J . Proof. Using the pseudo-Euclidean metric, one can identify the spaces R.u.p; q// and R.gl.n; R// with subspaces of S 2 .u.p; q// and S 2 .gl.n; R//, respectively, which obviously consist of J -invariant tensors. The following proposition, which gives an infinitesimal description of pseudoKähler and para-Kähler symmetric spaces, is a specialization of Theorem 1. Proposition 3. A simply connected symmetric space .M D G=K; g/ associated with an effective symmetric decomposition g D h C m of a Lie algebra g and an adh -invariant pseudo-Euclidean metric g0 on m is a pseudo-Kähler symmetric space (respectively, a para-Kähler symmetric space) if there exists a skew-symmetric adh invariant endomorphism J on m with J 2 D Id (respectively, J 2 D Id). We will call a triple .g D hCm; g0 ; J / described in the proposition an infinitesimal pseudo-Kähler symmetric space (respectively, an infinitesimal para-Kähler symmetric space) if J 2 D Id (respectively, J 2 D Id). So the classification of simply connected pseudo-Kähler symmetric spaces (respectively, para-Kähler symmetric spaces) reduces to the classification of infinitesimal pseudo-Kähler symmetric spaces and (respectively, infinitesimal para-Kähler symmetric spaces).


711

2 Structure of para-Kähler symmetric spaces 2.1 Curvature bi-form of a para-Kähler symmetric space Now we describe infinitesimal para-Kähler symmetric spaces in terms of admissible 3-graded Lie algebras. Definition 1. A 3-graded Lie algebra g D g1 C g0 C g1 ;

Œgi ; gj giCj

(2.1)

(and its underlying gradation) is called admissible if the subalgebra g0 has no nonzero ideal of g and g0 -modules g1 and g1 are conjugate, i.e., there is a g0 -invariant non-degenerate bilinear pairing g1 g1 ! R;

.X; / 7! .X/

1

which allows to identify g with the dual module .g1 / . We put h D g0 , V D g1 and V D g1 . An admissible graded Lie algebra g D g1 C g0 C g1 D V C h C V

(2.2)

is called minimal if ŒV; V D h. Proposition 4. There is a natural 1-1 correspondence between minimal admissible graded Lie algebras (2.2) (up to an isomorphism) and simply connected para-Kähler symmetric spaces .M D G=H; g; J / (up to an isometry which preserves the paracomplex structure). A minimal admissible graded Lie algebra defines the para-Kähler symmetric space .M D G=H; g; J / associated with the minimal admissible symmetric decomposition g D h C .V C V / D g0 C .g1 C g1 / and the natural pseudo-Euclidean metric g0 on m D V C V , defined by the pairing g0 .X C ; Y C / D .Y / C .X/: The para-complex structure J in .M; g/ is obtained by the parallel extension of the adh -invariant para-complex structure J0 2 so.m/ defined by J0 jg˙1 D ˙Id. The holonomy algebra of .M; g/ is equal to adh jm . The symmetric space .M D G=H; g/ is Ricci-flat (or, equivalently, has holonomy algebra h sl.n/) if and only if g is a solvable Lie algebra. Then adh is a nilpotent linear Lie algebra, which consists of nilpotent endomorphisms. Hence the problem of the classification of para-Kähler symmetric spaces reduces to the description of minimal admissible graded Lie algebras. Proof. The proof of the first part follows from Proposition 3. Ricci-flatness is equivalent to the vanishing of the Killing form B of g. This implies that the Lie algebra g

712


is solvable. Conversely, assume that g is solvable. Then h D Œm; m is a nilpotent subalgebra such that adh consists of nilpotent endomorphisms. Then Lemma 1 shows that the Ricci tensor vanishes. Let g D V C h C V be an admissible graded Lie algebra and .M D G=H; g; J / the associated para-Kähler symmetric space. We identify m D V C V with the tangent space To M . Then V adh jm V ^ V ' gl.V / so.m/ ' 2 m is the holonomy algebra of .M; g/. We identify h with a subalgebra of gl.V / D V ^ V so.m/. Then the space R.gl.V // can be identified with the space of symmetric endomorphisms R 2 S 2 .gl.V // End.gl.V // which satisfy the following Bianchi identities: R.X ^ /Y D R.Y ^ /X and R.X ^ / D R.Y ^ / for X; Y 2 V and ; 2 V . The Bianchi identities take this form since R.V; V / D R.V ; V / D 0. Lemma 3. The space R.gl.V // D R.V ^ V / of curvature tensors of type gl.V / is naturally isomorphic to the space S 2 V _ S 2 V , where _ is the symmetric product, hence, also to the space S 2 V ˝ S 2 V of bi-forms. Proof. The contraction of a bi-form R 2 S 2 V ˝S 2 V with a vector X and a covector defines a symmetric endomorphism R W gl.V / D V ^ V ! gl.V /;

X ^ 7! R.X ^ /

which satisfies the Bianchi identities, and hence is a curvature tensor.

For example, if e, f are vectors and ˛, ˇ covectors, then the bi-form R D .e _ f / ˝ .˛ _ ˇ/; corresponds to the curvature tensor R D .e ^ ˛/ _ .f ^ ˇ/ C .e ^ ˇ/ _ .f ^ ˛/: We will often identify the curvature tensor with the corresponding curvature endomorphism and curvature bi-form. We choose a basis ei of the vector space V and the dual basis e i of the dual vector space V and identify elements from V with vector columns, elements from V with vector rows and elements from gl.V / D V ˝ V ' V ^ V with matrices. Then the action of an element A 2 gl.V / D V ^ V is given by A.X/ D AX; A./ D A;

X 2 V; 2 V :

Since V and V are commutative subalgebras of g, the structure of an admissible graded Lie algebra in g D V C h C V is defined by an h-invariant curvature endomorphism R 2 R.h/h of type h. We get


713

Proposition 5. Let h gl.V / D V ^ V be a linear Lie algebra. A structure of an admissible graded Lie algebra in g D V ChCV is defined by an h-invariant bi-form R of type h, that is, an element R 2 .S 2 V ˝ S 2 V /h such that R.V ^ V / h. The Lie bracket in g associated with the bi-form R is defined by V V ! h;

ŒX; D Œ; X D R.X ^ /:

The admissible graded Lie algebra g is minimal if and only if the corresponding curvature endomorphism R of type h is surjective, i.e. R.V ^ V / D h: Two curvature bi-forms R, R0 define isomorphic admissible graded Lie algebras if and only if they belong to the same orbit of the normalizer NGL.V / h in the space .S 2 V ˝ S 2 V /h . Proof. The last claim follows from the observation that an isomorphism 'W V ChCV ! V ChCV of admissible graded Lie algebras is canonically defined by a linear transformation C 2 GL.V / such that the induced linear transformation of the space V ˝V preserves h. This means that C belongs to the normalizer of h in GL.V /. This proposition reduces the classification of simply connected para-Kähler symmetric spaces with holonomy algebra h V ˝ V so.V C V / to a description of tensors R 2 .S 2 V ˝ S 2 V /h such that spanfR.X ^ /; X 2 V; 2 V g D h. We have the following corollary. Corollary 1. A bi-form R 2 S 2 V ˝ S 2 V defines a minimal admissible graded Lie algebra g D V C h C V if and only if it is invariant under the linear Lie algebra h generated by endomorphisms R.X ^ / for X 2 V , 2 V , that is, it satisfies the equations R.X ^ / R D 0: Remark 1. Obviously, the results remain true for complex Lie algebras g, their complex admissible gradations g D V C h C V (where V; h and V are complex subspaces) and for complex curvature tensors R 2 S 2 V ^ S 2 V . All admissible gradations of semisimple Lie algebras can be easily described in terms of Satake diagrams. It implies the classification of para-Kähler symmetric spaces of semisimple Lie groups, see [Kan]. Another approach to the classification is based on the fact that the holonomy algebra h of a para-Kähler symmetric space has non-zero first prolongation h.1/ since the contraction R./ of the curvature tensor R with a covector belongs to h.1/ . Now we prove a version of the Levi decomposition for admissible graded Lie algebras.

714


Consider a minimal admissible graded Lie algebra (2.2). We denote by D the derivation of g defined by Djgi D i Id, and by yCV g y D V C .h C RD/ C V D V C h

(2.3)

the extended admissible graded Lie algebra. Proposition 6. There exists a Levi decomposition g y D s C rO of an extended graded Lie algebra (2.3) which is consistent with the gradation, that is, g y D s1 C s0 C s1 C rO 1 C rO 0 C rO 1 ; where si D s\gi , rO i D rO \gi , and the element D can be decomposed as D D d CD 0 , where d 2 s0 , D 0 2 rO 0 , Œs; D 0 D 0. The derivation add js defines a minimal admissible gradation of s and the derivation adD defines an admissible gradation of rO . Moreover, the derivation Djr defines an admissible gradation r D r1 C r0 C r1 of the radical r of g, where rO 0 D r0 C RD. If the derivation D is inner, we may take g y D g, D 0 D 0 and D D add , d 2 s. Corollary 2. Any minimal admissible graded Lie algebra g is a semidirect sum of a minimal admissible graded semisimple Lie algebra s and an admissible graded solvable ideal r D r1 C r0 C r1 . y i.e., a maximal nilpotent Proof of the proposition. Let a be a Cartan subalgebra of h, y y subalgebra of h which coincides with its normalizer in h. It contains D and is a Cartan subalgebra of g y since Djg˙1 D ˙Id. It is known [GOV] that there is a Levi decomposition g y D s C rO which is consistent with the Cartan subalgebra a such that a D aS C aR , where aS D a \ s is a Cartan subalgebra of s and aR is a Cartan subalgebra of the centralizer ZrO .aS /. Since the s-component d of D D d C D 0 is semisimple and commutes with the r-component D 0 , the element D 0 is also semisimple. Using the theory of splitting, see [SLie], we may assume (changing the Levi subalgebra, if needed) that Œs; D 0 D 0. Then adD js D add js defines a minimal admissible gradation of s and adD defines an admissible gradation of rO and r. Since r0 Œg; r, it is in the nilradical of g. If the derivation D is inner, D 0 is a semisimple element of r0 , hence is equal to 0.

2.2 Ricci-flat para-Kähler symmetric spaces Now we will study Ricci-flat para-Kähler symmetric spaces. According to Proposition 4, they correspond to minimal admissible graded solvable Lie algebras g D V C h C V , where the subalgebra h D ŒV; V is identified with a nilpotent subalgebra of gl.V / D V ^ V which consists of nilpotent elements. If h gl.V / is fixed, the structure of the graded Lie algebra in g is defined by an h-invariant bi-form R 2 .S 2 V ˝ S 2 V /h with R.V ^ V / h. We choose a basis e1 ; : : : ; en of V such that elements of h are represented by upper triangular matrices with zero diagonal

715


and denote by e 1 ; : : : ; e n the dual basis of V . We will identify h with a subalgebra of the Lie algebra n.n/ of upper triangular nilpotent matrices. Then a curvature endomorphism R W gl.V / ! h gl.V / of type h can be written as 0

0 a21 .X; / a31 .X; / a41 .X; /

B0 B B B0 B B R.X ^ / D B0 B: B: B: B @0

a32 .X; / a42 .X; /

0

0

a43 .X; /

0

0

0 :: :

0 :: :

0 :: :

:: :

0

0

0

0

0

0

an1 .X; /

1

an2 .X; / C C C 3 an .X; / C C C an4 .X; / C ; C :: C C : C ann1 .X; /A

(2.4)

0

where aji .X; / are bilinear functions of X 2 V and 2 V . Lemma 4. Bilinear functions aji .X; / define a curvature endomorphism R of type h D spanfR.X; /; X 2 V; 2 V g given by (2.4) if and only if ŒA; R.X ^ / D R.AX; / R.X; A/

(2.5)

for any A D R.Y ^ /, X; Y 2 V , ; 2 V , the expression X X R.X ^ /Y D aj1 .X; /Y j e1 C aj2 .X; /Y j e2 j >1

C

X

j >2

aj3 .X; /Y j e3 C C ann1 .X; /Y n en1 :

(2.6)

j >3

is symmetric with respect to any X D X i ei , Y D Y i ei 2 V for any 2 V and the expression R.X ^ / D 1 a21 .X; /e 2 C .1 a31 .X; / C 2 a32 .X; //e 3 C C

n2 n1 X X i .i an1 .X; //e n1 C i ani .X; //e n i D1

(2.7)

iD1 i

i

is symmetric with respect to D i e ; D i e 2 V for any X D X i ei 2 V .

Proof. Follows from Proposition 5.

Note that any upper triangular nilpotent matrix A D kaji k 2 n.n/ defines an obvious solution of equations (2.6) and (2.7) given by aji .X; / D aji X j i

(no summation).

(2.8)

The following lemma describes when the corresponding map R is equivariant and hence is a curvature endomorphism.

716


Lemma 5. The map R W gl.V / ! gl.V / associated with an upper triangular nilpotent matrix A D kaji k, aji D 0 for i j , by formulas (2.8), (2.4) is a curvature endomorphism R of type n.n/ if and only if the following conditions hold: ak` ¤ 0 H) aik D a`j D 0 for all i , j .

(2.9)

h D R.gl.V // D spanfE`k D e` ^ e k ; ak` ¤ 0g

(2.10)

In this case the image of the endomorphism R is a subalgebra of the commutative algebra VC ^ V n.n/, where V D VC C V is a direct sum decomposition of V and V D VC C V is the dual decomposition of V . Proof. It is clear that the image of the linear map R defined by a matrix A is given by (2.10). The map R is a curvature endomorphism if it is R.gl.V //-equivariant, that is, ŒE`k ; R.X ^ / D R.ŒE`k ; X ^ / for X 2 V , 2 V and for all .k; `/ such that ak` ¤ 0: It is sufficient to write these conditions for basic vectors X D ei and D e j . Since R.Eij / D aij Eji , we get the following conditions: i D k: ŒE`k ; R.Ekj / D ŒE`k ; akj Ejk D 0 D R.ŒE`k ; Ekj / D R.E`j / D aj` E`j H) aj` D 0I i D `: ŒE`k ; R.E`j / D ŒE`k ; a`j Ej` D a`j Ejk D R.ŒE`k ; E`j // D 0 H) a`j D 0I j D k: ŒE`k ; R.Eik / D ŒE`k ; aik Eki D aik Eì D R.ŒE`k ; Eik / D 0 H) aik D 0I j D `: ŒE`k ; R.Ei` / D ŒE`k ; ai` Eì D 0 D R.ŒE`k ; Ei` // D R.Eik / D aik Eki H) aik D 0: These are exactly conditions (2.9). They imply A2 D 0. We put VC D AV and denote by V a complementary subspace. Then hV D VC , hVC D 0 and h VC ^ V . In the next section we describe para-Kähler symmetric spaces with holonomy algebra h VC ^ V :


717

2.3 Para-Kähler symmetric spaces with holonomy algebra h VC ^ V Let V D VC C V be a direct sum decomposition of a vector space V and V D VC C V the dual decomposition of the dual space V . We will describe all para-Kähler symmetric spaces with holonomy algebra h VC ^ V so.m/ D ƒ2 .V C V /: Proposition 7. A curvature endomorphism R of the type h D VC ^ V is h-invariant. Such endomorphisms form the vector space S 2 VC _ S 2 V ' S 2 VC ˝ S 2 V . Proof. The proof is straightforward.

A curvature endomorphism R 2 S 2 VC _ S 2 V can be written as R D a1 _ ˛1 C C ak _ ˛k ; where ai 2 S 2 VC and ˛i 2 S 2 V are symmetric bilinear forms on VC and V , respectively. We will call the minimal possible number k the rank of R . The following theorem gives a description of simply connected para-Kähler symmetric spaces with holonomy h VC ^ V . Theorem 2. There exists a natural 1-1 correspondence between curvature endomorphisms R 2 S 2 VC _ S 2 V (up to a transformation from GL.VC / GL.V /), admissible graded Lie algebras of the form g D V C VC ^ V C V (up to an isomorphism) and simply connected para-Kähler symmetric spaces with the holonomy algebra h VC ^ V (up to an isometry). The following conditions are equivalent: a) The symmetric space has the holonomy algebra h D VC ^ V . b) The admissible graded Lie algebra g is minimal. c) The curvature endomorphism R W V ^ V ! h D VC ^ V is surjective. Proof. The proof follows from Propositions 4, 5 and 7. We put n˙ D dim V˙ .

Corollary 3. Any rank one surjective curvature endomorphism R of type h D VC ^V has the form R D a _ ˛, where a, ˛ are non-degenerate symmetric bilinear forms of signature .k; nC k/ and .`; n `/ on VC and V , respectively. With respect to an j appropriate basis eiC of VC and e of V , it can be written as X X j 2 "j .e / ; RD "i .eiC /2 _ j where eiC and e is a basis of VC and V , respectively, and "i D 1 for i k and "i D 1 for i > k, "j D 1 for j ` and "j D 1 for j > `. The structure of the associated minimal admissible graded Lie algebra g D V C VC ^ V is given by j ŒeiC ; e D "i "j ıij :

718


2.4 Ricci-flat para-Kähler symmetric spaces of dimension 6 Proposition 4 shows that any 2-dimensional Ricci-flat para-Kähler symmetric space is flat. Now we describe all Ricci-flat para-Kähler symmetric spaces of dimension 4 and 6. According to Proposition 5 and Lemma 4, a 4-dimensional Ricci-flat para-Kähler symmetric space is determined by a curvature endomorphism R.X; / V ^ V , V a 2-dimensional space, of the form 0 a.X; / R.X; / D ; (2.11) 0 0 where a.X; / is a bilinear function on V V , such that a.X; /Y 2 D a.Y; /X 2 ;

a.X; /1 D a.X; /1

for all X D X 1 e1 C X 2 e2 ;

Y D Y 1 e1 C Y 2 e2 ;

D 1 e 1 C 2 e 2 ;

D 1 e 1 C 2 e 2 :

Here e1 , e2 is a basis of V and e 1 , e 2 the dual basis of V . The conditions (2.6), (2.7) imply that a.X; / D aX 2 1 , where a is a constant. The condition (2.5) is satisfied Changing the basis e1 , e2 we can always attain a D " D ˙1. The corresponding curvature bi-form is R D "e1 ˝ e1 ˝ e 2 ˝ e 2 . We get Theorem 3. Any non-flat Ricci flat para-Kähler symmetric space of dimension 4 has the commutative holonomy algebra ² ³ 0 a h D 0 0 ; a 2 R gl.2; R/: (2.12) and is defined (up to an isometry) by the curvature bi-form R D "e1 ˝ e1 ˝ e 2 ˝ e 2 (or corresponding curvature tensor R D 14 ".e1 ^ e 2 / _ .e1 ^ e 2 //), where " D ˙1, e1 , e2 is a basis of V and e 1 , e 2 the dual basis of V . In the 6-dimensional case we have the following Theorem 4. Any 6-dimensional para-Kähler indecomposable Ricci-flat symmetric space has commutative holonomy algebra h isomorphic to 80 80 80 19 19 19 < 0 c1 c2 = < 0 c1 c2 = < 0 0 c2 = h1 D @0 0 c1 A ; h2 D @0 0 0 A ; or h3 D @0 0 c1 A ; ; ; ; : : : 0 0 0 0 0 0 0 0 0 (2.13) where c1 ; c2 ; 2 R and is determined by the curvature endomorphism (or curvature bi-forms and curvature tensor) given as follows.


For h1 :

719

80 19 < 0 a1 .X; / a2 .X; / = 0 a1 .X; /A ; R.X; / D @0 : ; 0 0 0 a1 D 21 X 3 ;

a2 D 21 X 2 C 22 X 3 ;

R.X; / D 4.12 X 2 X 3 C 1 2 .X 3 /2 / (the curvature bi-form), R D .e1 ^ e 3 / _ .e1 ^ e 2 C e2 ^ e 3 / (the curvature tensor). For h2 :

80 19 < 0 a1 .X; / a2 .X; / = 0 0 A ; R.X; / D @0 : ; 0 0 0 a1 D "2 1 X 2 ;

a2 D "3 1 X 3 ;

where "2 ; "3 D ˙1,

R D 1 .a1 X 2 C a2 X 3 / D 12 Œ"2 .X 2 /2 C "3 .X 3 /2 (the curvature bi-form), RD

1 Œ"2 .e1 ^ e 2 / _ .e1 ^ e 2 / C "3 .e1 ^ e 3 / _ .e1 ^ e 3 (the curvature tensor). 4

For h3 :

80 19 < 0 0 a2 .X; / = R.X; / D @0 0 a1 .X; /A ; : ; 0 0 0 a1 D "2 2 X 3 ;

a2 D "1 1 X 3 ;

where "1 ; "2 D ˙1,

R D ."1 12 C "2 22 /.X 3 /2 (the curvature bi-form), RD

1 Œ"1 .e1 ^ e 3 / _ .e1 ^ e 3 / C "2 .e2 ^ e 3 / _ .e2 ^ e 3 (the curvature tensor). 4

Proof. Lemma 4 reduces the classification of 6-dimensional Ricci-flat para-Kähler symmetric spaces to the description of three bilinear forms a3 D a21 .X; /, a2 D a31 .X; /, a1 D a32 .X; / 2 V V , where V is the 3-dimensional space with a basis e1 , e2 , e3 , which satisfy the conditions (2.5), (2.6), (2.7). Solving equations (2.6), (2.7), we get a1 D .˛1 C ˛ 0 2 /X 3 ; a3 D 1 .X 3 C 0 X 2 /; a2 D 1 X 2 C ˇ1 X 3 C ˛2 X 3 for some ˛; ˛ 0 ; ˇ; ; 0 2 R: Let

0

0 c3 C D @0 0 0 0

1 c2 c1 A 0

(2.14)

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be an element from the holonomy algebra h D spanfR.X; /; X 2 V; 2 V g. Then the condition (2.5) (with A D C ) gives ˛ 0 c3 D 0;

0 c1 D 0;

. a3 /c2 D .˛ a1 /c3 :

(2.15)

Assume that c1 ¤ 0, c3 ¤ 0 for some C 2 h. Then ˛ 0 D 0 D 0 and the formulas a1 D ˛1 X 3 ;

a3 D 1 X 3 ;

a2 D 1 X 2 C ˇ1 X 3 C ˛2 X 3

give a general solution. Using an appropriate change of the basis of the form e1 ! b1 e1 ;

e2 ! b2 e2 C ae1 ;

e3 ! b3 e3 ;

we can transform the coefficient ˇ into zero and ˛ into 2. Then the curvature bi-form can be written as R D 4Œ12 X 2 X 3 C 1 2 .X 3 /2 . The coefficients c1 , c3 cannot be both identically zero, since the manifold is indecomposable. Assume that c1 0. Then the equations (2.15) give a1 D 0;

a2 D 1 X 2 C ˇ1 X 3 ;

a3 D 1 .X3 C 0 X 2 /

and the bi-form R can be written as R D 1 .a3 X 2 C a2 X 3 / D 12 Œ 0 .X 2 /2 C 2X 2 X 3 C ˇ.X 3 /2 : Using the appropriate change of the basis ei , we may assume that D 0, 0 ; ˇ 2 ˙1. The proof in the case c3 0 is similar. The following proposition, which we will use later, describes the stability subalgebra of the full algebra of infinitesimal automorphisms of the para-Kähler symmetric space .M; g; J / associated with the curvature tensor R D .e1 ê 3 /_.e1 ê 2 Ce2 ê 3 /: It consists of all endomorphisms C 2 End.V / (naturally extended to V C V ) which preserve the curvature tensor R. Proposition 8. The group D Aut.R/ of automorphisms of the space V which preserve the curvature tensor R D .e1 ^ e 3 / _ .e1 ^ e 2 C e2 ^ e 3 / consists of automorphisms of the form V 3 X 7! CX; where 2 R and

1 1 c c0 C D @0 1 c A ; 0 0 1 0

c; c 0 2 R:

(2.16)

Proof. Since the curvature tensor R determines the flag Ce1 Ce1 C Ce2 V , the matrix kcji k of an automorphism C which preserves R with respect to the basis e1 ; e2 ; e3 has the upper triangular form. We denote by kcOji k the inverse matrix and put fi D C ei D cij ej ;

f i D C.e i / D cOji e j :


721

Then f1 D c11 e1 ;

f 1 D cO11 e 1 C cO21 C cO31 e 3 ;

f2 D c21 e1 C c22 e2 ;

f 2 D cO22 e 2 C cO32 e 3 ;

f3 D c31 e1 C c32 e2 C c33 e3 ; f 3 D cO33 e 3 : The condition C.R/ D R can be written as C.e1 ^ e 3 / D f1 ^ f 3 D c11 cO33 e1 ^ e 3 D e1 ^ e 3 ; C.e1 ^ e 2 C e2 ^ e 3 / D f1 ^ f 2 C f2 ^ f 3 D c11 cO22 e1 ^ e 2 C c22 cO33 e2 ^ e 3 C .c11 cO32 C c21 cO33 /e1 ^ e 3 D .e1 ^ e 2 C e2 ^ e 3 / for some 2 R . We get the system of equations D c11 cO22 D c22 cO33 D c11 cO33 ;

0 D c11 cO32 C c21 cO33 ;

which together with the equalities cOii D .cii /1 ;

cO32 D

c11 c32 c11 c22 c33

gives c11 D c22 D c33 , c21 D c32 :

Remark 2. Note that the given classification of admissible graded Lie algebras of dimension n 6 remains valid also for complex Lie algebras.

3 Structure of pseudo-Kähler symmetric spaces Now we reduce the classification of infinitesimal pseudo-Kähler symmetric spaces to a description of admissible anti-involutions of a complex admissible graded Lie algebras g D V C h C V . Let .g D h C m; g0 D h ; i; J / be an infinitesimal pseudo-Kähler symmetric space. We consider the complexification gC D hC C mC of the symmetric decomposition g D h C m and extend the endomorphism J to an endomorphism JO of g putting JO h D 0. Lemma 6. JO is a derivation of the Lie algebra g with eigenvalues ˙i , 0. Proof. The result follows from Lemma 2.

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The derivation D D i JO of gC has eigenvalues ˙1 and 0, and defines an admissible gradation gC D g1 C g0 C g1 D V C hC C V of the complex Lie algebra gC . Here mC D V CV is a decomposition of the complex Euclidean space into a direct sum of two g0C -isotropic dual subspaces V , V and the complex metric g D g0C of mC induced by g0 is identified with the natural metric defined by the pairing. Like in the real case, the admissible gradation is minimal and C it is determined by a surjective curvature endomorphism R 2 .S 2 V ˝ S 2 V /h of type hC . Note that the complex conjugation of gC with respect to g is an antilinear involutive automorphism of gC (considered as a real Lie algebra) which satisfies the following properties: 1) W V ! V , W V ! V , W hC ! hC ; 2) the restriction of the complex Euclidean metric g to the fixed point space m D .V C V / of is a (real) pseudo-Euclidean metric. The real Lie algebra g is the fixed point set g D .gC / of and the symmetric decomposition of g is induced by the gradation of gC as follows: g D h C m D g D .hC / C .V C V / : Definition 2. Let g D V C h C V be an admissible complex graded Lie algebra and D is the derivation associated with the gradation: Dh D 0;

DjV D Id;

DjV D Id:

An antilinear involutive automorphism of g which satisfies properties 1) and 2) is called an admissible anti-involution of g. Proposition 9. An admissible anti-involution of a complex admissible graded Lie algebra g D V C h C V defines an infinitesimal pseudo-Kähler symmetric space .g D hs C m D h C .V C V / /; g0 ; J / where g0 D gjm is the restriction of the complex metric g to m and the complex structure J D iDjm . Moreover, any infinitesimal pseudo-Kähler symmetric space can be obtained in this way. Proof. The proof is straightforward.

We will show that any admissible anti-involution is determined by an antilinear bijective map 0 W V ! V which satisfies certain conditions. We will identify hC with a complex subalgebra of the Lie algebra V gl.V / D V ^ V 2 .V C V / D so.V C V /: Note that an admissible anti-involution is determined by its restriction 0 to V0 since jV D .s0 /1 jV . The action of on h is the restriction of the action on V ^ V


723

induced by the action of on V C V . Indeed, since is an automorphism of gC as a real Lie algebra, we get ŒA; X D .AX/ D Œ .A/; .X / D .A/. .X //

for all A 2 hC ; X 2 V:

This shows that .A/ D B A B . The following proposition gives necessary and sufficient conditions that an antilinear bijective map W V ! V defines an admissible anti-involution of a given complex admissible graded Lie algebra. With respect to a basis e1 ; : : : ; en of V and the dual basis e 1 ; : : : ; e n of V , an antilinear map 0 W V ! V is defined by a complex n n matrix B D kbij k where 0 .ei / D bij e j : The vectors fj WD ej C ej D ej C bj k e k ;

fj0 WD i.ej ej / D i.ej bj k e k /

form a basis of the space m D .V C V / where is the extension of 0 to an antilinear involution of V C V . The Gram matrix of the induced symmetric bilinear form g0 WD gjm with respect to this basis is given by gij D g.fi ; fj / D bij C bj i ; gi 0 j 0 D g.fi0 ; fj0 / D bij C bj i ; gi 0 j D g.fi0 ; fj / D i.bij bj i /: In particular, g induces a pseudo-Riemannian metric on m if B is a non-degenerate Hermitian matrix. This implies Proposition 10. Let g D V C h C V be a complex minimal admissible graded Lie algebra defined by a (complex) curvature endomorphism R 2 .S 2 V _ S 2 V /h . Let 0 W V ! V be an antilinear map defined by a matrix B D kbij k. It defines an admissible anti-involution of g if and only if the following conditions hold: (1) The matrix B is Hermitian (bNij D bj i ) and non-degenerate. (2) The induced antilinear involution W V C V ! V C V (acting naturally on tensors) preserves the tensor R 2 S 2 V _ S 2 V and, hence, the complex subalgebra h V ^ V . Proof. Assume that 0 satisfies the conditions. The second condition implies that the induced antilinear involution is an involutive automorphism of g as a real Lie algebra and the first condition shows that the induced metric on m D .V C V / is real and non-degenerate. Hence, is an admissible anti-involution. The converse claim is clear.

724


3.1 A construction of pseudo-Calabi–Yau symmetric spaces with commutative holonomy In this section we describe pseudo-Riemannian Calabi–Yau symmetric spaces with commutative holonomy algebra h such that the complexification gC D V ChC CV of the corresponding infinitesimal symmetric space .g D h C m; g0 ; J / is an admissible minimal graded complex Lie algebra described in Section 2.3. According to Proposition 10, the problem reduces to the description of admissible anti-involutions of the complex admissible graded Lie algebra of the form gDV ChCV

(3.1)

defined by a curvature endomorphism R 2 S 2 VC _ S 2 V where V D VC C V is a decomposition of a complex vector space into a direct sum of subspaces, V D VC C V the dual decomposition and h D R.V ^ V / VC ^ V . Lemma 7. An admissible anti-involution of a complex admissible graded Lie algebra of the form (3.1) may exist only when dim VC D dim V . Proof. Since .V / D V , .V / D V , the condition .h/ D .VC / ^ .V / D V C ^ V implies that .VC / D V , .V / D VC . Hence dim VC D dim V D dim V :

Let be an admissible anti-involution. Then .V / D V ;

.V / D V;

.h/ D .h/ D VC ^ V :

Hence, interchanges VC and V . Conversely, if an antilinear bijective map 0 W V ! V transforms VC into V , then the induced involutive map preserves VC ^ V . Now we find conditions that preserves the curvature bi-form R 2 S 2 VC _ S 2 V . We set W D spanfR.X; Y /; X; Y 2 V g S 2 VC ; W 0 D spanfR.; /; ; 2 VC g S 2 V ; where R.; / and R.X; Y / denote the corresponding contractions. It .R/ D R, then W D W 0 . We choose a basis a1 ; : : : ; ak of the space W and denote by a10 D .a1 /; : : : ; ak0 D .ak / the corresponding basis of W 0 . Then the curvature bi-form can be written as X RD hij ai _ aj0 ;


725

where H D khij k is a non-degenerate complex matrix. The condition X R D .R/ D hNj i aj _ ai0 means that the matrix C is Hermitian, i.e., C D C . Now we can describe admissible anti-involutions of admissible graded Lie algebras of the form (3.1), where V D VC C V is a decomposition of a complex 2m-dimensional vector space V into a sum of two m-dimensional subspaces and V D VC CV is the dual decomposition. We denote by ei a basis of V and by e j the dual basis of V . Theorem 5. Let 0 W V ! V ;

ei 7! bij e j ;

be an antilinear map with 0 .VC / D V which is defined by a non-degenerate Hermitian matrix B D kbij k and W V C V ! V C V is the associated antiinvolution of V C V . Let a1 ; : : : ; ak be linear independent elements from S 2 VC and a10 D .a1 /; : : : ; ak0 D .ak / corresponding linear independent elements from S 2 V . Then any non-degenerate Hermitian k k matrix H D khij k defines a -invariant curvature bi-form X RD hij ai _ aj0 (3.2) of rank k and hence a structure of complex admissible Lie algebra in g D V C VC ^ V C V : Moreover, is an admissible anti-involution of this algebra and it defines an infinitesimal pseudo-Kähler symmetric space .g D h C .V C V / ; g0 ; J D iDjV CV /; where h is the Lie subalgebra of VC ^ V spanned by curvature operators R.u; v/, u; v 2 .V CV / , and g0 is the restriction of the natural complex metric g on V CV to .V C V / . Conversely, any infinitesimal pseudo-Kähler symmetric space whose complexified holonomy algebra is a subalgebra of VC ^ V so.V C V / can be obtained by this construction. Proof. The tensor (3.2) defines the structure of admissible graded Lie algebra in g D V C VC ^ V C V and is invariant under the anti-involution . Hence is an admissible anti-involution and by Proposition 10 it defines an infinitesimal pseudoKähler symmetric space. It is clear that any admissible anti-involution of an admissible graded algebra (3.1) can be obtained by this construction.

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3.2 Pseudo-Calabi–Yau symmetric spaces of dimension n 6 Here we classify all pseudo-Calabi–Yau symmetric spaces of dimension n 6. Since any such space of dimension 2 is flat, it is sufficient to consider the spaces of dimension 4 and 6. Theorem 6. There exist two (up to an isometry ) 4-dimensional indecomposable pseudo-Kähler symmetric spaces. They correspond to the real forms g of the complex infinitesimal para-Kähler symmetric space g D V C h1 C V , V C 2 with the curvature tensor R D .e1 ^ e 2 / _ .e1 ^ e 2 / defined by the admissible anti-involution D " W e1 ! "e 2 ;

e2 ! "N e 1 ;

e 1 ! "N e2 ;

e 2 ! "e1 ;

where " D 1; i. Proof. By Proposition 10, admissible anti-involution is defined by a Hermitian matrix B D kbij k such that ei D bij e j . Then ej D b j k ek where B 0 D kb j k k D Bx1 . The condition R D R is equivalent to the condition .e1 ^ e 2 / D ˙e1 ^ e 2 : We may assume that

Then

(3.3)

0 bN BD : b 1 B D kbk2 0

bN

b ; 0

where 2 R. The equation (3.3) gives b 2 D ˙kbk2 . Changing the basis e1 , e2 , we may assume that kbk D 1 and D 0. Then b D ˙1; ˙i . Moreover, multiple e1 to 1, if needed, we may assume that b D 1 or b D i . Theorem 7. There exist two (up to an isometry) 6-dimensional indecomposable pseudo-Kähler symmetric spaces. They correspond to the real forms g of the complex infinitesimal para-Kähler symmetric space g D V C h1 C V with the curvature tensor (3.4) R D .e1 ^ e 3 / _ .e1 ^ e 2 C e2 ^ e 3 / defined by admissible anti-involutions D " W e1 ! "e 3 ;

e2 ! "e 2 ;

e3 ! "e 1 ;

where " D ˙1, e1 , e2 , e3 is a basis of the complex 3-dimensional space V and e 1 , e 2 , e 3 the dual basis of V . Proof. We have to describe admissible anti-involutions of the complex admissible graded 6-dimensional Lie algebras g D V C hi C V , i D 1; 2; 3 described in Theorem 4. Note that h2 e1 ˝ V and h3 V ˝ e 3 . Then any admissible


727

anti-involution must map V into Ce1 , or, respectively, V into Ce 1 . This implies that the rank of B is one, which is impossible. It remains to describe an admissible anti-involution which preserves R D .e1 ^ e 3 / _ .e1 ^ e 2 C e2 ^ e 3 /. For such an anti-involution we have .e1 ^ e 3 / D ".e1 ^ e 3 /; .e1 ^ e 2 C e2 ^ e 3 / D ".e1 ^ e 2 C e2 ^ e 3 /; where " D ˙1. Hence transforms the flag spanfe1 g spanfe1 ; e2 g V into the flag spanfe 3 g spanfe 3 ; e 2 g V . In other words, the matrix B of jV W V ! V and the matrix B 0 of jV W V ! V have the form 1 0 0 0 1 0 0 b ı c0 b3 B D @0 ˇ c A and B 0 D @ cN 0 ˇ 0 0 A : b1 0 0 bN cN ı Since 2 D Id, we have B Bx0 D Id. This gives 1) bb 0 D 1, 2) ˇˇ 0 D 1, 3) ˇc 0 C cˇ 0 D 0, N 0 C cc N 0 C ıb 0 D 0, 4) b N 0 D 0. 5) bN cN 0 C cˇ The condition that preserves R can be written as 1) b bN 0 D ", N 0 C b 0 cN D 0, 2)0 bc 0 N 0 3) bˇ D "; 4)0 b 0 ˇ D ". The conditions 1/, 10 /, 30 /, 40 / give " D 1, b D bN D ˇ D ˇ10 D b10 . Then the conditions 3/, 20 / show that c D cN D where is a real number. Hence the matrix B is real and symmetric. Consider a new basis fj D Cji ei of V defined by a transition 0

j

matrix of the form (2.16) with real coefficients c, c 0 and the dual basis f j D .C 1 /i e i . By Proposition 8, the curvature tensor R has the same expression with respect to the new basis fi , f j . Then fj D .Cji ei / D Cji Bik e k D Cji Bik Clk f l D Bzj l f l . So the matrix Bz of jV with respect to the new basis is given by Bz D C t BC . The calculations show that the entries of Bz are ˇQ D ˇ, Q D C 2ˇc, ıQ D ı C 2c C ˇc 2 C 2ˇc 0 . Hence we can choose c, c 0 such that Q D ıQ D 0. Similar, under a scalar change of the basis fi ! afi , f j ! a1 f j , the matrix B of the anti-involution will transform as B ! aaB. N This shows that in an appropriate basis, the matrix B of the anti-involution has the form 0 1 0 0 " @0 " 0A : " 0 0

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Chapter 22

Prehomogeneous affine representations and flat pseudo-Riemannian manifolds Oliver Baues

Contents 1

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Flat pseudo-Riemannian and flat affine manifolds . . . . . . 1.2 Flat Riemannian manifolds . . . . . . . . . . . . . . . . . . 1.3 Flat manifolds of indefinite signature . . . . . . . . . . . . . 1.4 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The development map and holonomy . . . . . . . . . . . . . 2.2 Flat manifolds are .X; G/-manifolds . . . . . . . . . . . . . The group of affinities . . . . . . . . . . . . . . . . . . . . . . . 3.1 Affine vector fields . . . . . . . . . . . . . . . . . . . . . . 3.2 Development representation of affinities . . . . . . . . . . . 3.3 Affinities of compact volume preserving affine manifolds . . 3.4 Lie groups of isometries . . . . . . . . . . . . . . . . . . . 3.5 Lie groups of symplectic transformations . . . . . . . . . . Homogeneous model spaces . . . . . . . . . . . . . . . . . . . . 4.1 The development map of a homogeneous space . . . . . . . 4.2 Compact homogeneous affine manifolds . . . . . . . . . . . 4.3 Holonomy of homogeneous affine manifolds . . . . . . . . Flat affine Lie groups . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Left-invariant geometry on Lie groups . . . . . . . . . . . . 5.2 The development map of a flat affine Lie group . . . . . . . 5.3 Flat pseudo-Riemannian Lie groups . . . . . . . . . . . . . 5.4 Étale affine representations . . . . . . . . . . . . . . . . . . Affinely homogeneous domains . . . . . . . . . . . . . . . . . . 6.1 Prehomogeneous affine representations . . . . . . . . . . . 6.2 Some examples . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Characteristic map for a prehomogeneous representation . . 6.4 The automorphism group of an affine homogeneous domain 6.5 Tube like domains . . . . . . . . . . . . . . . . . . . . . . . 6.6 Pseudo-Riemannian affine homogeneous domains . . . . . . 6.7 Symplectic affine homogeneous domains . . . . . . . . . . A criterion for transitivity of prehomogeneous representations . . 7.1 Transitivity for prehomogeneous groups . . . . . . . . . . . 7.2 The fundamental diagram . . . . . . . . . . . . . . . . . . . 7.3 Transitivity of nilpotent prehomogeneous groups . . . . . .

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Characteristic cohomology classes associated with affine representations 8.1 Construction of the characteristic classes . . . . . . . . . . . . . . . 8.2 Significance of the relative classes . . . . . . . . . . . . . . . . . . 8.3 Affine representations of nilpotent groups . . . . . . . . . . . . . . 8.4 Prehomogeneous representations with reductive stabiliser . . . . . . 9 Compact affine manifolds and prehomogeneous algebraic groups . . . . 9.1 Holonomy of compact complete affine manifolds . . . . . . . . . . 9.2 Holonomy of compact volume preserving affine manifolds . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Linear algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Definition of linear algebraic groups . . . . . . . . . . . . . . . . . A.2 Structure theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Orbit closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B Lie algebra cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 Definition of Lie algebra cohomology . . . . . . . . . . . . . . . . B.2 Relative Lie algebra cohomology . . . . . . . . . . . . . . . . . . . C Invariant measures on homogeneous spaces . . . . . . . . . . . . . . . . C.1 Semi-invariant and invariant measures . . . . . . . . . . . . . . . . C.2 The unimodular character of X D G=H . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction 1.1 Flat pseudo-Riemannian and flat affine manifolds A pseudo-Riemannian manifold .M; h ; i/ is a smooth manifold M which is endowed with a possibly indefinite metric h ; i on its tangent bundle TM . We let the expression s.h ; i/ D .nC ; n /, nC C n D dim M , denote the signature of h ; i. A positive definite metric h ; i has signature .n; 0/ and is traditionally called a Riemannian metric. If the signature of h ; i is .n 1; 1/, h ; i is called a Lorentzian metric. Every pseudo-Riemannian manifold .M; h ; i/ has a unique torsion-free connection rh ; i on its tangent bundle TM which has the property that the metric tensor h ; i is parallel for rh ; i . The connection rh ; i is called the Levi-Civita connection for h ; i. Given vector fields X; Y; Z on M , the tensorial expression, Rr .X; Y /Z D rX rY Z rY rX Z rŒX;Y Z is called the curvature tensor for a connection r. The metric h ; i is called flat if the curvature Rrh ; i vanishes everywhere on M . Thus, in particular, flat pseudo-Riemannian manifolds carry a torsion-free and flat connection on their tangent bundle. A manifold M together with a torsion-free and flat connection r is called a flat affine manifold. Naturally, flat affine manifolds .M; r/ share many of their geometric properties with the more restricted class of flat pseudo-Riemannian manifolds.

Chapter 22. Prehomogeneous affine representations

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1.1.1 Global models and their quotients. There does exist an abundance of examples of simply connected flat pseudo-Riemannian and flat affine manifolds. Indeed, every open subset U of Euclidean space Rn , defines a flat pseudo-Riemannian manifold (of any signature), and, every simply connected flat pseudo-Riemannian manifold z ; h ; i/ or flat affine manifold .M z ; r/ is obtained by pulling back the flat structures .M z ! U onto an open subset of Rn . along a local diffeomorphism ˆ W M z ; h ; i/ will be called a global A simply connected pseudo-Riemannian manifold .M model space for flat pseudo-Riemannian manifolds. By elementary covering theory, every pseudo-Riemannian manifold .M; h ; i/ is obtained as a quotient space of a z ; h ; i/ by a properly discontinuous group of isometries of .M z ; h ; i/. global model .M Similarly, any flat affine manifold .M; r/ is a quotient of a simply connected manifold z ; r/ by a group of connection preserving diffeomorphisms. .M In principle, the study of flat pseudo-Riemannian manifolds breaks into two parts, z ; h ; i/ (apart from the stannamely the determination of interesting global models .M dard complete model Es ), and the study of the quotient spaces .M; h ; i/ which are z ; h ; i/. modelled on .M A particular interesting class of model spaces will be furnished by homogeneous flat pseudo-Riemannian manifolds. These are flat spaces which admit a transitive group of isometries. More generally, homogeneous flat affine manifolds, that is, flat affine manifolds with a transitive group of affine transformations constitute a natural class of models. 1.1.2 Completeness and pseudo-Euclidean space forms. A pseudo-Riemannian manifold .M; h ; i/ is called complete if every geodesic curve for the connection rh ; i can be extended to infinity. If the flat manifold .M; h ; i/ is complete, the Killing–Hopf z ; h ; i/ for theorem asserts, that the universal pseudo-Riemannian covering space .M s n .M; h ; i/ is isometric to the pseudo-Euclidean space E D .R ; h ; is / of signature s D s.h ; i/, where h ; is denotes the standard representative for a scalar product of signature s. In particular, for fixed signature s, there exists, up to isometry, a unique simply connected complete model space Es for flat pseudo-Riemannian manifolds of signature s. As a further consequence, every complete flat pseudo-Riemannian manifold .M; h ; i/ of signature s is isometric to a quotient of Es by a properly discontinuous subgroup of isometries, acting without fixed points on Es . Such quotient manifolds are also called pseudo-Euclidean space forms.

1.2 Flat Riemannian manifolds By the Hopf–Rinow theorem (which holds solely in Riemannian geometry), the compactness of a Riemannian manifold implies its completeness. Similarly, the equivalence of metric and geodesic completeness for Riemannian manifolds implies that

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every homogeneous Riemannian manifold is complete. In fact, every homogeneous flat Riemannian manifold is a quotient of Rn by a group of translations. In particular, every compact flat Riemannian manifold or homogeneous flat Riemannian manifold is a quotient of Euclidean space by a discontinuous group of isometries. Thus, the theory of flat Riemannian manifolds concerns mostly the study of complete space forms, and it is roughly equivalent to the study of discontinuous subgroups of the isometry group E.n/ of Euclidean space. The structure of discontinuous subgroups of E.n/ is rather well understood, by the famous three theorems of Bieberbach [13], [14], dating from around 1910. According to Bieberbach, every discrete subgroup of E.n/ is finitely generated and contains an abelian subgroup of finite index. If acts with compact quotient space, then the subgroup of translations is of finite index in . In particular, every compact Euclidean space form is finitely and isometrically covered by a flat torus Rn =ƒ, where ƒ is a lattice of translations. Moreover, Bieberbach proved, that in every dimension n, there exist only finitely many compact flat Riemannian manifolds up to affine equivalence. Thus, the class of compact flat Riemannian manifolds is rather restricted from a topological point of view. The determination of compact Riemannian space forms and their geometric properties has a long tradition, and remains a subject of geometric and algebraic (it is related to the study of integral representations of finite groups) interest. Recent contributions concern, for example, isospectrality phenomena [28], [77], [93], and spin structures [27], [86] on flat Riemannian manifolds. The theory of almost flat Riemannian manifolds as developed by Gromov [44], in a sense, extends the theory of Bieberbach to a much more general context. See [44], [23]. Here, the role of flat tori is taken over by compact nilmanifolds (these are quotient spaces of nilpotent Lie groups). Incidentally, nilmanifolds also appear (at least conjecturally) as the fundamental building blocks for compact flat pseudoRiemannian manifolds.

1.3 Flat manifolds of indefinite signature Much of the theory of flat pseudo-Riemannian manifolds aims to construct an analogy to the theory of Euclidean space forms. But, as is it turned out, many new phenomena and principle difficulties arise. Many of them constitute still open and difficult research questions. For example, it is widely believed, that, as in the Riemannian case, only the pseudoEuclidean spaces Es admit compact quotient manifolds. This conjecture is so far verified only for Lorentzian manifolds, see [25]. Likewise, the determination of simply connected homogeneous model spaces z ; h ; i/, and more generally the determination of all homogeneous flat pseudo.M Riemannian manifolds of a given signature s, is an unsolved problem.


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1.3.1 Structure theory of compact flat pseudo-Riemannian manifolds Compactness and completeness. A first stumbling stone for a structure theory is created by the general lack of understanding about the relationship of compactness and completeness for flat pseudo-Riemannian and affine manifolds. In fact, although there are known examples of compact (non flat) Lorentzian manifolds, which are incomplete, it is expected (“Conjecture of Markus”) that, as in the Riemannian case, a flat compact pseudo-Riemannian manifold is indeed also complete. This conjecture was proved by Carriére [25] for the case of flat Lorentzian manifolds. But it remains unverified for flat pseudo-Riemannian manifolds of arbitrary signature. More generally, the Markus conjecture (attributed to [74]) suggests that an orientable compact affinely flat manifold is complete if and only if it admits a parallel volume. Beside the result of Carriére, this conjecture is verified only in a few special cases. For example, compact homogeneous affine manifolds satisfy Markus’ conjecture. Note that, by dividing out a cyclic group of linear dilatations on the affine space with the point 0 removed, one constructs a simple example of a compact affinely flat and incomplete manifold. In general, the theory of compact flat affine manifolds is considered as wild and possibly untractable, although some results on the topology of such manifolds have been obtained in low dimensions. See [94] for wild examples. Group theoretic structure of the fundamental group. Assuming completeness, new obstacles appear to generalise Bieberbach’s theory of discontinuous Euclidean groups and Euclidean crystallographic groups to a theory of affine crystallographic groups. These concern the group theoretic structure of the fundamental group of a complete pseudo-Riemannian manifold. Conjecturally, (“Auslander’s conjecture”), an affine crystallographic group has a solvable subgroup of finite index. The conjecture arose from a paper [4] of Louis Auslander, which contained a flawed proof of the even stronger claim that every finitely generated discontinuous subgroup of affine transformations has a solvable subgroup of finite index. The solvable group replaces the finite index abelian subgroup in the Euclidean case. In this sense, the Auslander conjecture serves as a weak analogue to Bieberbach’s first theorem. Auslander’s conjecture is verified in low dimensions, in the Lorentzian case, and in some other cases (see [1] for a survey). But it remains one of the main open questions of the subject. Also, if the assumption of compactness is dropped, examples of flat complete three dimensional Lorentzian manifolds with a free non-abelian fundamental group (cf. [29], [73]) give a counter example to the original claim of Auslander. In this case, the analogy with the Euclidean theory breaks down. The classification theory of pseudo-Euclidean space forms. Assuming completeness and solvability of the fundamental group, new difficulties and phenomena arise for a possible classification program. For instance, the finiteness part of the Bieberbach theory breaks down, as well. But it admits a weak and rather subtly defined replacement, as is described in [46].

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The main achievement of the theory, so far, gives a rather precise and strong link of the theory of pseudo-Euclidean and affine crystallographic groups with the theory of left-invariant flat pseudo-Riemannian metrics on Lie groups. (See [36], [46] for an exposition.) For example, a classification theory of compact flat Lorentzian manifolds is developed in [7], [35], [43], [45], [48]. However, despite the achievements in the Lorentzian case, the structure theory for compact complete flat pseudo-Riemannian manifolds remains widely open. 1.3.2 Homogeneous flat pseudo-Riemannian manifolds. Unlike the Riemannian case, a homogeneous pseudo-Riemannian manifold need not be complete. The simplest example of a non-complete homogeneous flat pseudo-Riemannian manifold is an open orbit of the two-dimensional non-abelian solvable simply connected Lie group in E1;1 (see [103, §11]). In general, the classification of non-complete homogeneous flat pseudo-Riemannian manifolds is not fully understood. Only if the additional assumption of compactness is made then homogeneity implies completeness for pseudo-Riemannian manifolds (in fact, even without the assumption of flatness), see [51], [75]. Furthermore, in some cases, the group theoretical structure of a homogeneous space is directly linked to its completeness properties. For example, a flat pseudo-Riemannian Lie group, more generally a volume preserving flat affine Lie group, is complete if and only if it is unimodular. In this chapter, we introduce the related new result that a flat affine homogeneous space of a nilpotent Lie group is complete if and only if the action preserves a parallel volume form. In particular, a flat homogeneous pseudo-Riemannian manifold or a flat symplectically homogeneous affine manifold of a nilpotent group is complete. See Section 4 and Section 7 for further discussion of these results. The structure and classification of complete homogeneous flat pseudo-Riemannian manifolds is more accessible than the general case. Every Riemannian homogeneous flat manifold is obtained by dividing out a group of translations in Rn . In the pseudoRiemannian case interesting phenomena do occur. For example, contrasting the Riemannian case, there exists a large class of (noncompact) complete homogeneous flat pseudo-Riemannian manifolds with abelian but non-translational holonomy groups. See [102], [107], [108] for a detailed study of this examples. In general, the holonomy group of a homogeneous flat pseudo-Riemannian manifold must be a two-step nilpotent group. (See Section 4.3.1 for details on this.) Nonabelian fundamental groups occur, in particular, as fundamental groups of compact homogeneous flat pseudo-Riemannian manifolds. In fact, there exists a large class of compact two-step nilmanifolds, which admit an (essentially unique) homogeneous flat pseudo-Riemannian metric. The first examples, which are not homotopy equivalent to a torus arise in dimension six. See Section 4.2.2 for more details and proofs. For general homogeneous affinely flat manifolds similar results hold. By [42], the affine holonomy group of a compact affine manifold with parallel volume does not preserve any proper algebraic subsets in affine space An . As a corollary, a compact


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homogeneous affine manifold with parallel volume is complete. (An independent proof is given in Section 4.2.) More generally, if a volume preserving homogeneous affine manifold admits a compact Clifford–Klein form then it must be complete (see Section 9.2 for further discussion). In particular, if the universal covering of a compact flat pseudo-Riemannian manifold is homogeneous then it must be complete. It follows that non-complete homogeneous flat pseudo-Riemannian manifolds do not occur as models of compact flat pseudo-Riemannian manifolds.

1.4 Overview This chapter aims to exhibit several aspects, which link the theory of flat affine and pseudo-Riemannian manifolds with the topic of representations of Lie groups and algebraic groups on affine space An . In Section 1 we started with an overview on some of the main achievements and open problems in the topic of flat affine and pseudo-Riemannian manifolds. In the following Section 2 we discuss flat manifolds from the point of view of Thurston’s theory of locally homogeneous .X; G/-manifolds. Here we introduce our notation, as well as basic definitions and methods. In Section 3, we describe the structure of the group of affinities and of the Lie algebra of affine vector fields on flat affine manifolds. By the development process, these groups are represented as subgroups of the affine group, and relate to certain associative matrix algebras. Compactness poses strong restrictions on the symmetries of volume preserving, in particular, of pseudo-Riemannian and symplectic affine flat manifolds, which we describe in detail. Similarly, the holonomy groups of flat homogeneous manifolds are determined by the centralisers of prehomogeneous representations. They turn out to be a nilpotent, if the homogeneous manifold is complete. The following Section 4 is devoted to the basic properties of homogeneous affine manifolds. Their automorphism groups develop to prehomogeneous subgroups of the affine group. We show that the Markus conjecture is satisfied for compact homogeneous affine manifolds, and we describe the structure of compact homogeneous flat pseudo-Riemannian, and also of compact symplectically homogeneous affine flat manifolds. We also introduce a result which states that a homogeneous affine manifold of a nilpotent group is complete if and only if the group is volume preserving. In particular, a homogeneous flat pseudo-Riemannian or symplectic affine flat manifold of a nilpotent group is always complete. The proof of these results depends on methods which are developed in Section 7. In Section 5, we review the relationship between the geometry of a flat affine Lie group and the behaviour of its left and right Haar measures. The main result shows that the completeness of a flat affine Lie group is determined by the interaction of its unimodular character with the volume character which is defined by the affine structure. These results build on the study of étale affine representations of Lie groups.

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In Section 6, we discuss the basic properties of affinely homogeneous domains and of prehomogeneous affine representations of Lie groups. In the following Section 7, we develop a criterion for the transitivity of prehomogeneous affine representations, which extends corresponding results for étale affine representations. The main application shows that every volume preserving nilpotent prehomogeneous group of affine transformations is transitive. In Section 8, we explain how the geometry of invariant measures on an affine homogeneous space and the transitivity properties of its associated prehomogeneous representation are linked by certain naturally defined characteristic classes of the affine representation. In Section 9, we study properties of the Zariski closure A./, where is the holonomy group of a compact affine manifold M . One of the main objectives of the subject is to characterise the groups A./ in relationship with the geometric properties of M . An important result of Goldman and Hirsch states that A./ acts transitively on affine space if M is a volume preserving compact flat affine manifold. This has strong consequences for the geometry of compact volume preserving flat affine manifolds. We explain some of the applications of this result, and we also explain how its proof relates to the methods developed in the previous sections. Acknowledgement. I thank Joseph A. Wolf, Vicente Cortés and Wolfgang Globke for detailed comments, numerous helpful suggestions, and careful reading of a first draft of this exposition. I also wish to thank the anonymous referee for suggesting several bibliographical references.

2 Foundations Flat pseudo-Riemannian and affine manifolds constitute actually a particular class of locally homogenous manifolds. This point of view allows to express many geometric properties of flat manifolds in an elegant and transparent way.

2.1 The development map and holonomy We start by briefly recalling the fundamental notion of an .X; G/-manifold. For further and more detailed reference on .X; G/-structures, see, for example, [24], [88], [95].

2.1.1 .X; G /-manifolds. Let X be a homogeneous space for the Lie group G. A manifold M is said to be locally modelled on .X; G/ if M admits an atlas of charts with range in X such that the coordinate changes are restrictions of elements of G. A maximal atlas with this property is then called a .X; G/-structure on M , and M is called a .X; G/-manifold, or locally homogeneous space modelled on .X; G/.


