SUPERGRAVITY AND SUPERSTR·INGS A Geometric Perspective Vol. 3 : Superstrings
Leonardo Castellani Istituto Nazionale df Fisica Nuclears Sezione dl Torino
Riccardo 0' Auria Dipartimento di Fisica UnivsfSita di Padova
Pietro Fre International School tor Advanced Studies, Trieste
\\h World Scientific .. Singapore. New Jersey London •
•
Hong Kong
P"Wished I1y World Scientific Publishing Co. Pte. Ltd. POBox 128. Faner Road. S"Ulgapore 9128 USA qJJice: fKI Hartwell Street. Teaneck, NJ 00666 UK offu:e: 73 Lynton Mead. TOIIeridge. London N20 80H
SUPERGRAVITY AND SUPERSTRINGS - AGeomelrlc Perspective Copyright © 1991 by World Scientific Publishing Co. Pte. Ltd.
All righls reserml. This book, or ptUl$/hereof. rnay /lOt be reprodllCtd in any form or byanYl7Ulans. eketronic or I7U1Chanic41. includingphOlocopying ,recordillg oratry information storage and reWval sy:Iem now known or 10 be invented. wilhoul written permission /r0lll the Publisher.
ISBN 9971·5O..(l37·X (set) ISBN 9971-50-038·8 pbk (set)
Printed in Singapore by I.Di Printing Pte. Ltd.
v
CONTENTS
Preface
v
VolUme 1 PART ONE. GRAVITY AND DIFFERENTIAL GEOMETRY Chapter 1.0. Introduction Chapter 1.1. Exterior Calculus I. 1.1. Exterior forms on vector spaces 1.1.2. Mappings and operations on forms 1.1.3. Differentiable manifolds. vector fields and dilferential forms 1.1.4. Functions. vector fields and differential forms 1.1.5. Exterior differentiation and behaviour under mappings 1.1.6. The vielbein basis 1.1.7. Lie derivative, coordinate transformations and invariance Appendix: The 6 Operator and the Hodge Decomposition
3
9 10 23 28 37 47 53 59 72
Chapter 1.2. Riemannian Manifolds 1.2.1. Introduction 1.2.2. Geometry 01 the linear spaces 1.2.3. The geometry of general Riemannian manifolds in the vielbein ballis 1.2.4. Relation with the standard world-tensor formalism
75
Chapter 1.3. Group Manifolds and Maurer-Cartan Equations 1.3.1. Introduction 1.3.2. Lie groups as manifolds: left and right invariant vector fields 1.3.3. Maurer-Cartan equations 1.3.4. Adjoint representation and Killing metric; 1.3.5. KiRing metric 1.3.6. Riemannian geometry of semisimple groups 1.3.7. Soft group manifolds 1.3.S. The example 01 Poincare and anti de Sitter soft group manifold
97
75 76 SO 91
97 98 104 107 113 116 119 131
Chapter 1.4. Poincare Gravity 1.4.1. Poincare gravity 1.4.2. Extension to the soft group manifold 1.4.3. BuHding rules for the gravity Lagrangians 1.4.4. Gravily in de Siner and anti de Sitter space
141
Chapter 1.5. Coupling of Gravity to Matter Fields 1.5.1. Geometrical Lagrangian for scalar fields on a rigid background 1.5.2. Extension to the Poincare group manifold and interpl'etation of Ihe lorentz transformation rules as variational equations
170 170
141 152 157 166
174
vi 1.5.3. 1.5.4.
1.5.5. 1.5.6.
The interaction of the scalar fields with gravity and the effective cosmological constant The field equation of a massless scalar field in anti de Sitter space (in general in a curved space) Geometrical Lagrangian for spin 1 fields Geometrical Lagrangian for spin 112 fields
Chapter 1.6. Differential Geometry of Coset ManHolds 1.6.1. \.6.2. 1.6.3. 1.6.4. 1.6.5. 1.6.6. 1.6.7.
1.6.8. 1.6.9. 1.6.10. 1.6.11. 1.6.12. 1.6.13. 1.6.14. 1.6.15.
Introduction Classification of coset manifolds Coordinates on GlH and finite G-transformations FlIlite lransformations on GIH Infinitesimallransformations and KiUing vectors Vl9lbeins and metric on GlH Covariant Ue derivative Geodesics invariant measure Connection and curvature Rescalings A Note on the isometries of GIH Some examples Elemants of algebraic topology Homotopy and (co)homology of coset spaces
Chapter 1.7. Applications of the Formalism and Miscellaneous Examples 1.7.1. I. 7.2. 1.7.3.
TheBrans-Dicketheory Minimal coupling of pseudoscalars through a torsion mechanism The Schwarzschild solution
Bibliography
176 181 185 188
190 190 195 197 204 210 212 219 222 225 226 231 235 240 254 262
272 272 278 283
296
PART lWO. THE ALGEBRAIC BASIS OF SUPERSYMMETRY Chapter 11.1. Introduction
301
Chapter 11.2. Super lie algebras, Supermanifolds and Supergroups 11.2.1. The definition of superalgebras and !he example of N-extended
310
super Poincare algebra Classification of the simple suparalgebras whose Ue algebra is reductive Grassmann algebras Supermanifolds Supergroups and graded malrices Osp(41N) as the N-extended supersymmetry algebra in anti de
310
Sitter space
352
11.2.2. 11.2.3. 11.2.4. 11.2.5. 11.2.6.
323 333 338 345
Chapter 11.3. Super Maurer-Cartan Equations and the Geometry ofS~
11.3.1.
11.3.2.
Maurer-Cartan equations of supergroups ~roup manifolds Maurer-Cartan equations of OSP(41N) and Osp(41N)
360 360 364
vii 11.3.3.
11.3.4.
0sP(4/N} Maurer-Canan equations as the structural equations of rigid superspace Kimng vectors on superspace,!hat is the generators of !he supersymme1ly algebra of suPerisometries
Chapter 11.4. Poincare Supermultiplets 11.4.1. How 10 construct the unitary irreducible representations of the 11.4.2. 11.4.3. 11.4.4.
N-extended Poincare superalgebra Massive muhiplets wilhout central charges Massive multiplets with central charges Massless multiplets
Chapter 11.5. Supermultiplets in Anti de Sitter Space 11.5.1.
11.5.2. 11.5.3. 11.5.4. 11.5.5.
Free field equations and the concept of mass in anti de Sitter space Unitary irreducible representations of SO{2,3) Unitary irreducible representations of 0sp(4/N} Osp(411) supermulliplets Remarks about the N-extended case and !he example of the OsP(412) multiplets
Chapter 11.6. Supersymmetric Field Theories: The Example of the Wess-Zumino Multiplet Supersymmetric field-theories corresponding to an irreducible representation of lhe supersymmelly algebra 11.6.2. The Wess-Zumino model: the simplest example of a supersymmetl'ic field lheory 11.6.3. Superlield interpretation of the Wess-Zumino model and rheonomy 0.6.4. The integrability of the rheonomic conditions and the Bianchi identities 11.6.5. The rheonomic action principle
370
3SO
390 390 395 411 416
425 425 435 448 454 4S4
473
11.6.1.
473
4n 489
500 508
Chapter 11.7. r -matrix Algebra and Spinors in 4 ~ Ds 11
519
11.7.1. The construction of r-matriees 11.7.2. The c:harge conjugation matrix 11.7.3. Majora1l8, Wayl and Majorana-Weyl spioors 11.7.4. Useful formulae in r -matrix algebra
519 523 526 530
Chapter II.S. Rerz Identities and Group Theory 11.8.1. 11.8.2. 11.8.3. 11.8.4.
11.8.5. u.s.6. 11.8.7. 11.8.8.
Introduction The structure of forms on N-extended D,. 4 superspace Fiea decompositions in the N. 1. D= 4 superspace The N.. 2, D,. 4 ease The N .. 3, D,. 4 case The Nit 2, D,. 5 case Systematics of Fiea identities in eleven dimensions Irreducible representations of SO{t,9} and !he irreducible basis 01 the D = 10 superspace
535 535 537 545 547
552 555 563 567
viii
Chapter 11.9. Super Yang-Mills Theories 11.9.1. 11.9.2. 11.9.3.
Introduction Super Yang·MilIs Iheories in D= 4 The action principle for N ",, D= 10 superYang-Mills theory
Historical Remarks and References
582 582 684 594
597
ix
Volume 2 PART THREE. SUPERGRAVlTY IN THE RHEONOMY FRAMEWORK Chapter 111.1. Introduction
607
Chapter IIl2. Supergravity in the Standard Component Approach IIl2.1. Local supersymmetly and gravity 111.2.2. Spaclrtime Lagrangian of D= 4. N .. 1 supergravity 111.2.3. The equations of motion of D.. 4, N.. 1 supergravity 111.2.4. Supersymmetry transformations and action invariance 111.2.5. On-sheH supersymmetry invariance 111.2.6. The linearized theory of supergravity Appendix 1II.2.A. Commutator of Two Supersymmetries on the
611
Gravitino Field
Chapter III.S. Supergravity in Superspace and the Rheonomy Principle 111.3.1. From space·time 10 superspace 111.3.2. Geometry of superspace 111.3.3. The rheonomy principle 111.3.4. An extended action principle IU.3.5. D'" 4, N .. 1 supergravity and rheonomy 111.3.6. Rheonomic constraints and Bianchi identities 111.3.7. On-shell supersymmetly 1II.3.S. Ac:Iion invarianee and off·shell supersymmelry 111.3.9. BuDding rules for supergravity Lagrangians 111.3.10. Retrieving N =1. D .. 4 supergravity from lhe buUding rules IU.3.11. Extension 10 anti de Siner supergravity 111.3.12. Building rules for supergravily theories using rheonomy and Bianchi identities
Chapter 111.4. D= 4. N = 2 Simple Supergravity 111.4.1. Introduction 111.4.2. Rheonomic solution of the N", 2. D '" 4 Bianchi identities 111.4.3. The Lagrangian of N .. 2. D", 4 supergravity
Chapter 111.5. The D.. 5, N. 2 Supergravity Theory 111.5.1. Introduclion 111.5.2. Identification of lhe supergroup and construction of its curvatures 111.5.3. Construction of the Lagrangian 111.5.4. Superspace equations of motion and on·sheD supersymmetry 111.5.5. The second order formulation and lhe contraeled version of the theory
Chapter 111.6. The Theory of Free Differential Algebras and Some Applications m.s.1. Introduction 111.6.2. The concept of free differential algebra 111.6.3. The structure of free differential algebras and some theorems
611 615 618 624 629 633 636
641 641 643
649 661 665 672 677 680 686 706 709 716
726 726 728 737
755 755 758 767 777 783
794 794 795 796
x 111.6.4. Gauging 01 the free differential algebras and the building rules revisited 111.6.5. The Sohnius-West model (new minimal N = 1 supergravity): the on·shell formulation III.S.S. The Sohnius-West model: off-shel! extensions 111.6.7. The building rules in their final form
Chapter 111.7. Supergravity in 6 Dimensions 111.7.1. Introduction 111.7.2. D.. 6 Weyl spinors and selfduallensors 111.7.3. The free differential algebra of D .. 6 supergravity 111.7.4. Construction of the model 111.7.5. Non·invariance of the space-time action and how to cure it
Chapter 111.8. D= 11 Supergravity III.S.1. Introduction 111.8.2. Free differential algebra of D.. 11 supergravity III.B.3. Extended F.O.A. and the introduction of a &-form 111.8.4. The gauging of F.D.A. revisited UI.8.5. Constructing the theory from Bianchi identities 111.8.6. The action of 0 = 11 supergravity 111.8.7. The comple~on of the action and the equations of motion Historical Remarks and References
806
810 817 826
832 832 834 841 844
855
861 861 863 866 868 873 881
897
911
PART FOUR. THE ROLE OF THE SCALAR FIELDS: cr·MODEL AND SUPERHIGGS PHENOMENON IN SIMPLE AND EXTENDED SUPERGRAVITY Chapter IV.l. Introduction
919
Chapter IV.2. Kahler manifolds IY.2.1. a-models of supergravity and complex manifolds IV.2.2. Almost complex and complex structures on a 2n-dimensional
926
manifold
IY.2.3. Hermitean and Kahler metrics IV.2.4. The differential geometry of Kahler manifolds Chapter IV.3. Coupling of N .. 1 $upergravity to n Scalar Mukiplets IV.S.1. Kahler geometry for the N", 1 coupling IV.S.2. Solution of the Bianchi identities and auxiliary fields
Iv.a.a.
Construction of the action: generalities
IV.3.4. Construction of ;flirt) IV.3.S. Construction of t:.;f
926 928
934 937
943 943 950
967 969 976
Chapter IVA. The Vector Multiplets and the Gauging of the Kahler
Manifold Isometries IV.4.1. Killing vectors and isomelries of the scalar manifold IV.4.2. The vector multiplet
983 983 987
xi
Chapter IV.S. The Super Higgs Phenomenon IV.S.t Introduction IV.S.2. The mass relation in the minimal coupling case IV.S.3. Examples and Hat potentials'
Chapter IV.S. Dualny Transformations and the Coset Structure of Extended Supergravkies IV.6.l. How to extend the symmetries of the non linear a-model to the vector fields IV.6.2. The coset structure of extended supergravities in D", 4
Chapter IV.7. The Example of the N", 3 Theory IV.7.1. Introductory remarks IV.7.2. The N", 3 vector multiplet and the G!H slrUcture of the supergravity coupling IV.7.3. SU(3,n}/SU(3) ® SU(n} ® U(l) formalism and the solution of Bianchi identities IV.7A. The Lagrangian IV.7.S. The scalar field potential Appendix A: The Scalar Potential Appendix B: The AIII Matrix and the Embedding of SU{3.NJ into the Symplectic Group
Chapter IV.S. The Supersymmelry Breaking in the N = 3 Theory and a Short Account of the N = 4 Theory IV.S.1. IV.8.2. IV.8.3. IV.8.4.
Introduction 10 partial supersymmetry breaking Features of the N", 3 theory and of its potential A short discussion of simple N", 4 supergravity A short discussion of matter coupled N", 4 supergravity
Chapter IV.9. The Directory of Supergravity Theories and the N.. 8 Model IV.9.1. Introduction IV.9.2. Classification of D", 4 supergravities and guide to the related literature IV.9.3. The N = 8 Theory IV.9.4. Results for matter coupling in D", 4 supergravities
References
997 997 1003 1013
1018 1018 1030
1033 1033 1034 1042 1051 1064 1067 1069
1077 1077 1085 1087 1093
1106 1106 1107 1115 1131
1139 PART FIVE. KALUZA·KLEIN SUPERGRAVITY
Chapter V.l. Introduction
1147
Chapter V.2. Spontaneous Compactification of D=5 Pure Gravity
1157 1157 1162 1166 1169
V.2.1. V.2.2. V.2.3. V.2.4
Spontaneous compactification of D '" 5 pure gravity Symmetries in D= 4 A preliminary example: the spectrum of M. x 5' Maxwell theory The Spectrum of M. x SI gravity
xii
Chapter V.3. Harmonic Expansions on Coset Manifolds V.3.1. V.3.2. V.S.3.
H-harmonics on GiH
Harmonic expansions in Kaluza-Klein theories Yang-Mills fields from M. x MK compactilicalions
Chapter V.4. Compactifying Solutions of D=11 Supergravity V.4.1. The D.. 4 vacuum: maximal symmelly V.4.2. Ad~ x Af1 solutions (Freund-Rubin) V.4.S. Propel1ies of !he internal space W: Kimng spinors and Weyl
holonomy V.4.4. OsP(41N} formulalion V.4.S. Diflerential operators on M7 Appendix V.4.1. SO(7) r -matrices
ChapterV5. The D= 4 Mass Spectrum in AdS' x M7 Backgrounds V.S.l. The linearized field equations of D.. 11 supergravity V.S.2. Fermion masses V.5.3. Boson masses V.5.4. Supersymmetric mass relations V.5.S. Vacuum Slabi6ty
Chapter V.S. ClassifICation of Compact Homogeneous D =7 Einstein Spaces V.S.1. Homogeneous 7-manifolds V.S.2. V.6.3.
The spaces $U(S) x8U(2) x U(1)ISU(2) x U(1) x U(1) The other D= 7 Einstein spaces GIH
Chapter V.7. The Spectra of Specific Solutions: The Seven-Sphere V.7.1. V.7.2. V.7.3.
How to compUle ~ on the G/H harmonics The spectrum of !he round $7: harmonic analysis The spectrum of the round seven-sphere: Osp(418) analysis
Chapter V.S. The M"'" spaces V.S.t V.8.2. V.8.3. V.8.4. V.8.S.
The M"" spaces Harmonics on the M""spaces The spectrum of the 8U(S) x SU(2) x U(l) irreps in the Mpqr spinor expansion Conjugation in !he longitudinal spectrum Calculation of the longitudinal mass eigenvalues
1175 1175 1182 1186
1189 1189 1192 1194 1199 1204 1217
1221 1221 1224 1230 1232 1241
1249 1249 1251 1254
1259 1259 1262 1277
1302 1302 1306 1313 1322 1329
Chapter V.9. Other Classical Solutions of D", 11 Supergravity
1343
V.9.l. Introduction V.9.2. Nonvanishing internal photon (Englert-type solutions) V.9.S. Symmetries of Englert-type solutions V.9.4. Stretched and warped solutions
1343 1343 1347 1347
Chapter V.l O. The Embedding of D = 4 S.G. into D = 11 S.G.
1355
Chapter V.11. The Chirality Problem
1360
Bibliographical Note
1367
xiii
Volume 3
PART SIX. HETEROTIC SUPERSTRINGS AND SUPERGRAVITY Chapter V1.1. Introduction
1375
Chapter V1.2. Elements of Two·dimensional Differential Geometry and of Riemann Surface Theory
1391
VI.2.1. Introduction VI.2.2. Definition of a Riemann surface: metrics. complex structures and moduli space VI.2.3. The simply connected Riemann surfaces and the unilormization theorem Vi.2.4. Deformation of the metric. quadratic differentials and the complex structure of TeichmOlier space VI.2.5. Homology bases. abelian differentials and the period matrix VI.2.S. Dehn twists. the mapping class group and its homomorphism onto Sp(2g. Z) VI. 2.7. The group of divisors and the Riemann·Roch theorem VI. 2.8. The Jacobian variety: Riemann theta functions and spin structures
Chapter VJ.S. The Classical Action of the Heterotic Superstrings and Their Canonical Quantization . VI.3.1. Introduction VI.3.2. N", 1. D = 2 conformal supergravity and the heterotic superspace geometry VI.3.3. Classica! superconformal theories and the WZW-action VI.3.4. Heterotic (i-model on a general target space and the choice of M..."", leading to a classical superconformal theory VI.3.S. Canonical quantization 01 the heterotic WZW-mode! and the superconformal algebra Appendix: Rules forthe Wick Rotation of Spinors
Chapter VIA. The BRST Charge and the Ghost Fields VI.4.1. VI.4.2. VI.4.3. V1.4.4.
Introduction BRST quantization. Abstract properties of Q Construction of Q The BRST invariant hamiltonian and the Fradkin-Vilkovski theorem VI.4.S. BRST quantization of string theories
Chapter V1.5. Quantum Determination of the Target Manifold and Kac-Moody Algebras
1391 1393 1400 1416 1428 1449 1458 1472
1501 1501 1511 1518 1531 1541 1556
1558 1558 1559 1561 1565 1566
1582
V!. 5. 1. Introduction V!'S.2. The BRST charge: cancellation of the conformal anomaly. boundary conditions and intercepts VI.S.3 Twisted Kac-Moody algebras and massless target fermions
1600
Chapter VI.S. The Polyakov Path Integral and the Partition Function of String Models
1629
VI.6.t Introduction
1582 1585
1629
xiv VI.6.2. The cosmological constant,the partition function and the Polyakov path integral VI.6.S. Operatorial evaluation of the bosonic siring partition function VLS.4. The Polyakov integration measure for the bosonic siring VI.S.S. Functional evaluation of the bosonic string partition function in the case of the torus VI.S.S. Functional determinants of the Laplacian and of the Dirac operator on the torus VI.S.7. The gravitino ghost Appendix. A Oerailed Treatment of Conformal KiUing Vectors Chapter 1V.7. Modular Invariance, Fermionization and the Particle Spectrum of Heterotic Superstrings VI.7.1. Introduction VI.7.2. Modular invarianca and GNO fermionization VI.7.3. Modular invariance and spin s1rUClUres VI.7.4. An example in 0 .. 10: the SO(32} superstring VI.7.5. A second example in 0 .. 10: the E.@E;andSO(16)@$O{16) heterotic strings VI.7.6. Examples in 0 .. 4 Chapter V1.8. Ouantum Conformal Field Theories, Vertex Operators and String Tree Ampl~udes VI.S.1. InlrOduction VI.S.2. Quantum conformal field theories and emission vertices VI.8.3. Bosonjzation, vertex operators and spin fields in the matter sector VI.8.4. b-c systems, superg!lost bosonization and the background charge VI.8.5. The covariant lattice for 0 .. 10 superstrings VI.8.S. Conjugacy classes and GSO projectors: the 50(32) example in 0 = 10 VI.S.7. Massless emission vertices and the effective theory of 0 .. 10 superslrings Chapter V1.9. Effective Supergravity Theories and the Coupling of the Lorentz Chern-Simons Term VI.9.1. Introduction VI.9.2. The algebraic basis of N .1,0 .. 10 matter-iXlUpied supargravity VI.9.3. The general solution of the 0 .. 10 super PoiIlCalli Bianchi identities VI.9.4. The H·Bianchi identity in the {O,4}- and (1,3}-sectors: determination of the H.paramelrization VI.9.5, The (2,2)- and (3,1 )·sectors of the Jf.Bianehi identity and the equations of motion VL9.S. The lagrangian of N"" 1, 0 .. 10 matter-c;oupled supergravity at 'Y =0 VI.9.7. Retrieving the superspaca constraints from the" symmetry of the Green-Schwatz string formulation VI.9.S. Bianchi identilies and off·sheU formulations of N =1, 0 =4 supergravily revisited VI.9.9. Chiral multiplets, the rmear multiplet and the geometrical interpretation of R-symmetry VI.9.10. 0 =4 Chern-Simons cohomology and the linear multiplet
1632 1646 1654 1672
1677 1692 1697
1702 1702 1707 1730 1745 1754 1760
1766 1766 1769 1787 1804 1819
1832 1838
1854 1854 1859
1865 1891 1915 1942
1959 1982
1997 2012
xv Chapter VI.l0. (2,2) Superconformal Field Theories and the Class~ication of N.. 1, D.. 4 Heterotic Superstring
Vacua ,.-/ Introduction Type II superstrings on SU(2P groupfolds Construction of modular invariants and GSO projectors for the type II superstring VI. 10.4. SU(2)3 groupfolds and superconformal field theories VI.l0.5. The h·map VI.l0.S. Emission vertices of \he massless multiplets in an N = 1 heterotic model based on a {2.2).,t intemaltheory VI. 10.7. Emission vertices of \he massless multiplets in N =2 heterotic: models based on a (4,4)... (& (2,2~ internal theory VI. 10.8. Embedding of a {2.~... into the direct sum (4.4)... fa (2.2)3,3 VI. 10.9. Classification of the 5U(2)3 groupfold reafizalions of the internal conformal field theory Vt10.l0. Details of the SU(2P groupfold construction with emphasis on bosonization Appendix Vl.l0.A. A bosonizable (2.2) vacuum of type A: VI.l0.1. VI. 10.2. VI. 10.3.
