Quaternionic Structures in Mathematics and Physics: Proceedings of the Second Meeting Rome, Italy 6 - 10 September 1999

Proceedings of the Second Meeting

Quaternionic Structures in Mathematics and Physics

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Editors

Stefano Marchiafava Paolo Piccinni Massimiliano Pontecorvo

i2=j2=k2=-l, ij=-ji=kjk=-kj=i, ki=-ik=j

Proceedings of die Second Meeting

Quaternionic Structures in Mathematics and Physics

Proceedings of the Second Meeting

Quaternionic Structures in Mathematics and Physics Rome, Italy

6-10 September 1999

Editors

Stefano Marchiafava University degli Studi di Roma "La Sapienza "

Paolo Piccinni Universita degli Studi di Roma "La Sapienza "

Massimiliano Pontecorvo Universita degli Studi Roma Tre

Printed with a partial support of C.N.R. (Italy)

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World Scientific Singapore • New Jersey • London • Hong Kong

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On the cover: William R. Hamilton

QUATERNIONIC STRUCTURES IN MATHEMATICS AND PHYSICS Proceedings of the Second Meeting Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-02-4630-7

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This volume is dedicated to the memory of Andre Lichnerowicz and Enzo Martinelli FOREWORD Five years after the meeting "Quaternionic Structures in Mathematics and Physics", which took place at the International School for Advanced Studies (SISSA), Trieste, 5-9 September 1994, we felt it was time to have another meeting on the same subject to bring together scientists from both areas. The second Meeting on Quaternionic Structures in Mathematics and Physics was held in Rome, 6-10 September 1999. Like in 1994, also this time the Meeting opened a semester at The Erwin Schrodinger International Institute for Mathematical Physics of Vienna, that in Fall 1999 was dedicated to "Holonomy Groups in Differential Geometry", and many participants of this ESI program were invited speakers at the Quaternionic Meeting. We thank D.V. Alekseevsky, K. Galicki, P. Gauduchon, S. Salamon, members of the Scientific Committee of this Second Meeting, and all the speakers for their contribution. We gratefully acknowledge financial support from Progetto Nazionale di Ricerca MURST "Proprieta' Geometriche delle Varieta' reali e complesse" (both with local and national funds), Universita' di Roma "La Sapienza", Universita' di Roma Tre, Comitato Nazionale per la Matematica - C.N.R.. We acknowledge also the hospitality of both Departments of Mathematics of Universities " Roma La Sapienza" and "Roma Tre". We hearty thank the valuable collaboration of Elena Colazingari in the organization of the Meeting. We are also grateful to Angelo Bardelloni for the technical preparation of the electronic version of these Proceedings, to Tiziana Manfroni and Paolo Marini for setting up the web site and to our colleague Francesco Pappalardi for TeX-nical support. Finally, it gives us great pleasure to thank all the participants of the meeting for their interest and enthusiasm, and of course all the contributors of the present Proceedings. Roma, March 2001 Stefano Marchiafava Paolo Piccinni Massimiliano Pontecorvo

I N T R O D U C T I O N TO T H E C O N T R I B U T I O N S

During the five years, which passed after the first meeting "Quaternionic Structures in Mathematics and Physics " interest in quaternionic geometry and its applications continued to increase. A progress was done in constructing new classes of manifolds with quaternionic structures (quaternionic Kahler, hyper-Kahler, hypercomplex etc.), studying differential geometry of special classes of such manifolds and their submanifolds, understanding relations between the quaternionic structure and other differential-geometric structures, and also in physical applications of quaternionic geometry. Some generalizations of classical quaternionic-like structures (like HKT -structures and hyper-Kahler manifolds with singularities) naturally appeared and were studied. Some of these results are published in this proceedings. A new simple and elegant construction of homogeneous quaternionic pseudoKahler manifolds is proposed by V. CORTES. It gives a unified description of all known homogeneous quaternionic Kahler manifolds as well as new families of quaternionic pseudo-Kahler manifolds and their natural mirror in the category of supermanifolds. Generalizing the Hitchin classification of Sp(l)-invariant hyper-Kahler and quaternionic Kahler 4-manifolds, T. NITTA and T. TANIGUCHI obtain a classification of S'p(l) n -invariant quaternionic Kahler metrics on 4n-manifold. All of these metrics are hyper-Kahler. I.G. DOTTI presents a general method to construct quaternionic Kahler compact flat manifolds using Bieberbach theory of torsion free crystallographic groups. Using the representation theory , U. SEMMELMANN and G. WEINGART find some Weitzebock type formulas for the Laplacian and Dirac operators on a compact quaternionic Kahler manifold and use them for eigenvalue estimates of these operators. As an application, they prove some vanishing theorems, for example, they prove that odd Betti numbers of a compact quaternionic Kahler manifold with negative scalar curvature vanish. A hyper-Kahler structure on a manifold M defines a family (Jt,u>t) of complex symplectic structures, parametrized by t € C P 1 . R. BIELAWSKI gives a generalization of the hyper-Kahler quotient construction to the case when a holomorphic family Gt, t £ C P 1 of complex Lie group is given, such that Gt acts on M as a group of automorphisms of (J t ,wt). The existence of a canonical hyper-Kahler metric on the cotangent bundle T*M of a Kahler manifold M was proved independently by D.Kaledin and B.Feix. In the present paper D. KALEDIN presents his proof in simplified form and obtains an explicit formula for the case when M is a Hermitian symmetric space. A toric hyper-Kahler manifold is defined as the hyper-Kahler quotient of the quaternionic vector space I F by a subtorus of the symplectic group Sp(n). H. KONNO determines the ring structure of the integral equivariant cohomology of a toric hyper-Kahler manifold. M. VERBITSKY gives a survey of some recent works about singularities in hyper-Kahler geometry and their resolution. It is shown that singularities in a singular hyper-Kahler variety (in the sense of Deligne and Simpson) have a simple