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A map ˆ between two .X; G/-manifolds is called an .X; G/-map if it looks like the action of an element of G in the local charts. If the .X; G/-map ˆ is a diffeomorphism it is called an .X; G/-equivalence. Every .X; G/-manifold comes equipped with some extra structure, called the dez ! M denote the universal covering space velopment and the holonomy. Let W M of the .X; G/-manifold M . We fix x0 2 M . The development map of the .X; G/structure on M is the local diffeomorphism z !X DW M which is obtained by analytic continuation of a local .X; G/-chart of M in x0 . The z , there development map is an .X; G/-map, and, for any .X; G/-equivalence ˆ of M exists an element h.ˆ/ 2 G such that D B ˆ D h.ˆ/ B D:

(2.1)

z via covering transformaThe fundamental group 1 .M / D 1 .M; x0 / acts on M z . This induces the holonomy homomortions, which are .X; G/-equivalences of M phism h W 1 .M; x0 / ! G which satisfies D B D h. / B D;

(2.2)

for all 2 1 .M; x0 /. After the choice of the development map (which corresponds to a choice of a germ of a .X; G/-chart in x0 ), the holonomy homomorphism h is well defined. We note that the .X; G/-structure on M determines the conjugacy class of h under the action of G. z , and Clearly, the development map already determines the .X; G/-structure on M z is equivalent to specifying a development pair for the action of 1 .M; x0 / on M constructing an .X; G/-structure on M : z ! X which satisfies (2.2), for Proposition 2.1. Every local diffeomorphism D W M some h W 1 .M; x0 / ! G defines a unique .X; G/-structure on M , and every .X; G/structure on M arises in this way. Properly discontinuous actions. Let be a group of diffeomorphisms of a manifold M . Then is said to act properly discontinuously on M if, for all compact subsets K M , the set K D f 2 j K \ K ¤ ;g is finite. If acts properly discontinuously and freely on M then the quotient space M= is a smooth manifold, and the projection map W M ! M= is a smooth covering map. Now if acts by .X; G/-equivalences on an .X; G/-manifold M , then the quotient space M= inherits a natural .X; G/-manifold structure from M . In fact, the development maps of M and M= coincide (as well, as their universal coverings.) Example 2.1 (.X; G/-space forms). Assume that X is simply connected, and is a group of .X; G/-equivalences of X (that is, is a subgroup of G) acting properly

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discontinuously and freely on X . Then X= is an .X; G/-manifold, and the identity map of X is a development map for X= . See [24], [95] for further discussion of .X; G/-geometries and the properties of the development process. 2.1.2 Affine and projectively flat manifolds. The geometry of the pseudo-Euclidean space Es D .Rn ; E.s// is determined by the transitive action of the isometry group E.s/ of the standard scalar product of signature s. Similarly the geometry of affine space An D .Rn ; Aff.n// is determined by the action of the full affine group Aff.n/. Coordinate representation of the affine group. We view affine space An as a hyperplane An D f.x; 1/ j x 2 Rn g embedded in RnC1 . In this setting, the group of affine transformations Aff.n/ identifies naturally with a group of linear transformations of RnC1 . Namely, ³ ² g t ˇˇ Aff.n/ D A D ˇ g 2 GL.n; R/ ; 0 1 which is a subgroup of GL.n C 1; R/. Note that the affine group decomposes as a semi-direct product Aff.n/ D T .n/ Ì GLn .R/; where

²

ˇ

1 t T .n/ D A D 0 1

ˇ n ˇ t 2R

³

is the group of translations of Rn . The natural quotient homomorphism ` W Aff.n/ ! GLn .R/;

A 7! `.A/ D g

associates to the affine transformation A 2 Aff.n/ its linear part. The vector t .A/ D t , is called the translational part of A. The Lie algebra aff.n/ of Aff.n/ is ³ ² ' v ˇˇ aff.n/ D X D ˇ ' 2 gl.n; R/ ; 0 0 which is a Lie subalgebra of the matrix algebra gl.n C 1; R/. Note that the evaluation map of the affine action at x 2 An ox W Aff.n/ ! An ;

A 7! A x D g.x/ C t

is expressed by matrix multiplication, and so is its derivative at the identity, which is the map tx W aff.n/ ! Rn ; X 7! X x D '.X / C v: (2.3)


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Transitive subgroups of Aff.n/. We shall also consider various subgroups of Aff.n/. We let O.s/ D O.h; is / denote the group of linear isometries of the standard scalar product h ; is of signature s. The pseudo-Euclidean isometry group E.s/ is a semidirect product E.s/ D T .n/ Ì O.s/. Thus E.s/ embeds into the affine group as ³ ² g t ˇˇ E.s/ D A D ˇ g 2 O.s/ : 0 1 The group of volume preserving affine transformations is SAff.n/ D fA 2 Aff.n/ j `.A/ 2 SLn .R/g. The group of symplectic affine transformations is Aff.!n / D fA 2 Aff.2n/ j `.A/ 2 Sp.!n /g, where !n is a non-degenerate skew bilinear form on R2n . Fixed points for reductive affine actions. We call a subgroup G of a linear group reductive if every G-invariant linear subspace admits an invariant complementary subspace. The following lemma is a particular useful fact: Lemma 2.2. Let G Aff.n/ be a reductive subgroup. Then the affine action of G has a fixed point on An . Proof. Take a complementary line to the G-invariant subspace f.v; 0/ j v 2 Rn g RnC1 , and intersect with the hyperplane An . 2 Projective geometry. The space of lines through the origin in RnC1 is called real projective space, and it is denoted by P n R. The projective linear group is PGL.n; R/ D GL.n C 1; R/=f˙EnC1 g acting on the projective space P n R. Via the above coordinate representation, affine space An embeds as an open subset in P n R, and Aff.n/ embeds as a subgroup of PGL.n; R/. In particular, affine geometry (that is, geometry modelled on .X; G/ D .An ; Aff.n//), is a subclass of projective .P n R; PGL.n; R// geometry. Note also that a cone over any .P n R; PGL.n; R//-manifold becomes a .RnC1 ; GL.n C 1; R// manifold. (See [40], for an exposition about projectively flat manifolds, and their relation with affine flat manifolds.)

2.2 Flat manifolds are .X; G /-manifolds As already remarked, the geometry of the pseudo-Euclidean space Es is determined by the transitive action of its isometry group E.s/, and similarly the geometry of affine space An is determined by the action of the affine group Aff.n/. Theorem 2.3. A flat pseudo-Riemannian manifold .M; h ; i/ is a locally homogeneous space modelled on the standard pseudo-Euclidean space Es , s D s.h ; i/. A flat affine manifold .M; r/ is a locally homogeneous space modelled on affine space An .

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Proof. In fact, by flatness of .M; h ; i/, the exponential map expp W Tp M ! M allows to define an isometry of a neighbourhood of 0 in Es to a normal neighbourhood in .M; h ; i/, for every point p 2 M . This defines a compatible atlas for M , where all charts are local isometries, and with coordinate changes in the pseudo-Euclidean group E.s/. Evidently, the analoguous argument also works for a flat affine manifold 2 .M; r/. Note that the locally homogeneous Es -structure on .M; h ; i/ is uniquely determined by the compatibility condition that its charts are local isometries. The compatible An -structure for r is determined by the condition that its charts are affine maps. In z ! An and holonomy of a flat particular, we may speak of the development map D W M affine manifold, where D is then a local locally affine diffeomorphism (respectively, z ! Es is a local isometry, for a flat pseudo-Riemannian manifold). DW M The traditional point of view (pseudo-Riemannian metric and flat connection) and the Es -manifold point of view are completely equivalent. Namely, every locally homogeneous space modeled on Es has a unique flat pseudo-Riemannian metric h ; i which turns the local charts of the Es -structure into local isometries for h ; i. A map between two flat manifolds .M; h ; i/ and .M 0 ; h ; i0 / is a local isometry if and only if it is a map of Es -structures, and so on. Holonomy and parallel transport. Let h W 1 .M / ! E.s/ denote the holonomy homomorphism of an Es -structure on M . Definition 2.4. The homomorphism h is called the affine holonomy homomorphism of the flat manifold .M; h ; i/. The composition hol D ` B h W 1 .M / ! O.s/ is called the linear holonomy homomorphism. Recall that the parallel transport of a flat manifold .M; h ; i/ defines a homomorphism px W 1 .M; x/ ! O.Tx M; h ; ix / of the fundamental group of M into the group of linear isometries of the tangent space. The image px .1 .M; x// is called the holonomy group of .M; h ; i/ (at x). The following is easy to see: Proposition 2.5. In a local chart for the induced Es -structure on M , based at x 2 M , the parallel transport homomorphism px W 1 .M; x/ ! O.Tx M; h ; ix / corresponds to (that is, it is conjugate to) the linear holonomy homomorphism of the Es -structure. Of course, the analogous result holds for a flat affine manifold .M; r/ and its parallel transport. Note that also the affine holonomy may be interpreted as a parallel transport in a suitable associated bundle over .M; r/ (cf. [66]). For yet another interpretation of the affine parallel transport, see [23]. Affine structures of type A. We do not need to restrict our attention to the transitive groups E.s/ or Aff.n/. More generally, we may consider any subgroup A of the affine group Aff.n/ to define a flat model geometry. This gives rise to the following notion.


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Definition 2.6. A locally homogeneous space modelled on .An ; A/ is called a flat affine manifold of type A. The holonomy determines if the affine structure group for a flat affine manifold .M; r/ can be reduced to a subgroup A of Aff.n/. Definition 2.7. A flat affine manifold .M; r/ is called of type A if (a conjugate) of its affine holonomy homomorphism h W 1 .M / ! Aff.n/ takes image in A. If .M; r/ is of type A then it admits a compatible structure of an .An ; A/ manifold. The definition allows to consider various kinds of geometric flatness conditions for .M; r/. For example, the groups A D SAff.n/, Aff.!/, Aff.2n; C/, UE.2n/ describe the concepts of volume preserving, symplectic, complex or Kähler flat affine manifold, respectively. 2.2.1 Completeness and crystallographic groups. Geodesic completeness of a flat affine manifold is interpreted naturally by its development map. Theorem 2.8. A flat affine manifold .M; r/ is complete if and only if its development z ! An is a diffeomorphism. map D W M Proof. The development map D is a local affine diffeomorphism. By completeness z ; r/ it is onto An . Moreover, geodesics of An admit a lift along any preimage of .M for D. Therefore, D is a covering map. Hence, it must be a diffeomorphism. 2 Question 1. Does there exists a flat affine manifold with development map D, which is onto An , but not a diffeomorphism? The characterisation of completeness via the development implies the following: Corollary 2.9 (Killing–Hopf theorem). Let .M; r/ be a complete flat affine manifold of type A. Let D h.1 .M // A be its affine holonomy group. Then acts properly discontinuously and freely on An , and M is A-equivalent to the A-manifold An =. It follows that every complete pseudo-Riemannian manifold .M; h ; i/ of signature s, is isometric to a quotient of Es by a properly discontinuous group E.s/. Such a manifold M D Es = is called a pseudo-Euclidean space form. A complete flat affine manifold .M; r/ is called an affine space form. By Corollary 2.9, the study of affine space forms reduces to the study of properly discontinuous subgroups of Aff.n/.

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Example 2.2. Every discrete subgroup of the Euclidean group E.n/ acts properly discontinuously on Rn . If is in addition torsion-free then En = is a complete flat Riemannian manifold. In general, if A is properly discontinuous then it is discrete, but the converse may not hold. Crystallographic groups. A uniform discrete subgroup of E.n/ is traditionally called a (Euclidean-) crystallographic group, cf. [106]. This motivates the following Definition 2.10. A properly discontinuous subgroup A, A Aff.n/, is called an affine crystallographic group of type A if the quotient space An = is compact. Question 2. Is an affine crystallographic group (up to finite index) contained in the group of volume preserving affine transformations of An ? Or equivalently, has an orientable compact complete affine manifold always a parallel volume form? This is one direction of Markus’ conjecture, see Section 1.3.1). Note that a finite volume flat complete Riemannian manifold is necessarily compact. This is a consequence of the classification of discrete subgroups of the Euclidean group E.n/, as given by Bieberbach, see [106]. Question 3. Does there exist a non-compact, finite volume complete affine or pseudoRiemannian manifold? For example, does a complete flat Lorentzian manifold admit finite volume, without being compact? 2.2.2 The development image of a compact affine manifold. Let M be a compact z ! An its development map. The development image affinely flat manifold, D W M of M is an open domain z / An : UM D D.M It is conjectured that M is complete, if M admits a parallel volume (Markus’ conjecz / D An . ture). In particular, in this case, the conjecture claims that D.M Example 2.3. The development image of an affine two-torus UT 2 is one of the four affinely homogeneous domains which admit an abelian simply transitive group of affine transformations, see Corollary 6.6. See also [9] for construction methods of affine structures on T 2 , and visualisation of the development process. In general, the development image of a compact affinely flat, or projectively flat, manifold can have a very complicated development image. See [94], for the construction of examples.


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3 The group of affinities Let r be a torsion-free connection on M . We put Aut.M; r/ for the group of connection preserving diffeomorphisms of M . Recall (cf. [65]) that Aut.M; r/ is a Lie group.

3.1 Affine vector fields A vector field X on .M; r/ is called an affine vector field if its local flow preserves the connection r. The affine vector fields form a subalgebra aut.M; r/ of the Lie algebra Vect.M / of all C 1 -vector fields. A vector field on M is called complete if its local flow generates a one-parameter group of diffeomorphisms of M . The complete affine vector fields on M form a subalgebra autc .M; r/ of aut.M; r/. If M is compact or r is complete then autc .M; r/ D aut.M; r/ (see [66, VI, §2]). Note that the Lie algebra of complete affine vector fields is (anti-) isomorphic to the tangent Lie algebra of (left-invariant) vector fields on Aut.M; r/. 3.1.1 The associative algebra of affine vector fields on a flat manifold. Let .M; r/ be a flat affine manifold, and let X be a vector field on M . The covariant derivative of X defines an endomorphism field AX W Y 7! rY X on M . Since r is flat, the vector field X is affine if and only if AX is parallel, that is, X is affine if and only if, for all vector fields Y; Z, rZ AX Y D AX rZ Y:

(3.1)

Moreover, for an affine vector field X on the flat manifold .M; r/ we have, for all Y 2 Vect.M /, the relation AŒX;Y D ŒAX ; AY : (3.2) Induced algebra structure on Vect.M /. We declare the product of two vector fields X; Y 2 Vect.M / by X r Y WD AX Y D rY X: (3.3) In terms of the product r , condition (3.1), that X 2 Vect.M / is affine, is equivalent to X r .Y r Z/ D .X r Y / r Z; (3.4) for all Y; Z 2 Vect.M /. A short calculation, involving the associativity condition (3.4) shows: if X and 0 X are affine vector fields then AX X 0 D X r X 0 also satisfies (3.4), for all Y; Z 2

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Vect.M /. Hence, X r X 0 is an affine vector field. In particular, the Lie algebra of affine vector fields aut.M; r/ is a subalgebra of .Vect.M /; r / which is associative: Proposition 3.1. The Lie algebra of affine vector fields .aut.M; r/ forms an associative subalgebra of .Vect.M /; r /. Let us furthermore remark that the centralisers of affine vector fields form a subalgebra in .Vect.M /; r /: Lemma 3.2. Let Y 2 aut.M; r/ be an affine vector field. Let X; X 0 2 Vect.M / be vector fields on M , which centralise Y . Then also rX X 0 centralises Y . Proof. Using (3.1) for the affine field Y , as well as ŒX; Y D ŒX 0 ; Y D 0, we calculate ŒrX X 0 ; Y D rrX X 0 Y rY rX X 0 D rX rX 0 Y rY rX X 0 D rX rY X 0 rY rX X 0 D rŒX;Y X 0 D 0:

2

Remark. The fact that aut.M; r/ inherits the structure of an associative matrix algebra, has strong consequences, especially in a situation, where aut.M; r/ describes the tangent algebra for the group Aut.M; r/. For example, homogeneous affine manifolds, which are compact or complete are naturally related to associative matrix algebras. See, for example, [109] for applications. See also some of the results in Section 6 and Section 7 for further exploitation of this principle. 3.1.2 Right- and left-symmetric algebras. A bilinear product “ ” on a vector space which satisfies the identity X .Y Z/ .X Y / Z D Y .X Z/ .Y X / Z

(3.5)

is called a left-symmetric algebra. Complementary, a product is called right-symmetric if Z .Y X/ .Z Y / X D Z .X Y / .Z X / Y: (3.6) Both identities naturally generalise the associativity condition for algebras, and, by exchanging factors, right- and left symmetric products are in one-to-one correspondence. Moreover, as in the associative case, there is an associated Lie product which is defined by ŒX; Y WD X Y Y X: The associative subalgebra of elements X which is defined by equation (3.4) is called the associative kernel of the right-symmetric algebra. Example 3.1. Let .M; r/ be a flat affine manifold. As declared in (3.3), the flat torsion-free connection r induces an algebra product r on the Lie algebra Vect.M /


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of C 1 - vector fields on M . Note then that r is a compatible right-symmetric algebra product on the Lie algebra Vect.M /. The associative kernel of .Vect.M /; r / is finite dimensional, and it is precisely the subalgebra .aut.M; r/; r / of affine vector fields. Algebras satisfying (3.5) appeared and a play mayor role in the study of convex homogeneous cones [99]. Naturally finite dimensional LSAs over the real numbers play an important role in the study of left invariant flat affine structures on Lie groups, see Section 5. Left symmetric algebras also appear in several other mathematical and physical contexts, see [22] for a recent survey.

3.2 Development representation of affinities z ; r/ be the universal covering manifold of .M; r/. We put z Aut.M z ; r/ for Let .M z the group of covering transformations, and D h./ for the affine holonomy group of .M; r/. The development process provides a local representation of the group of affine transformations Aut.M; r/: We define

b but.M; r/ is a covering group of Aut.M; r/, and the development homoThe group A z ; ; z r/ D fˆ 2 Aut.M z ; r/ j ˆ z ˆ1 D g: z Aut.M; r/ D Aut.M

morphism (2.1) induces a homomorphism of Lie groups

b

h W Aut.M; r/ ! Aff.n/

b

(3.7)

b

into the affine group Aff.n/. Note that h has discrete kernel. Moreover, the image h.Aut.M; r// normalises , and h.Aut.M; r/0 / centralises . Representation of affine vector fields. We study now the tangent representation of the development homomorphism. Proposition 3.3. The development defines a a natural faithful associative algebra representation (3.8) hN W .aut.M; r/; r / ! aff.n/: Proof. We already remarked that aut.M; r/ forms an associative algebra with respect to r /. Via local affine coordinates, we can identify the tangent space Tp M with Rn . By the formula AX r Y D AAX Y D AX AY , which is deduced from (3.4), the map AXp Xp N (3.9) h W X 7! Xp D 0 0 is easily seen to be a faithful representation of .aut.M; r/; r / into the associative algebra of aff.n/ (where the associative algebra structure on aff.n/ is given by the 2 usual product of matrices).

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The development representation hN W aut.M; r/ ! aff.n/; as defined in (3.9), is associated to the local representation h of Aut.M; r/. In fact, on the subalgebra autc .M; r/, the development representation hN corresponds to the derivative of h. We further remark: Proposition 3.4. The representation hN identifies aut.M; r/ with the subalgebra aff.n/ of -invariant affine vector fields on An . Proof. If M is simply connected the map hN W aut.M; r/ ! aff.n/ is an isomorphism, since affine vector fields may be extended uniquely to all of M from any coordinate patch. For the general case, note that aut.M; r/ is isomorphic to the subalgebra z ; r/z of -invariant affine vector fields on M z . Since hN W aut.M z ; r/ ! aff.n/ aut.M z 2 is equivariant with respect to h W ! , aut.M; r/ is mapped onto aff.n/ .

3.3 Affinities of compact volume preserving affine manifolds Recall that a compact volume preserving affine manifold is conjectured to be complete. Now if .M; r/ is a compact complete affine manifold then the centraliser of the affine crystallographic group D hol.1 .M // in Aff.n/ is unipotent (see for example [46], or Section 9). Since the centraliser of hol.1 .M // is the development of Aut.M; r/0 , the latter group is a simply connected nilpotent Lie group, which is faithfully represented by unipotent matrices. In this section we show that the analogous facts do indeed hold for the affinities of a compact volume preserving affine manifold, without assuming its completeness. Let .aut.M; r/; r / denote the Lie algebra of affine vector fields on M with the associative algebra structure induced by r. We note:

b

Proposition 3.5. Let .M; r/ be a flat affine manifold with parallel volume, which is compact (or with finite volume). Then .aut.M; r/; r / is isomorphic to an associative algebra of nilpotent matrices. In particular, aut.M; r/ is a nilpotent Lie algebra. Proof. Define the divergence of X 2 Vect.M / relative to the parallel volume on M by the formula div X D LX : Since M is compact, by Green’s theorem (cf. [66, Appendix 6]) we have, Z div Xd D 0;

(3.10)

M

and, moreover, since the volume form is parallel, we also have (again according to [66, Appendix 6]), div X D trace AX .D trace r X /:

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In particular, if X 2 aut.M; r/ then div X is a constant function, which implies div X D 0. Thus, via the linear representation X 7! Xp of .aut.M; r/; r / constructed in Proposition 3.3, aut.M; r/ is isomorphic to an associative algebra of linear operators of trace zero. By Lemma 6.13, Xp is a nilpotent operator, for all X 2 aut.M; r/. In the finite volume case, we use an analogous argument, noting that (3.10) holds 2 for an affine vector field X on finite volume M as well. Corollary 3.6. Let .M; r/ be a flat affine manifold with parallel volume which is compact (or with finite volume). Then the centraliser of the affine holonomy group of M is a connected unipotent group, which is isomorphic to Aut.M; r/0 under the development.

b

N defined in (3.8), Proof. In appropriately chosen coordinates, the homomorphism h, corresponds to the differential of the holonomy representation h W Aut.M; r/ ! Aff.n/. By Proposition 3.3, the tangent Lie algebra of the holonomy image is the associative algebra aff.n/ , which is a subalgebra in aff.n/. By Proposition 3.5, the elements of aff.n/ are nilpotent. If A 2 ZAff.n/ ./ is an affine transformation which centralises then A EnC1 2 aff.n/ . Thus, A is a unipotent affine transformation. It follows that ZAff.n/ ./ is an algebraic subgroup of Aff.n/ consisting of unipotent elements. In particular, ZAff.n/ ./ must be connected, and simply connected. It follows that the development homomorphism (3.7),

b

b

h W Aut.M; r/0 ! ZAff.n/ ./ is a covering map onto, and, in fact, it is an isomorphism.

2

The following is actually an application of Theorem 9.3. Corollary 3.7. Let .M; r/ be a flat affine manifold with parallel volume, which is compact. If an affine vector field on M has a zero, it must be trivial. In particular, dim Aut.M; r/ n D dim M . Proof. In fact, by Theorem 9.3, the Zariski closure of acts transitively on An . This implies that the unipotent group ZAff.n/ ./ acts without fixed points on An . 2 In particular, any locally faithful connection preserving action on a compact volume preserving affine manifold is locally free (the stabiliser of any point is discrete).

3.4 Lie groups of isometries The group of isometries Isom.M; h ; i/ of a pseudo-Riemannian manifold is a finitedimensional Lie group, and it is a subgroup of the group of affine (connection preserving) transformations for the Levi-Civita connection of h ; i. If M is compact and h ; i is Riemannian then Isom.M; h ; i/ is compact. The identity components of the

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automorphism groups of compact Lorentzian manifolds are known to be of rather restricted type. (Compare [2], [110], [111]). In fact, by [111], non-compact semisimple factors in the isometry group of a compact Lorentzian manifold are locally isomorphic to SL.2; R/. Moreover, a connected solvable subgroup of isometries of a compact Lorentzian manifold is a product of an abelian group with a 2-step nilpotent group of Heisenberg type (see [110]). In the presence of a compatible flat torsion-free connection, particular strong restrictions hold for the group of isometries of a compact pseudo-Riemannian manifold .M; h ; i/. Proposition 3.5 already implies that the identity component of Isom.M; h ; i/ is a nilpotent Lie group. Here we show, more specifically, that, in case .M; h ; i/ is a compact flat pseudo-Riemannian manifold, Isom.M; h ; i/ is a two-step nilpotent group of rather restricted type. If .M; h ; i/ is a compact flat Lorentzian manifold then the identity component Isom.M; h ; i/0 is abelian and consists of translations only. 3.4.1 Killing vector fields. Let .M; h ; i/ be a pseudo-Riemannian manifold and r the Levi-Civita connection for h ; i. A vector field X on M is called a Killing vector field if its flow preserves the metric h ; i. Equivalently, we have LX h ; i D 0. Thus, X is Killing if and only if LX hU; V i D hŒX; U ; V i C hU; ŒX; V i;

(3.11)

for all vector fields U; V on M . Since h ; i is parallel, (3.11) holds if and only if AX is skew with respect to h ; i. Namely, (3.11) is equivalent to hrU X; V i C hU; rV X i D 0:

(3.12)

Let o.M; h ; i/ denote the Lie algebra of Killing vector fields for .M; h ; i/. Since Killing vector fields preserve the Levi-Civita connection, every Killing vector field is affine. We have the inclusion o.M; h ; i/ aut.M; r/: Lemma 3.8. Let X; Y 2 o.M; h ; i/ be Killing vector fields, where Y centralises X. Then (1) hrY X; Xi D 0. (2) hrX X; Y i D 0. Proof. Since Y commutes with X , rX Y D rY X . Because Y is Killing, AY is skew. Hence, hrY X; Xi D hrX Y; Xi D hX; rX Y i: Thus (1) follows. We note next that LXhY; X i D 0, since ŒX; Y D 0. Since h ; i is parallel, this implies 0 D hrX Y; Xi C hY; rX Xi D hrY X; X i C hY; rX X i: By (1), hrY X; X i D 0, and, hence, (2) holds.

2


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Proposition 3.9. Let X be a Killing vector field whose flow centralises a group G of isometries of .M; h ; i/, which has an open orbit on M . Then (1) rX X D 0. If .M; h ; i/ is also flat then: (2) AX AX D 0. Proof. The Killling fields Y corresponding to the action of G span the tangent spaces on an open subset of M . Thus, rX X D 0 is a consequence of Lemma 3.8. If .M; h ; i/ is flat then AX is parallel. Therefore, AX AX Y D AX rY X D rY AX X D rY rX X D 0: Hence, AX AX D 0.

2

We deduce a few consequences: Proposition 3.10. Under the assumptions of Proposition 3.9, the following holds for all Killing vector fields X , X 0 , X 00 , which are commuting with G: (1) ŒX 0 ; X D 2AX 0 X D 2AX X 0 . If .M; h ; i/ is also flat then: (2) AX 0 AX 00 X D AX 00 AX 0 X . (3) ŒŒX 0 ; X 00 ; X D 0. (4) AŒX;X 0 D ŒAX ; AX 0 D 2AX AX 0 . (5) AX AX 0 AX 00 D 0. Proof. Note that X C X 0 is Killing and commutes with G. Therefore, by (1) of Proposition 3.9 , rX CX 0 X C X 0 D rX 0 X C rX X 0 D 0. It follows that ŒX; X 0 D rX X 0 rX 0 X D 2AX X 0 . Therefore (1) holds. Next (using (1)) we note that AX 0 AX 00 X D AX 0 AX X 00 . Since AX 0 is parallel, AX 0 AX X 00 D AAX 0 X X 00 . We remark that the Killing vector fields centralising G form a Lie algebra with respect to the bracket of vector fields. Thus, by (1) AX 0 X is a Killing vector field, and also centralises G. Therefore, AX 0 AX 00 X D AX 00 AX 0 X . Thus, (2) holds. Now ŒŒX 0 ; X 00 ; X D 2AŒX 0 ;X 00 X D 2.AX 0 AX 00 X AX 00 AX 0 X / D 0 follows. Thus, (3) holds. Using polarisation, (2) of Proposition 3.9 implies that AX AX 0 D AX 0 AX . Hence, AŒX;X 0 D ŒAX ; AX 0 D 2AX AX 0 . Using these facts, we can compute, AX AX 0 AX 00 D 2 .AX AX 0 /AX 00 D AX 00 AX AX 0 D AX AX 00 AX 0 D AX AX 0 AX 00 . Corollary 3.11. Let .M; h ; i/ be a flat pseudo-Riemannian manifold which admits a group G of isometries, which has an open orbit on M . Then:

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(1) The Lie algebra o.M; h ; i/G of Killing vector fields on .M; h ; i/ which centralise G forms a subalgebra of the associative algebra of affine vector fields .aut.M; r/; r /. (2) The Lie algebra o.M; h ; i/G is (at most) two step nilpotent. The result implies that the centraliser of a prehomogeneous group of isometries on Es is a unipotent group, see Corollary 6.25. The main applications concern the automorphism groups of compact flat pseudo-Riemannian manifolds (see Section 3.4.2 below) and the holonomy groups of homogeneous flat pseudo-Riemannian manifolds (see Section 4.3). 3.4.2 Isometries of compact flat pseudo-Riemannian manifolds. The following result gives a rough description of the possible connected groups of isometries of compact flat pseudo-Riemannian manifolds. It is another consequence of Theorem 9.3: Theorem 3.12. Let .M; h ; i/ be a compact flat pseudo-Riemannian manifold, and r the Levi-Civita connection. Let X 2 o.M; h ; i/ be a Killing vector field. Then the following hold: (1) If X has a zero then X D 0. (2) AX AX D 0. (3) The Lie algebra o.M; h ; i/ of Killing vector fields on .M; h ; i/ forms a subalgebra of the associative algebra of affine vector fields .aut.M; r/; r /. (4) The Lie algebra o.M; h ; i/ is (at most) two step nilpotent. Proof. Let be the holonomy group of .M; h ; i/. By Theorem 9.3, the Zariski closure A./ of the holonomy group acts transitively on An . This transitive group of isometries commutes with the development of the Killing vector fields on M . Therefore, X cannot have a zero. Moreover, Proposition 3.9 and Proposition 3.10 imply the next three claims. 2 Remark. The development homomorphism (3.8), hN W aut.M; r/ ! aff.n/, maps the subalgebra o.M; h ; i/ to an associative subalgebra of aff.n/, which is contained N / D X , X 2 o.M; h ; i/, satisfy in o.h ; is /. By the theorem, the elements h.X p 2 the conditions Xp … Im AXp and AXp D 0. Subalgebras of linear maps satisfying both conditions appear in the context of complete homogeneous pseudo-Riemannian manifolds, as well. (See Section 4.3.) Abelian algebras of this type have been further investigated in [107], [108]. For the construction of non-abelian examples, see Section 5.3.2. On a compact Riemannian manifold with non-positive Ricci curvature, every Killing vector field must be parallel, see [66]. This also holds in the flat Lorentzian case:


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Corollary 3.13. Let .M; h ; i/ be a compact flat pseudo-Riemannian manifold. If h ; i is Riemannian or h ; i has Lorentzian signature then every Killing vector field on M is parallel. Proof. Let X be Killing. By Proposition 3.12, the linear operator AX develops to a two-step nilpotent element contained in o.h ; i/. In the Riemannian case o.h ; i/ has no non-zero nilpotent elements. For the Lorentzian case, note that every two-step nilpotent element of o.h ; i/ is zero. This implies AX D 0, in both cases. 2 In Section 5.3.2, we construct a compact flat manifold .M; h ; i/ of dimension six, with signature s.h ; i/ D .3; 3/, and six-dimensional non-abelian algebra o.M; h ; i/.

3.5 Lie groups of symplectic transformations Let ! be a non-degenerate 2-form on M which is closed. Then .M; !/ is called a symplectic manifold. A diffeomorphism of M which preserves ! is called a symplectomorphism of .M; !/. In general, the group of symplectomorphisms Diff ! .M / of a symplectic manifold is very large, and it is not a Lie group, even if M is compact. However, if M is compact, there do exist strong restrictions on the finite-dimensional Lie subgroups of Diff ! .M /. For example, Zwart and Boothby [112] proved that a compact symplectically homogeneous manifold of a solvable Lie group S is diffeomorphic to a torus T 2n . More generally, by Guan’s work [47], for any compact symplectic manifold .M; !/, every connected solvable Lie subgroup of Diff ! .M / is 2-step solvable, with a compact adjoint image. Moreover, any finite dimensional connected subgroup of Diff ! .M / is a semi-direct product of a compact group and a group S of the above type. Let .M; !/ be compact symplectic manifold, which admits a compatible flat affine connection r. We show below (cf. Theorem 3.18) that every Lie subgroup of Diff ! .M /, which also preserves r, is abelian. Remark. The above mentioned restrictions do not apply if M is non-compact. For example, there do exist plenty solvable Lie subgroups of Diff.Rn /, not 2-step solvable, which preserve the standard symplectic structure on R2n . Such examples may be constructed using solvable Lie groups with left-invariant symplectic structure (symplectic Lie groups). See Section 6.5.2, for a particular construction method for such groups, which also produces non-solvable examples. Further examples are discussed, for example, in [97]. 3.5.1 Symplectic vector fields. A vector field X on .M; !/ is called a symplectic vector field if its flow preserves !. Equivalently, X satisfies LX ! D 0, which means that LX !.U; V / D !.ŒX; U ; V / C !.U; ŒX; V /; for all vector fields U; V on M .

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Lemma 3.14. Let X; Y; Z be symplectic vector fields on .M; !/. Then !.X; ŒY; Z/ C !.Z; ŒX; Y / C !.Y; ŒZ; X / D 0:

(3.13)

Proof. Since d! D 0, for all vector fields X; Y; Z on M , the relation LX !.Y; Z/ C LZ !.X; Y / C LY !.Z; X / C !.X; ŒY; Z/ C !.Z; ŒX; Y / C !.Y; ŒZ; X / D 0

(3.14)

is satisfied. If X; Y; Z are symplectic, LX ! D LY ! D LZ ! D 0. It follows that LX !.Y; Z/ C LY !.Z; X / C LZ !.X; Y / D 2 .!.X; ŒY; Z/ C !.Z; ŒX; Y / C !.Y; ŒZ; X // : Substituting the right hand side in (3.14), the lemma follows.

2

Proposition 3.15. Let .M; !/ be a symplectic manifold which admits a Lie group G Diff ! .M /, which acts transitively on M . Then every connected Lie subgroup of Diff ! .M /, which centralizes G, is abelian. Proof. Let X; Y be vector fields on M , whose flows preserve !, and which centralise G. Let Z be a vector field tangent to G. Since X , Y , Z are symplectic, equation (3.13) holds. Since ŒX; Z D ŒY; Z D 0, it follows that !.Z; ŒX; Y / D 0: Since G acts transitively on M , the tangent spaces at every point x 2 M , are spanned by vectors Zx , where Z is tangent to G. Hence, ŒX; Y D 0. This proves the 2 proposition. The following special case of Proposition 3.15 is well known: Corollary 3.16. Let .G; !/ be a symplectic Lie group, where ! is biinvariant. Then G is abelian. Symplectic affine vector fields. Let .M; r; !/ be a flat affine manifold with parallel symplectic form !. As in the metric case, a vector field X is symplectic if and only if AX is skew with respect to !. An affine vector field X on .M; r/ which is also symplectic is called a symplectic affine vector field. The symplectic affine vector fields form a Lie subalgebra s.M; r; !/ of aut.M; r/. The following gives a useful analogue of Lemma 3.8 for the symplectic case: Lemma 3.17. Let X be a symplectic affine vector field on .M; r; !/, and let Y; Z be symplectic affine vector fields which commute with X. Then 2 !.AX AX Y; Z/ D !.rX X; ŒY; Z/:

(3.15)

Proof. The proof is a straightforward computation. See [10, proof of Theorem 9] for 2 a special case.

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3.5.2 Automorphisms of compact symplectic affine manifolds. On a symplectic affine manifold M the group of affine symplectic transformations forms a Lie subgroup of Diff ! .M /. If M is compact, we show that this group must be abelian. In fact, the following holds: Theorem 3.18. Let .M; r/ be a compact flat affine manifold with parallel symplectic structure !. Then the identity component of the group of symplectic affine transformations is an abelian group which develops to a unipotent subgroup of symplectic affine transformations of Rn . Proof. Let Aff.!n / be the holonomy group of M . The group of symplectic affine transformations Aut.M; r; !/0 develops onto a subgroup h.Aut.M; r; !/0 / of Aff.!n / (compare Section 3.2). This group centralises , and therefore also the Zariski closure A./ Aff.n/. The Zariski closure A./ is a group of symplectic affine transformation which acts transitively on affine space A2n (by Theorem 9.3). Thus, by Proposition 3.15, h.Aut.M; r; !/0 / must be abelian. Hence, so is Aut.M; r; !/0 . 2

b

b

b

It follows that a compact affine manifold which is homogeneous under the group of symplectic affine transformations must be diffeomorphic to a torus: Corollary 3.19. Let .M; r/ be a compact flat affine manifold with parallel symplectic structure which is homogeneous for the group of symplectic affine transformations. Then M is diffeomorphic to a torus. Let s.M; r; !/ denote the Lie algebra of symplectic affine vector fields on M . As shown above, if M is compact then s.M; r; !/ is abelian. We further note: Proposition 3.20. Let .M; r/ be a compact flat affine manifold with parallel symplectic structure !. Then the following hold: (1) The Lie algebra of symplectic affine vector fields s.M; r; !/ is abelian, and dim s.M; r; !/ dim M . (2) For all X; Y 2 s.M; r; !/, AX AX Y D 0. Proof. The Lie algebra s.M; r; !/ is abelian, by Theorem 3.18. Let Xx be the development image of X and Y . Let Z 2 s.An ; !n / be a symplectic affine vector field on An centralising Xx and Yx . By Lemma 3.17, we obtain !n .AXx AXx Yx ; Z/ D 0. Since A./ acts transitively, the vector fields Z commuting with Yx and Xx span all tangent 2 spaces. Therefore, AX AX Y D 0. If .M; r; !/ is homogeneous then s.M; r; !/ forms a subalgebra of the algebra of affine vector fields: Proposition 3.21. If M is homogeneous under the group of symplectic affine transformations then the following hold:

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(1) For X; Y 2 s.M; r; !/, AX AY D 0. (2) s.M; r; !/ forms a subalgebra of the associative algebra .aut.M; r; r /. Proof. Since M is homogeneous, the vector fields Y 2 s.M; r; !/ span all tangent spaces. Therefore, AX AX D 0, for all X 2 s.M; r; !/. Since s.M; r; !/ is abelian, also AX AY D 0. 2 Remark. The structure and classification of algebras which are appearing in Proposition 3.21 have been investigated in [10].

4 Homogeneous model spaces Let G be a Lie group, and H G a closed subgroup. Then X D G=H is a homogeneous space for G. If X admits a flat affine connection r, which is G-invariant, then .X; r/ is called a flat affine homogeneous space. Let G be a discrete subgroup, such that the quotient space M D nG=H is a manifold. Since r is invariant by , M inherits a flat connection r, such that the natural map .X; r/ ! .M; r/ is an affine covering map. Then M is called a Clifford– Klein form for X D G=H , and the (simply connected) covering space .X; r/ is called a global model for .M; r/. The Auslander and Markus conjectures suggest a strong relationship between the geometry of a flat affine homogeneous space .X; r/, and the existence of compact Clifford–Klein forms for X . The following result a consequence of Theorem 9.3: Theorem 4.1. Let .X D G=H; r/ be a flat affine homogeneous space. If X has a compact Clifford–Klein form .M; r/ with parallel volume then .X; r/ is complete. In particular, if the universal covering .X; r/ of a compact volume preserving flat affine manifold .M; r/ is homogeneous under its group of affine transformations, then .M; r/ is complete. In general, it seems difficult to understand the precise conditions on an arbitrary flat affine homogeneous space .X; r/, which ensure the completeness of .X; r/. For flat affine Lie groups .G; r/, a simple characterisation is known (see Section 5). An approach to the general problem will be introduced in Section 7 and Section 8. In particular, we will prove the following result: Theorem 4.2. Let .M; r/ be a homogeneous flat affine manifold for a nilpotent Lie group G. Then M is complete if and only if G preserves a parallel volume form on M .


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The theorem is a direct consequence of a corresponding result for affine homogeneous domains of nilpotent groups (see Section 7.3). In what follows, we will then mainly investigate compact homogeneous affine manifolds. Before doing so, we discuss the development map of a homogeneous space and indicate the proofs of Theorem 4.1 and Theorem 4.2 in Section 4.1. Next we provide a structure theorem for volume preserving compact homogeneous affine manifolds in Section 4.2.1. This result implies a direct proof (independent of Theorem 4.1) that every compact volume preserving homogeneous affine manifold is complete. As another consequence of the structure theorem, we derive a classification theorem for compact flat pseudo-Riemannian homogeneous manifolds. We also show that a compact symplectic homogeneous affine manifold is diffeomorphic to a torus. In Section 4.3 we further discuss the properties of the holonomy groups of homogeneous manifolds.

4.1 The development map of a homogeneous space Recall that a group action on An is called prehomogeneous if it has a Zariski-dense open orbit. A subset of An is semi-algebraic if it is defined by polynomial equations and inequalities. The following is a basic observation: Proposition 4.3. Let .M; r/ be a homogeneous flat affine manifold. Then the development map is a covering map, and its development image is a semi-algebraic subset in An . Let G be a group which acts transitively on .M; r/. Then its universal covering group acts prehomogeneously on An by the development homomorphism. z Aff.M z / be Proof. Let G be a Lie group which acts transitively on .M; r/, and G z Aff.n/ a covering group, which lifts the action of G. Then the development h.G/ z /. In fact, it follows that acts transitively on the development image U D D.M n U A is an affinely homogeneous domain. By Proposition 6.8, every affinely homogeneous domain U is a semi-algebraic subset of Rn . The development map z ! U is a covering map, because it identifies with a covering of homogeneous DW M z D G= z H z ! h.G/=L z spaces M D U , which is induced by the locally faithful z homomorphism h W G ! Aff.n/. 2 Proof of Theorem 4.1. Since X has a compact Clifford–Klein form with parallel volume, its development image is a homogeneous affine domain U , which is divisible by volume preserving affine transformations. Hence, by Corollary 9.4, U D An . Since D is a covering, .X; r/ must be complete. 2 Proof of Theorem 4.2. Assume M has a transitive volume preserving action of some nilpotent Lie group. Then the universal covering X of M is a homogeneous space for a nilpotent Lie group N of affine transformations, which preserves a parallel volume form. The development of N preserves the parallel volume on An , and acts prehomogeneously. By Corollary 7.8, the development of N acts transitively on An ,

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and therefore D.X/ D An . Since D is a covering, it must be a diffeomorphism. Therefore, M is complete. 2 The converse is a direct consequence of Corollary 7.8.

4.2 Compact homogeneous affine manifolds Robert Hermann [52] observed that a compact pseudo-Riemannian manifold which admits a transitive group of isometries must be complete (see [75] for a complete proof). 2

Example 4.1. Consider GL.n; R/ as an open subset in Rn . Then the natural affine structure on GL.n; R/ is invariant under left- and right-translations on the group. As a consequence of a result of Borel [16], there exist cocompact lattices GL.n; R/. The compact manifold M D GL.n; R/= inherits an affine flat structure, and GL.n; R/ acts as a transitive group of affine transformations on .M; r/. Thus .M; r/ is a compact affinely homogeneous manifold, which is not complete. Markus’ conjecture asserts that an (orientable) compact flat affine manifold is complete if and only if it admits a parallel volume form. The conjecture is known to hold for compact affinely homogeneous affine manifolds. (This is proved in [42]. See Section 9.2 for further discussion.) We obtain an independent proof in Corollary 4.5 below. 4.2.1 Compact homogeneous manifolds with parallel volume. We have the following structure result for compact homogeneous flat affine manifold with parallel volume: Theorem 4.4. Let .M; r/ be a compact homogeneous flat affine manifold with volume preserving affine structure. Then the following hold: (1) .M; r/ is affinely diffeomorphic to a quotient space of a simply connected nilpotent affine Lie group .N; r/ with biinvariant affine structure r by a discrete subgroup N . (2) .M; r/ is geodesically complete. Furthermore, every parallel tensor field on .M; r/, which is preserved by the action of N , pulls back to a biinvariant (and parallel) tensor field on N .

b

Proof. By Proposition 3.5, the group of affinities Aut.M; r/0 develops to a connected unipotent Lie group. Moreover, M is a compact homogeneous space of the simply connected nilpotent Lie group N D Aut.M; r/0 . Since N acts almost effectively on M , if follows that dim N D dim M (see [72]). Therefore, M D N= is a quotient of N by a uniform discrete subgroup N . Pull back r to obtain a left-invariant affine

b


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connection r on N , which is right-invariant under the lattice . Since is Zariskidense in the adjoint representation (see [87]), r is a biinvariant connection on N . (In fact, r is determined by the associative product structure r on the Lie algebra aut.M; r/ D n.) The volume preserving left-invariant connection r on the nilpotent 2 Lie group N is complete. (See Section 5.2.2.) Therefore, M is complete. Remark. By a result of Mostow [80] (see also, [87], [96]), a finite volume homogeneous space of a solvable Lie group is compact. Thus, the above proof also shows that a finite volume homogeneous affine manifold is compact and geodesically complete. The Markus conjecture holds for compact homogeneous affine manifolds: Corollary 4.5. Let .M; r/ be a compact homogeneous flat affine manifold. Then .M; r/ is complete if and only if .M; r/ has parallel volume. Proof. If .M; r has parallel volume Theorem 4.4 applies. Assume now that .M; r/ is complete. Then, as already remarked in the beginning of Section 3.3, Aut.M; r/0 is a nilpotent Lie group, which develops to a unipotent subgroup of Aff.n/. Therefore, as in the proof of Theorem 4.4, M D N= , where N is a nilpotent Lie group with complete left invariant affine connection r, such that .N; r/ ! .M; r/ is an affine covering. Since N is nilpotent and r is complete, the left-multiplications on N are volume preserving affine transformations (see Section 5.2.2). Since N is nilpotent, also the right-multiplications preserve the left-invariant parallel volume on N . Since hol.1 .M // D is contained in the rightmultiplications on N , preserves the parallel volume on N . This shows that .M; r/ 2 has parallel volume. Note also that Theorem 4.4 strengthens the completeness result in [52], [75] considerably, since it is only required that the group of affinities Aut.M; rh ; i / acts transitively on M , to ensure completeness. 4.2.2 Pseudo-Riemannian examples. Let .M; h ; i/ be a homogeneous flat pseudoRiemannian manifold, which is compact. Then, by the above, .M; rh ; i / is complete. Moreover, the following structure theorem holds: Theorem 4.6. Let M be a compact (or finite volume) homogeneous flat pseudoRiemannian manifold. Then M is isometric to a quotient of a flat pseudo-Riemannian Lie group N with biinvariant metric. Proof. In fact, by Theorem 4.4, .M; h ; i/ is isometric to a quotient of a nilpotent flat pseudo-Riemannian Lie group .N; h ; i/ with biinvariant metric h ; i, such that the 2 natural (orbit-) map N ! M D N= is a pseudo-Riemannian covering. Theorem 4.6 implies the following strong geometric rigidity property for compact homogeneous flat pseudo-Riemannian manifolds:

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Corollary 4.7. Let M and M 0 be compact homogeneous flat pseudo-Riemannian manifolds with isomorphic fundamental groups. Then M and M 0 are affinely diffeomorphic. Proof. As above, we write M D N= and M 0 D N 0 = 0 , where , 0 are lattices in the simply connected nilpotent Lie groups N and N 0 , respectively. By Malcevrigidity, cf. [72], every isomorphism W ! 0 extends to an isomorphism of Lie groups ˆ W N ! N 0 . We contend that ˆ is an affine isomorphism of metric Lie groups .N; h ; i/ ! .N 0 ; h ; i0 /. In fact, both h ; i and h ; i0 are biinvariant metrics, and therefore (see [83, Proposition 11.9]), the Levi-Civita connection r (respectively r 0 ) is the canonical torsion-free connection on the Lie group N (respectively N 0 ). That is, for left-invariant vector fields X and Y , rX Y D 12 ŒX; Y . In particular, it follows that the isomorphism of Lie groups ˆ is affine with respect to r and r 0 . Therefore, x W M ! M 0 is affine. the induced diffeomorphism ˆ 2 In particular, the fundamental group 1 .M / determines the Levi-Civita connection on a compact homogeneous flat pseudo-Riemannian manifold M up to affine equivalence. Example 4.2. If 1 .M / is abelian then 1 .M / Š Zn , and M D T n is diffeomorphic to a torus. Moreover, by Corollary 4.7, every homogeneous flat manifold .T n ; h ; i/ is isometric to a quotient of Es by a lattice of translations. The affine structure of .T n ; h ; i/ is the uniquely determined translational structure on T n . Remark. In fact, every flat metric on the torus T n is homogeneous, and thus isometric to a quotient of Es by a lattice of translations. To the contrary, the set of affine equivalence classes of volume preserving complete affine structures on T n is rather large. See [9], [11] for further discussion. A nilpotent Lie group N , which acts transitively by isometries on a compact flat pseudo-Riemannian manifold M is of rather restricted type: Lemma 4.8. Let M D N= be a compact homogeneous flat pseudo-Riemannian manifold. Then N is a two-step nilpotent Lie group. Proof. Remark (see [83, Proposition 11.9]), that the curvature of the Levi-Civita connection r of the biinvariant flat metric h ; i on N is computed as, Rr .X; Y /Z D 1 ŒX; ŒY; Z, for all left-invariant vector fields X; Y; Z. Since h ; i is flat, this implies 4 2 ŒX; ŒY; Z D 0. It follows that N is two-step nilpotent. Note also that every quotient space M D N= of a flat pseudo-Riemannian Lie group N with biinvariant metric by a discrete subgroup is a homogeneous flat pseudo-Riemannian manifold. In fact, left-multiplication in N induces a transitive isometric action of N on M D N= .


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In Section 5.3.1, we will discuss the structure of flat pseudo-Riemannian Lie groups with biinvariant metric in detail. In particular, see Corollary 5.14, we explicitly construct a family of such Lie groups, which gives rise to an interesting class of examples of compact (complete) homogeneous flat pseudo-Riemannian manifolds with nonabelian fundamental group. This is summarised in Corollary 4.10 below. First, we state a few general consequences concerning the classification of compact homogeneous flat pseudo-Riemannian manifolds: Corollary 4.9. Let M be a compact homogeneous flat pseudo-Riemannian manifold. Then the following hold: (1) If dim M 5 then M is isometric to a translation torus (that is, M is a quotient of Es , by a lattice of translations). (2) The fundamental group of M is two-step nilpotent. The first examples of compact homogeneous flat pseudo-Riemannian manifolds with non-abelian fundamental group arise in dimension six. More generally, there is the following interesting class of examples: Corollary 4.10. Let N= be a compact two-step nilmanifold (where is a lattice in the two-step nilpotent Lie group N , n D dim N ). Then there exists a flat rank n torus bundle T n ! M ! N= such that M admits a homogeneous flat pseudo-Riemannian metric of signature s D .n; n/ with totally geodesic and isotropic fibers T n . Proof. Let A W N ! GL.n / denote the coadjoint representation of N on the dual n of its Lie algebra n. By Corollary 5.3.1, the corresponding semidirect product N ÌA n admits a flat biinvariant pseudo-Riemannian metric of signature .n; n/, n D dim M . Now is a lattice in N , and under the representation A, maps to a finitely generated subgroup of the group of unipotent upper triangular matrices with rational coefficients relative a basis of n. In particular, A./ preserves a lattice ƒ Š Zn in n . The semidirect product 0 D ÌA ƒ is a lattice in N ÌA n , and the manifold M D N ÌA n = 0 has the required properties. 2 In particular, let be a torsion-free finitely generated two-step nilpotent group. Then arises as a quotient D 0 =ƒ, where 0 is the holonomy of a compact homogeneous flat pseudo-Riemannian manifold, and ƒ is an abelian normal subgroup of 0 , with rank ƒ D rank . Example 4.3. In dimension n D 6, there exist up to affine equivalence two classes of compact homogeneous flat pseudo-Riemannian manifolds. The six-dimensional translation torus T 6 , and torus bundles T 3 ! M6; ! H3 = , where is a lattice in the three-dimensional Heisenberg group H3 .

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4.2.3 Symplectic examples. As noted before, and contrasting the pseudo-Riemannian situation, only tori admit the structure of a homogeneous symplectic flat affine manifold. However, such a structure is, in general, not affinely equivalent to a translation torus: Corollary 4.11. Let M be a compact (or finite volume) homogeneous symplectic flat affine manifold. Then M is a compact abelian flat affine symplectic Lie group. In particular, M is diffeomorphic to a torus. Proof. By Theorem 4.4, .M; r/ is a quotient of an affine symplectic Lie group N with biinvariant symplectic structure. By Proposition 3.16, N is abelian. (To show that M is a torus, we might have applied the classification results for connected groups of 2 symmetries of compact symplectic manifolds given in [112], [47], as well.) Note that the space of affine equivalence classes of flat symplectic connections on the torus T 2n is very large. In fact, as is shown in [10], the set of simply connected abelian flat affine symplectic Lie groups forms an algebraic variety of (real) dimension n3 . For the precise classification result and further geometric properties of abelian flat affine symplectic Lie groups, see [10].

4.3 Holonomy of homogeneous affine manifolds The fundamental group of a homogeneous flat Riemannian manifold is abelian, and its holonomy acts by translations. Starting in dimension five, there do exist complete flat homogeneous pseudo-Riemannian manifolds with abelian but non-translational holonomy group, see [102]. As Example 4.1 shows, the holonomy of a (compact) homogeneous affine manifold can be far away from being abelian or nilpotent. This example is however a non-complete manifold. Theorem 4.12. Let Aff.n/ be the holonomy group of a complete affinely homogeneous affine manifold. Then is a unipotent subgroup of Aff.n/. Proof. Since M is homogeneous, the development G of Aff.M; r/ acts transitively on the development image of M , which is An . Moreover, centralises G. By Proposition 6.15, the centraliser of the transitive group G is a connected unipotent 2 group. In particular, is unipotent. In particular: Corollary 4.13. The fundamental group of a complete affinely homogeneous affine manifold is nilpotent.


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4.3.1 Holonomy of homogenous flat pseudo-Riemannian manifolds. The above result can be strengthened considerably for homogeneous flat pseudo-Riemannian manifolds even without the assumption of completeness. Proposition 4.14. Let E.s/ be the holonomy group of a homogeneous flat pseudoRiemannian manifold. Then is a unipotent group, and is nilpotent of nilpotency class at most two. Proof. Let .M; h ; i/ be a homogeneous flat pseudo-Riemannian manifold. The development image of M is a pseudo-Riemannian domain U in Es , which is homogeneous with respect to the development G of Isom.M; h ; i/. The holonomy group preserves the domain U and is contained in the centraliser of G in the full isometry group E.s/. By Corollary 6.25, the centraliser is a connected unipotent group of nilpotency class 2 at most two. In particular, has this property. Remark. The proposition leads to a classification theory of flat homogeneous pseudoRiemannian manifolds, which are complete. Wolf studied the classification of complete homogeneous pseudo-Riemannian manifolds with abelian holonomy , discovering interesting phenomena, see [102], [107], [108]. In Section 4.2.2, we proved the existence of a six-dimensional compact (complete) homogeneous pseudo-Riemannian manifold with non-abelian holonomy . Moreover, all examples constructed in Section 4.2.2 relate to certain two-step nilpotent Lie groups. It remains to join these results to a full structure theory for complete flat homogeneous pseudo-Riemannian manifolds. The determination of all homogeneous flat pseudo-Riemannian manifolds (that is, including non-complete examples) seems not at hand, because of the difficulties associated with the classification of homogeneous pseudo-Riemannian domains. (See Section 6.6.) We mention further that Proposition 4.14 admits an analogous version for symplectically homogeneous affine manifolds. This is a consequence of Proposition 6.28 below: Proposition 4.15. Let Aff.!n / be the holonomy group of a symplectically homogeneous affine manifold with parallel symplectic form. Then is an abelian group of unipotent symplectic affine transformations.

5 Flat affine Lie groups We consider a particular tractable class of homogeneous spaces, namely those which admit a simply transitive group of equivalences. Many phenomena can be illustrated, and in many cases examples constructed from flat affine Lie groups serve as the basic building blocks of the theory.

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5.1 Left-invariant geometry on Lie groups Let G be a Lie group. Recall that a geometric structure on G is called left-invariant if it is preserved by all left-multiplication maps lg W G ! G, g 2 G. A connection r on G is called a left-invariant connection if, for all g 2 G, lg is a connection preserving diffeomorphism. A pseudo-Riemannian metric is called left-invariant if all left-multiplications are isometries. More generally, if G has the structure of an .X; A/manifold then the structure on G is called left-invariant if all left multiplications are .X; A/-equivalences. Definition 5.1. Let G be a Lie group and r a connection on G. If r is flat affine and left-invariant, then .G; r/ is called a flat affine Lie group. Moreover, r is called a flat left-invariant connection of type A, where A Aff.n/, if .G; r/ has a compatible left-invariant locally homogeneous structure of type .An ; A/. In the latter situation we shall call the pair .G; r/ a flat affine Lie group of type A. If h ; i is a flat left-invariant metric, then .G; h ; i/ is called a flat pseudo-Riemannian Lie group. Example 5.1. Let rh ; i be the Levi-Civita connection of a flat pseudo-Riemannian Lie group. Then .G; rh ; i / is a flat affine Lie group of type E.s/. 5.1.1 Left-symmetric algebras. There is a one-to-one correspondence of left-invariant flat affine connections r on the simply connected group G, with left-symmetric algebra structures on the Lie algebra g of left-invariant vector fields on G. In fact, if X; Y are left-invariant then rX Y D Y r X is a left-invariant vector field. Hence r defines a left-symmetric product on g. (Compare Section 3.1.2.) The flat connection r is a biinvariant connection if and only if the induced product is associative. Example 5.2. Let N be a two-step nilpotent Lie group. The Lie algebra of leftinvariant vector fields n thus satisfies Œn; Œn; n D f0g. It follows that the canonical torsion-free connection on N , defined by 1 ŒX; Y ; X; Y 2 n; 2 is flat. Therefore, N is naturally a flat affine Lie group with biinvariant flat connection r. The bilinear product X Y WD rX Y defines an associative algebra structure on n. (See [91], [92] for more constructions of left-invariant flat connections on nilpotent Lie groups.) rX Y D

5.2 The development map of a flat affine Lie group Let W G ! Aff.n/ be a homomorphism. Recall that is called an étale affine representation (see Definition 6.3) if there exists x 2 Rn such that the orbit map at x, ox W G ! An ;

g 7! .g/ x;


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is a local diffeomorphism onto an open subset U of An . The representation is simply transitive if ox is a diffeomorphism for one (and, hence, for all) x 2 An . Example 5.3. Let W G ! A be an étale affine representation, and x 2 An such that ox W G ! An is a local diffeomorphism. By pull back of the standard flat connection on An along ox , we obtain a flat left invariant connection of type A on G. Moreover, every flat left invariant connection of type A arises in this way: Proposition 5.2. Let G be a simply connected Lie group, and r a torsion-free flat connection on G. Then the following are equivalent: • r is a left-invariant connection of type A. • There exists an étale affine representation W G ! A, and x 2 Rn , such that r and its compatible .An ; A/-structure are a pullback along the orbit map ox . Moreover, the connection r is complete if and only if is simply transitive. Proof. Let D W G ! An be any development map for the compatible left-invariant .An ; A/-structure. Let h denote the holonomy homomorphism with respect to D. Since the locally homogeneous structure is compatible with r, the map D is also an affine map .G; r/ ! An . Since the maps lg W G ! G are equivalences for the locally homogeneous .An ; A/-structure on G, .g/ D h.lg / defines a representation W G ! A. We put x D D.1/. Then, by (2.2), D.g/ D D.lg 1/ D .g/D.1/ D ox .g/: Thus, the orbit map for at x D D.1/ is the development map D. This shows that is an étale affine representation, and the left-invariant .An ; A/-structure on G is 2 obtained by pullback along ox . 5.2.1 The boundary of the development image. An étale affine representation W G ! A, which induces a flat left-invariant connection r of type A, as in Proposition 5.2, will be called a compatible étale affine representation for r. Corollary 5.3. The development map of a flat affine Lie group .G; r/ is a covering map onto its image, and it is an orbit-map for any compatible étale affine representation z r/. of its universal covering group .G; z ! An is a local diffeomorphism and an orbit-map, it is a covering Proof. Since D W G 2 map. Let Ad W G ! GL.g/ denote the adjoint representation of G. Furthermore, if .G; r/ is affine we put det r l W G ! R¤0 for the character which is obtained by the left-action of G on parallel volume forms.

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Theorem 5.4. Let .G; r/ be a flat affine Lie group. Then the following hold: (1) Either .G; r/ is complete or the development image of .G; r/ is a connected component of the complement of a non-empty hypersurface in Rn , n D dim G. In particular, the development image is a semialgebraic subset of Rn . (2) Assume that G is simply connected. Then the connection r is complete if and only if det Ad.g/1 detr lg D 1, for all g 2 G. z Proof. In fact, the development image of G is the orbit .G/x, for a compatible affine z z D étale representation for the induced flat structure on G. By Section 5.4, D.G/ z is a connected component of the open semi-algebraic subset Uı D fx 2 Rn j .G/x ı.x/ ¤ 0g Rn , where ı D ı./ is the relative invariant for . This implies (1). Moreover, since D is a covering map, r is complete if and only if D is surjective. If G is simply connected, the volume character of G satisfies det r lg D det r .g/, for z ! Aff.n/. Thus (2) follows from any compatible étale affine representation r W G Theorem 5.19 below. 2 5.2.2 Completeness and unimodularity of flat affine Lie groups. Let .G; r/ be a flat affine Lie group. If r is complete then, by Proposition 5.2, the universal covering Lie group of G acts simply transitively on affine space. This, more or less, reduces the study of complete affine Lie groups to a study of simply transitive representations on affine space. (See [5], for a seminal paper on simply transitive groups of affine transformations.) Corollary 5.5. Let .G; r/ be a flat affine Lie group. If r is complete then G is a solvable Lie group. z splits as a semi-direct product of a semisimple group H Proof. By Levi’s theorem G z can not have a reductive Levi-component H since and a solvable Lie group S . Now G z D S must be solvable. every affine representation of H has a fixed point. Hence, G 2 Recall that a Lie group G is called unimodular if det Ad.g/ D 1, for all g 2 G. If a flat Lie group .G; r/ has a left-invariant parallel volume form, we call it volume preserving. Corollary 5.6. Let .G; r/ be a volume-preserving flat affine Lie group. Then .G; r/ is complete if and only if G is unimodular. Corollary 5.7. Let .G; r/ be a flat affine Lie group, where G is unimodular. Then .G; r/ is complete if and only if .G; r/ is a volume-preserving flat affine Lie group. Example 5.4 (Semi-simple Lie groups are not affine.). Let S be a semisimple Lie group. Then S does not admit a flat left-invariant connection. For further generalisation, see Section 5.4.2.