A1(1.2.3.4)t!13
2028 2028 2032
2037 2039 2050
2056 2068
2076 2078
2091
Appenidx Vl.l O.B. ABosonizable (2.2) Vacuum of Type B:
B5(1.2,3.4)fu Appendix VI.l0.C. An LRP (2.2) Vacuum of Type B: 826(1.2.3)
2094 2097
Bibliographical Note
2102
Historical Remarks and References
2117
Index
2129
PART SIX
HETEROTIC SUPERSTRINGS
and SUPERGRAVITY
... c'est une chose merveflleuse q'en laisant I'addition d'un, de deux, de trois et de quatre, on trouve Ie nombre de dlx, qui est 10 lin, Ie terme et la perfection de I'unlte Voltaire, Essal sur les Moeurs et I'Esprlt des Nations, 1756
1375 CHAPTER VI. 1
INTRODucrION
The topics treated in the conclusive part of this book are, at the time of writing, the subject of current research. Hence a systematic illustration of all the results is premature: indeed the field is in rapid evolution and our understanding of string theory deepens and broadens by the day. Nonetheless we feel that a few guidelines are by now well established and liable to no further renormalization. These guidelines constitute the modern viewpoint on string theory whose illustration we have set as our main goal in the following pages. We should also stress that our presentation is far from comprehensive. although we made our best to make it self-contained. Indeed there are many trends in superstring theory and many different viewpoints regarding what, in the theory, should be considered most important. Each viewpoint naturally leads to a different emphasis on the various mathematical aspects involved by, and on the various models contained in the theory.
1376
Our viewpoint, which inspires the presentation of the following chapters, will now be explained. It was seen in PART fOUR of this book that the low-energy phenomenology of elementary particle interactions can be described, in its main features, by an N=l chiral supergravity, coupled to a suitable set of Yang-Mills and Wess-Zumino supermultiplets. Such a theory cannot be regarded as fundamental since, at the
quantum level, it is neither finite, nor renormalizable. Furthermore its beautiful geometrical structure contains a rather large variety of data that are not fixed from internal consistency but correspond to free choices. To be specific, the free "parameters" involved in the construction of an N=l supergravity are the following: i)
the gauge group G
ii) the G-representation assignments of the Wess-Zumino multiplets (Zi,X i ). iii)
the Kahler potential G(z.z) of the scalar manifold M(scalar) spanned by the wess-Zumino multiplets.
iv) the analytic function ~(z) entering the gauge kinetic term. From this situation originates the intellectual urgency of finding a fundamental "microscopic" quant\.UII theory of which the phenomenologically viable N=l supergravity could be viewed as the low energy "macroscopic" effective Lagrangian. The dream underlying the unification programme is that the free parameters i), iil. iii) and iv) of the "macroscopiC" theory could be fixed, or at least severely constrained, by the quantum consistency requirements of the microscopic one. A proposal in this direction was provided by the Kaluza-Klein interpretation of higher-dimensional supergravities.
1371
Its success would have explained the origin of the gauge group G in terms of the isometries of the compactified manifold Md in the decomposition : (Vl. 1. 1)
Unfortunately the KaluzawKlein programme fails for various reasons of which the most prominent are: a) the persistence of a cosmological constant of enormous size, b) the absence of chiral fel'lDions in the KaluzawKlein massspectrum. c) The nonwfiniteness of the 0=11 quantum theory. This failure suggests - at least so it does to us - that the very idea of tracing back the origin of the gauge group G to the isometries of the compactified space (VI.I.I) is probably wrong. The main virtue of heterotic superstrings is, in our opinion, that they provide an alternative and completely different explanation for the origin of G. Here with a typical change of perspective the chiral nature of the effective theory is built in from the very beginning. Indeed it can eventually be traced back to the Majorana-weyl nature of the twOwdimensional supercharge, generating the worldwsheet supersymmetry against which the theory is invariant. Let us explain what we mean by this. The basic idea is that of constructing a locally supersymmetric theory in a 1+1-dimensional space, named the world-sheet (WS), whose quantum excitations are perceived as elementary particles in a D~dimensional target space Mtarget' The intuitive picture underlying such a construction identifies the ~~rld~ sheet WS with the surface spanned by a one-dimensional object (the string) while moving in the target space Mtarget' In two as in ten dimensions, Majorana-~~yl spinors do exist. Hencefore the smallest supersymmetry algebra contains just one Majoranaweyl supercharge and we call it the N=l superalgebra. Some people give
1378
the same algebra the name N=~ but we think that such a nomenclature is confusing and completely dissimilar to the classification of supersymmetry algebras utilized in all the other dimensions. Hence we stick to the N=l notation. If we go to the N=2 case there are two possibilities. Either the two supercharges have the same chirality or their chiralities are opposite and they fuse into a single nonchiral Majorana spinor. (The latter case is named N=l supersymmetry by the same authors who cal] N=~ what we call N=1.) We shall simply distinguish between a chiral (2,0) and a nonchiral (1,1) N=2 superalgebra. The general case is that of a (p,q) superalgebra containing p lefthanded supercharges and q right-handed ones. It turns out that the quantum consistency requirement corresponding to the nilpotency of the BRST charge (see Chapters VI.4 and VI.8) reduces the number of the target space dimensions D = dim Mtarget
(VI. 1. 2)
to unphysical values (VLI. 3)
D ::; 2
unless the number of local supercharges with the same chirality is equal or less then one: p$I;
q$l
(VI. 1. 4)
Hence we have essentially two kinds of viable superstrings: the (I,D), or heterotic superstrings and the (1,1), or nonheterotic s~perstrings. The nonheterotic superstrings suffer from all the diseases which plagued Kaluza-Klein supergravity. In the simplest case Mtarget is flat, its dimension is 0=10, and the corresponding effective theory is the 0=10 nonchiral N=2
1379 supergravity; this theory is nothing else but the dimensional reduction of 0=11 supergravity on the compactified manifold (VI. 1.5)
In view of this it should not be too surprising that the nonheterotic superstrings are faced with the problem of nonchiral representations in the fermion spectrum and with the problem of too small gauge groups, which, as in Kaluza-Klein theories, originate only through the compactification of the extra dimensions. For these reasons we discard nonheterotic superstrings and focus only on the heterotic ones. Here the space-time and the internal symmetries are nicely separated from the very beginning. The left-handed world-sheet fermions, which transform nontrivially under the world-sheet supersyrnmetry, are responsible for the existence of the target space gravitino, while the right-handed world sheet fermions, which behave as singlets under the world-sheet supersymmetry, are responsible for the existence of the gauge bosons and carry the internal quantum numbers of the gauge group
G. G is no longer interpretable as the isometry group of any space but it is nonetheless determined by quantum consistency conditions. These correspond to the cancellation of the potential anomalies of the quantum field theory which has the two dimensional world-sheet as support. The local symmetry whose anomalies we want to cancel is the diffeomorphism group Diff (1: ) g
of the compact, g-handled, Riemann surface Lg into which the string world-sheet WS is turned by the Wick rotation t ...
it
(VI. 1.6)
1380
the number of handles, that is the genus of the surface. being equal to the number of loops of the corresponding quantum amplitude. The group DirE (E) has a connected part OiffO (1:g), containg lng those diffeomorphisms that are continuously deformable to the identity map and a disconnected part
(Vl.1. 7)
which turns out to be a finitely generated, nonabelian, discrete group: the mapping class group. Correspondingly there are two kind of anomalies: the local conformal anomaly and the global or modular anomalies. The cancellation of the local anomaly is obtained by enforcing the nilpotency of the BRST charge and this fixes, among other things the number of the right-handed heterotic fermions. The cancellation of the global anomaly requires the exact invariance of the multiloops functional integral under the mapping class group (VI.l.7) (modular invariance). Among other things, this fixes the allowed boundary conditions of the heterotic fermions and finally decides the possible choices for the gauge group G, whose gauge bosons are created from the vacuum by the heterotic fermions. As one sees, the origin of the gauge group is still geometrical, but it is the global geometry of the world-sheet that plays a crucial role rather than the local geometry of the target space. In view of this discussion we have selected the issue of modular invariance as the central one in our presentation of string theory, the final goal being the classification of the consistent heterotic superstrings that display N=l target supersymmetry in 0=4 space~time dimensions. Each of these theories corresponds to a unique choice of the gauge group G and of the other three items appearlng in the identification card of an N=l supergravity (see Part IV). Such a classification programme has been considered in the literature by various authors with the discouraging result that the number of viable theories seems too large (of the order of billions of billions)
1381 leading clearly to an untreatable problem and. what is most disappointing, to very little predictive power. Indeed when the chapters (VI.IVI.9) were written (1987-1988) the situation concerning the plethora of 0=4 superstring models was still rather confused. At the time of checking the proofs (December 1989) the perspective is much more clear and it can be summarized as follows. From an abstract view-point a heterotic superstring vacuum with N~l space-time supersymmetry is described by the following superposition of conformal field-theories:
Vacuum .. (c -
6!c " 4)Mink. $
(c" 9, n = 21e
..
22)rnternal (VLI.8)
where c, c denote the left (right) central charges (see Chapters VI.4, VI.S, VI.8) and n, n denote the number of world-sheet (global) supersymmetries in the left (right) sector of the second addend. The first conformal field theory in the above decomposition Corresponds to the degrees of freedom associated with the propagation of the
string on the uncompactified Minkowski space-time. The quantum fields of M4 and their superpartners world-sheet fermions. As a both XIJ and WIJ are free a solvable theory.
this theory are the coordinates Xl.l of wIJ. that behave as chiral left-handed consequence of the Riemann flatness of M4, quantum fields, so that (c=6!C=4)Mink is
the other hand the theory (c .. 9, n .. 21 C" 22) Internal can be either a free or an interacting one. One just fixes the central charges (c =9, C=22) from BRST invariance, the number of left-moving supercharges (n=2) from an argument that relates such a number to the required number of target supersymmetries (N-I) and finally one imposes modular invariance. A linear combination of the two global supersymmetries is the local one. Any superconformal field theory fulfilling these requirements can be used as a viable superstring vacuum. On
However if one wants to keep the geometrical interpretation of superstring theory and i f one demands the existence of a classical
1382 world-sheet action, then the superconformal field theory (c" 9, should describe propagation on a suitable intemal manifold Mint plus the current algebra of a suitable number of heterotic fermions (see Chapters VI.3, VI.S, VI.7) coupled to gauge fields living on the same internal manifold M.Int'
n;; 21e" 22) Internal
In particular if one thinks in terms of compactification of the ten-dimensional heterotic superstring, then it is natural to assume that Mint is six-dimensional and that the target manifold (VI.I.9) is a solution of ten-dimensional anomaly free supergravity (see Chapter VI.9) i.e. of the effective theory of the 10D heterotic superstring model. Exact solutions of anomaly free supergravity leading to an Wlbroken
N=l supergravity with gauge group G" E6 • ES and chiral families of
ferrnions in the 27 (27) of E6 are obtained by choosing for Mint a Calabi-Yau 3-fold that is a manifold with 3 complex dimensions and a vanishing first Chem class, 1. e. with SU(3) holonomy. The corresponding N.. l, D=4 supergravity Lagrangian can be retrieved from the 0..10 Lagrangian utilizing Kaluza-Klein techniques and harmonic expansions on the Calabi-Yau 3-fold. A particularly nice feature of this effective theory is that its structure seems to be completely topological, namely it depends only on the topology of the compact manifold and not on its metric. It was a question debated for some time whether Calabi-Yau 3-folds could be regarded also as exact solutions of superstring theory. In order to establish an affirmative answer to such a question it was necessary to rewrite Eq. (VI.I.S) in the following way:
Vacuum
=
(c .. $
(c
6Jc = 4}Mink
$
(c
= 9,
= Ole" 13)50(10) x £
n
= 21c = 9, n = 2)C.Y. (Vl.l.lO)
8
1383
where (c .. 0 Ie .. 13)50(10) )( E
denotes the right-handed confol'lDai field
8
theory spanned by an 50(10) x £8 ~rent algebra, generated by 26 heterotic fermions and where (c=§, n::zlc::9, n=2)c.¥. is a leftright symmetric (2,2) superconformal field theory describing propagation on th~ Calabi-Yau 3-fold. In other words one had to prove that the supersymmetric a-model on a Calabi-Yau 3-fold with the SU(3) Lie algebra valued spin-connection identified with an SU(3) gauge connection embedded in fS &E 8, is superconformal invariant at the quantum level and leads to an exact superconformal quantum theory. Gepner showed that such is the case by constructing directly examples of the theory (c:: 9, n::: 21e .. 9, il .. 2)C. Y. that reproduce all the properties of known Calabi-Yau 3-folds. In this way it became evident that Calabi-Yau 3-folds shoUld be in correspondence with appropriate (c::: 9, n = 21e::: 9, ii .. 2) superconformal field-theories. Since Calabi-Yau 3-folds are not isolated spaces and fall into families labeled by continuous parameters (the moduli) it follows that also (2,2)-superconformal field theories should admit continuous deformations parametrized by the moduli. This is indeed the case with most conformal field theories: they are not isolated and admit a moduli space whose geometry has been extensively investigated by many authors over the last two years. From the point of view of the effective D=4 N=l supergravity Lagrangian the moduli are just a subclass of the full set of massless Wess-Zumino multiplets, namely those that correspond to flat directions of the scalar potential. Each point in the moduli space corresponds to a different extremum of the potential and to a different superconformal field theory. A very "important point that has been clarified in the last two years is the following: it happens very often that the moduli spaces of very different target manifolds are connected. This implies that those superconformal field theories that sit on the junction of different moduli spaces admit a multiple interpretation in terms of 2-dimensional a-models.
1384
In particular it may happen that the number of dimensions of the target manifold in the various geometrical interpretations is different. If dim Mint ~ 6 it seems that we cannot think in terms of com~ pactifications of the 0=10 heterotic superstring. yet it may happen that the superconformal field theory corresponding to our choice of Mint lies also at a specific point in the modUli space of some Calabi~Yau 3~ fold or of some other 6-dimensional variety that is a solution of N=1, D=10 anomaly free supergravity. In this case the reinterpretation of our 0=4 string model as a compactification of the D=10 heterotic superstring is possible. In the years 1986-1987, prior to the work by Gepner, the emphasis was mostly on the direct construction of 0:4 superstring models, namely on the direct construction of superconformal field theories that fulfill the necessary requirements (modular invariance, in particular) to be identified as Viable superstring vacua. In dealing with this problem a major simplification was introduced by restricting one's attention to free field theories, namely either to free bosons or to free fermions. The first choice led to the so-called covariant lattice approach, while the second led to the free fermion constructions (see Chapters VI. 7 and VI. 8). In both approaches modular invariance is dealt with in a systematic way, but the geometrical interpretation in terms of target manifOlds is lost. Furthermore the plethora of models is largely overcounted since one counts as separate cases theories that possibly sit in the same moduli space and are therefore connected by continuous deformations. Tbe deeper understanding provided by the concept of deformations and moduli space suggests that one shOuld require a geometrical interpretation in terms of suitable target manifolds also for the free boson and free fermion construction keeping in mind the possibility that such an interpretation might be a multiple one. In connection with this point we. mention that one of us (P.F.) in collaboration with F. Gliozzi has developed an approach that leads to a
1385
uniform geometrical interpretation of a large, yet manageable class of free fermion rodels as describing propagation on curved target varieties that admit a non-abelian Lie group as covering space. Specifically in this approach one starts from a (1.0) locally supersymmetric a-model on a target space: M
target
.. M
lO-2p
e [SU(2)]P 8
(VI.l.ll)
where M10 _2p is Minkowski space in lO-2p dimensions and 8 c [SU(2)J2p is a discrete subgroup of the isometry group of the group-manifold [SU(2)]P. To the choice (VI. I. II) one arrives by implementing the requirement that the target manifold should have a Lie group as covering space and yet should be compatible with space-time supersymmetry. (Actually different choices for Mtarget could be allowed in view of a recent result obtained by P. Bouwknegt and A. Ceresole (see bibliographical note).} In the following chapters we utilize this approach to construct explicit superstring models. Our purpose is indeed paedagogical and we want to show the conceptual path that leads from a 2-dimensional world-sheet Lagrangian to an effective D.. 4 Lagrangian passing through modular invariance and the explicit construction of a superconformal field theorY. In this respect the free fermion approach with its group-manifold interpretation provides a very useful and versatile tool. However the reader shOUld be aware that the group-manifold interpretation is not the only possible one. Indeed in many cases the same superconformal field theory obtained through the free fermion construction admits also an alternative interpretation in terms of orbifolds of tori or of Calabi-Yau 3-fOlds. (By orbifold one means the quotient with respect to some discrete group acting with fixed points.)
1386
Concerning the main goal of the whole programme, namely the classification of the set of N-l superstring models and the determination of the corresponding Kahler potentials we must say that, at the time of correcting the proofs (December 1989), we are much closer to it than we were at the time of writing (1987-1988): this is particularly due to the better understanding of the moduli space. However the goal is not yet achieved since there are still many loose ends. In this book the geometry of the moduli space is not discussed since its proper understanding is too recent. For a similar reason the Calabi-Yau aspects of the compactification and the Gepner's construction of the (2,2) internal theory are also not presented. In the last chapter VI.IO, which was added while correcting the proofs, we shall present the relation between the free fermion approach utilized in the previous chapters and the compactification framework based on abstract superconformal field theories. In the same chapter we shall present, with explicit examples, the classification of (2,2) compactifications one obtains by utilizing the SU(2)3 group-manifold approach. The plan of PART SIX is the following. In Chapter VI.2 we have COllected, mostly without proofs, all the elements of Riemann surface theory that are necessary to understand the functional approach to superstrings and to correctly formulate the problem of modular invariance. In Chapter VI.S we write the classical action of heterotic superstrings as matter-coupled N-1 conformal supergravity in D=2. The matter fields span a a-model characterized by a target space endowed with a suitable metric, a suitable torsion and a suitable gauge field coupled to the heterotic fermions. The supergravity fields are non-propagating and play the role of Lagrangian multipliers for a set of first-class constraints whose algebra is the so-called superconformal, or super-Virasoro algebra. we define classical superconformal theory any cHnodei of this type (nalJled heterotic
1387
in the following) whose equations of motion for a complete set of fields i ~i J and J take the form: ;.
(Vl.l.12a) (Vl.1.12b) the variable z,i being z = exp (T + fer)
i
= exp
(1: -
(VI. 1. 13a) (VI. 1. 13b)
ia)
where a is the parameter labeling the string points and Wick rotated intrinsic time.