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INTRODUCTION TO THE CONTRIBUTIONS

structure and admit canonical desingularization to a smooth hyper-Kahler manifold. Some results can be extended to the case of the hypercomplex geometry. Hypercomplex manifolds (which is the same as 4n-manifolds with a torsion free connection with holonomy in GL(n,M) ) are studied by H. PEDERSEN. He describes three constructions of such manifolds: 1) via Abelian monopoles and geodesic congruences on Einstein-Weyl 3-manifolds , 2) as a deformation of Joyce homogeneous hypercomplex structures on G x Tk where G is a compact Lie group and 3) as a deformation of the hypercomplex manifold Vp(M), associated with a quaternionic Kahler manifold M and an instanton bundle P —>• M by the construction of Swann and Joyce. M.L. BARBERIS describes a construction of left-invariant hypercomplex structures on some class of solvable Lie groups. It gives all left-invariant hypercomplex structures on 4-dimensional Lie groups. Properties of associated hyper-Hermitian metrics on 4-dimensional Lie groups are discussed. D. JOYCE proposes an original theory of quaternionic algebra, having in mind to create algebraic tools for developing quaternionic algebraic geometry. Applications for constructing hypercomplex manifolds and study their singularities are considered. Properties of hyperholomorphic functions in I 4 are studied by S-L. ERIKSSONBIQUE. Hyperholomorphic functions are defined as solutions of some generalized Cauchy-Riemann equation, which is defined in terms of the Clifford algebra C1(M°' 3 )«H©H. Other more general notion of hyper-holomorphic function on a hypercomplex manifold M is proposed and discussed in the paper by ST. DIMIEV , R. LAZOV and N. MILEV. O.BIQUARD defines and studies quaternionic contact structures on a manifold. Roughly speaking, it is a quaternionic analogous of integrable CR structures. Generalizing the ideas of A. Gray about weak holonomy groups, A.SWANN looks for G-structures which admit a connection with "small" torsion, such that the curvature of these connections satisfies automatically some interesting conditions, for example, the Einstein equation. G. GRANTCHAROV and L. ORNEA propose a procedure of reduction which associates to a Sasakian manifold S with a group of symmetries a new Sasakian manifold and relate it to the Kahler reduction of the associated Kahler cone K{S). The geometry of circles in quaternionic and complex projective spaces are studied by S. MAEDA and T. ADACHI. The main problem is to find the full system of invariants of a circle C, which determines C up to an isometry, and to determine when a circle is closed. Special 4-planar mappings between almost Hermitian quaternionic spaces are defined and studied by J. MIKES, J. BELOHL'AVKOV'A and O. POKORN'A. Some generalization of the flat Penrose twistor space C 2,2 is constructed and discussed by J. LAWRYNOWICZ and 0 . SUZUKI. M. PUTA considers some geometrical aspects of the left-invariant control problem on the Lie group Sp(l). Quaternionic representations of finite groups are studied by G. SCOLARICI and L. SOLOMBRINO. Quaternionic and hyper-Kahler manifolds naturally appear in the different physical models and physical ideas produce new results in quaternionic geometry. For

INTRODUCTION TO THE CONTRIBUTIONS

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example, Rozanski and Witten introduce a new invariant of hyper-Kahler manifold as the weights in a Feynman diagram expansion of the partition function of a 3-dimensional physical theory. A variation of this construction, proposed by N. Hitchin and J. Sawon , gives a new invariant of links and a new relations between invariants of a hyper-Kahler manifold X, in particular, a formula for the norm of the curvature of X in terms of some characteristic numbers and the volume of X. These results are presented in the paper by J. SAWON. The classical Atiyah-Hitchin-Drinfeld-Manin's monad construction of anti-selfdual connections over 5 4 = H P 1 was generalized by M. Capria and S. Salamon to any quaternionic projective space H P " . Using representation theory of compact Lie groups, Y. NAGATOMO extends the monad construction to any Wolf space (i.e. compact quaternionic symmetric space). A quaternionic description of the classical Maxwell electrodynamics is proposed in the paper by D. SWEETSER and G. SANDRI. A. PRASTARO applies his theory of non commutative quantum manifolds to the category of quantum quaternionic manifolds and discusses the theorem of existence of local and global solutions of some partial differential equations. A hyper-Kahler structure on a 4n-manifold M is defined by a torsion free connection V with holonomy group in Sp(n). A natural generalization is a connection V with a torsion T = (T^) and the holonomy in Sp(n). If the covariant torsion r = g(T) = (gijT^), where g is the V-parallel metric on M, is skew-symmetric, then the manifold (M, V,