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5.3 Flat pseudo-Riemannian Lie groups Corollary 5.8. Let .G; h ; i/ be a flat pseudo-Riemannian Lie group. Then .G; rh ; i / is complete if and only if G is unimodular. In particular, a unimodular flat pseudoRiemannian Lie group is solvable. Example 5.5. A flat pseudo-Riemannian Lie group .G; h ; i/, with biinvariant metric h ; i is complete and G is a two-step nilpotent Lie group. (See Section 5.3.1 below.) The above corollary already excludes a large class of Lie groups from carrying flat left-invariant metrics: 2

Example 5.6. Let G D GL.n; R/. Since G is an open subset of Rn , it has a natural structure of a flat affine Lie group. However, G does not admit a volume preserving flat left-invariant connection. In particular, G does not admit a flat pseudo-Riemannian metric. The latter argument holds for all reductive Lie groups G. Here we call Lie group reductive if it is an almost semi-direct product of its center and a semisimple Lie group. Such a group is necessarily unimodular. Hence: Corollary 5.9. Let G be reductive. Then G does not admit a volume preserving flat left-invariant connection. In particular, G does not admit a left-invariant flat pseudo-Riemannian metric. Remark. Similar restrictions, as stated here for the pseudo-Riemannian case, also hold for symplectic affine Lie groups. Concerning the existence of flat pseudo-Riemannian Lie groups .G; h ; i/, where G is not solvable, we observe: Example 5.7. Every flat Lorentzian Lie group G is solvable. This is a consequence of the classification of homogeneous domains in En1;1 . In fact, every such domain is diffeomorphic to Rn . (See Section 6.6, respectively [30].) Therefore, the universal z of G is diffeomorphic to Rn . It can not have a reductive part, since covering group G SL.2; R/ does not admit a faithful representation.

C

Not every flat pseudo-Riemannian Lie group is solvable, as the following example shows: Example 5.8. Let U D GL.2; R/ A4 . Then the dual tube domain T{ .U / E4;4 is a homogeneous pseudo-Riemannian domain, which has a simply transitive isometric action of the group GL.4; R/ Ë .R4 / . (Compare Section 6.5.2.)

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Question 4. Are flat pseudo-Riemannian Lie groups of signature s D .n 2; 2/ or .n 3; 3/ always solvable? Milnor [78] gives the classification and structure of simply connected flat Riemannian Lie groups. These are automatically complete. In [45] (see also [3] for the case of nilpotent Lie groups) the classification of all simply connected complete flat Lorentzian Lie groups is described. 5.3.1 Flat biinvariant metrics on Lie groups. A left-invariant metric h ; i on G is called biinvariant if the right-multiplications of G are isometries as well. If G is connected, h ; i is biinvariant if and only if the induced scalar product h ; i on the Lie algebra g is skew with respect to the Lie bracket. That is, h ; i is biinvariant iff, for all X; Y; Z 2 g, hŒX; Y ; Zi D hY; ŒX; Zi: (5.1) The following fact was already used in the proof of Corollary 4.7: Lemma 5.10. Let .G; h ; i/ be a pseudo-Riemannian Lie group with biinvariant metric. Then the Levi-Civita connection r D rh ; i is the canonical torsion-free connection on G, that is, rX Y D 12 ŒX; Y , for all X; Y 2 g. Moreover, .G; h ; i/ is flat if and only if G is two-step nilpotent. Corollary 5.11. Every flat pseudo-Riemannian Lie group with biinvariant metric is complete. 5.3.2 Construction of flat biinvariant metric Lie algebras. We describe now a method, which allows to construct examples of flat biinvariant metric Lie algebras .g; h ; i/. This, of course, implies the construction of flat Lie groups with biinvariant metric. Coadjoint extensions. Let n denote a Lie algebra, and n its dual vector space. By the coadjoint representation ad W n ! gl.n /, n is a module for n. Let ! 2 Z 2 .n; n / be a two-cocycle (cf. Appendix B). To ! there belongs a Lie algebra extension 0 ! n ! tn;! ! n ! 0; where n is an abelian ideal in tn;! . Put W D n ˚ n for the direct sum of vector spaces. Explicitly, the Lie algebra tn;! arises from the Lie product on W , which is declared by Œ. ; v/; . 0 ; v 0 / D .ad .v/ 0 ad .v 0 / C !.v; v 0 /; Œv; v 0 /: (See, for example, [71] for the general theory of Lie-algebra extensions.) Next we define a natural scalar product h ; io of signature s D .n; n/ on W , where n D dim n. With respect to h ; io , n and n are dual totally isotropic subspaces under


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h ; io . Moreover, if v 2 n and 2 n , we put hv; io D .v/: V We put V D n, and we define a three-form F! 2 2 V ˝ V by putting F! .u; v; w/ D h!.u; v/; wio : The following lemma is easily verified by direct calculation: Lemma 5.12. Let n be two-step nilpotent, and ! 2 Z 2 .n; n /. Then the following hold: (1) The canonical scalar product h ; io on tn;! satisfies (5.1) if and only if F! is V alternating (that is, if F! 2 3 V ). (2) The Lie algebra tn;! is two-step nilpotent, iff, for all u; v; w; z 2 n, F! .u; Œv; w; z/ D F! .v; w; Œu; z/:

(5.2)

Proof. Note, since n is two-step nilpotent, triple commutators in tn;! satisfy Œ. ; u/; Œ.; v/; . ; w/ D .adu !.v; w/ C !.u; Œv; w/; 0/: This proves (2).

2

Thus we have, in particular: Proposition 5.13. Let n be two-step nilpotent, and ! 2 Z 2 .n; n /. If F! is alternating and satisfies (5.2), then the metric Lie algebra .tn;! ; h ; io / is flat and h ; io is biinvariant. Proof. Since F! is alternating the metric h ; io is biinvariant. Since , (5.2) is satisfied, 2 tn;! is two-step nilpotent, and therefore, the biinvariant metric h ; io is flat. Split extensions. We may always choose ! D 0. In this case, the Lie algebra tn;! is the semi-direct product n ˚ad n of n with its representation space n . Thus, a particular interesting and rich class of examples arises as follows: Corollary 5.14. Let n be a two-step nilpotent Lie algebra. Then the metric Lie algebra .n ˚ad n; h ; io / is flat with biinvariant metric. The corollary shows in particular that the class of Lie algebras with flat biinvariant metrics is as rich as the class of two-step nilpotent Lie algebras. Twisted extensions with abelian base. If n D a is abelian, examples may be constructed using alternating three-forms on a.

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Example 5.9. Let n D a3 be the three-dimensional abelian Lie algebra. Let det 2 V 3 a3 be a non-degenerate alternating three-form, and define !det .X; Y / D det.X; Y; /: Then .ta3 ;!det ; h ; io / is a non-abelian biinvariant flat metric Lie algebra. The following result states that every flat metric Lie algebra .g; h ; i/ with biinvariant metric arises in this way: Theorem 5.15. Let .g; h ; i/ be a flat metric Lie algebra with biinvariant metric. Then V there exists an abelian Lie algebra a and an alternating three-form F! 2 3 a , and an abelian Lie algebra z such that .g; h ; i/ can be written as a direct product of metric Lie algebras .g; h ; i/ D .z; h ; i/ ˚ .ta;! ; h ; io /: Proof. We show that it is possible to choose n as an abelian Lie algebra, as in the previous example. Note that the commutator subalgebra Œg; g of .g; h ; i/ is an isotropic ideal in .g; h ; i/, and its orthogonal complement with respect to h ; i is the center z.g/ of g. This shows that we can choose an isotropic subspace a of .g; h ; i/, such that 2 there is a direct decomposition of subspaces g D z ˚ a ˚ Œg; g, where z z.g/. Example 5.10. Let .g; h ; i/ be a flat biinvariant metric Lie algebra of dimension six. Then g is abelian or .g; h ; i/ D .ta3 ;!det ; h ; io /. In particular, in the second case g, is a semidirect product of the three-dimensional Heisenberg algebra h3 with its coadjoint representation. Remark that a version of Theorem 5.15 already appeared in [104, Proposition 7.5]. A similar result is stated recently also in [26].

5.4 Étale affine representations Let W G ! Aff.n/ be an étale affine representation, and let g W g ! aff.Rn / be the corresponding Lie algebra representation. 5.4.1 The relative invariant for an étale affine representation. Choose a basis X1 ; : : : ; Xn for g, and x 2 Rn . Let us consider the linear map x ./ W Rn ! Rn , which is defined by n X ˛i g .Xi / :

x ./ W .˛1 ; : : : ; ˛n / 7! tx iD1

Here, tx is the derivative of the evaluation map at x, see (2.3). We easily see that x ./ is an isomorphism if and only if the orbit map ox W G ! Rn is non-singular, that is, if is étale at x.


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If A 2 Aff.Rn /, we put A .g/ D A .g/A1 . The following formulas are easily verified by direct calculation: Lemma 5.16. Let A 2 Aff.Rn /, with `.A/ 2 GL.n; R/ its linear part. Let g 2 G, and Ad.g/ W g ! g its adjoint action on the Lie algebra g of G. We put B D .g/. Then the following hold: det x .A / D .det `.A// det A1 x ./; det x .B / D .det Ad.g// det x ./: We define the polynomial function ı D ı./ W Rn ! R;

x 7! ı.x/ D det x ./

which (by the above lemma) satisfies the transformation rule ı..g/x/ D det Ad.g/1 det `..g//ı.x/:

(5.3)

Definition 5.17. We call ı D ı./ the relative invariant for the étale affine representation . Its character is the function D ./ W G ! R>0 , which is defined as .g/ D det Ad.g/1 det `..g// Note that an open orbit .G/x of an affine étale representation is contained in (and it is actually a connected component of) the open semi-algebraic subset Uı D fx 2 Rn j ı.x/ ¤ 0g Rn : By a theorem of Whitney [101], a real semi-algebraic set Uı has only finitely many connected components in the standard Euclidean topology on Rn . Thus we have proved the following result: Corollary 5.18. Let W G ! Aff.n/ be an affine étale representation. Then G has finitely many open orbits on Rn . Moreover, either G is transitive or each orbit is a connected component of the complement of a hypersurface of degree n. Example 5.11. The list of étale affine representations of Lie groups G with dim G D 2 is contained in Example 6.2. The relative invariant ı may also be interpreted in terms of growth of the rightinvariant volume on G, relative to a parallel volume on An , see Example 9.1. 5.4.2 The relative invariant of a simply transitive representation is constant. The following theorem is actually a direct consequence of Proposition 7.7. For transparency, we provide a direct proof. (For a different proof see [42], [50], [62] for the case of nilpotent groups.)

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Theorem 5.19. Let G be a Lie group, and W G ! Aff.n/ be an affine étale representation. Then the following conditions are equivalent: (1) is simply transitive. (2) The relative invariant ı./ is a non-zero constant function. (3) ./ 1. Proof. Clearly, if ı./ is non-vanishing, every orbit of G is open. Hence, there can be only one orbit, and .G/ is a simply transitive group of affine transformations. Conversely, assume that .G/ is simply transitive. Then the relative invariant ı./ is a nowhere vanishing polynomial function. Let G denote the Zariski closure of .G/ in Aff.Rn /. Then G is a transitive group of affine transformations on Rn , and ı./ remains a relative invariant, for some polynomial homomorphism G W G ! R>0 , which extends . Note that also the unipotent radical U of G acts transitively on Rn (even simply transitively, see [5]), and the polynomial character G is trivial on U . Therefore, by formula (5.3), evaluated on U , ı./ must be constant. The latter fact clearly implies that ./ is constant. Conversely, if ./ is constant, by (5.3), ı./ is constant and non-zero on an open subset of Rn . Since, ı./ is a 2 polynomial it must be constant. Remark. In the case of an étale affine representation,Vthe relative invariant ı./ may be n n g (as defined in Section 6.3), identified with the characteristic Vn map ˆ./ W R ! g D R. See Section 8.4 for further discussion, and, in using an isomorphism particular, Proposition 8.27. We call volume preserving if its linear part satisfies `..G// SL.n; R/. Corollary 5.20. Let G be unimodular, and W G ! Aff.n/ an étale affine representation. Then G is simply transitive on An if and only if .G/ is volume preserving. Here is an immediate application which shows that a large class of Lie groups does not admit étale affine representations. Consider the connected abelian group H 1 .G/ D G=ŒG; G; over which every homomorphism of G to an abelian group factorises. Corollary 5.21. If the group H 1 .G/ is compact then G does not admit an étale affine representation. Proof. By our assumption, for every étale representation of G, its character ./ must be trivial. Therefore, by the above theorem, must be simply transitive, and G is simply connected solvable. This shows that, in fact, H 1 .G; R/ D f1g, and since G 2 is solvable, G must be trivial. A contradiction.


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Example 5.12. The criterion is satisfied, for example, if H 1 .g; R/ D g=Œg; g D f0g: Hence, no semisimple Lie group admits an étale affine representation. Also no compact group admits an étale affine representation. Corollary 5.22. If the group G is compact then G does not admit an étale affine representation. Example 5.13. Let G be a reductive Lie group with one-dimensional center. Then every étale affine representation of G is linear. Since G is unimodular, ı./ is a nonzero homogeneous relative invariant polynomial for G of degree n, with non-trivial character det D ./. Such representations exist, for example, for G D GL.n; R/. There do exist other examples as well. See [8], [22], [63] for further reference on this problem. 5.4.3 The dual tube representation. The dual tube, as discussed in Section 6.5, is a useful tool in the construction of pseudo-Riemannian and symplectic affine étale representations. This is based on the following simple observation: Lemma 5.23. Let W G ! Aff.n/ be an étale affine representation, with open orbit U . Let `./ W G ! GL.n/ denote its linear part. Then the associated semi-direct product Lie group G Ë`./ Rn has an étale affine representation on A2n , which has the dual tube domain T{ .U / as open orbit. Moreover, G Ë`./ Rn preserves the natural flat pseudo-Riemannian metric of signature .n; n/, and also the natural symplectic form on T{ .U /. (Here, `./ denotes the dual representation for `./.) The lemma is a particular case of Proposition 6.17 applied to étale affine representations.

6 Affinely homogeneous domains Let U An denote an open subset of An . We put Aff.U / D fA 2 Aff.n/ j A.U / D U g for the affine automorphism group of U . Definition 6.1. Let U An be a connected open subset. Then U is called an affinely homogeneous domain if Aff.U / acts transitively on U .

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A domain U in Es is called homogeneous, or a pseudo-Riemannian affine homogeneous domain if its isometry group Isom.U / D Aff.U / \ E.s/ acts transitively on U . Similarly, a domain U in symplectic affine space .A2k ; !/ is called a symplectic affine homogeneous domain, if the group of symplectic affine automorphisms of U , Aff.U; !/ D Aff.U / \ Aff.!/ acts transitively. An affine domain U is called divisible if there exists a discrete subgroup Aff.U / such that U= is compact.

6.1 Prehomogeneous affine representations Let G be a Lie group and W G ! Aff.n/ a homomorphism. The homomorphism is called an affine representation of G. Composition with the natural homomorphism Aff.n/ ! GL.n/, defines a linear representation ` D ` W G ! GL.n/ which is called the linear part of . Definition 6.2. The affine representation is called a prehomogeneous affine representation, if .G/ acts transitively on an open set U An . If G Aff.n/ is a subgroup with an open orbit in An then G is called a prehomogeneous subgroup. We let U An denote the union of all open orbits for G. Remark. If G admits a fixed point on An , then the homomorphism is conjugate by a translation to the linear representation ` on the vector space V D Rn . Such a representation will be called linear. A vector space V with a linear G-action, which has an open orbit, is traditionally called a prehomogeneous vector space (see [63]). Quite opposite to a linear representation is the situation that the affine representation is transitive on An . In this case, U D An . The following notion plays a special role. Definition 6.3. A prehomogeneous representation on An is called an étale affine representation if dim G D n.

6.2 Some examples There is a wealth of prehomogeneous affine representations , and associated affinely homogeneous domains. We consider some simple examples.


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Example 6.1. (1) The vector group Rn acts simply transitively by translations on An . (2) G D GL.n; R/ acts by left- and also by right-multiplication on the vector space 2 of matrices Mat.n n; R/. Both actions have open orbit U D GL.n; R/ Rn D Mat.n n; R/, and both are simply transitive on U . (3) SL.n; R/ acts transitively on U D Rn f0g. Also SL.n; R/ SL.n; R/ has an open orbit U U in R2n . Note that all the homogeneous domains in the previous example are divisible by affine transformations. The first two examples admit transitive étale affine groups, the first one being the role model for a simply transitive affine representation. The second and third examples come from prehomogeneous linear representations of reductive groups, a topic which is studied extensively in the literature, see [63] and the references therein. Two-dimensional affinely homogeneous domains. In dimension two there are up to affine equivalence six homogeneous affine domains. The following list also exhausts the set of all two-dimensional étale affine representations up to affine equivalence. (Compare [82], [33].) Note that each two-dimensional affinely homogeneous domain admits a simply transitive group of affine transformations. Example 6.2 (Étale affine representations in dimension 2). (1) The plane U D A2 . R2

and

80 19 < 1 v u C 12 v 2 = A Aff.2/ U D @0 1 v : ; 0 0 1

are simply transitive abelian groups of affine transformations. The group

80 19 < exp.t / 0 s = 1 t A Aff.2/ A D @ 0 : ; 0 0 1

is a simply transitive solvable, non-abelian, group of affine transformations, for ¤ 0. (2) The halfspace U D H 2 . 80 < exp.t/ 0 1 BD @ 0 : 0 0

Let H 2 be the halfspace x > 0. Then 80 9 1 19 0 = < ˛ 0 0 ˇ = ˇ v A Aff.H 2 / D @ z ˇ v A ˇ ˛ > 0 : ; ; 0 0 1 1

is an abelian group of affine transformations. The half-spaces .x; y/, x > 0 and x < 0 are open orbits.

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The groups C;

80 19 0 s < exp.t / = exp.t / .exp.t / 1/ A Aff.2/ D @ 0 : ; 0 0 1

are solvable, non-abelian, groups of affine transformations, with two halfspaces as open orbits, ¤ 0. The group of upper triangular matrices ² ³ exp.t / b CD GL.2/ 0 exp.t / is a solvable, non-abelian, linear group of transformations with two halfspaces as open orbits. (3) The sector U D Q2 . Aff.Q2 /0 D

² ˇ ³ a 0 ˇ ˇ a > 0; b > 0 GL.2/ 0 b

is an abelian, linear group of transformations, which has the four open sectors as open orbits. (4) The parabolic domains U D P2C ; P2 . convex domain enclosed by a parabola. 80 2 10 1 < 0 0 Aff.P2˙ /0 D @ 0 0A @0 : 0 0 1 0

Let P2C D f.y; x/ j y > 12 x 2 g be the v 1 0

1 2 v 2

1

9

= ˇ ˇ v A ˇ > 0 Aff.2/ 1

;

is solvable, non-abelian, with open orbits P2C and the exterior P2 of a parabola.

(5) The plane with a point removed, U D A2 0. ² ³ cos sin E D exp.t/ GL.2/ D Aut.U / sin cos is an abelian linear group, with open orbit R2 f0g. We remark:

Corollary 6.4. Every two-dimensional affinely homogeneous domain U A2 admits a simple transitive étale group of affine automorphisms in Aff.U /. Corollary 6.5. Every two-dimensional affinely homogeneous domain is divisible, except for the parabolic domains P ˙ . Proof. If U A2 admits a simply transitive abelian group G of affine transformations, then U is divisible by a lattice G, Š Z2 . This is the case for all domains, but P ˙ . The automorphism group G D Aff.P ˙ / is a solvable non-unimodular étale


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group of affine transformations. Therefore, every which divides P ˙ is a lattice in G. However, the existence of a lattice implies that G is unimodular (see Appendix C); 2 a contradiction. Corollary 6.6 ([82]). Let U A2 be the development image of a compact affine twotorus. Then U is an affinely homogeneous domain which admits an abelian simply transitive group of affine transformations. This also implies: Corollary 6.7. Every two-dimensional divisible domain U A2 is convex and homogeneous. There are attempts to classify étale affine representations in low dimensions, see [82], [38], [36], [12], for some results. In general, it is a difficult problem to decide which Lie groups G admit an étale affine representation. See Section 5.4, for further discussion.

6.3 Characteristic map for a prehomogeneous representation Let W G ! Aff.n/ be an affine representation, and let x 2 An . We define the orbit map at x as ox W G ! Rn ; g 7! .g/x: Let g be the Lie algebra of G, and let

x D x ./ W g ! Rn denote the differential of ox at the identity of G. We remark: (1) The kernel of the linear map x is the Lie algebra h D gx of the stabiliser H D Gx G of x under the action . (2) The orbit .G/x is open if and only if ox W G ! U D .G/x is a submersion of G onto the open subset U of An . Moreover, G has an open orbit at x if and only if x has maximal rank, that is, if x is onto. Let us fix a parallel volume form on An . We construct the characteristic map for V ˆ D ˆ W An ! n g ; ˆ.x/ D x ; by taking the pull back of along x . Its image V ˆ.An / n g is called the characteristic image of . The map ˆ and the characteristic image ˆ.An / carry fundamental information about and the associated homogeneous domain U . This theme will be further developed in Section 7.2.

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Note, in particular, that the map ˆ is a polynomial map in the natural coordinates of An . Moreover, by the above considerations, G has an open orbit at x, if and only if the form ˆx D ˆ.x/ does not vanish. As a first application, we have two basic remarks: Proposition 6.8. (1) Every affinely homogenous domain U is an open semi-algebraic subset of Rn . (2) Let W G ! Aff.n/ be an affine representation. Then G has only finitely many open orbits. Proof. Let G be a connected group which is transitive on U . The set U D fx j ˆx ¤ 0g is a Zariski open subset of An . Its connected components are precisely the open orbits for G. By [101, Theorem 4], there are only finitely many components in the complement of a real algebraic variety. Therefore, U has only finitely many 2 components. Each component is a semi-algebraic subset of Rn .

6.4 The automorphism group of an affine homogeneous domain Let U An be an affine homogeneous domain, and Aff.U / its affine automorphism group. The following explains the prominent role which algebraic groups play in the study of affinely homogeneous domains. (For a review on linear algebraic groups and basic definitions, consult Appendix A.) Lemma 6.9. Let U An be an affine homogeneous domain. Then the automorphism group Aff.U / of U is a finite index subgroup of a real algebraic group. In particular, Aff.U / has finitely many connected components. Proof. Let Uz An be the domain, where the Lie group Aff.U / has open orbits. Then U is a connected component of Uz . The complement W D An Uz is an algebraic subset of An , by Proposition 6.8. Since Aff.Uz / D Aff.W /, we conclude that Aff.Uz / is a real algebraic subgroup of Aff.n/. Since Uz has only finitely many components 2 (cf. Proposition 6.8), Aff.U / has finite index in Aff.Uz /. Let G Aff.n/ be a prehomogeneous Lie subgroup. The real Zariski closure x R Aff.n/ of G, is clearly also a prehomogeneous real linear algebraic group. G Corollary 6.10. Let W G ! Aff.n/ be a prehomogeneous representation. Then the x R Aff.n/ preserves the maximal domain U . Zariski closure G 6.4.1 Automorphisms of complex affine domains. A connected domain U C n is called a complex affine homogeneous domain, if its group of complex affine transformations Aff C .U/ D fA 2 Aff.C n / j A U D Ug acts transitively on U. We mention:


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Lemma 6.11. Let U C n be a complex affine homogeneous domain. Then U is a Zariski-open, hence irreducible and Zariski-dense connected subset of C n . The automorphism group Aff C .U/ Aff.C n / is a complex linear algebraic group. Proof. The proof follows along the lines of the proof of Lemma 6.9. Note that any open and connected orbit U D G 0 x, where G is an algebraic subgroup of Aff.C n /, is Zariski-dense in C n . By Appendix A.3, G x is also dense in the Euclidean topology z D U is a connected of C n . Therefore, there is only one such orbit. In particular, U z 2 domain, and Aff C .U/ D Aff C .U/. 6.4.2 Centralisers of prehomogeneous representations. Let W G ! Aff.n/ be a prehomogeneous representation. We let ZAff.n/ .G/ D fA 2 Aff.n/ j A .g/ D .g/A for all g 2 G g Aff.n/ denote the centraliser of G in Aff.n/. Since the elements of ZAff.n/ .G/ permute the open orbits of G, ZAff.n/ .G/ preserves the maximal set U . In particular, it follows that ZAff.n/ .G/ Aff.U /: Note that ZAff.n/ .G/ is an algebraic subgroup of Aff.n/. Moreover, we have the following fact: Lemma 6.12. The Lie algebra z .G/ of ZAff.n/ .G/ forms an associative subalgebra of aff.n/. Proof. Let ' 2 aff.n/. Clearly, ' 2 z .G/ if and only if A'A1 D ', for all 2 A 2 .G/. Thus, if '; 2 z .G/ then A.' /A1 D ' 2 z .G/. The following lemma is useful in this context: Lemma 6.13. Let A Mat.n n/ be an associative subalgebra of linear operators, such that trace D 0, for all 2 A. Then every element 2 A is nilpotent. Moreover, the Lie algebra which belongs to A (by taking commutators in A) is nilpotent. Corollary 6.14. Let W G ! Aff.n/ be a prehomogeneous representation such that every element of the centraliser algebra z .G/ contains only elements of trace zero. Then the centraliser ZAff.n/ .G/ contains only unipotent elements, and, in particular, it is a connected nilpotent Lie group. Proof. Let A 2 ZAff.n/ .G/. Viewing A as an element of GL.n C 1; R/, we obtain A EnC1 2 z .G/. Thus, A EnC1 is nilpotent, by Lemma 6.13. This implies that 2 the real algebraic group ZAff.n/ .G/ has only unipotent elements. This situation occurs, for example, if is transitive:

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Proposition 6.15. Let be a transitive affine representation. Then the centraliser of .G/ in Aff.n/ is a connected unipotent group of dimension n. Proof. If G Aff.n/ is a transitive subgroup, ZAff.n/ .G/ acts without fixed points on An . The centraliser ZAff.n/ .G/ is an algebraic subgroup of Aff.n/. Its reductive 2 part T has a fixed point, by Lemma 2.2. Therefore, T must be trivial. We shall show below (see Proposition 6.26 and Proposition 6.28) that the centralisers of pseudo-Riemannian and symplectic prehomogeneous groups in the group of isometries of Es , respectively symplectic affine transformations of .An ; !/ are unipotent groups. Such results have immediate geometric implications. As the following example shows, the latter phenomena depend on the orthogonal and symplectic structure: Example 6.3. G D SL.n; R/ SL.n; R/ is a volume preserving prehomogeneous subgroup of Aff.Rn ˚ Rn /, as in Example 6.1 (2). For 2 R, define t .u; v/ D . u; 1 v/. Then T D ft j > 0g SL.Rn ˚ Rn / is an abelian group of volume preserving linear maps, consisting of semisimple elements, and T is contained in the centraliser of G. Note also that the volume preserving group T in the example leaves invariant the natural symplectic form ! on Rn ˚ Rn .

6.5 Tube like domains Affinely homogeneous domains play an important role in the theory of Hermitian symmetric spaces, by the tube construction, which itself is an important source for the construction of complex homogeneous domains from affine homogeneous domains. An analogous dual construction plays a role for the theory of pseudo-Riemannian and symplectically affine homogeneous domains. We develop its basic properties and draw some consequences concerning the classification of pseudo-Riemannian and symplectically affine homogenous domains. 6.5.1 Complex tube domains. The classical construction of tube domains goes as follows. Let U Rn be a connected open set. Then T .U / D U C i Rn C n is called the complex tube domain associated to U . Every A 2 Aff.Rn / extends to a complex affine transformation AC 2 Aff.C n /, where AC .x C iy/ D Ax C i `.A/y: If A 2 Aff.U / is an affine automorphism of U then AC 2 Aff C .T .U //. Also the group of purely imaginary translations i Rn C n acts on T .U /, and this group is


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normalised by the transformations AC . In particular, if U is a homogeneous affine domain then T .U / admits a transitive group of complex affine transformations. That is, T .U / is a complex affine homogeneous domain. Example 6.4. Riemannian Hermitian symmetric spaces (cf. [49], Chapter VIII) are obtained by using the Bergman metric on a complex homogeneous domain T .U /, where U Rn is a convex non-degenerate homogeneous selfdual cone. Complex domains T .U /, where U is a convex non-degenerate cone are also called Siegel-domains of the first kind. Here a convex cone is called non-degenerate if it does not contain any straight line. See [69], [70] and Matsushimas expository paper [76] for results on homogeneous tube domains over convex non-degenerate cones, and their relation with bounded domains in C n . See [34] for a generalisation in the context of pseudo-Hermitian symmetric spaces. 6.5.2 Dual tube domains. A process which is dual to the construction of the tube domain T .U / C n , A 7! AC , allows to construct pseudo-Riemannian and symplectically affine homogeneous domains from arbitrary homogeneous affine domains. Perhaps surprisingly, this construction suggests that the determination of homogeneous pseudo-Riemannian and symplectic affine domains is at least as complicated as the classification of all affinely homogeneous domains. The dual tube domain T{ .U /. Let V D Rn denote a real vector space, and V the dual space of V . If ' is a linear map of V , we let ' denote the dual (transposed) map of V . If U V is a connected open subset we call T{ .U / D U V V ˚ V the dual tube domain over U . Let A 2 Aff.Rn /. We extend A to an affine map AL 2 Aff.V ˚ V /, by declaring AL .u C / D Au C .`.A/1 / : If A is an affine automorphism of U then AL preserves T{ .U /. Note also that the group L In particular, of translations V Aff.T{ .U // is normalised by the transformation A. if U is a homogeneous affine domain then T{ .U / is a homogeneous affine domain, as well. Natural pseudo-Riemannnian metric on T{ .U /. The vector space V ˚ V admits a naturally defined scalar product h} ; i of signature .n; n/. This is defined by

W

hu C ; u0 C 0 i D .u0 / C 0 .u/: L A 2 Aff.n/. Thus: The scalar product h} ; i is evidently preserved by A,

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Proposition 6.16. The map A 7! AL defines a faithful homomorphism Aff.U / ! Isom.T{ .U /; h} ; i/: In particular, we remark: Proposition 6.17. Let U An be an affine homogeneous domain. Then the tube domain .T{ .U /; h} ; i/ is a pseudo-Riemannian homogeneous domain of signature .n; n/. Proof. In fact let, G Aff.n/ be a subgroup which acts transitively on U . Then { E.n; n/ acts transitively on T{ .U / D U V . 2 VÌG Note that V is a maximal isotropic subspace in Rn;n D V ˚ V . The following geometric characterisation of pseudo-Riemannian domains of tube type is straightforward: Lemma 6.18. Let D En;n be a homogeneous pseudo-Riemannian domain. If Isom.D/ contains a maximal isotropic subgroup of translations then D D T{ .U /; for some affinely homogeneous domain U An . The following notion is related: Definition 6.19. A domain U Es is called translationally isotropic if the group of translations t.U / in Isom.U / satisfies t .U /? t .U /. Every tube domain .T{ .U /; h} ; i/ En;n is translationally isotropic, since V t .U / is a maximal isotropic subspace, and thus t .U /? V . Proposition 6.20 ([31][Theorem 5.3]). Every translationally isotropic homogeneous pseudo-Riemannian domain D Em;l is of the form D D T{ .U / Emk;lk ; for some affinely homogeneous domain U Ak , l; m k. Question 5. By [30], the Lorentzian homogeneous domains are En1;1 , and the translationally isotropic domains T{ .R>0 / En2 . Is every homogeneous pseudo-Riemannian domain translationally isotropic? The question suggests, in particular, that homogeneous pseudo-Riemannian domains admit large groups of translations.


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Natural symplectic geometry on T{ .U /. On the vector space V ˚ V there is also a naturally defined symplectic form !, L which is given by !.u L C ; u0 C 0 / D .u0 / 0 .u/: L A 2 Aff.n/. Most considerations The symplectic form !L is evidently preserved by A, for the pseudo-Riemannian case carry over analogously to the symplectic case. We just remark: Proposition 6.21. The map A 7! AL defines a faithful homomorphism Aff.U / ! Aff.T{ .U /; !/: L Thus: Corollary 6.22. Let U be an affine homogeneous domain. Then .T{ .U /; !/ L is a symplectic affine homogeneous domain. Moreover, we mention: Lemma 6.23. Suppose that D A2n is a symplectic affine homogeneous domain. If Aff.T{ .U /; !/ L contains a Lagrangian subgroup of translations then D D T{ .U /; for some affinely homogeneous domain U An . As in the pseudo-Riemannian case, one might wonder which role translationally isotropic symplectically affine homogeneous domains play in the classification of all symplectically affine homogeneous domains. However, here we have Example 6.5. .U D R2 f0g; ! D dx ^dy/ is a symplectically affine homogeneous domain, and Aff.U; !/ D SL.2; R/. In particular, U is not translationally isotropic. Para-Kähler geometry of T{ .U /. We remark that the natural geometry on T{ .U /, L is actually determined by the symplectic which is preserved by the transformations A, } form !L together with the metric h ; i. It may be expressed in a manner which is analogous to classical Kähler geometry as follows. We define a linear operator JL 2 GL.V ˚ V /, which is skew with respect to !L and satisfies JL 2 D id by the relation g.; L / D !. L JL ; /: It is computed by the formula L C / D .u C /: J.u Note, in particular, that the transformations AL are JL -linear, that is, L JL D JL `.A/: L `.A/

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A geometric structure .!; L JL / of this kind, is thus a sort of analogue of complex Kähler geometry. Such structures have attracted recent interest of mathematicians and also physicists under the names of para-Kähler or bi-Lagrangian geometry. See, for example, § 5.2 in [20] and, in particular, the work of Kaneyuki et al. on para-Kähler symmetric spaces [59], [60], [61].

6.6 Pseudo-Riemannian affine homogeneous domains As follows from Proposition 6.17, there are as many pseudo-Riemannian affine homogeneous domains as there are homogeneous affine domains. In particular, the classification of pseudo-Riemannian affine homogeneous domains might be a quite untractable problem. However, for small index s, the possible types of homogeneous domains in Es seem rather restricted. Recall that a homogeneous Riemannian manifold is geodesically complete, since it is complete with respect to the Riemannian distance. Thus: Example 6.6. There is only one Euclidean homogeneous domain U En , namely, Euclidean space En itself. As follows from Example 6.2, we note for the case n D 2: Example 6.7. The two-dimensional pseudo-Riemannian homogeneous domains are E2 , E1;1 , and the half-space H 2 D T .R>0 / E1;1 . In the Lorentzian case, we have the following result: Proposition 6.24 ([30]). Let U En1;1 be a homogeneous Lorentzian domain. Then U D En1;1 or U D H 2 En2 En1;1 . In particular, every Lorentzian affine homogeneous domain is translationally isotropic. We call a pseudo-Riemannian domain irreducible if it does not admit a decomposition as in Proposition 6.20. Example 6.8. The (translationally isotropic) irreducible domains in E2;2 are the dual tube domains .T{ .U /; h} ; i/, where U D A2 f0g, H 2 , Q2 , and P2˙ . (Compare Example 6.2.) It seems unknown, whether (up to products) the latter list exhausts all homogeneous domains in En2;2 . 6.6.1 Centralisers of prehomogeneous groups of isometries. Let W G ! E.s/ be a prehomogeneous group of isometries. We consider the subgroup of isometries in E.s/, which centralise .G/.


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Corollary 6.25. Let U Es be a pseudo-Riemannian affine homogeneous domain. Then the centraliser of any prehomogeneous subgroup G Isom.U / in the group E.s/ is a connected unipotent group. Moreover, it is a nilpotent group of nilpotency class at most two. Proof. By Corollary 3.11, the Lie algebra of ZAff.n/ .G/ \ E.s/ consists of nilpotent elements of aff.n/, and it is a nilpotent Lie algebra of nilpotency class at most two. Therefore, the identity component .ZAff.n/ .G/ \ E.s//0 is a unipotent group, and it is nilpotent of nilpotency class at most two. We remark next that ZAff.n/ .G/ \ E.s/ is connected. (To prove that ZAff.n/ .G/ \ E.s/ is connected, we can argue as in the 2 proof of Proposition 6.28.) The following observation is due to Wolf [106] (for transitive groups), see also [31] for the general case: Proposition 6.26. The elements in g 2 ZAff.n/ .G/ \ E.s/ are unipotent, and satisfy .g EnC1 /2 D 0. Proof. By the above, ZAff.n/ .G/ \ E.s/ is a unipotent linear algebraic group. Using the exponential representation of g 2 ZAff.n/ .G/ \ E.s/, Proposition 3.9 implies that g EnC1 represents a Killing vector field in aff.n/, and .g EnC1 /2 D 0. 2 The previous two results play a central role in the determination of all flat pseudoRiemannian homogeneous manifolds, which are complete. (See Section 4.3.1 for further discussion.) Another application was observed in [31]: Corollary 6.27. Every nilpotent prehomogeneous group N of isometries of Es is transitive on Es . Proof. The set of semisimple elements in a nilpotent algebraic group forms a central subgroup, see [17]. Thus, by Proposition 6.26, N must be a unipotent group. By 2 Corollary 7.5, N acts transitively on An . The latter result generalises to nilpotent groups of volume preserving affine transformations, see Corollary 7.8. For the proof, other methods are required.

6.7 Symplectic affine homogeneous domains As for pseudo-Riemannian domains, the tube construction produces a symplectic affinely homogeneous domain .T{ .U /; !/ L A2k from each affinely homogeneous k domain U A . However, we already noted in Example 6.5 that this construction does not exhaust the set of symplectic affinely homogeneous domains: Example 6.9. The two-dimensional symplectic affine homogeneous domains of tube type are A2 and the halfspace H2 D T{ .R>0 /.

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On the other hand, from Example 6.2, we note: Example 6.10. The two-dimensional symplectic affinely homogeneous domains are A2 f0g, and the tube type domains A2 , H2 . Here is another similarity to the pseudo-Riemannian situation (compare Corollary 6.25): Proposition 6.28. Let U A2k be a symplectic affine homogeneous domain. Then the centraliser of any prehomogeneous subgroup G Aff.U; !k / in the group of symplectic affine transformations of A2k is a connected unipotent group (and it is also abelian). (The connected component of the symplectic centraliser of a prehomogeneous group of symplectic transformations is abelian, by Proposition 3.15.) Hence, we get: Corollary 6.29. Every nilpotent prehomogeneous subgroup of Aff.!k / is transitive on A2k . For the proof of Proposition 6.28, let us first formulate a lemma: Lemma 6.30. Let T be an abelian subgroup of semisimple elements in Sp.V; !/, and G Sp.V; !/ a subgroup which centralises T . Let V T denote the subspace of invariants for T . Then there exists a decomposition V D V T ˚ .W1 ˚ W2 /, which is invariant by G and T , such that W1 and W2 are isotropic subspaces, and the restriction of ! to the subspaces V T and W1 ˚ W2 is non-degenerate. Moreover, V T and W1 ˚ W2 are orthogonal with respect to !. The lemma is a direct consequence of the decomposition of V in eigenspaces with respect to T . Proof. Proof of Proposition 6.28. Let G Aff.U; !k / be a prehomogeneous subgroup with orbit U . For g 2 Aff.n/, t .g/ 2 V D Rn denotes the translation part of g. Let T denote a subgroup of semisimple elements in ZAff.!k / .G/. After a change of origin, we may assume that T is linear, and, in particular, T Sp.!k /. Since T centralises G, t .g/ 2 V T , for all g 2 G, and the subspace V T is invariant by `.G/. Let x D x0 C x1 2 U , where x0 2 V T and x1 2 W1 ˚ W2 , as in Lemma 6.30. Let 2 W V ! W1 ˚ W2 be the corresponding projection operator. We conclude that 2 .U / D 2 .Gx/ D 2 .Gx1 / D 2 .`.G/x1 / D `.G/x1 . Since `.G/ also preserves a Lagrange decomposition of W1 ˚ W2 , it cannot have an open orbit in W1 ˚ W2 . (In fact, the dual pairing of W1 and W2 with respect to ! produces a non-trivial `.G/invariant polynomial.) Unless T D f1g, this is a contradiction, since 2 .U / is open 2 in W1 ˚ W2 .


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7 A criterion for transitivity of prehomogeneous representations A distinguished class of prehomogeneous affine representations is formed by those representations, which are transitive on affine space An . Our main concern in this section will be to formulate a criterion, which ensures that a given prehomogeneous affine representation W G ! Aff.n/ is transitive on An . In Section 7.1, we recall how transitivity depends on the unipotent radical of the Zariski closure. A closer analysis of the characteristic map ˆ./, for a prehomogeneous representation , is given in Section 7.2. This allows to interpret transitivity for in terms of the characteristic image V ˆ./.An / n g : An immediate application (see Corollary 7.8) is the following result: Theorem 7.1. Let U be an affine homogeneous domain, which admits a nilpotent transitive subgroup of volume preserving affine transformations. Then U D An . In particular, for nilpotent prehomogeneous groups, transitivity is equivalent to volume preservation. Various special cases of this result have been treated in the literature before. For the case of étale affine representations see [62], [37], and for pseudo-Riemannian prehomogeneous affine representations of nilpotent groups see [31, Corollary 6.27].

7.1 Transitivity for prehomogeneous groups As a first remark, we note that transitivity may be recognised by looking at the real Zariski closure: Corollary 7.2. Let G Aff.n/ be a prehomogeneous subgroup. Then G is transitive x R Aff.n/ is transitive. on An if and only if its Zariski closure G The corollary is a direct consequence of Corollary 6.10. Transitivity for prehomogeneous algebraic groups is determined by the unipotent radical, as follows: Proposition 7.3. Let G Aff.n/ be a prehomogeneous real algebraic group. Then the following are equivalent: (1) G is transitive on An . (2) The unipotent radical U.G/ is transitive on An . Proof. By Section A, G D U.G/H , where H is reductive. Moreover, H has a fixed point on An , by Lemma 2.2. Thus, if G is transitive, U.G/ must be transitive as well. 2

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x Z Aff.C n / denote Let G Aff.n/ be a prehomogeneous subgroup. Let G D G the complex Zariski closure of G. Then G acts prehomogeneously on complex affine space C n . Corollary 7.4. Let G Aff.n/ be prehomogeneous. Then G is transitive on An if and only if its complex Zariski closure G is transitive on C n . x R . The Proof. If G is transitive, so is the unipotent radical U of its real closure G n Zariski closure of U , U G is prehomogeneous on C and unipotent. By the closed orbit property, U must be transitive. Therefore, G is transitive. Conversely, assume G is transitive. Note that Proposition 7.3 holds analogously for complex algebraic actions on C n . We deduce that U acts transitively on C n . Note that U D UR is Zariski-dense in U (see [17]). In particular, dimR U D dim U . x R acts transitively on An . By Corollary 7.2, G is Thus U , and, in particular, also G transitive, as well. 2 Transitivity of nilpotent groups. We remark that unipotent prehomogeneous groups are always transitive: Corollary 7.5. Let N Aff.n/ be a prehomogeneous group which is nilpotent. Then N is transitive on An if and only if N is unipotent. Proof. In fact, if N is unipotent every orbit on An is closed (cf. Appendix A, Propo2 sition A.2). The converse is a consequence of Proposition 6.15.

7.2 The fundamental diagram We let N D NAff.n/ ..G// Aff.n/ denote the normaliser of G in Aff.n/. Then N acts by conjugation on the image .G/. Assuming that is faithful, this gives also rise to an action C W N ! Aut.G/, where, for A 2 N , C.A/ W G ! G is defined by the relation .C.A/.g// D A.g/A1 : Let further c W N ! Aut.g/ denote the induced representation on the Lie algebra g. We pick up the notation used in Section 6.3. Recall, in particular, that x W g ! Rn denotes the differential of the orbit map ox W G ! An at x 2 An . For all x 2 An , A 2 N , the following diagram is commutative: g c.A/

x

g

/ Rn

Ax

`.A/

/ Rn .

(7.1)


Now consider the characteristic map ˆ D ˆ./ W An !

Vn

g ;

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ˆ.x/ D x

for . The commutative diagram (7.1) implies the following relation, which holds for all A 2 NAff.n/ ..G//: (7.2) Âx D det `.A/ .c.A/1 / ˆx : By Section 6.3, the following is evident: Proposition 7.6. Let W G ! Aff.n/ be a prehomogeneous affine representation. Then the following conditions are equivalent: i) G acts transitively on An . ii) 0 is not contained in the characteristic image ˆ.An /. For an algebraic representation, we note the following refinement: Proposition 7.7. Let W G ! Aff.G/ be a prehomogeneous algebraic representation with maximal domain U . Then the following are equivalent: (i) G acts transitively on An . (ii) 0 is not contained in the closure of the characteristic image ˆ.An /. V (iii) The image ˆ.U / n g is Zariski-closed. V (iv) The image ˆ.U / n g is closed in the Euclidean topology. Proof. We show first, if G acts transitively on An then ˆ.An / is Zariski closed in V n g . In fact, if G acts transitively then its unipotent radical U.G/ acts transitively as well. Thus, using relation (7.2), we deduce that ˆ.An / D ˆ.U.G/x/ D c.U.G//ˆx is the orbit of a unipotent linear algebraic group. Therefore, ˆ.An / is Zariski closed. Hence, (i) implies (ii) and also (iii). In particular, by Proposition 7.6, (i) is equivalent to (ii). Now, assume condition (iii). Then ˆ.U / is Zariski-closed. Note that the open set U is Zariski-dense in An . Since ˆ is a polynomial map (a morphism of varieties), it preserves the Zariski closure. Thus, ˆ.An / D ˆ.U R / ˆ.U /R D ˆ.U /: Since ˆ is non-zero on U , 0 is not contained in ˆ.An /. This shows that U D An , and, hence, G acts transitively. We proved that (i) is equivalent to (iii). Since ˆ.U / is a finite union of orbits, ˆ.U / is Zariski closed if and only if it is closed in the Euclidean topology. (See the remarks in Appendix A.) Hence, (iii) is 2 equivalent to (iv).

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n Remark. Vn It might well happen that the characteristic image ˆ.A / is Zariski-closed g and also contains 0. in

For étale affine representations the above criterion implies: Example 7.1. Let n D dim G. Then G acts simply transitively on An if and only if ˆ is constant, and non-zero. In particular, ˆ.An / D fˆ0 g, where ˆ0 ¤ 0.

7.3 Transitivity of nilpotent prehomogeneous groups As a direct application of Proposition 7.7, we can deduce that, for nilpotent groups, transitivity is equivalent with volume preservation: Corollary 7.8. Let W G ! Aff.n/ be a prehomogeneous affine representation, where G is a connected nilpotent Lie group. Then the following are equivalent: (i) is transitive. (ii) .G/ is unipotent. (iii) is volume preserving (that is, `..G// SL.n/). Proof. Suppose G is acting transitively on An . Without loss of generality, we may assume that G is an algebraic subgroup of Aff.n/. A connected nilpotent linear algebraic group is a direct product G D U.G/T , where U.G/ is unipotent and T is a connected abelian group of diagonalisable matrices, which is contained in the center of G, cf. [17]. Since, U.G/ acts transitively, its centraliser is unipotent. Hence, T D f1g and G D U.G/ is unipotent. Thus, i) implies ii). Since every algebraic character of a unipotent group is trivial, ii) implies iii). Suppose now that .G/ is volume preserving. Let x0 2 U . Then the characteristic orbit ˆ.G x0 / D c.G/ ˆx0 is (Zariski-) closed, since the adjoint action of G on g is unipotent. Then ˆ.An / D ˆ.G x0 Z / ˆ.G x0 /Z D c.G/ ˆx0 does not contain 0. 2 Hence, G acts transitively. Thus, iii) implies i).

8 Characteristic cohomology classes associated with affine representations Let W G ! Aff.n/ be an affine representation. The characteristic map ˆ gives rise to certain characteristic cohomology classes, which carry information about ˆ , and, in particular, about the transitivity properties of .


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8.1 Construction of the characteristic classes Let N W g ! aff.n/ denote the differential of , and `N W g ! gl.n/ the corresponding linear part. The representation `N turns Rn into a g-module, which we denote by RnN . ` Furthermore, we let R`N denote the one-dimensional g-module, which is induced by N We shall also consider the Lie algebra cohomology groups H n .g; R N /, the trace of `. ` n H .g; h; R`N /, where n D dim g=h. (See Appendix B for definitions and notation on Lie algebra cohomology.) n N Lemma 8.1. Let Vnx 2 A and h D gx the stabiliser of x under the action . n Let ˆ W A ! g be the characteristic map for . Then ˆx is an element of C n .g=h; R`N /h C n .g; R`N /. In particular,

d`N ˆx D 0: Moreover, if x 2 U then ˆx ¤ 0, and h acts trivially on C n .g=h; R`N /. Proof. Clearly, ˆx is in C n .g=h; R`N /. By the relation (7.2), the form ˆx is stabilised by the twisted adjoint action of H D Gx . Taking derivatives shows that ˆx 2 C n .g=h; R`N /h is an invariant for h. Since C .g=h; R`N /h is a subcomplex for d`N and C k .g=h; M / D f0g, k > n, (for any module M ), we deduce that d`N ˆx D 0. Since C n .g=h; R`N / is one-dimensional and spanned by ˆx , h acts trivially. 2 The absolute class. As a consequence of relation (7.2), we deduce: Proposition 8.2. The forms ˆx , x 2 An , are elements of the cocycle vector space Z n .g; R`N /, and represent a unique cohomology class cN D Œˆx 2 H n .g; R`N /, which does not depend on x. Proof. By Lemma 8.1, ˆx is a cocycle. We show now that the associated cohomology classes Œˆx 2 H n .g; R`N / do not depend on x. We argue as follows. Recall that the representation ` defines a twisted adjoint action of G on the cohomology groups H k .g; R`N /, whose derivative is induced by the operators LX , X 2 g, cf. Appendix B. Assuming G is connected, this inner action of G on the cohomology is trivial, cf. [85]. If is not prehomogeneous then ˆx D 0, for all x 2 An . Hence, Œˆx D 0, and the claim is proved. Otherwise, let x0 2 U . The transformation formula (7.2) shows that the forms ˆx , x 2 G x0 , form a twisted adjoint orbit in Z n .g; R`N /. By the above remark, they represent the same cohomology class c 2 H n .g; R`N /. Therefore, the polynomial map An ! H n .g; R`N /, x 7! Œˆx is constant on the Zariski dense 2 subset G x0 . Hence, it is constant. Definition 8.3. We call cN D c./ N 2 H n .g; R`N / the (absolute) characteristic class of the affine representation . The geometric importance of the absolute class c./ N stems from:

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Proposition 8.4. Let W G ! Aff.n/ be an affine representation. If the absolute characteristic class c./ N is different from 0 then is transitive. Proof. Assume that is not transitive. Then, by Proposition 7.6, ˆ.An / contains 0. 2 By Proposition 8.2, c./ N D 0. However, the class cN may vanish for a transitive affine representation. The following example of a nilpotent transitive group is taken from [42]: Example 8.1. Consider 8 0 1 0 t v s ˆ ˆ < B0 0 u t C C g D X.s; t; u; v/ D B @0 0 0 uA ˆ ˆ : 0 0 0 0

9 > ˇ > = ˇ ˇ s; t; u; v 2 R aff.3/ ˇ > > ;

Then g is the tangent algebra to a transitive action of a four-dimensional unipotent group G Aff.n/. The tangent algebra of the stabiliser at 0 2 A3 is h D fX.0; 0; 0; v/ j v 2 Rg. We put ; ; !; , for the left invariant one-forms dual to S D X.1; 0; 0; 0/, T D X.0; 1; 0; 0/, U D X.0; 0; 1; 0/, V D X.0; 0; 0; 1/ 2 g. Then d D d! D 0, and d D ^ !, d D ! ^ . Thus ˆ0 D ^ ^ ! D d. ^ /. But note that H 3 .g; h; R/ D hŒˆ0 i ¤ f0g. The example motivates the following construction of the relative classes. The relative classes. Let x 2 An . We put h D gx for the stabiliser of x under N x/ 2 the action . N By Lemma 8.1, the form ˆx represents a cohomology class c.; H n .g; h; R`N /. Definition 8.5. We call c.; N x/ D Œˆx 2 H n .g; h; R`N / the relative characteristic class of the affine representation , at x 2 An . Note that the forms ˆ.; x/ vanish precisely outside the union of all open orbits of G. Hence: Lemma 8.6. If c.; N x/ ¤ 0, for some x 2 An , then is prehomogeneous. In fact, we also have: N x/ ¤ 0 if and only if H n .g; h; R`N / ¤ 0. Lemma 8.7. Let x 2 U . Then c.; As example 8.1 shows, there exist transitive representations , where G is nilpotent, and the absolute class c./ N is zero, while all relative versions c.; N x/ are non-zero. In fact, in Section 8.2 below we show that c.; N x/ is non-zero, whenever is transitive. We will also show that, for nilpotent groups G, the non-vanishing of the class c.; N x/, n for some x 2 A , implies transitivity of .


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Naturality properties (1) By construction, the relative classes c.; N x/ map to the absolute class c./ N under the natural homomorphism H n .g; h; R`N / ! H n .g; R`N /: (2) Let L G be a closed subgroup, and let L W L ! Aff.n/ denote the restriction of to L. Then in the following commutative diagram (vertical arrows denoting restriction homomorphisms) H n .g; h; R`N /

/ H n .g; R N / `

H n .l; l \ h; R`N /

/ H n .l; R N / `

the classes c.; N x/, c./, map to c. N L ; x/, c.L / respectively. (3) Let A 2 NAff.n/ ..G//, and c.A/ W g ! g the induced automorphism of g. If h D gx then c.A/h D gAx . Consider the map of cohomology groups c.A/ W H n .g; gAx ; R`N / ! H n .g; gx ; R`N /: Then, by (7.2), c.A/ c.; N Ax/ D det `.A/c.; N x/: 8.1.1 Alternative construction of the characteristic classes. Let be a group. To any affine representation W ! Aff.n/ the translation part t W ! Rn defines a cohomology class Œt 2 H 1 .; Rn` /, which vanishes if and only if has a fixed point on An . (See [53, 41] for some applications.) Analogously, for a Lie algebra representation N W g ! aff.n/, the translation part t W g ! Rn defines a class u 2 H 1 .g; RnN /, which is called the radiance obstruction. Goldman ` and Hirsch [41], [42] considered the exterior powers V ui D ƒi u 2 H i .g; i RnN /: `

i

By [42], if u ¤ 0 then every orbit of G has at least dimension i . Note that the fixed n-form defines an isomorphism of g-modules Vn n R N Š R`N : `

Under this isomorphism, the n-th exterior power of u identifies with c./: N c./ N D ƒn u: The representation of c./ N as an exterior product of a class in degree one, leads to an important application for compact affine manifolds, see Section 9.2.

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8.2 Significance of the relative classes Let W G ! Aff.n/ be a prehomogeneous representation. We show below that the cohomology groups H n .g; h; R`N / are computed by a characteristic character G=H ./, which is associated to and G. We then explain how the group H n .g; h; R`N / is linked to the geometry of the semi-invariant measure, which is induced by on the homogeneous space X D G=H . As a first application, we deduce that, for a transitive affine representation , all relative cohomology classes c.; N x/ are non-zero (contrary to what may happen for the absolute class c./). N 8.2.1 The character G=H ./. Let x 2 U . We put NG .H / for the normaliser of H D Gx in G. The adjoint representation Ad W N.G/ D NAff.n/ .G/ ! GL.g/ induces a quotient representation AdG=H W NG .H / ! GL.g=h/: We then define the unimodular character for G=H and as >0 G=H ./ D det `./ det Ad1 G=H W NG .H / ! R :

The group N.G/ acts on C n .g; R`N / by the twisted adjoint action as in equation (7.2). Since NG .H / normalises H and h, this action preserves the subspace C n .g=h; R`N / C n .g; R`N /: Then NG .H / acts on the one-dimensional space C n .g=h; R`N / Š Hom.ƒn g=h; R`N / with the character G=H ./. For all A 2 NG .H /, equation (7.2) turns to Âx D G=H ./.A/ˆx :

(8.1)

In particular, this implies that G=H ./ factorises over H , that is, G=H ./jH 1: Associated Lie algebra cohomology group. By Proposition C.3, the character G=H ./ computes the cohomology group H n .g; h; R`N /: Proposition 8.8. The following conditions are equivalent: (1) G=H ./ W .NG .H /=H /0 ! R>0 1. (2) H n .g; h; R`N / ¤ f0g. Example 8.2. If NG .H /0 D H 0 then H n .g; h; R`N / ¤ f0g. For example, the automorphism group G D Aut.H 2 / of a halfspace has H 2 .g; h; R`N / D R.


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Induced semi-invariant measure on X D G=H . We put X D G=H for the associated homogeneous space, where H D Gx . The pull back of the parallel volume form on An defines a semi-invariant measure on X . Its properties strongly interact with the geometry of the representation . The following geometric interpretation of the characteristic character G=H ./ is a consequence of Lemma C.2: Proposition 8.9. Let W G ! Aff.n/ be a prehomogeneous representation, x 2 U and X D G=H the associated homogeneous space. Then the following conditions are equivalent: (1) jG=H ./j W NG .H / ! R>0 1. (2) The semi-invariant measure on X, which is induced from a parallel volume form on An , is invariant by the right-action of NG .H / on X . Homogeneous domains with non-vanishing relative class. We say that the affinely homogeneous domain U has a non-vanishing relative class, if the relative cohomology groups for the group G D Aff.U / are non-vanishing. Corollary 8.10. Let U be a homogeneous affine domain, G D Aff.U /, and H D Gx , for x 2 U . If H n .g; h; R`N / ¤ f0g then the center Z.G/ of Aff.U / is a connected unipotent group. Proof. Clearly, Z.Aff.n/; G/0 NG .H /. Thus, by our assumption, the character G=H ./ is trivial on Z.Aff.n/; G/0 . Since Z.Aff.n/; G/0 is in the kernel of the representation AdG=H , this implies that the elements of Z.Aff.n/; G/0 are volume preserving on An . By Corollary 6.14, Z.Aff.n/; G/ is a connected unipotent group. Thus, Z.G/ D Z.Aff.n/; G/. 2 Example 8.3. Among the two-dimensional homogeneous domains, precisely U D R2 f0g, and the sector U D Q2 have vanishing relative class. 8.2.2 Transitive representations. As remarked before, G=H ./ is trivial on H , for any prehomogeneous affine representation . If is transitive then G=H ./ is trivial on all of NG .H /: Corollary 8.11. If is a transitive affine representation then jG=H ./j 1 on NG .H /. Proof. By Proposition 7.7, the orbit Âx , A 2 NG .H / must be bounded away from 0. By formula (8.1), this can only be if G=H .A/ 2 f1; 1g. 2 We have the following consequence:

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Theorem 8.12. Let W G ! Aff.n/ be a transitive affine representation. For x 2 An , N x/ 2 H n .g; h; R`N / is put h D gx . Then, for all x 2 An , the cohomology class c.; non-zero. Proof. Since G is transitive, the character G=H is trivial, by Corollary 8.11. This implies, H n .g; h; R`N / Š R, by Proposition 8.8. More specifically, the natural map induces an isomorphism of the group of relative n-cocycles C n .g; h; R`N / with H n .g; h; R`N / Š R. Since ˆx ¤ 0, Œˆx D c.; N x/ 2 H n .g; h; R`N / is a non-zero 2 generator of H n .g; h; R`N /. Remark that, conversely, if, for all x 2 An , c.; N x/ ¤ 0 then G acts transitively. Corollary 8.13. Let W G ! Aff.n/ be a transitive affine representation. Then H n .g; h; R`N / ¤ f0g. We thus note (compare Proposition 8.9): Corollary 8.14. Let W G ! Aff.n/ be a transitive affine representation. Then the semi-invariant measure on X D G=H with character det `./ is right-invariant by NG .H /. For a transitive algebraic representation, we add the following observation: Proposition 8.15. Let W G ! Aff.n/ be a transitive algebraic affine representation. Then the following are equivalent: (1) is volume preserving. (2) det Adg=h 1 on H . (3) det Adg=h 1 on NG .H /0 . Proof. Put D det `./. If is volume preserving then H ker . Moreover, G=H ./ 1 on H , as for any prehomogeneous representation. Thus, det Adg=h 1 on H . Conversely, assume det Adg=h 1 on H . Then H ker . Since the unipotent radical of .G/ acts transitively, ker acts transitively. Thus .G/ D ..ker / H / D .ker /. Therefore, 1. It follows the equivalence of (1) and (2). By Corollary 8.11, the analogous argument may be used to show the equivalence 2 of (1) and (3). Corollary 8.16. Let W G ! Aff.n/ be a transitive affine representation. Then is volume preserving if and only if X D G=H admits a G-invariant measure.