1:
is its
In a classical·superconformal theory all the on-shell fields are either analytic or antianalytic and, upon quantization, they give rise to an exactly solvable quantum field theory, which is characterized by a pair of c-numbers called the conformal anomalies (c,c), the first relative to the analytic fields, the second relative to the antianalytic ones. As already stated, string vacua correspond to the quantum superconformal field theories whose conformal anomalies turn out to be zero if the ghost contributions are included. The quantum superconformal field theories are made out of analytiC and antianalytic fields but, in prinCiple, there is no guarantee that they correspond to the canonical quantization of a classical superconformal field theory. The counterexample is provided by the Gepner's construction of exactly solvable (2.2) theories that correspond to the quantization of o-models on CalabiYau 3-folds: these latter are not classical superconformal according to the previous definition. Indeed the nontrivial spin-connection of SUeS) holonomy forbids that the classical equations of motion may take the form (VI. 1. 12) •
1388 In our presentation we restrict our attention to thQse string vacua that are obtained by canonical quanti~ation of Classical superconformal field theories. This guarantees that the corresponding quantum theory will be exactly solvable and, furthermore, makes much easier the illustration of the path connecting the 2-dimensional Lagrangian with the N~l, 0=4 effective Lagrangian. In Chapter VI.S we show that the only classical superconformal theories have as target manifold the D-dimensional Minkowski space-times Gr/B, where Gr is a semisimple Lie group and B c Gr is a discrete subgroup of the isometry group G.r ® Gr. The classical superconformal theory associated with M
target -- G
(VI.l.14)
G being a simple group is called the heterotic Wess-Zumino-Witten model. In Chapter VI.S ~~ proceed to its canonical quantization. In Chapter VI.4 we introduce the BRST charge and the related ghost fields. The nil potency of the latter implies the cancellation of the conformal anomalies of the matter fields against those of the ghosts. This is the necessary background for the study of the next chapter where we finally arrive at the already antiCipated result (VI.I.II). In Chapter VI.S we show that the conformal anomaly cancellation condition leads to fifteen possible choices of the group manifold ~ in the case 0=4. Next we enter a detailed study of the twisted Kac-Moody algebras and of their relation with the homotopy subgroup B C Gr ® ~. The requirement that massless target fermions should exist leads. essentially, to the selection of (VI.I.II) as M .(*) target (,oj
A recent result by A. Ceresole reveals that changing the normal ordering prescription of the Sugawara stress-energy tensor a few more of the IS solutions mentioned above have massless target fermions, besides the solution (~I.l.ll). (See bibliographical note. )
1389 Chapter VI.6 is a general introduction to tne concept of partition function in string models. We define the Polyakov path integral and we show how it reduces to an integral over the finite dimensional moduli space of the g~handled Riemann surfaces. Utilizing the ~~ function regularization scheme we perform the explicit calculation of the functional determinant for the Laplacian and for the Dirac operator in the case of the torus (g =1). These are the main ingredients of the partition function in any heterotic superstring. We compare the results of the functional and operatorial approaches and we bdefly discuss the problem of the gravitino ghost determinant. In Chapter VI.7 we reduce the classification of the heterotic super~ strings we have introduced and of their particle spectra to an algebraic problem whose solution can be obtained through a well~defined algorithm. First we show that the Kac-MDody characters associated with the group-manifold degrees of freedom can be replaced by suitable Dirac determinants. In this way the partition function is completely expressed in terms of Riemann theta functions. whose transformation properties under the mapping class group (VI.l.7) are known. The rules for the construction of multi loop modular invariants are derived and some examples in 0=10 and D=4 are given. Chapter VI.8 contains an introduction to quantum superconformal theories and to their use in the construction of superstring tree amplitudes. In particular we discuss spin fields. vertex operators for the massless states and the lattice approach naturally linked to these concepts. Finally we compute some elementary processes and show the emergence of the Lorentz~Chern·Simons term in the effective theory. Chapter VI.9 is devoted to a study of the effective supergravity theories associated with the heterotic superstrings. Here the main new feature is the presence of the Lorentz·Chern-Simons terms and the higher curvature interactions. We give a detailed treatment of these problems in 0=10 and also in 0=4. our tools being. as usual. Bianchi identities in superspace
l~O
and rheonomy. In addition we discuss the microscopic origin of superspies constraints within the framework of K-symmetry. Finally the content of Chapter VI.lO, added while correcting the proofs, has already been described. In that chapter we show how the SU(2)3_approach leads to the construction of specific (2,2)-compactifications. Let us finally make some remarks on bibliography. Departing from the rule adopted in the previous five parts, here each chapter is equipped with a very few references which are actually quoted in the text, where some proofs or arguments are omitted. The only exception is the bibliographical note at the end of Chapter VI.lO. It was added while correcting the proofs and it is meant to be a guide to the most recent literature, particularly in relation with the advances in the study of the moduli spaces for (2.2) and (2,0) superconformal field theories and of their relation with the effective N=l Supergravity Lagrangian. At the end of PART SIX, the reader can also find the usual section on references and historical remarks. The bibliography included in this section is meant to be a gUide for the reader putting the history of superstring theory in the perspective we have adopted. In no way is it meant to be comprehensive or accurate. For a general bibliography and background in superstring theory up to the derivation of the 0=4 effective supergravity models, we refer the reader to the book by Green. Schwarz and Witten.
1391 CHAPTER VI.2
ELEMENTS OF TWO-DIMENSIONAL DIFFERENTIAL GEOMETRY AND OF RIEMANN SURFACE THEORY
VI.2.l Introduction In the next chapter it will be shown that the proper fxamework for the description of superstring theories is conformal supergravity in two space-time dimensions. The intuitive explanation of this statement relies on the observation that two i~ the dimension of the worldsheet (WS) spanned by a one-dimensional object while propagating in an external space-time, hereafter named target manifold (Mtarget)' The embedding of the world-sheet into the target manifold is descxibed by functions XU (~a) (XU e Mtarget' ~a e WS) which are treated as quantum fields living in a 2-dimensional world. Requiring 2-dimensional local supersymmetry leads one to introduce additional quantum fields living on the world-sheet and completing supermultiplets of which XU (~a) is the first component. These new fields are anti commuting world-sheet spinors ~U (~a).
1392 Furthermore one needs the 2-dimensional vielbein and the 2dimensional gravitino required to make the supersymmetry algebra local. At this point the beautiful peculiarities of 2-dimensional mani~ folds come into play and are responsible for the fundamental structure of superstring theory. The basic fact in a D=2 world is that the Einstein action reduces to a pure divergence. Indeed the spin connection wah has just one component (VL2.1)
the Riemann curvature becomes abelian (VI. 2.2)
and the Einstein-Cartan action of Eq. ([.4.1) reduces to
(VI. 2 .3)
The same happens in the supergravity action. Hence in two dimensions the gravitational field is not dynamical; rather, it is to be interpreted as the Lagrangian mUltiplier for a set of constraints corresponding to the vanishing of the stress-energy tensor of the matter fields. Due to Eq. (VI.2.3) the whole theory of gravitation reduces, in 0=2, to a tneory of boundary conditions. Indeed since the action of the gravitational field is the integral of a coboundary, i.e. a surface integral, the only thing that matters is its topology, which is a sophisticated word for boundary conditions. Specifically, using a path integral quantization scheme, the functional integral oveT the 2~dimensional 'metries becomes, after division
1393
by the group of Diffeomorphism (Diff), a discrete sum over the topologies, labelled by a positive integer number g (the genus of the surface), and, at fixed topology, a multiple integral over a finitedimensional complex parameter space M (the moduli space) whose g coordinates label the conformal classes of the world-sheet. The latter are the equivalence classes of world-sheet shapes with respect to conformal transformations. Performing a Wick rotation of the time variable, the superstring world-sheet becomes a Riemann surface and the whole machinery of algebraic geometry can be applied. The present chapter is meant to be a self-contained, although very much simplified, illustration of the differential and algebraic geometry of Riemann surfaces needed in the later chapters (VI.3-VI.6). In particular it is propaedeutic to Chapters VI.6 and vr.7 where it is shown how the physical mass spectrum of superstrings is determined by the cancellation of global diffeomorphism anomalies, i.e. by the implementation of the so-called multi loop modular invariance.
VI.2.2 Definition of a Riemann surface: metrics, complex structures and moduli space Let us begin by defining the main object of our study. 6.2.1 Definition. A Riemann surface r is a complex connected onedimensional analytic manifold. This means that r can be covered by a finite atlas A{U} of 0: open subsets UCL a.
(Va.eL)
(VI.2.4)
(the charts) which are diffeomorphic to open subsets of the complex plane C. A point p e U(l is therefore labelled by a complex coordinate z(a)(p) e (.
If P belongs to the intersection of two charts:
1394 (VI.2.S) then the relation between its complex coordinates in the chart U« and in the chart Ua is an analytic transition function
faa: (VI.2.6)
As
Io'e
are going to discuss further in later chapters. the relation
between the complex coordinate z on the Riemann surface t
and the
description of the string evolution can be given in the following way. At a given instant T = TO of its proper time a closed string is a loop in the target manifOld: S ... M
(VI.2.7)
target
1
described by a periodiC funCtion XIJ{O,T O) =x" (0 + 211.T O) of a suitable parameter (]. goes on the loop moves into an adjacent one so that we finally get a two-dimensional world-sheet described by the embedding As time
T
funCtion x\l(o,t). The two dimensional equations of motion imply that is made of left-moving and right-moving waves, namely, it is
aTXlJeO,T)
the superposition of a function of After Wick rotation nate
z, t
= exp
('J - '[
and of a function of
(J
+ T.
(t "'it) we can identify the complex coordi-
with the following combinations: (1 + ia)
(VI. 2. 8a)
i '" exp (1 - ia)
(VI.2.8b)
z
In this way analytic and antianalytic fields on the Riemann surface t
correspond, respectively. to left-moving and right-moving modes
on the world-sheet WS.
As discussed at length in Chapter IV.2 any n-
dimensional complex manifold can be viewed, to begin with, as a Zn-
1395
dimensional real one. Correspondingly the Riemann surface 0=2 real manifold whose line element is given by (a .. 1,2)
~
is a
(VI.2.9)
the me'tric gaB (f,;) being a symmetric 2 x 2 matrix
(VI.Z.lO)
which contains three independent components. Relying on a theorem aheady proved by Riemann [1] we know that, in a given open chart of an n-dimensional manifold, n components of the metric tensor can be written in the form of specified functions by means of a coordinate transforma~ tion. This implies that in every open subset U c r we Can dispose of two of the three functions gll' gI2 and g2Z' In particular we can always find a suitable coordinate system where (VI.2.l1a) (VI.2.lib)
Defining (VI.2.l2)
the above result means that in every open subset U C L the line element can be reduced to the following form: (V1.2.13)
In the coordinate frame ~~ere ds Z has the form (VI.2.I3) we can introduce the following complex coordinates (adapted to the metric gaaCf,;)):
1396 (VI.2.14a) -
~
z '"
1
-
. Z
l~
(VI.2.14b)
and obtain (VI.2 .15)
where 4>* .. 4>.
In any other coordinate frame related to the above one
by an analytic transformation: Z '"
f(zl)
Z" f"O metrics g!~)(~) and g!~)(~) which are not equivalent under (Vl.2.24) can nevertheless be equivalent with respect to a suitable globally defined diffeomorphism. not necessarily analytic. In other words, a coordinate transfoImation, globally defined over the topological space t. ·might be found which maps the metric g~) into the metric g (2) CIS •
In such a case we say that gel) and g(2) are confoImally equivalent and define the same complex structure. The space of complex structures is also called the moduli space and can now be precisely defined. Consider a 2~dimensional compact differentiable manifold without boundary but with a fixed topology. As we discuss in the next
1399
section. this topology is fully characteri~ed by a single positive integer number g (the genus of the surface) related to the Euler characteristic X by the follo~ing classical formula:
x=
(VI.2.26)
2 - 2g •
Let Eg denote the genus g topological space and let Met (Lg) be the space of metrics defined on r. Calling Diff (E) the group of g g diffeomorphisms on r and Weyl the group of transformations (VI.2.24J, g the moduli space Mg corresponding to genus g (i.e. the space of inequivalent complex structures on r) is defined by the formula g
Met (1: )
M .. g
(VI.2.27)
g
Diff (L ) g
®
Weyl
This space is finite-dimensional and its relevance in string theory is utmost. To appreciate this point let us stress that, as we are going to see in detail in later chapters, the classical action of any string theory has the following general form: (VI.2.28)
{~i(~)l being suitable matter fields defined on the world-sheet, and
the Lagrangian density (VI.2.29)
being invariant both under the diffeomorphisms and the \'Ieyl transformation (VI.2.Z4). These local invariances have the fOllowing consequence. If one considers the quantum generating functional
(j (0
are the
external sources):
~(j)
"
Jgig Z(g;j)
(VI. 2.30)
1400
where (VI. 2.31a) (VI. 2. 3Ib) one concludes that the partition function Z(g,j) defined by Sq. (VI.2.31a) depends only on the orbits of the space of metrics tmder the local invariance group. This means that at fixed topolOgy g the partition function is just a function on moduli space (Vl.2.27). In particular, if the matter fields ti(~) are free the functional integral (VI.2.31) is gaussian and can be evaluated in closed form; then the next functional integral (VI.2.30) over the 2-dimensional metrics reduces to an ordinary multiple integral: a tmique situation peculiar to string theory. From these observations it is obvious that we need more information on mOduli-space. in particular on its dimensionality and on its invariant volume element which is the COl'l'ect functional integration measure to be used in Eq. (VI. 2.30). To answer such questions we have to develop the theory of Riemann surfaces a little further.
VI.2.S The simply connected Riemann surfaces and the tmiformization theorem The most powerful result in Riemann surface theory is probably the uniformization theorem of Klein. Poincare and Koebe which states that up to conformal transformations there are just three simply connected Riemann surfaces, namely The compactified complex plane C v {oo}, whose topology is that of the two·sphere 52' i)
ii) The complex plane ( itself. iii)
The upper complex plane Ii: {z eH .Im z >O}.
1401 This theorem implies that any Riemann surface admits one of the above listed surfaces as universal covering space. In other words, if L is a Riemann surface and ~l(E) is its fundamental group (i.e. its first homotopy group) then one and only one of the following three is a true equation:
i)
C u {oo} E=-1T 1(1:)
ti)
C E=-1T 1(E)
iii)
H E=-1T 1(E)
Relying on this result, the classification of Riemann surfaces can be reduced to the classification of the possible representations of the homotopy group 1T 1el:) by means of fixed point free subgroups of the analyti c automorphism group of either C u {oo} or II: or H. Indeed by definition the fundamental group of any space has a fixed point free action on the corresponding universal covering space; furthermore in order to preserve the natural complex structure of the covering space the homotopy group must be represented by analytic automorphisms of the latter. In view of this let us first describe the fundamental group of a genus g surface ig and then briefly discuss the automorphism groups and the invariant metrics of ( v {co}, C and H. Topologically Eg is a 4g-sided polygon with the sides identified in pairs as in the following picture.
1402
(VI.2.32)
Each pair of identified sides represents a closed curve on 1.: which is not homotopic to zero. If we fold the polygon as prescribed by the identifications of picture (VI.2.32) we obtain a surface with g handles like the one depicted below
(VI. 2. 33)
Each of the closed curveS ai
or bi
generates a nontrivial homotopy
class of 111 (1.:). However if one considers the curve (VI. 2. 34)
1403
we see by inspection that c is homotopically trivial. Indeed c is nothing else but the boundary of the 4g·sided polygon (VI.2.32). Shrinking c continuously to is a point of the surface Eg)
an
interior point of the polygon (which shows that c is trivial as claimed.
Hence nICE) is a group with 2g generators rCa.). feb.) g 1 1 just one relation: g IT
i=l
{.1.1 r (b.) f (ai)r(b. )f(a.) 1 ,,1 1
1
It is this group which
we
1
and
(VI.2.35)
Tepresent by means of fixed point free elements of the automorphism groups of (; u {co}. C or H. We note that we have three distinct cases: IJIUSt
if g=O then 'ltl(EO) .. 0. This is obvious since EO is a two-sphere 52 whose fundamental group is indeed trivial. i)
ii) if &=1 then '!f1(E l ) is abelian with two generators. Indeed for g" I the fundamental relation (VI. 2. 3S) implies
r(a)f(b)
~
r(b)f(a)
showing that fCa}
and reb)
(VI. 2. 36)
commute.
Not t~o surprisingly these three cases match with the three choices of simply connected Riemann surfaces. Indeed, as we shall show. C u {co}, I!: and H aTe the miversal covering spaces of the Riemann surfaces of
genuses g" O. g" 1 and g:: 2, respectively. Let us describe the structUTes of II: u {co}, C and H and of their automorphism groups. Recalling Eq. (VI.2.IS) the metric of any Riemann surface can always be written as follows ds 2 .. g _dzdi zz
= exp[t(z,z)]
dzdi
(VI. 2.37)
1404 where ~(z,i) is a sUitable function. This statement applies in particular to the simply connected surfaces. We have to decide which function ~(z, i) is proper to [v {co}, to [ and to H. To solve this question we begin by observing that, at least locally, any Riemann surface can be thought of as a Kahlerian manifold. Indeed it suffices to solve the equation (VI.2.38) in order to obtain a description of the metric (VI.2.37) in terms of a Kahler potential G(z,i). Applying the results of Chapter IV.2 we can then compute the analytic Christoffel connection (see Eqs. (IV.2.S7): { Z}
zz
~Z = gag _ = d $(z,z) z zz z
i 1 zi {--I zz ::;: g 3-g z zz-
=
a-w(z,z) z
(VI.2.39a)
(VI. 2. 39b)
and the Riemann curvature tensor: R-
Z
zz·z
::;:
az{zzZ }
(VI. 2 .40)
As it happens in the case of any Kahler space a generiC covariant tensor has n unstarred indices and m starred indices (VI. 2.41)
In our one-dimensional case, however, there is only one available value for each i and only one available value for each j*. We named these values z and z respectively. This explains the notation of Eqs. (VI.2.39) and (VI.2.40). Furthermore a do~n z index can always be converted into an up z index through multiplication by g~z:
1405 (VI.2.42)
or vice versa. So any tensor can be brought to the following standard f0110 t (n,m) = t
._
(VI. 2. 43)
z... z z.... z n m
The covariant derivatives are then defined by
(VI.2.44a)
(n,m)
( a.
z - m3-41)t z z... z,z-...-z
and their commutator yields the curvature tensor.
(VI.2.44b)
For instance, on a
pure (n,O)-tensor we find
! [V_,v 2
:t
jt(n,O) t
Z .. Z
= -n R.
Z
t(n,O)
Zt.Z Z.. Z
(VI. 2 .45)
The curvature scalar can be defined as follows
R" -g
zz
Z
Riz.z
(VI. 2 .46)
and reads (VI.2.47)
1406
We
conside~
now three special choices for the function
~(z.z)
(VI.2.48a) (VI.2.48b) (VI.2 .48c) If we compute the curvature scalar for the above metrics (VI.2.48) we obtain (VI. 2 .49a)
(V1.2.49b)
(VI.2.49c)
Hence the three line elements
(VI.2.S0a)
ds~O) 2
ds(_)
"
Idzl 2
1 Idzl 2 = '4 (1m z)2
(VI. 2. SOb)
(VI. 2.S0c)
corresponding to the three choices (VI.2.48) of ~(z,f) describe the three available maximally symmetric manifolds whose curvature is constant and respectively positive, zero and negative. These maximally symmetric manifolds are C u {w}, C and H respectively.
1407
Indeed the three simply connected Riemann surfaces can be identified with the following three coset manifolds 52 '" SU(2) = IC v {"'} U(l)
(VI. 2.51a)
150(2) t=SO(2)
(VI. 2.51b)
H = SU(l,l) U(I)
= SL(2.
R)
(VI.2.S1c)
U(I)
of which (VI.2.S0) are the invariant metrics. metry groups SU(2). ISO(2) and SU(1,1) of different 3-parameters subgroups contained in SL(2.C). Indeed let us recall that SL(2.C) modular cOlllplex matrices.
Note that the three isoEqs. (VI.2.S1) are three the 6-parameter group is the set of 2 x2 uni-
(VI. 2.52)
Then the additional conditions y,,-13 (VI. 2 .53)
select the maximal compact subgroup SU(2) c SL(2.C). On the other hand, the non compact subgroup SU(I,1);; SL(2.R) c
SL(2.t) corresponds to the subset of matrices (VI.2.52) whose entries are real (I "
a ;
(VI. 2.54)
Y" Y;
Finally the subgroup 150(2) c SL(2.C) is given by the unimodular matrices having the following special form:
1408
e 150(2)
B )
(VI. 2.55)
e- i6 / 2
where 6 is an angle and 8 is any complex number. The meaning of these subgroups anticipated in Eqs. (VI.2.SI) becomes clear if we recall that SL(2,() is the automorphism group Aut {( u {co}) of the extended complex plane and that it acts on q; u {oo} by means of Mobius transfonnations az + B
z~_=
yz + Ii
z'
(VI.2.56)
(By Aut (E) we mean the group of bijecti ve analytic maps of the Riemann surface r into itself.)
If we specialize (VI.2.57) to the matrices (VI.2.53), (VI.2.54)
and (VI.l.SS) we obtain the following results: The subgroup SU(2) c SL(2,«:) acts transitively on (u roo} as much as the full group SL{2,«:), but it has the additional property that i)
it leaves the metric (VI.2.S0a) invariant. transformation (VI.2.56) we have
Idz'I 2 (1 + Iz' 12)2 and when the parameters we obtain
ct,
Indeed, under any Mobius
Idzl 2
cla.z + 61 2 + Illz + (1 2)2
(VI. 2. 57)
B, y, Ii fuHU the coilst raint (VI. 2.53)
(VI. 2.58)
Hence the isometry group Isom (R" 2) of the constant curvature metric (VI.2.S0a) naturally defined on the extended complex plane ( v {~} is a proper subgroup (SU(2») of the automorphism group (SL(2,C) of such a Riemann surface
1409 Isom (R = 2) c Aut (( u {co}) •
(VI.2.59)
ii) The subgroup ISO(2) c SL(2,C) maps the complex plane ~ into itself leaving the point at infinity fixed. Furthermore 150(2) is the isometry group of the flat metric (VI.2.S0b) naturally defined on C. These statements are evident from the form of the Mobius transformation corresponding to the matrix (VI.2.5S):
z-ze i6 +B.