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8.3 Affine representations of nilpotent groups In the case of nilpotent groups, we can summarise as follows: Theorem 8.17. Let W G ! Aff.n/ be an affine prehomogeneous representation, where G is a connected nilpotent Lie group. Let x 2 U , and put H D Gx . Then the following are equivalent: (1) is transitive on An . (2) G=H ./ 1 on NG .H /0 . (3) H n .g; h; R`N / ¤ 0. (4) The semi-invariant measure induced by a parallel volume on X D G=H is invariant by the right-action of NG .H /0 . (5) The induced semi-invariant measure on X is a G-invariant measure. (6) .G/ is volume preserving. (7) .G/ is unipotent. Proof. If is transitive then H n .g; h; R`N / ¤ 0, by Theorem 8.12. By Proposition 8.8 and Proposition 8.9, this implies that G=H ./ 1 on NG .H /0 , and also that the induced semi-invariant measure is invariant by NG .H /0 . Since G is nilpotent this implies (cf. Example C.1) that the character det `./ is trivial on NG .H /0 . Moreover .G/ .G/R D T U , where T is a central subgroup of diagonalisable elements, and U is unipotent. By functoriality we may assume now that .G/ D .G/R . It follows that det `./ D 1 on the central subgroup T . Since det D 1 on U , .G/ is volume preserving, and, in particular, the induced measure is an invariant measure. 2 By Corollary 7.8, .G/ is volume preserving implies that is transitive. Moreover, transitivity of a nilpotent affine representation is determined by the relative class c. N L ; x/ at an arbitrary point x 2 An : Corollary 8.18. Let W G ! Aff.n/ be an affine representation, where G is a connected nilpotent Lie group, and let x 2 An . Then is a transitive representation if and only if 0 ¤ c.; N x/ 2 H n .g; h; R`N /: Proof. Assume that c.; N x/ 2 H n .g; h; R`N / is different from zero. In particular, this implies that 0 ¤ ˆx 2 C n .g; h; R`N /. Thus, the representation is prehomogeneous 2 at x. By Theorem 8.17, is transitive. Note, that in the situation of Theorem 8.17, R`N D R is the trivial g-module. Theorem 8.17, together with Corollary 8.18 characterise transitivity for nilpotent affine actions. In particular, the results summarise various special cases, which have been obtained in the literature before: see [62] for étale affine representations of nilpotent groups, [42] for relation with the absolute class c./, N and [31] for nilpotent representations with invariant scalar product (compare also Sections 6.6, 6.7).

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8.3.1 Minimal classes for algebraic group actions. Let be an algebraic representation. Definition 8.19. A relative class c.x; N / will be called minimal if NG .H / contains a maximal R-split torus in G, where H D Gx . For example, if G is nilpotent then c.x; N / is minimal, for all x 2 An . Every reductive subgroup of G fixes a point x 2 An . Therefore, every has minimal relative classes c.x; N /. The following result generalises Corollary 8.18: Theorem 8.20. Let W G ! Aff.n/ be an affine representation, where G is algebraic. N x/ is minimal, and put h D gx . Then is transitive if and Let x 2 An such that c.; only if c.; N x/ 2 H n .g; h; R`N / is different from zero. Proof. Assume c.; N x/ ¤ 0. Then, in particular, is prehomogeneous at x. Since H n .g; h; R`N / is different from zero, G=H ./ 1, on NG .H /0 . In particular, by (8.1), NG .H /0 is contained in the stability group of ˆx . By Proposition A.3, the orbit c.G/ ˆx is closed in C n .g; h; R`N /. Hence, Proposition 7.7 implies that is transitive. 2

8.4 Prehomogeneous representations with reductive stabiliser A homogeneous space G=H is called reductive if the adjoint action of H on g is reductive. In particular, the Lie algebra h acts reductively on g, and there is a hinvariant direct decomposition g D h ˚ p, where p is a vector subspace isomorphic to g=h, and h acts reductively on g=h. Let W G ! Aff.n/ be a prehomogeneous affine representation. We call reductive if, for some x 2 U , the isotropy algebra h for the homogeneous space G=H , H D Gx is reductive in g.1 In particular H 0 is a reductive subgroup in Aff.n/. Also is reductive if h D f0g. This covers the important special case of étale affine representations, as well. The character . Let W G ! Aff.n/ be a prehomogeneous representation, where G is connected. We define the characteristic character D ./ W G ! R>0 by putting

./.g/ D det `.g/ det Ad.g/1 :

Let Adh denote the adjoint representation of NG .H / restricted to h. On NG .H / we have ./ D .det Adh /1 G=H ./. 1A

particular important class of examples appears in the work of Sato and Kimura [90] on regular reductive prehomogeneous vector spaces, where both G and H are assumed reductive. See also [63], and Example 8.4.


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Lemma 8.21. Let be a reductive representation, where G is connected. Then ./ D G=H ./ on NG .H /0 . Moreover, ./ 1 if and only if G=H ./ 1 on NG .H /0 . Proof. Since h is a reductive Lie Algebra det Adh .h/ D 1, for all h 2 H 0 . Let ph be the elements of p, centralised by h. Since h acts reductively, n.g; h/ D h C ph

(8.2)

g D n.g; h/ C Œp; h:

(8.3)

and It follows from (8.2) that det Adh .n/ D 1, for all n 2 NG .H /0 . Therefore, ./ D G=H ./ on NG .H /0 . In particular, ./ 1, implies G=H ./ 1 on NG .H /0 . For the converse, assume that G=H ./ 1 on NG .H /0 . Then the corresponding infinitesimal character N G=H W n.g; h/ ! R vanishes. Since N G=H is the restriction of the infinitesimal character N W g ! R, which belongs to ./, the decomposition 2 (8.3) implies that N 0. Hence, the character ./ is constant. Note, in particular, that ./ 1 on H . Corollary 8.22. Let W G ! Aff.n/ be a reductive prehomogeneous representation, which is transitive. Then j./j 1. Recall (cf. [68], [55]) that h is called not homologous to zero in g if the restriction homomorphism H q .g; R/ ! H q .h; R/ is surjective, for all q. For example, if g D j˚h, where j is an ideal in g then h is not homologous to zero. If h is reductive in g, then surjectivity of H q .g; R/ ! H q .h; R/, q D dim h, is sufficient for h being not homologous to zero. If h is not homologous to zero then (cf. [55, Theorem 12]) it follows, in particular, that the natural map on cohomology H .g; h; R / ! H .g; R / is injective, for any one-dimensional module R . We summarise: Corollary 8.23. Let W G ! Aff.n/ be a reductive prehomogeneous representation. If h D gx is not homologous to zero in g, then the following are equivalent: (1) is transitive. (2) j./j 1. (3) H n .g; h; R`N / ¤ f0g. (4) H n .g; R`N / 3 c./ N ¤ 0. Remark. If h is reductive in g, we could (in lieu of Lemma 8.21) have applied Poincaré duality for relative cohomology (see [64]). Thereby, H k .g; h; R / Š H nk .g; h; Rad ˝ R / ;

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where n D dim g=h. The module Rad ˝ RN is determined by the character . N ` Moreover, H 0 .g; h; Rad ˝ R`N / D H 0 .g; h; RN / D RgN is non-zero if and only if N D 0. 8.4.1 Prehomogeneous domains of reductive algebraic representations Corollary 8.24. Let W G ! Aff.n/ be a reductive prehomogeneous representation, which is algebraic. Then is transitive if and only if the absolute class c./ N 2 H n .g; R` / is non-vanishing. Proof. If is a transitive algebraic representation then .G/ D U T , where T is maximal reductive. In particular, T D Gx fixes a point x 2 An . By Theorem 8.12, 0 ¤ c.; N x/ 2 H n .g; t; R`N /. Since t is not homologous to zero in g, its image 2 c./ N 2 H n .g; R` / is non-zero. Example 8.4 (Regular prehomogeneous vector spaces). Let be a linear prehomogeneous algebraic representation. If H is reductive then the set U is the complement of a hypersurface, and it is also the set of real points of an affine algebraic variety. (See, for example, [63, p. 41ff], for reference.) In particular, any open orbit .G/x is (a connected component) of the set of real points of an affine algebraic variety. If furthermore .G/ is reductive then the representation space ..G/; Rn / is called a regular prehomogeneous vector space. By [63, Proposition 2.26] there exists a non-degenerate relative invariant for .G/ which has character .det /2 . In particular, D det ¤ 1. It follows, in particular, that H n .g; h; R` / D f0g, for a regular prehomogeneous vector space. Using arguments as in [63, loc. cit.], we note: Proposition 8.25. Let be a prehomogeneous algebraic representation. If H is reductive then the affine homogeneous domain U is the complement of a hypersurface, and it is also the set of real points of an affine algebraic variety. In particular, in the situation of the proposition, there exists a relative polynomial invariant ı W An ! R; satisfying ı .g/ D .g/ ı, for some character W G ! R¤0 , and U D Uı D fx 2 An j ı.x/ ¤ 0g: One can show that there is a one-to-one correspondence between algebraic characters of G which factor over H , and the irreducible components of the complement of U . Hence, the group of characters, which factor over H is of rank one if U ¤ An . We conclude that ./ must be a power of .


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Corollary 8.26. Let W G ! Aff.n/ be an algebraic reductive prehomogeneous representation. Then the following are equivalent: (1) is transitive. (2) j./j 1. (3) H n .g; h; R`N / ¤ f0g.

N ¤ 0. (4) H n .g; R`N / 3 c./

Proof. By the previous corollary, we have the equivalence of (1), (3) and (4) Assume that (1) holds. By the above remarks, this implies that the character j./j is constant. Conversely, if j./j is constant, then G=H ./ is constant on NG .H /0 . By 2 Proposition 8.8, this implies (3). By Corollary 7.8, the equivalences (1)–(3) of Corollary 8.23 and Corollary 8.26 also hold if G is nilpotent. However, in general, ./ 1 does not imply transitivity: Example 8.5. Consider the 2-dimensional linear prehomogeneous representation of the group SL.2; R/ on R2 . It has open orbit R2 f0g. The stabiliser H D Gx , x ¤ 0, is a unipotent subgroup. Moreover, SL.2;R/ 1, but SL.2;R/=H ¤ 1. In particular, H 2 .sl.2; R/; h; R/ D f0g. Also, the character may not be trivial for a transitive representation, as the following example shows: Example 8.6. Let G D R2 Ì H Aff.2/, where H SL.2; R/ denotes the 2-dim. solvable group of upper triangular matrices, and R2 is the group of translations. Then G is transitive and G ¤ 1. 8.4.2 The case of étale affine representations. If the affine representation is étale then h D f0g, and there are no relative versions of the characteristic class c./. As a special case of Corollary 8.23, we obtain: Proposition 8.27. Let G be a Lie group, and W G ! Aff.n/ be an affine étale representation. Then the following are equivalent (1) is transitive. (2) ./ 1. (3) H n .g; R`N / ¤ f0g. This result (in particular, the equivalence of (1) and (2)) appears in the work of Helmstetter [50], Kim [62], whereas the relation with the cohomology group H n .g; R`N / is introduced in [42]. It also plays a role in the theory of prehomogeneous vector spaces, see [63]. As shown in Section 5.4, if is an étale affine representation, the character relates to a relative invariant ı of . This may also be interpreted as follows:

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Example 8.7. Any right-invariant volume form on G induces a volume form on U . By comparison with the parallel volume on An , we can write D f ; for some function f on U . It follows that ı D f 1 is a relative invariant for G with character , satisfying ı D . Choose linear independent affine vector fields Xx1 ; : : : Xxn , whose flows generate .G/, and such that .Xx1 ; : : : ; Xxn / D 1. Then ı.x/ D .Xx1 .x/; : : : ; Xxn .x//: It follows that the function ı is a polynomial on Rn , and the zeros of ı describe the boundary of U .

9 Compact affine manifolds and prehomogeneous algebraic groups Here, we develop a theme, which is contained in a series of papers [53], [41], [42] by Goldman and Hirsch. Their ideas show a strong link of the parallel volume conjecture with the theory of prehomogeneous affine representations, as presented in the previous two sections.

9.1 Holonomy of compact complete affine manifolds A basic remark concerning compact complete affine manifolds is: Theorem 9.1. Let M be a compact complete affine manifold, and D h.1 .M // its affine holonomy group. Then the Zariski closure A./ of acts transitively on An . Proof. Let U be the unipotent radical of A./. Every orbit Ux is a contractible closed submanifold of An which is preserved by . Since acts properly discontinuously and freely on Ux, the quotient nUx is a compact manifold. Considering the cohomological dimension cd./ of the group over Z (see [19]), we have 2 n D dim An D dim M D cd./ D dim nUx. Thus Ux D An . Polynomial volume forms. A volume form on M is called a polynomial volume form if it is expressed by a polynomial function in the affine coordinates of M . The following observation is thus an immediate consequence of Theorem 9.1 and Corollary 8.16: Proposition 9.2. Let M be a compact complete affine manifold. Put G D A./, and H D Gx , for some x 2 An . Then the following conditions are equivalent:


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(1) M has a parallel volume form. (2) M has a polynomial volume form. (3) The homogeneous space G=H admits a G-invariant volume. In particular, every polynomial volume form on a compact complete affine manifold is parallel. It seems natural to conjecture that on a compact affine manifold M a polynomial volume form must be parallel. In [37], this is proved under the assumption that the holonomy group of M is nilpotent. Not much seems to be known in the general case. Question 6. Let M be a compact affine manifold. Is every polynomial volume form on M parallel? Example 9.1. Let W G ! Aff.n/ be an étale affine representation. Assume that G admits a uniform lattice . Every polynomial volume form on M D nG lifts to a -left-invariant volume form on G, which is polynomial. We may integrate with respect to a finite G-invariant volume on nG, to obtain a polynomial volume form , which is left-invariant on G. Explicitly, if Xi are vector fields on G, Z lg .X1 ; : : : ; Xn / d : .X1 ; : : : ; Xn / D nG

Since is uniform, G is unimodular and, therefore, is also right-invariant on G. As in Example 8.7, we can write D f ; where ı D f 1 is a polynomial. Since is polynomial, also f is a polynomial. It follows that f must be constant. Hence, the parallel volume on G is left-invariant. This implies that G and M are complete (see Corollary 5.7). By the above, since M is complete, the polynomial volume form is parallel.

9.2 Holonomy of compact volume preserving affine manifolds Let M be a compact affine manifold, and D hol.1 .M // its affine holonomy group. We put A./ for the Zariski closure of in Aff.n/. The following striking observation is the main result of [42]. Theorem 9.3 (Goldman-Hirsch, [42]). Let M be a compact affine manifold with parallel volume. Then A./ acts transitively on An . We put G D A./, and H D Gx , for some x 2 An . The proof of Theorem 9.3, which we will explain below, then shows:

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Corollary 9.4. Let M be a compact affine manifold with parallel volume. Then the absolute characteristic class c.G/ N 2 H n .g; R/ is non-zero. We also obtain the following strong restriction on the homogeneous spaces G=H , which may appear as Zariski closures of the holonomy groups of compact volume preserving affine manifolds. Corollary 9.5. Let M be a compact affine manifold with parallel volume. Then dim H n .g; h; R/ D 1; and the natural homomorphism H n .g; h; R/ ! H n .g; R/ is injective. Proof. By Theorem 9.3, A./ acts transitively, and by Corollary 9.4, the absolute class c.G/ N 2 H n .g; R/ is non-zero. Since, as described in Section 8.1, c.G/ N is the n 2 image of a relative class c.G; N x/ 2 H .g; h; R/, the claim follows. Here is an important consequence of Theorem 9.3 concerning the divisibility of homogeneous domains. Corollary 9.6. Let U be a homogenous domain which is divisible by a properly discontinuous group of affine transformations. Then either U D An or there exists an element 2 with det > 1. Proof. Follows immediately, since A./0 Aff.U / preserves U . (See Section 6.4.) 2 More generally, Theorem 9.3 implies that the development image of a compact volume preserving affine manifold M is not contained in a proper semi-algebraic subset of An . (See [42]). One can also use Theorem 9.3 to deduce: Corollary 9.7 ([42]). A compact homogeneous affine manifold M with parallel volume is complete. Proof. In fact, the universal cover X of a homogeneous affine manifold M develops onto an affine homogeneous domain U , and the development map is a covering map. If M is volume preserving U is divisible by a volume preserving properly discontinuous group of affine transformations. Hence, U D An . It follows that the development map of M is a covering map onto An . Therefore, it is a diffeomorphism. Thus, M is 2 complete. See Section 4.2, for an independent proof of the latter fact. For an application of Theorem 9.3 concerning the structure of the groups of isometries of compact flat pseudo-Riemannian manifolds, see Section 3.3.


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9.2.1 Proof of Theorem 9.3. We outline the proof of Theorem 9.3 using the main ideas of [42], which relate nicely to the concepts discussed in Section 8.1. The basic new tools which we require stem from the cohomology theory of discrete groups (as documented for example in [19]), and the cohomology of algebraic linear groups G, as developed in [54]. Cohomology of algebraic linear groups. Let G D U T be an algebraic linear group, where U is the unipotent radical and T is a maximal reductive subgroup. We let g, u, t denote their corresponding Lie algebras, which are subalgebras of aff.n/. Let V i .G; V / are defined in be a rational G-module. The algebraic cohomology groups Halg [54]. For any subgroup of G, there exists then a natural restriction homomorphism i r W Halg .G; V / ! H i .; V / i .G; V / may be into the ordinary cohomology groups of . Moreover, the groups Halg computed by the Hochschild isomorphism i hs W Halg .G; V / ! H i .u; V /T ; i .G; V / with the T -invariants of the Lie algebra cohomology of which identifies Halg the unipotent radical. (See [54] for reference.)

Discrete cohomology of , and cohomology of M . Let M be a manifold, D 1 .M /. There exists a natural homomorphism q W H i .; V / ! H i .M; V / where V is the local coefficient system on M induced by V . If M has contractible universal cover X then q is an isomorphism. Moreover q is an isomorphism on H 1 . (See for example [19, VII, §7] or [71, IV, §11].) The proof of Theorem 9.3 then builds on the following two remarks: Functoriality of the characteristic class un . Let D hol.1 .M // be the holonomy of a compact volume preserving affine manifold, and G D A./ D U T its Zariski closure in Aff.n/. As remarked in Section 8.1.1, the translation part t , defines aV characteristic class un , within the n-th cohomology groups with coefficients in R D n Rn . The characteristic class u (and therefore un ) is naturally defined in the discrete group-, algebraic group- and Lie algebra cohomology theory, and, as observed in [42], it is compatible with the restriction homomorphisms and the Hochschild isomorphism hs. Representation of un in de Rham cohomology. It is proved in Œ41 that q.un / 2 H n .M; R/ is represented by the parallel volume form in the de Rham cohomology n .M; R/ of M . group HdR Proof of Theorem 9.3. Since M is compact, the de Rham cohomology class q.un / of the parallel volume form on M does not vanish. Therefore, in particular un ¤ 0.

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By the Hochschild isomorphism this also holds for un 2 H n .u; R/. As remarked N which therefore is non-vanishing. in Section 8.1.1, un corresponds to the class c.u/, 2 Hence, U acts transitively on An .

Appendices A Linear algebraic groups We provide some background material and basic facts from the theory of linear algebraic groups and algebraic group actions.

A.1 Definition of linear algebraic groups A subgroup G GL.n; C/ is a linear algebraic group if it is the zero locus of polynomial equations in its matrix entries. A subgroup G GL.n; R/ is said to be a real linear algebraic group if it is closed with respect to the Zariski topology on GL.n; R/. A real linear algebraic group G is thus defined as the real zero locus of a set of real polynomials in its matrix entries. In particular, G is of the form G D G .R/, for some linear algebraic group G , which is defined over the reals. A homomorphism W G ! L between linear algebraic groups is called a morphism of algebraic groups or algebraic homomorphism if it may be expressed by polynomials in the matrix entries of G . x R GL.n; R/ denote its real Zariski If G GL.n; R/ is any subgroup, we let G closure, which is the smallest real algebraic subgroup containing G. Correspondingly, x Z GL.n; C/ is the smallest linear let G GL.n; C/. Then its Zariski closure G algebraic group containing G.

A.2 Structure theory Unipotent groups. A linear algebraic group is called unipotent if it is conjugate to a subgroup of the group of all upper triangular matrices which have only 1 as eigenvalue. All its elements are unipotent matrices. Every connected subgroup of unipotent matrices in GL.n; R/ is a unipotent real linear algebraic group. Also every unipotent linear algebraic group is connected. (See [17] or [84] for more details.) Reductive groups. A subgroup G of GL.n; R/ is called reductive if every G-invariant subspace in Rn has a G-invariant complement in Rn . Main examples are compact subgroups, semisimple groups and (complex-) diagonalisable groups. A linear algebraic group G is reductive if and only if its unipotent radical U is trivial. A connected abelian group of semisimple elements is called an algebraic torus.


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Levi-splitting. Every linear algebraic group G GL.n; C/ splits as a semidirect product of linear algebraic groups G D H U , where H is reductive and U is the maximal unipotent normal subgroup of G . The group U is called the unipotent radical of G . The splitting induces a corresponding splitting of the groups of real points. In particular, every real linear algebraic group G GL.n; R/ splits as a semidirect product of linear algebraic groups G D H U.G/, where H is reductive and U.G/ is the unipotent radical of G. See [17], [84].

A.3 Orbit closure Every orbit G x of a linear algebraic group G on a vector space (or algebraic variety) contains a closed orbit in its Zariski closure G x Z , and the orbit G x is open in its closure. This is called the closed orbit lemma, see [17, 1.8]. By the latter fact and [81, I.10], orbit closure in the Zariski-topology and in the Euclidean (Hausdorff ) topology coincide. Concerning the topology of real algebraic group actions we have: Proposition A.1. If G is a linear algebraic group defined over R, then the real points XR of the orbit X D G x form a finite union of orbits of GR . Moreover, if X is closed, the orbits of GR in the real algebraic variety XR are closed in the Euclidean topology. For this result, see [18, Proposition 2.3]. If G is reductive also the converse holds, cf. [15]. Proposition A.2 (See [15], [17], [89]). Let U be a unipotent linear algebraic group, which acts on a vector space (or on an affine algebraic variety). Then every orbit of U is closed. The analogous result holds for actions of unipotent real linear algebraic groups. That is, every orbit of a unipotent real algebraic group is closed in the real Zariskitopology. For algebraic actions, which are defined over R, the preceding result generalises as follows. Let G GL.n; C/ be a linear algebraic group which is defined over R. Recall that a torus T G is called R-split if it can be diagonalised with respect to a real basis. Proposition A.3 (See [15]). If G is defined over R, and the stabiliser Gx , of x 2 Rn , contains a maximal R-split torus of G then the orbit G x is Zariski closed. A corresponding result holds for real algebraic group actions. Namely, if Gx contains the maximal connected diagonalisable subgroup A of G, then Gx is closed in the Euclidean topology. (This can be proved directly as a consequence of Proposition A.2 and the Iwasawa decomposition (see [84]) G D KAN D KNA, where K is compact, N is unipotent.)

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B Lie algebra cohomology Here we introduce Lie algebra cohomology, and relative Lie algebra cohomology. We compute the top cohomology group in relative Lie algebra cohomology with onedimensional coefficients.

B.1 Definition of Lie algebra cohomology Let g be a Lie algebra over the reals. (We shall use the real numbers as ground field, for convenience.) Let V D V be the g-module, corresponding to an action W g ! gl.V /. Let Vk g ˝V C k .g; V / D be the module of k-forms on g with values in V . The Lie algebra g acts on C k .g; V / V by the adjoint action on k g twisted with . For X 2 g, the corresponding operator LX W C k .g; V / ! C k .g; V / is called the Lie derivative. For ! 2 C k .g; V /, we put X ! 2 C k1 .g; V / to denote the contraction with X. Then the following commutation formula holds: (B.1) Y LX D LX Y ŒX;Y : The boundary operator dV W C k .g; V / ! C kC1 .g; V / is a differential of degree one, which commutes with the operators LX . It is defined inductively by the relation X dV C dV X D LX :

(B.2)

Recall, that dV may be explicitly computed as dV ! .Y1 ^ ^ Yk / D

k X .1/lC1 .Yl / !.Y1 ^ ^ Yyl ^ ^ Yk / lD1

C

X

.1/rCs !.ŒYr ; Ys ^ Y1 ^ ^ Yyr ^ ^ Yys ^ ^ Yk /:

r<s

As usual, Z .g; V / D f! 2 C k .g; V / j dV ! D 0g denotes the group of k-cocycles, and B k .g; V / D fdV j 2 C k1 .g; V /g Z k .g; V / the group of coboundaries. The complex .C .g; V /; dV / is called the Koszul-complex. Its cohomology vector spaces H k .g; V / D Z k .g; V /=B k .g; V / are called the cohomology groups of g with coefficients in V . k

B.2 Relative Lie algebra cohomology Let h g be a subalgebra. We shall also consider the relative cohomology groups H k .g; h; V /. Let C k .g=h; V / denote the subspace of cochains in C k .g; V / which


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vanish if one argument is contained in h. Note that C k .g; h; V / Š ƒk .g=h ; V /. As follows from (B.2), the h-invariants C .g=h; V /h Š Homh .ƒ g=h; V /; are preserved by dV , and thus form a subcomplex C .g; h; V / of the Koszul complex .C .g; V /; dV /. This subcomplex is called the complex of relative cochains. Its cohomology groups are the relative cohomology groups H k .g; h; V /. Note, in particular, that the inclusion of cochain complexes, induces natural homomorphisms H k .g; h; V / ! H k .g; V /: See [55], [64], [68] for detailed reference on Lie algebra (co-) homology. B.2.1 Top cohomology group with one-dimensional coefficients. Observe that H k .g; h; V / D f0g, k > n D dim g=h . Let W g ! R be a one dimensional representation. We shall compute the top cohomology group H n .g; h; R /. Let n.g; h/ denote the normaliser of h in g. Let ad W g ! gl.g/ denote the adjoint representation, adg=h W n.g; h/ ! gl.g=h/ the quotient representation. Note, in particular, that (induced by the adjoint action) n.g; h/ acts on the one dimensional module C n .g=h; R/. Since C n .g=h; R/ Š ƒn .g=h/ , n.g; h/ acts with the character trace adg=h . The Lie derivatives LX , for X 2 n.g; h/, preserve C n .g=h; R / C n .g; R /, and, thereby, n.g; h/ acts with the character N D trace adg=h

(B.3)

on C n .g=h; R /, that is, for all 2 C n .g=h; R /, LX D N .X / :

(B.4)

Now the following holds: Proposition B.1. Let n D dim g=h. Then the group H n .g; h; R / is non-zero if and only if N 0. Proof. Assume that H n .g; h; R / ¤ f0g. Then, in particular, there exists a non-zero generator of the module C n .g; h; / D .ƒn .g=h/ /h . We compute the boundary h operator d W C n1 .g; h; / ! C n .g; h; /: Let ! 2 ƒn1 .g=h/ be a relative n 1 cochain. By duality in C .g=h; R/, there exists X 2 g, such that X D !. Moreover, by (B.1), 0 D LH ! D LH X D ŒH;X , for all H 2 h. In fact, this implies that X 2 n.g; h/. Using (B.2), we compute d ! D d X D LX D N .X / :

(B.5)

Therefore, if N ¤ 0, H n .g; h; R / D f0g. For the converse, assume H n .g; h; R / D f0g. Then either .ƒn g=h /h D f0g, or d ¤ 0. In the first case, we must have N .H / ¤ 0, for some H 2 h. In the second 2 case, we may use the computation of the boundary (B.5), to conclude N ¤ 0.

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The special case, H n .g/ ¤ 0, n D dim g, if and only if g is unimodular, is due to Koszul [68]. Example B.1. If g is nilpotent then the group H n .g; h; R / is non-zero if and only if 0 on n.g; h/. For an interpretation of H n .g; h; R / concerning invariant measures on homogeneous spaces, see Proposition C.3.

C Invariant measures on homogeneous spaces We briefly give some standard background material on the existence of invariant measures on homogeneous spaces. One may consult [87], Chapter I, or [100] for more detailed reference. We then proceed to show that the top relative Lie algebra cohomology group with one-dimensional coefficients, which is associated to a semi-invariant measure on a homogeneous space, carries information about the measure preserving automorphisms of the space. This extends a well known non-vanishing result of Koszul [68] on the top cohomology of unimodular Lie groups.

C.1 Semi-invariant and invariant measures Let G be a Lie group, H G is a closed subgroup. We put X D G=H for the associated G-homogeneous space, n D dim X. Let Lg W X ! X denote left-multiplication with g 2 G. A Borel measure on X is called semi-invariant with character if there exists a continuous homomorphism W G ! R>0 such that Lg D .g/, for all g 2 G. The measure is called invariant if 1. Haar measure and unimodular character. Every locally compact group G has a (up to scalar multiple) unique (left)-invariant measure D G , which is called the Haar measure of G. Let Rg W G ! G denote right-multiplication with g 2 G. Then Rg D G .g/ is another Haar measure for G. The homomorphism D G W G ! R>0 is called the unimodular character of G. If G 1 the Haar-measure is also rightinvariant. Therefore, G has a biinvariant measure if and only if G 1. In this case, G is called unimodular. Since G is a Lie group, the Haar measure can be computed by integration relative to a left-invariant n-form ! ¤ 0 on G, n D dim G. We have G .g/ D j det Ad.g/j;


811

where det Ad.g/ is the determinant of the adjoint representation. In particular, G is unimodular if and only if the determinant of the adjoint representation has absolute value one. Existence of semi-invariant measures. Not every homogeneous space admits an invariant measure. A precise criterion is as follows: For h 2 H , define G=H .h/ D G .h/H .h/1 . Proposition C.1 ([87], Lemma 1.4). The homogeneous space X D G=H admits a (unique up to scalar) semi-invariant measure with character if and only if, for all h 2 H, .h/ D G=H .h/: In particular, X admits an invariant measure if and only if G=H 1 on H . Let AdG=H denote the adjoint representation of H on g=h. Then, for all h 2 H , G=H .h/ D j det AdG=H .h/j.

C.2 The unimodular character of X D G=H Let NG .H / denote the normaliser of H in G. The adjoint representation G ! GL.g/ induces a quotient representation AdG=H W NG .H / ! GL.g=h/: Note that the restriction of AdG=H to H corresponds to the isotropy representation of H on the tangent space of X at H . For every g 2 NG .H /, define C.g/ D Lg Rg1 W X ! X; where Rg denotes right-multiplication on X . Let be a semi-invariant measure on X with character . The relation C.g/ D G=H .g/; defines a unimodular character, G=H W NG .H / ! R>0 ; independently of . In fact, G=H D jdet AdG=H j. Obviously, we have: Lemma C.2. The semi-invariant measure with character is right-invariant by .g/ D 1. g 2 NG .H / if and only if .g/ 1 G=H In addition, we obtain the following:

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Proposition C.3. Let X D G=H be a homogeneous space and a semi-invariant measure with (smooth) character . Let N W g ! R denote the derivative of . Then the following conditions are equivalent: (1) is invariant by the right action of NG .H /0 on X. W NG .H /0 ! R>0 1. (2) D 1 G=H (3) H n .g; h; RN / ¤ f0g. Proof. This a direct consequence of the above lemma and Proposition B.1.

2

Example C.1. Let G be nilpotent. Then G=H 1. Therefore, H n .g; h; RN / ¤ f0g is equivalent to Š 1 on NG .H /0 . Corollary C.4. Let X D G=H be a homogeneous space which admits an invariant measure . Then is right-invariant by the action of NG .H /0 if and only if H n .g; h; R/ ¤ f0g. As a special case of the corollary, we have the well known theorem of Koszul [68], which states that H n .g; R/ ¤ 0 if and only if G is unimodular.

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Part G

Conformal geometry

Chapter 23

The conformal analog of Calabi–Yau manifolds Helga Baum

Contents 1 2 3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cartan connections and holonomy groups . . . . . . . . . . . . . . . . . . . . . . . Holonomy groups of conformal structures . . . . . . . . . . . . . . . . . . . . . . . 3.1 Cartan connections in conformal geometry . . . . . . . . . . . . . . . . . . . . 3.2 The normal conformal Cartan connection in invariant form . . . . . . . . . . . 3.3 The normal conformal Cartan connection in its metric form . . . . . . . . . . . 3.4 The tractor connection and its curvature . . . . . . . . . . . . . . . . . . . . . 3.5 Conformal holonomy and Einstein metrics . . . . . . . . . . . . . . . . . . . . 3.6 Conformal holonomy and conformal Killing spinors . . . . . . . . . . . . . . 4 Fefferman spin manifolds – conformal structures with holonomy group in SU.1; m/ 4.1 CR geometry and Fefferman spaces . . . . . . . . . . . . . . . . . . . . . . . 4.2 The conformal holonomy group of Fefferman spin spaces . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

821 824 834 834 837 839 841 844 847 853 854 857 859

1 Introduction In this survey we intend to introduce the reader to Cartan connections, their holonomy theory and applications to conformal supersymmetries. The concept of nowadays called ’Cartan connections’ was introduced and successfully applied by E. Cartan during the first decades of 20th century. Cartan studied geometric structures associated with solution spaces of classical ordinary differential equations using moving frames1 . Thereby, Cartan found a uniform concept that generalizes F. Klein’s approach to geometry by studying invariants of group actions as well as B. Riemann’s concept of curved Riemannian manifolds to a broad class of geometries, such as conformal, projective, Einstein–Weyl, 3-dimensional CR and so on. The renewed interest in conformal and quaternionic structures as well as relations to complex analysis via CR structures brought Cartan connections back to general interest during the last decade. In particular, the systematic development of parabolic differential geometry in modern language of principle fibre bundles by the group in Vienna, Brno and Prag (A. Cap, J. Slovak, V. Soucek and others) over the last years turned out to be an 1 Cf.

[Nu05] for a reformulation and extension of some of Cartan’s work.

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extremely useful tool in studying geometric problems and structures in an unified and systematic way. Let us mention here for example the classification of different conformal and CR invariants, the classification of conformally invariant operators, the classification of higher symmetries of differential operators or the progress in understanding T. Branson’s Q-curvature (cf. for example [FG85], [CSS01], [CD01], [E05], [J06]). The common feature of many different kinds of geometries is the existence of a distinguished (normal) Cartan connection which characterizes the geometry in question, similarly as the Levi-Civita connection in Riemannian geometry. An example is conformal geometry, which we will focus on in this chapter. Many symmetries and super-symmetries in conformal geometry (conformal vector fields, conformal Killing forms, conformal Killing tensors, conformal Killing spinors) have an interpretation as invariant objects for the holonomy group of the normal conformal Cartan connection and can be classified in this way. We will explain this here for the case of conformal Killing spinors. It is well known that the existence of parallel spinors on (pseudo)-Riemannian spin manifolds .M n ; g/, i.e., spinors ' with rX ' D 0

for all vector fields X,

(1)

is characterized by the holonomy group of .M; g/ – the group of all parallel displacements with respect to the Levi-Civita connection along curves which are closed in a fixed point. By the holonomy principle, any of these holonomy groups correspond to a special geometric structure. Using the known classification of Riemannian holonomy groups, McK. Wang ([Wan89]) showed that the holonomy group of simply connected indecomposable Riemannian manifolds with parallel spinors is SU.n=2/, Sp.n=4/, G2 or Spin.7/. The corresponding special geometric structures are Ricci-flat Kähler metrics, hyper-Kähler metrics, a special parallel 3-form (n D 7) or a special parallel 4-form (n D 8), respectively. In 2003, T. Leistner ([Lei03]) classified the holonomy groups of indecomposable simply connected Lorentzian manifolds. As an application he showed that this holonomy group in presence of a parallel spinor is G Ë Rn2 , where G is a product of f1g, SU.k/, Sp.l/, G2 and Spin.7/. Another already ‘classical’ kind of spinor fields occurred in the 80th in studying eigenvalues of the Dirac operator (cf. [Fr81]) as well as in supergravity theories (cf. [DNP86]). Let D be the Dirac operator on a compact Riemannian spin manifold .M n ; g/ of positive scalar curvature scalg . All eigenvalues of D are bounded from below by 2

n min scalg : 4.n 1/

(2)

The eigenspinors ' in the limiting case (equality in (2)) are geometric Killing spinors, that is, they satisfy the stronger equation rX ' C ˛X ' D 0

for all vector fields X ;

(3)

Chapter 23. The conformal analog of Calabi–Yau manifolds

823

where ˛ is a real number related to the eigenvalue . This initiated an intensive search for geometric structures with geometric Killing spinors (cf. [BFGK91], [Kat99]). In 1993 Ch. Bär observed that there is a Cartan connection of type SO.n C 1/=SO.n/ behind (3) and gave a reinterpretation of geometric Killing spinors as invariants of the holonomy group of this Cartan connection. In particular, this Cartan connection is the same as the Levi-Civita connection of the Riemannian cone over .M; g/, hence Killing spinors on .M; g/ correspond to parallel spinors on the cone. This observation together with Wang’s result led him to the complete classification of all geometric structures with Killing spinors of type (3) (cf. [Bär93]). There are a lot of different spinor field equations in mathematics as well as in physics (string theory), generalizing (1) and (3). In the present article we deal with conformal Killing spinors, spinors satisfying rX ' C

1 X D' D 0 n

for all vector fields X.

(4)

These spinors occur in conformal geometry as the solutions of the ‘twistor equation’, one of the two conformally invariant spinor field equations on 1/2 spinors. They were first studied by R. Penrose and his collaborators in General Relativity (cf. [PR86]), later on in mathematics (cf. [BFGK91]). In [ACDS98], D. Alekseevski, V. Cortés et all gave an interpretation of conformal Killing spinors as infinitesimal symmetries of supermanifolds. J. Figueroa O’Farrill and H. Rajaniemi used it to give geometric constructions of superalgebras (cf. [R06]). It is natural to ask for a classification of conformal structures which admit conformal Killing spinors. The aim of the present survey is to explain the holonomy theory behind (4), which turns out to be the holonomy theory of the normal conformal Cartan connection of the conformal manifold .M; Œg/. We call its holonomy group, the ‘conformal holonomy group’. The conformal holonomy group of a manifold .M; g/ with a metric of signature .p; q/ is contained in SO.p C 1; q C 1/. In Riemannian and algebraic geometry as well as in string theory Calabi–Yau manifolds play a special role. A Calabi–Yau manifold is a compact 2mdimensional Riemannian manifold with holonomy group SU.m/ SO.2m/. Such a manifold is Ricci-flat, Kähler and admits a 2-parameter family of parallel spinors. In view of that we will focus attention in particular to Lorentzian manifolds with conformal holonomy group in SU.1; m/ SO.2; 2m/, which can be viewed as conformal analog of Calabi–Yau manifolds. Such Lorentzian manifolds have an interesting geometric structure. They are Fefferman spaces, which occur as S 1 -bundles over strictly pseudoconvex CR spin manifolds, thus relating conformal Lorentzian geometry to CR geometry. Furthermore, they admit a 2-parameter family of conformal Killing spinors. The chapter is organized as follows. In Section 2 we will introduce the notion of a Cartan connection and of its holonomy group. In particular, we relate this holonomy group to the holonomy group of covariant derivatives on associated tractor bundles. Section 3 contains an outline of the construction of the normal conformal Cartan connection in its invariant form as well as its description in terms of the metrics in the

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conformal class (3.1.–3.3.). This leads to the ‘metric’formula for the associated normal derivative on tractor bundles, usually used in conformal geometry (Section 3.4). In Section 3.5. we give a review on recent classification results for conformal holonomy groups. In particular we explain the relation to the existence of Einstein metrics in the conformal class. In 3.6. we introduce conformal Killing spinors and give the reinterpretation as parallel sections in the spin tractor bundle with respect to the covariant derivative induced by the normal Cartan connection. We close this section with the recent classification result of F. Leitner ([L05a], who used the methods of Cartan geometry to give a local conformal classification of all Lorentzian geometries admitting generic conformal Killing spinors. In Section 4 we deal with Lorentzian manifolds with conformal holonomy in SU.1; m/, the ‘conformal analog of Calabi– Yau-manifolds’. For this aim we explain in 4.1. the construction of Fefferman spaces, which are S 1 -bundles over strictly pseudoconvex CR-manifolds. Finally, using our former results about conformal Killing spinors on Fefferman spaces, we explain in 4.2. why the conformal holonomy group of Fefferman spin spaces is contained in SU.1; m/. Acknowledgement. This work was supported by the SFB 647 ‘Space-Time-Matter’ of the DFG, the Erwin Schrödinger Institute for Mathematical Physics (ESI) in Vienna, and the IRMA Strasbourg.

2 Cartan connections and holonomy groups In this section we give a short introduction to Cartan connections and their holonomy groups. In particular, we explain the relation to holonomy groups of principal fibre bundle connections and to holonomy groups of covariant derivatives in associated vector bundles. Details can be found in [KN63] and [S97]. Let M be a smooth manifold and let L.M; x/ denote the set of all piecewise smooth loops in a point x 2 M . Usually, holonomy groups are defined by parallel displacements along loops in x with respect to a connection form on a principal fibre bundle or with respect to a covariant derivative in a vector bundle over M given by the geometric structure in question. Let us shortly recall this in order to fix some notation. First, we consider a vector bundle E over M with covariant derivative r E . For any piecewise smooth curve W Œ˛; ˇ R ! M the covariant derivative gives rise to a parallel displacement E

Pr W e 2 E.˛/ 7! 'e .ˇ/ 2 E.ˇ / ; where 'e W Œ˛; ˇ ! E is the unique parallel section in E along defined by r E 'e D0 dt

and

'e .˛/ D e:

(5)


825

The holonomy group of .E; r E / with respect to the base point x is the Lie group of all parallel displacements along loops in x: Holx .E; r E / WD fPr

E

j 2 L.M; x/g GL.Ex /:

Next, let .P; ; M I B/ be a principal fibre bundle over M with structure group B. We consider a connection form A on P , i.e., a 1-form A 2 1 .P; b/ with values in the Lie algebra b of B, which satisfies the following invariance conditions. (1) Rb A D Ad.b 1 / B A for all b 2 B, and z D X for all X 2 b; (2) A.X/ where Rb denotes the right action of B on the total space P of the principal fibre bundle and Xz is the fundamental vector field of the B-action on P defined by the element X 2 b: d z u exp.tX / jt D0 : X.u/ D dt The connection form A defines a right invariant horizontal distribution T hA P on P by T hA W u 2 P ! T hA u P WD Ker Au Tu P: Any piecewise smooth curve W Œ˛; ˇ ! M admits a unique horizontal lift u W Œ˛; ˇ ! P with fixed initial point u 2 Px WD 1 .x/, defined by the conditions . .t // D .t /; .˛/ D u: P .t/ 2 T hA .t/ P; Thus, we obtain a parallel displacement along by PA W u 2 Px 7! u .ˇ/ 2 P.ˇ / :

(6)

If is a loop in x, then the points u and PA .u/ lie in the same fibre Px . Hence there is a unique element holA u ./ 2 B with u D PA .u/ holA u . /; called the holonomy of the loop with respect to A and u. The holonomy group of .P; A/ with respect to the base point u is the Lie group of all of these holonomies, Holu .P; A/ WD fholA u . / j 2 L.M; x/g B: Now, let W B ! GL.V / be a representation of the Lie group B over a vector space V . There is a standard way to associate a vector bundle E over M to the principal fibre bundle P by means of . The total space E is defined to be the orbit space of the right action of B on P V given by .u; v/ b WD .u b; .b 1 /v/;

.u; v/ 2 P V; b 2 B:

We denote this orbit space by E WD P B V WD .P V /=B and the elements of E by Œu; v WD f.u b; .b 1 /v/ j b 2 Bg. Obviously, E is a vector bundle over M

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with fibre type V and projection E .Œu; v/ WD .u/. Any point u in the fibre Px of P over x 2 M gives rise to a linear isomorphism Œu W v 2 V ! Œu; v 2 Ex WD E1 .x/

(7)

between the fibre type V and the fibre Ex of E. A smooth section in the vector bundle E can locally be represented in the form jU D Œs; v, where s W U M ! P is a smooth local section in the principal fibre bundle P and v 2 C 1 .U; V / is a smooth function on U with values in the vector space V . Any connection form A 2 1 .P; b/ on P induces a covariant derivative r A W .E/ ! .T M ˝ E/ by rXA

jU

WD Œs; X.v/ C .A.ds.X ///v;

where jU D Œs; v:

(8)

The formulas (5), (6) and (8) show that the holonomy groups of .E; r A / and .P; A/ are related by (9) Holx .E; r A / D Œu B .Holu .P; A// B Œu1 : Hence, if is faithful, the holonomy groups Holx .E; r A / and Holu .P; A/ are isomorphic. Now, let us come to the object of our interest, to Cartan connections and their holonomy groups. For that we consider in addition a Lie group G that contains the structure group B of P as a closed subgroup. A Cartan connection of type G=B on the B-principal fibre bundle P is a 1-form ! 2 1 .P; g/ on P with values in the Lie algebra g (!!) such that (1) Rb ! D Ad.b 1 / B ! for all b 2 B G, z D X for all X 2 b g, and (2) !.X/ (3) ! is a global parallelism, i.e., !p W Tp P ! g is an isomorphism for all p 2 P: Note that the third condition implies that the base manifold M has the same dimension as the homogeneous space G=B. Furthermore, any vector Z 2 g induces a global nowhere vanishing vector field Z! defined by Z! .p/ WD .!p /1 .Z/: Hence, the manifold P is parallelizable, i.e., it has a global basis field. A Cartan connection is called complete, if all these vector fields Z! are complete (meaning that the integral curves of Z! are defined for all parameters t 2 R). Let us consider two examples of Cartan connections. Example 1. Let G be a Lie group and !G its Maurer–Cartan form: !G W X 2 Tg G 7! dLg 1 .X / 2 Te G D g: If B G is a closed subgroup, then the projection W G ! G=B gives rise to a principal fibre bundle with structure group B over the homogeneous space M D G=B. The Maurer–Cartan form !G is a complete Cartan connection of type G=B on this homogeneous bundle.


827

Example 2. Let M n be a smooth manifold and P be the frame bundle of M . Then the tangent bundle of M is given by TM D P GL.n/ Rn . Let A 2 1 .P; gl.n// be an arbitrary connection form on P and let 2 1 .P; Rn / be the displacement form of P :

u .X/ WD Œu1 d u .X /; u 2 P; X 2 Tu P: (10) Then the 1-form given by ! WD A C 2 1 .P; a.n// is a Cartan connection of type A.n/=GL.n/, where A.n/ WD GL.n/ Ë Rn is the affine group and a.n/ its Lie algebra. Although, contrary to usual connections, Cartan connections do not allow to distinguish a right invariant horizontal distribution on P , one can define the notion of a holonomy group also for Cartan connections using the development of the connection form along curves. By the development of a 1-form along a curve we understand the following: Let N be a manifold and 2 1 .N I g/ a 1-form with values in the Lie algebra g of a Lie group G. For any piecewise smooth curve W Œ˛; ˇ ! N there is an unique curve Œ W Œ˛; ˇ ! G starting in the unit element e 2 G such that 0Œ .t/ D dLŒ .t / !. 0 .t //;

(11)

where Lg denotes the left action on G by the element g 2 G. We call Œ W Œ˛; ˇ ! G the development of along and its endpoint hol . / WD Œ .ˇ/

(12)

the holonomy of with respect to . Example 3. Let G D .Rn ; C/ be the additive group. Then the development of a 1-form on N along a curve in N is the usual curve integral of along the curve , Z t Z 0 Œ .t/ WD . .s//ds and hol . / D : ˛

Example 4. Let !G be the Maurer–Cartan form of a Lie group G and a curve in G from g0 to g1 . Then the development of !G along is the curve Œ!G given by Œ!G .t / D g01 .t /: The holonomy is hol!G . / D g01 g1 . In particular, the holonomy of any closed curve with respect to !G is trivial. Now, let us come back to the B-principal fibre bundle P with a given Cartan connection ! 2 1 .P; g/. We fix a point u in the fibre Px over x 2 M and denote by .L.P; u// the set of all loops in L.M; x/ which admit a closed lift in L.P; u/. One observes that hol ! .ı1 / D hol ! .ı2 / for loops ı1 ; ı2 2 L.P; u/ with the same projection B ı1 D B ı2 2 L.M; x/. Thus we can define the holonomy of a loop 2 .L.P; u// L.M; x/ with respect to ! and u by holu! ./ WD hol ! .Q / 2 G;

where Q 2 L.P; u/ and B Q D :

(13)

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The holonomy group of the Cartan connection ! with respect to the base point u is the group Holu .P; !/ WD fholu! . / j 2 .L.P; u// L.M; x/g G: Holu .P; !/ is an immersed Lie subgroup of G with the reduced holonomy group Hol0u .P; !/ WD fholu! . / j 2 L.M; x/ null homotop g as connected component of the unit element. Holonomy groups for different reference points u are conjugated in G. Therefore, we often omit the base point and understand the holonomy group as a class of conjugated subgroups in the group G in question. Example 5. Consider the Maurer–Cartan form !G as Cartan connection on the homogeneous principal fibre bundle .G; ; G=BI B/. Then, by Example 4, the holonomy group is trivial: Holu .G; !G / D feg. Now, let W G ! GL.V / be a representation of the larger group G on a vector space V . Of course, by restriction, this gives a representation of B on V and an associated vector bundle E WD P B V: Bundles of this form (defined using a representation of the larger group G) are called tractor bundles and play a crucial role in Cartan geometry – since, contrary to the case of arbitrary associated vector bundles, on a tractor bundle there is a covariant derivative associated to the Cartan connection !. To see this, we extend the B-principal fibre bundle P to a G-principal fibre bundle Px WD P B G. Let prB W P G 7! Px be the natural projection. We use the Cartan connection ! 2 1 .P; g/ and the Maurer–Cartan form of G to define a 1-form !O 2 1 .P G; g/ by !O .u;g/ WD Ad.g 1 / B .P !/.u;g/ C .G !G /.u;g/ ;

(14)

where P and G are the projections from P G onto P and G, respectively. !O is invariant under the B-action on P G and projects to a 1-form !N 2 1 .Px ; g/, prB !N D !, O which is a usual connection form on the G-principal fibre bundle Px . For the canonical embedding W u 2 P 7! Œu; e 2 Px obviously holds !N D !. Since W G ! GL.V / is a G-representation, E D P B V ' Px G V: Hence, any Cartan connection ! 2 1 .P; g/ defines a covariant derivative r ! on E via its extension to the principal fibre bundle connection !N on Px , r ! WD r !N . According to formula (8), r ! is given by .rX! /jU D Œs; X.v/ C .!.ds.X ///v;

(15)

where jU D Œs; v 2 .EjU / for a local section s W U ! P and a smooth function v W U ! V . By definition, r ! is a metric covariant derivative with respect to any


829

metric h ; i on the bundle E, which is induced by a G-invariant inner product h ; iV on V , i.e., X.h'; i/ D hrX! '; i C h'; rX! i: Example 6. We consider the situation of Example 2. Let P be the frame bundle over M n with Cartan connection ! D A C , where A is a connection form and the displacement form on P . We realize the affine group A.n/ as a subgroup of the linear group GL.n C 1/ by A x 2 GL.n C 1/ .A; x/ 2 A.n/ D GL.n/ Ë Rn 7! 0 1 and consider its matrix representation on RnC1 . The restriction of this representation to GL.n/ decomposes into the sum Rn ˚ R, where the first factor is the matrix representation of GL.n/ on Rn and the second the trivial representation. Hence, the tractor bundle T associated to P by is T D P GL.n/ RnC1 TM ˚ R; where R denotes the trivial line bundle on M . For the tractor connection r ! on T induced by ! we obtain from (8), (10) and (15) A rX Z C f X ! Z ; D rX f X.f / where f is a function and Z a vector field on M and r A the covariant derivative on the tangent bundle induced by A. Now, let us discuss the relation between the holonomy groups of !, !N and r ! which we defined above. Proposition 1. (1) The connected components of the holonomy groups coincide Hol0u .P; !/ D Hol0Œu;e .Px ; !/ N

(16)

(2) If the structure group B of P is connected or if the base space M of P is simply connected, then N G; Holu .P; !/ D HolŒu;e .Px ; !/ 1

.Holu .P; !// D Œu

!

B Holx .E; r / B Œu GL.V /:

(17) (18)

Proof. Let 2 L.M; x/ be a loop in x which has a closed lift Q 2 L.P; u/. First we will show that the holonomy of such curves with respect to the Cartan connection ! 2 1 .P; g/ and the point u 2 P coincide with the holonomy of with respect to the connection form !N 2 1 .Px ; g/ and the point uN WD Œu; e 2 Px . N lift of in Px with initial Let uN W Œ˛; ˇ ! Px WD P B G be the !-horizontal x Q g, point u. N By definition of P we can represent the curve uN in the form uN D Œ;

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where g W Œ˛; ˇ ! G and g.˛/ D e. Then uN .ˇ/ D Œ.ˇ/; Q g.ˇ/ D Œu; g.ˇ/ D Œu; e g.ˇ/ D uN g.ˇ/: Hence, by definition, holu!NN ./ D g.ˇ/1 . Since uN is !-horizontal, N we obtain from the definition of !N and formula (14) that 0 D !N uN .t / .PuN .t //

PQ /; g.t D !O ..t P // Q /;g.t // ..t 1 PQ // C !G .g.t P //: D Ad.g .t //!..t

Therefore, PQ // D Ad.g.t // !G .g.t !..t P // D dRg 1 .t / g.t P / D dLa.t /1 a.t P /; where a.t / WD g.t/1 . Thus, a WD g 1 W Œ˛; ˇ ! G is the development of ! along Q and by definition of the holonomy of with respect to ! (cf. (11), (12) and (13)) follows that holu! ./ D g.ˇ/1 . Thus, holu! . / D holu!NN . / for all loops 2 L.M; x/ which admit a closed lift Q 2 L.P; u/. It remains to show that L.M; x/ D .L.P; u// if B is connected or if M is simply connected. From the homotopy lifting lemma for fibrations follows that .L.P; u// D f 2 L.M; x/ j Œ 2 ] .1 .P; u//g: The exact homotopy sequence of the fibration .P; ; M I B/ shows that the induced map ] on the fundamental groups is surjective if the structure group B is connected. Therefore, L.M; x/ D .L.P; u// if the B is connected or if M is simply connected. The statements (16) and (17) follow then from the definition of the holonomy groups in question. Statement (18) is a consequence of (9). This coincidence between the holonomy groups allows us to translate the generally known properties of holonomy groups of connection forms on principal fibre bundles (resp. covariant derivatives) into properties of holonomy groups of Cartan connections. In particular, the holonomy algebra is generated by the curvature. In general, the curvature of a 1-form 2 1 .N; g/ on a manifold N with values in a Lie algebra g is the 2-form 2 2 .N; g/ defined by 1 WD d C Œ; : 2 Hence, the curvature ! of a Cartan connection ! 2 1 .P; g/ is a 2-form on P with values in the Lie algebra g. The invariance properties of a Cartan connection yield that its curvature is horizontal and of type Ad, i.e., T vP ³ ! D 0; Rb ! D Ad.b 1 / B !

for all b 2 B;

where T vP TP is the vertical tangent bundle of P . The torsion of a Cartan connection is the 2-form Tor ! , which arises from the curvature form by projection


831

onto the factor space g=b: Tor ! WD projg=b B ! 2 2 .P; g=b/: We call a Cartan connection ! torsion free, if its torsion is zero, that is, if the curvature ! takes its values in the subalgebra b g. We call ! flat, if its curvature is zero. For example, the Maurer–Cartan form !G of a Lie group G considered as Cartan connection on the homogeneous bundle .G; ; G=BI B/ over the homogeneous space G=B is flat. The Uniformization Theorem in Cartan geometry says that up to discrete groups this is the only situation, where flat connections appear. More precisely: Proposition 2 (cf. [S97], Chapter 5.5). Let .P; ; M I B/ be a principal fibre bundle with connected structure group B over a connected manifold M which admits a complete flat Cartan connection ! 2 1 .P; g/. Then, there exists a connected Lie group G with Lie algebra g, containing B as closed subgroup, such that M is diffeomorphic to the locally homogeneous space n G=B, where G is a discrete subgroup. Moreover, the B-bundle . n G; pr; n G=BI B/ with the Cartan connection !nG given by the Maurer–Cartan form of G, is isomorphic to .P; !/. The next proposition shows, how the Lie algebra of the holonomy group of a Cartan connection is determined by its curvature. Proposition 3. Let ! 2 1 .P; g/ be a Cartan connection on the B-principal fibre bundle P over M and u 2 Px . Then the Lie algebra of the holonomy group Holu .P; !/ is given by ˚ holu .P; !/ D span Ad.hol ! . // B p! .X; Y / j p 2 P; X; Y 2 Tp P; a path in P from u to p g: Furthermore, if W G ! GL.V / is a G-representation, E WD P B V the associated tractor bundle, r ! the covariant derivative on E induced by the Cartan connection ! ! its curvature endomorphism. Then ! and Rr .X; Y / WD ŒrX! ; rY! rŒX;Y ˚ ! ! ! holx .E; r ! / D span Pr1 B Ryr .X; Y / B Pr j X; Y 2 Ty M; a path in M from x to y : Proof. Let !N be the connection form on the extended G-bundle Px , induced by the Cartan connection ! (cf. formula (14)). Furthermore, let uN WD Œu; e 2 Px . According to (16) the Lie algebras holu .P; !/ and holuN .Px ; !/ N coincide. By the Ambrose– Singer Theorem (cf. [KN63], Chapter II, Theorem 8.1) the holonomy algebra of the connection form !N is given by ˚ !N x Yx / j there exists a !-horizontal holuN .Px ; !/ N D span Œp;g .X; N path in Px from uN to Œp; gXx ; Yx 2 TŒp;g Px :

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Let ı.t / D Œ .t/; g.t/ be a path in Px starting in uN D Œu; e. As in the proof of Proposition 1 follows that ı.t / D Œ .t/; g.t / is !-horizontal N if and only if is an arbitrary path in P starting from u and a.t / WD g.t /1 is the development of ! along . Thus ˚ !N ! x x holuN .Px ; !/ N D span Œp;a 1 .X ; Y / j p 2 P; a D hol . / for a path in P from u to p, Xx ; Yx 2 TŒp;a1 Px : Using formula (14), we obtain for the curvatures of the Cartan connection ! on P and the induced connection form !N in Px that !N ! x x Œp;a 1 .X; Y / D Ad.a/ p .X; Y /

for appropriate X; Y 2 Tp P . Hence, holu .P; !/ D holuN .Px ; !/ N ˚ D span Ad.hol ! . // B p! .X; Y / j p 2 P; X; Y 2 Tp P;

a path in P from u to p :

The second formula in the proposition comes from the relation between the curvature ! form !N on the principal bundle Px and the curvature endomorphism Rr on E: !

.q!N .Xx ; Yx // D Œq1 B Ryr .X; Y / B Œq; x D X; .Yx / D Y 2 Ty M . For the holonomy algebra we where q 2 Pxy and .X/ have to take into account all points q that arise by parallel displacement from u along ! curves connecting x with y in M , i.e., all points q with Œq D Pr B Œu. This gives the second statement. In special situations there is a more appropriate formula for the holonomy algebra of a Cartan connection. Let us denote by r End the covariant derivative in the endomorphism bundle End.E/ induced by r ! : rXEnd F WD ŒrX! ; F D rX! B F F B rX!

for F 2 .End.E//:

Furthermore, we fix an arbitrary covariant derivative D on TM . For a 2-form on M with values in End.E/ we define by induction .rQ V1 /.X; Y / WD rVEnd . .X; Y // .DV1 X; Y / .X; DV1 Y / 1 and .rQ Vk1 :::Vk /.X; Y / WD rVEnd ..rQ Vk1 /.X; Y // 1 :::Vk1 k

/.DVk X; Y / .rQ Vk1 /.X; DVk Y / .rQ Vk1 1 :::Vk1 1 :::Vk1

k1 X iD1

z k1 .r V1 :::DV

k

Vi :::Vk1 /.X; Y /:


833

Proposition 4. Let r ! be the covariant derivative on the tractor bundle E defined by the Cartan connection ! and let x 2 M . Consider the following subspace in GL.Ex /: hol0x .E; r ! /

˚ k r! z WD span .r /x .X; Y / j X; Y; V1 ; : : : ; Vk 2 Tx M; 0 k 1 : V1 ::::Vk R

If the dimension of hol0x .E; r E / is constant in x, then this vector space coincides with the holonomy algebra holx .E; r ! /. In particular this is the case if all data are analytic. Proof. This is a translation of the corresponding statements for the infinitesimal holonomy group of affine connections in [KN63], vol. 1. Finally, let us state the basic relation between the holonomy group of a Cartan connection ! and r ! -parallel tractors. Proposition 5. Let ! be a Cartan connection on the B-principal fibre bundle P over M , E D P B V a tractor bundle associated to P , r ! the induced covariant derivative on E and u 2 Px a fixed point. Let B be connected or M be simply connected. Then there is a 1-1 correspondence between the space of parallel sections f' 2 .E/ j r ! ' D 0g and the space of holonomy invariant vectors Œu

fv 2 V j .Holu .P; !// v D vg ' fe 2 Ex j Holx .E; r ! / e D eg: Proof. For v 2 V we define a section 'v 2 .E/ by the parallel transport of the element e D Œu; v 2 Ex with respect to r ! : !

'v .y/ WD Pr .e/ 2 Ey ; where is a curve in M from x to y. The section 'v does not depend on the choice of , is parallel and the map v 7! 'v is bijective between the space of holonomy invariant vectors and the space of parallel sections. This proposition shows on one hand, how useful the knowledge of the holonomy group is if one wants to determine the spaces of parallel sections. The task to solve the differential equation r ! ' D 0 reduces to a problem of linear algebra. On the other hand, if one has a special kind of geometry (for example conformal geometry) which is characterized by the Cartan connection !, then the holonomy group – by its fixed vectors on a representation space – defines distinguished geometric invariants (r ! -parallel object) of the manifold.

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3 Holonomy groups of conformal structures In this section we consider Cartan connections which are associated to conformal manifolds. A conformal manifold of signature .p; q/ is a pair .M; c/, where M is a smooth manifold of dimension n D p C q and c is an equivalence class of metrics of signature .p; q/, where two metrics g and gQ are called equivalent if gQ D fg for a positive smooth function f on M . We always assume that n D p C q 3. For a metric g there is a distinguished covariant derivative on the tangent bundle and a corresponding distinguished connection form on the frame bundle of M , the Levi-Civita connection. Contrary to that, there cannot be a distinguished covariant derivative on the tangent bundle or a connection form on the frame bundle of M depending only on the conformal structure c itself, since one needs derivatives up to second order to determine conformal diffeomorphisms uniquely. In order to establish a conformally invariant differential calculus for conformal manifolds one uses techniques of Cartan geometry. In this approach the bundle of frames on M is replaced by a larger bundle P 1 , which is obtained from the conformal frame bundle by a process called 1. prolongation. This first prolongation comes with a distinguished Cartan connection, the normal conformal Cartan connection ! nor . The normal conformal Cartan connection plays in conformal geometry the same role the Levi-Civita connection is playing in (pseudo)Riemannian geometry. It is used to define conformally invariant differentiations on all tractor bundles defined on conformal manifolds. We define the holonomy group of the conformal manifold .M; c/ to be the holonomy group of this normal conformal Cartan connection Hol.M; c/ WD Hol.P 1 ; ! nor /: For the convenience of the reader and to fix the notation we recall shortly some basic facts on conformal Cartan geometry, in particular the definition of the normal conformal Cartan connection (for details cf. for example [K72], [CSSI], [CSSII], [F05]).

3.1 Cartan connections in conformal geometry First, we want to describe the groups B and G (compare Section 2) which are used in conformal geometry. For this purpose let us consider the isotropic cone C p;q WD fx 2 RpC1;qC1 n f0g j hx; xipC1;qC1 D 0g in the pseudo-Euclidean space .RpC1;qC1 ; h ; ipC1;qC1 /. The projectivization Qp;q WD P C p;q can be equipped with a conformal structure c of signature .p; q/: c WD Œ h ; ipC1;qC1 ;


835

where W Qp;q ! CCp;q is an arbitrary section from the projectivization into a fixed component CCp;q of the cone. The conformal manifold .Qp;q ; c/ is called Möbius sphere of signature .p; q/. For Riemannian conformal structures (signature .0; n/) the Möbius sphere is just the sphere S n equipped with the conformal class of the round metric. It is the conformal boundary of the hyperbolic space H nC1 as well as the conformal compactification of the Euclidean space. For Lorentzian conformal structures (signature .1; n 1/) the Möbius sphere appears in physics literature under the name ‘Einstein universe Einn ’. It is the conformal boundary of the Anti de Sitter space AdSnC1 as well as the conformal compactification of the Minkowski space. The group G we are working with in conformal geometry is G WD SO.pC1; qC1/. It acts transitively and conformally on the Möbius sphere. Moreover, G (if n is odd) resp. G=˙I (if n is even) is isomorphic to the group of orientation preserving conformal diffeomorphisms of the Möbius sphere .Qp;q ; c/. For the group B G we take the stabilizer of a fixed isotropic line p1 2 Qp;q . Hence, the Möbius sphere can be identified with the n-dimensional homogeneous space G=B. This explains the choice of these groups. Let b g WD so.p C 1; q C 1/ be the Lie algebra of B and let us denote by b1 b the n-dimensional abelian ideal in b obtained as the orthogonal complement of b g with respect to the Killing form of g. Then B1 WD exp b1 B is a normal abelian subgroup in B and the factor group B=B1 is isomorphic to the conformal group CO.p; q/ WD fA 2 GL.n/ j there exists a > 0 such that hAx; Ayip;q D hx; yip;q for all x; y 2 Rn g ' RC O.p; q/ ' R SO.p; q/: For short, we will denote B0 WD CO.p; q/. The projection from B to the linear conformal group B0 corresponds geometrically to the mapping f 2 Conf C .Qp;q / 7! dfp1 2 Conf.Tp1 Qp;q /. The sequence b1 b g is an Ad.B/-invariant filtration of g. The Killing form of g gives an Ad.B/-equivariant identification between b1 and the dual Lie algebra .g=b/ . In the following we will consider g D so.p C 1; q C 1/ as j1j-graded Lie algebra, i.e., we decompose g into a sum of subspaces g D b1 ˚ b0 ˚ b1

with Œbi ; bj g biCj ;

where furthermore, b D b0 ˚ b1 . Let us describe such a splitting in term of matrices. pC1;qC1 n We fix a basis .f0 ; e1 ; : : : ; en ; fnC1 / in R 0 0 1 D R ˚ R ˚ R such that the Ip 0 Gram matrix of h ; ipC1;qC1 has the form 0 J 0 where J D and Ik is 0 Iq 1 0 0

the identity matrix with k rows. For a row vector z 2 .Rn / let z ] WD J z > . For a column vector x 2 Rn let x [ WD x > J . With respect to the basis .f0 ; e1 ; : : : ; en ; fnC1 /

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the Lie algebra g D so.p C 1; q C 1/ is given by the matrices ³ ˇ ² a z 0 ˇ x 2 Rn ; z 2 .Rn / ] : so.p C 1; q C 1/ D M.x; .A; a/; z/ WD x A z ˇ a 2 R; A 2 so.p; q/ 0 x [ a We set b1 WD fM.x; .0; 0/; 0/ j x 2 Rn g ' Rn ; (19) b1 WD fM.0; .0; 0/; z/ j z 2 .Rn / g ' Rn ; b0 WD fM.0; .A; a/; 0/ j A 2 so.p; q/; a 2 Rg ' co.p; q/ D so.p; q/ ˚ R For elements x 2 b1 , .A; a/ 2 b0 , z 2 b1 we have Œ.A; a/; x D ax C Ax 2 b1 ; Œ.A; a/; z D az zA 2 b1 ;

(20)

] [

Œx; z D .xz z x ; zx/ 2 b0 : Clearly, b D b0 ˚ b1 . We identify b1 and b1 using the Ad.B0 /-equivariant isomorphism given by the Killing form Bg of g 1 Bg .x; z/; x 2 b1 ; z 2 b1 : 2n This corresponds to the standard dual pairing between Rn and .Rn / . The adjoint representation of B0 on b1 yields an isomorphism between B0 and the group of linear conformal isomorphisms of .b1 D Rn ; h ; ip;q /. Now, let .M n ; c/ be a conformal manifold. We call a frame .xI s1 ; : : : ; sn / on M conformal frame, if the vectors .s1 ; : : : ; sn / form a pseudo-orthonormal basis in Tx M with respect to a metric g in the conformal class c. We denote by 0 W P 0 ! M the bundle of all conformal frames of .M; c/. This bundle is a reduction of the frame bundle of M to the linear conformal group B0 D CO.p; q/ GL.n/. The basic B-principal fibre bundle over M , which is used in conformal geometry, is obtained from the conformal frame bundle P 0 by 1. prolongation. There are several equivalent models for the first prolongation of P 0 . We will describe the first prolongation using horizontal subspaces in the tangent space of P 0 . For this purpose, let u 2 P 0 be a conformal frame in x 2 M . The vertical tangent space of P 0 in u is the tangent space of the fibre Px0 in u: .z/x WD

T vu P 0 WD Tu .Px0 /: By a horizontal subspace H Tu P 0 we mean a subspace which is complementary to the vertical tangent space T vu P 0 Tu P 0 . To any horizontal subspace H Tu P 0 we can associate a 2-form t .H / 2 ƒ2 .b1 / ˝ b1 called the torsion of H , which we want to define now: First note that the identification between the matrix representation of B0 on Rn and the adjoint representation of B0 on b1 allows us to represent the tangent bundle of M in the form TM D P 0 .B0 ;Ad/ b1 :


837

In this representation the displacement form (cf. Example 2) is a 1-form on P 0 with values in b1 :

u .X/ WD Œu1 d u0 .X/ 2 b1

for u 2 P 0 ; X 2 Tu P 0 :

In particular, jH W H ! b1 is an isomorphism for any horizontal space H Tu P 0 . Thus we can define the 2-form t .H / 2 ƒ2 .b1 / ˝ b1 as t .H /.v; w/ WD d u . jH /1 .v/; . jH /1 .w/ 2 b1 ; v; w; 2 b1 : A horizontal space H is called torsion-free, if t .H / D 0. With this preparations we can define the first prolongation P 1 by P 1 WD fH Tu P 0 j u 2 P 0 and H horizontal and torsion-free g: There are natural projections 1

0

W P 1 ! P 0 ! M;

H Tu P 0 7! u 2 Px0 7! xI

W P 1 ! M is a principal fibre bundle with structure group B, whereas 1 W P 1 ! P 0 is a principal fibre bundle with structure group B1 . Let us explain the action of B on P 1 . Any element b 2 B D B0 Ë B1 can be represented by b D b0 exp.Z/, where b0 2 B0 and Z 2 b1 . An element b0 2 B0 ‘transports’ the horizontal space H Tu P 0 into the point ub0 using the right action of B0 on P 0 : H b0 WD dRb0 .H / Tub0 P 0 : An element exp Z 2 B1 acts on P 1 by ‘rotating’ H inside Tu P 0 : z / .u/ j X 2 H g: H exp Z WD fX C ŒZ; .X g The bundle .P 1 ; ; M I B/ is called the first prolongation of the conformal frame bundle P 0 . It is the basic bundle that is used in conformal differential geometry. Here the Cartan connections live, in particular the distinguished normal Cartan connection of conformal geometry.

3.2 The normal conformal Cartan connection in invariant form In order to distinguish a special Cartan connection, we first describe a certain affine subspace of the space of all Cartan connections of type G=B on P 1 , the space of admissible Cartan connections. Let ! 2 1 .P 1 ; g/ be a Cartan connection on P 1 . With respect to the grading g D b1 ˚ b0 ˚ b1 the 1-form ! splits into the components ! D !1 ˚ !0 ˚ !1 : The invariance properties of a Cartan connection show that the 1-forms !1 and !0 are horizontal with respect to the B1 -bundle 1 W P 1 ! P 0 . Moreover, the right

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translation of !1 and !0 with respect to b1 D exp Z 2 B1 is given by Rb1 !1 D 0;

Rb1 !0 D !0 C ŒZ; !1 g :

We call the Cartan connection ! admissible if its b1 -part and its b0 -part are given by the following conditions: (1) !1 D . 1 /

C

1 (2) d H ./ .!0 /H ./.u/ 2 H Tu P 0 D H ˚ T vu P 0

for all 2 TH P 1 :

Clearly, two admissible Cartan connections differ only by the b1 -component. Let us note that any admissible Cartan connection ! is torsion-free, i.e., the curvature ! takes its values in the subalgebra b g. To see this, remember that the 2-form ! is horizontal with respect to the B-bundle W P 1 ! M and check for the b1 -part of the curvature the formula .! 1 /H .; / D t.H /.v; w/;

where !H ./ D v; !H ./ D w 2 b1 :

The normal conformal Cartan connection is an admissible Cartan connection which is determined by an additional condition on its curvature. The curvature of a Cartan connection ! can obviously be identified with the curvature function ! 2 C 1 .P 1 ; ƒ2 .b1 / ˝ g/, ! 1 1 ! .H /.v; w/ WD H .!H .v/; !H .w//;

v; w 2 b1 :

Let @ W ƒ2 .b1 / ˝ g ! b1 ˝ g be defined by

@ .X / WD

n X

Œvi ; .vi ; X/g ;

X 2 b1 ;

iD1

where .v1 ; : : : ; vn / is a basis in b1 and .v1 ; : : : ; vn / is the dual basis in b1 ' b1 . The normal conformal Cartan connection ! nor on P 1 is characterized in the following way: Proposition 6. There is a unique admissible Cartan connection ! nor 2 1 .P 1 ; g/ such that nor @ ! D 0: ! Proof. Admissible Cartan connections ! satisfy 1 D 0. Hence, for admissible ! Cartan connections the condition @ D 0 is equivalent to @ 0! D 0, where i! denotes the projection of ! onto bi . Then, using the Lie algebra structure of g, one shows that the condition @ 0! D 0 determines !1 uniquely. For details cf. [K72] Theorem IV.4.2, [CSSII], or [F05], Chapter 6.


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We defined the normal conformal Cartan connection starting with a conformal manifold .M; c/. In Cartan geometry often the opposite viewpoint is used. One works with an arbitrary B-principal fibre bundle .P; ; M I B/ over a manifold M and a Cartan connection ! 2 1 .P; g/ satisfying @ ! D 0, where G D SO.p C1; q C1/ and B D SO.p C 1; q C 1/p1 as above. In this case, P 0 WD P =B1 is a B0 -principal fibre bundle over M . The b1 -part !1 of the Cartan connection projects down to a 1-form 2 1 .P 0 ; b1 / of type Ad with the vertical tangent bundle T vP 0 as kernel. Such a pair .P 0 ; / defines a reduction of the frame bundle of M to the linear conformal group CO.p; q/ D B0 and therefore determines a conformal structure c on M . Let P 1 be the 1. prolongation of the conformal frame bundle of .M; c/ for this conformal structure c. Then there exists a principal fibre bundle isomorphism W P ! P 1 such that ! nor D !.

3.3 The normal conformal Cartan connection in its metric form For many applications it is useful to describe the invariantly defined normal conformal Cartan connection ! nor and the induced covariant derivative on tractor bundles in terms of the metrics g in the conformal class c. In order to state these formulas we need various curvature tensors of the pseudoRiemannian manifold .M; g/. In the following we will denote by Rg the curvature endomorphism given by the Levi-Civita connection r g of g, g Rg .X; Y / WD rXg rYg rYg rXg rŒX;Y ;

by Ricg the Ricci tensor and by scalg the scalar curvature, Ricg .X; Y / WD trace.Z 7! Rg .Z; X /Y /; scalg WD traceg Ricg : Further, K g denotes the Schouten tensor: K g WD

1 1 scalg g Ricg : n 2 2.n 1/

We often consider the Schouten tensor as endomorphism K g W TM ! T M , using the identification K g .X/.Y / WD K g .X; Y /. The skew-symmetric derivative of the endomorphism K g gives the Cotton–York tensor C g : C g .X; Y / WD rXg .K g /.Y / rYg .K g /.X /: Finally, W g is the Weyl tensor considered as 2-form with values in the g-skewsymmetric endomorphism on TM : W g .X; Y / WD Rg .X; Y /X [ ˝K g .Y /] K g .X /˝Y CK g .Y /˝X CY [ ˝K g .X /] ; where X [ for a vector field X denotes the dual 1-form X [ .Z/ WD g.X; Z/ and ] for a 1-form denotes the dual vector field .Z/ D g.] ; Z/.

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Before we describe the normal conformal Cartan connection in its metric form, we show how torsion-free connection forms on the conformal frame bundle P 0 over .M; c/ induce admissible Cartan connections on the first prolongation P 1 . Let A 2 1 .P 0 ; b0 / be a connection form on the conformal frame bundle P 0 and

2 1 .P 0 ; b1 / the displacement form as above. The torsion of A is the 2-form T A 2 2 .P 0 ; b1 / defined by T A WD d C ŒA; : The connection form A defines a right invariant distribution of horizontal spaces Hu WD Ker Au Tu P 0 with torsion t .Hu /. .X/; .Y // D TuA .X; Y /;

X; Y 2 Tu P 0 :

Therefore, any torsion-free connection form A on P 0 defines a B0 -equivariant smooth section A W P 0 ! P 1 ;

u 7! Ker Au ;

in the B1 -bundle 1 W P 1 ! P 0 (and vice versa). Proposition 7. For any torsion-free connection form A 2 1 .P 0 ; b0 / there exists a unique admissible Cartan connection ! A 2 1 .P 1 ; g/ such that . A / ! A D C A:

(21)

Proof. Let A W P 0 ! P 1 be the B0 -equivariant section defined by A. We consider the map W P 1 ! b1 given by H D . 1 .H // exp..H //

for all H 2 P 1

and define ! A 2 1 .P 1 ; g/ by A !.u/ WD .. 1 / /.u/ C .. 1 / A/.u/ C d .u/ ; u 2 P 0 ; A A !H WD Ad.b11 / B !.u/ B dRb 1 ;

H D .u/ b1 2 P 1 :

1

A direct calculation shows that ! A is an admissible Cartan connection. Obviously, . A / ! A D C A. Now, let us consider a metric g in the conformal class c. We denote by P g P 0 the subbundle of all g-orthonormal frames and by Ag the Levi-Civita connection of g on P g (which extends to a torsion-free connection on P 0 ). The choice of g 2 c gives us two data: (1) A reduction of the B-bundle P 1 to the bundle of g-orthonormal frames P g g

A

i

g W P g ,! P 0 ! P 1 : g

g

(2) An admissible Cartan connection ! A on P 1 with . g / ! A D C Ag .


841

g

Now, we compare the admissible Cartan connections ! nor and ! A . For this purpose we fix an orthonormal basis .e1 ; : : : ; en / in .b1 D Rn ; h ; ip;q / and denote by .e1 ; : : : ; en / the dual basis in b1 ' b1 . Let u D .s1 ; : : : ; sn / be a conformal frame in a point x 2 M . Recall that we identified the tangent bundle TM with the associated bundle P 0 B0 b1 in such a way that sj D Œu; ej . If u 2 P g P 0 , then .s1 ; : : : ; sn / is g-orthonormal. Proposition 8. Let g be a metric in the conformal class c. Then the normal conformal Cartan connection is given by g

nor A !H ./ D !H ./ C

n X

Kxg .d H ./; sj / ej for 2 TH P 1 ;

(22)

j D1

where x D .H / and .s1 ; : : : ; sn / D 1 .H /. In particular, if g W P g ! P 1 is the O.p; q/-reduction of P 1 given by the metric g, then .. g / ! nor /u .X/ D u .X/ C Agu .X / C

n X

Kxg d u0 .X /; sj ej for X 2 Tu P g ;

(23)

j D1

where u D .s1 ; : : : ; sn / 2 P g and x D 0 .u/. g

Proof. The calculation of the difference ! nor ! A , which is a 1-form on P 1 with values in b1 , is straight forward. Details can be found in [CSSII] or [F05], Satz 6.8., Formula (23) follows from (21) in Proposition 7.

3.4 The tractor connection and its curvature As we explained in Section 2, any Cartan connection on P 1 defines a covariant derivative on an associated tractor bundle. In this section we want to express the covariant derivative induced by the normal conformal Cartan connection and its curvature in terms of a metric g in the conformal class c. Let W G ! GL.V / be a representation of the group G D SO.p C 1; q C 1/ on a vector space V , E D P 1 B V the associated tractor bundle over the conformal manifold .M; c/ and r nor W .E/ ! .T M ˝ E/ the covariant derivative on E induced by the normal conformal Cartan connection ! nor according to (15). Let us fix again a metric g in the conformal class c. As we saw in Section 3.3, the choice of g allows us to reduce the bundles P 0 and P 1 to the bundle of g-orthonormal frames P g . Hence, the vector bundles E, TM , T M and the bundle so.TM; g/ of skew-symmetric endomorphisms on .TM; g/ can be expressed as bundles associated

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to the O.p; q/-bundle P g : E D Pg TM D P g T M D P g so.TM; g/ D P g

.O.p:q/;/ V; .O.p;q/;Ad/ b1 ; .O.p;q/;Ad/ b1 ; .O.p;q/;Ad/ so.p; q/:

The Levi-Civita connection of g defines a covariant derivative r g W .E/ ! .T M ˝ E/ (cf. (8)). Moreover, the representation induces g-dependent maps g W TM ! End.E; E/; g W T M ! End.E; E/; g W so.TM; g/ ! End.E; E/: To define these map, choose a frame u 2 Pxg and set

g .‡ /' WD u; Œu1 ‡ Œu1 ' 2 Ex ;

(24)

where ' 2 Ex and ‡ is a tangent vector, a covector or a skew-symmetric map in x. It is easy to check that the right hand side does not depend on the choice of u 2 Pxg . We obtain the following ‘metric’ formula for the normal tractor derivative r nor and nor . its curvature endomorphism Rnor .X; Y / WD ŒrXnor ; rYnor rŒX;Y Proposition 9. Let W G ! GL.V / be a representation of G WD SO.p C 1; q C 1/ and E WD P 1 B V the associated tractor bundle. Then, for any metric g 2 c the normal tractor derivative on E and its curvature are given by rXnor D rXg C g .X / C g .K g .X //; Rnor .X; Y / D g .W g .X; Y // C g .C g .X; Y //:

(25) (26)

Proof. Let 2 .E/ be locally represented by jU D Œu; v D Œ g B u; v, where u W U ! P g is a local smooth section in P g and v W U ! V is a smooth function. Then, from the formulas (8),(15), (23) and (24) we obtain

(15) rXnor D u; X.v/ C ! nor .d. g B u/.X // v (23)

h

g

D u; X.v/ C .du.X// C A .du.X // C

(8)

D

rXg

j D1

i K g .X; sj /ej v

C u; Œu1 X v C u; Œu1 .K g .X / v

D rXg C g .X/ C g .K g .X //:

(24)

n X


843

Inserting (25) into the formula for the curvature endomorphism Rnor gives g

Rnor .X; Y / D Rr .X; Y / C Œg .X /; g .Y / C Œg .K g .X //; g .K g .Y // C Œg .X/; g .K g .Y // C Œg .K g .X //; g .Y / C ŒrXg ; g .Y / C Œg .X /; rYg g .ŒX; Y / C ŒrXg ; g .K g .Y // C Œg .K g .X //; rYg g .K g .ŒX; Y //: Using (24) for the endomorphism g , formula .20/ for the Lie bracket Œb1 ; b1 g and taking into account that the subalgebras b1 and b1 of g are abelian, we obtain g

Rnor .X; Y / D Rr .X; Y / C Œg .X /; g .K g .Y // Œg .Y /; g .K g .X // C g .C g .X; Y // D g .W g .X; Y // C g .C g .X; Y //:

As a next step, we apply these formulas to the standard representation W G ! GL.RpC1;qC1 / given by the matrix action. We denote the associated tractor bundle by T .M / D P 1 B RpC1;qC1 and call it the standard tractor bundle. The scalar product h ; ipC1;qC1 on RpC1;qC1 induces a bundle metric h ; i on T .M /. The normal covariant derivative r nor on T .M / is metric X.h; i/ D hrXnor ; i C h; rXnor i: We fix again a metric g in the conformal class c and reduce P 1 to the frame bundle P . If we restrict the representation to the subgroup O.p; q/ we obtain the splitting g

RpC1;qC1 ' R ˚ Rp;q ˚ R;

˛f0 C y C ˇfnC1 7! .˛; y; ˇ/

into three O.p; q/-representations, where O.p; q/ acts trivial on the both 1-dimensional summands and by matrix action on Rp;q . Hence, any metric g in the conformal class c defines the following splitting of the standard tractor bundle g

T .M / ' R ˚ TM ˚ R

(27)

where R denotes the trivial line bundle over M . In this identification the bundle metric is given by h.˛1 ; Y1 ; ˇ1 / ; .˛2 ; Y2 ; ˇ2 /i D ˛1 ˇ2 C ˛2 ˇ1 C g.Y1 ; Y2 /: Let X be a vector field and a 1-form on M and a g-skew-symmetric endomorphism on TM . Using (19) and (24) we obtain for the endomorphisms g .X /, g ./ and g ./ the following formulas on sections of T .M /, represented as a triple

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.˛; Y; ˇ/ of two functions ˛, ˇ and a vector field Y on M : 1 0 1 0 1 0 1 0 .Y / ˛ 0 ˛ g .X/ @Y A D @ ˛ X A ; g ./ @Y A D @ˇ ] A ; ˇ g.X; Y / ˇ 0 0 1 0 1 ˛ 0 g @ A @ Y .Y /A : . / D ˇ 0 Using these formulas in Proposition 9 we obtain for the normal derivative r nor on the standard tractor bundle the following g

Proposition 10. Suppose that g is a metric in the conformal class c and let T .M / ' R ˚ TM ˚ R be the corresponding splitting of the standard tractor bundle. The smooth sections of T .M / are identified with triples .˛; Y; ˇ/, where ˛ and ˇ are smooth functions and Y is a smooth vector field on M . Then the tractor derivative r nor induced on T .M / by the normal Cartan connection ! nor is given by 1 0 1 0 X.˛/ C K g .Y; X / ˛ C B C B (28) rXnor @Y A D @rXg Y C ˛X ˇK g .X /] A : ˇ X.ˇ/ g.X; Y / The curvature endomorphism of r nor satisfies 1 0 1 0 C g .X1 ; X2 /Y ˛ C B C B Rnor .X1 ; X2 / @Y A D @W g .X1 ; X2 /Y ˇ C g .X1 ; X2 /] A : ˇ 0

(29)

3.5 Conformal holonomy and Einstein metrics The conformal holonomy group is an invariant of the conformal structure and it is natural to study which information about the conformal structure it contains. In this section we shortly review the relation between the conformal holonomy group of .M; c/ and the existence of Einstein metrics in the conformal class c. For different proofs we refer to the papers of S. Armstrong [A05] (Riemannian case), F. Leitner [L05a] (arbitrary signature) and T. Leistner [Lei05] (Lorentzian case). To begin with let us state the following direct application of the ‘metric’ formula for the tractor connection r nor on T .M / (cf. Proposition 10). Proposition 11. Let .M; c/ be a conformal manifold. If there is an Einstein metric g in the conformal class c, then there exists a nontrivial parallel section 2 T .M /: r nor D 0: On the other hand, if we suppose that there is a nontrivial r nor -parallel z M and an Einstein section 2 T .M /, then there is an open dense subset M


845

metric g in the conformal class cjMz . In both cases, for the scalar curvature scalg of g holds: • scalg > 0 () is timelike. • scalg < 0 () is spacelike. • scalg D 0 () is lightlike. Proof. Let g 2 c be an Einstein metric. Then the Schouten tensor K g is a multiple of the metric g, 1 Kg D scalg g: 2n.n 1/ Hence, decomposing T .M / with respect to g as it was described in the latter section, we obtain 1 0 1 0 scalg Y[ d˛ 2n.n1/ ˛ C B C B g scalg C r nor @Y A D B @r Y C .˛ C ˇ 2n.n1/ /IdA : ˇ dˇ Y [ If scalg D 0, then D .0; 0; 1/? is a lightlike parallel section in .T .M /; r nor /. If scalg scalg 6D 0, then D . 2n.n1/ ; 0; 1/? is parallel and timelike if scalg > 0 resp. g spacelike if scal < 0. Now, let us suppose that there is a nontrivial parallel section 2 .T .M //. We fix an arbitrary metric g 2 c. Then with respect to this metric g we represent as D .˛; Y; ˇ/? . The parallelity condition r nor D 0 implies Y D gradg ˇ; d˛ D K g .gradg ˇ/; ˛g D ˇK g Hessg .ˇ/:

(30)

z WD fx 2 M j ˇ.x/ 6D 0g is an open dense subset in M , otherwise Therefore, M would vanish on an open set, hence on M , since is parallel. Now, consider the z . With the general transformation formulas (cf. [Be87], p. 58) metric gQ WD ˇ 2 g on M z and (30) we obtain for the Schouten tensor of gQ on M 1 K gQ D K g C ˇ 2 jdˇj2 g ˇ 1 Hessg ˇ 2 1 2 D ˛ˇ C jdˇj g: Q 2 Therefore, gQ is an Einstein metric and the scalar curvature is given by

1 1 1 1 scalgQ D ˛ˇ C jdˇj2 D ˛ˇ C g.Y; Y / D h; i: 2n.n 1/ 2 2 2

This concludes the proof.

This result can be generalized to the following statements: Let .M p;q ; c/ be a simply connected conformal manifold of dimension n D p C q and let us denote by H D Hol.M; c/ the conformal holonomy group of .M; c/.

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(1) Holonomy characterization of conformal Einstein spaces. If there is a 1-dimensional H -invariant subspace V 1 RpC1;qC1 , then there exists an Einstein z M . The scalar metric g in the conformal class c over an open dense subset M curvature of g is positive, negative or zero, if V is timelike, spacelike or lightlike, respectively. (2) Local splitting theorem. If there is a non-degenerate H -invariant subspace V k RpC1;qC1 of dimension k 2. Then, locally outside a singular set, .M; c/ is a product .M1k1 M2nkC1 ; Œg1 g2 /, where .Mi ; gi / are Einstein manifolds with scalar curvature k.k 1/ scalg2 : scalg1 D .n k/.n k 1/ The conformal holonomy group of .M; c/ is the product of the conformal holomomy groups of .M1 ; Œg1 / and .M2 ; Œg2 /. (3) Holonomy of conformal Einstein spaces. Let g 2 c be an Einstein metric in the conformal class c. Then the conformal holonomy can be expressed by the metric holonomy of a certain ambient space. In case of non-zero scalar curvature scalg z ; g/ we consider the Ricci-flat pseudo-Riemannian manifold .M Q and the cone .CM; gC / over .M; g/: z WD R M RC ; M CM WD RC M;

n.n 1/ .dt 2 ds 2 / C t 2 g; scalg n.n 1/ 2 gC WD d C 2 g: scalg gQ WD

We obtain z ; g/ Q D Hol.1;x/ .CM; gC /: Holx .M; c/ D Hol.1;x;1/ .M In case scalg D 0, we consider the Ricci-flat pseudo-Riemannian manifold z WD R M RC ; M for which

gQ WD 2dsdt C t 2 g

z ; g/: Holx .M; c/ D Hol.1;x;1/ .M Q

These statements can be used to give a complete classification of the conformal holonomy groups of simply connected Riemannian conformal structures (cf. [A05]). To understand this, remember that for a Riemannian conformal structure the holonomy group Hol.M n ; c/ is a subgroup of SO.1; n C 1/. It is known that any connected irreducible acting subgroup of SO.1; n C 1/ equals SO0 .1; n C 1/ (cf. [DO01]). Hence, if the conformal holonomy group of a simply connected Riemannian conformal manifold .M n ; c/ is not the full group SO0 .1; n C 1/, it admits a holonomy invariant subspace. But then, either c contains an Einstein metric on an open dense set or (locally) a product of Einstein metrics. Therefore, one has only to classify the conformal holonomy groups of indecomposable Einstein spaces. The identification of the conformal holonomy group of an Einstein spaces with the metric holonomy group of a


847

Ricci-flat ambient space allows then to apply known classification results for metric holonomy groups. Proposition 12 ([A05]). Let .M n ; Œg/ be a simply connected indecomposable Riemannian conformal Einstein space of dimension n 4 and let Hol.M; Œg/ SO0 .1; n C 1/ be its conformal holonomy group. (1) If scalg < 0, then Hol.M; Œg/ D SO0 .1; n/. /, (2) If scalg > 0, then Hol.M; Œg/ is one of the groups: SO.n C 1/, SU. nC1 2 nC1 Sp. 4 /, G2 if n D 6 or Spin.7/ if n D 7. (3) If scalg D 0, then Hol.M n ; Œg/ is one of the groups: SO.n/ Ì Rn , SU. n2 / Ì Rn , Sp. n4 / Ì Rn , G2 Ì R7 if n D 7 or Spin.7/ Ì R8 if n D 8. For indefinite metrics and indefinite conformal structures the classification of holonomy groups is open and much more difficult than in the Riemannian case. Let us consider the case of a Lorentzian conformal manifold .M 1;n1 ; c/. Then the conformal holonomy group Hol.M; c/ is contained in SO.2; n/. If it acts nonirreducible, than – locally – c contains an Einstein metric (1-dimensional invariant subspace), a product of 2 Einstein metrics with related scalar curvatures (k-dimensional non-degenerate invariant subspace, k 2) or a metric with totally isotropic Ricci tensor and recurrent lightlike vector field (2-dimensional totally isotropic invariant subspace) (see [Lei05]). The classification of Lorentzian conformal structures with irreducible holonomy group is not completed yet. In the Section 4 we will describe even-dimensional Lorentzian conformal manifolds with conformal holonomy group in the irreducible subgroup SU.1; n2 / SO.2; n/.

3.6 Conformal holonomy and conformal Killing spinors In this section we want to explain the relation between conformal holonomy groups and solutions of the conformally invariant twistor equation on spinors (conformal Killing spinors). To begin with, we recall the definition of conformal Killing spinors. Afterwards we show that conformal Killing spinors correspond to parallel sections in the spin tractor bundle for the normal tractor derivative. Therefore, they are directly related to conformal holonomy groups. For an detailed introduction to spin geometry of Riemannian and pseudo-Riemannian manifolds we refer to [B81] or [Fr00], for basic results about conformal Killing spinors see [PR86], [BFGK91], [Lew91], [KR96], [KR96], [B99], [L01], [BL04], [L05a] and the references therein. Let .M p;q ; g/ be a space- and time-oriented semi-Riemannian spin manifold of dimension n D p C q 3. We denote by S g the spinor bundle with respect to the metric g and by g W T M ˝ S g ! S g the Clifford multiplication. The spinor bundle S g is equipped with a natural inner product h ; i, which is non-degenerate (but indefinite, if g is indefinite). The bundle of 1-forms with values in the spinor

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bundle decomposes into two subbundles T M ˝ S g D V g ˚ Ker g ; where V g , being the orthogonal complement to the subbundle Ker g , is isomorphic to S g . Obviously, we obtain two differential operators of first order by composing the g spinor derivative r S induced by the Levi-Civita connection of g with the orthogonal projections onto each of these subbundles, the Dirac operator D.g/, rS

g

pr S g

D.g/ W .S g / ! .T M ˝ S g / D .S g ˚ Ker g / ! .S g /; and the twistor operator P .g/, rS

g

pr Ker g

P .g/ W .S g / ! .T M ˝ S g / D .S g ˚ Ker g / ! .Ker g /: Both operators are conformally invariant. More precisely, if gQ D e 2 g is a conformal change of the metric, the Dirac and the twistor operator change by D.g/ Q D e P .g/ Q De

nC1 2

2

D.g/e

P .g/e

2

n1 2

;

:

This relates the study of solutions of the Dirac equation D.g/' D 0 and of the twistor equation P .g/' D 0 to conformal geometry: it is enough to study these equations for an appropriate "simple" metric in the conformal class of g. Studying these two equations more closely, it turns out that they are rather different in nature. The latter one, the twistor equation, is of so-called twistor type, that means it is equivalent to parallel sections of a certain covariant derivative on a certain vector bundle. To see this, let us first remark that a spinor field ' 2 .S / is a solution of the twistor equation P .g/' D 0 if and only if it satisfies the conformal Killing spinor equation 1 X D.g/ ' D 0 for all vector fields X ; (31) n where the denotes the Clifford multiplication. It is obvious that parallel spinors g g (r S ' D 0) and Killing spinors (rXS ' C X ' D 0 for some 2 C n f0g) are special solutions of (31). The name conformal Killing spinor for solutions of (31) reflects the fact that the Dirac current of ', i.e., the vector field V' on M associated to the spinor ' by g

rXS ' C

g.V' ; X/ D i pC1 hX '; 'i

for all vector fields X ;

(32)

is a conformal Killing vector field. Using (31) one obtains for any conformal Killing spinor ' the integrability condition n g (33) rXS D.g/' D K g .X / '; 2 where K g is the Schouten tensor of g (cf. [BFGK91], Theorem 1.3). Then, a direct calculation using (31) and (33) yields the following result.


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Proposition 13 ([BFGK91], Theorem 1.4). Let E g be the vector bundle E g D S g ˚ g S g and r E the covariant derivative on E g defined by ! g rXS X Eg rX WD : (34) g 12 K g .X/ rXS ' g 1 D 0. Conversely, if Then, for any conformal Killing spinor ' holds r E n D.g/' ' Eg is r -parallel, then ' is a conformal Killing spinor and D n1 D.g/'. The latter proposition shows that conformal Killing spinors, i.e., the solutions of g the twistor equation P .g/' D 0, are in 1-1-correspondence with the r E -parallel sections in the double spinor bundle E g . Hence, by the general philosophy which we explained in Section 1, we can use the holonomy group of the covariant derivative g r E to study the existence of solutions of the twistor equation on spinors. We now want to give a reinterpretation of the situation in the framework of conformal Cartan geometry. We will show that the bundle E g is isomorphic to the spin tractor bundle g over the conformal manifold .M; c D Œg/ and that the covariant derivative r E is the normal covariant derivative on this tractor bundle. Hence, the holonomy group of g r E is just given by the holonomy group of the conformal structure c D Œg of the metric g in question. Therefore, the conformal holonomy group Hol.M; c/ contains the full information about the existence of conformal Killing spinors. To explain this in more detail, we first extend the basics of conformal Cartan geometry which we described in Sections 3.1–3.4. to the spin case in the obvious way. Let .M; c/ be a time- and space-oriented conformal manifold of signature .p; q/. In this section we denote by P 0 the bundle of all conformal frames which respect the given time- and space-orientation. The structure group is the connected component CO0 .p; q/ WD RC SO0 .p; q/ of the linear conformal group. Let CSpin0 .p; q/ be the conformal spin group CSpin0 .p; q/ WD RC Spin0 .p; q/ and W CSpin0 .p; q/ ! CO0 .p; q/ the 2-fold covering of the linear conformal group given by the identity in the first component and the usual 2-fold covering of the orthogonal group by the spin group in the second component, which we will denote for simplicity also by . A conformal spin structure of .M; c/ is a CSpin0 .p; q/principal fibre bundle Q0 over M with a smooth map f 0 W Q0 ! P 0 which respects the bundle projections and the group actions f 0 .q A/ D f 0 .q/ .A/

for all q 2 Q0 ; A 2 CSpin0 .p; q/;

0 B f 0 D 0: We call the conformal manifold .M; c/ a spin manifold, if it admits a conformal spin structure. This is equivalent to the existence of spin structures for any metric g 2 c. In fact, any conformal spin structure .Q0 ; f 0 / of .M; c/ induces a metric spin structure

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.Qg ; f g / for the metric g 2 c by reduction: Qg WD .f 0 /1 .P g /;

f g WD f 0 jQg ;

where P g P 0 denotes the subbundle of time- and space-oriented g-orthonormal frames. On the other hand, since P 0 D P g SO0 .p;q/ CO0 .p; q/, any spin structure .Qg ; f g / of a metric g 2 c gives rise to a conformal spin structure for the conformal class c by enlargement: Q0 WD Qg Spin0 .p;q/ CSpin0 .p; q/;

f 0 WD f g :

Hence, due to the orientability assumption, .M; c/ is spin if and only if the second Stiefel–Whitney class of M vanishes: w2 .M / D 0. Now, let G D SO0 .p C 1; q C 1/ be the connected component of the pseudoorthogonal group and B G the stabilizer of the isotropic line p1 of the Möbius z WD Spin0 .p C 1; q C 1/ acts transitively on Qp:q sphere Qp:q . Then the spin group G z If Bz1 WD exp. 1 z z z z with p1 -stabilizer B. .b1 // B G then B=B1 is isomorphic 0 0 z to the conformal spin group B0 WD CSpin0 .p; q/. Let .Q ; f / be a conformal spin structure of .M; c/. We define the first prolongation of the .Q0 ; f 0 / as a reduction of P 1 with respect to W Bz ! B. For that, consider the set Q1 WD fH Tq Q0 j q 2 Q0 and dfq0 .H / Tf 0 .q/ P 0 horizontal and torsion-free g z with the B-action 0 Q H bQ WD .dfqb /1 .dfq0 .H / .b//; 0

H Q1 \ Tq Q0 ; bQ D bQ0 bQ1 2 Bz D Bzo Bz1

z and the natural projections 1 .H / WD q and .H / WD 0 .q/. This gives us a Bz over M and a 2-fold covering principal bundle .Q1 ; ; M I B/ f 1 W H 2 Q1 7! df 0 .H / 2 P 1 which commutes with the group actions and the projections and 1 . We call the pair .Q1 ; f 1 / the first prolongation of the conformal spin structure .Q0 ; f 0 /. Now, let ! 2 1 .P 1 ; g/ be a Cartan connection on P 1 . Since and dfH1 are isomorphisms, the 1-form 1 1 1 !Q WD 1 Q B ! B df 2 .Q ; g/

is a Cartan connection on Q1 with values in gQ WD spin.p C1; q C1/. In particular, the normal conformal Cartan connection ! nor on P 1 defines a normal conformal Cartan connection !Q nor on Q1 . Any representation Q W Spin.p C 1; q C 1/ ! GL.V / of the spin group defines a tractor bundle E WD Q1 Bz V with covariant derivative r nor induced by the normal Cartan connection !Q nor in the same way as we explained in Section 3.4. Now, we fix again a metric g in the conformal class c and reduce Q1 to the spin bundle Qg . Then E ' Qg Spin0 .p;q/ V . Let F be one of the bundles TM , T M or


851

so.TM; g/. Analogously to (24) the choice of g allows us to define the maps Qg W F ! End.E; E/; 1

1 Qg .‡/' WD u; Q Q 1 Q ' 2 Ex .Œu ‡ / Œu

(35)

Qxg .

where ‡ 2 Fx , ' 2 Ex and uQ 2 By r g we denote the covariant derivative on E, which is defined by the Levi-Civita connection of g. In the same way as in Proposition 9 we obtain z ! GL.V / be a representation of the spin group G z D Proposition 14. Let Q W G 1 Spin0 .p C 1; q C 1/ and E D Q Bz V the associated tractor bundle. For any metric g in the conformal class c the normal tractor connection on E is given by rXnor D rXg C Qg .X / C Qg .K g .X //:

(36)

The curvature endomorphism of r nor is Rnor .X; Y / D Qg .W g .X; Y // C Qg .C g .X; Y //:

(37)

Let us apply this to the standard spinor representation W Spin0 .p C 1; q C 1/ ! pC1;qC1 : We denote the corresponding tractor bundle on a conformal spin manifold .M; c/ by .M / WD Q1 Bz pC1;qC1 and call it spin tractor bundle of .M; c/. In the same way as for metric spinor bundles we define a Clifford multiplication W T .M / .M / ! .M / and a bundle metric h ; i on .M / using the corresponding Spin0 -invariant data on the fibers. Now, we will prove that any metric g 2 c defines a canonical isomorphism between the spin tractor bundle .M / and the double metric spin bundle E g D S g ˚S g in such a way that the normal tractor connection r nor corresponds to the covariant derivative g r E defined in (34). For this purpose, let us denote by W˙ the following Spin.p; q/-invariant subspaces of the spinor module pC1;qC1 : W WD fv 2 pC1;qC1 j f0 v D 0g; WC WD fv 2 pC1;qC1 j fnC1 v D 0g: As Spin.p; q/-representation, WC is isomorphic to p;q . If we restrict the spin representation to Spin.p; q/ we obtain the decomposition of pC1;qC1 into the sum of two Spin.p; q/-representations WC ' p:q : pC1;qC1 ' WC ˚ WC ;

w1 C f0 w2 7! .w1 ; w2 /; w1 ; w2 2 WC :

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Therefore, .M / D Qg Spin0 .p;q/ pC1;qC1 ' Qg Spin0 .p;q/ .WC ˚ WC / ' S g ˚ S g D E g : The inverse of W spin.pC1; q C1/ ! so.pC1; q C1/ applied to x 2 b1 ' Rp;q , z 2 b1 ' .Rp;q / and .A; a/ 2 b0 D co.p; q/ is 1 1 .x/ D x fnC1 ; 2 1 1 .z/ D C z f0 ; 2 a 1 1 ..A; a// D .A/ C .f0 fnC1 fnC1 f0 /: 4 Furthermore, if we represent v 2 pC1:qC1 in the form v D w1 C f0 w2 with w1 ; w2 2 WC , then fnC1 v D fnC1 f0 w2 D f0 fnC1 w2 2hf0 ; fnC1 iw2 D 2w2 ; f0 v D f0 w1 C f0 f0 w2 D f0 w1 : With these formulas we obtain for the action g .‡ / on .M / D S g ˚ S g defined in (35) in our special case ' .X g D ; .X/ 0 0 ' D g ./ ; 12 ' 1 ' ' g ; D . / 2 where '; 2 .S g /, X is a vector field, a 1-form on M and a skew-symmetric endomorphism 2 .so.TM; g// D .ƒ2 .T M //. The application of Proposition 14 to this special case gives Proposition 15. Let g be a metric in the conformal class c. Then the spin tractor bundle .M / can be identified with S g ˚ S g , and in this identification the normal tractor derivative on .M / is given by ! ! g g rXS rXS ' C X X ' ' nor rX D D (38) 1 g 1 g Sg Sg rX 2 K .X / rX 2 K .X / ' The curvature of r nor is ! W g .X; Y / ' 1 ' nor D R .X; Y / 2 W g .X; Y / C g .X; Y / '

(39)

Recently, studying invariants of the conformal holonomy group, F. Leitner (cf. [L05a]) was able to prove a local classification result for generic conformal Killing


853

spinors on Lorentzian manifolds. To state this result, let us recall that any spinor field ' 2 .S g / on a Lorentzian manifold .M; g/ induces a causal (i.e., time- or lightlike) vector field V' on .M; g/ by g.V' ; X/ D hX '; 'i

for all X 2 X.M /:

If ' is a conformal Killing spinor, then V' is conformal as well. We call the spinor ' generic if it has no zeros, V' does not change the causal type and the dual 1-form V'[ has constant rank, where the rank of a 1-form is the maximum of all k with ^ .d/k 6D 0. Proposition 16 ([L05a]). Let .M; g/ be a Lorentzian spin manifold with a generic conformal Killing spinor. Then .M; g/ is locally conformal equivalent to one of the following spaces: (1) .R; dt 2 / .N1 ; h1 / .Nr ; hr /, where .Nj ; hj / are Ricci-flat Kähler, hyper-Kähler, G2 - or Spin.7/-manifolds; (2) Lorentzian Einstein–Sasaki manifold; (3) .N1 ; h1 / .N2 ; h2 /, where .N1 ; h1 / is a Lorentzian Einstein–Sasaki manifold and .N2 ; h2 / a Riemannian Einstein–Sasaki manifold, a 3-Sasaki-manifold, a nearly Kähler manifold or a Riemannian sphere; (4) Fefferman space; (5) Brinkmann space with parallel spinor. All Lorentzian manifolds which appear in the list, admit global solutions of the conformal Killing spinor equation. In the final section we will describe Fefferman spin manifolds and use the global existence of conformal Killing spinors on it to prove that the conformal holonomy group of Fefferman spin manifolds is contained in SU.1; n2 /.

4 Fefferman spin manifolds – conformal structures with holonomy group in SU.1; m/ Fefferman metrics are Lorentzian metrics on S 1 -bundles over strictly pseudoconvex CR manifolds. Such metric was first discovered by Ch. Fefferman in [F76]. Fefferman studied the boundary behaviour of the Bergman kernel of a strictly pseudoconvex domain C n and in connection with this the solution u of the Dirichlet problem for the complex Monge–Ampère equation on : u @u=@Nzk n D 1 in ; u D 0 on @: (40) .1/ det @u=@zj @2 u=@zj @zN k For existence, uniqueness and regularity of the solution cf. [CY77]. Let u be a solution x C ! R the function of (40) and H W H.z1 ; : : : ; zn ; z0 / D jz0 j2=.nC1/ u.z1 ; : : : ; zn /:

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Then the tensor field GD

@2 H dz˛ ˇ d zN ˇ @z˛ @ zNˇ

x C is non-degenerate and the pull back g D j G for j W @ S 1 ,! x C on 1 1 is a Lorentzian metric on @ S . If D arg.z0 / is the coordinate on S , the metric g is given by gD

2 i N ˇ d C j . @ u dzj ˇ d zN k /: j .@u @u/ nC1 @zj @Nzk

(41)

Any biholomorphism of C n lifts to a smooth conformal diffeomorphism of .@ S 1 ; g/. This indicated a close and interesting link between CR geometry and Lorentzian geometry. Of course, a solution u of (40) and hence the metric (41) is not explicitly given. Fefferman used a formal approximation procedure to describe the x Some years later, Farris ([F86]) and Lee ambient metric G on a neighborhood of . ([Lee86]) gave an explicit intrinsic description of the Fefferman metric g as Lorentzian metric on the canonical S 1 -bundle over the strictly pseudoconvex CR manifold @ and extended the construction to abstract CR manifolds. Graham ([Gra87]) characterized the Fefferman metric locally by curvature properties. Other approaches studied the Fefferman construction and its relation to conformal geometry in the frame work of Cartan geometry (see [BDS77], [Ku97], [C02] and [C05]). Inspired by the close link and the analogies the Fefferman construction provides between CR geometry and conformal Lorentzian geometry there is now a growing number of geometric and analytic investigations of Fefferman spaces. We refer for details to [DT06]. In [Lew91], J. Lewandowski studied the twistor equation for spinors on 4-dimensional space-times and found local solutions on Fefferman spaces. In the following section we will explain the intrinsic geometric construction of Fefferman spaces in CR geometry. In particular, we will describe Fefferman spin manifolds, which admit global solutions of the conformal Killing spinor equation. Details of the proofs can be found in [B99]. As a consequence, we will conclude that the conformal holonomy group of any Fefferman spin manifold is contained in SU.1; m/.

4.1 CR geometry and Fefferman spaces Let us first explain the necessary notations from CR geometry. Let N 2mC1 be a smooth oriented manifold of odd dimension 2m C 1. A CR structure on N is a pair .H; J / where 1. H T N is a real 2m-dimensional subbundle. 2. J W H ! H is an almost complex structure on H , i.e., J 2 D Id. 3. If X; Y 2 .H /, then ŒJX; Y C ŒX; J Y 2 .H / and J.ŒJX; Y C ŒX; J Y / ŒJX; J Y C ŒX; Y 0 (integrability condition).


855

Example 7 (Real hypersurfaces of complex manifolds). Any real hypersurface N Q N / and Nz of a complex manifold .Nz ; JQ / is a CR-manifold. Set H WD T N \ J.T Q J D JjH . Example 8 (The complex Möbius space). Let L WD fz 2 C mC1 nf0g j hz; zi1;m D 0g be the isotropic cone in the complex vector space C 1;m with hermitian form of signature .1; m/. The projectivization P L is diffeomorphic to the sphere S 2m1 C m ' f1g C m C 1;m and has therefore a canonical CR-structure. P L is the boundary of the complex hyperbolic space HCm D fz 2 C mC1 j hz; zi1;m D 1g: Example 9 (Heisenberg manifolds). Let He.m/ be the Heisenberg group ´ μ ! 1 x? z ˇ ˇ 0 Im y ˇ x; y 2 Rm ; z 2 R : He.m/ WD 0

0

1

and a discrete subgroup in He.m/. Than the manifold He.m/= is a CR manifold with the canonical CR structure induced by the Lie algebra structure of He.m/. Example 10 (Sasaki manifolds). Let .N; g/ be an odd-dimensional Riemannian manifold with a Killing vector field and let ' WD r g . .N; g; / is called Sasakimanifold if g.; / D 1; ' 2 .X / D X C g.X; /; D g.X; Y / g.Y; /X:

.rXg '/.Y /

A Riemannian manifold .N; g/ admits a Sasaki structure if and only if the cone .CN D RC N; gC D dt 2 C t 2 g/ is a Kähler manifold. Obviously, H WD ? and J WD 'jH is a CR structure on the Sasaki manifold .N; g; /. Now, let .N; H; J / be an oriented CR manifold. In order to define Fefferman spaces we fix a contact form 2 1 .N / on N such that jH D 0. Let us denote by T the Reeb vector field of . In the following we suppose that the Levi form L W H H ! R L .X; Y / WD d .X; J Y / is positive definite. In this case, .N; H; J; / is called a strictly pseudoconvex manifold. The tensor field g WD L C ˇ defines a Riemannian metric on N . There is a special metric covariant derivative on a strictly pseudoconvex manifold, the Tanaka–Webster connection r W W .T N / ! .T N ˝ T N /. Since r W is metric (r W g D 0) it is uniquely defined by its torsion tensor Tor W which is given by Tor W .X; Y / D L .JX; Y / T; 1 Tor W .T; X / D .ŒT; X C J ŒT; JX / 2

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for X; Y 2 .H /. Let us denote by T10 T N C the eigenspace of the complex extension of J on H C to the eigenvalue i. Then L extends to a Hermitian form on T10 by L .U; V / WD i d .U; Vx /; U; V 2 T10 : For a complex 2-form ! 2 ƒ2 N C we denote by trace ! the -trace of !: trace ! WD

m X

x ˛ /; !.Z˛ ; Z

˛D1

where .Z1 ; : : : ; Zm / is a unitary basis of .T10 ; L /. Let RW be the .4; 0/-curvature tensor of the Tanaka–Webster connection r W on the complexified tangent bundle of N W x RW .X; Y; Z; V / WD g ..ŒrXW ; rYW rŒX;Y /Z; V /: and let us denote by WD RicW WD trace.3;4/

m X

x˛ / RW . ; ; Z˛ ; Z

˛D1

the Tanaka–Webster Ricci curvature and by scalW WD trace RicW the Tanaka–Webster scalar curvature. We now suppose in addition that N has a spin structure. This spin structure defines p mC1;0 a square root ƒ N of the canonical line bundle ƒmC1;0 N WD f! 2 ƒmC1 N C j V ³ ! D 0 for all V 2 Tx10 g: p We denote by .F; ; N I S 1 / the S 1 -principal bundle associated to ƒmC1;0 N . Let AW denote the connection form on F defined by the Tanaka–Webster connection r W and A the modified connection form A WD AW

i scalW ; 4.m C 1/

which we call Fefferman connection. Then 8 ˇ A mC2 is a Lorentzian metric on F . Note that the Fefferman connection and the constants in the Fefferman metric are chosen in this way, in order to ensure that the conformal class Œh does not depend on the choice of . Hence, the conformal structure Œh on F is an invariant of the CR structure .N; H; J / itself. We call .F 2mC2 ; h / with its canonically induced spin structure Fefferman spin space of the strictly pseudoconvex spin manifold .N; H; J; /. Let us describe some further geometric properties of the Lorentzian manifold .F; h /. By construction, the metric h is S 1 -invariant, the fibres of the S 1 -bundle are lightlike. We denote by T the horizontal lift of the Reeb vector field T with h WD L i


857

respect to the Fefferman connection A and by the vertical fundamental vector field normalized by h .; T / D 1. Then is a lightlike Killing field and T is lightlike as well. Furthermore, for the Schouten tensor K h , the Cotton–York tensor C h and the Weyl tensor W h the following hold: 4K h .; Y / D h .T ; Y / 4K

h

for all Y 2 X.F /;

.; / D 1;

³ W h D 0; ³ C h D 0: Moreover, if H denotes the horizontal lift of H with respect to A and J X WD .JX / for X 2 H , then h

J X D 2rX : In [B99] we studied the twistor equation on Fefferman spin manifolds. We gave explicit formulas for the solutions and studied their properties. Taking into account results of [BL04] we obtain Proposition 17 ([B99]). Let .N; H; J; / be a strictly pseudoconvex spin manifold with the Fefferman spin space .F; h /. Then there exists a 2-parameter family of conformal Killing spinors ' on .F; h / which satisfy a) V' is a regular lightlike Killing field, and h

b) rVS' ' D i c ', where c 2 Rnf0g. Conversely, if .M; g/ is an even dimensional Lorentzian spin manifold with a conformal Killing spinor satisfying a) and b), then there exists a strictly pseudoconvex spin manifold .N; H; J; / such that its Fefferman space is locally isometric to .M; g/.