(VI.2.60)
We might be tempted to conclude that 150(2) is the automorphism group of the complex plane C. This, however, ~~uld be wrong since, in addition to (VI.2.60), there is another one-parameter subgroup of Mobius transformations which maps ~ into itself and leaves the point at infinity fixed. It is the subgroup of dilatations:
z-
~z,
Ae R •
(VI.2.61)
Hence the full automorphism group of the complex plane is the following semidirect product Aut (C) ,. 1SO(2)
0
lR
(VI. 2.62)
Differently from the transformations (VI.2.60) the dilatations do not leave the metric (VI.2.S0) invariant. 50 also in this case we have Isom (R =0) c Aut (G:) •
(VI.2.63)
iii) Finally one can show that SL(2.R) =SU(l, 1) c SL(2,C) is the largest subgroup which maps the upper complex plane into itself. It happens that it is at the same time the isometry group of the metric (VI.2.50c). Indeed one can easily verify that for a, a, Y. 0 e R and
z' defined by (VI.2.56) we have
1410 1m
Zl
> 0 if Im z > 0 (VI. 2.64)
- - - . --(1m z,)2
(1m z)2
This shows that SL(2,R) '" SUO,I) is the automotphism group of the upper complex plane H and furthermore that Isom (R =-2)
= Aut
(H) .
(VI.2.65)
Having singled out Aut (t u (~}). Aut (C) and Aut (H) let us discuss the possible representations of w1 (Lg) by means of fixed point free subgroups of these automotphism groups. The first observation is that every MObius transformation (VI.2.56) has three fixed points on the extended complex plane (u {oo}, as one can easily check by direct computation. Hence there are no nontrivial fixed point free subgroups of Aut (C u {oo}). COnsequently only the trivial homotopy group wI(LO) can be represented into Aut (t u {~}). On the other hand 4: v (..} is the only one that is compact among the three Simply connected Riemann surfaces. Hence C u {oo} is the universal covering space of all genuses g" 0 compact Riemann surfaces. Let us next consider the automotphism group of the plane 11:. Aut «() is the semidirect product of a rotation-dilatation (VI,2.64a)
and a trans lation subgroup (VI. 2 .64b)
All the rotation-dilatations have a fixed point at z,. 0, while the translations are all fixed-point-free. It follows that n1 (L g) can be represented into Aut (C) only by means of translations. On the other hand the translation group is abelian so that the only n1(Lg) which
1411 admits a representation inside Aut «() genus
g '" 1.
is the one corresponding to
Hence G: is a universal covering space only for genus
g" 1 Riemann surfaces, namely f0'r tori. Let us then identify ~l(rl) by two elements
with a translation group generated
(VI.2.6Sa)
where n l , n2 e Z are integers and whose ratio is not real 101/10 2 i R.
10 1 ,102
e G: aTe two complex numbers
Every choice of wI' 10 2 defines a g" 1 Riemann surface (a torus) as the set of equivalence classes of points of G: under the following equivalence relation (VI. 2. 6Sb)
We are interested in the g" 1 moduli space, namely in the space of conformally inequi valent representations of
11 1
0:1) inside Aut
(a:).
First of all we note that by an automorphism (a rotation) we can choose axes for the lattice ZWI + 1102 so that WI is positive real. Furthermore by another automorphism (a dilatation) we can choose wl =1. Finally by a reflection we can arrange that 1m T > 0 where we have defined
T ==
wzlw1•
In this way for each ehoi ce of a complex number
T
in the upper complex plane we have a torus realized as a parallelogram in C with vertices at
0, T, 1,1 + 1 and identified opposite sides:
't + 1
:----
(VI. 2.66)
o
1412
Recalling the discussion of the previous section it remains to be seen whether, by means of a diffeolJlOrphism, two tori corresponding to two different values of T might still be equivalent. The cOlDplete answer to this question involves knowledge of the structure and of the action on t of the mapping class group (defined few lines below) and ~111 be given in the following sections. For the time being We can conclude that the g =1 moduli space 141 is some space of complex dimension one whose covering space is the upper complex plane of the parameter T (called hereafter HT). Indeed Ml is to be identified with the following quotient (VI.2.67)
where Diff denotes the full diffeomorphism group (global and local). In the next sections ~~ will show that the variable T is insensitive to the local diffeolJlOrphism (those which can be continuously deformed into the identity map) but feels the global ones (those which are not homotopic to the identity map). Hence calling DiffO (tg) c Diff (kg) the normal subgroup of the diffeomorphisms connected to the identity and defining the !ll8PPing class gro!:!!? by: M(L ) g
=
Diff ('1: ) S Diff (I:)
o
(VI. 2.68)
g
Eq. (VI.2.67) can be rewritten as (VI.2.69)
Then it turns out that M(k1) is a discrete group with a finite number of generators, whose action on H, has some fixed points. This implies that Ml is a compact space with a nontrivial topology and also some singular conical points. The covering space "T' however, is fairly Simple and has a complex structure which, being respected by the mapping class group M(k 1) , is inherited by the moduli space MI'
1413
In the next section we show that the pattern we have outlined for the g =1 case is cOlDlllon to all genuses. Indeed for any genus g the moduli space M is given by the quotient g
M =
g
Teich(g) M(kg )
(VI. 2.70)
where Telch(g) (the Teichmuller space) is a simply connected complex manifold whose points correspond to those deformations of the metric which are unequivalent under DiffO (t), and where M(L) is the g g mapping class group of the genus g surface. The complex structure of Telch(g) is inherited by Mg' Actually on Teich (g) one can even define a KShlerian metric which is invariant under the action of the mapping class group M(r) (the Weil-Petersson metric). As a conseg quence of its invariance this Kahler metric is well defined on the moduli space Mg and can be utilized to define its natural integration measure. The most basic information on the moduli space is obviously its dimensionality which, in view of our previous discussion, coincides with the dimensionality of the Teichmuller space dim Mg
= dim Teich (g)
•
(VI. 2.71)
In the g = 1 case we were able to deduce the value of dim Mg by counting the number of parameters needed to define the embedding of the fundamental group R1(r 1) into Aut ((). up to conjugation:
The same method can be applied to the higher genus surfaces. Let us then complete our discussion of the uniformization theorem in the case g ~2.
Since C v {~} and ( have been shown to be the covering spaces for the g =0 and g;: 1 surfaces, respectively, it follows that the
1414
higher genus surfaces are all covered by the hyperbolic upper complex plane H whose automorphism group is SL(2.:R). Let us then consider the group of MObius transformations (VI.2.56) where the parameters a, 13, Y. «5 are taken to be real. To signify this fact they will be denoted by the corresponding Latin letters:
Z .... 1: 1
az + b =--. cz + d
(V1.2.73)
Furthermore since the common sign of a, b, c, d is ilDlllaterial we can actually restrict our attention to the group PSL(2,R)" SL(2,R)/{1, -I}. The elements
T -_(a.b) c , d
e SL(2.R)
(VI. 2. 74)
can be divided into three classes according to the following rule: i) T is said to be elliptic if by means of a Similarity transformation (T'" ATA-I, A e SL(2,R» it can be reduced to the form
e Sin e
COS
(
-
sin cos
e) e
(VI.2.75)
ii) T is said to be parabolic if it is similar to a translation
(VI. 2. 76)
iii) T is said to be hyperbolic if it is similar to a dilatation
(VI. 2. 77)
1415 One can establish a simple criterion to decide the elliptic. parabolic or hyperbolic character of any group element T by looking at the trace of T which is a similarity invariant. One easily finds the fOllowing result: If
Itr 11 '" la +dl ..
IE
d2f;
Ig(~) gaB(s)t~mt~m
(VI. 2. 85)
g
then we consider the action of an infinitesimal diffeomorphism on the metric. We find (VI.2.86)
where (VIo2.87)
is the covariant derivative. The action of an infinitesimal Weyl transformation on instead given by
gaB(~}
is
where 6.(f;) is an infinitesimal function. My deformation 6gae can always be deCOillposed into a trace (or Weyl) part and into a traceless part (VI.2.89) where
(VI.2.90a)
{VI. 2. 9Gb)
If we apply this decomposition to the deformation induced by an infinitesimal diffeomorphism we get:
1419 (VI.2.91a) (VI.2.91b)
where PI is the following operator (VI.2.92)
mapping the space of vectors into the space of traceless symmetric tensors. We can illllllediately verify that the Weyl deformations of the metric
are orthogonal to the image of the operator P1 : (VI. 2 .93)
Indeed we have
'e
have shown is the following identification (VI. 2. 117)
In other words the space of quadratic holomorphic differentials is identical with the space of those infinitesimal deformations of the metric which can be obtained neither from a Weyl transformation nor from a diffeomorphism. This shows that H(2) (1:) is nothing else but g TeichmUller space. Indeed combining Eqs. (VI.2.97), (VI.2.99) and (VI.2.ll7) with the obvious identification 1m PI
= r(Oiff)
(VI.2.ll8)
we obtain T{Met) " TeWeyl)
6)
T(01ff)
G)
H(2)
(VI.2.1l9)
\>nich composed with (VI.2.80) proves our statement. In the previous section, by use of the uniformization theorem, we sho\>'ed that Teichmuller space has 6g - 6 real coordinates for g ~ 2. The identification Teich(r) g
= H(2)(L) g
(VI.2.120)
implies that the linear space H(l)(r) should have complex dimension 3g-3 when g~2. Similarly, for g8=I, H(2) (Lg) should have complex dimension one and it should vanish for g" o.
1425 We will show that this is indeed the case by applying a very fundamental theorem to be discussed in later sections of this chapter: the Riemann-Roch theorem. The identification of Teich (L)
with the OQmplex linear space
H(2l (Lg) provides the natural COmpl!X stl'Ucture of Teichmilller space
annotmced in the previous section. Before closing this section. let us once more revert to the real coordinate notation and consider the variation of the curvature scalar R tmder an arbitrary variat:i:on
ogaa
of the metric gaS'
Given the Riemann tensor
(VI. 2.121)
one can easily veri f1 that in 0=2 the Ricci tensor
(VI. 2.122)
is proportional to the curvature scalar (VI.2.l2S)
where (VI. 2 .124)
The proof Qf this statement is very easy.
Since
(VI. 2 .125)
is antisYlDllletric in
(8
++
a)
and
(y ++ 6)
we can write as follows
1426 {VI. 2. 126)
from which Eq. (VI.2.l2S) immediately follows.
Incidentally Eq.
(VI.2.123) shows that in D=2 every metric is an Einstein metric corresponding to the triviality of the Einstein action (VI. 2. 3). Under the replacement gaa ..... ga/3 + 6ga/3 the variation of R is given by (Vl.2.l27) where, furthennore.
(VI. 2 .128)
With straightforward manipulations one obtains the final formula for the variation of R
This fo:rmula has some very important consequences.
Let us consider a
constant curvature metric R{~) " k
(VI. 2. 130)
and the variation of R induced by a diffeomorphism (VI.2.131)
Utilizing the identity (VI.2.132) and inserting (VI.2.1SI) into (VI.2.129) we find
1427
oR = - lkVot 2 k - - Vot 2
+
VIlV Vot - !'rprl'v to - lv¥v t " 11 2 (XI" 2 }.I IX
k k;- k V·t + (- - -)'\lot" 0 2 4 4
(V1.2.133)
+ -
which shows that the space of constant curvature metrics is invariant under diffeomorphisms.
Similarly if
ogaB
is a traceless divergence-
less TeichmUller deformation, fol'lllllla (VI.2.123) shows that oR= 0 also in this case. Consequently the subspace of constant curvature metrics is invariant under both the diffeomorphism and the Teichmllller deformations.
The
deformations which move away from the constant curvature metries are just the Weyl transfomations. Calling MetR=k the slice of the space of metrics selected by the condition (VI. 2.130) we can draw the following picture
TtWey11
(VI.Z.134)
In the path integral approach to string theories (discussed in Chapter VI.S) one bas the problem of constructing the orbits of the space of metrics Met under the gauge group \'Ieyl
®
Diff.
The above discussion shows that the We),i gauge is easily fixed by
the condition (VI.2.I30).
1428 In particular one can choose k,. 2, k =0 and k .. -2 for the three cases g" 0, g =1 and g ~ 2. In this way in the standard complex coordinate z the metric is always given by Eqs. (VI.2.S0) and the only freedom which is left resides in the relation between the standard complex coordinate z and the real ones ~l. ~2 (VI.2.13S) Under a diffeomorphism f.(j .... F.fl + t fl the fmctioRs according to the rule (Lie derivative)
F and G change
(VI. 2. 136a)
(VI.2.136b)
However not every deformation of(F.)
and oG(F.)
can be traced
back to a diffeomorphism. Those deformations of F and G which do not admit the representation (VI.2.136) are the Teichmlliler deformations: they change the complex structure of the surface.
~
Homology bases. abelian differentials and the period matrix
So far we have studied the metric and the homotopy properties of the Riemann surfaces. We consider next their homology and cohomology. In local real coordinates a complex valued l·form on 1:g is given by the following formula (VI.2.137) Its Hodge dual related to those of
*w is the I-form whose components *w(l (F.) are
w by the following equation:
1429 (V!.2.138)
where gBY is the metric and ea8 is the Levi-Civita tensor (VI.2.139)
When the coordinates are eomplexified the local expression for w becomes
w = to:t
dz +
w-z di
(VI. 2. 140)
where w
z
,,!2 (w 1 - iw2)
(VI. 2 . 141a)
(VI.2.141b) Furthermore if the metric gaB is conformally flat the dual form 'w is given by *w " -i wz dz + i
z di .
to-
(VI. 2.142)
Since we can always choose conformally flat metries Eq. (VI.2.142) can be used as the definition of the Hodge dual. We can now compute dw and d*w and obtain dw" ( 0
(VI. 2.194a) (VI. 2. 194b )
which express the positivity and symmetricity of r. Since 'Y I is again a harmonic 1-form we can write it as a linear combination of YJ : (VI.2.19S) The matrix K whose entries are kIJ represents the Hodge duality operation in the canonical homology basis so that it necessarily fulfils the following condi tion (VI, 2 .196)
In this way we obtain (VI. 2.197)
which, in matrix notation, can be written as follows
r " - KJ
(VI. 2. 19B)
Writing K in block form
(VI.2.199)
1442 we obtain
(VI.2.200)
and we have the identifications: (VU.20l) which combined with Eq. (VI.2.l96) imply
(VJ.2.202)
Let us now introduce the following 2g holomorphic I-forms: (VI.2.203) Defining the matrix (VI.2.204) and by explicit calculation we obtain (VI. 2.205) Defining then the following set of g holomorphic differentials {VI.2.206) We can easily veri fy that (VI. 2.207a)
1443
(VI.2.207b) where the matrix
n =:
n is defined as follows:
(.a+ i ~)9J.l
(V1.2.208)
in terms of the blocks (VI.2.194) of the matrix (VI.2.191).
6.2.15 Definition. IT is called the period matrix of the Riemann surface I: . g
n
The fundamental properties of the period matrix are the following: is symmetric and its imaginary part is positive definite, i.e.
n.. ; ; n..
(VI.2.209a)
1m n > 0
(VI. 2. 209b)
1.J
)1
Both properties (VI.2.209) are an immediate consequence of Eqs. (VI.2.194), (VI.2.202) and the definition (VI.2.20B). What we have shown is that given a canonical homology basis we can always choose g holomorphic differentials whose "periods" along the a-cycles are given by Eq. (VI.2.207a) Once this normalization has been chosen the period matrix along the b-cycles is uniquely defined (in the chosen homology basis) and reflects intrinsic properties of the Riemann surface under considera· tion. An obvious question is the following.
If we perform a transformation from a canonical homology basis to another one, how does the period matrix IT change under the corresponding symplectic modular group matrix A e Sp(2g,Z)? The answer is simple. In the notations of Sq. (VI.2.187) we have:
TI' : (C
+
Dll)(A
+
BIT)-l
(VI.2.210)
1444 Let us prove the above result.
into a 2g-vector Yr
(CLi'Si)
ing cycles
We have arranged the hatlJlOIlic I-fol1llS
Similarly we can arrange the correspond-
(al.b i ) into another 2g-vector CI , We have the normaliza-
tion condition (VI.2.21l)
The new cycles
(ar bi) are given by the Sp(2g.Z) transformation
(VI.2.212)
Then we obtain (VI.2.213)
where
An
are the matrix elements of the Sp(2g.Z) matrix (see Eq.
(VI. 2.187)) . T yi =(A-1 )IR 'YR
If we redefine
JC·
Yj
= °IJ
we get (VI. 2.214)
.
r
Correspondingly since
fa.'),. = Akl'aJ It
r
'j
+
Bk
J ~.J
l'b
r
= (A
+
SU)k'
(VI.2.21S)
J
in order to rest01.'e the standard normalization condition (VI.2.216)
we must set
1445 (VI.2.217) so that
fbi k
~j " Ckr fa
tj
tj"
+ Dkr Ib
r
r
[(C + Dfi)(A + Bfl)-1I k;
(VI.2.218)
which proves Eq. (VI.2.2l0). It is clear that two period matrices II'
and II which are
related by an Sp(2g,Z) transformation are equivalent and describe the same Riemann surface.
The next natural question which arises is then
the following. If we call upper Siegel plane M(g) the set of complex gxg matrices fulfilling conditions (VI.2.209) and we divide it by the action of the symplectic modular group (VI.2.219a) we may ....,onder whether the space Ag one obtains in this way is not by any chance the moduli space M of the genus g Riemann surfaces. g
In
other words the question is whether, given, a matrix IT which fulfills Eq. (VI.2.209) we can always find a suitable Riemann surface I
and a g suitable homology basis such that IT is the corresponding period matrix. On a simple dimensional argument the answer to this question is clearly negative. for genuses g > 2.
Indeed the complex dimension
of the upper Siegel plane and hence of Ag is dim A "dim H g
",'bien for
g
".!..2 g(g + 1)
(VI. 2. 219b)
g > 2 is always bigger than the dimension of the moduli space
Mg' the latter being 3g - 3. Hence for g >2,
M is injected into a proper subspace of g
1446 M ..... A
g
g
= Hg/Sp(2g,Z)
(VI. 2 .220)
whose characterization has been, for a very long time. an outstanding problem in Algebraic Geometry (the SChottky problem). The solution was apparently fOWld in the course of last year [2].
For g =1 and
g = 2. however. the upper Siegel plane has exactly the same dimension as the moduli space. (VI. 2. 221a)
dim M(l) = dim H(l) = 1
dim N(2) = dim H(2)
=3 .
(VI.2. 221b)
Hence in these cases the entries of the period matrix can be utilized as a parametrization of moduli space. The case of the torus
(g =1)
can be worked out in details and
it is very instructive. In picture (VI.2.66) we saw that a torus can be viewed as a parallelogram in C with vertices at 0,
'!.
I,
'! + 1
and identified
opposite sides. The complex number Let ~l e lO.2~] of the torus.
identifies the complex structure of the
'!
torus and is its modulus.
This is seen in the following way.
and
(2 e
[O.2~]
be the two real coordinates
We can introduce the complex variable z through the
following relation (VI.2.222)
The a and b cycles of the canonical homology basis are easily identified.
i_____7 a
(VI.2.223)
1447 Consider the fOllowing I-forms: 1 dE" a .. '01 211
1 (d= --.1 t
a = -2111 d'o2co
1 " --_- Cdz - dZ)
(1-1)
./
j. -d - 1 z)
(VI. 2.224a)
(VI.2.224b)
(1-1)
using the definition (VI.2.142) of the Hodge dual and the obvious relation d2z =d2z=O. we see that (l and B are both closed and coclosed. Hence they are hamonic. Furthermore when we integrate a along the cycle a:
z
= ~l'
~1 e [0,211]
and when we integrate
(Vr:2.22S)
B along the cycle b: (VI. 2.226)
in both cases the result is one.
Hence a and B are the representa-
tives of the canonical cohomology basis.
Considering their duals: (VI. 2. 227a)
*6 .. -
~
(VI. 2 .227b)
(dz + di)
(T - T)
we can compute the matrix
r
of Eq. (VI.2.193):
(VI. 2.228a) T -1
a"
(a.S) ..
If
(l",
*6"
l(l+_f~ (1 - t)
If
dz"dz" -i 1+~ T - '[
(VI. 2 .228b)
1448 (VI.2.228c)
610. .. :II'
(8.8) =
II fL. *8 = ~
21 (T - 1')2
J dZ"dz- = -2i- .
(VI.2.228d)
(T - 'f)
Inserting these results into Sq. (VI.2.20S) we find (VI. 2.229) Hence in the case of the torus the llIOdulus paramet ri dng the complex structure coincides exactly with the period mattix. This identification has an immediate implication for the problem which was left unsolved in Section VI.2.3. There we considered the question whether all values of T corresponded to inequivalent tori or whether there were some identifications in the upper T-plane induced by suitable diffeomorphisms. We anticipated the result that the variable T is insensitive to the local diffeomorphisms (those which can be continuously deformed into the identity map) but that it feels the global ones (those which are not homotopic to the identity map). We concluded that the moduli space M1 is equal to the upper complex plane eli vided by the action of the mapping class group MeL!). The structure of M(Ll ). however. was not discussed. The identification (V1.2.229) of the modulus T with the period v tells us a lot about MeLl). Indeed two periods 11 and '1ft differing by a symplectic modular group transformation (see Eq. (VI.2.210» '1ft
dll +_ C =_
(VI.2.2lOa)
b1l + a
b)
a , ( c d
e Sp(2,Z)
= SL(2,1)
(VI.2.230b)
are equivalent since they just correspond to two different canonical homology bases on the same surface.
1449 Correspondingly, values of T differing by the modular transformation (VI.2.230) must be equivalent under suitable global diffeomorphisms. This shows that M(L 1} must at least contain SL(2,Z). This leads to the topic of the next section where we consider the general structure of the mapping class group and its representation on the homology bases.
VI.2.6 Oehn twists, the mapping class group and its homomorphism onto Sp(2g,l) The mapping class group was defined in Eq. (VI.2.68). A deep theorem which we do not prove here [31 states that every nontrivial equivalence class in Di Ef/Di ffO can be represented by a Dehn twist around a suitable non contractible loop eeL. The Dehn twist is g constructed as follows. Given c, let us consider a neighborhood of c that is topologically equivalent to a cylinder. Let us now cut Lg along c and keeping one of the edges of the cut fixed let us twist the other by a 2~ rotation and then glue the edges together once more. In this way every pOint of the original surface is associated to a point of the new surface in a way which is smooth and yet clearly not continuously related to the identity map. A useful set of generators of M(L) is given by the Dehn twists around the 2g + g - 1 cycles g displayed in the following picture:
(VI. 2.231)
The cycles ai' bi correspond to the standard canonical homology basis and wind around the i-th handle. The cycles
1450 (VI. 2.232)
connect instead two adjacent handles. This choice of the mapping class group generators shows that there is nothing qualitatively new in M(E) beyond g =2. g This observation will be very important in Chapter VI. 6 when we discuss multiloop modular invariance of the heterotic superstrings. Let now D
c
be the Oehn twist associated to a cycle c.
D
c
is
a diffeomorphism and as such it leaves the intersection matrix (VI.2.169) invariant. Indeed the entries of J. being integrals of l·form wedge products, are manifestly diffeomorphic invariant. Hence the action A(D) of D on the homology basis must be
c
c
given by a symplectic unimodular matrix
A{Dc) E Sp(2g,I) .
(VI. 2.233)
In fact the set of matrices A(De ) generate all of Sp(2g.I). However a lot of info:nnation is lost when passing from 0 to c its matrix representation A(Dc )' Indeed, a Oehn twist around a homo· logically trivial curve, although nontrl vial as a global diffeomorphism.
yet does not affect the homology class of any curve and, as such, it maps to the unit matrix. The twists around the hOlllOlogically trivial cycles generate a nOl'Jllal subgroup of the mapping class group Tor(E ) c M{l: )
g
g
(VI. 2.234)
called the "Torelli group". The period matrix IT is insensitive to the action of the Torelli group and it just feels the transformations of Sp(2g,Z) which is nothing else but the following quotient:
Sp(2g.Z) ='
M(l: ) g
Tor (Eg)
(VI.2.23S)
1451 At. we are going to see in later Shapters. in most of the applications to string theory the structures one considers (spin structures for fermions and bosons) are
insensit~ye
to the Torelli group and one has
just to implement the invariance under Sp(2g,Z). This is a very fortunate situation since the structure of the Torelli group is not known for general surfaces. As an illustration of the general theory and also in view of its relevance in later chapters. let us now discuss in some detail the g" 1 mapping class group. In the previous section we pointed out that the identification of the period n with the modulus T implied the relation (VI. 2. 236)
where PSL(2,Z)
is defined by
PSL(2.Z)
SL(2.1)
The normal subgroup 12 generated by the matrix
(VI. 2.237)
(-1o -10)
has to be
factored out since the two matrices
(ac b), (-a-c ,, -b) d
(VI. 2.238)
-d
correspond to the same MObius-like transformation (VI.2.230) on the period n.
In this section we want to show that, actually, PSL(2,Z) is the full mapping class group in the g" 1 case. First we observe
that if the matrix (VI. 2.239)
is an element of SL(2,lj the same is true of the matrix
1452
-b)
a -c
A'1 '" (
(VI.Z .240)
d
and of the matrix
(VI. 2 .241)
Therefore the transformation (VI. 2.230a) can be viewed as a MObius transformation associated to the element (VI.2.241) of PSL(2,Z) according to the standard rule (VI.2.56). Next we state a theorem whose proof. which we omit, can be foWld in the mathematical literature (see for instance Ref. [4]):
6.2.16 Theorem.
The group PSL(2,Z)
is generated by the follOl>1ng
two elements:
(VI.2.242)
which obey the following obvious relations: 5
2 = 1,
(TS)
3 '" 1 .
(VI. 2.243)
From the point of view of Mobius transformations respectiv~ly
S :
S and T correspond
to the following substitutions:
1. .... -
T : 1. ....
(VI. 2.244a)
llr. ::: r. S
1: +
1 ;: 1.r
(VI.2. 244b)
•
Taking this into account. the division of the upper i-plane 1m i > O} by the action of PSL(2,1)
(t e lit ....
can be easily performed.
1453
First, since PSL(2,Z) c PSL(2,1R)
(VI.2.24S)
it is obvious from the discussion following Eq. (VI.Z.73) that H, is mapped into itself by PSL{2,Z). Secondly. with rather simple manipulations one can show that the following region A c H T
(V 1. 2. 246a) (VI.2. 246b)
(VI.2.246c)
is a fundamental region for the modular group PSL(2,Z).
By fundamental
region one means the fOllowing. Given any two interior points " 1, "2 e A no element Al e PSL(2,Z) can be found with respect to which '1 is equivalent to '2' That is, no matrix (~~) e PSL(2,1) can be found such that ar z fob 1:1 " - - . c'2
(VI, 2.247)
+d
Conversely, if T e Hr is any point outside A then we can always map it into a point ,e A belonging to the fundamental region by means of a suitable PSL(2,Z) transformation. Because of its very definition the fundamental region A is the set of equivalence classes H/PSL(2,Z). In Section VI. 2.3 we argued that H, is the Teichmiiller space for the tori. Hence if PSL(2,Z) is the mapping class group it follows that the fundamental region A, (shaded in the picture (VI.2.248») is the g" 1 moduli space MI'
1454
(VI. 2. 248)
-1
-t
o
To be precise we still have to prove that HT is the Teichmuller space Teich (b l ). Indeed, what we were able to show in Section VI.2.3 is that each T € HT defines a lattice in the complex plane C and hence a torus, the lattice group being the representation of ~l(rl) inside Aut (C). We still had to check whether any two different T'S could be equivalent with respect to some diffeomorphism. In Section VI.2.3 we stated that this is not the case if the diffeomorphism belongs to DiffO and hence we concluded that H1 :; Tei ch (1.). Let us prove that the above is indeed a true statement. This is fairly easy by using the notion of the quadratic differential introduced in Section VI.2.4. Consider an infinitesimal displacement of the variable
T
1455
(VI. 2.249) and insert it into Eq. (VI.2.222).
ot
-
i --(1.-1.) 1.1-1.-2 Im T
We get (VI.2.2S0)
from which follows dz I
,.
d1.
-!. .l!... (dz - di) 2 Im
t
(VI.2.251)
and (VI.2.252) where \.I
zz
6T = - -i2 1mT
(vr. 2 .253)
As we see the variation of the metric associated to an infinitesimal T-shift is a quadratic differential which is orthogonal to any variation induced by local diffeomorphislllS. Hence Hr=Teich (E I ) as claimed.
To show that PSL(2,Z) is the mapping class group we must simply show that its t~~ generators can be associated to the two Dehn twists existing on the torus (see picture (VI.2.23I». We note that in view of Eq. (VI.2.243) TST and T form an equally good basis of generators as the basis spanned by S and T. Indeed we have (VI. 2.254)
and hence
1456 . (V1.2. 255)
The Dehn twists associated to the cycles a and b of picture (VI.2.223) are respectively associated to the generators T and 1ST. This is easily seen on the homology basis. Consider the torus marked with the cycles a and b
(VI. 2. 256)
and let us perform the Dehn twist along the cycle a. picture (Vr.2.2S6) becomes the following:
After the twist,
(VI. 2.257)
This means that under Da a goes into itself while b goes into . a + h. Hence the couesponding symplectic modular transformation on the homology basis is
1457
(V1.2.2S8)
which, recalling Eq. (VI.2.230). yields the following transformation on the period 1T =t : T' .. t +
1 .
(VI.2.259)
This is just the action of T. Similarly the Dehn twist along b yields
(:)' . (: :) (:) which on the period
11"
=T
(VI.2.260)
corresponds to the TST transformation:
t .... _ T _
(Vl.2.261)
t +1
In obtaining this result we have utilized the equality of the period 1T with the modulus r, proved in Eq. (VI.2.229). This is not necessary; Eqs. (VI.2.259) and (VI.2.260) can be obtained also independently. Consider once more Eq. (VI.2.222). The two Dehn twists correspond to the following two global diffeomorphlsms:
(VI. 2. 262a)
(VI.2.262b)
Under the first transformation ,,;e have (VI.2.263a)
1458
while under the second we find (VI.2.263b)
Hence if we call ds 2 (,) the line element in the metric (or complex structure) associated to , and denote by primes the new quantities after the global diffeomorphism has been performed, we can write
= dz' (,)dZ'(T) = e$("A)dZ(A(T)di(A(T») =
ds'2(T)
= e~(TJA)ds2(A('))
(VI. 2. 264)
where for the first transformation we have A(t)
=t
+
1
J
~(r,A)
=0
(VI.2.265a)
while for the second we find: T
A(T) = - -
T+l
Vf(r,Al
= 1911 + 11 2
.
(VI. 2. 265b)
In both cases we have shown that the metric associated to T + 1 or T is in the same conformal class as the metric associated to T. T+1 Indeed it is related to it hy a diffeomorphism plus a Weyl transformation.
VI. 2.7 The group of divisors and the Riemann-Roch theorem In this section we shall introduce the concept of divisor and state the Riemann-Roch theorem by means of which we can compute the dimension of the space of holomorphic q-differentials. In particular we verify that the number of Teichmiiller deformations (= quadratic differentials) is
1459
1/
Teichmi.iller deformations
,~f
o
for g" 0
1 for g" 1 3g - 3 for g?-2
(VI.2.266)
as claimed several times prevlously. The idea of divisors originates in the elementary properties of meromorphic functions on the compactified complex plane, that is on the g .. O Riemann surface.
rational function
As everybody knows a meromorphic function is a
R(z), that is, the ratio of two polynomials: (VI.2.267)
and it is determined up to a multiplicative constant when we assign the locations and orders of its zeros and poles. We can say that it is determined by its divisor (VI.2.268)
where
~1 ••• zn
are the values of z for which R has either a zero
or a pole. and (11 e Z are the corresponding orders. If Cli ~1 if (h) > 4'2' This shows that we have a linear injective mapping (V1.2.293)
of the vector space Let2) into the vector space L( 1). Hence we can write
~:
i (l) "
dim !l(I) ,. g •
(VI. 2. 303)
Combining Eq. (VI.2.303) with Eq. (VI.2.292) we have g = 1'(Z
-1
).
(VI.2.304)
On the other hand. from Eq. (VJ.2.292) we also obtain i (Z) = 1'(1) = 1 •
(VI. 2. 305)
Indeed, by definition. r(l} is the dimensionality of the space of functions everywhere holomorphic on r. Such functions are just the g
1467 constants so that 1'(1) '" 1.
Inserting Eqs. (VI. 2. 304) and (VI. 2.303)
into Eq. (VI.2.288) we obtain the proof of our theorem. We illustrate this important theorem with a couple of simple examples. Consider the g =0 Riemann surface. Le. the compactified complex plane C v {co}. and the following meromorphic differential: III
= _z_dz
•
(VI. 2. 306)
z;-a
It has a zero at a double pole.
Z =0
and a simple pole at z" a.
At infinity it has
Indeed performing the standard transformation
z=-t1
(VI.2.307)
we get
(VI. 2.308)
~ilich
shot;s that at z ="" (t '" 0) there is a double pole. Hence we can write the divisor of (11)
= co-2
III
a-1 01
as follows (VI.2.309)
and verify that deg (00) " -2 " 2g - 2
Consider now the g" 1 case, namely a torus.
(VI.2.310)
A meromorphic differential
on the torus is given by II)
(VI.2.311)
= P(z)dz
where pez) is a doubly periodic meromorphic function defined on C: P(z + 1)
= P(z TT)
" P{z) .
(VI.2.312)
1468 ,
being the modulus of the torus.
P(z)
=constant.
The simplest case corresponds to
In this case the abelian differential (VI. 2. 311) has
neither poles nor zeros so that (VI.2.313)
(w) '" 1
and hence
= 0 = 2g - 2
deg (w)
(VI. 2 .314)
•
Next we utilize the Riemann-Roch theorem to compute the dimensionality of ;rq (Eg)
that is the dimensionality of the space of holomorphic q-
differentials.
To this purpose we still need a few more preliminaries.
First we observe that (VI.2.31S)
r(1fl) '" 0 if deg tfI > 0 •
This can be easily proven.
Indeed if r(tfI) were not zero there would
be at least one meromorphic funCtion than tfI,
f
whose di visor (f)
is bigger
that is.
(f)OW- 1 >
(VI. 2. 316)
which implies
o < deg fef) 'Ii -1 )
This is clearly absurd since deg (f) '" O. i{q[) =
(VI.2.317)
= deg (f) - deg tfI •
Similarly we have
0 if deg tfI > (2g - 2)
The proof is completely analogous.
(VI.2.318) If i('II)
were not zero we could
find a meromorphic.differential w whose divisor fulfills the relation (w) of-I > 1
(VI. 2 • 319)
1469
so that we would come to the relation
o < deg
(w) - deg
~
(VI. 2.320)
which is absurd since the canonical class has degree 2g - 2. On
the other hand we have:
deg ttl
=0
00
r(li/l) ~ 1 .
(VI. 2.321)
The reason is Simple. If Ii/l is a principal divisor then, up to a multiplicative constant, there is just one meromorphic function fez) such that (f) = $fl. In this case r{'iI)", 1. If IiJI has degree zero but it is not principal then no meromorphic function can be found such that (f) = $fl. In this case r(Ifl) .. o. Finally we observe that the space of holomorphic q-differentials is isomorphic to the vector space of meromorphic functions whose divisor is bigger than z-q, Z being the canonical class. In formula
The explanation of this formula is Simple.
Let (VI. 2 .323)
q
be a holomorphic q-differential and let w be an abelian differential of the first kind, that is. a holomorphic I-form w = w dz z
(VI. 2.324)
The ratio (VI. 2.325)
1470 is a meromorphic flUlction.
Since by hypothesis the divisor of ll(q)
is integral, we have (VI. 2. 326) Hence f 6 L(Z~q). between L(Z~q)
Equation (VI.2.325) induces a linear isomorphism
and Jl"q 0;) g
which therefore have the same dimension.
Using this information we can write (VIo2.327) On the other hand the Riemann~Roch theorem, Eq. (VI.2.288), implies
(VI.Z.328) Inserting Eqs. (VI.2.292) and (VI.2.302) into Eq. (VI.2.328) we obtain (VI.2.329) From Eq. (VI.2.329) we get the desired dimensionality of the various ,(q) spaces. The Teichmuller defozmations were identified in Eq. (VI.2.117) with the quadratic differentials.
Let us then fix q .. 2.
From Eq.
(Vl.2.329) we get dim ,(2) 0: ) .. 3g. 3 + r(Z) g
(VI. 2.330)
When g ~ 2 the degree of the canonical class is positive so that, in view of Sq. (VI.2.31S),
w"e
have r{Z) ,,0.
dim ,(2) (I: ) .. 3g·3 as anticipated. g yields the relation
Hence for
g~2
we have
In the g .. 1 case Eq. (Vr.2.329)
(VI.2.331)
1471
Then we observe that in genus g .. 1 a holomorphic q-differential is not only free from poles but also from zeros since its divisor has a vanishing degree. (This fbllows from Eq. (VI.2.326) and Eq. (VI.2.302» Hence i f \l (q) is a holomorphic q-differential then l//q) is a holomorphic-q-differential. This shows that (VI. 2. 332)
which combined with Eq. (VI.2.33I) yields (VI-2.333a)
Since in any case r(l)" 1 we obtain (V q
(V!.2 .333b)
e I)
and, in particular (VI.2.334)
as already shown in Section (VI.2.6). -2
Finally. in the case g .. 0 we have deg Z .. -2 so that Z a positive degree.
dim.1l'
(2)
has
This implies
0:0)" r{Z -2 ) .. 0 •
(VI. 2. 335)
In this way we have completed the verification of formula (VI.2.266) for the number of Teichmililer deformations in the various genuses. Before closing this section let us come back once more to the Riemann-Roch theorem in the fOrmulation given by Eq. (VI.2.329). Using Eq. (VI. 2.327) and fixing q .. 2 we can write dim .11'(2) (E ) -
g
dim Jl"H) (E ) .. 3g - 3 . g
(VI.2.336)
1472 The holomorphic quadratic differentials correspond. as we have seen, to the infinitesimal Teichmuller deformations. The holomorphic l-differentials are, instead, associated with the conformal Killing vectors. Indeed, recalling Eqs. (VI.2.114) we see that a confotmal Killing vector t z fulfills the condition
Raising the index with the metric gz1: we have
which shows that t Z is a holomorphic-l-differential. Hence the Riemann~Roch theorem states that 3g - 3 equals the difference between the number of moduli and the number of conformal Killing vectors. /I
moduli -
/I
conf. Kill. vectors
= 3g - 3 .
(VI. 2.339)
At genus g =0 we have no moduli and three conformal Killing vectors (the six real generators of the automorphism group SL(2,C).) At genus g'" 1 we have one modulus and one conformal Killing vector (the translation generator z,... Z + oa) . For g ~ 2 we have 3g - 3 moduli and 110 conformal Killing vectors. As we will see in Chapter VI.8, the moduli and the conformal Killing vectors can also be identified with the zeromodes of the antighost and ghost fields respectively.
VI.2.8 The Jacobian variety:
Riemann theta functions and spin
structures In the closing section of this chapter we introduce the objects which play an essential role in the construction of fermionic string theories: the Riemann theta functions. They naturally appear as building blocks of the partition function whenever the list of two-dimensional fields {~(~)}, which define the
1473 string model under consideration, includes two-dimensional fel'lllions. On the other hand, lD-fermions are a necessary ingredient if the string spectrum is required to contain space-time fermions. Hence theta-ftmctions are a vital part of superstring theories. The partition function Z(g,j) was mentioned in Section VI.2.2 and was defined as the result of functionally integrating the exponential of minus the classical action (coupled to the external sources j(~» over all the two dimensional fields {!I} living on the Riemann surface Eg" At zero external Sources j (~) =0, such an operation corresponds to calculating the detel'lllinant of the kinetic operator acting on the free fields {~i(~)}. As already pointed out in Section VI.2.2, the invariances of the classical action imply that the partition function Z(g,j) should not depend on the choice of the metric gall (~) but only on its confoTmal class. In other words, the partition function should be a function of the moduli. Correspondingly the functional integral over the two-dimensional metrics (see Eq. (VI. 2. 30)) must be restricted to the orbit-space and becomes an integral over the moduli-space M. g These statements are not complete in one respect: the boundary conditions to be assigned to the two-dimensional fields ~i(~) have not been considered. Geometrically every field in {ti(~)} is to be viewed as a cross-section of a suitable bundle constructed over the surface Eg. If the corresponding bundle is uniquely defined, like the canonical tangent bundle, then the functional integration over that particular field leads to a unique function of the moduli. If. on the other hand, we deal with cross-sections of bundles which are not unique and are specified by additional characterizations, then the partition function depends not only on the moduli, but also on the additional parameters classifying the bundles involved. These parameters are, essentially, a set of boundary conditions imposed on the field {i(~)} and require, in order that they may be given, the specification of the homology basis. So it happens that the partition function depends on the choice of the homology basis and, at a fixed choice of the latter, on a set of
1474
boundary conditions specifying a particular bundle within a .certain class of bundles. As we know from Section Vl.2.6, the mapping class group acts
nontrivially on the homology bases (to be specific. as the group Sp(2g;Z)) and. to nobody's surprise, it also transfol'llls the available choices of boundary conditions one into another. Eventually, however, the quantum generating functional (VI.2.30) must be diffeomorphic invariant. We obtain this invariance by means of a summation over different sets of boundary conditions (bundles). each weighted with an appropriate coefficient. Although each of the terms in the sum is not Sp(2g,Z)invariant, the Sum can be made such. This is the programme of modular invariance (addressed in Chapter VI.6) which leads to a determination of the possible superstring spectra, the classification of the available superstring models being in correspondence with the classification of Sp(2g,Z) invariants. The possibility of carrying through such a programme clearly depends on our control over the Sp(2g,Z) transformation properties of the partition functions. As we mentioned at the beginning of this section, in most of the cases relevant to our purposes, the fields ~i(~) corresponding to cross-sections of a non-canonical bundle are 2-dimensional spinors. We will show that on a genus g surface there are 22g spinor bundles labelled, in a given homology basis by a couple of g-dimensional vectors called a spin structure:
[~ 1
(VI. 2. 340)
whose entries are .Iz-elements (VI. 2 . 341)
1475
The spin structures [ :ba ~
1
are in
one~to-one
correspondence wi th what
is known as the characteristics of the Riemann-theta function. Indeed it turns out that there is a deep relation between the set of zeros of the Riemann theta, known as the theta-divisor, and the set of spin bundles (or spin structures) one can construct on the surface r. g
This intimate relation between theta functions and spin bundles h'; , ,"ural
[ b1 is
""A1phys> have zero norm and are orthogonal to the physi cal states. To construct these zero norm states, we need to introduce negative nonn states called "ghost states". This is efficiently taken care of by BRST quantization.