4.2 The conformal holonomy group of Fefferman spin spaces In this section we will show that the conformal holonomy group Hol.F; Œh / of an n-dimensional Fefferman spin manifolds is contained in SU.1; n2 / SO.2; n/. In the first step we show that Hol.F; Œh / U.1; n2 /. For this purpose, we will define an isometric almost complex structure on the standard tractor bundle T .F /, which is parallel with respect to the normal tractor connection r nor . The further reduction to SU.1; n2 / is then a consequence of the existence of two global conformal Killing spinors. Let T .F / be the standard tractor bundle of the Fefferman spin manifold .F; c D Œh /. As we saw in Section 3.4, the choice of a metric h 2 c gives rise to a splitting of T .F / into h

T .F / ' R ˚ TF ˚ R:

858

Helga Baum

Using this splitting, we define an almost complex structure J T on T .F / by 1 1 0 0 2ı ˛ C C B B J T @T C ı C X A WD @2˛ C ˇ2 T C 2rXh A ; ˇ 2

(42)

where ˛, ˇ, , ı are functions on F and T , , X are the vector fields on F defined in the previous section. Using the geometric properties of h described above, one obtains (by a direct but lengthy calculation) the following proposition. For a more conceptional proof see [L05b]. Proposition 18. The endomorphism J T W T .F / ! T .F / satisfies (1) .J T /2 D IdT .F / ; (2) hJ T 1 ; J T 2 iT D h1 ; 2 iT for all 1 ; 2 2 T .F /; (3) r nor .J T / D 0. According to Proposition 5, the parallel almost complex structure J T defines a hermitian almost complex structure J0 on the fibre R2;n of T .F /, which is invariant under the action of the conformal holonomy group Hol.F; h /. This means that Hol.F; h / U.1; n2 / SO.2; n/: Now, in order to reduce the situation further to SU.1; n2 /, we use the following Lemma, which can be proved considering the different Jordan normal forms of a matrix in U.1; m/. Lemma 1. Let A 2 U.1; m/ SO.2; 2m/ and let AQ 2 1 .A/ Spin.2; 2m/ be one of the two elements of the spin group that cover A. If there are two linearly Q then independent spinors in the spin-module 2;2m , that are fixed by the action of A, A 2 SU.1; m/. Now, according to Proposition 17 there exists at least 2 linearly independent conformal Killing spinors on the Fefferman spin manifold .F; h /. Hence, by Proposition 15 there are 2 linearly independent r nor -parallel sections in the spin tractor bundle .F /. Again Proposition 5 shows us that the conformal holonomy group Hol.F; Œh / has 2 linearly independent fixed spinors in 2;n . Since we already know that any element of this holonomy group is unitary, Lemma 1 yields Proposition 19. The conformal holonomy group of a 2m-dimensional Fefferman spin manifold is special unitary: Hol.F; Œh / SU.1; m/. Using the local curvature characterization of Fefferman spaces given by R. Graham ([Gra87]) and further, more detailed investigations in conformal Cartan geometry one proves the converse of Proposition 19 (cf. for example [L06]):


859

Proposition 20. Let .M; g/ be a 2m-dimensional Lorentzian manifold and let the connected component of the conformal holonomy group be special unitary: Hol0 .M; Œg/ SU.1; m/: Then .M; g/ is locally conformal equivalent to a Fefferman space. Finally, let us close with a recent result of F. Leitner ([L06]) which is a bit surprising. Using curvature arguments in Cartan geometry, he proved that for any conformal manifold with holonomy group Hol.M; c/ U.1; m/ the connected component Hol0 .M; c/ is already contained in SU.1; m/.

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Chapter 24

Nondegenerate conformal structures, CR structures and quaternionic CR structures on manifolds Yoshinobu Kamishima

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Geometric structures on hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Model geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Pseudo K-hyperbolic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Model flat space forms of type .c.p C 1/ 1; cq/ . . . . . . . . . . . . . . . . . . 3.1 Invariant subbundle of T †c.pC1/1;cq . . . . . . . . . . . . . . . . . . . . . K 4 Conformal structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Weyl tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Nondegenerate CR geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Existence of CR structure on †2pC1;2q . . . . . . . . . . . . . . . . . . . . . C 5.2 Existence of flat CR structure on S 2pC1;2q . . . . . . . . . . . . . . . . . . . 5.3 Conformal viewpoint of pseudo-Hermitian geometry and uniformization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 CR structures from the viewpoint of Cartan geometry . . . . . . . . . . . . . . 6 Bochner flat pseudo-Kähler geometry . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Bochner flat pseudo-Kähler structures on C p;q and CR structures on the Heisenberg space N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Bochner flat structures on weighted pseudo-Kähler projective orbifolds . . . . 7 Pseudo-conformal quaternionic CR geometry . . . . . . . . . . . . . . . . . . . . . 7.1 Construction of the qCR structure on †3C4p;4q . . . . . . . . . . . . . . . . H 7.2 Construction of the flat p-c qCR structure on S 4pC3;4q . . . . . . . . . . . . 7.3 Pseudo-conformal quaternionic Heisenberg geometry . . . . . . . . . . . . . . 7.4 Curvature forms in R.Sp.p; q/ Sp.1// . . . . . . . . . . . . . . . . . . . . . 8 Remarks and problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Conformally flat Lorentz metrics . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Exotic quaternionic invariant closed 4-forms on Hn . . . . . . . . . . . . . . . 8.3 Global rigidity of compact nondegenerate geometric manifolds with noncompact geometric flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

864 865 865 866 867 868 869 871 872 872 873 875 876 876 877 880 881 883 885 886 888 889 889 889 891 892

864

Yoshinobu Kamishima

Introduction The Weyl curvature tensor is a conformal invariant of Riemannian manifolds and the Chern–Moser curvature tensor is a CR invariant of strictly pseudo-convex CR-manifolds. The geometric significance of the vanishing of these curvature tensors is the appearance of the finite dimensional Lie group G with homogeneous space X, where .G ; X/ D .PO.n C 1; 1/; S n / or .PU.n C 1; 1/; S 2nC1 /, respectively. On the other hand, the complete simply connected .n C 1/-dimensional quaternionic hyperbolic nC1 with the group of isometries PSp.n C 1; 1/ has a natural compactification space HH inside the quaternionic projective space HP nC1 . The subgroup PSp.n C 1; 1/ of the group of quaternionic projective transformations PGL.n C 2; H/ acts smoothly nC1 HP nC1 . The action of PSp.n C 1; 1/ on the boundary sphere S 4nC3 of HH 4nC3 , and we obtain a flat (spherical) quaternionic CR geometry is transitive on S .PSp.nC1; 1/; S 4nC3 /. Combined with conformally flat and spherical CR geometries, this exhibits parabolic geometry on the boundary of the compactification of a rankone symmetric space of noncompact type over R, C or H. We note that these model spaces are hyperspheres in the K-projective spaces KP nC1 .K D R; C; H/. This observation naturally raises the following problems. (1) Construction of (positive definite) conformal, CR and quaternionic CR structures on hyperquadrics in KP nC1 . More precisely, a generalization to (possibly indefinite) nondegenerate conformal, CR, quaternionic CR structure on manifolds. Notion of curvature forms, construction of flat model spaces, uniformization. (2) Construction of a nondegenerate integrable geometric structure on a .4n C 3/dimensional manifold M and of the curvature form of a geometric invariant whose vanishing implies that M is locally equivalent to the flat quaternionic CR geometry .PSp.p C 1; q C 1/; S 4pC3;4q /. For (1), we will review and generalize the known results from the conformal viewpoint. For (2), we have introduced the notion of a pseudo-conformal quaternionic CR (p-c qCR) structure .D; f!˛ g˛D1;2;3 / on a .4n C 3/-dimensional manifold M ([3]). This geometric structure is a special case of an almost pseudo-conformal quaternionic CR structure (almost p-c qCR structure) .D; Q/, which is discussed in [4]. (3) Which geometry relates to the above (indefinite) geometric structures? (4) Which compact manifolds admit a nondegenerate conformal structure, nondegenerate CR structure, pseudo-conformal quaternionic CR structure, respectively? For (3), we have introduced pseudo-Sasakian structures (also pseudo-Sasakian 3-structures). From the fibration associated to a nondegenerate spherical CR structure, we obtain Bochner flat pseudo-Kähler structures and also locally conformal Kähler structures. The question (4) is not well understood so far in the indefinite case, but we will give a few compact examples. In this chapter of the Handbook, we shall discuss these problems and explain the current results. In Section 2 we will introduce model spaces. In order to study flatness, we construct principal bundles as open dense subsets

Chapter 24. Nondegenerate conformal, CR, quaternionic CR structures on manifolds 865

of the model space in Section 3. Section 4 is concerned with conformal structures on pseudo-Riemannian manifolds. In Section 5 we will discuss nondegenerate CR geometry. In Section 6 we will study Bochner flat pseudo-Kähler geometry as a related geometry. We will explain the notion of pseudo-conformal quaternionic CR geometry in Section 7. It is introduced in [3]. For the viewpoint of G-structures of parabolic geometry of semisimple type, we refer to [5], [13] and references therein for the Weyl structures and Cartan connections. In the final Section 8, we discuss some problems related to these geometries.

2 Geometric structures on hypersurfaces 2.1 Model geometry Let K be the field of real numbers, complex numbers or quaternions, R, C, H, and denote by KnC2 D KpC1;qC1 the n-dimensional K-vector space .p C q D n/. Let GL.n C 2; K/ be the group of all invertible .n C 2/ .n C 2/-matrices with entries in K. Put 8 ˆ .K D R/; q D 1, where O.p C 1; 2/ is a lift of PO.p C 1; 2/ associated to the universal covering S p R. For p D q D 1, .O.2; 2/; R R/, where Z Z ! O.2; 2/ ! O.2; 2/ is a central group extension. Note that O.2; 2/0 Š SL.2; R/ SL.2; R/. See [6] for related results.

B

B

2.2 Pseudo K-hyperbolic geometry pC1;q Put Vc.nC2/ D fz 2 KnC2 j B.z; z/ < 0g. The pseudo K- hyperbolic space HK c.nC2/ c.nC2/ is defined to be P .V /, where the action O.p C 1; q C 1; K/ on V induces pC1;q an action on H K . The kernel of this action is the center Z=2 D f˙1g if K D R or H, and S 1 if K D C whose quotient is the pseudo K-hyperbolic group PO.p C 1; q C 1; K/. When q D 0; p D n, remark that P .Vc.nC2/ / is the real hyperbolic space nC1 nC1 , complex Kähler hyperbolic space HC or quaternionic Kähler hyperbolic HR nC1 space HH with the group of isometries PO.n C 1; 1/, PU.n C 1; 1/, PSp.n C 1; 1/, pC1;q respectively. The K-projective compactification of HK is obtained by taking the pC1;q nC1 x x pC1;q D HpC1;q [ closure H in KP . Then it is easy to check that H K K K S c.pC1/1;cq . From this viewpoint, the pseudo hyperbolic action of PO.p C 1;

Chapter 24. Nondegenerate conformal, CR, quaternionic CR structures on manifolds 867 pC1;q q C 1; K/ on HK (for K D R; C; H, respectively) extends to a conformal action on S p;q , CR-action on S 2pC1;2q and a smooth action on S 4pC3;4q D P .V04nC7 / as projective transformations.

3 Model flat space forms of type .c.p C 1/ 1; cq/ Suppose that p C q D n and c D 1; 2; 4. Let KnC1 be the K-number space with the nondegenerate quaternionic Hermitian form hx; yi D xN 1 y1 C C xN pC1 ypC1 xNpC2 ypC2 xN nC1 ynC1 :

(3.1)

If Rehx; yi denotes the real part of hx; yi, observe that Reh ; i is a nondegenerate symmetric bilinear form on Rc.nC1/ D KnC1 . For K D R; C denote by O.p C 1; qI K/ the subgroup of GL.n C 1; K/ preserving the nondegenerate form Reh ; i. Then O.pC1; qI K/ D O.pC1; q/, U.pC1; q/, respectively. The group Sp.pC1; q/Sp.1/ is a subgroup of GL.n C 1; H/ GL.1; H/ preserving the nondegenerate bilinear form Reh ; i. Denote by †c.pC1/1;cq the .c.n C 1/ 1/-dimensional quadric space: K ˚ P P jz j2 qkD1 jwk j2 D 1 : .z1 ; : : : ; zpC1 ; w1 ; : : : ; wq / 2 KnC1 j pC1 kD1 k invariant .c D 1; 2/ In particular, the group O.p C 1; qI K/ leaves †c.pC1/1;cq K 3C4p;4q and Sp.p C 1; q/ Sp.1/ leaves †H invariant. Let h ; ix be the nondegenerate quaternionic Hermitian inner product on the tangent space Tx KnC1 obtained from the parallel translation of h ; i to the point x 2 KnC1 . Recall that J0 is the standard complex structure on C nC1 acting as J0 z D zi , and fI; J; Kg is the standard hypercomplex structure on HnC1 which operates as I z D zi , J z D zj , or Kz D zk. As the action of J0x on Tx C nC1 , fIx ; Jx ; Kx g also acts on Tx HnC1 at each point x. Then it is easy to see that, for all X; Y 2 Tx KnC1 , gxK .X; Y / D RehX; Y ix is the standard pseudo-euclidean metric of signature .p C 1; q/ on KnC1 which is invariant under J0 and fI; J; Kg, respectively. (gxR .X; Y / is the usual pseudo-euclidean metric of type .p C 1; q/ on RnC1 .) Restricting g K to the quadric †c.pC1/1;cq in KnC1 , we obtain K a nondegenerate pseudo-Riemannian metric of signature .c.p C 1/ 1; cq/ which we also denote by g K . ; g K / is referred to as the pseudo K-RieDefinition 3.1. The quadric .†c.pC1/1;cq K mannian space form of signature .c.p C 1/ 1; cq/ of constant curvature 1 endowed with transitive group of isometries O.p C 1; q/; U.p C 1; q/, Sp.p C 1; q/ Sp.1/ for 2pC1;2q D U.p C 1; q/=U.p; q/, †4pC3;4q D which †p;q R D O.p C 1; q/=O.p; q/, †C H Sp.p C 1; q/ Sp.1/=Sp.p; q/ Sp.1/. Here O.p; q/, U.p; q/ and Sp.p; q/ Sp.1/ are, respectively, the stabilizers at .1; 0; : : : ; 0/. Compare [35], [26] for this definition. When .†3C4p;4q ; g H / is viewed as a real H pseudo-Riemannian space form, then the full group of isometries is O.4p C 4; 4q/.

868

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Notice that the intersection of O.4p C 4; 4q/ with GL.n C 1; H/ GL.1; H/ is Sp.p C 1; q/ Sp.1/. (Similarly, O.2p C 2; 2q/ \ GL.n C 1; C/ D U.p C 1; q/.) Now, O.1; K/ D O.1/; U.1/; Sp.1/ acts freely on †c.pC1/1;cq as right translations K . 2 O.1; K//: N : : : ; zpC1 ; N w1 ; N : : : ; wq /: N .; .z1 ; : : : ; zpC1 ; w1 ; : : : ; wq // D .z1 ; =O.1; K/ is said to be the pseudo-K Definition 3.2. The orbit space †c.pC1/1;cq K p;q of type .cp; cq/. projective space KP Note that CP p;q is a pseudo-Kähler manifold provided that 2n 4, and HP p;q is a pseudo-quaternionic Kähler manifold provided that 4n 8. When p D n, q D 0, then KP n;0 is the standard K-projective space KP n . When p D 0, q D n, then KP 0;n n is the quaternionic hyperbolic space HK . It is easy to see that KP p;q is homotopic to p q the canonical K -bundle over KP . There are the equivariant principal bundles ! PO.p C 1; q/; RP p;q ; O.1/ ! O.p C 1; q/; †p;q R (3.2) ! PU.p C 1; q/; CP p;q ; U.1/ ! U.p C 1; q/; †2pC1;2q C ! PSp.p C 1; q/; HP p;q : Sp.1/ ! Sp.p C 1; q/ Sp.1/; †3C4p;4q H c.pC1/1;cq 3.1 Invariant subbundle of T †K

If Nx is the normal vector at x 2 †c.pC1/1;cq KnC1 , it then follows that K D Nx? Tx †c.pC1/1;cq K

with respect to g K :

(3.3)

, then J0 N 2 T †2pC1;2q , where If N is a normal vector field on †2pC1;2q C C D fJ0 N g ˚ fJ0 N g? : T †2pC1;2q C

(3.4)

, then I N; JN; KN 2 T †3C4p;4q such that If N is a normal vector field on †3C4p;4q H H we have the decomposition T †3C4p;4q D fI N; JN; KN g ˚ fI N; JN; KN g? : H

(3.5)

We denote DC D fJ0 N g?

and

DH D fI N; JN; KN g?

(3.6)

the 2n-dimensional subbundle on †2pC1;2q and the 4n-dimensional subbundle on C 4pC3;4q †H , respectively. Lemma 3.3. DC is J0 -invariant. Similarly, DH is fI; J; Kg-invariant. Proof. Note that g C jDC is J0 -invariant and g H is an fI; J; Kg-invariant metric on


DH . As Tx †2pC1;2q D Nx? D fJ0 Nx g ˚ fJ0 Nx g? , C [

DC D

fXx 2 Tx C nC1 j g C .Xx ; Nx / D g C .Xx ; J0 Nx / D 0g:

2pC1;2q

x2†C

It is obvious that DC is J0 -invariant. Similarly, it follows that [ ˚ Xx 2 Tx HnC1 j g H .Xx ; Nx / D g H .Xx ; I Nx / DH D 3C4q;4q

x2†H

D g H .Xx ; JNx / D g H .Xx ; KNx / D 0 :

We shall see later that .D; J0 / is a nondegenerate spherical CR structure and .DH ; fI; J; Kg/ is a flat pseudo-conformal quaternionic CR structure.

4 Conformal structures The notion of conformal structure was introduced as the conformal equivalence class of Riemannian metrics on smooth manifolds. The Weyl conformal curvature tensor W is an invariant of conformal equivalence on pseudo-Riemannian metrics and its vanishing implies that the class locally contains an indefinite euclidean metric (flat pseudo-Riemannian metric). A nondegenerate conformally flat manifold is a pseudoRiemannian manifold which admits a pseudo-Riemannian flat metric in its conformal class. As a G-structure, the conformal structure on an n-manifold M is equivalent to the reduction of the structure group of the principal frame bundle over M to a subgroup of O.p; q/RC .p Cq D n/. An integrable nondegenerate O.p; q/RC -structure is said to be a conformally flat structure on M , if each frame contains a basis consisting of coordinate frames on a neighborhood of M . We shall explain this equivalence between flatness of pseudo-Riemannian metrics and integrable O.p; q/ RC -structures. Recall that g0 D ds 2 D dx12 C Cdxp2 dy12 dyq2 is the pseudo-euclidean metric on Rn of signature .p; q/ with p C q D n. Proposition 4.1. There exists the standard pseudo-Riemannian metric g1 on S p;q which is locally conformal to the euclidean metric g0 on Rn .p C q D n 2/. Proof. First of all, we embed Rn into S p;q in the following way: 2 p p jxj2 jyj2 jxj jyj2 1; 2x; 2y; C1 ; (4.1) W .x; y/ 7! 2 2 q q x D .x1 ; : : : ; xp /, y D .y1 ; : : : ; yq /, jxj D x12 C C xp2 , jyj D y12 C C yq2 . Note that the embedding W Rn ! S p;q is equivariant with respect to the subgroups Rn .O.p; q/ RC / PO.p C 1; q C 1/. Put U˛ D f˛1 ..Rn // for each

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f˛ 2 PO.p C 1; q C 1/ and define the indefinite metric g˛ D f˛ g0 on U˛ . Since the open sets U˛ cover S p;q , suppose that U˛ \ Uˇ ¤ ;. Recall that the elements of PO.p C 1; q C 1/ preserve the integrable O.p; q/ RC -structure by definition (cf: Section 4.1). As gˇ ˛ D fˇ B f˛1 W f˛ .U˛ \ Uˇ / ! fˇ .U˛ \ Uˇ / is a local conformal change of .Rn / . S p;q /, it extends uniquely to gˇ ˛ W .Rn / ! .Rn / as an element of PO.p C 1; q C 1/. Note that, by definition, gˇ ˛ g0 D ˇ ˛ g0 for some positive function ˇ ˛ on .Rn /. Since g˛ D gˇ gˇ ˛ on U˛ \ Uˇ \ U by the definition, ˛ g0 D g˛ g0 D gˇ ˛ gˇ g0 D gˇ ˛ .ˇ g0 / D gˇ ˛ ˇ ˇ ˛ g0

on .Rn /, we obtain that ˛ D gˇ ˛ ˇ ˇ ˛ : Define a local function fg˛;ˇ 2ƒ to be ˇ ˛ D ˇ ˛ B f˛ W U˛ \ Uˇ ! RC

.˛; ˇ 2 ƒ/:

Then, fg˛;ˇ 2ƒ satisfy on U˛ \ Uˇ \ U that ˛ D ˛ B f˛ D gˇ ˛ ˇ ˇ ˛ B f˛ D .ˇ .gˇ ˛ / ˇ ˛ / B f˛ D ˇ .gˇ ˛ B f˛ / ˇ ˛ .f˛ / D ˇ .fˇ / ˇ ˛ .f˛ / D ˇ ˇ ˛ : This implies that fg˛;ˇ 2ƒ defines a 1-cocycle on S p;q . Since RC is a fine sheaf viewed as the germ of local continuous functions, recall that the first cohomology H 1 .U; RC R / D 0. (Here U is a chain complex of covers running over all open covers of S p;q .) Therefore there exists a local function fhg˛2ƒ such that ıh.ˇ; ˛/ D ˇ ˛ , D ˇ ˛ on U˛ \ Uˇ . We obtain that i.e., h˛ h1 ˇ fˇ g0 D f˛ gˇ ˛ g0 D f˛ .ˇ ˛ g0 / D ˇ ˛ B f˛ f˛ g0 D ˇ ˛ f˛ g0 D h˛ h1 ˇ f˛ g0 :

In particular, hˇ fˇ g0 D h˛ f˛ g0 on U˛ \ Uˇ . As S p;q D

S ˛2ƒ

U˛ , setting

g1 jU˛ D h˛ f˛ g0 ; this defines a pseudo-Riemannian metric g1 on S p;q which is locally conformal to g0 . Theorem 4.1 (Coincidence with the classical definition). An n-dimensional (n 3) smooth manifold M is uniformizable over .PO.p C 1; q C 1/; S p;q / if and only if M admits a pseudo-Riemannian metric locally conformal to the euclidean metric g0 . Proof. H): Given a conformally flat structure f.U˛ ; '˛ /g˛2ƒ on M , there are elements gˇ ˛ ; gˇ ; g˛ 2 PO.pC1; qC1/ defined on the intersection U˛ \Uˇ \U ¤ ;


such that U˛ \ U ˇ

'˛ %

˚

& 'ˇ

S p;q # gˇ ˛ S p;q ;

Uˇ \ U

'ˇ %

˚

& '

S p;q # gˇ S p;q ;

U˛ \ U

'˛ %

˚

& '

S p;q # g ˛ S p;q :

Let g1 be the standard pseudo-Riemannian metric on S p;q which is locally conformal to g0 on .Rn / S p;q by Proposition 4.1. As gˇ ˛ is a conformal transformation of S p;q , there is a function ˇ ˛ > 0 such that gˇ ˛ g1 D ˇ ˛ g1 on S p;q . Putting ˇ ˛ D ˇ ˛ B'˛ W U˛ \Uˇ ! RC , the same argument as in the proof of Proposition 4.1 applies to show that fˇ 'ˇ g1 D f˛ '˛ g1 on U˛ \ Uˇ for some local functions ff g˛2ƒ . Setting gjU˛ D f˛ '˛ g1 , this defines a pseudo-Riemannian metric g on M which is locally conformal to g1 . By our choice of g1 , .M; Œg/ is a nondegenerate conformally flat manifold. (Note that this proof works for n 2.) (H: There is a coordinate chart .U; '/ at each point of M such that U is an open subset and ' W U ! Rn is a homeomorphism onto its image. Suppose that gjU D ' g0 and W U ! RC is a function. For an open subset V , suppose that U \V ¤ ; and W V ! Rn is a coordinate function such that gjV D g0 . Then the local change of Rn , ' 1 W '.U \ V / ! .U \ V / satisfies that . ' 1 / g0 D v g0 where v D 1 .' 1 / .' 1 /. As .Rn / S p;q , ' is viewed as an immersion ' W U ! S p;q such that ' 1 is a locally conformal transformation of S p;q . By Liouville’s theorem (n 3), a local conformal change can be extended to a conformal transformation of S p;q , i.e., there exists an element h 2 PO.p C 1; q C 1/ such that ' 1 D hj'.U \ V /:

(4.2)

Hence a collection of such coordinate neighborhoods .U; '/ satisfying (4.2) gives a nondegenerate conformally flat structure on M .

4.1 Weyl tensor A nondegenerate conformal (O.p; q/ RC -) structure is a conformal class of nondegenerate pseudo-Riemannian metric g on a manifold M . Since we know that O.p; q/ RC is of finite type (depth 1), an integrable O.p; q/ RC -structure is equivalent to the existence of a conformally flat pseudo-Riemannian metric, i.e., g is locally conformally equivalent to the standard euclidean metric g0 . As a matter of fact, the existence of an integrable O.p; q/ RC -structure is equivalent with vanishing Weyl curvature tensor of M provided that dimM 4. The vanishing of Weyl curvature tensor on a Riemannian metric on M gives a uniformization over the standard pseudo-sphere S p;q with respect to the group of conformal transformations PO.p C 1; q C 1/. The pair .PO.p C 1; q C 1/; S p;q / is said to be the nondegenerate (indefinite) conformally flat geometry. A conformally flat manifold is a smooth n-manifold locally modeled on S p;q with coordinate changes belonging to PO.p C 1; q C 1/. By the monodromy argument,

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z into the universal there is a developing map dev from the universal covering space M p;q which is equivariant under the holonomy homomorphism , covering space of S i.e., z / ! .PO.p f C 1; q C 1/; Szp;q /; .; dev/ W .1 .M /; M dev. x/ D . /dev.x/ for all x 2 M , 2 1 .M /, where 1 .M / is the fundamental group of M .

5 Nondegenerate CR geometry A nondegenerate CR structure on a .2n C 1/-manifold M is a pair .D; J / consisting of a codimension 1 subbundle of TM and a complex structure on it. D is a contact bundle such that ŒD; D [ D D TM at each point of M . When J is an almost complex structure on D and D ˝ C D T 1;0 ˚ T 0;1 is the eigenspace decomposition, J is a complex structure if and only if ŒT 1;0 ; T 1;0 T 1;0 . If ! is a 1-form whose kernel represents D, it satisfies that (1) d!.JX; J Y / D d!.X; Y / for all X; Y 2 D, (2) g D .X; Y / D d!.JX; Y / is a nondegenerate symmetric bilinear form on D with type .2p; 2q/. If q D 0, i.e., g D is positive definite, then the CR structure .D; J / is said to be strictly pseudoconvex. 2pC1;2 q 5.1 Existence of CR structure on †C

As before, let †2pC1;2q be the complex pseudo-Riemannian space form of signature C .2p C 1; 2q/. Put x1 dw1 w xq dwq /: !0 D .zN1 dz1 C C zNpC1 dzpC1 w

(5.1)

Then it is easy to check that !0 is an Im C-valued 1-form on †2pC1;2q . Let 1 be the C i vector field on †2pC1;2q induced by the one-parameter subgroup fe g 2R . Note that C nC1 if J0 is the standard complex structure on C (cf: Section 3), then 1 D J0 N . A calculation shows that !0 . 1 / D i . Let S 1 be the center of U.pC1; q/. As U.pC1; q/ leaves !0 invariant, it follows that Ra !0 D !0 . Therefore, !0 is a connection form of

! CP p;q of (3.2). Put !0 D !1 i . Then !1 the principal bundle S 1 ! †2pC1;2q C 2pC1;2q is a nondegenerate contact form on †C because !1 ^ .d!1 ^ ^ d!1 / ¤ 0 ƒ‚ … „ n-times

at any point of †2pC1;2q from (5.1). (Compare [19], [29], for example). Noting that C !1 . 1 / D 1, we have that L1 !1 D 1 d!1 D 0 and so 1 is the characteristic vector


field for !1 . From Section 3.1 it follows that DC D Null !1 ;

(5.2)

where Null !1 D fX 2 T †2pC1;2q j !1 .X / D 0g. It is easily checked that C C d!1 .X; Y / D g .X; J0 Y / .X; Y 2 DC /. (Compare Lemma 7.8.) In particular, if ˛ is a coframe field of T 1;0 D fe1 ; : : : ; en g and g˛ˇN D g C .e˛ ; eˇN /, then this gives the following structure equation of the bundle (3.2): N

d!1 D i g˛ˇN ˛ ^ ˇ :

(5.3)

; J0 ; g C / is a pseudo-Sasakian space form of constant posIt is shown that .†2pC1;2q C itive curvature with signature .2p C 1; 2q/. (See [2], [20].) We note that g C .X; Y / D !1 .X / !1 .Y / C d!1 .J0 X; Y / which is called the pseudo-Sasakian metric of †2pC1;2q . In particular, g C induces a C pseudo-Kähler metric gO 1 of constant positive holomorphic curvature on CP p;q , see [31], [32] for details. ; !1 ; J0 ; g C / is a .2n C 1/-dimensional homogeneous CRCorollary 5.1. .†2pC1;2q C manifold of signature .2p C 1; 2q/ equipped with the nondegenerate CR structure .DC ; J0 / .p C q D n 1/. Moreover, there exists the equivariant principal bundle of the pseudo-Riemannian submersion over the (homogeneous) pseudo-complex projective space CP p;q of signature .2p; 2q/:

; g C / ! .PU.p C 1; q/; CP p;q ; g/: O S 1 ! .U.p C 1; q/; †2pC1;2q C Remark 5.2. It is generally difficult to find compact nondegenerate CR-manifolds. There are discrete cocompact subgroups of U.1; n/ that act properly and freely on 2nC1 D V1 . We obtain compact nondegenerate CR-manifolds. (1) The pseudo†1;2n C 2nC1 Riemannian standard space form V1 = of signature .1; 2n/ with constant sectional n 1 curvature 1 which is an S -bundle over the Kähler hyperbolic orbifold HC = of constant negative sectional curvature. ( PU.1; n/ is a discrete subgroup.) (2) The indefinite Heisenberg nilmanifold of signature .1C2p; 2q/ which is a compact nondegenerate CR-manifold whose quotient is the complex euclidean orbifold (i.e., pseudo-Kähler orbifold of zero Ricci curvature), see Section 6.1.

5.2 Existence of flat CR structure on S 2pC1;2 q Recall that if Nx is the normal vector at x 2 †2pC1;2q C nC1 , then .Null !1 /x D C Dx D fJ0 Nx g? Tx †2pC1;2q D Nx? with respect to g C . If we note that C C fNx ; J0 Nx g D xC and g D Reh ; i, it follows that Dx D xC ? with respect to h ; i. In particular, this shows that Dx is J0 -invariant. Let W C nC1 ! C nC2 be

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Yoshinobu Kamishima

the embedding defined by .z/ D .z; 1/ so that P B .†2pC1;2q / P .V02nC3 / D S 2pC1;2q ; C where

(5.4)

/ D fŒ.z1 ; : : : ; zpC1 ; w1 ; : : : ; wq ; 1/g: P B .†2pC1;2q C

As †2pC1;2q has the transitive isometry group U.p C 1; q/, this embedding imC plies that U.p C 1; q/ is identified with the subgroup P .U.p C 1; q/ U.1// of PU.p C 1; q C 1/. We define a 1-form 1 on P B .†2pC1;2q / by C .P B / 1 D !1 :

(5.5)

Fix x 2 †2pC1;2q . As the inner product satisfies C h. ..Null !1 /x /; .x/i D h..Null !1 /x ; 0/; .x; 1//i D h.Null !1 /x ; xi h0; 1i D 0; we obtain that

..Null !1 /x / D .x/? D ..x/C/? :

Since z ? is obviously J0 -invariant for any z 2 V02nC3 , J0 induces a complex structure J on P .z ? / for all P .z/ 2 S 2pC1;2q . (5.5) implies that .Null 1 /Œ.x/ D P ..Null !1 /x / D P ...x/C/? /:

(5.6)

If f 2 PU.p C 1; q C 1/, then f ..Null 1 /Œ.x/ / D f P ..x/? / D P .f ..x/? // D P ..f .x//? /: As

(5.7)

f ..Null 1 /Œ.x/ / D .Null f 1 1 /Œf ..x//

//, .Null f 1 1 /Œf ..x// is J0 -invariant. Furthermore, if holds on f .P B .†2pC1;2q C h.Œ.x// D Œ.x/, then h..x// D .x/ for 2 C . It follows that h ..Null 1 /Œ.x/ / D h .P ...x/C/? // D P ...x/C/? / D .Null 1 /Œ.x/ : (5.8) / is an open dense subset in S 2pC1;2q , obviously ff .P B Since P B .†2pC1;2q C 2pC1;2q .†C //gf 2PU.pC1;qC1/ covers S 2pC1;2q . For a 2 S 2pC1;2q , choose f 2 PU.p C 1; q C 1/ such that f .Œ.x// D a. In view of (5.8), we obtain the welldefined subspace Ha D .Null f 1 1 /Œf ..x// . Hence the distribution H D f.Null f 1 1 /Œf ..x// gf 2PU.pC1;qC1/ endowed with J is a nondegenerate CR structure on S 2pC1;2q . Theorem 5.3 (cf: Definition 5.5). There exists a pseudo-Hermitian pair .!; J / on S 2pC1;2q such that H D Null !: Moreover, the group of CR transformations AutCR .S 2pC1;2q / is PU.p C 1; q C 1/.


Proof. Put U˛ D h˛ .P B .†2pC1;2q // for each h˛ 2 PU.p C 1; q C 1/ and the form C 2pC1;2q , suppose that U˛ \Uˇ ¤ ˛ D h1 ˛ 1 on U˛ . Since the open sets U˛ covers S 1 1 ;. Consider the local change hˇ ˛ D hˇ B h˛ W h˛ .U˛ \ Uˇ / ! h1 .U˛ \ Uˇ / ˇ

which extends to P B .†2pC1;2q /. Then since hˇ ˛ preserves the CR structure on C P B.†2pC1;2q /, .h / Null D Null 1 , which implies that .hˇ ˛ /1 1 D uˇ ˛ 1 1 ˇ˛ C for some uˇ ˛ > 0 on U˛ \ Uˇ . As in the proof of Proposition 4.1, there is the set of local functions fug˛2ƒ such that uˇ ˇ D u˛ ˛ on U˛ \ Uˇ . Setting !jU˛ D u˛ ˛ ; this defines a nondegenerate 1-form ! on S 2pC1;2q for which H D Null !.

Remark 5.4. From (2.2), the flat nondegenerate CR space form S 2pC1;2q D S 2pC1 S 2qC1 =S 1 has the principal circle bundle

T 1 D S 1 S 1 =S 1 ! S 2pC1 S 2qC1 =S 1 ! CP p CP q : There is the connection !2 of this bundle such that . p q / D d!2 where Œ p Œ q is the Kähler class of the product CP p CP q . In particular, .Null !2 ; J / gives a CR structure on S 2pC1;2q of signature .2p; 2q/. Here J is a complex structure which is the pullback of the complex structure under the isomorphism W Null !2 ! T .CP p CP q / at each point. The restriction of this structure C1;2q provides a nondegenerate CR structure to the embedding P B .†2P / of (5.4). C However, this CR structure is different from . 1 ; J /. For this, the above T 1 -action C1;2q restricted to P B .†2P / is described by C t Œ.z1 ; : : : ; zpC1 ; w1 ; : : : ; wq ; 1/ D Œ.t z1 ; : : : ; t zpC1 ; w1 ; : : : ; wq ; 1/ C1;2q which is different from the principal action of S 1 on P B .†2P / because these C circle actions are linear on C nC2 and are non-equivalent. Hence, this structure .Null !2 ; J / is not flat.

5.3 Conformal viewpoint of pseudo-Hermitian geometry and uniformization There is no canonical choice of contact forms ! which represents D leaving J fixed. If is another form, then there exists a positive function u on M such that

D u !:

(5.9)

Definition 5.5. A pair .!; J / is said to be a nondegenerate pseudo-Hermitian structure on M .

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Another nondegenerate pseudo-Hermitian structure . ; J / gives a nondegenerate CR structure .D; J / D .Null ; J / on M . So to say, a CR structure is the conformal class of the pseudo Hermitian forms with J fixed. Chern and Moser have defined a curvature tensor S.!; J / for the pseudo-Hermitian pair .!; J /. From this conformal viewpoint it is a CR invariant, as proved by the same authors and Webster (cf: [15], [34], [33]). S D S.!; J / D S. ; J /: (5.10) The vanishing of Chern–Moser curvature tensor S provides a uniformization of a nondegenerate CR-manifold .M; fD; J g/ with respect to flat CR geometry .PU.p C 1; q C 1/; S 2pC1;2q /. When q D 0, i.e., in the strictly pseudokonvex case, .PU.n C 1; 1/; S 2nC1 / is usually called spherical CR geometry. Theorem 5.6. If S D 0 identically on a .2n C 1/-dimensional nondegenerate CRmanifold .n 2/, then M admits a developing pair .; dev/ up to composition with elements of PU.p C 1; q C 1/: z /; M z / ! .PU.p C 1; q C 1/; S 2pC1;2q /; .; dev/ W .Aut CR .M z //; dev.h x/ D .h/ dev.x/ .h 2 AutCR .M z / is the group of CR-automorphisms of the universal covering space M z. where Aut CR .M

5.4 CR structures from the viewpoint of Cartan geometry Given a nondegenerate CR structure .H; J / on a .2n C 1/-dimensional manifold M , there is an su.p C 1; q C 1/-valued 1-form called a Cartan connection whose associated curvature form … vanishes if and only if M is locally isomorphic to PU.p C 1; q C 1/=PC .C/ where PC .C/ denotes the maximal parabolic subgroup .p C q D n/. (See [3].) The fourth-order Chern–Moser CR curvature tensor S D .S˛ ˇ / is the coefficient of the component ˆˇ˛ of the curvature matrix …. Observed by Webster (cf: [34], [33]), the other components are obtained from S by further covariant differentiation for n > 1. In the CR case, the Chern–Moser curvature tensor S vanishes on M if and only if so does the su.p C 1; q C 1/-valued Cartan curvature form …

6 Bochner flat pseudo-Kähler geometry Webster [33] has proved that the formula of the Chern–Moser CR curvature tensor coincides with the Bochner curvature tensor of a Kähler manifold. In fact, suppose that a nondegenerate CR-manifold M admits a Seifert fibration

T ! M ! Y;


where T is either S 1 or R acting properly on M as CR-transformations. (In general, this works locally and Y is an orbifold.) Let ! be a contact form representing the CR structure D (keeping J fixed) and the CR-vector field which generates T . If !. / ¤ 0 on a domain X of M , then we can define a contact form as follows: D

1 ! !. /

on X:

(6.1)

Then becomes a characteristic CR-vector field for . In other words, generates T which is a 1-parameter subgroup of pseudo-Hermitian transformations of , i.e., . / D 1; L D d D 0:

(6.2)

When we put W D X=T , the complex structure J on D induces a complex structure JO on W . Letting D d ; (6.3)

gives a JO -invariant closed 2-form on W . As is a contact form, becomes a Kähler form on W for which g.; O / D .JO ; / becomes a pseudo-Kähler metric (indefinite metric) on .W; JO /. Note that the Bochner curvature tensor B.g; O JO / of the pseudo-Kähler metric gO on W coincides with S.!; J / of X by (6.3), (5.10). As a consequence, if .M; fD; J g/ is a flat nondegenerate CR-manifold, then .W; fg; O JO g/ is a Bochner flat pseudo-Kähler manifold.

6.1 Bochner flat pseudo-Kähler structures on C p;q and CR structures on the Heisenberg space N Let Aut CR .S 2pC1;2q / D PU.p C 1; q C 1/ be the group of all automorphisms preserving the flat CR structure of S 2pC1;2q . We consider the stabilizer of the point at / S 2pC1;2q . Recall the notion infinity f1g D Œ1; 0; : : : ; 0; 1 2 P B .†2pC1;2q C of the (indefinite) Heisenberg nilpotent Lie group N D N .p; q/ from [19]. It is the product R C n with group law .a; y/ .b; z/ D .a C b Imhy; zi; y C z/: Here h ; i is the Hermitian inner product of signature .p; q/ on C n as in (2.1) and Imh ; i is the imaginary part .p C q D n/. It is nilpotent because the commutator subgroup ŒN ; N D R coincides with the center of N consisting of elements of the form .a; 0/. In particular, there is the central group extension 1 ! R ! N !C n ! 1:

(6.4)

Denote by Sim.N / the semidirect product N Ì .U.p; q/ RC / where the action .A; t / 2 U.p; q/ RC on .a; y/ 2 N is given by .A; t / B .a; y/ D .t 2 a; t Ay/:

(6.5)

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Yoshinobu Kamishima

Denote the origin by O D Œ1; 0; : : : ; 0; 1 2 P B .†2pC1;2q / f1g. The stabilizer C AutCR .S 2pC1;2q /1 is isomorphic to Sim.N / (cf: [21]). The orbit N O is an open dense subset of S 2pC1;2q . The embedding is defined by 3 2 kz k2 kz k2 C 1 C ia 2 p 7 6 7 6 p2zC (6.6) .a; .zC ; z // 2 N ! 6 7: 2z 5 4 kzC k2 kz k2 2

C 1 C ia

Then the pair .Sim.N /; N / is said to be a flat Heisenberg CR geometry which is a subgeometry of the flat CR geometry .AutCR .S 2pC1;2q /; S 2pC1;2q /. The rest of this section is devoted to the construction of Bochner flat pseudo-Kähler structures on C n from N . The contact form !N on N is described as follows. If .t; .z1 ; : : : ; zn // is the coordinate of N D R C n , then !N D dt C

p X

.xj dyj yj dxj /

j D1

n X

.xj dyj yj dxj /

j DpC1

(6.7)

D dt C Imhz; dzi: The subgroup Psh.N / D N Ì U.p; q/ leaves the form !N invariant. In fact, if D ..a; w/; A/ 2 N Ì U.p; q/, then ..a; w/; A/ .t; z/ D .a C t Imhw; Azi; w C Az/; and so !N D dt d Imhw; Azi C Imhw C Az; d.w C Az/i: Since d Imhw; Azi D Imhw; dAzi, it is easy to see that !N D dt C Imhz; dzi D !N : Let .Null !; J / be the flat CR structure on S 2pC1;2q (cf: Theorem 5.5). Restricted to N S 2pC1;2q f0g, we have a flat CR structure .Null !N ; J / on N . In general, for h 2 Aut CR .N /, there exists a positive function u on N such that h !N D u !N : Moreover, h is holomorphic (Cauchy–Riemann) on Null !N , so every element h of Aut CR .N / preserves the CR structure .Null !N ; J /. On the other hand, we have the canonical principal fibration P

R ! .N ; !N / ! .C n ; 0 /;

(6.8)

where d!N D P 0 such that

0 D 2

p X j D1

dxj ^ dyj 2

n X j DpC1

dxj ^ dyj

(6.9)


is the standard (indefinite) Kähler form of C n and g0 D 0 .JC ; / is the pseudocomplex euclidean metric of signature .p; q/: ds 2 D jdz1 j2 C Cjdzp j2 jdzpC1 j2 jdzn j2 .p C q D n/. In other words, the CR structure J on Null !N is obtained from the standard complex structure JC on C n by the following commutative diagram: Null !N J

Null !N

P

/ T .C n /

P

(6.10)

JC

/ T .C n /.

Let W R ! R T n be the representation defined by .t / D ..t; 0/; .e i ta1 ; : : : ; e i t ap ; e i t bpC1 ; : : : ; e i t bn // such that bpC1 bn 0 a1 ap :

(6.11)

Note that if all ai D 0; bj D 0, then .R/ is the center of N . In general, .R/ is a closed subgroup of R T n Psh.N / isomorphic to R. As R T n acts properly on N , .R/ acts properly and freely on N . Let

.R/ ! N ! N =.R/

(6.12)

be the principal bundle. As maps Null !N isomorphically onto T .N =.R// at each point, J induces a complex structure JO on N =.R/. We note the following proposition whose proof is the same as that of [20]. Proposition 6.1. The orbit space .N =.R/; JO / is biholomorphic to .C n ; JC /. Put A t D .e i t a1 ; : : : ; e i t ap ; e i t bpC1 ; : : : ; e i t bn /. Recall that .R/ acts on N by .t/.s; z/ D .s C t; A t z/ ..s; z/ 2 N /: Let be the vector field on N induced by .R/. Then, n

D

X d d d aj xj yj C dt dyj dxj j D1

Pp

n X j DpC1

bj xj

d d yj : dyj dxj

Pn

Using (6.7), !N . / D 1 C . j D1 aj jzj j2 j DpC1 bj jzj j2 /. By the hypothesis (6.11), !N . / > 0 everywhere on N . We have the contact form .Z/ D

1 !N .Z/ !N . /

for all Z 2 T N .

(6.13)

d . ; X / D 0 for all X 2 T N .

(6.14)

As in (6.2), it follows that . / D 1;

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Yoshinobu Kamishima

From (6.3), we have an indefinite Hermitian metric on .N =.R/; JO / D .C n ; JC /: g. O Xy ; Yy / D d .JX; Y /; y Xy ; Yy / D where X; Y 2 Null such that .X / D Xy , .Y / D Yy . Let . n y O y g. O X; J Y / be the fundamental two-form on N =.R/ D C . Using (6.14), it foly D d , i.e., d y D 0. Therefore, y is a pseudo-Kähler form on lows that n y JC / on C n . For O ; C . Thus we obtain a Bochner flat pseudo-Kähler structure .g; y D d and .a; b/ D .a1 ; : : : ; ap ; bpC1 ; : : : ; bn /, we put gO D ga;b . Since .Null ; J / is a nondegenerate spherical CR structure (i.e., S. ; J / D 0), we note that B.ga;b ; JC / D 0. Hence, .C n ; ga;b / is a Bochner flat pseudo-Kähler manifold. Problem 6.2. We do not know whether the pseudo-Kähler metric ga;b is geodesically complete whenever bpC1 bn 0 a1 ap . Remark 6.3. Put ! D !N . The new pseudo-Kähler metric gO D ga;b on C n has been obtained by deforming ! to D u! where u D 1=!. /. The effect under this change is to produce a globally conformal pseudo-Kähler metric gO ! starting from the standard pseudo-complex euclidean metric g0 of (6.9). Because d D u d! on Null !, the y D uO ‚ on C n where ‚. X; Y / D d!.X; Y /. Let pushforward by gives y C A; B/ D g.A; O B/. Hence gO ! .A; B/ D ‚.JC A; B/. Then uO gO ! .A; B/ D .J the nondegenerate Hermitian metric gO ! is globally conformal to the pseudo-Kähler metric g. O This deformation is carried within the locally conformal (indefinite) Kähler geometry.

6.2 Bochner flat structures on weighted pseudo-Kähler projective orbifolds There is an equivariant principal bundle from (3.2):

/ ! .PU.p C 1; q/; CP p;q /: U.1/ ! .U.p C 1; q/; †2pC1;2q C The maximal compact Lie group of U.p C 1; q/ is conjugate to U.p C 1/ U.q/ which contains the maximal torus T nC1 . Choose a representation of R such that .t / D .e it a1 ; : : : ; e it apC1 ; e it bpC2 ; : : : ; e it bnC1 /;

(6.15)

where the ai ; bj ’s are rational numbers .p C q D n/. So the image .R/ D S 1 . In particular, .R/ acts properly on †2pC1;2q . It acts freely if and only if a1 D D C apC1 D bpC2 D D bnC1 . It is easy to see that .R/ induces the vector field D

pC1 X j D1

aj

d d xj yj dyj dxj

C

nC1 X j DpC2

bj

d d xj yj dyj dxj

(6.16)


on †2pC1;2q . For the 1-form !0 of (5.1), let 1 Di !0 D Imhz; dzi which is a real C 1-form. Calculate that 1 . / D

pC1 X j D1

nC1 X

aj jzj j2

bj jzj j2 :

(6.17)

j DpC2

Note that 1 . / > 0 whenever bpC2 bnC1 < 0 < a1 a2 apC1 : As remarked, .R/ D S 1 acts properly but not necessarily freely. When all ai ; bj are mutually distinct, the quotient is an orbifold. The quotient orbifold CP p;q .a1 ; : : : ; apC1 I bpC2 ; : : : ; bnC1 / D †2pC1;2q =he ita1 ; : : : ; e it apC1 ; e it bpC2 ; : : : ; e itbnC1 i C

(6.18)

is called the weighted pseudo-Kähler projective space. It is not necessarily compact (cf: Remark 6.5). In the same manner we obtain the following result. Corollary 6.4. Suppose that all ai , bj are mutually distinct positive rational numbers. Then an orbifold CP p;q .a1 ; : : : ; apC1 I bpC2 ; : : : ; bnC1 / admits a Bochner flat pseudoKähler (singular) metric. Remark 6.5. If p D n, q D 0, then the above compact quotient is the weighted complex projective space CP n .a1 ; : : : ; anC1 /. If p D 0, q D n, the above noncom2nC1 pact quotient V1 =he ita1 ; e it b2 ; : : : ; e itbnC1 i is the complex hyperbolic orbifold 2nC1 n D fz 2 C nC1 j B.z; z/ D 1g. HC .a1 I b2 ; : : : ; bnC1 /. Here V1

7 Pseudo-conformal quaternionic CR geometry Definition 7.1. A 4n-dimensional orientable subbundle D equipped with a quaternionic structure Q is called an almost pseudo-conformal quaternionic CR structure (almost p-c qCR structure) on M if the following properties are satisfied. (i) D [ ŒD; D D TM . (ii) At any point p 2 M , the 3-dimensional quotient bundle TM=D is isomorphic to the Lie algebra Im H D Ri C Rj C Rk. (iii) There exists an Im H-valued 1-form ! D !1 i C !2 j C !3 k locally defined on T a neighborhood of M such that D D Null ! D 3˛D1 Null !˛ and d!˛ jD is nondegenerate. Here each !˛ is a real valued 1-form .˛ D 1; 2; 3/. (iv) The endomorphism J D .d!ˇ jD/1 B .d!˛ jD/ W D ! D constitutes the quaternionic structure Q on D: J 2 D 1; J˛ Jˇ D J D Jˇ J˛ ( D 1; 2; 3) etc.

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The form ! is called an admissible 1-form and fJ˛ g˛D1;2;3 are hypercomplex structures on D associated to !. We have the following (cf. [3]): Lemma 7.2. If we put ˛ D .d!˛ jD/ on D, then the equality 1 .J1 X; Y / D 2 .J2 X; Y / D 3 .J3 X; Y / holds for all X; Y 2 D. Moreover, the form (7.1) g D D ˛ B J˛ is a nondegenerate Q-invariant symmetric bilinear form on D. In general, there is no canonical choice of ! which annihilates D. The fiber of the quotient bundle TM=D is orientable and isomorphic to Im H by ! on a neighborhood U by (ii). If ! 0 is another admissible 1-form such that Null ! 0 D D on U 0 , then ! 0 is uniquely determined on U \ U 0 by ! 0 D u a ! aN J˛0 D aˇ ˛ J˛ :

for some functions a 2 Sp.1/; u > 0;

(7.2)

Here fJ˛0 g are the associated hypercomplex structures to ! 0 for which A D .a˛ˇ / 2 SO.3/ is obtained from the conjugate by a 2 Sp.1/. Thus Definition 7.1 does not depend on the choice of ! 0 satisfying (7.2). By Lemma 7.2, we may assume that g D on DjU has signature .4p; 4q/ .pCq D n/. As above put g 0D .X; Y / D d ! 0 ˛ .J 0 ˛ X; Y / .X; Y 2 D/. Corollary 7.3. If ! 0 D u a ! aN on U \ U 0 , then g 0D D u g D . As a consequence, the signature .4p; 4q/ is constant on U \ U 0 (and hence everywhere on M ) under the N change ! 0 D u a ! a. We shall consider an integrability condition on the almost p-c qCR structure D. Definition 7.4. Put the skew symmetric 2-forms ˛ locally on a neighborhood: ˛ D d!˛ C 2!ˇ ^ ! ;

(7.3)

where .˛; ˇ; / .1; 2; 3/ up to cyclic permutation. If ˛ satisfies that Null 1 D Null 2 D Null 3 ;

(7.4)

then the pair .!; Q/ is a (local) quaternionic CR structure (qCR structure) on M . (See [7], [2].) When ! is entirely defined on M , it is called a quaternionic CR structure. Using the two Definitions 7.1 and 7.4, we arrive at the following notion due to Libermann [28]. Definition 7.5 (p-c qCR structure). The pair .D; Q/ on M is said to be a pseudoconformal quaternionic CR structure if there exists locally an admissible 1-form ! with Null ! D D on a neighborhood U at each point of M such that ! is a qCR structure on U .


Note that a quaternionic CR structure .!; fI; J; Kg/ gives an almost pseudoconformal quaternionic CR structure .Null !; fI; J; Kg/ on M . We can choose three (characteristic CR) vector fields f 1 ; 1 ; 3 g which constitute the distribution of (7.4) such that !˛ . ˇ / D ı˛ˇ . Then the almost complex structure J˛ naturally extends to an almost complex structure JN˛ on D ˚ f ˇ ; g D Null !˛ such that JN˛ jD D J˛ ;

JN˛ ˇ D ;

JN˛ D ˇ :

Contrary to the nondegenerate CR structure, the almost complex structure on D is not assumed to be integrable. However, by the requirement of structure equations defining the qCR structure, we can prove the integrability of the quaternionic structure. (Compare [3], [2] for the proof.) Theorem 7.6. Each almost complex structure JN˛ of the quaternionic CR structure is a complex structure on the codimension-1 contact subbundle Null !˛ .˛ D 1; 2; 3/. This implies that the integrability of three almost complex structures fJN˛ g˛D1;2;3 is equivalent with the condition that .M; g/ is a pseudo-Sasakian 3-structure, i.e., the notion of a nondegenerate quaternionic CR structure is equivalent to the notion of a pseudo-Sasakian 3-structure (cf: [2]). The 3-Sasakian metric on M is defined by g.X; Y / D

3 X

!˛ .X / !˛ .Y / C d!1 .JN1 X; Y /

for all X; Y 2 TM .

(7.5)

˛D1

Note that d!1 .JN1 ; / D d!2 .JN2 ; / D d!3 .JN3 ; /. Remark 7.7. Consider the structure equation of the nondegenerate CR case. Let ! be a 1-form which represents a CR structure .Null!; J /. The CR structure has no such equality (7.4). There corresponds only one structure equation (5.3). Here J is assumed to be integrable. However, by the coincidence (7.4) in the p-c qCR case, Theorem 7.6 shows that each almost complex structure JN˛ .˛ D 1; 2; 3/ is integrable. Moreover, each characteristic vector field ˛ of !˛ is a CR-vector field which never occurs only from (5.3) for the nondegenerate CR structure in general. 3C4p;4 q 7.1 Construction of the qCR structure on †H

We now consider the quaternionic pseudo-Riemannian space form †4pC3;4q of sigH nature .4p C 3; 4q/ (cf: Section 3). As in (5.1), we put x1 dw1 w xq dwq /: !0 D .zN1 dz1 C C zNpC1 dzpC1 w

(7.6)

. Let 1 , 2 , 3 Then it is easy to check that !0 is an sp.1/-valued 1-form on †3C4p;4q H 3C4p;4q be the vector fields on †H induced by the one-parameter subgroups fe i g2R , fe j g2R , fe k g2R , respectively, which is equivalent to 1 D I N , 2 D JN , 3 D

884

Yoshinobu Kamishima

KN . Note that f 1 ; 2 ; 3 g generates the fields of Lie algebra of Sp.1/. A calculation shows that (7.7) !0 . 1 / D i ; !0 . 2 / D j ; !0 . 3 / D k: By the formula for !0 , if a 2 Sp.1/, then the right translation Ra on †3C4p;4q satisfies H Ra !0 D a !0 a: N

(7.8)

Therefore, !0 is a connection form of the principal bundle given in (3.2). Note that Sp.p C 1; q/ leaves !0 invariant. From (3.6) we have the 4n-dimensional bundle satisfies the condiDH D fI N; JN; KN g? . For this, we shall check that †4pC3;4q H tions (i), (ii), (iii) and (iv) of Definition 7.1 and (7.4). In order to prove (iv) and (7.4), we quote the following lemma from [4]. Lemma 7.8. Put J1 D I , J2 D J , J3 D K. Then d!˛ .X; Y / D g H .X; J˛ Y / .X; Y 2 DH ; ˛ D 1; 2; 3/: Theorem 7.9. .†3C4p;4q ; !0 ; fI; J; Kg; g H / is a .4nC3/-dimensional homogeneous H quaternionic CR-manifold of signature .3 C 4p; 4q/ such that Null !0 D DH .p C q D n 1/. Moreover, there exists the equivariant principal bundle of the pseudoRiemannian submersion over (homogeneous) pseudo-quaternionic Kähler projective space HP p;q of signature .4p; 4q/:

; g H / ! .PSp.p C 1; q/; HP p;q ; gO H /: Sp.1/ ! .Sp.p C 1; q/ Sp.1/; †3C4p;4q H Proof. First of all, let !0 D !1 i C !2 j C !3 k. Then it follows that !0 3 ^ d !0 2n D !0 ^ !0 ^ !0 ^ .d!0 ^ d!0 / ^ ^ .d!0 ^ d!0 / ƒ‚ … „ n-times D 6!1 ^ !2 ^ !3 ^ .d!1 2 C d!2 2 C d!3 2 /n ¤ 0 . (Compare, for example, [19], [29].) This calculation shows at any point of †3C4p;4q H (i) and (ii). In particular, each !a is a nondegenerate contact form on †3C4p;4q . Using H (7.8) and the fact that 1 generates fe i g 2R Sp.1/, this implies that L1 !1 D 0:

(7.9)

(Similarly L2 !2 D L3 !3 D 0.) Since !a . a / D 1 from (7.7), it follows that La !a D a d!a D 0:

(7.10)

In particular, each a is the characteristic vector field for !a . This implies that DH D T3 3C4p;4q D f 1 ; 2 ; 3 g ˚ DH . aD1 Null!a for which there is the decomposition T †H H This shows (iii). From Lemma 7.8, d!˛ .X; Y / D g .X; J˛ Y / D g H .X; Jˇ J Y / D d!ˇ .X; J Y /, this shows (iv), i.e., J D .d!ˇ jDH /1 B .d!˛ jDH /. If fei giD1;:::;4n is the orthonormal basis of DH , then the dual frame i is obtained as i .ej / D ıji and i . 1 / D i . 2 / D i . 3 / D 0. Then d!a .ei ; ej / D g.ei ; Ja ej / D Jija .


Since f 1 ; 2 ; 3 g generates Sp.1/ of the bundle (3.2), we obtain d!a C 2!b ^ !c D Jija i ^ j . Applying to J; K similarly, we obtain the following structure equation of the bundle (3.2): d!0 C !0 ^ !0 D .Iij i C Jij j C Kij k/ i ^ j :

(7.11)

From this equation, condition (7.4) is easily checked and hence Null ˛ D f 1 ; 2 ; 3 g .˛ D 1; 2; 3/. Remark 7.10. It is shown in [3] that .†3C4p;4q ; fI; J; Kg; g H / is a pseudo 3-Sasakian H space form of constant positive curvature with type .4p C 3; 4q/. Using O’Neill’s formula, the pseudo-quaternionic Kähler projective space HP 4p;4q is of constant positive quaternionic curvature.