The absence of anomalies is equi-
valent to the nilpotency of the BRST operator (see next section).
Vl.4.2 BRST quantization.
Abstract properties of Q
Historically. BRST invariance was discovered as a global invariance of the gauge-fixed quantum action for gauge theories [1,2].
1560 Here we discuss it in the more general context of constrained systems. We want to find an operator Q such that Qlphys>
=0
(VI.4.S)
The operator Q generates gauge transformations, since by definition physical states are is sufficient to project out all the physical states. insensitive to its action. If we
require Q to be a gauge generator fOr all states (physical
and nonphysical), we find the condition:
(VI. 4.6)
(henniticity) . Indeed,
Q is a gauge generator if and only if
0Q < phys I!/I >
= is a generic
state) . Consistency of (VI.4.5) implies also the crucial condition:
(VI.4.8)
(nilpotency) Two physical states
!phys 1>
and
Iphys 2>
are then gauge equivalent
if
Iphys 1>
= Iphys
2> + Ix>.
Ix> =: Q!tp>
(VI.4.9)
since, in virtue of (VI.4.6) and (VI.4.8). < physlx> = 0
(VI.4.10)
<xlx>=O.
(VI.4.11)
1561 A physical operator A is defined to transform a physical state into another physical state: Alphys 1> .. Iphys 2 >
V Iphys 1> •
(VI.4.12)
QD if At is physical) •
(VI. 4.13)
This implies (=
A gauge operator B transforms a physical state into another physical state of the form Qil/I>: Blphys> :: Qil/I> In particular. operator. Theorem 1.
•
(VI. 4.14)
B is a physical operator, and Q is a (trivial) gauge
every operator G= [Q,C):t is a gauge operator.
The proof is trivial. Theorem 2.
(A,B!:!: is a gauge operator.
Proof: (AB ± BA) Iphys:> .. AQ!l/I> ± BIphys' > ..
.. Qlt4I' >
+
QDll/I:. ± Qli/I">
= Q!l/I'" >
VI.4.3 Construction of Q
The first step is to enlarge the original phase space to include odd Grassmann variables. i.e. the ghosts nA and their conjugated momenta lIS'
satisfying the (symmetric) Poisson bracket couunutations
1562 (VI.4.1S) Suppose now that the commutations (VI.4.1a) have the form
c = constants
C AS
(VI.4.16)
as is the case for gauge theories. The corresponding BRST operator Q is constructed as follows
[1,81 (VI.4.17)
Its nilpotency is due to the Jacobi identities of the structure constants
c'\c
(Exercise:
verify Q2 =0).
Moreover, the operators (VI. 4. 18)
are gauge operators (see Theorem 1 of previous section), and are some· times called "improved" generators.
They close on the same algebra
(VI.4.16) as the 4lA• It is an easy exercise to show that (VI.4.19) \'Ie note at this juncture that all the considerations of this section have
been classical, i.e. using Poisson brackets rather than (anti) commuta· tors. In the quantum theory, however, the commutations (VI.4.16) lIIay develop Schwinger terms, and these will show up as deviations from Q2 =O.
l'I'hen we deal with an infinite number of constraints +A'
as in
the case of the string (+A =Virasoro generators), normal ordering in the ghosts may cancel the anomaly in Q2 for sOllIe critical values of parameters (e.g. the spacetime dimension 0). In that case we have a consistent quantum theory only for those critical values (e.g. D.. 26 for the bosonic string, see Sect. VI.4.5). Theorem: The ghosts nA are gauge generators.
1563 ~:
consider the gauge generator (VI.4.20)
where XA are the gauge fixings of Vl.4.2. Then (VI. 4. 21)
can be inverted to yield (VI. 4.22)
proving the theorem, since ~B are linear combinations of gauge generators. In an appropriate gauge one can therefore eliminate the canonical A variables n, nA• The construction of Q can be generalited to non constant (field dependent) structure functions UCAS in (VI.4.23) In this case the algebra of the infinitesimal transformations generated by the .A t s is ~ (it closes only on the shell 4>A = 0). Indeed defining
F ~ arbitrary function
(VI.4.24)
we find (V1.4.2S) Algebras that close only on-shell do appear in the supergravity theories discussed throughout this book. The interested reader may study their BRST quantitation by using the appropriate generalization of (VI.4.17) (see Ref. [7] for a review on the BRS! operator for open algebra~).
1564 We have now the necessary lore to characterize physical states as defined in (VI.4.3). The incorporation of ghosts in the quantum theory requires an enlarged Fock space, containing ghost and anti ghost excitations. The ghost number operator U is defined as (Vr.4.26)
The physical states are then the BRST invariant states with ghost mnnber U" O. This holds in the case of finite dimensional Lie algebras. For infinite-dimensional Lie algebras, for example the Virasoro algebra, the ghost number operator U contains a normal-ordering constant, so that physical states have actually U# 0 (see later). Since two states differing by QA are gauge eqUivalent, the physical states are given by equivalence classes, where W and ~' belong to the same class if l/I- $' =QA. The equivalence classes of BRSTinvariant states X, for which Qx ,. 0
(VI.4.27)
with ghost number n such that
Ux" nx
(VI.4.28)
are the cohomology classes Hn(G,R) of the Lie algebra G (see Chapter 1.6). Q plays the role of the exterior derivative, an analogy that has deep consequences in string field theory. The BRST-invari>int states X with ghost number U", 0, Le. the states satisfying
Qx If
=0
X A
=0
(VL4.29) (VI.4.30)
1565 are of special interest. In fact. they are by themselves cohomology classes HO(G,R). Indeed, these states cannot be written as QA. since then ). should have U .. ·1 (Q increases the ghost number by 1). U has eigenvalues U=O.1,2 •.•. dim G, at least for dim GO
(",a tWa(z) + fa ta(z))} :
(VI,5.71d)
An explicit expression for the homotopy generator y e (y,y) e B can now be easily written. First note that, since B is a subgroup of Gr, any of its generators is a direct product (Vl.S.72)
The group B will. then be specified by listing its generators: B .. {()',y), (Y',y')' ... }
(VI. 5. 73)
1603 Hence we can just focus on a simple factor G. (as we have 1 already done in the formulae above) and on an element y. e G. whose 1 1 N-th power must be equal to unity: Without loss of generality we Can assume that y is the exponential of a CSA-element. So if we denote by NO the Cartan generators we can write (VI. 5. 74)
The
r~component
vector t which identifies y is named the twist
~.
Since the possible eigenvalues of NO are the weight vectors A lying in the weight lattice Aw' it follows that =1 if and only if there exists an integer N such that
l
Nt"A€Z.
(VLS.75)
The roots a are particular weight vectors, so that Eq. (VI.S.75) is true if we write a in place of A. However. there may also exist a smaller integer N< N such that
Nt •
(l
€
Z
(for every root ex) •
(VLS.76)
The number N is the same numbe! defined by Eqs. (VI.5.38) and N (VI.5.39). Indeed if Eq. (VI.5.76) holds true the group element y acts as the identity on all the states belonging to the adjoint representation. The twist vector t is used to provide an explicit representation of the boundary matriX rBA. Indeed, utilizing the Cartan basis for the JA(z} the definition (VI.5.74) for y,
Eqs. (VI.S.34) become
currents and
1604 (VI. S. 77a)
..a
~
(ze
21Ti
. a ),. exp(-2lTlt·a)C (z)
(VI. 5. 77b)
(VI .5. 77c)
(VI.5.77d)
and lead to the following mode-expansions:
(VI. 5. 78a)
(VI. 5. 78b)
As already pOinted out, the boundary conditions on the group fermions are not independent, rather, they follow from those imposed on the Kac-Moody currents. Indeed, combining Eqs. (VI.5.77) with Eq. (VI.5.44) and Eq. (VI.S.71) we obtain (VI. S. 79a)
a
1/1
2rri
(z e
.
);; exp(21Tl
("21 w - t
•
a))
0.
1j!
(z)
(VI.S.79b)
which yield the following mode-expansions:
(VI. S. 80a)
(VI.5.80b)
1605
Comparing Eqs. (VI.S.BO) with Eqs. (VI.S.63) we see that the group fermions subdivide into rank G Majorana fermions of frequency \I" w plus as many complex fermions as there are positive roots, each with frequency aa" }w~t.a. So, by comparison Iqith Eq. (VI.S.65) we can immediately write down the coboundary bF of the Virasoro algebra associated with the fermionic part of the energy~momentum tensor (see Eq. (VI.S.71c». We find
2
"
~ 16
dim G +!
r t· aCt· a ~ w)
(VI. S. 81)
2 Ct>O
We still need bB(t,w), that is the coboundary of the Virasoro algebra associated with the bosonic energy-momentum tensor (VI.S.71b). In order to calculate this number we begin by inserting the modeexpansions (VI.S.78) into the OPE (VI.S.6B) and in the following: B iIi T (z)1-I (w) " - - - Ii (w)
+ --
1
"a TB(z)Cex (w) " - -l - 2 (; (w)
+ - - ()
(z_w)2
(1. ~ 1'1)
B
B
z-w
1
Z ~
1k.
1'1
i d Ii (1'1) + reg. terms 1'1
,,(1 (j
(1'1) +
reg. terms
1 a TB(1'1) z - 1'1 1'1
+ --
1
2
(z~w)4
(z~w)2
+
(VI. 5. 82b)
W
B
T (z)T (1'1) " - - - dlm G - - - + - - - T (1'1) 2 k+CV
(VI. 5. 82a)
reg. terms .
+
(VI. 5. B2c)
The result is (VI. 5. 83a) (VI. 5 .83b)
1606
(VI. 5. 83c) (VI. 5. 83d) (VI. S. S3e)
(Vl.S.83f)
+
1- _k_
12 k + Cy
dim G(m 3 • m)15
m+n,
0 + 2bt m15
m+n.
O.
(VI.S.83g) Eqs. (VI.S.83) describe the semidirect product of a twisted Kac· Moody algebra with the associated Sugawara realization of the Virasoro algebra. The coboundary bt is the number we want to calculate. To this effect we must introduce a little bit of Kac-Moody algebra representation theory, which will be of use also in Chapter VI.7. We focus on the untwisted case where the twist vector t is zero. From the point of view of representation theory this is not a restriction since the twisted algebra is isomorphic to the untwisted one [2]. Indeed if we perform the following change of basis: (VI. 5. 84a)
(VI.5.84b) ~
n
t
o n +t H+b6n, 0
L "L
(VI.S.85)
we find that the hatted quantities fulfill the algebra (VI.S.83) with t" 0 and
b" O.
1607
Hence. every unitary representation of the untwisted algebra is a unitary representation of the twisted algebra as well. and vice versa. For proof. we refer the reader to the mathematical literature r1]. Hel'e we just list our conventions and notations. Let a e ~ be the roots of the finite Lie Algebra G c G. We denote by p the semisum of positive roots
..
p
= 1:.
ra
(VI.5.86)
2 fulfill Eqs. (VI.5.l0S) with h .. CgI (k + Cy) (see Eq. (VI. 5. 99) ) . Hence they are good highest-weight states also for the Virasoro algebra and can be used to calculate b. Since the singlet representation jl is compatible with any value of k (see Eq. (VI.5.98)) we can just take this choice which is also the simplest. We find L 110; (O,.!k,O) > "
-
2
°"'" b(t"
0) .. 0
(VI. 5.111)
The coboundary of the Virasoro algebra is therefore zero in the case of the untwisted algebra. Let us nol; consider the twisted case. A highest-weight state of the twisted algebra is characterized by the analogues of conditions (V[.5.99) and (Vl.5.102):
(VI. 5. 112 a)
tl jl,-k,O> t 1 It I HOIl,-k,O>=1l 2 2 (n + t • 0. > 0)
(VLS.112b)
(VI. S. 112c)
(VI. 5. 112d)
(VI. 5. 112e) where 0.1 (the orthogonal roots) are the roots which have zero scalar product with the twist vector. The isomorphism (VI.S.8S) fixes the allowed values of the vector. Indeed from Eq. (VI.S.85a) we see that the vector
jlt
1613
jJ.
= JJ t
1 ... - kt
(VI. S.113)
2
must be a highest-weight fulfilling Eq. (VI.5.98b). In this way the state ljJ.t+ tkt.ik.O > = Ill.ik.O > is a good highest-weight state for the untwisted algebra generated by the hatted operators of Eq. (VI.5.84). The only meaningful difference between the twisted and untwisted
algebra is due to the nOl'lll&l ordering of the L~ operator. Indeed. the result depends on whether we normal order with respect to one way of moding the Kac-Moody operators or to the other. This is the mechanism which leads to a non-vanishing bt for t ~ O. To see this let us write the explicit form of Lo:
(VI.5.114) and let us choose (which is always possible) the singlet representation 11 '" 0 corresponding to
JJ
t
"'.
'21 kt •
(V1.S.l1S)
With this choice of the vacuum we obtain (VI.S.1l6)
Consider next the state L~ 11 0 > • For
t" 0 this is not zero but
reads:
Lt 10> '" _1_ {2H *H + ~ la I-a. +ra ga lID> -1 k+C -1 0 l. -l+t*a -toa -t·a -l+t*a
V
a>O
(VI.S.117)
1614 If we denote by Pfl the semisum of all positive roots which are not
orthogonal to t:
PII "
1
I"...
(YI.5.118)
2 (pO
a·t~O
the state (VI.S.117) can be rewritten as follows: Lt1IO> -
= - _1_ kTC y
( 2P
II
+ kt).
(VI.S.1l9)
H 110 > -
and we immediately obtain t I 112 = 2k 1 ~+CV~ 1 2 I2Pn IlL_l 0>
+ kt
[2
(VI.5.120)
which combined with Eqs. (VI.S.109) and (VI.S.116) yields the desired value of bt : (VI.S.I2l) Putting together Eqs. (VI.S.S1) and (VI.S.121) we have the final expression for b(G.k):
1
+"4
k +kCv {I k + Cv I2 P [I + kt 12 - k It
[2}
•
(VI.S.122)
Inserting Eqs. (VI.S.S1), (VI.S.57) and (VI.S.122) into (VI.5.42) and using at the same time Eqs. (VI.S.2S) to eliminate the parameter d (dimension of space-time) we obtain the final expression for the intercepts:
1615
(Vl.S.123a)
~
1
2
1
a=l--Iv-16 p=l P
ND
Ie
2
(VI. S.123b)
2 p=NM+I p
which can be used to investigate the question of massless target fermions. We just observe that the operator L~Mink) is of the form (Mink) 1 2 Lo ="2 P
.. (Mink)
(VLS.124)
+1'
where p~ is the 4-momentum and tor.
/oJ
is a positive-valued number opera-
Similarly L~G.k) is of the form (G k)
L •
o
t
t
Cg(~) (G) =--+14 k+ Cy
(VLS.12S)
where the twisted Casimir C~(~t) is given by t t g
C (lJ )
= lJt
t
• (J.I + 2 p ) J.
(VLS.126)
p being the semisum of the positive roots orthogonal to t and pt being related to a weight-vector by Eq. (VI.S.IIS). The additional term N(G) appearing in (VI.S.I2S) is a positivevalued number operator. Taking this into account the mass-shell equations (VI.S.29) become the following:
1616 1 2 -p TN" - Am2 (w)
= a(w}
2
12
Z,.e%-
~
-p +N=-Am Z
(VI. 5. 127a)
n (Cg(~))
~ -i=1 k + Cv
(VI.S.127b)
.
1
Since target fermions sit in the Ramond sector the necessary condition for the existence of massless fermionic excitations is that the mass-shift 6m2 (l) should be non-positive: 2
lim (1) 5 0 .
(VI. 5.128)
To analyze Eq. (VI.S.I2S) we observe that in view of Eqs. (VI.S.l21), (VI.S.123) and (VI.S.126) 6m 2(!) can be rewritten as a sum of contributions from the different simple groups: 2
Am (1):
n
L
i=1
2
llmG (k.,t,U) i
(VI.S.129)
1
where for each group we have 2 Cv dimG 1 \ Am (k,t,V) , , - - - - t. t· a(l- t· a} + G k + Cv 24 2 0 -k- (12 P 1' +ktl 2 -k 112 t ) + k + Cv 4 k+ Cv I
+ -1- {I -
1 I + (J.!- -kt)· (v- -kt+2p) 2
2
.I.
1•
(VI. 5.130)
Looking at Tables VI.S.I, II, III where we listed all solutions of Eqs. (VI.S.28), we see that the only simple groups of relevance to us are SU(Z}, SU(3) and 50(5). We consider then the explicit structure of the mass-shifts for these three groups. A)
The SU(2) mass-shift We write the twist vector in the following form
1617 t
= qA
(VI.5.131)
A= 1/11 being the fmdamental weight and q a positive parameter smaller than one, 0 < q < 1. In this way t is not a weight vector. Since the SUeZ) root diagram is one-dimensional we always have P=PI! and PJ. =0.
Writing l! = ZJA as in Eq. (VI.5.104) and allowing the same spectrum of J-values as in Eq. (VI.S.10S) we obtain: 2
AmsU(2)(k.q.J)
=
=_4_lk+2 z(_1_q_I2)2 k+41k+4 11 4
+!(J_~q)21 2
(VI.S.132)
4
2
Inspection of Sq. (VI.S.132) reveals that AmSU (2) is a non-negative function which has just one zero at
q=I1
(VI.S.133a)
k k J=-q=4
8
(VI. S.133b)
The last equation can be fulfi lied if and only if k = 4n is a multiple of four. Looking at Tables VI.S.I, II, III we consider. to begin with, the solutions of the anomaly cancellation equation where the target group Gr is just a product of SUe2}'s and U(I)'s. There are ten of them corresponding respectively to case I), 2}, 3), 4), 7), 8). 9), 10), 11) and 12) if we number the solutions from the first of Table 1 to the last of Table III. Cases 9), 10), 11) and 15) can be immediately ruled out since they include level k = 2 SU(2)Kat-Moody algebras whiCh, according to the previous discussion, contribute a strictly positive mass-shift in the fermion sector. In the remaining cases 1), 2), 3). 4). 7), 8) massless target fermions can be constructed by twisting the SU(2) algebra with q = 1/2
1618
and choosing the representation J .. 1/2 in the It .. 4 case and the representation J '" 3/2 in the It .. 12 case. At)
Digression on symmetric twists
Recalling now Eqs. (VI.S.37), (VI.5.S8) and (IV.5.76) we see that the choice q'" 1/2 corresponds to the choice of a boundary subgroup By e B charaCterized by N", 2, and hence
By
(VI. 5.134)
.. Z. Z(G) 2
Eq. (VI.5.134) defines what we call a symmetric twist of the Kac· Moody algebra G. Indeed, when Eq. (VI.5.134) holds true the matrix rBA represent· ing the generator y inside the adjoint representation becomes an involutory automorphism r of the Lie Algebra G and induces the following orthogonal decomposition: G '" H $ l .
[H. I} .. I.
[H • U] ,. H
[a: • IJ .. H
(VI. 5. 135)
where the subalgebra H is the eigenspace belonging to the eigenvalue r .. 1 while the subspace ( is the eigenspace belonging to the eigenvalue f'" ·1. Correspondingly the Kac·Moody generators with H-indices are integer moded while those with (-indices are half-integer moded. In this case the twisted version of the Kac-Moody algebra (VI.3.160) can be written in a somewhat simpler way than in Eqs. (VI.5.83). If we distinguish between the H and , indices by means of the convention A = (i.a) where A e G, i e H,
~
e,
(VI.S.136)
1619
we obtain (VI.5.137a)
(VI. S.l37b)
(VI. 5.137c)
and the associated Virasoro generators take the explicit form
I
L ,,_1_ m k + Cv n"-
(i
Ji
m-n
+
n
If.
lJ.) .
m-n-li
(VI.S.l38)
n+% .
The vacuum state Jo> is characterized by the equation
LoIO> " llO > k +C v where. on COllIparison with Sq. (Vl •.S.l26). the value of
(VI.5.l39)
112 leads to the following value for the coboundary b in the case of a symmetric twist:
bSymm .. _k_
k +C V
where by dim (G/H)
116 dim (G/H)
(VI.5.14l)
we mean the dimension of the symmetric space
associated to the orthogonal decomposition (VI.S.13S).
1620
This expression can be cross-checked, in the SU(2) case, with the expression for general twists (VI. 5.121). Putting q'" 1/2 in (VI.S.l31) and inserting it into (VI.S.I2!) we get b=_k_ 8(k + 4)
which is the right value since, in this case, the symmetric space G/H is the 2-sphere 52 =SU(2)/U(1) " SO(3)/SO(2). If we restrict our attention to symmetric twists the expression (VI.5.130) for the fermionic mass-shift simplifies considerably. Indeed we get 2 LimG(symm) = _C_h_
+
k +C V
Cy
(dim G-
24(k +CV)
~
dim
2
2.) H
Going through the list of symmetric spaces we can check that (dim G- dim G/H) is a non~negative number which becomes zero only in one case, name ly. for the choice G", SOC 3) and H=50(2) . Hence massless fermions can be produced through symmetric twists only in the case of SUeZ) groups.