7.2 Construction of the flat p-c qCR structure on S 4pC3;4 q nC1 Put D D DH . If Nx is the normal vector at x 2 †3C4p;4q HH , recall that H 3C4p;4q ? ? .Null !0 /x D Dx D fI Nx ; JNx ; KNx g Tx †H D Nx with respect to g H D Reh ; i. If we note that fNx ; I Nx ; JNx ; KNx g D xH, then Dx D xH? with respect to h ; i. Let W HnC1 ! HnC2 be the embedding defined by .z1 ; : : : ; znC1 / D .z1 ; : : : ; znC1 ; 1/ so that

/ V04nC7 ; .†3C4p;4q H

/ P .V0 / D S 4pC3;4q P B .†3C4p;4q H

on the quotient:

(7.12)

has the transitive isometry group Sp.p C1; q/Sp.1/ (cf: Definition 3.1), As †3C4p;4q H this embedding implies that Sp.p C 1; q/ Sp.1/ is identified with the subgroup P .Sp.p C 1; q/ Sp.1// of PSp.p C 1; q C 1/ leaving the last component znC2 invariant in V04nC7 HnC2 . As usual, we define a 1-form 0 on P B .†3C4p;4q / H by .P B / 0 D !0 . As h ..Null !0 /x /; .x/i D h..Null !0 /x ; 0/; .x; 1//i D 0, it follows that ..Null !0 /x / D .x/? D ..x/H/? and hence .Null 0 /Œ.x/ D P ..Null !0 /x / D P ...x/H/? /:

(7.13)

In particular, .Null 0 /Œ.x/ is invariant under fI; J; Kg. If f 2 PSp.p C 1; q C 1/, then it follows that f ..Null 0 /Œ.x/ / D .Null f 1 0 /Œf ..x// D P ..f ..x//H/? / //. Suppose that h.Œ.x// D Œ.x/. Then h..x// D .x/ for on f .P B .†3C4p;4q H 2 H . Then we have h ..Null 0 /Œ.x/ / D P ...x/H/? / D .Null 0 /Œ.x/ :

(7.14)

886

Yoshinobu Kamishima

For any point b 2 S 4pC3;4q , we have f .Œ.x// D b for some element f 2 PSp.p C 1; q C 1/. From (7.14), Hb D f.Null f 1 0 /Œf ..x// gf 2PSp.pC1;qC1/ is the well-defined distribution at b. S Put H D b2S 4pC3;4q Hb . As above, it is invariant under fI; J; Kg. Obviously //gf 2PSp.pC1;qC1/ covers S 4pC3;4q and each f 1 0 is a qCR ff .P B .†3C4p;4q H structure on U D f .P B .†3C4p;4q //. By Definition 7.5, we get: H Theorem 7.11. S 4pC3;4q supports a pseudo-conformal qCR structure, i.e., there exists a qCR structure ! locally on each neighborhood U such that H jU D Null !: Moreover, the automorphism group of pseudo-conformal qCR transformations Autpc qCR .S 4pC3;4q / with respect to this p-c qCR structure on S 4pC3;4q is PSp.p C 1; q C 1/. Proof. We have shown in [4] that there is a canonical pseudo-conformal qCR structure H can on S 3C4p;4q invariant under PSp.p C 1; q C 1/. In fact, we have H D H can . As for a proof of the second part of the theorem, we will explain the pseudo-conformal quaternionic Heisenberg geometry in the next section. This geometry is used to prove Aut.S 4pC3;4q / D Autpc qCR .S 4pC3;4q / D PSp.p C 1; q C 1/ (cf. [4]).

7.3 Pseudo-conformal quaternionic Heisenberg geometry Let PSp.p C 1; q C 1/ be the group of all automorphisms preserving the flat p-c qCR structure of S 4pC3;4q D PSp.p C 1; q C 1/=P C .H/ (cf: [3]). We consider the stabilizer of the point at infinity f1g D Œ1; 0; : : : ; 0; 1 2 S 4pC3;4q . Recall the (indefinite) Heisenberg nilpotent Lie group M D M.p; q/ from [19]. It is the product R3 Hn with group law .a; y/ .b; z/ D .a C b Imhy; zi; y C z/: Here h ; i is the Hermitian inner product of signature .p; q/ on Hn as in (2.1) and Imh ; i is the imaginary part .p C q D n/. It is nilpotent because the commutator subgroup ŒM; M D R3 which is the center of M consisting of elements of the form .a; 0/. In particular, there is the central extension 1 ! R3 ! M!Hn ! 1:

(7.15)

Denote by Sim.M/ the semidirect product M Ì .Sp.p; q/ Sp.1/ RC / where the action .A g; t / 2 Sp.p; q/ Sp.1/ RC on .a; y/ 2 M is given by .A g; t / B .a; y/ D .t 2 gag 1 ; t Ayg 1 /: 4pC3;4q

(7.16)

f1g. The stabilizer Denote the origin by O D Œ1; 0; : : : ; 0; 1 2 S 4pC3;4q /1 is isomorphic to Sim.M/ (cf: [21]). The orbit M O is a dense Aut.S


open subset of S 4pC3;4q . The embedding is defined by 3 2 kz k2 kz k2 C 1 C i a C j b C kc 2 p 7 6 7 6 p2zC ..a; b; c/; .zC ; z // 2 M ! 6 7: 2z 5 4 kzC k2 kz k2 2

(7.17)

C 1 C i a C j b C kc

The pair .Sim.M/; M/ is said to be a pseudo-conformal quaternionic Heisenberg geometry which is a subgeometry of flat p-c qCR geometry .Aut.S 4pC3;4q /;S 4pC3;4q /. Proposition 7.12. There exists a global Im H-valued 1- form !M on the (indefinite) Heisenberg nilpotent group M which represents the pseudo-conformal qCR structure H jM . As in (6.7), !M is obtained explicitly. If .t1 ; t2 ; t3 ; .z1 ; : : : ; zn // is the coordinate of M D R3 Hn , then !M D dt1 i C dt2 j C dt3 k C Imhz; dzi: Note that !M is not an admissible 1-form by the definition (because f ˛ g˛D1;2;3 generates R3 ) but it is locally conformal to !0 , i.e., !M D !0 N for some H -valued S 4pC3;4q . It follows that Null !M D Null !0 D H jM function on M\†3C4p;4q H under the identification †3C4p;4q D P B .†3C4p;4q /. So M is a pseudo-conformal H H quaternionic CR-manifold but not a qCR-manifold. Taking a discrete uniform subgroup M Ì .Sp.p/ Sp.q// Sp.1/ and an infinite cyclic subgroup Z from .Sp.p/ Sp.q// RC , we obtain that Corollary 7.13. There exist compact flat pseudo-conformal quaternionic CR-manifolds M= and S 4nC2 S 1 . Here the universal covering of the latter manifold is M fOg S 4nC2 RC ( S 4nC3;4q fO; 1g/. Remark 7.14. Pseudo-conformal qCR-manifolds contain the class of 3-pseudo-Sasa= , where is a subgroup of Sp.p C 1; q/ kian manifolds. For example, †3C4p;4q H Sp.1/ that acts properly and freely. But it is very difficult whether there exist compact 3-pseudo-Sasakian manifolds in general. As a special case, when q D 0 or p D 0, we can find discrete cocompact subgroups from Sp.n C 1/ Sp.1/ or Sp.1; n/ Sp.1/ that 4nC3 act properly and freely on †3C4n;0 D S 4nC3 or †3;4n D V1 D Sp.1; n/=Sp.n/, H H respectively. Thus, we obtain compact 3-pseudo-Sasakian manifolds (nondegenerate qCR-manifolds); (i) The spherical space form S 4nC3 =F which is Sp.1/ or SO.3/bundle over the quaternionic Kähler projective orbifold HP n =F of positive scalar curvature. .F Sp.nC1/Sp.1/ is a finite group./ (ii) The pseudo-Riemannian stan4nC3 = of signature .4n; 3/ with constant sectional curvature 1 dard space form V1 n = of which is an Sp.1/-bundle over the quaternionic Kähler hyperbolic orbifold HH

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Yoshinobu Kamishima

negative scalar curvature. . PSp.1; n/ is a discrete uniform subgroup./ It is unknown whether there exist compact 3-pseudo-Sasakian manifolds (or qCR-manifolds) whose pseudo-quaternionic Kähler quotients (orbifolds) have zero Ricci curvature. However, (iii) the indefinite quaternionic Heisenberg nilmanifold M= is a compact p-c qCR manifold whose quotient has the zero Ricci curvature. Some finite cover of M= is a Heisenberg nilmanifold which is a principal 3-torus bundle over the flat quaternionic n-torus THp;q of signature .4p; 4q/ .p C q D n/. Obviously those compact manifolds are not positive-definite 3-Sasakian manifolds. .See [19], [21] more generally./ Refer to [8], [10], [9], [11], [31], [32] for .positive definite/ Sasakian 3-structures. Remark that there is a noncompact pseudo-quaternionic Kähler manifold with zero Ricci curvature .non-isometric to Hn /.

7.4 Curvature forms in R.Sp.p; q/ Sp.1// Concerning geometric invariants, there is the curvature tensor T of a p-c qCR structure in [3]. This is thought of as a quaternionic analogue of Chern–Moser’s CR curvature tensor. On a .4n C 3/-dimensional almost p-c qCR-manifold .M; D; Q/, there is an sp.pC1; qC1/-valued Cartan form whose associated curvature form … has zero curvature if and only if .M; D/ is locally isomorphic to .PSp.p C 1; q C 1/=P C .H/; H /. We do not know whether the curvature tensor T on M could be derived only from the Cartan form … and its curvature form of the almost p-c qCR structure D because D lacks the structure equations representing the integrability conditions unlike the p-c qCR structure, see (7.11). However, with the aid of the pseudo-Riemannian connection of the pseudo-Sasakian 3-structure, which is locally equivalent to a p-c qCR structure, we can define a quaternionic CR curvature tensor R. Based on the construction of this tensor R, we shall obtain a curvature tensor T D R RHP p;q :

(7.18)

Here RHP p;q is the curvature tensor of the pseudo-quaternionic Kähler projective space HP p;q in dimensional 4n .p C q D n/. Remark that if T vanishes for a p-c qCR structure, then … also vanishes with respect to the underlying almost p-c qCR structure. Let R.Sp.p; q/ Sp.1// be the space of all curvature tensors. (See, for example, [1].) It decomposes into the direct sum R.Sp.p; q//˚RHP .Sp.p; q/Sp.1// .n 2/. Here R.Sp.p; q// is the space of those curvatures with zero Ricci forms and RHP R is the space of curvature tensors of HP p;q . Since T is locally defined by our Definition 7.5, we have to show that it becomes a global invariant on M , i.e., T is invariant under the equivalence of p-c qCR structures. In [3] we have shown the following result. Theorem 7.15. The fourth-order curvature tensor T D .Tjik` / 2 R.Sp.p; q// exists globally on any pseudo-conformal quaternionic CR-manifold of dimension 4n C 3 .n 1/.


When n D 1, then T D W 2 R.Sp.1// where W is anti-self-dual .W C D 0/. We now state the uniformization theorem from [3]. Theorem 7.16. (1) Let M be a .4n C 3/-dimensional p-c qCR manifold .n 1/. If the curvature tensor T vanishes, then M is locally modeled on S 4pC3;4q whose coordinate changes belong to the group PSp.p C 1; q C 1/. (2) If M is a 3-dimensional p-c qCR manifold, then there is a global Riemannian metric g so that .M; g/ is locally isometric to the standard 3-sphere S 3 .

8 Remarks and problems 8.1 Conformally flat Lorentz metrics In Section 4, we have obtained the standard pseudo-Riemannian metric g1 on S p;q in Proposition 4.1. It is locally conformal to the euclidean metric g0 of signature .p; q/ on Rn .p C q D n 2/. When p D 2n C 1, q D 1, the metric g1 is a conformally flat Lorentz metric on the .2n C 2/-manifold S 2nC1;1 D S 2nC1 S 1 =Z2 with the group of conformal transformations O.2n C 2; 2/. Note that U.n C 1; 1/ O.2n C 2; 2/ is the natural embedding. Problem 8.1. How does g1 on S 2nC1;1 relate to the Fefferman Lorentz metric obtained from the standard CR-manifold S 2nC1 ?

8.2 Exotic quaternionic invariant closed 4-forms on Hn Let M be the (positive definite) quaternionic Heisenberg Lie group as before (cf. Section 7.3). The abelian Lie subgroup R3 T n consists of the direct product of the center R3 of M and the maximal torus T n of Sp.n/. Note that R3 T n is the subgroup of the similarity group Sim.M/ D M.Sp.n/Sp.1/RC /. Let s D .s1 ; s2 ; s3 / 2 R3 and z D .z1 ; : : : ; zn / 2 Hn . We introduce a representation W R3 ! R3 T n such that .t1 /..s1 ; s2 ; s3 /; .z1 ; : : : ; zn // D ..s1 C t1 ; s2 ; s3 /; .e ita1 z1 ; : : : ; e it an zn //; (8.1) .t2 /..s1 ; s2 ; s3 /; z/ D ..s1 ; s2 C t2 ; s3 /; z/; .t3 /..s1 ; s2 ; s3 /; z/ D ..s1 ; s2 ; s3 C t3 /; z/: See (7.16) for the action of R3 T n on M. As the .R3 /-action on M is proper, we have the principal bundle

R3 ! M ! M=.R3 /:

(8.2)

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Identifying Hn with R4n , the group .R3 / induces the three vector fields

d d d d d C a1 x1 x2 C a 1 x3 x4 1 D dt1 dx2 dx1 dx4 dx3 d d C C an x4n3 x4n2 x4n2 dx4n3 d d C an x4n1 x4n ; dx4n dx4n1 d d 2 D : and 3 D dt3 dt2

(8.3)

We define real 1-forms on M as follows: !1 D dt1 C .x1 dx2 x2 dx1 / C .x3 dx4 x4 dx3 / C C .x4n3 dx4n2 x4n2 dx4n3 / C .x4n1 dx4n x4n dx4n1 /; !2 D dt2 C .x1 dx3 x3 dx1 / .x2 dx4 x4 dx2 / C (8.4) C .x4n3 dx4n1 x4n1 dx4n3 / .x4n2 dx4n x4n dx4n2 /; !3 D dt3 C .x1 dx4 x4 dx1 / C .x2 dx3 x3 dx2 / C C .x4n3 dx4n x4n dx4n3 / C .x4n2 dx4n1 x4n1 dx4n2 /: Setting ! D !1 i C !2 j C !3 k;

(8.5)

! is an Im H-valued 1-form on M. Then it follows that !1 . 1 / D 1 C a1 jz1 j2 C C an jzn j2 . Suppose that 0 a1 an : (8.6) Then !1 . 1 / > 0, so we introduce new 1-forms 1 D

1 !1 ; !1 . 1 /

2 D !2

and

3 D !3 :

(8.7)

We obtain that i . j / D ıij ; d i . i ; X/ D 0 for all X 2 T M.

(8.8)

Then the 4-form on M, (8.9)

D d 1 ^ d 1 C d 2 ^ d 2 C d 3 ^ d 3 T3 is fI; J; Kg-invariant on D D iD1 Null !i . As the projection induces an isomorphism of D onto T .M=.R3 // at each point from (8.2), we can show that Lemma 8.2. The quaternion structure Q D SpanfI; J; Kg on D induces the quatery D SpanfI; O J; O Kg y on the quotient M=.R3 /. In particular, M=.R3 / nion structure Q y is isomorphic to the standard quaternionic number space Hn . endowed with Q


O J; O Kg-invariant y y on Now using (8.8), the 4-form induces an fI; 4-form 3 n y .M=.R /; Q/ D .H ; Q/ such that y D :

(8.10)

y D 0. Since is closed, d y of signature .4p; 4q/ on Hn which is Proposition 8.3. There exists a closed 4-form not isomorphic to the standard closed 4-form of signature .4p; 4q/ on Hn .pCq D n/. Proof. Let 0 be the canonical 4-form on M which induces the standard quaternionic y 0 on Hn . If y is isometric to y 0 , then it is easy to see that the isometry Kähler 4-form extends to an isometry between and 0 on M. Then this isometry conjugates the group .R3 / to the standard central group R3 inside PSp.p C 1; q C 1/. However, by construction of the representation , this is impossible. On the other hand, setting gO . X; Y / D d 1 .IX; Y /

for all X; Y 2 D,

(8.11) n

gO is a well-defined fI; J; Kg-invariant Riemannian metric on H . However the equality that d 1 .IX; Y / D d 2 .JX; Y / D d 3 .KX; Y / .X; Y 2 D/ does not hold although d!1 .IX; Y / D d!2 .JX; Y / D d!3 .KX; Y / by (8.4). So gO is not a quaternionic Kähler metric. Obviously when we set gO ! D d 2 .J ; / D d 3 .K; /, it is the standard quaternionic Kähler metric on Hn by the above equality. Problem 8.4. Does there exist a nonstandard quaternionic Kähler metric on Hn ?

8.3 Global rigidity of compact nondegenerate geometric manifolds with noncompact geometric flow We pose the following vague conjecture whose supporting results are the Obata and Lelong-Ferrand’s theorem, and the analogue of compact strictly pseudoconvex CRmanifolds with noncompact CR flow, and also compact positive definite p-c qCR manifolds with noncompact p-c qCR flow. See [18]. (We refer to [16] for the related works and references therein, especially [27], [30].) Conjecture 8.5. If a compact nondegenerate, conformal (respectively CR, pseudoconformal quaternionic CR) manifold M supports a noncompact closed R-flow consisting of conformal (respectively CR, pseudo-conformal quaternionic CR) transformations, then M is flat, moreover M is isomorphic to a geometric manifold covered by a domain of S c.pC1/1;cq .c D 1; 2; 4/.

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References [1] [2]

[3]

[4] [5]

[6]

[7] [8] [9]

[10] [11] [12] [13] [14]

[15] [16] [17]

D. A. Alekseevsky, Riemannian spaces with exceptional holonomy groups. Funct. Anal. Appl. 2 (1968), 97–105. 888 D. V. Alekseevsky and Y. Kamishima, Quaternionic and para-quaternionic CR structure on .4n C 3/-dimensional manifolds. Cent. Eur. J. Math. 2 (2004), no. 5, 732–753. 873, 882, 883 D. V Alekseevsky and Y. Kamishima, Pseudo-conformal quaternionic CR structure on .4n C 3/-dimensional manifolds. Ann. Mat. Pura Appl. (4) 187 (2008), no. 3, 487–529. 864, 865, 876, 882, 883, 885, 886, 888, 889 D. V.Alekseevsky andY. Kamishima,Almost pseudo-conformal quaternionic CR structure on .4n C 3/-dimensional manifolds. In preparation. 864, 884, 886 D. V. Alekseevsky and A. F. Spiro, Prolongations of Tanaka structures and regular CR structures. In Selected topics in Cauchy-Riemann geometry, Quad. Mat. 9, Department of Mathematics, Seconda Università di Napoli, Caserta 2001, 1–37. 865 T. Barbot, V. Charette, T. Drumm, W. Goldman, and K. Melnick, A primer on the .2 C 1/ Einstein universe. In Recent developments in pseudo-Riemannian geometry, ESI Lect. Math. Phys., European Math. Soc. Publishing House, Zürich, 2008 179–229. 866 O. Biquard, Quaternionic contact structures. In Quaternionic structures in mathematics and physics, Rome 1999, World Scientific Publishing, River Edge, NJ, 2001, 23–30. 882 D. E. Blair, Riemannian geometry of contact and symplectic manifolds. Progr. Math. 203, Birkhäuser, Boston, MA, 2002. 888 C. Boyer and K. Galicki, 3-Sasakian manifolds. In Surveys in differential geometry: essays on Einstein manifolds, Surv. Differ. Geom. VI, International Press, Boston, MA, 1999, 123–184. 888 C. P. Boyer, K. Galicki, and B. M. Mann, The geometry and topology of 3-Sasakian manifolds. J. Reine Angew. Math. 455 (1994), 183–220. 888 C. P. Boyer, K. Galicki, and P. Piccinni, 3-Sasakian geometry, nilpotent orbits, and exceptional quotients. Ann. Global Anal. Geom. 21 (2002), no. 1, 85–110. 888 D. M. Burns and S. Shnider, Spherical hypersurfaces in complex manifolds. Invent. Math. 33 (1976), 223–246. ˇ and J. Slovak, Weyl structures for parabolic geometries. Math. Scand. 93 (2003), A. Cap no. 1, 53–90. 865 S. S. Chen and L. Greenberg, Hyperbolic spaces. In Contributions to analysis (A collection of papers dedicated to Lipman Bers), ed. by L. Ahlfors et al., Academic Press, New York 1974, 49–87. S. S. Chern and J. Moser, Real hypersurfaces in complex manifolds. Acta Math. 133 (1974), 48–69. 876 S. Dragomir and G. Tomassini, Differential geometry and anaysis on CR manifolds. Progr. Math. 246, Birkhäuser, Basel 2006. 891 N. J. Hitchin, The self-duality equations on a Riemannian surface. Proc. London Math. Soc. 55 (1987), 59–126.

[18] Y. Kamishima, Geometric flows on compact manifolds and global rigidity. Topology 35 (1996), 439–450. 891

Chapter 24. Nondegenerate conformal, CR, quaternionic CR structures on manifolds 893 [19] Y. Kamishima, Geometric rigidity of spherical hypersurfaces in quaternionic manifolds. Asian J. Math. 3 (1999), no.3, 519–556. 872, 877, 884, 886, 888 [20] Y. Kamishima, Heisenberg, spherical CR geometry and Bochner flat locally conformal Kähler manifolds. Int. J. Geom. Methods Mod. Phys. 3 (2006), 1089–1116. 873, 879 [21] Y. Kamishima and T. Udono, Three dimensional Lie group actions on compact (4n+3)dimensional geometric manifolds. Differential Geom. Appl. 21 (2004), 1–26. 878, 886, 888 [22] S. Kobayashi, Remarks on complex contact manifolds. Proc. Amer. Math. Soc. 10 (1959), 164–167. [23] S. Kobayashi, Transformation groups in differential geometry. Ergeb. Math. Grenzgeb. 70, Springer-Verlag, Heidelberg 1972. [24] S. Kobayashi and K. Nomizu, Foundations of differential geometry I, II. John Wiley & Sons, New York 1963, 1969. [25] R. Kulkarni, On the principle of uniformization. J. Differential Geom. 13 (1978), 109–138. [26] R. Kulkarni, Proper actions and pseudo-Riemannian space forms. Adv. Math. 40 (1981), 10–51. 867 [27] J. M. Lee, CR manifolds with noncompact connected automorphism groups. J. Geom. Anal. (1) 6 (1996), 79–90. 891 [28] P. Libermann, Sur les structures presque complexes et autres structures infinitésimales régulières. Bull. Soc. Math. France 83 (1955), 195–224. 882 [29] S. Sasaki, Spherical space forms with normal contact metric 3-structure. J. Differential Geom. 6 (1972), 307–315. 872, 884 [30] R. Schoen, On the conformal and CR automorphism groups. Geom. Func. Anal. (2) 5 (1995), 464– 81. 891 [31] S. Tanno, Killing vector fields on contact Riemannian manifolds and fibering related to the Hopf fibrations. Tôhoku Math. J. 23 (1971), 313–333. 873, 888 [32] S. Tanno, Remarks on a triple of K-contact structures. Tôhoku Math. J. 48 (1996), 519–531. 873, 888 [33] S. Webster, On the pseudo conformal geometry of a Kähler manifold. Math. Z. 157 (1977), 265–270. 876 [34] S. Webster, Pseudo-Hermitian structures on a real hypersurface. J. Differential Geom. 13 (1978), 25–41. 876 [35] J. Wolf, Spaces of constant curvature. McGraw-Hill, Inc., Sydney 1967. 867

Part H

Other topics of recent interest

Chapter 25

Linear wave equations on Lorentzian manifolds Christian Bär

Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . . Distributional sections in vector bundles . . . . . 2.1 Preliminaries on distributional sections . . . 2.2 Differential operators acting on distributions 2.3 Supports . . . . . . . . . . . . . . . . . . . 2.4 Convergence of distributions . . . . . . . . 3 Globally hyperbolic Lorentzian manifolds . . . . 4 Wave operators . . . . . . . . . . . . . . . . . . 5 The Cauchy problem . . . . . . . . . . . . . . . 6 Fundamental solutions . . . . . . . . . . . . . . 7 Green’s operators . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

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897 898 898 900 900 901 902 905 907 908 910 914

1 Introduction In General Relativity spacetime is modelled by a Lorentzian manifold, see e.g. [8], [15]. Many physical phenomena, such as electro-magnetic radiation, are described by solutions to certain linear wave equations defined on this spacetime manifold. Thus a good understanding of the theory of wave equations is crucial. This includes initial value problems (the Cauchy problem), fundamental solutions, and inverse operators (Green’s operators). The classical textbooks on partial differential equations contain the relevant results for small domains in Lorentzian manifolds or for very special manifolds such as Minkowski space. In this chapter we summarize the global analytic results obtained in [4], see also Leray’s unpublished lecture notes [13] and Choquet-Bruhat’s exposition [7]. In order to obtain a good solution theory one has to impose certain geometric conditions on the underlying manifold. The situation is similar to the study of elliptic operators on Riemannian manifolds. In order to ensure that the Laplace–Beltrami operator on a Riemannian manifold M is essentially self-adjoint one may make the natural assumption that M be complete. Unfortunately, there is no good notion of completeness for Lorentzian manifolds. It will turn out that the analysis of wave operators works out nicely if one assumes that the underlying Lorentzian manifold be globally hyperbolic.

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Completeness of Riemannian manifolds and global hyperbolicity of Lorentzian manifolds are indeed related. If .S; g0 / is a Riemannian manifold, then the Lorentzian cylinder M D R S with product metric g D dt 2 C g0 is globally hyperbolic if and only if .S; g0 / is complete. We will start by collecting some material on distributional sections in vector bundles. Then we will summarize the theory of globally hyperbolic Lorentzian manifolds. Then we will define wave operators, also called normally hyperbolic operators, and give some examples. After that we consider the basic initial value problem, the Cauchy problem. It turns out that on a globally hyperbolic manifold solutions exist and are unique. They depend continuously on the initial data. The support of the solutions can be controlled which is physically nothing than the statement that a wave can never propagate faster than with the speed of light. In the subsequent section we use the results on the Cauchy problem to show existence and uniqueness of fundamental solutions. This is closely related to the existence and uniqueness of Green’s operators. The author is very grateful for many helpful discussions with colleagues including Helga Baum, Olaf Müller, Nicolas Ginoux, Frank Pfäffle, and Miguel Sánchez. The author also thanks the Deutsche Forschungsgemeinschaft for financial support.

2 Distributional sections in vector bundles Let us start by giving some definitions and by fixing the terminology for distributions on manifolds. We will confine ourselves to those facts that we will actually need later on. A systematic and much more complete introduction may be found e.g. in [9].

2.1 Preliminaries on distributional sections Let M be a manifold equipped with a smooth volume density dV. Later on we will use the volume density induced by a Lorentzian metric but this is irrelevant for now. We consider a real or complex vector bundle E ! M . We will always write K D R or K D C depending on whether E is real or complex. The space of compactly supported smooth sections in E will be denoted by D.M; E/. We equip E and the cotangent bundle T M with connections, both denoted by r. They induce connections on the tensor bundles T M ˝ ˝ T M ˝ E, again denoted by r. For a continuously differentiable section ' 2 C 1 .M; E/ the covariant derivative is a continuous section in T M ˝ E, r' 2 C 0 .M; T M ˝ E/. More generally, for ' 2 C k .M; E/ we get r k ' 2 C 0 .M; T M ˝ ˝ T M ˝E/. ƒ‚ … „ k factors

We choose an auxiliary Riemannian metric on T M and an auxiliary Riemannian or Hermitian metric on E depending on whether E is real or complex. This induces metrics on all bundles T M ˝ ˝ T M ˝ E. Hence the norm of r k ' is defined at all points of M .

Chapter 25. Linear wave equations on Lorentzian manifolds

899

For a subset A M and ' 2 C k .M; E/ we define the C k -norm by k'kC k .A/ WD max

sup jr j '.x/j:

j D0;:::;k x2A

(1)

If A is compact, then different choices of the metrics and the connections yield equivalent norms k kC k .A/ . For this reason there will be no need to explicitly specify the metrics and the connections. The elements of D.M; E/ are referred to as test sections in E. We define a notion of convergence of test sections. Definition 2.1. Let '; 'n 2 D.M; E/. We say that the sequence .'n /n converges to ' in D.M; E/ if the following two conditions hold: (1) There is a compact set K M such that the supports of ' and of all 'n are contained in K, i.e., supp.'/; supp.'n / K for all n. (2) The sequence .'n /n converges to ' in all C k -norms over K, i.e., for each k 2 N k' 'n kC k .K/ ! 0: n!1

We fix a finite-dimensional K-vector space W . Recall that K D R or K D C depending on whether E is real or complex. Denote by E the vector bundle over M dual to E. Definition 2.2. A K-linear map F W D.M; E / ! W is called a distribution in E with values in W or a distributional section in E with values in W if it is continuous in the sense that for all convergent sequences 'n ! ' in D.M; E / one has F Œ'n ! F Œ'. We write D 0 .M; E; W / for the space of all W -valued distributions in E. Note that since W is finite-dimensional all norms jj on W yield the same topology on W . Hence there is no need to specify a norm on W for Definition 2.2 to make sense. Note moreover, that distributional sections in E act on test sections in E . Example 2.3. Pick a bundle E ! M and a point x 2 M . The delta-distribution ıx is a distributional section in E with values in Ex . For ' 2 D.M; E / it is defined by ıx Œ' D '.x/: Example 2.4. Every locally integrable section f 2 L1loc .M; E/ can be regarded as a K-valued distribution in E by setting for any ' 2 D.M; E / Z '.f / dV: f Œ' WD M 1

Here '.f / denotes the K-valued L -function with compact support on M obtained by pointwise application of '.x/ 2 Ex to f .x/ 2 Ex .

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2.2 Differential operators acting on distributions Let E and F be two K-vector bundles over the manifold M , K D R or K D C. Consider a linear differential operator P W C 1 .M; E/ ! C 1 .M; F /. There is a unique linear differential operator P W C 1 .M; F / ! C 1 .M; E / called the formal adjoint of P such that for any ' 2 D.M; E/ and 2 D.M; F / Z Z .P '/ dV D .P /.'/ dV: (2) M

M

2 If P is of order k, then so is P and (2) holds for all ' 2 C k .M; E/ and C k .M; F / such that supp.'/ \ supp. / is compact. With respect to the canonical identification E D .E / we have .P / D P . Any linear differential operator P W C 1 .M; E/ ! C 1 .M; F / extends canonically to a linear operator P W D 0 .M; E; W / ! D 0 .M; F; W / by .P T /Œ' WD T ŒP ' where ' 2 D.M; F /. If a sequence .'n /n converges in D.M; F / to 0, then the sequence .P 'n /n converges to 0 as well because P is a differential operator. Hence .P T /Œ'n D T ŒP 'n ! 0. Therefore P T is indeed again a distribution. The map P W D 0 .M; E; W / ! D 0 .M; F; W / is K-linear. If P is of order k and ' is a C k -section in E, seen as a K-valued distribution in E, then the distribution P ' coincides with the continuous section obtained by applying P to ' classically. An important special case occurs when P is of order 0, that is, if P lies in C 1 .M; Hom.E; F //. Then P 2 C 1 .M; Hom.F ; E // is the pointwise adjoint. In particular, for a function f 2 C 1 .M; K/ we have .f T /Œ' D T Œf ':

2.3 Supports Definition 2.5. The support of a distribution T 2 D 0 .M; E; W / is defined as the set supp.T / WD fx 2 M j for all neighborhoods U of x there exists ' 2 D.M; E/ with supp.'/ U and T Œ' 6D 0g. It follows from the definition that the support of T is a closed subset of M . In case T is a L1loc -section this notion of support coincides with the usual one for sections. If for ' 2 D.M; E / the supports of ' and T are disjoint, then T Œ' D 0. Namely, for each x 2 supp.'/ there is a neighborhood U of x such that T Œ D 0 whenever supp. / U . Cover the compact set supp.'/ by finitely many such open sets U1 ; : : : ; Uk . Using a partition of unity one can write ' D 1 C C k with j 2 D.M; E / and supp. j / Uj . Hence T Œ' D T Œ

1

C C

k

D TŒ

1

C C TŒ

k

D 0:


901

Be aware that it is not sufficient to assume that ' vanishes on supp.T / in order to ensure T Œ' D 0. For example, if M D R and E is the trivial K-line bundle let T 2 D 0 .R; K/ be given by T Œ' D ' 0 .0/. Then supp.T / D f0g but T Œ' D ' 0 .0/ may well be nonzero while '.0/ D 0. If T 2 D 0 .M; E; W / and ' 2 C 1 .M; E /, then the evaluation T Œ' can be defined if supp.T / \ supp.'/ is compact even if the support of ' itself is noncompact. To do this pick a function 2 D.M; R/ that is constant 1 on a neighborhood of supp.T / \ supp.'/ and put T Œ' WD T Œ ': This definition is independent of the choice of since for another choice 0 we have T Œ' T Œ 0 ' D T Œ. 0 /' D 0 because supp.. 0 /'/ and supp.T / are disjoint. Let T 2 D 0 .M; E; W / and let M be an open subset. Each test section ' 2 D.; E / can be extended by 0 and yields a test section ' 2 D.M; E /. This defines an embedding D.; E / D.M; E /. By the restriction of T to we mean its restriction from D.M; E / to D.; E /. Definition 2.6. The singular support sing supp.T / of a distribution T 2 D 0 .M; E; W / is the set of points which do not have a neighborhood restricted to which T coincides with a smooth section. The singular support is also closed and we always have sing supp.T / supp.T /. Example 2.7. For the delta-distribution ıx we have supp.ıx / D sing supp.ıx / D fxg.

2.4 Convergence of distributions The space D 0 .M; E/ of distributions in E will always be given the weak topology. This means that Tn ! T in D 0 .M; E; W / if and only if Tn Œ' ! T Œ' for all ' 2 D.M; E /. Linear differential operators P are always continuous with respect to the weak topology. Namely, if Tn ! T , then we have for every ' 2 D.M; E / P Tn Œ' D Tn ŒP ' ! T ŒP ' D P T Œ': Hence P Tn ! P T: Remark 2.8. Let Tn ; T 2 C 0 .M; E/ and suppose kTn T kC 0 .M / ! 0. Consider Tn and T as distributions. Then Tn ! T in D 0 .M; E/. In particular, for every linear differential operator P we have P Tn ! P T .

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3 Globally hyperbolic Lorentzian manifolds Next we summarize some notions and facts from Lorentzian geometry. More comprehensive introductions can be found in [2] and in [14]. By a Lorentzian manifold we mean a semi-Riemannian manifold whose metric has signature .; C; : : : ; C/. We denote the Lorentzian metric by g or by h; i. A tangent vector X 2 TM is called timelike if hX; X i < 0, lightlike if hX; X i D 0 and X 6D 0, causal if it is timelike or lightlike, and spacelike otherwise. At each point p 2 M the set of timelike vectors in Tp M decomposes into two connected components. A timeorientation on M is a choice of one of the two connected components of timelike vectors in Tp M which depends continuously on p. This means that we can find a continuous timelike vector field on M taking values in the chosen connected components. Tangent vectors in the chosen connected component are called future directed, those in the other component are called past directed. Let M be a timeoriented Lorentzian manifold. A piecewise C 1 -curve in M is called timelike, lightlike, causal, spacelike, future directed, or past directed if its tangent vectors are timelike, lightlike, causal, spacelike, future directed, or past directed respectively. M .x/ of a point x 2 M is the set of points that can be The chronological future IC reached from x by future directed timelike curves. Similarly, the causal future JCM .x/ of a point x 2 M consists of those points that can be reached from x by causal curves M .A/ WD and of x itself. The chronological future of a subset A M is defined to be IC S S M M M I .x/. Similarly, the causal future of A is J .A/ WD J .x/. The x2A C x2A C C chronological past IM .A/ and the causal past JM .A/ are defined by replacing future M directed curves by past directed curves. One has in general that I˙ .A/ is the interior M M M of J˙ .A/ and that J˙ .A/ is contained in the closure of I˙ .A/. The chronological future and past are open subsets but the causal future and past are not always closed even if A is closed. M JC .A/

M .A/ IC

A

IM .A/ M .A/ J

Figure 1. Causal and chronological future resp. past of A.


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We will also use the notation J M .A/ WD JM .A/ [ JCM .A/. A subset A M is called past compact if A \ JM .p/ is compact for all p 2 M . Similarly, one defines future compact subsets.

A p

JM .p/

Figure 2. Past compact subset.

Definition 3.1. A subset S of a connected timeoriented Lorentzian manifold is called achronal if each timelike curve meets S in at most one point. A subset S of a connected timeoriented Lorentzian manifold is called acausal if each causal curve meets S in at most one point. A subset S of a connected timeoriented Lorentzian manifold is a Cauchy hypersurface if each inextendible timelike curve in M meets S at exactly one point.

M

S

Figure 3. Cauchy hypersurface.

Obviously every acausal subset is achronal, but the reverse is wrong. Any Cauchy hypersurface is achronal. Moreover, it is a closed topological hypersurface and it is

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hit by each inextendible causal curve in at least one point. Any two Cauchy hypersurfaces in M are homeomorphic. Furthermore, the causal future and past of a Cauchy hypersurface is past and future compact respectively. Definition 3.2. A Lorentzian manifold is said to satisfy the causality condition if it does not contain any closed causal curve. A Lorentzian manifold is said to satisfy the strong causality condition if there are no almost closed causal curves. More precisely, for each point p 2 M and for each open neighborhood U of p there exists an open neighborhood V U of p such that each causal curve in M starting and ending in V is entirely contained in U .

U

V forbidden!

p

Figure 4. Strong causality condition.

Obviously, the strong causality condition implies the causality condition. In order to get a good analytical theory for wave operators we must impose certain geometric conditions on the Lorentzian manifold. Here are several equivalent formulations. Theorem 3.3. Let M be a connected timeoriented Lorentzian manifold. Then the following are equivalent: (1) M satisfies the strong causality condition and for all p; q 2 M the intersection JCM .p/ \ JM .q/ is compact. (2) There exists a Cauchy hypersurface in M . (3) There exists a smooth spacelike Cauchy hypersurface in M . (4) M is foliated by smooth spacelike Cauchy hypersurfaces. More precisely, M is isometric to RS with metric ˇdt 2 Cg t where ˇ is a smooth positive function, g t is a Riemannian metric on S depending smoothly on t 2 R and each ft g S is a smooth spacelike Cauchy hypersurface in M . That (1) implies (4) has been shown by Bernal and Sánchez in [5, Theorem 1.1] using work of Geroch [11, Theorem 11]. See also [8, Proposition 6.6.8] and [15, p. 209] for earlier mentioning of this fact. The implications .4/ ) .3/ and .3/ ) .2/ are trivial. That (2) implies (1) is well known, see e.g. [14, Corollary 39, p. 422]. Definition 3.4. A connected timeoriented Lorentzian manifold satisfying one and hence all conditions in Theorem 3.3 is called globally hyperbolic.


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Remark 3.5. If M is a globally hyperbolic Lorentzian manifold, then a nonempty open subset M is itself globally hyperbolic if and only if for any p; q 2 the intersection JC .p/ \ J .q/ is compact. Indeed non-existence of almost closed causal curves in M directly implies non-existence of such curves in . Remark 3.6. It should be noted that global hyperbolicity is a conformal notion. The definition of a Cauchy hypersurface requires only causal concepts. Hence if .M; g/ is globally hyperbolic and we replace the metric g by a conformally related metric gO D f g, f a smooth positive function on M , then .M; g/ O is again globally hyperbolic. Examples 3.7. Minkowski space is globally hyperbolic. Every spacelike hyperplane is a Cauchy hypersurface. One can write Minkowski space as R Rn1 with the metric dt 2 C g t where g t is the Euclidean metric on Rn1 and does not depend on t . Let .S; g0 / be a connected Riemannian manifold and I R an interval. The manifold M D I S with the metric g D dt 2 C g0 is globally hyperbolic if and only if .S; g0 / is complete. This applies in particular if S is compact. More generally, if f W I ! R is a smooth positive function we may equip M D I S with the metric g D dt 2 C f .t /2 g0 . Again, .M; g/ is globally hyperbolic if and only if .S; g0 / is complete. Robertson–Walker spacetimes and, in particular, Friedmann cosmological models, are of this type. They are used to discuss big bang, expansion of the universe, and cosmological redshift, compare [14, Chapter 12]. Another example of this type is deSitter spacetime, where I D R, S D S n1 , g0 is the canonical metric of S n1 of constant sectional curvature 1, and f .t / D cosh.t/. But Anti-deSitter spacetime is not globally hyperbolic. The interior and exterior Schwarzschild spacetimes are globally hyperbolic. They model the universe in the neighborhood of a massive static rotationally symmetric body such as a black hole. They are used to investigate perihelion advance of Mercury, the bending of light near the sun and other astronomical phenomena, see [14, Chapter 13]. Lemma 3.8. Let S be a Cauchy hypersurface in a globally hyperbolic Lorentzian manifold M and let K; K 0 M be compact. Then J˙M .K/ \ S, J˙M .K/ \ JM .S /, and JCM .K/ \ JM .K 0 / are compact.

4 Wave operators Let M be a Lorentzian manifold and let E ! M be a real or complex vector bundle. A linear differential operator P W C 1 .M; E/ ! C 1 .M; E/ of second order will be called a wave operator or a normally hyperbolic operator if its principal symbol is given by the metric P ./ D h; i idEx

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for all x 2 M and all 2 Tx M . In other words, if we choose local coordinates x 1 ; : : : ; x n on M and a local trivialization of E, then P D

n X

g ij .x/

i;j D1

n

X @2 @ C Aj .x/ j C B.x/ i j @x @x @x j D1

where Aj and B are matrix-valued coefficients depending smoothly on x and .g ij /ij is the inverse matrix of .gij /ij with gij D h @x@ i ; @x@j i. Example 4.1. Let E be the trivial line bundle so that sections in E are just functions. The d’Alembert operator P D D div B grad is a wave operator. Example 4.2. Let E be a vector bundle and let r be a connection on E. This connection together with the Levi-Civita connection on T M induces a connection on T M ˝ E, again denoted r. We define the connection-d’Alembert operator r to be minus the composition of the following three maps r

C 1 .M; E/ ! C 1 .M; T M ˝ E/ tr ˝idE

r

! C 1 .M; T M ˝ T M ˝ E/ ! C 1 .M; E/; where tr W T M ˝ T M ! R denotes the metric trace, tr. ˝ / D h; i. We compute the principal symbol, r ./' D .tr ˝idE / B r ./ B r ./.'/ D .tr ˝idE /. ˝ ˝ '/ D h; i ': Hence r is a wave operator. Example 4.3. Let E D ƒk T M be the bundle of k-forms. Exterior differentiation d W C 1 .M; ƒk T M / ! C 1 .M; ƒkC1 T M / increases the degree by one while the codifferential ı W C 1 .M; ƒk T M / ! C 1 .M; ƒk1 T M / decreases the degree by one. While d is independent of the metric, the codifferential ı does depend on the Lorentzian metric. The operator P D d ı C ıd is a wave operator. Example 4.4. If M carries a Lorentzian metric and a spin structure, then one can define the spinor bundle †M and the Dirac operator D W C 1 .M; †M / ! C 1 .M; †M /; see [1] or [3] for the definitions. The principal symbol of D is given by Clifford multiplication, D ./ D ] : Hence D 2 ./

D D ./D ./

Thus P D D 2 is a wave operator.

D ] ]

D h; i :


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5 The Cauchy problem We now come to the basic initial value problem for wave operators, the Cauchy problem. The local theory of linear hyperbolic operators can be found in basically any textbook on partial differential equations. In [10] and [12] the local theory for wave operators on Lorentzian manifolds is developed. The results of this section are of global nature. They make statements about solutions to the Cauchy problem which are defined globally on a manifold. Proofs of the results of this section can be found in [4, Section 3.2]. Theorem 5.1 (Existence and uniqueness of solutions). Let M be a globally hyperbolic Lorentzian manifold and let S M be a smooth spacelike Cauchy hypersurface. Let be the future directed timelike unit normal field along S . Let E be a vector bundle over M and let P be a wave operator acting on sections in E. Then for each u0 ; u1 2 D.S; E/ and for each f 2 D.M; E/ there exists a unique u 2 C 1 .M; E/ satisfying P u D f , ujS D u0 , and r ujS D u1 . It is unclear how to even formulate the Cauchy problem on a Lorentzian manifold which is not globally hyperbolic. One would have to replace the concept of a Cauchy hypersurface by something different to impose the initial conditions upon. Here are two examples which illustrate what can typically go wrong. Example 5.2. Let M D S 1 Rn1 with the metric g D d 2 C g0 where d 2 is the standard metric on S 1 of length 1 and g0 is the Euclidean metric on Rn1 . The universal covering of M is Minkowski space. Let us try to impose a Cauchy problem on f0 g Rn1 which is the image of a Cauchy hypersurface in Minkowski space. Such a solution would lift to Minkowski space where it indeed exists uniquely due to Theorem 5.1. But such a solution on Minkowski space is in general not time periodic, hence does not descend to a solution on M . Therefore existence of solutions fails. The problem is here that M violates the causality condition, i.e., there are closed causal curves. Remark 5.3. Compact Lorentzian manifolds always possess closed timelike curves and are therefore never well suited for the analysis of wave operators. Example 5.4. Let M be a timelike strip in 2-dimensional Minkowski space, i.e., M D R .0; 1/ with metric g D dt 2 C dx 2 . Let S WD f0g .0; 1/. Given any u0 ; u1 2 D.S; E/ and any f 2 D.M; E/, there exists a solution u to the Cauchy problem. One can simply take the solution in Minkowski space and restrict it to M . But this solution is not unique in M . Choose x in Minkowski space, x 62 M , such that JCMink .x/ intersects M in the future of S and of supp.f /. The advanced fundamental solution w D FC .x/ (see next section) has support contained in JCMink .x/ and satisfies P w D 0 away from x. Hence u C w restricted to M is again a solution to the Cauchy problem on M with the same initial data.

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JCMink .x/

supp.f /

x S

supp.u0 / [ supp.u1 / M Figure 5. Nonunique solution to Cauchy problem.

The problem is here that S is acausal but not a Cauchy hypersurface. Physically, a wave “from outside the manifold” enters into M . The physical statement that a wave can never propagate faster than with the speed of light is contained in the following. Theorem 5.5 (Finite propagation speed). The solution u from Theorem 5.1 satisfies supp.u/ J M .K/ where K D supp.u0 / [ supp.u1 / [ supp.f /. The solution to the Cauchy problem depends continuously on the data. Theorem 5.6 (Stability). Let M be a globally hyperbolic Lorentzian manifold and let S M be a smooth spacelike Cauchy hypersurface. Let be the future directed timelike unit normal field along S. Let E be a vector bundle over M and let P be a wave operator acting on sections in E. Then the map D.M; E/˚D.S; E/˚D.S; E/ ! C 1 .M; E/ sending .f; u0 ; u1 / to the unique solution u of the Cauchy problem P u D f , ujS D uj0 , r u D u1 is linear continuous. This is essentially an application of the open mapping theorem for Fréchet spaces.

6 Fundamental solutions Definition 6.1. Let M be a timeoriented Lorentzian manifold, let E ! M be a vector bundle and let P W C 1 .M; E/ ! C 1 .M; E/ be a wave operator. Let x 2 M . A fundamental solution of P at x is a distribution F 2 D 0 .M; E; Ex / such that PF D ıx :


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In other words, for all ' 2 D.M; E / we have F ŒP ' D '.x/: If supp.F .x// JCM .x/, then we call F an advanced fundamental solution, if supp.F .x// JM .x/, then we call F a retarded fundamental solution. Using the knowledge about the Cauchy problem from the previous section it is now not hard to find global fundamental solutions on a globally hyperbolic manifold. Theorem 6.2. Let M be a globally hyperbolic Lorentzian manifold. Let P be a wave operator acting on sections in a vector bundle E over M . Then for every x 2 M there is exactly one fundamental solution FC .x/ for P at x with past compact support and exactly one fundamental solution F .x/ for P at x with future compact support. They satisfy (1) supp.F˙ .x// J˙M .x/, (2) for each ' 2 D.M; E / the maps x 7! F˙ .x/Œ' are smooth sections in E satisfying the differential equation P .F˙ ./Œ'/ D '. Sketch of proof. We do not do the uniqueness part. To show existence fix a foliation of M by spacelike Cauchy hypersurfaces S t , t 2 R as in Theorem 3.3. Let be the future directed unit normal field along the leaves S t . Let ' 2 D.M; E /. Choose t so large that supp.'/ IM .S t /. By Theorem 5.1 there exists a unique ' 2 C 1 .M; E / such that P ' D ' and ' jS t D .r ' /jS t D 0. One can check that ' does not depend on the choice of t . Fix x 2 M . By Theorem 5.6 ' depends continuously on '. Since the evaluation map C 1 .M; E/ ! Ex is continuous, the map D.M; E / ! Ex , ' 7! ' .x/, is also continuous. Thus FC .x/Œ' WD ' .x/ defines a distribution. By definition P .FC ./Œ'/ D P ' D '. Now P P ' D P ', hence P . P ' '/ D 0. Since both P ' and ' vanish along S t the uniqueness part which we have omitted shows P ' D '. Thus .PFC .x//Œ' D FC .x/ŒP ' D P ' .x/ D '.x/ D ıx Œ': Hence FC .x/ is a fundamental solution of P at x. It remains to show supp.FC .x// JCM .x/. Let y 2 M n JCM .x/. We have to construct a neighborhood of y such that for each test section ' 2 D.M; E / whose support is contained in this neighborhood we have FC .x/Œ' D ' .x/ D 0. Since M is globally hyperbolic JCM .x/ is closed and therefore JCM .x/ \ JM .y 0 / D ; for M .y/ and y 00 2 IM .y/ so close that all y 0 sufficiently close to y. We choose y 0 2 IC M 00 S M M 0 JC .x/ \ J .y / D ; and JC .y / \ t t 0 S t \ JCM .x/ D ; where t 0 2 R is such that y 0 2 S t 0 .

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S

M JC .y 00 / \ .

t t 0

St /

M JC .x/

y0

St 0

y x

y 00

M J .y 0 /

Figure 6. Construction of y, y 0 and y 00 .

Now K WD JM .y 0 / \ JCM .y 00 / is a compact neighborhood of y. Let ' 2 D.M; E / be such that supp.'/ K. By Theorem 5.1 supp. ' / JCM .K/ [ JM .K/ JCM .y 00 / [ JM .y 0 /. By the independence of ' of the choice of t > t 0 S S we have that ' vanishes on t >t 0 S t . Hence supp. ' / JCM .y 00 / \ t t 0 S t [ JM .y 0 / and is therefore disjoint from JCM .x/. Thus FC .x/Œ' D ' .x/ D 0 as required. For a complete proof see [4, Section 3.3].

7 Green’s operators Now we want to find “solution operators” for a given wave operator P . More precisely, we want to find operators which are inverses of P when restricted to suitable spaces of sections. We will see that existence of such operators is basically equivalent to the existence of fundamental solutions. Definition 7.1. Let M be a timeoriented connected Lorentzian manifold. Let P be a wave operator acting on sections in a vector bundle E over M . A linear map GC W D.M; E/ ! C 1 .M; E/ satisfying (i) P B GC D idD.M;E / ,


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(ii) GC B P jD.M;E / D idD.M;E / , (iii) supp.GC '/ JCM .supp.'// for all ' 2 D.M; E/, is called an advanced Green’s operator for P . Similarly, a linear map G W D.M; E/ ! C 1 .M; E/ satisfying (i), (ii), and (iii0 ) supp.G '/ JM .supp.'// for all ' 2 D.M; E/ instead of (iii) is called a retarded Green’s operator for P . Fundamental solutions and Green’s operators are closely related. Theorem 7.2. Let M be a globally hyperbolic Lorentzian manifold. Let P be a wave operator acting on sections in a vector bundle E over M . Then there exist unique advanced and retarded Green’s operators G˙ W D.M; E/ ! C 1 .M; E/ for P . Proof. By Theorem 6.2 there exist families F˙ .x/ of advanced and retarded fundamental solutions for the adjoint operator P respectively. We know that F˙ .x/ depend smoothly on x and the differential equation P .F˙ ./Œ'/ D ' holds. By definition we have P .G˙ '/ D P .F ./Œ'/ D ' thus showing (i). Assertion (ii) follows from the fact that the F˙ .x/ are fundamental solutions, G˙ .P '/.x/ D F .x/ŒP ' D P F .x/Œ' D ıx Œ' D '.x/: To show (iii) let x 2 M such that .GC '/.x/ 6D 0. Since supp.F .x// JM .x/ the support of ' must hit JM .x/. Hence x 2 JCM .supp.'// and therefore supp.GC '/ JCM .supp.'//. The argument for G is analogous. We have seen that existence of fundamental solutions for P depending nicely on x implies existence of Green’s operators for P . This construction can be reversed. Then uniqueness of fundamental solutions in Theorem 6.2 implies uniqueness of Green’s operators. Lemma 7.3. Let M be a globally hyperbolic Lorentzian manifold. Let P be a wave operator acting on sections in a vector bundle E over M . Let G˙ be the Green’s the Green’s operators for the adjoint operator P . Then operators for P and G˙ Z Z .G˙ '/ dV D ' .G / dV (1) M

M

holds for all ' 2 D.M; E / and

2 D.M; E/.

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Proof. For the Green’s operators we have P G˙ D idD.M;E / and P G˙ D idD.M;E / and hence Z Z '/ .P G / dV .G˙ '/ dV D .G˙ M M Z .P G˙ '/ .G / dV D M Z ' .G / dV: D M

Notice that supp.G˙ '/ \ supp.G / J˙M .supp.'// \ JM .supp. // is compact in a globally hyperbolic manifold so that the partial integration in the second equation is justified. Notation 7.4. We write Csc1 .M; E/ for the set of all ' 2 C 1 .M; E/ for which there exists a compact subset K M such that supp.'/ J M .K/. Obviously, Csc1 .M; E/ is a vector subspace of C 1 .M; E/. The subscript “sc” should remind the reader of “space-like compact”. Namely, if M is globally hyperbolic and ' 2 Csc1 .M; E/, then for every Cauchy hypersurface S M the support of 'jS is contained in S \ J M .K/ hence compact by Lemma 3.8. In this sense sections in Csc1 .M; E/ have space-like compact support. Definition 7.5. We say that a sequence of elements 'j 2 Csc1 .M; E/ converges in Csc1 .M; E/ to ' 2 Csc1 .M; E/ if there exists a compact subset K M such that supp.'/ J M .K/

and

supp.'j / J M .K/

for all j , and k'j 'kC k .K 0 ;E / ! 0 for all k 2 N and all compact subsets K 0 M . If GC and G are advanced and retarded Green’s operators for P respectively, then we get a linear map G WD GC G W D.M; E/ ! Csc1 .M; E/: Much of the solution theory of wave operators on globally hyperbolic Lorentzian manifolds is collected in the following theorem. Theorem 7.6. Let M be a globally hyperbolic Lorentzian manifold. Let P be a wave operator acting on sections in a vector bundle E over M . Let GC and G be advanced and retarded Green’s operators for P respectively. Then P

G

P

0 ! D.M; E/ ! D.M; E/ ! Csc1 .M; E/ ! Csc1 .M; E/ is an exact sequence of linear maps.

(2)

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Proof. Properties (i) and (ii) in Definition 7.1 of Green’s operators directly yield G B P D 0 and P B G D 0, both on D.M; E/. Properties (iii) and (iii’) ensure that G maps D.M; E/ to Csc1 .M; E/. Hence the sequence of linear maps forms a complex. Exactness at the first D.M; E/ means that P W D.M; E/ ! D.M; E/ is injective. To see injectivity let ' 2 D.M; E/ with P ' D 0. Then ' D GC P ' D GC 0 D 0. WD Next let ' 2 D.M; E/ with G' D 0, i.e., GC ' D G '. We put GC ' D G ' 2 C 1 .M; E/ and we see supp. / D supp.GC '/ \ supp.G '/ JCM .supp.'// \ JM .supp.'//. Since .M; g/ is globally hyperbolic JCM .supp.'// \ JM .supp.'// is compact, hence 2 D.M; E/. From P . / D P .GC .'// D ' we see that ' 2 P .D.M; E//. This shows exactness at the second D.M; E/. Finally, let ' 2 Csc1 .M; E/ such that P ' D 0. Without loss of generality we M may assume that supp.'/ IC .K/ [ IM .K/ for a compact subset K of M . Using M .K/; IM .K/g write ' as a partition of unity subordinated to the open covering fIC M .K/ ' D '1 C '2 where supp.'1 / IM .K/ JM .K/ and supp.'2 / IC M M M JC .K/. For WD P '1 D P '2 we see that supp. / J .K/ \ JC .K/, hence 2 D.M; E/. We check that GC D '2 . For all 2 D.M; E / we have Z Z Z Z .GC P '2 / dV D .G /.P '2 / dV D .P G /'2 dV D '2 dV M

where G

M

M

M

is the Green’s operator for the adjoint operator P according to Lemma 7.3. Notice that for the second equation we use the fact that supp.'2 / \ supp.G / JCM .K/ \ JM .supp. // is compact. Similarly, one shows G D '1 . Now G D GC G D '2 C '1 D ', hence ' is in the image of G. Proposition 7.7. Let M be a globally hyperbolic Lorentzian manifold, let P be a wave operator acting on sections in a vector bundle E over M . Let GC and G be the advanced and retarded Green’s operators for P respectively. Then G˙ W D.M; E/ ! Csc1 .M; E/ and all maps in the complex (2) are continuous. Proof. The maps P W D.M; E/ ! D.M; E/ and P W Csc1 .M; E/ ! Csc1 .M; E/ are continuous simply because P is a differential operator. It remains to show that G W D.M; E/ ! Csc1 .M; E/ is continuous. Let 'j ; ' 2 D.M; E/ and 'j ! ' in D.M; E/ for all j . Then there exists a compact subset K M such that supp.'j / K for all j and supp.'/ K. Hence supp.G'j / J M .K/ for all j and supp.G'/ J M .K/. From the proof of Theorem 6.2 we know that GC ' coincides with the solution u to the Cauchy problem P u D ' with initial conditions ujS D .r u/jS D 0 where S M is a spacelike M .S /. Theorem 5.6 tells us that if 'j ! ' Cauchy hypersurface such that K IC

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in D.M; E/, then the solutions GC 'j ! GC ' in C 1 .M; E/. The proof for G is analogous and the statement for G follows.

References [1]

H. Baum, Spin-Strukturen und Dirac-Operatoren über pseudoriemannschen Mannigfaltigkeiten. Teubner-Texte Math. 41, Teubner Verlagsgesellschaft, Leipzig 1981. 906

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J. K. Beem, P. E. Ehrlich, and K. L. Easley, Global Lorentzian geometry. Second edition, Marcel Dekker, New York 1996. 902

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C. Bär, P. Gauduchon, and A. Moroianu, Generalized cylinders in semi-Riemannian and spin geometry. Math. Z. 249 (2005), 545–580. 906

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C. Bär, N. Ginoux, and F. Pfäffle, Wave equations on Lorentzian manifolds and quantization. ESI Lect. Math. Phys., Europ. Math. Soc. Publ. House, Zürich 2007. 897, 907, 910

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A. N. Bernal and M. Sánchez, Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes. Comm. Math. Phys. 257 (2005), 43–50. 904

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A. N. Bernal and M. Sánchez, Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions. Lett. Math. Phys. 77 (2006), 183–197.

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Y. Choquet-Bruhat, Hyperbolic partial differential equations on a manifold. In Battelle Rencontres, 1967, Lect. Math. Phys., Benjamin, New York 1968, 84–106. 897

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G. F. R. Ellis and S. W. Hawking, The large scale structure of space-time. Cambridge Monogr. Math. Phys. 1, Cambridge University Press, London 1973. 897, 904

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F. Friedlander, Introduction to the theory of distributions. Second edition, Cambridge University Press, Cambridge 1998. 898

[10] F. Friedlander, The wave equation on a curved space-time. Cambridge University Press, Cambridge 1975. 907 [11] R. Geroch, Domain of dependence. J. Math. Phys. 11 (1970), 437–449. 904 [12] P. Günther, Huygens’ principle and hyperbolic equations. Academic Press, Boston, MA, 1988. 907 [13] J. Leray, Hyperbolic differential equations. Unpublished Lecture Notes, Princeton 1953. 897 [14] B. O’Neill, Semi-Riemannian geometry. Academic Press, San Diego 1983. 902, 904, 905 [15] R. M. Wald, General relativity. University of Chicago Press, Chicago, IL 1984. 897, 904

Chapter 26

Survey of D-branes and K-theory Daniel S. Freed

Contents Introduction . . . . . . . . . . . . . . . . . . . . Couplings and anomalies in the low energy theory Worldsheet considerations . . . . . . . . . . . . Tachyon condensation . . . . . . . . . . . . . . . Torsion charges . . . . . . . . . . . . . . . . . . Relation to M-theory . . . . . . . . . . . . . . . Concluding remarks . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .

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Introduction K-theory was introduced into mathematics by Grothendieck in the mid-1950s as part of his generalization of the Riemann–Roch–Hirzebruch theorem in algebraic geometry [BS58]. A few years later Atiyah and Hirzebruch [AH61] adopted these ideas in a purely topological context, where they proved a powerful tool in longstanding geometric problems. The most far-reaching application was to linear analysis, the index of elliptic operators [AS63], and this drove subsequent development of the subject. One source of delight for topologists over the past decade has been the reemergence of K-theory in a new context: superstring theory. Namely, the charge of a D-brane in a superstring background – the Cartesian product of a spatial 9-manifold Y and the time line R – is measured in the topological K-theory of Y . There are many variations, depending on the details of the string theory, and these have led to developments on the mathematical side as well. We will not review the purely mathematical developments – see [Ati01] for one account – but will instead recount the justification for positing K-theory as the home for D-brane charges. We mention straightaway the seminal work of Polchinski [Pol95] which demonstrates that D-branes are a source of electric and magnetic current for Ramond-Ramond gauge fields. In a classical gauge theory the charges, which are the de Rham cohomology classes of the currents, vary continuously. Dirac charge quantization asserts that these charges are quantized in the quantum theory: the possible values form a lattice. Even better, charges lie in an abelian group which is part of a generalized

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cohomology theory. The physical question is: which cohomology theory? The traditional answer, integral cohomology, applies to abelian gauge fields in many theories. (In more concrete terms a current quantized by integral cohomology has integer periods when integrated over cycles in space.) The novelty with Ramond-Ramond fields is that the fluxes and charges are quantized by K-theory. This chapter is a survey of justifications for this assertion. There are many answers to “Why K-theory?”: couplings and anomalies in the low energy 10-dimensional theory, anomalies in the 2-dimensional worldsheet theory, tachyon condensation, torsion charges, and the relation to M-theory. We review each in turn. More evidence can be marshaled in support of K-theory from the worldsheet point of view; see [Moo04] for an extensive account, which includes the example of D-branes in the WZW model. We have written expository texts elsewhere [Fre00], [Fre02, Part 3] which elaborate on some aspects in detail. In addition to the survey [Moo04], we also recommend the survey [Wit01]. The list of references here is by no means exhaustive. We have endeavored to open a path into the literature, which may be further pursued via the bibliographies in the references.

Couplings and anomalies in the low energy theory Low energy Type II superstring theory, in its Euclidean version, is formulated on a 10-dimensional spin manifold X. The bosonic fields are a Riemannian metric; a gauge field, the B-field, whose field strength is a global 3-form; a scalar field, the dilaton field; and the Ramond-Ramond field. There are two theories, Types IIA and IIB, and the Ramond-Ramond field strength is an inhomogeneous closed differential form of even degree in IIA and odd degree in IIB. There are also fermionic fields – spinor fields and Rarita–Schwinger fields. These are the fields of a supergravity theory, and the starting point for a supersymmetric action functional is that determined by supergravity. In both cases the supergravity theory is free of anomalies [AGW83]. A D-brane, in the worldsheet string theory, is a Dirichlet boundary condition for an open string. It is modeled by a submanifold W Y on which the ends of the open string are constrained. One should think of W as analogous to the worldline of a particle – a 1-dimensional oriented submanifold – in the ordinary Maxwell theory of electromagnetism. In the classical Maxwell theory the gauge field is locally a 1-form A, and the classical action has an electric coupling Z Z qA D qW ^ A; (1) W

X

where q 2 R is the electric charge of the particle represented by the worldline and W is a differential form Poincaré dual form to W . (It is a distributional form, or current in the sense of de Rham.) The form jE D qW is the electric current in the Maxwell

Chapter 26. Survey of D-branes and K-theory

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d F D jE ;

(2)

equation where the 2-form F D dA is the field strength. The other Maxwell equation asserts that F is closed: dF D 0: (3) Now from the canonical quantization of the open string one computes the low energy theory on the D-brane. The fields on W are a scalar field (which controls deformations of the position of the submanifold W ) and a gauge field whose field strength is a 2-form F ; there are also fermions. The fermions lead to anomalies in the quantum theory. The anomaly inflow mechanism, first introduced in another context by Callan and Harvey [CH85], cancels these anomalies by simultaneously introducing a term in the action of the form (1) and modifying the Maxwell equation (3), termed the “Bianchi identity” in this context, by postulating a nonzero right hand side.1 This scheme was carried out for Ramond-Ramond fields and D-branes in [GHM97], [MM98], and [CY98]. The result is that the Hirzebruch AO genus, introduced in the 1950s as part of the developments surrounding the Riemann–Roch theorem, appears in the electric coupling. Minasian and Moore [MM98] use the Atiyah–Hirzebruch topological version of Grothendieck–Riemann–Roch, and input from Graeme Segal and Maxim Kontsevich, to interpret this coupling in K-theory. They then postulate that Ramond-Ramond charge induced by the D-brane is an element of topological K-theory. The inflow mechanism is closely related to what is known as the Green–Schwarz mechanism [IT94]. From the latter point of view, the right hand side in the amended Maxwell equation (3) is a magnetic current jB and the anomaly Z jB ^ jE (4) X

is due to the simultaneous electric and magnetic currents. This anomaly formula (4) is for a family X ! S of spacetimes parametrized by a base space S (of other fields in the theory on which the currents depend), and the integral in (4) is over the fibers in that family. The resulting 2-form on S is the local anomaly, or in physics language, the local anomaly is the descent mechanism applied to the form jB ^ jE . In more geometric terms, the anomaly is measured by a hermitian line bundle with connection over a space of parameters, and (4) is the formula for the curvature of that connection. The next step was the author’s realization that if charges are quantized by Ktheory then so are the fluxes of the Ramond-Ramond fields. Thus Ramond-Ramond fields are geometric objects which on the one hand have a field strength which is a global closed differential form and on the other hand represent a class in topological K-theory. Analogous geometric objects are well-known: a connection on a circle 1 We remark that physicists often write formulas which don’t quite make sense: gauge potentials are treated as global, yet with charge quantization they cannot be; the geometric ideas which enable sensible semiclassical formulas are mentioned below.

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bundle has a curvature which is a global closed 2-form and the circle bundle itself represents a class in degree two integral cohomology. Higher degree generalizations of connections on circle bundles, or more precisely equivalence classes of such, were introduced into mathematics by Cheeger and Simons [CS85], and their use in physics was pioneered by Gaw¸edski [edz88]. A geometric generalization for degree three integral cohomology, called a gerbe with connection, appears in [Bry93], [Hit01] and many other places. At that time, in the late 1990s, a very general theory of such geometric objects was being developed by Hopkins and Singer [HS05]. This gave the language to propose [FH00] that a Ramond-Ramond field is a “cocycle” or “object” in a category representing differential K-theory. The electric coupling formula of Minasian–Moore and others has a natural, and simple, expression in this theory. Furthermore, the anomaly cancellation now works on the nose – not just at the level of differential forms. For example, in (4) the currents are refined to objects in differential K-theory and the result of the integral is mapped to degree two differential cohomology. The anomaly cancellation is for Type IIA is more delicate than for Type IIB as the anomalies are torsion of order two. In differential K-theory the Euler class of an odd rank real vector bundle shows up as the anomaly (whereas the image of this Euler class in topological K-theory vanishes). The quantization of RamondRamond flux by K-theory and anomaly cancellations are also discussed in [MW00]. An important property of the Ramond-Ramond field is its self-duality. Witten explained special features of self-dual fields in [Wit97a], [Wit00] and the partition function of Ramond-Ramond fields is further described in [DMW03]. Mathematical aspects of self-dual fields are developed in [HS05] from the Lagrangian point of view and in [FMS] from a Hamiltonian point of view. It is an open problem to integrate these points of view and so construct a complete theory of a self-dual field. For a self-dual field the magnetic current determines the electric current and the bilinear pairing (4) between magnetic and electric currents is refined to a quadratic form on the single current, all this on the level of objects in the differential theory so that charge and flux quantization is taken into account. For the Ramond-Ramond field the self-dual current – think of it as magnetic – may be represented by a vector bundle E x and the with connection. Then the corresponding electric current is represented by E, x quadratic form lifts the tensor product E ˝ E to a real bundle, then integrates it over the 10-manifold X.2 Type I supergravity is inconsistent due to an anomaly [AGW83], and it was a great advance in 1984 when Green and Schwarz realized how low energy effects of the Type I superstring modify supergravity to an anomaly-free theory [GS84]. This is the original example of the Green–Schwarz mechanism. They worked at the level of differential forms – the local anomaly – and so the question arises to incorporate charge quantization. Already Moore and Minasian [MM98] recognized that charges induced by D-branes in Type I should be quantized by real K-theory, that is, by KO-theory. Then, following the logic above, the Ramond-Ramond field in Type I should be an object in differential KO-theory and must be treated as a self-dual field. The necessary 2 By

x. the index theorem that integral is the pfaffian line bundle of a Dirac operator coupled to E ˝ E


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quadratic form was supplied in [Fre00] and it was used to refine the Green–Schwarz anomaly cancellation to a statement about both local and global anomalies. We summarize as follows. The anomaly due to a fermion is measured by a line bundle with connection over the space of bosonic fields. This bundle is defined as the pfaffian line bundle of a family of Dirac operators. The Atiyah–Singer index theorem computes the topology of this bundle in terms of topological K-theory, and there is a presumed geometric refinement to differential K-theory which computes the pfaffian line bundle with its geometry. (See [Fre02, Part 2] for details.) There is a general anomaly (4) due to simultaneous electric and magnetic current, and if RamondRamond fluxes and charges are quantized by topological K-theory, then this formula too lies in differential K-theory. This makes possible the cancellation of fermion anomalies against the electric coupling anomaly, termed the Green–Schwarz mechanism. We remark that there are examples in lower dimensions, such as 2-dimensional worldsheet heterotic string theory, where there is a Green–Schwarz mechanism and yet the gauge field is quantized in ordinary integer cohomology. This is possible since in low dimensions the index formula can sometimes be expressed in integer cohomology rather than K-theory; see [Fre87, §5], for example.

Worldsheet considerations As mentioned in the introduction, the article [Moo04] by Greg Moore motivates Ktheory as the home for Ramond-Ramond charge using more worldsheet considerations than we recount here. We simply mention in passing one simple harbinger of K-theory which dates from the early days of string theory, when strings were thought to be a theory of hadrons. Namely, in the elementary theory of an open string one may attach “Chan–Paton factors” on the ends of an open string [GSW87, §1.5.3]. In Type II this is a U.n/ gauge field, so it is natural as the boundary of the string moves over a D-brane W in a spacetime X to postulate that there is a complex vector bundle with connection on W and the Chan–Paton factor is the pullback to the boundary of the open string. A complex vector bundle E ! W determines a class ŒE 2 K.W /. Suppose the normal bundle to W X is oriented for K-theory, that is, carries a spinc structure. Then the inclusion i W W ,! X induces a pushforward3 i W K.W / ! K.X /, and the associated charge of the D-brane is i ŒE. Of course, the charge depends on the orientation of the normal bundle, and one should find that orientation choice in the physics. Where? Anomalies in the worldsheet theory, analyzed in [FW99], provide the answer. Let † be a compact oriented 2-manifold with boundary, the worldsheet of an open string. The worldsheet theory includes fermions with local boundary conditions; the partition 4 over the function of the fermions is then a section of a line bundle with connection space of bosonic fields, which includes the space Map .†; @†/I .X; W / of maps of † 3 with

a degree shift we omit from the notation.

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to X which take the boundary @† into the D-brane W . The main result is that this line bundle is flat and holonomies are measured by w2 ./, the second Stiefel–Whitney class of the normal bundle to W in X. Here W is assumed oriented. This anomaly is canceled by a Green–Schwarz mechanism for the gauge field on W , assumed here to be abelian; the generalization to the nonabelian case was taken up by Kapustin [Kap00]. Assume the B-field on X vanishes. The gauge field on W is locally a 1-form A, and there is an “electric” coupling Z A

(1)

@†

to the boundary of the worldsheet. This becomes anomalous if we postulate a magnetic current for this gauge field. Precisely, we take the Dirac quantization law for the gauge field A to be given by integer cohomology, so that a magnetic current is a cocycle for a degree three differential cohomology class on W . There is a natural choice, the gerbe which obstructs the existence of a spinc structure on the normal bundle . It is a flat gerbe whose holonomies are given by w2 ./. The gauge field A is postulated to be a (non-flat) trivialization of this gerbe. Topologically, then, it defines a spinc structure on and so the pushforward map i necessary to define the charge in Ktheory. Geometrically, the gauge field A is usually termed a spin c connection. This argument supposed that the B-field on X vanishes, and there is a natural generalization to include nonzero B. In fact, B is also quantized by integer cohomology, so is a cocycle for a degree three differential cohomology class on X. This is exactly the geometric object which twists complex K-theory, and this led Witten [Wit98] to propose that the proper home for Ramond-Ramond charges is twisted K-theory in case B 6D 0, at least if the flux of B is torsion. (It was later proposed for all B fields; see [BM00], for example.) The worldsheet story with B 6D 0 is consistent with that proposal.

Tachyon condensation Suppose now that W is a D-brane and W 0 an anti-D-brane. So open strings have both endpoints on W , both on W 0 , or one on each. The analysis of low energy states in the quantized open string with both endpoints on the same brane gives a gauge field, as analyzed above. If one endpoint is on each, then there is a sole low energy mode which is tachyonic, that is, not physical. A physical mechanism of brane– antibrane annihilation via tachyon condensation was proposed by Sen [Sen98] and others. Witten [Wit98] interprets those arguments in terms of K-theory and so gives more evidence that Ramond-Ramond charge lies in K-theory. Suppose that E ! W and F ! W 0 are the Chan–Paton vector bundles. The low energy modes come into 4 In [FW99] doubling is used to replace the Dirac operator on † with an operator on the double. The recent Ph.D. thesis of Matthew Scholl [Sch06] analyzes determinants directly for the operator on † with local boundary conditions.


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play when the length of the open string goes to zero, and for that to happen we must take W D W 0 . Then Witten interprets the tachyon field as a linear map T W E ! F . Brane–antibrane annihilation occurs if T is an isomorphism; then the configuration is equivalent to zero. This annihilation harks back to the difference construction in topological K-theory, introduced in [AH62, §3] and [Bot69] and elaborated in [ABS64, Part II]. Witten interprets the annihilation as an equivalence relation on configurations of branes and antibranes, and so identifies the equivalence classes with K-theory.

Torsion charges The charges associated to an abelian gauge field take values in an abelian group, which may include torsion. The quotient by the torsion, a free abelian group, naturally sits in a real vector space and periods of the current around cycles in space define points in this free group. There may also be configurations which carry torsion charge; these cannot be measured by periods. The possible torsion charges depend heavily on the choice of cohomology theory with which to quantize charge. In [Wit98], following a suggestion of [MM98], this is used as further evidence of the correctness of K-theory for Ramond-Ramond charge. He works with the Type I theory, in which the charges are quantized by KO-theory. Taking space to be flat R9 the charge group in Type I is KO0c .R9 /, where the subscript denotes classes of compact support. (Charges are always assumed compactly supported in spatial directions – no charge “leaks out to infinity”.) By Bott periodicity this group is isomorphic to KO9 .point/ Š Z=2Z. Witten identifies the Type I 0-brane as a D-brane whose associated charge is the nontrivial element of this group. For more on torsion charges, see [BD02] and [BDM02].

Relation to M-theory The Type IIA superstring in 10 dimensions is identified with a certain limit of 11dimensional M-theory. More precisely, Type IIA on a 10-dimensional spacetime X is M-theory on X S 1 (or a circle bundle over X) in the limit where S 1 is very small. Thus one can ask to recover facts about Type IIA, in particular the quantization of Ramond-Ramond charge by K-theory, from facts about M-theory. Diaconescu, Moore, and Witten [DMW03] analyze the partition function of M-theory on X S 1 to test if the partition functions agree. Their analysis reveals explicit indications that the Ramond-Ramond field is quantized by K-theory. We review a few of the main ideas. The subtleties in the analysis already start with the gauge field, sometimes called the C -field, in M-theory. Note that its dimensional reduction to 10 dimensions is (part of) the Ramond-Ramond field. The C -field has field strength a closed form of

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degree 4, so the magnetic current as a differential form vanishes. However, there is a torsion magnetic current – its quantization law is shifted [Wit97b] – and this affects the computations. Explicit geometric models can be given for the C -field which make this shift manifest. One particularly nice one, following Witten, is in terms of E8 -connections [DFM]. Now in the limit of large metric on X which DiaconescuMoore-Witten consider, the contribution to the Type IIA partition function of the Ramond-Ramond field strength, which is a 0-form C 2-form C 4-form, is dominated by the 4-form component. This sets up the possibility of a comparison with the contribution of the M-theory 4-form field strength to its partition function. The first bit of analysis concerns the topological Chern–Simons term in the Mtheory action. On an 11-manifold of the form X S 1 it reduces to a topological invariant that depends only on the topological class of the C -field in H 4 .X /. (This class lives on X S 1 , but is presumed pulled back from X since the analysis assumes that the Neveu–Schwarz B-field in the Type IIA theory vanishes.) This invariant is a Z=2Z-valued quadratic function on H 4 .X / whose underlying bilinear form was studied by Landweber–Stong [LS87]. The key step is to compute the integral over C -fields in M-theory by first averaging over shifts by flat C -fields. The Steenrod square Sq 3 appears in this computation. This same operation is the first nontrivial differential in the spectral sequence relating cohomology and K-theory. Careful analysis leads to the conclusion that the topological class of Ramond-Ramond fields which contribute to the partition function are best regarded in K-theory. There are refinements to the argument in [DFM].

Concluding remarks There are variations on the arguments recounted here which further match features of Ramond-Ramond fields in superstring theory with topological K-theory. For example, in case the B-field in Type II is nonzero Witten [Wit98, §5.3] suggests that a twisted form of K-theory quantizes Ramond-Ramond charges and fluxes. This fits well with the worldsheet anomaly, as mentioned above. There is also a precise geometric home for the Ramond-Ramond field in this case: it is an object in twisted differential K-theory, the twisting being defined by the B-field, which itself is an object in differential cohomology. For more on twistings by B-fields, see [BM00], [MMS03] and many other references. Other variations include orbifolds and orientifolds, which involve both equivariant K-theory and further twistings of K-theory (including KR-theory [Guk00], [BS]). There is also a vast literature on D-branes in topological string theory, much of which postulates that a D-brane is a complex of sheaves on a Calabi–Yau manifold. In this context we are back to Grothendieck’s original picture of K-theory. The reemergence of K-theory in the context of D-branes has been the catalyst for mathematical developments in several directions. The theory of twistings, for


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K-theory and also other generalized cohomology theories, has been worked out on a variety of spaces – manifolds, orbifolds, smooth stacks, algebraic stacks, etc. – in various contexts – topological K-theory, K-theory of operator algebra, K-theory of sheaves, etc. Old ideas are reinterpreted in terms of twistings, and there are new occurrences as well, for example the identification of the Verlinde ring as a twisted equivariant K-theory ring. There remain mysteries in this area, perhaps the most pressing being the seeming incompatibility of the K-theory home for Ramond-Ramond charges with S -duality in the Type IIB theory [DMW03, §11]. Acknowledgement. The author gratefully acknowledges support by NSF grant DMS0603964.

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[Wit01]

E. Witten, Overview of K-theory applied to strings. Internat. J. Modern Phys. A 16 (2001), no. 5, 693–706. 916

List of Contributors Ilka Agricola, FB 12 / Mathematik und Informatik, Philipps-Universität Marburg, Hans-Meerwein-Straße / Campus Lahnberge, 35032 Marburg, Germany email: [email protected] DmitriV.Alekseevsky, School of Mathematics and Maxwell Institute for Mathematical Sciences, The Kings Buildings, JCMB, University of Edinburgh, Mayfield Road, Edinburgh, EH9 3JZ, UK e-mail: [email protected] Christian Bär, Universität Potsdam, Institut für Mathematik, Am Neuen Palais 10, 14469 Potsdam, Germany e-mail: [email protected] Oliver Baues, Institut für Algebra und Geometrie, Universität Karlsruhe, 76128 Karlsruhe, Germany e-mail: [email protected] Helga Baum, Humboldt-Universität Berlin, Institut für Mathematik, Rudower Chaussee 25, 12489 Berlin, Germany e-mail: [email protected] Charles P. Boyer, Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, U.S.A. e-mail: [email protected] Jean-Baptiste Butruille, Université de Cergy-Pontoise, Département de Mathématiques, site de Saint-Martin, 2, avenue Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France e-mail: [email protected] Daniel S. Freed, Department of Mathematics, University of Texas, 1 University Station C1200, Austin, TX 78712-0257, U.S.A. e-mail: [email protected] Anton Galaev, Department of Mathematics and Statistics, Faculty of Science, Masaryk University in Brno, Kotláˇrská 2, 611 37 Brno, Czech Republic Email: [email protected] Nigel Hitchin, Mathematical Institute, 24–29 St Giles, Oxford OX1 3LB, UK e-mail: [email protected] Stefan Ivanov, University of Sofia “St. Kl. Ohridski”, Faculty of Mathematics and Informatics, Blvd. James Bourchier, 5, 1164 Sofia, Bulgaria e-mail: [email protected]

928

List of Contributors

Yoshinobu Kamishima, Department of Mathematics, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji, Tokyo 192-0397, Japan e-mail: [email protected] Ines Kath, Institut für Mathematik und Informatik, Ernst-Moritz-Arndt-Universität Greifswald, Robert-Blum-Straße 2, 17487 Greifswald, Germany e-mail: [email protected] Alexei Kotov, Université du Luxembourg, Laboratoire de Mathématiques, Campus Kirchberg, G 213, 6, rue Richard Coudenhove-Kalergi, 1359 Luxembourg, GrandDuchy of Luxembourg e-mail: [email protected] Matthias Krahe, Sömmerringstraße 21, 60322 Frankfurt, Germany e-mail: [email protected] Thomas Leistner, School of Mathematical Sciences, The University of Adelaide, SA 5005, Australia e-mail: [email protected] Ulf Lindström, Department of Theoretical Physics, Uppsala University, Box 516, 75120 Uppsala, Sweden, and HIP-Helsinki Institute of Physics, 00014 University of Helsinki, P.O. Box 64, Finland e-mail: [email protected] María A. Lledó, Departamento de Física Teòrica, Universidad de València e IFIC (Centro mixto CSIC-UVEG), C/ Dr. Moliner, 50, 46100 Burjassot (València), Spain e-mail: [email protected] Óscar Maciá, Departamento de Física Teòrica, Universidad de València e IFIC (Centro mixto CSIC-UVEG), C/ Dr. Moliner, 50, 46100 Burjassot (València), Spain e-mail: [email protected] Costantino Medori, Dipartimento di Matematica, Università di Parma, Viale G. P. Usberti, 53/A, 43100 Parma, Italy e-mail: [email protected] Ivan Minchev, University of Sofia “St. Kl. Ohridski”, Faculty of Mathematics and Informatics, Blvd. James Bourchier, 5, 1164 Sofia, Bulgaria, and Philipps-Universität Marburg, Fachbereich Mathematik und Informatik, Hans-Meerwein-Straße, 35032 Marburg, Germany e-mail: [email protected] Thomas Mohaupt, Theoretical Physics Division, Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool L69 7ZL, UK e-mail: [email protected] Paul-Andi Nagy, Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand email: [email protected]


929

Thomas Neukirchner, Institut für Mathematik, Humboldt-Universität Berlin, Unter den Linden 6, 10099 Berlin, Germany e-mail: [email protected] Martin Roˇcek, C. N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY 11794-3840, U.S.A., and Institute for Theoretical Physics, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands e-mail: [email protected] Antonio J. Di Scala Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy e-mail: [email protected] Lars Schäfer, Institut für Differentialgeometrie, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany e-mail address: [email protected] Thomas Strobl, Université de Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France e-mail address: [email protected] Fabian Schulte-Hengesbach, Department Mathematik, Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany e-mail: [email protected] Andrew Swann, Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark e-mail: [email protected] Adriano Tomassini, Dipartimento di Matematica, Università di Parma, Viale G. P. Usberti, 53/A, 43100 Parma, Italy e-mail: [email protected] Rikard von Unge, Institute for Theoretical Physics, Masaryk University, 61137 Brno, Czech Republic e-mail: [email protected] Cumrun Vafa, Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, U.S.A. e-mail: [email protected] Stefan Vandoren, Institute for Theoretical Physics and Spinoza Institute, Utrecht University, 3508 TD Utrecht, The Netherlands e-mail: [email protected] Antoine Van Proeyen, Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven, Belgium e-mail: [email protected]

930


Veeravalli S. Varadarajan, Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, U.S.A. e-mail: [email protected] Konrad Waldorf, University of California, Berkeley, Department of Mathematics, 970 Evans Hall, Berkeley, CA 94720-3840, U.S.A. e-mail: [email protected] Gregor Weingart, Universidad Nacional Autónoma de México, Instituto de Matemáticas (Unidad Cuernavaca), Avenida Universidad s/n, Col. Lomas de Chamilpa, 62210 Cuernavaca, Morelos, Mexico e-mail: [email protected] Maxim Zabzine, Department of Theoretical Physics, Uppsala University, Box 516, 75120 Uppsala, Sweden e-mail: [email protected] Simeon Zamkovoy, University of Sofia “St. Kl. Ohridski”, Faculty of Mathematics and Informatics, Blvd. James Bourchier, 5, 1164 Sofia, Bulgaria e-mail: [email protected]

Index acausal subset, 903 achronal subset, 903 action (functional), 209, 213, 242, 248, 253, 256, 260 adjoint, 250 BV-, 244, 246 classical, 243, 247 Lie algebroid, 235, 242 of Lie algebra, 242 Yang–Mills, 255 action Lie algebroid, 235, 242 admissible anti-involution, 722 basis, 478 Cartan connection, 838, 840 AdS/CFT-correspondence (or duality), 75, 296 affine crystallographic group, 743 affine manifold, 735 compact complete, 802 compact homogeneous, 758, 759 with parallel volume, 758, 803 affine representation characteristic classes of, 790, 793 étale, 764, 770, 774 of nilpotent group, 797 of reductive group, 800 of tube type, 773 prehomogeneous, 757, 774 reductive, 798 simply transitive, 771 transitive, 796 volume preserving, 796 affine special Kähler manifold see also rigid special Kähler manifold, 153 affine special para-Kähler manifold, 153, 555

affine transformation, 114 infinitesimal, 115 affine vector fields associative algebra of, 745 development representation of, 747 on compact affine manifolds, 748 AKSZ (Alexandrov–Kontsevich– Schwarz–Zaboronsky), 244, 247, 259 algebroid Atiyah, 260 Courant, 211, 214, 219, 221, 226, 230, 237, 239, 244, 247, 248, 252 Lie, 215, 218, 221, 235, 239, 240, 250, 258, 260 Lie 2-, 222 Loday, 214, 218, 238 almost complex linear connections, 91 manifold, 89 structure, 301 structures on the twistor space, 480, 485 almost contact structure, 293, 309 almost Hermitian manifold, 290, 313, 402 almost hypercomplex, 456 Hermitian manifold, 456 paracomplex structure, 478 almost Kähler manifold, 92, 315 almost para-complex induced by a stable three-form, 430 structure, 428 structures on the reflector space, 480, 485 almost para-Hermitian manifold, 428

932 almost para-quaternionic Hermitian, 479 structure, 478 almost p-c qCR structure, 881 almost product structure, 478 almost pseudo-Hermitian manifold, 428 almost quaternionic structure of the second kind, 478 Aloff–Wallach space, 295, 321 Ambrose–Singer connection, 381 theorem, 290 angular momentum, 281 annihilation (brane anti-brane), 921 anomaly, 917 Anti de Sitter space(time), 835, 905 antiholomorphic form, 90 anti-self-dual, 479, 491, 493 associated vector bundle, 825 automorphism group of pseudo-conformal qCR structure, 886 Bär’s correspondence, 58 Bérard-Bergery, L., 582 Berger algebra, 584, 596 (holonomy) theorem, 42, 279, 297 list, 392, 586 Berger, M., 278 Betti numbers, 36 B-field, B-field, 186, 669 263, 278, 284, 916 BF-theory, 253 Bianchi identity (first), 584 identity (first and second), 94 identity (with torsion), 298 map, 352 bi-form , 712 bi-harmonic function, 461

Index

bihermitian geometry, 263 bi-Lagrangian manifold, 709 bidegree splitting, 350 big bang, 905 bihermitian geometry, 265 biinvariant metric, 768 bilagrangian, 478 Bilinear form Ad.G/-invariant, 648 G-invariant, 629 BiLP, 271 Bismut connection, 292, 457 bivector, 192 black hole, 11, 150, 157, 905 attractor equations, 151, 167, 169, 173 entropy, 150, 159–161, 167, 169, 171 free energy, 173, 174 partition function, 173, 176 variational principle, 151, 168 Bochner flat pseudo-Kähler manifold, 877, 880 Borel–Lichnerowicz property, 586, 594 boundary of complex hyperbolic space, 614 of real hyperbolic space, 591 BPS black hole, 150, 162, 165 multiplet, 162 representation, see BPS multiplet soliton, 162 state, see BPS multiplet bracket, 211, 214, 215, 219, 235, 239, 249 BV-, 243, 246 Courant, 212, 219, 224, 225, 230, 237 derived, 218, 225, 236, 239 Dorfman, 212, 215, 218 Lie, 211, 215, 218 Lie algebroid, 239 Poisson, 236, 243 Schouten–Nijenhuis, 210, 232, 236 Brinkmann wave, 605

Index

933

connection, 291, 309, 314, 317, 361 foliation, 46 torsion, 309 charge quantization, 915 Calabi conjecture, 280 Chern form, 380 Calabi–Yau, Chern–Moser curvature tensor, 864 compactification (of string theory), Chern–Simons, 243, 252, 259, 922 3, 10, 164 Chern–Weil cone, 70 formalism, 258, 259 conformal analog of CY manifold, map, 259 823 Christoffel symbols, 94 manifold, 43, 278, 322 chronological metric, 458 future, 902 canonical past, 902 connection, 290, 400, 402–405, classification of metric connections with 419 torsion, 286 element of a Lie algebra, 527 Cleyton–Swann theorem, 381 isotropic ideal, 691 Clifford algebras, 477 para-Hermitian connection, 435 c-map, 4, 150, 154, 477 pseudo-Hermitian connection, 435 cohomology, 462, 691, 916 Cartan d-bar, 462 connection, 826, 876 Lie algebra, 692 decomposition, 483, 484 quadratic, 693 formula, 203 compact Lie group, 459, 470 Cartan, E., 281, 285, 296 587, 596 compactification Casimir operator, 280, 327, 328 conformal, 835, 864 Cauchy hypersurface, 903 Kaluza–Klein, 76 Cauchy problem, 907 of eleven-dimensional supergravity, existence and uniqueness of 74 solutions, 907 of string theory, 3 stability of solutions, 908 compactly supported smooth sections, 898 causal comparison formula, 368 future, 902 compatible metric, 456 past, 902 complete causality condition, 904 Cartan connection, 826 central symmetry, 706 linear connection, 116 Chan–Paton factors, 919 vector field, 115 character, 22, 30 completeness, 897 characteristic and development map, 743 class, 229, 250, 258, 259 of compact affine manifold, 735, 804 class of affine representation, 790, of compact homogeneous affine 793 manifold, 759 of flat affine Lie group, 766 class of compact affine manifold, 805 Bryant, R. L., 604 bundle gerbe, 659 BV (Batalin–Vilkovisky) formalism, 242

934

Index

of pseudo-Riemannian manifold, 735 complex conjugation, 89 Grassmannian, 129 hyperbolic space, 614 manifold, 89 tangent bundle, 96 product structure, 478 projective space, 95, 139 cone construction, 284, 297 conformal Calabi–Yau manifold, 823 calculus, 131–135 Cartan connection, 837 class, 870 compactification, 835, 864 Einstein metric, 846 frame bundle, 836 geometry, 455, 467 holonomy groups of Lorentzian manifolds, 847 holonomy of Riemannian manifolds, 847 Killing spinor, 848, 853 manifold, 834 spin structure, 849 structure, 869 symmetry, 462 (tensor) calculus, see superconformal calculus connection, 898 Bismut –, 292 canonical –, 290 characteristic –, 291, 309 d’Alembert operator, 906 first canonical Hermitian –, 291 form, 825 Hermitian –, 291 spin –, 281, 288 Tanaka–, 294 volume-preserving –, 299 with skew-symmetric torsion, 287, 289

with skew torsion, 188, 189 with torsion, 285 with vectorial torsion, 287, 288 connective structure, 190, 658 contact manifold, 44, 294 metric structure, 45 structure, 44 structure (para-holomorphic), 514 contour integral, 6, 7 cosmological redshift, 905 cotangent bundle, 458, 460 Courant algebroid, 205, 211, 214, 219, 221, 226, 230, 237, 239, 244, 247, 248, 252 bracket, 186, 219, 224, 225, 230, 237 sigma model, 248, 264 covariant derivative, 145 associated by a connection form, 826 induced by a Cartan connection, 828 covariant differential, 98 CR manifold, 864 structure, 560, 854 cross-product, 483 current, 915, 917 curvature, 888 metric, 468–471 of a Cartan connection, 830 tensor, 352, 372, 839 2-form, 482 curve causal, 902 future directed, 902 lightlike, 902 past directed, 902 spacelike, 902 timelike, 902 curving, 191, 203 d’Alembert operator, 906 d’Atri space, 299

Index

D-brane, 673, 915 Darboux coordinates, 87, 92 Darboux theorem for para-holomorphic contact structures, 515 for para-holomorphic symplectic structures, 514 de Rham’s theorem, 298, 302 de Rham–Wu decomposition, 586, 589 decomposition of curvature, 355, 372 of spin bundle, 301 Deligne cohomology, 656 delta-distribution, 899 de Sitter spacetime, 905 developing maps, 872 development image boundary of, 765 of affine manifold, 744, 802 development map, 738 of flat affine Lie group, 764 of homogeneous space, 757 development of a 1-form along a curve, 827 differential cohomology, 918 differential K-theory, 918 dilatation symmetry, 131 dilaton, 283 dimensional reduction, 921 Dirac current, 603 Dirac operator, 281, 848, 906 Kostant’s cubic –, 278 Weitzenböck formulas, 319 Dirac, P., 320 displacement form, 837 distributional section, 899 singular support of, 901 support of, 900 Dorfman bracket, 203 double structure, 198 D.2; 1I ˛/, 464–466 dual pair, 15, 26

935

effective action, 158 Eguchi–Hanson metric, 458, 460 Einstein, A., 469 Einstein manifold, 593 space, 290 universe, 835 Einstein–Cartan theory, 281, 283 Einstein–Sasaki manifold, 281, 294, 297 electric coupling, 916 electric-magnetic duality, see symplectic rotations electro-magnetic radiation, 897 elliptic operator, 897 energy-momentum tensor, 281 Engel, F., 303 essentially self-adjoint, 897 -Einstein Sasaki manifold, 318 étale affine representation, 764, 774 character of, 771 in dimension two, 775 of compact or semisimple group, 773 of reductive group, 773 relative invariant of, 770, 771 transitivity criterion for, 771, 801 Euclidean supersymmetry, 150, 153, 156 event horizon, 150, 157 exceptional Riemannian holonomy, 284 expansion of the universe, 905 extension double, 688 quadratic, 691 T -, 694 Fefferman connection, 856 metric, 853, 856, 889 spin space, 856 Fernández classification, 318 fiber metric, 97 finite propagation speed, 908 first prolongation, 597, 837

936 flat Cartan connection, 831 coordinates, 100 frame, 99 symplectic coordinates, 107 trivialization, 190 flat manifold affine, 732, 740 and .X; G/-manifolds, 741 projective, 740 pseudo-Riemannian, 732 Riemannian, 734 flat pseudo-Riemannian manifold, 732 compact, 804 compact homogeneous, 763 fundamental group of, 735 homogeneous, 736, 763 structure theory of, 735 flux compactification (of string theory), 168 form Kähler, 25 Killing, 23 formula branching, 31, 33 Kostant, 31 Weyl, 31 Friedmann cosmological model, 905 Friedrich inequality, 281 full prolongation, 563 fundamental solution, 908 advanced, 909 retarded, 909 fundamental vector field, 119, 484 future compact subset, 903 gauge field, 915 general holonomy principle, 278, 283, 299, 309 general relativity, 897 generalised momenta, 462 monopole, 459

Index

Calabi–Yau manifold, 196 complex geometry, 264 complex manifold, 191 complex structure, 191 geometry, 212 Hopf structure, 315 hypercomplex structure, 479 Kähler geometry, 266 Kähler potential, 271 Kähler structure, 193 geometric manifold, 891 gerbe, 189, 655, 920 connective structure (on a gerbe), 190, 658 holonomy, 667 nonabelian, 221, 253, 256, 260 global geometric flow, 891 global rigidity, 891 Goldberg conjecture, 315 G1 -manifold, 436 G1 -structures, 368 gradation, 561 graded Lie algebra fundamental, 561 minimal, 711 non-degenerate, 562 of finite type, 564 of semisimple type, 564 transitive, 562 grading element, 561 gravity, 281 Gray, A., 278, 280, 291, 425 Gray–Hervella class, 361 classification, 313, 430 Green’s operator advanced, 911 continuity of, 913 retarded, 911 Green–Schwarz mechanism, 917, 919 Grothendieck, A., 915, 922 group Lorentz, 26

Index

maximal compact, 26 maximal torus, 22, 29 ring, 30 of conformal diffeomorphisms of the Möbius sphere, 835 of transvections, 706 Sp.n/, 17 G-structure, 307, 309 G2 , 302 G2 -holonomy manifold, 44 G2 -manifold, 278, 304, 316 G2 (subgroups), 304 Gualtieri’s map, 266 half-flat structure, 439 SU.3/-structure, 313 Hamilton, W. R., 455 Hamiltonian, 462 harmonic function, 458 Hawking effect, 159 radiation, see Hawking effect temperature, 160 Heisenberg group, 290, 311 group (para-complex), 539 manifold, 855 rotation, 614 similarity transformation, 614 space, 614 translation, 614 Hermitian fiber metric, 96 Killing form, 363 metric, 91 Ricci tensor, 379 symmetric space, 131, 460 vector bundle, 97 Hermitian connection, 291, 349, 357 Hesse potential, 155, 169, 172 HKT, see hyper-Kähler manifold with torsion

937

cone, 465 cylinder, 465 potential, see potential, HKT HKT structure, 285, 292 Hodge star operator, 199 holomorphic cubic form, 112–114, 137 form, 90 map, 96 prepotential, 4, see prepotential vector bundle, 96 holonomy, 458, 685, 686, 688, 689, 696, 697, 699, 738 algebra of a Cartan connection, 831, 833 around a closed surface, 667 decomposition, 16, 28, 34 differential forms, 18 harmonic forms, 36 exceptional, 284 group, 279, 296 of a curve, 827 of compact affine manifold, 803 of homogeneous affine manifold, 762 principle, 278, 283, 299, 309 representation, 381, 400, 419 system, 382 unipotent, 762 weak, 280, 284 Zariski-closure of, 803 holonomy group Abelian, 606 connected, 584 Lorentzian, 631 non-closed, 602 of a Cartan connection, 828 of a covariant derivative, 825 of a curve, 825 of a Lie group with left-invariant metric, 617, 629, 824 of a linear connection, 584 of a Lorentzian manifold, 588 of a principal bundle connection, 825

938 of a pseudo-Kählerian manifolds of index 2, 610 of a pseudo-Riemannian manifold of dimension 4, 619 of a pseudo-Riemannian manifold of index 2, 608, 619 of a pseudo-Riemannian manifold of neutral signature, 617, 619 of a special pseudo-Kählerian manifold, 613 of a torsion free connection, 639 of space-times, 606 homogeneous domain, 773 automorphisms of, 778 examples, 775 of nilpotent group, 787 pseudo-Riemannian, 781, 784 symplectic, 781, 783, 785 translationally isotropic, 782 tubelike, 780 homogeneous manifold affine, 755 flat and symplectic, 762 flat pseudo-Riemannian, 736 symplectic, 755 homogeneous space, 629 Clifford–Klein form of, 756 development map of, 757 flat affine, 756 isotropy irreducible, 631 of nilpotent Lie group, 756 of reductive group, 800 unimodular character of, 794, 796 homothetic closed homothetic Killing vector, 117 holomorphic homothetic Killing vector, 118, 120 Killing vectors, 116 transformation, 116 homothety special, 465–470 Hopf manifold, 315

Index

horizontal distribution, 484, 516 subspace, 484 lift, 125 projection, 126 space, 126, 837 horosphere, 290 hyperHermitian structure, 457 Kähler, 457, 460, 470 Kähler cone, 3, 5 Kähler manifold, 43, 156 paracomplex structure, 478 paraHermitian metric, 478 paraHermitian structure, 478 paraKähler, 479 potential, 3, 5, 8 symplectic, 479 with torsion, 292, 456, 463, 465–471 strong, 457–461, 470 weak, 456 hyperbolic-complex structure, see para-complex structure hyperbolic nearly Kähler structure, 427 hyperbolic space, 835 hypercomplex structure, 457, 459, 470, 882 hypermultiplet, 135, 154, 477 Ikemakhen, A., 582, 608 indecomposable, 586 indefinite Heisenberg nilmanifold, 873 infinitesimal para-Kähler symmetric space, 710 pseudo-Kähler symmetric space, 710 initial value problem, 907 instanton type, 468, 469, 471 integrability, 489 integrable complex structure, 457, 463 geometry, 278, 308 quaternionic structure, 467 systems, 477

Index

intrinsic connection, see canonical connection intrinsic torsion, 278, 307 of almost contact structure, 310 tensor, 359 isometries Lie group of, 750 of nearly Kähler structures, 442 of pseudo-Riemannian manifolds, 750, 752 isometry, 5, 116 special, 465, 466, 469, 470 tri-holomorphic, 458 isometry group, 279 isoparametric surface, 312 isotropic algebra, 383 group, 308 representation, 399, 412–414, 418, 421 subspace, 383 Kac–Moody Killing vector, 271 Kähler–Einstein manifold, 43 Kähler form, 25, 350 Kähler–Hodge manifold, 87, 114, 124 Kähler manifold, 43, 92 holonomy, 92 special, see special Kähler manifold nearly, see nearly Kähler manifold Kähler potential, 4, 93, 198 Kaluza–Klein compactification, 76 Killing equation, 281 Killing form, 23, 362, 377, 459 Killing horizon, 157 Killing spinor, 55, 281, 284, 316, 318 imaginary, 55 parallel, 55 real, 55 type, 59 Killing vector, see Killing vector fields Killing vector fields, 116, 365, 750 Lie algebra of, 752

939

of nearly Kähler structures, 442 on flat Lorentzian manifold, 752 on flat Riemannian manifold, 752 Kirichenko’s theorem, 315 Kodaira embedding theorem, 124 Kostant’s cubic Dirac operator, 278, 324 Kostant–Parthasarathy formula, 324 K-theory, 915 K3 surface, 458 label vector, 567 Laplace–Beltrami operator, see also Laplacian, 897 Laplacian, 462 on spinors, 319 laws of black hole mechanics, 150, 157 leaves of a foliation, 119 Lee form, 315 left-symmetric algebra, 746 and flat affine Lie groups, 764 Legendre transform, 4, 6, 11 Levi form, 855 Levi-Civita connection, 94, 187, 189 Lichnerowicz, Andre, 281 Lie algebra sl2 , 20 so4;1 R, 21 sp.n/, 15 Lie algebroid, 215, 218, 221, 235, 239, 240, 250, 258, 260 BF, 254 bracket, 239 representation, 221 Yang–Mills, 255 Lie group flat affine, 763, 764 flat Lorentzian, 767 flat pseudo-Riemannian, 767 flat Riemannian, 768 left-invariant geometry on, 764 reductive, 767 solvable, 768 symplectic, 754

940

Index

unimodular, 766 Lie 2-algebroid, 222, 240, 256 linear differential operator, 900 acting on distributions, 900 formal adjoint of, 900 local frame, 145 locally conformally Kähler manifold, 315 loop space, 296 Lorentzian cylinder, 898 Lorentz metric, see Lorentzian metric manifold, 897, 902 globally hyperbolic, 904 metric, 889, 902 Möbius sphere, 835 Majumdar–Papapetrou solution, 165 manifold base, 211, 214, 220, 240, 241, 258 complex, 222 graded, 212, 216, 235, 238, 241, 244, 245, 257 hyperkähler, 18 P -, 236 Poisson, 210, 212, 215, 236, 240, 247 PQ-, 236, 239, 243, 246, 253, 259 Q-, 212, 234, 239, 243, 246, 257, 258 quaternionic Kähler, 18 Riemannian, 209, 220, 252 source, 210, 212 super-, 235 symplectic, 233, 245, 253 target, 210, 245 Maurer–Cartan form, 525, 826 maximally homogeneous, 565 Maxwell equations, 917 metric connection, 97 metric Lie algebra, 687, 768 Minkowski space, 905 monodromy, 871

M -theory, 282, 921 Mukai pairing, 194 multi-instanton metric, 458 multi-Taub-NUT metric, 458 multisymplectic structure, 304 naturally reductive space, 289, 297, 309 nearly half-flat structure, 439 nearly Kähler manifold, 66, 278, 281, 315, 377, 399–402, 405, 408, 412, 415, 419, 425 automorphisms, 442 left-invariant, 445 of constant type, 436 S 3 S 3 , 426 nearly para-Kähler manifold, 433 defined by exterior system, 440 Ricci-flat, 437 nearly parallel G2 -manifold, 281 nearly parallel G2 -structure, 62 nearly pseudo-Kähler manifold, 433 defined by exterior system, 440 SL.2; R/ SL.2; R/, 426, 445 nearly-Kähler, see nearly Kähler manifold neutral hyper-Kähler, 479 metric, 491 pseudo-Riemannian manifolds, 480 signature, 478 surfaces, 479 Newlander–Nirenberg theorem, 91 Nijenhuis tensor, 290, 294, 348, 357, 429, 457, 478, 489 mixed, 463 totally skew-symmetric, 433, 438, 439 nilmanifold, 314, 315 nilpotent orbit, 460 non-integrable geometry, 278, 307, 308 nondegenerate conformally flat geometry, 866 flat pseudo-conformal qCR geometry, 866

Index

spherical CR geometry, 866 nonlinear sigma model, 193 normal conformal Cartan connection, 838, 841 normal tractor derivative, 842, 844 normally hyperbolic operator, 905 null subspace, 383 Obata connection, 457 one-sheeted hyperboloid, 478, 482 orbifold, 467 orientation, 920 orthogonal part, 590, 595 OSV conjecture, 151, 173–175 para-conformal structure, 479 para-quaternionic connection, 478, 484 Hermitian, 479 identities, 477 Kähler, 156, 479 Kähler manifold, 510 Kähler manifolds with torsion, 493 Kähler symmetric spaces, 538 projective space, 518 structure, 479, 509 para-quaternions, 477, 481 para-complex Dolbault operator, 504 flag manifold, 535 Grassmannian, 535 Heisenberg group, 539 Kähler manifold, 509 projective space, 518 structure, 153, 500, 559, 709 para-CR algebra, 562 integrable, 562 manifold, 560 structure, 560 of finite type, 564 of semisimple type, 564 para-holomorphic contact structure, 514

941

differential form, 506 function, 503 symplectic structure, 511 para-hyper Kähler manifold, 156 para-Kähler geometry on tubelike domains, 783 manifold, 709 potential, 619 structure, 618 para-pluriharmonic map, 520, 553 associated family, 522 isotropic, 526 twistor space, 528 para-quaternions, 509 para-symplectic group, 510 para-unitary group, 508 parabolic geometry, 864 parabolic subgroup, 531 parallel displacement see also parallel transport, 584, 824 objects, 299 spinor, 321, 329 torsion, 309, 315, 318, 332 transport, 296 Parthasarathy formula, 280 past compact subset, 903 p-c qCR structure, 881 Penrose–Hawking singularity theorem, 282 '-symmetric space, 290 Poisson bivector, 215, 225, 233, 247, 248 bracket, 236, 243 manifold, 192, 210, 212, 215, 236, 240, 247 sigma model, 210, 247, 249, 260 structure, 210, 233, 240 twisted, 210, 249 potential, 460 HKT, 460–462, 465, 467, 469 Kähler, 460 pp-wave, 605

942

Index

pr-waves, 606 prehomogeneous representation, 774, 778 by isometries, 785 characteristic character of, 798 characteristic image of, 789 characteristic map of, 777 of nilpotent group, 787, 788, 790 reductive, 798 transitivity criterion, 788, 799 transitivity criterion for, 787, 790 prehomogeneous vector space, 774 regular, 800 prepotential, 86, 106–109 presymplectic, 212, 222 principal bundle, 92 connection, 145 projective Kähler manifold, 114–131 definition, 119 projective special Kähler geometry, 86 projective special Kähler manifold, 136–142, 164 curvature, 138 definition, 136 pseudo 3-Sasakian space form, 885 pseudo K-hyperbolic geometry, 866 pseudohyper-complex, 478 Riemannian manifold, 491 pseudo-conformal quaternionic CR structure, 864 pseudo-Hermitian structure, 875 pseudo-Kähler manifold, 610, 709 pseudo-Kählerian manifold, see pseudo-Kähler manifold pseudo-Riemannian metrics, 869 pseudo-Riemannian special Kähler manifold, 86 pseudo-Sasakian 3-structure, 883 pseudo-Sasakian metric, 873 QKT, see quaternionic Kähler manifold with torsion quadratic form, 918

quantum mechanics one-dimensional, 456, 462 quasicomplex manifold, 89 quasicomplex structure torsion, 91 quasi-Kähler manifold, 92 quasi-Sasakian manifold, 311 quaternion-Kähler manifold, see quaternionic Kähler manifold quaternionic CR structure, 883 quaternionic geometry, 455 quaternionic Heisenberg nilpotent Lie group, 886 quaternionic Kähler manifold, 43, 156, 295, 469 with torsion, 456, 467–471 quaternionic metric, 5 quaternionic unitary group, 647 quaternions, 455, 463, 477 quotient, 263, 272 quotient metric, 204 Ramond-Ramond field, 915 rank of a bi-form, 717 real Heisenberg dilation, 614 real hyperbolic space, 591 recurrent vector field, 598 reduced holonomy group, 828 reduction (of the frame bundle), 308 Reeb vector field, 44, 293 reflector space, 479, 484 regular distribution, 562 Reichel, W., 303 Reissner–Nordström solution, 165 representation real type, 632 complex type, 633 irreducible, 630 quaternionic type, 634 self-dual, 643 restricted holonomy group, 92 Ricci curvature, 839 Ricci-flat, 458, 593

Index

Ricci form, 482, 489, 491 Ricci-isotropic, 605 Ricci tensor, 95 Riemannian manifold, 897 Riemannian submersion, 393 Riemann–Roch–Hirzebruch, 915 rigid special Kähler manifold, see also affine special Kähler manifold, 98–114 curvature, 114 definition, 99 Robertson–Walker spacetime, 905 root, 30 positive, 30, 32 R-symmetry, 464 Salamon operator, 461 Sasaki–Einstein manifold, 49 toric, 51 Sasaki manifold,46, 294, 297, 855 Sasakian manifold, see Sasaki manifold Sasakian structure, 46 irregular, 46 negative, 48 null, 48 positive, 48 quasi-regular, 46 regular, 46 Satake diagram, 570 scalar curvature, 839 Schimming, R., 599, 605 Schouten bracket, 192 Schouten tensor, 839 Schouten, J. A., 299 Schrödinger, E., 320 Schrödinger–Lichnerowicz formula, 320 Schur functor, 16, 26 Schur’s lemma, 632 Schwarzschild spacetime, 905 screen holonomy, 604 screw dilations, 592 screw isometries, 592 self-dual, 455, 467, 469, 471

943

self-duality, 918 sigma model, 209, 293 AKSZ, 212, 241, 247, 251, 259 Courant, 247, 248 Dirac, 248 (higher) Yang–Mills type, 253 nontopological, 213 Poisson, 210, 247, 249, 260 topological, 210, 212, 241, 248, 251, 259 similarity transformation, 591 of C n , 615 sine-cone lemma, 70 skew torsion, 188 skew-symmetric torsion, 287 SL.3; R/-structure, 431 half-flat, 439 induced by a pair of stable forms, 433 nearly half-flat, 439 SO.3/-structure, 312 solvmanifold, 323 space-like compact support, 912 space of curvature tensor of type h, 709 spacetime, 897 special coordinates, 86, 87, 107, 110 special geometry, 149 special homothety, see homothety, special special Kähler manifold, 4, 426, 555 affine (or rigid), 153, 98–114 projective, 136–142, 164 geometry, 86 special subspace, 383, 387 spin bundle, 301, 305 connection, 281, 288 Laplacian, 319 representation, 301, 305, 630 spin tractor bundle, 851 Spin.7/, 305 Spin.7/-holonomy manifold, 44 Spin.7/-structure, 318 spinors, 194 parallel, 631

944 split complex structure, see para-complex structure split quaternions, see para-quaternions splitting theorem for conformal holonomy, 846 squashed 7-sphere, 285 stabilizer, 283, 300, 308 stable form, 430, 432 standard horizontal vector field, 484 standard tractor bundle, 843 strict nearly-Kähler, 378, 379 strictly pseudoconvex manifold, 855 string theory, 477, 916 compactification, 3 Strominger model, 278, 284, 330 strong causality condition, 904 strong geometry, 456–459 structure algebroid, 218, 221, 230, 235, 238, 250 complex, 212, 227, 233 constants, 234, 244 Dirac, 210, 212, 222, 225, 248 functions, 235, 254 generalized complex, 210, 212, 221, 230, 248 group, 258 Lie algebra, 215, 239 Lie algebroid, 210, 218, 235, 239, 248 Loday algebroid, 221 orthogonal quaternionic, 17 P -, 236 Poisson, 210, 233, 240 twisted, 210 Q-, 238, 258 real, 19, 227 sheaf, 235 symplectic, 212, 233, 236, 238 SU.n/, 300 superalgebra, 464 simple, 464 superconformal, see D.2; 1I ˛/

Index

superconformal calculus, 163 symmetry, 455, 456, 463, 465 supergravity, 283, 916 N D 2, 3, 86 superhorizontal bundle, 529 superspace action, 463 superstring theory, 73, 278, 282, 330 supersymmetric quantum mechanics, 456 black hole, see BPS black hole supersymmetry, 456, 463 algebra, 151 breaking (N D 2 to N D 1), 87 Euclidean, 150 local, 142, 150 N D 1, 74, 153 N D 2, 74, 86, 152 477 rigid, 150 sigma model with .2; 2/, 193, 265 sigma model with .4; 0/, 293 transformation, 278, 283 SU.p; q/-structure, 431 half-flat, 439 induced by a pair of stable forms, 433 nearly half-flat, 439 surface holonomy, 667 SU.3/-structure, 401, 406, 408, 416, 417, 421 Swann bundle, 3, 5, 469, 471 symmetric decomposition, 706 effective, 706 minimal, 706 symmetric pair, 26 dual, 26 symmetric space, 280, 706 extrinsic, 700 hyper-Kähler, 697–700 hypersymplectic, 697 Lorentzian, 687, 690 para-Hermitian, 697 para-hyper-Kähler, 697, 700 para-quaternionic Kähler, 697, 698

Index

pseudo-Hermitian, 697 pseudo-Riemannian, 685, 686 quaternionic Kähler, 697–698 symmetric triple, 689 symplectic, 212, 222, 230, 233, 236, 245, 259 bundle, 86 connection, 99 covariance, see symplectic rotations form, 632 rotations, 152–154, 163, 164 structure, 92 structure (para-holomorphic), 511 transformation, 753, 755 transformations, see symplectic rotations152 symplectic manifold, 92, 192, 754 compact, 762 homogeneous, 762 symplectic vector fields, 753 affine, 754 Lie algebra of, 755 tachyon, 921 Tanaka connection, 294 Tanaka structure, 564 of finite type, 564 of semisimple type, 564 Tanaka–Webster connection, 855 tangent vector causal, 902 future directed, 902 lightlike, 902 past directed, 902 spacelike, 902 timelike, 902 target manifolds, 477 tautological bundle, 130 T-duality, 458, 466, 470, 471 test sections, 899 convergence of, 899 theorem de Rham’s, 298, 302

945

of Ambrose–Singer, 290 of Kirichenko, 315 of Wang (on G-connections), 289 of Wang (on parallel spinors), 307 3-graded Lie algebra, 711 3-graded Lie algebra admissible, 711 3-Sasakian manifold, 43, 49 295, 297 3-symmetric space, 399, 401, 403–406, 417–419 S 3 S 3 , 426 SL.2; R/ SL.2; R/, 426 locally 3-symmetric space, 405 timeorientation, 902 topological field theories, see also topological sigma models, 251, 499, 656 topological free energy, 174 topological string, 10, 11, 173, 174 topological-antitopological fusion, 426 torsion, 278, 285, 921 characteristic –, 309 of a Cartan connection, 830 of a connection form, 840 of a horizontal space, 837 of almost Hermitian manifold, 291, 314 of G2 -structure, 317 parallel –, 309, 436 skew-symmetric –, 287 tensor, 349, 360, 381 totally skew-symmetric –, 435 three-form, 456, 457, 467 2-form, 482 vectorial –, 287, 288 torsion-free Cartan connection, 831 torsion-free horizontal space, 837 torus, 458, 470 tractor, r ! -parallel, 833 tractor bundle, 828 transvection group, 689 triholomorphic

946

Index

circle symmetry, 458 isometries, 6 tt*-bundle, 553 twisted cohomology, 195 twistor equation, 848 twistor geometry, 43 twistor operator, 848 twistor space, 6, 9, 349, 393, 400, 411, 417, 419, 479, 483 associated to a para-pluriharmonic map, 528 birational equivalence, 541 Kähler potential of, 9 of a para-quaternionic Kähler manifold, 516, 539 twistor theory, 278 two-point homogeneous space, 290 two-sheeted hyperboloid, 478, 482 type change, 192 uniformization, 870 unit ball, 139 unitary type, 597 U.n/, 300, 309 upper half plane, 109 U.p; q/-structure, 428 quasi-integrable, 437 U.3/-structure, 313 vector multiplet, 135, 152, 163 vectorial torsion, 287, 288 Verlinde ring, 923 vertical distribution, 484 space, 484 volume form and characteristic homomorphism, 804 and completeness, 766, 804 left-invariant, 766 parallel, 802

polynomial, 802 right-invariant, 801 Wang’s theorem, 307 wave operator, 905 weak Berger algebra, 594–597 geometry, 456 holonomy, 280, 284, 400, 414, 419 PSU.3/-structure, 306 topology, 901 weight, 29 diagram, 22 dominant, 30 highest, 22, 27, 31 positive, 30 space, 22 weighted pseudo-Kähler projective space, 881 Weitzenböck formula, 278, 319 Wess–Zumino term, 293 Weyl chamber, 22 character formula, 31 curvature tensor, 839, 864, 871 denominator, 31 group, 30 orbit, 33 Weyl tensor see Weyl curvature tensor Wolf space, 470 worldsheet, 919 W1 C W4 structure, 371 .X; G/-structure, 738 left invariant, 743 Yang–Mills field equations, 213 (gauge) theory, 214, 252, 257 Lie algebroid, 255 theory, 278

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