On the other hand, as we are going to see, the non-symmetric twists needed to produce massless fermions in the case of the SU(3) and SO(S) groups require values of the central charge k which are not available in our list of solutions (see Tables VI.5.1, II, III). Hence, eventually. we will be forced to restrict ourselves to symmetrically twisted SU(2)'s. Such a choice has important consequences in the further development of the theory when one comes to the question of fermionization and modular invariance. This is postponed to Chapter VI. 7.
For the time being let us see how we groups manifolds. B)
e~clude
the SU(3) and SO(5)
The SU(3} mass-shift For SU(3} the simple roots a 1 and aZ can be chosen as follows:
1621 (VI. S.142)
and the corresponding fundamental weights are (VI.5.143)
In addition to ~1 and Ctz one has the third positive root (%1 + a2 which is to be identified with the highest root e.
e
is also equal to the vector p:
e " al
+
(%Z
= (12 /Z,
f3fi) .
(VI. 5.144)
A highest -weight of SU(3) is a vector
(VI. 5.145)
where mI , mZ are positive integers. The twist vector t has a similar expression
with ql and qz non-negative and less than 1/2. t· a < for all roots).
°
(In this way
We must distinguish two cases: i)
p1
° and
=
Both ql and qz are different from zero. In this case PII '" p =a l + a 2 .
ii) Either ql or q2 is zero. Without loss of generality we can say Ql" 0. In this case we have PJ." a l and PI! '" tal + a 2, What happens is that we have an untwisted subgroup SU(2) ® U(l) c SU(3) and it is convenient to label states with SUeZ) ® U(l) quantum numbers, The fundamental weight of SU(2) is A= (1/12 ,0) and it is related to >'1 and >"z by
1622
(VI.5.147) Correspondingly we can identifY the integers m1 and m2 with the isospin and hypel'chllrge by (VI.S.148) With these notations, in the first case, the SU(3) mass-shift reads as follows: 2
&1SU (3) (k,t,lJ) .. .. - 6 k+6
f3 -k+ 3 It - -1 pi 2 + -1 III ~ -1 kt 12 J k+6 3 6 2
(VI.5.149)
while in the second case it takes the form
I
2 k + 3 [4k + 27 e.msU(3) (k,qA2 ,J, Y) "' k 6+ 6 W '4k712 +
'61 J(J + 1)
+
'92
-
2q(1 - q) J ..
1
(Y - 2'kq)
21J
•
(VI. 5. 150)
In both cases the positivity properties are evident by inspection.
~U(3) as given by Eq. (VI.S.lS0) is non-negative and never reaches zero for any value of q. J or Y, while ~U(3) as given by Eq. (VI.S.149) is non-negative and has an absolute minimum Am;U(3)"' 0 at (VI. 5 .15 la) 1 JJ .. -kt .
2
(VI.S.lSlb)
Eq. (VI.S.lSlb) implies (VI. 5. 152)
1623 which can be fulfilled with integer values of m1 and m2 if and only if the level of the SUeS) algebra is a multiple of six: k =6n.
This is sufficient to rule out all the solutions of our Tables with an SUeS) factor (cases 6), 9) and 16) of Tables VI.5.I, II. III). Indeed,
C)
kSU(S) < 24
in all cases.
The sotS} mass-shift In the SO(5) case the simple roots are
III ; (1,0),
~2"
(-1,1)
(VI. 5.153)
to which we must adjoin the other two positive roots (11 + ttz and Za1 +CtZ'
The vector p is given by (VI.S.154)
whose length square is
Ipl 2 =5/2.
The fundamental weights are
A]
= (1/2.
1/2).
A2
= (O,l)
(VI. 5.155)
In terms of these we can write the twist vector t and the highest weight vector ~ exactly as in the SU(3) case (see Eqs. (VI.5.145) and (Vl.S.146». AS in the SUeS) case one should distinguish the cases where t is not orthogonal to any root and where Pl '10. The result is the same as in the previous example. For P1" Q we obtain
(VI. 5 .156)
1624 which is clearly non-negative and has a zero at (VLS.157a) (VI. S.157b)
Sq. (VI.S.157b) shows that in order to get massless fermions the level of the SQ(S) Kac-Moody algebra should be a multiple of six: kSO(S) .. 6n. This observation suffices ·to exclude all the 50(5) -solutions found in Tables VI.S.l. II, IrI.
In this way the list of viable group-manifolds has been reduced to those of Table I plus the two cases 7) and 8) of Table II. In Chapter VI.7 we shall exclude also these last two cases on the ground of fermionization and modular invariance. Before closing this section let us make some remarks on the general pattern which has emerged from our case-study. Extrapolating our results we can say that the existence of massless target fermions imposes the following conditions on the twist vector t and the highest weight 1.I of the fermionic vacuum: 2 p Cv
(VI. 5. 158a)
t .. -
k
II ,,-p
(VL5.158b)
Cv
So the necessary and sufficient condition for the existence of massless fermions seems to be a condition on the level k, which must be such that ; p belongs to the weight lattice. This happens if and V
only if (VI. 5.159)
1625
The smallest possible value of k fulfilling Sq. (VI.S.l59) is obviously k =",. In this case the contribution of the super WZW~system to the conformal anomaly is c(G,k) :: _k_ dim G + 1 dim G :: 2 k +C V
dim G
(Vr.S.160)
that is, the same as if we had just a set of free fermions in the adjoint representation of the group Gx G. This will be a very important observation while addreSSing the topiC of modular invariance (see Chapter VI. 7). Let us now anticipate from later discussions that the presence of U(l)-factors in the target manifold is incompatible with chiral fermions. Adding this information to the material presented in this section suffices to show that the target groups of chiral superstrings are those listed in Eq. (VI.S.4). The level of the SU(2). groups is always k.:: 4, 1 1 and since we need synnuetric twists the boundary groups Bi are products of lz ~ factors as claimed in Eq. (VI. 5.5) • This will be further clarified in Chapter VI. 7. References for Chapter VI.S [1] [2]
V.G. lac and D.H. Peterson, Advances in Mathematics S3 (1984) 125; O. Gepner, E. Witten, Nucl. Phys. B278 (1986) 493. P. Goddard and O. Olive, J. Mod. Phys. Al (1986) 303.
1626 TABLE VI. 5.1 l'ermionizable Group-Manifnlds for 0:4 Superstrings Target Group: G,-
Nhet
Massless
Target
GNO Symmetri c Space
Fel'lllions
SUSY
GlG,-
Femionization Group GF : SU(2)6
r
fSU (2)k:4] 3
3S
Yes
1 SN $4
[SU{2} It SU(2) SU(2J
ISIJ(2)k:i ~ [U(1)]2
34
Yes
2:(NH
(SU(Z)ltSU(2)r It [SO(3)f SU(2) 50(2)
lSU(2\~41 It lU(I))4
33
yes
N ,,4
SU(2) " SU(2) 0[S()(3)f su(2) 50(2)
[U(1)]6
32
Yes
N =4
[SO(Slt 50(2)
1627 TABLE VI. S.l! Quasi Fermionhable Group-Manifolds for 0=4 Superstrings Target Group. Gr
lihet ·
Massless
(;NO Symmet ri c Space
Fennions
Gp/Gr
Fermionization Group
GF~SU(4) '" Ileft >
®
Iright >
(VI. 6. 7)
where Ileft > e ffL and Iright > e ffR. However, not all such combinations are phYSical closed string states. The fundamental question we try to answer in the present and the next chapters is the following. Given the space: (VI. 6. 8)
of all left-moving and all right-moving states, which subspace Jf of ff .Yfcfl
(VI. 6. 9)
is the Hilbert space of closed string states?
It turns out that the answer to this question is not unique; rathel', we have a finite set of Hilbert subspaces Jfi c ff (i" 1, ••. ,n) corresponding to a finite number of superstring models described by the same two-dimensional Lagrangian and associated with the propagation on the same target manifold. The proper tool to study and classify these Hilbert spaces is the associated partition function (VI.6.6) which can be rewritten as follows
1632 (VI, 6. 10)
where Pi is the projection operator on the
i~th
Hilbert subspace Jl"i:
(VI.6.11)
Clearly a claSSification of the Z.(8) partition functions amounts to 1 a classification of the projection operators Pi and of the associated superstring models. The reason why Zi (8) is a useful object to consider is due to its relation with the Polyakov functional integral (VI.2.30). (VI.2.31) which leads to an identification of the inverse temperature B with the imaginary part of the modulus T of a torus. This identification implies that the partition function (VI.6.10) should be an invariant against the action of the mapping class group and eventually allows for a group-theoretical classification of the available projection operators Pi (Glioul, OliVe and Scherk projectors (GSO)) in terms of symplectic modular group invariants. The present chapter is devoted to establishing the precise relation between the partition fUnction (VI.6.10) and the Polyakov path integral posing the prob lem of modular invariance. The next chapter will be concerned with the solution of this problem and with the classification of the GSO projectors; this programme involves a separate study of the fermionization of group-bosons and of the Kac~Moody algebra characters.
VI.6.2 The cosmological constant, the partition function, and the Polyakov path integral The first step in establiShing the advocated relation between the thermodynamical partition function (VI.6.10) and the integrand (VI.2.31a) over moduli space of the Polyakov path integral is the following observation.
1633 In a closed string the origin of the a-axis should not matter; indeed a rigid a-translation corresponds to a meaningless renaming of the base point 0"0 of the loop: (VI.6.1Z)
Invariance under a-translations is achieved if all the closed string states are eigenstates of the P generator (VI.6.S) corresponding to some universal eigenvalue cZ: (LO -
Lo) Iclosed
string>
= ezi closed
string>
(VI. 6.13)
In this case the closed string states transform with a phase under the shift 0 ~ cr + const, which is therefore a symmetry of the theory. The condition (VI.6.13) can be implemented inserting an integral representation of the delta function o(LO - Lo - c Z) in the Fock-space trace (VI.6.10)(*)
(VI. 6. 14)
With the use of (VI.6.14), Eq. (VI.6.10) can be rewritten as follows:
Zi (B) ,.
f d ReT Tr (I) {lPi exp[ 21Th (LO - a)} • exp [- 21Ti i (i.o
- a) ]l
• (VI. 6.15)
where we have defined the new variables
(*)
Here and in the following we do not bother about overall normalization constants.
1634 1
(x + is)
(VI.6.16a)
a" -2(c1+2 c )
(VI. 6. 16b)
12 (c 1 ~
(VI.6.16c)
T " -
21£ 1
(i "
c ) 2
The combinations (VI.6.17b,c) have been named a and a for a very simple reason: it will turn out precisely through the identification with the Polyakov path integral that they are nothing else but the intercepts (or Virasoro coboundaries) defined by Eqs. (VI.5.123). tion
Considering Eq. (VI.6.1S) we are led to introduce a complex funcof the complex parameter T defined as
~ (t)
!'i (T) " ImT Tr(I){lPi q
La-a
_to-a}
q
(VI. 6. 17)
where the normalization 1m T is chosen for later convenience and where we have defined (VI.6.l8)
q '" exp(21li T}
In terms of this complex function the thermodynamical partition function (VI.6.1S) can be rewritten as Z. (21f ImT " B) = 1
I
dReT IN - ... (1) ImT 1
(VI.6.19)
The transcription (VI.6.19) suggests a rather natural question: if the thermodynamical partition function Zi(S} is the integral of ~i(l} on the real part of T. then what is the meaning of the complete twodimensional integral of ~i (t) in
lr " dTdi' " dRet
d ImT
?
1635 Furthermore, what is the meaning of the complex parameter l and what is the appropriate integration domain in the complex T-plane? The answer is given by the following propositions. 6.6.1 Proposition. Let .!'i (ll be defined by Eg. (VI.6.17)! being the intercepts (VI.S.123), and consider the integral
(l
and
(VI.6.20)
The constant All) is the one-loop contribution of the string-model characterized by the GSO-projector IP i to the effective cosmological constant. 6.6.2 Proposition. The integral (VI.6.20) can be viewed as the vacuum to vacuum amplitude corresponding to the emission and reabSOrption of a closed string, which, in its virtual propagation, sweeps a toroidal world-sheet
(VI.6.21)
The parameter l in the integral (VI.6.20) is the modulus of the world-sheet torus (VI.6.21). Hence for the integration domain in (VI.6.20) one must take the PSL(2,Z) fundamental region a (see Eqs. (VI.2.246) and (V!.2.248»), which is nothing else but the g= 1 moduli space. Furthermore, since the measure
iT
d(Weil-Petersson) = --"-2(Imt)
(VI.6.22)
(i
1636 is in variant under the action of the mapping class group pst (2 ,ll , the partition function fl. (tl I\IUSt also be PSL(2,Z)-invariant. Recalling 1 theorem 6.2.16 and Eqs. (VI.2.244), flf(t) is invariant, if and only if it fulfills the following conditions:
1r.(1
1) = fl.(t) t 1
(VI.6.23a) (VI,6.23b)
Proposition 6.6.2 is more precisely stated as the following theorem. 6.6.3 Theorem. The integral Ai of Eg. (V1.6.20) can be identified 1) on genus g= 1 Riemann surfaces: with the Polyakov path integral
9'i
A~l) .. ~ ~1) . 1
(VI. 6. 24)
1
The higher lOjP contributions given by ~~g .
A1g)
to the cosmological constant are
1
6.6.4 Definition. In general the PolyaKov path integral ~!g) ~ genus g Riemann surfaces is given by (VI .6. 25a)
(VI. 6. 25b) where the above symbols have the following meaning. S(g.C.~) is the classical action (VI.3.64) which depend~ on the metric gaB' the gravitino ~a and the matter fields ~l, bc is a shorthand for the boundary conditions specifying the appropriate bundles on Lg of which . the matter fields {~l(~)} and the gravitino Ca(~) are cross-sections.
1637
j'bc(gQS) denotes the fUnctional integral on all the fields ~a and " 0
(VI. 6. 48a)
lon-shell> '" 0
(VI. 6. 48b)
where, recalling Eqs. (VI.5.124) and (VI.5.125), the oscillator number operators N and N are defined by
N " N(Mink)
n +
(G.)
I
N
(VI.6.49a)
1
i=1
N" N(Mink)
+
I N(G
N
i ) + het
(VI. 6. 49b)
i .. 1
In simple words N(G), (N(G»
is that part of LO (respectively Lo)
which is not due to the zero mode Kac-Moody currents:
(VI. 6. 50)
For the Minl could be tepresented as a path integral on a cylinder of circumference 211
Tf
and length 21f 1m T as shown below.
rm 1
ooE
)0
(J
()
(VI.6.56)
i
f
Correspondingly the trace Tr exp{ - 21f lm T H}
can be represented as a
path integral on a torus by gluing the ends of the cylinder as follows: f: i
(VI.6.S7)
Actually in the trace (VI.6.17) we have also the factor exp[i21l ReT
ii] .
This operator rotates the closed string by an angle
211 ReT.
Hence the
trace (VI.6.17) corresponds to a torus constructed by gluing together the ends of the open cylinder (VI.6.56) with a relative twist
21f Re T:
(VI. 6.58)
1646
This shows that or is indeed the modulus of a torus represented by the parallelogram (VI.2.66J with identified opposite sides. This is the heuristic argument behind Proposition 6.6.2 and hence behind the more precise Theorem 6.6.3. In the next three sections this crucial theorem which relates the functional and operatorial approaches will be proved for the case of the purely bosonic string. In Section VI.6.6 the proof will be extended to the fermionic strings by considering the relation between fermion determinants and Riemann theta functions already introduced in Chapter VI.2 (see Sq. (VI.2. 342)).
VI.6.3 Qperational evaluation of the bosonic string partition function
For pedagogical reasons, in the next three sections we focus on the bosonic string. Consider Sq. (VI.5.11) and set both ~ and the gravitino l-form I',; to zero. The result is the classical action of the bosonic string in a Minkowski space, which for quantum consistency tums out to be 26dimensional: (VI.6.59) The auxiliary fields n~ can be eliminated through their own equation of motion: (VI. 6.60) and (VI.6.59) becomes the following action
(VI.6.6l)
1647
where instead of the vielbein e±(~) we utilize the metric a (VI.6.62) and its detel'lllinant (VI.6.63) The inverse vielbein (V1.6.64a) (V1.6.64b) is also utilized to define the inverse metric (VI.6.65) and to relate the intrinsic derivatives a±xiJ. to the ordinary ones (VI.6.66) With these definitions Eq. (VI.6.61) is easily proved. Inserting (VI.6.60) into (VI.6.59) we get
(VI.6.67) and utilizing Eqs. (VI.6.63) and (VI.6.65) we obtain the result (VI.6.61). which will be OUr starting point for the discussion of the Polyakov path integral. Prior to this discussion, however, we want to show that the dimension of the target Minkowski space is D= 26.
1648
Recalling the definitions (VI.S.I2) and (VI.S.lS) we see that in the case of the bosonic string the stress-energy tensor is given by (VI.6.68a) ~
1
~j.I
-
1
j.I
T{f) "-P (i)PCz) " - -2 (a-x) 2 z
2
(VI. 6.68b)
and the corresponding central charges are c " D = number of space-time dimensions
(VI.6.69a)
c " D•
(VI.6.69b)
Since the world-sheet supersymmetry has been suppressed there is no superghost contribution and the analogues of Eqs. (VI.S.28) are the following conditions: 26 "D
(for left-movers)
(VI. 6. 70a)
26 "D
(for right-movers)
(VI. 6. 7Ob)
This shows that the target space has 26 dimensions. Recalling equations (VI.S.SS), in the conformal gauge we can write the follOWing mode expansion: (VI. 6. 71)
~~ere the operators X~. i~ obey the Heisenberg algebra (VI.S.S6), and the O-modes XO" XU = pU have been identified with one-half the momentum operator. Furthermore. recalling Eqs. (VI.5.S2) the complete LO' LO operators are given by
t
L(X) L0-- 0
+
L(ghost) 0
(VI.6.72a)
(VI.6.12b)
1649
where L(X)
o
,,!
r : X-nXn
+""
(VI. 6. 73a)
2 n=~w
L(X) ,,! r : X X : o 2 n=."" -n n +CO
(VI.6.73b)
and L(ghost)
o
= - ."" \' n • L' n=-
L~ghost) " _
c- c : n-n
r n : ~nC-n :
n
(VI. 6. 74a)
(VI. 6. 74b)
the ghost modes fulfilling the algebra (VI.5.48). Finally, recalling Eqs. (VI.S.123) the intercepts have the following values ~"
a"
(VI. 6. 75)
1 •
Indeed in the bosonic string there are no fermions and hence no spin structure to ,",'Ony about; thus the intercepts take a lUliversal value fixed once for all by Eq. (Vl.6.7S). For the same reason there are no c.1 [bel coefficients and consequently no GSO projection operator lP1•• This allows for an immediate calculation of the partition flUlction (VI. 6.17).
Inserting (VI.6.73) and (VI.6.74) into (VI.6.17) we find
(VI. 6. 76)
where we have defined (VI. 6. 77a)
1650 L(ghost) '!(gho t) (t) = Tr
ghost
S
= ['1__..._ (i)] * =
q0
(ghost)
~Lghost
= [ Tr q 0] * (8hOSt)
(VI. 6. 77b)
and where the variable q is given by equation (VI.6.18). The definition of the number operator follows from Eqs. (VI.6.Sl) and (VI.6.73). Its explicit form is (VI. 6. 78)
Redefining (VI.6.79a) (VI. 6. 79b) we obtain
(VI. 6. 80)
where as a consequence of Eq. (VI.5.56) the operators {a~tJ a~} form an infinite set of independent creation~absorption operators: (VI.6.81) Using this observation we can immediately calculate N(I) '1{X) (T) ., Tr q (X1
=
nD"n 1J=1 n=1
D
00
"n n
~(X)(T):
n alJt alJ Tr q
n
n
\.1'=1 n=1
n
2n + q3n + ••• ) " ( ..n
(1 + q + q
n _1)D
(1· q )
n=1
(VI.6.82)
1651 Comparing Eq. (VI.6.82) with the definition of the Dedekind eta function
net) :::
IT
ql/24
(1
~ qn)
(VI. 6. 83)
n=1 we conclude that
D/24 [ ~(x)(T) ::: q net) ]-D
.
The calculation of ~(ghost)(T)
(VI. 6. 84)
is similar.
From Eq. (VI.6.74) we
deduce (JO
L(ghost)
o
= L n N(ghost) n=l
(VI. 6.85)
(n)
where N(ghost) - (c
(n)
-
c + c-nc) n
(VI. 6. 86)
-n n
Hence we can write (JO
n
I(q)
(VI.6.87)
Nghost I(qn) ::: Tr(qn) (n)
(VI.6.8B)
~(ghost)::: IT
n=l
where
At each level n we have four possible ghost states 10,1>, 11,1>,
and
C_ncn
10,0>. 11,0>,
where the labels refer to the eigenvalues of c_ncn
respectively.
Starting from
10,0> they are given by
11,0 > ::: C-n 10,0 >
(VI.6.89a)
C-n 10,0 >
(VI.6.89b)
10,1 > :::
1652
11,1 > = c ~nc-n 10,0 > •
(VI.6.8k)
The states 10,0>, [1,1> have positive norm < 0.010,0> = < 1,111,1 > • 1
(VI. 6.90)
and belong to the eigenvalues 0 and 2 respectively of N~~ost). The states 11,0> and 10,1> have negative norm
= < 0,110,1 > = -1
(VI. 6. 91)
and span the eigenspace belonging to the eigenvalue -1 of
N~~ost).
With this information the trace I (qn) is immediately evaluated n
n
n
I (q ) " 1 - q - q
+
2n n2 q '" (1 - q) •
(VI,6.92)
Inserting this result into Eq. (VI.6.81) and comparing it with the definition of the Dedekind eta we get ~ghost(T)
= q-2/24
12
(VI.6.93)
In(r).
Inserting fUrthermore Eqs. (VI.6.84) and (VI.6.93) into (VI.6.76) and evaluating the gaussian integral in the momentum p, we obtain the final result D-2
D~2
!l(T) .. const e
4'1r Imt
_~
(qq)
(Iu)
-2
[n(t)
1-2(0-2) (VI.6.94)
At this point. recalling the modular transformation properties of the n-function: S : n( - lit) T : neT + 1)
= (- i
T)l/2 n(T)
= ei'lr/12 n('t)
(VI.6.95a) (VI.6.9Sb)
1653 we see that the function
Z~~~(t) =
D-2 (lmt) - -2 In(t)r 2 (O-2)
(VI.6.96)
is modular invariant for any value of the dimension D: (VI.6.97a)
(D)
Z(X) (t + 1)
(D) = Z(x) (t)
(VI. 6. 97b)
Hence, since Eq. (VI.6.94) can be rewritten as follows: (V1.6.98) we conclude that ~(t) is moduiar invariant only for that value of the parameter 0 which reduces the factor
to a constant. To no one's surprise this value is D= 26, the same number of space-time dimensions seleCted by the conformal anomaly cancellation (VI.6.70). Hence in 0 =26 the one-loop contribution of the bosonic string to the cosmological constant is given by (VI. 6.99)
In order to prove Theorem 6.6.3 we need to show that the same result (VI.6.99) is obtained by calculating the Polyakov path integral. We do this in the next sections.
1654 VI.6.4 The Polyakov integration measure for the bosonic string
In order to give a precise meaning to the functional integral (VI.6.25) we proceed as follows. In the bosonic string case we set (VI.6.I00)
where SIg.xl is the Euclidean analogue of the action (VI.6.61), in which the sign of the metric determinant has been flipped: (VI.6.101) and where a ' is a dimensionful parameter needed to make the exponent in (Vl.6.100) dimensionless. The action (VI.6.101) has the dimensions of a squared length since its value is the area of the world-sheet. Hence we have (VI.6.l02) The parameter (l' is the so-called Regge slope of the old dual models and provides the fundamental unit in terms of which all the masses of the string particle spectrum are measured. It can be reabsorbed in the embedding function X~(t). Indeed X~(~) has the dimension of a length and therefore we can use natural units defined by the following condi· tion 1 1 --=81Ia 2
(VI.6.l03)
l
With this convention, Eq. (VI.6.100) can be rewritten as follows: (VI.6.104)
1655
where
~
g
is the Laplacian on the surface L equipped with the metric
(VI.6.lOS) and where the scalar product of functions defined over the surface L is given by (VI.6.106) The scalar product (VI.6.106) induces the definition of a norm for the functions X~(;):
IIxIl g2 =<x,x> g
(VI. 6. 107)
and this norm can be utilized to define gaussian functional integrals. Formally we declare that the following path integral over all the functions x~(~) is equal to unity: (VI.6.108)
This implicitly defines the integration measure. Next we decompose the functions X~{~) in a constant mode x~ plus a deviation x~(;) whose average value is zero: X~(;)
= x~
+
x~(;)
J~ x~(;) rg i; " 0
(VI.6.109a) (VI. 6.109b)
Substituting (VI.6.109) into (VI.6.10B) we obtain (VI.6.110)
1656 Defining the volume Q(g) of the surface
E: (VI. 6.111)
we find 2n )1/2 J-~.. dXo exp[ - '21_2Xo Sl(g)} = ( Sl(g)
(VI. 6. 112)
and hence the normalization:
JDxg
)J
(~)
1 2 f'~(g) exp[--Ilxlll 2 g = - 211
(VI.6.113)
which implicitly defines the functional integration measure on the deviation functions
x)J(~).
Next we observe that. since E is a compact surface,
xIJ(~)
can
be decomposed into eigenfunctions of the Laplacian operator Il whose g spectrum is discrete: (VI.6.114)
n e N and (i) labels the possible degeneracy of the eigenvalue An' where
The eigenfunctions y(i)(~) n
can be orthonormalized: (VI.6.115)
and we can write (VI.6.1l6)
where cIJ( "J are numerical coefficients. Notice that the constant n~l mode xIJo belongs to the kernel of A and as such it is orthogonal to " g all the y~ (~) belonging to non-vanIshing eigenvalues. Hence in order
1657 for x~(;) to fulfill condition (VI.6.109b) it suffices that the summa~ tion in (VI.6.116) be restricted to the eigenfunctions belonging to nonzero An' In this way we obtain an explicit representation of the functional integration measure defined by Eq. (VI.6.113):
(VI. 6. 117)
Given these preliminaries we begin by calculating the functional integral on embedding functions at a given 2-dimensional metric gaa' Setting
.r (g) '"
J!2." Xll
g
exp{ ~ !. < X, A X > J g
2
(VI.6.118)
we observe that xi! .... X\.l + l(~ is a symmetry of the classical action corresponding to the invariance of the background metric n\.lV under the Poincare group. Hence .9"(g) is ill-defined and we must divide it by the (infinite) volume of the target space translation group Vol(transl) =
f
+CO
1
dX O •••
I+<X>
-c;o
This is easily done
by
D
dXO
(Vr. 6.119)
-00
replacing
.9"(g) ... ft· (g)
(VI.6.120)
where we have defined (VI.6.121) Substituting (VI.6.116) and (VI.6.117) into (VI.6.121) we get
1658 ~I (g) "
"
D IT \l=1
I(CO
deg An del! IT n (n.i) n=l i .. 1 f2i
exp[.! (e" 2
n
)2 A (n,U n
)
1/2
J (S2(g») " 2n
(VI. 6. 122)
where det t Ag
is the dete1'Jllinant of the Laplacian in which the contri-
bution of the O-modes has been OllIitted.
Using this result the Polyakov
path integral (VI.6.104) reduces to
9' ;;
f ~g (
12(g) 211'det' A g
'j>/2 •
(VI. 6. 123)
J
In order to make the above integral well defined we need to factor out the volumes of the symmetry groups of the classical Lagrangian. that is. the Diffeomorphism group and the Weyl group.
Clearly this will reduce
the integration domain in (VI. 6.123) from the space of all metrics Met (1:g) to the moduli space Mg (see Eq. (VL2.27)). The problem is to insert the correct Jacobian. The strategy we follow to solve this
problem starts from the follOWing consideration. Let M be a manifold and T (M)
be the tangent space to M.
Let
tql ••••• ~} be the coordinates of a point p E M and
be the integral we are interested in evaluating.
be a coordinate transformation.
Let
Then I becomes (VI. 6. 126)
1659
where F' " F
0
f
(VI. 6.127)
and J(q') is the Jacobian it f. '" det a.,f. aq! 1 J 1 J
J(q'} .. det -
(VI.6.128)
The transformation (VI.6.12S1 induces a transformation in the tangent space T(M). If (VI.6.129)
is a vector (veT (~I)), then after the transformat ion (VI. 6 .125) its new components will be given by the relation {VI. 6. 130)
It follows that. calling coordinate transformation integral performed on the
i K. (q) the matrix of the change induced by a J on the tangent vectors and considering an tangent vectors (VI. 6.131)
we have (VI. 6. 132)
where J(a') '" det K(q) " J(q)
(VI.6.133)
This means that the Jacobian calculated in the tangent space coincides with the Jacobian calculated directly on the manifold.
1660
Relying an this we define the integration measure an the tangent space to the manifold M by setting 1
=f
T (M)
q
[da] exp( _1 lIall 2 ) 2 q
(VI.6.134)
where Tq(M) denotes the tangent space{ai the p:i}nt q= {ql .....
2
= ai aj
(VI. 6. 135)
gij(q)
Writing [da]
n
dai
i=l
f2ii
= N(q) n ----
(VI.6.136)
we see that Eq. (VI.6.134) implies da f nn -ffi i
1 = N(q)
i=1
1 T exp( - - a g a) 2
N(q)
=- -
{detg'
(VI.6.137)
Hence N(q) =
Idetg .
(VI. 6.138)
Starting from Eq. (VI.6.134) we can calculate the Jacobian J(a'). which is to be identified with J(q'), by means of the following procedure. A tangent vector ai
is an infinitesimal change of the coordinate
(VI. 6. 139)
Hence we obtain
1661
1"
JT (M)
[daJ exp(.l II a11 2 ) = 2
q
=f
J(a')[da'] exp(.1IlKa'1I 2 ) =
T (M)
2
q
= J(q) J Tq{M)
= J(q)
q
q
1 T' T Ida'] exp(--a K gKa ' } 2
[det (KTg K)j -1/2 (det g)I/2
= J(q)
(VI.6.140)
det K which in view of Eq. (VI.6.133) is an identity as it should be. Let us now apply this general technique to the infinite-dimensional manifold Met (L) of all metries on a compact surface with g-handles. g
Recalling the results of Chapter VI.2 (see in particular picture (VI.2.134» we define a slice of Met (Lg) transverse to the action of the diffeomorphism and of the Weyl group in the following way. Let gaS ('l"""n) be a set of constant curvature metrics, parametrized by n Tei chmuller parameters (n :; 0 for g" 0; n" 1 for g" 1; n =3g-3 for g ~ 2) such that a g is orthogonal to the image of Lr
PI defined in Eq. (VI.2.92).
In other words, the metric variations (VI. 6.141)
are quadratic differentials for the metric gaS(L): 'i'i
e ker
pi
where we have utilized the notations of Eq. (VI.2.99). action of a diffeomorphism r:,fJ. ..... l\~)
(VI.6.142) Calling f* the
(VI. 6 .143)
1662 on the metric and on the embedding functions Xil (~) ,
the action of the
Weyl e Diffeomorphism group on the space of metrics e embeddings can be written as follows: (VI.6.I44)
Hence we can change coordinate system in the metrics.e embeddings manifold by writing (VI.6.14S) Given the metric gaa(t),
xP
denotes a slice of the space of embeddings
which is transverse to the action of the conformal Killing subgroup of the diffeomorphism group. In simple words. this means that Xli is not annihilated by any of " the conformal Killing vectors of the metric gaB (Tl'·.·.Tn.~): +
if
A e ker PI .
(V!.6.146)
We remind the reader that the conformal Killing vectors were defined in
Eq. (VI.2.100). Let uS introduce a basis for ker PI:
s • l •••. ,dim (ker Pi) •
(VI.6.147)
Given this basis every tangent vector (generator of the Diffeomorphism group) t(l(;)
can be written as follows: (VI.6.148)
where the
decompo~ition
is orthogonal with respect to the scalar product ~
(VI. 2.85) • In other words the tangent vector t
ker Pi:
is orthogonal to
1663 (V1.6.149)
Similarly the most general variation ogaa(t) of the metric can be decomposed as in Sq. (VI.2.89). Recalling fUrthermore Eq. (VI.2.99) we can write (VI. 6. 150)
pi
pi).
where 1/1~ provides a basis for ker (r:;; l ..... dim ker Tben utilizing the method previously explained we come to the implicit definition of our functional measures. We set 1 ..
IT-/
1 ..
JTg(Met)
1=
I
1 ::
JTx& [d(c5X)] g exp( --21 lI&ill g2 )
(VI.6.1S1d)
1 ..
IT !I [dt]g eXP(-i Ilt ll !)
(Vl.6.1S1e)
[d(6X)] exp(-t g
TIj)(r')
g
(VI.6.151a)
-i !loglI!)
(V1.6.1S1b)
Uc5cPU 2 )
(VI. 6. ISle)
g
[d(ogJ]g exp(
[d(6cP)]
lIoXll 2 )
1 exp(--2
g
f where Tx/. Tg (Met),
Tlt).
Tf~
are the tangent spaces to the
manifolds of the embeddings, of the metrics, of the conformal group and
of the diffeomorphism group, at the points I, g, cP and f respectively. (Similarly for the restricted embeddings X). Our goal is to obtain the Jacobian [d(6g)]g[d(oX)]g ..
:;; J(t,cP,f,X) • d(o't)[d(ocPJ] g[dt] gld(OX) 19
(VI.6.152)
1664
where we have called (VI.6.1S3)
The norms are defined by the scalar products (VI.2.82) and (VI.2.SS) for the metric and the tangent vectors, by Sq. (VI.6.106) for the embedding functions, and by the following scalar product for the Weyl transformation: (VI. 6. 154} Let US proceed as in Eq. (VI.6.140). We have (VI. 6. 155a) (VI.6.1S5b) where
...
f* t d • -;====;=====/ det H{pi) det H(P 1)
• (det'
t 1/2 1 ( 2n det Pl -1) vet)
I
llg )- 0/2
.
(VI. 6. 191)
!leg)
The interpretation of the various factors appearing in Eq. (VI.6.191) is the following: det < wSlx > -;====;;===r====- dnt ::; d(l'Ieil-Petersson) I det H(Pi) det H(P 1)
I det' (Pip 1) Vol (Aut
eE»
211 det' Ag ( --~ ) !'I (g)
- D/2
(0) = z(X) = ~
(VI.6.192a)
(VI. 6 . 192b)
d(Weil-Petersson) being the mapping class group invariant measure in moduli space and ~(X)(t) being the partition function associated to the XU-system, which in our case coincides with the full partition function.
1672 VI.6.S Functional evaluation of the bosonic string partition function in the case of the torus In this section we explicitly evaluate both the measure (VI.6.192a) and the partition function (VI.6.l92b) in the case where the world-sheet has the topology of a torus. We first retrieve the operatorial result (VI.6.99). Recallingpicture (VI.2.66) and Eqs. (VI.2.l06), (VI.2.222), on a torus the line element can be alternatively written as
2 _ 1 z~ ds '" dzdz - (gzz- = -2 ; g = 2)
(VI. 6.193)
utilizing complex coordinates, or as (VI. 6. 194a)
(VI. 6. 194b)
utilizing real coordinates. Note that in Eq. (VI.6.194) we have reabsorbed the factor 1/2~ into the definition of the coordinates ~a which now vary in the interval [0,11. From (VI.6.194b) we obtain
;g =
Im T
(VI. 6. 19Sa)
as = - -1 -
(VI.6.19Sb)
g
(Im T) 2
Q(g) "
ri; Ii = ImT
•
(VI. 6. 19Sc)
As we know from Chapter VI.2, the conformal automorphism group of the torus is given by the translations
1673 (VI.6.196)
Clearly to can vary on the torus itself, namely, it is defined up to lattice vectors. Hence the volume of the conformal Killing group coincides with the volume of the torus: (VI.6.197)
Furthermore, a conformal Killing vector, namely, a generator of the translation (VI.6.196), is given in real notation by (VI. 6.198) Correspondingly, recalling Eq. (VI.6.147) we can set (VI. 6 . 199a)
A:{t
• '" 1
(VI, 6. 199b)
The matrix H(Pl) is immediately evaluated. Substituting Eqs. (Vr.6.199) and (VI.6.19S) into Eq. (VI.6.159) we get
(VI. 6.200)
So
that det H (PI) rs
const' (Irnr)2 •
(V1.6.201)
Let us now consider the quadratic differentials. On the torus, as we know from Chapter VI.2, the space of these differentials has one complex dimension. Indeed, the solution of the equation:
1674
v.% h%Z ..
0
(VI. 6.202)
2 is provided by the constants which span H (I 1). Hence we can set (VI.6.203)
where
iu .. 1
2 .. i
(VI. 6. 204)
"'%%
is an explicit repnsentation of the basis
I/I~) utilized in Eq.
(VI.6.1S0). We obtain
t
Hrs (PI) ~ < l/Ir ll/ls > ..
(VI.6.20S)
and hence (VI.6.206)
~. These were alnady computed in Chapter VI.2. There we showed that under the infinitesimal shift (VI.2.249) the variation of the metric is given by Eqs. (VI.2.252) and We still need the slice deformations
(VI.2. 253). Hence we bave (VI.6.207)
wbere
X1 : :i u
IlIIt
2
Xzz
1 =--1m!
(VI.6.20B)
1675
With these identifications we obtain < X1'1 1/1s >..
I
5 r d2z ~~ g gti gzi" (Xl' 1/1-+ )(~-
zt
ZZ
.. const
(~
'~a
s
zz ) =
1/1
(VI. 6. 209)
:)
which yields (VI. 6. 210) Results (VI.6.201J, (VI.6.206) and (VI.6.210) inserted into Eq. (VI.6.192a) give the Correct Weyl-Petersson measure on the space of tori;
i t_ d(Weyl-Petersson) " const __ (lm'[ )2
.
(VI.6.211)
On the other hand, using Eqs. (VI.6.197) and (VI.6.195c) the partition function reduces to D-2 .2"(t) ..
const· (ImT)2 {det' PiPl)1 / 2 (det' 6gf D/2
(VI. 6. 212)
If Theorem 6.6.3 were true we should find that the function defined by Eq. (VI.6.212) coincides with the function (VI.6.9B). We will see in a moment that this happens in the critica.l dimensions D" 26. Indeed, irrespectively of the value of D, the partition function as defined by (VI.6.212) is equal to D-2 ~{'[J .. !reD) ..
(X)
(Imr)--2-ln(T)I- 2 (O-2)
(VI. 6. 213)
The proof of this result goes in two steps. First, recalling Eqs. (VI.2.39) and (VI.2.114) we note that on a torus, where the conformal factor ~(z,i) can be set everywhere equal to zero, we have t
(P1P1)t z : - 2(V-V)t zzz
=-
2(3-a )t zzz
(VI. 6. 214a)
1676 t (PIP1)tz = - 2(VzV~)t t z = - 2(a za·}t. z z
(Vl.6.214b)
so that, utilizing a matrix notation, we can write
(VI. 6. 215)
~
g
being the Laplacian on scalar functions f
~ f =- 4 g
az.. a..
..L Ii aa (gO$Ig all
f .. -
l"
f)
(VI. 6. 216)
Indeed Eq. (VI.6.216) can be verified using Eqs •. (VI.6.195) and their consequence :
a = _ iz
a-
z
21m.
a
(f l •
a
a2)
= - _ i- (t 1 21m!
(VI. 6. 217a)
a2)
(VI .6. 217b)
From Eq. (VI.6.21S) we obtain t
det PIP l
= (2"1 detAg) 2
det' ptp .. 1 1
..
(12 det' Ag )2 '
(VI.6.218)
Hence Eq. (VI.6.212) can be rewritten as 1)...2
!"(t)
= const.
(1m t)-2- (det t ~) g
_ (D-2)
2
(VI.6.219)
and the result (VI.6.213) follows by inserting into Eq. (VI.6.219) the value of the determinant det' Ag which is found to be det' Il = (Imt)2In(t) 14 g
•
(VI, 6.220)
1677
The proof of Eq. (VI.6.220) is the goal of the next section, where the functional determinant will be evaluated by means of the I;;-function regularization scheme. Actually while calculating dat' A we shall g also calculate the determinant of the Dirac operator » exhibiting its dependence on the spin structure. This will pave the way for the extension of our results to the superstring case.
VI.6.6 Functional determinants of the Laplacian and of the Dirac operator on the torus A scalar on the torus is a function in both coordinates ~1' ~2:
~(~1';2)
which is periodic
(VI. 6. 221)
As such it can be expanded in exponential functions: (VI.6.222)
(VI.6.223)
The
~(nl.n2)(~)
are eigenfunctions of the Laplacian
(VI .6. 224)
with eigenvalues (V1.6.225)
1682
where
1;:~a,bJ(S)" _1_ res)
idttS-l
10
L
{n 1,n 2 u}
expl_t[<W)+ )(a,b)]2 J . nl'n 2 {VI. 6.245)
In Eq. (VI.6.245) we exploited the property of the eigenvalues (VI.6.Z41) of being arranged in pairs of opposite sign. This implies
(VI. 6. 246)
which justifies the position (VI.6.Z45). We shall now proceed to evaluate the determinant (VI.6.244) in the case of the even spin st ruct ures : (VI. 6.247)
setting aside the case of the odd spin structure (V1.6.248) which involves some additional subtleties. As the reader can verify by inspecting Eqs. (VI.6.Z36), the spin structures are classified as even or odd depending on whether the corresponding theta is an even or odd function of the variable z sweeping the Jacobian variety. Clearly the odd thetas vanish at z = O. This is just a signal that for odd spin structures the Dirac operator has zero modes which make its determinant zero, while for even spin structures ~ is zero-mode~free. This connection between the parity of the theta functions and the existence of O-modes is manifest from Eq. (VI.6.Z41); indeed we see that A~ never vanishes for any n1, nZ e Z, unless
1683 WI .. tAl2 .. 0 .. a :: b :: 1 •
(VI. 6. 249}
In the case of the odd-spin structure we are obliged to remove the contribution of the ~ero-modes by excluding in the sum on "I' n2 the case (0,0). In that case we have (VI .6. 250)
the right hand side of Eq. (VI.6.250) being defined by Eqs. (VI.6.227) and (VI. 6. 228). Even Spin Structures
~:
Substituting Eq. (Vl.6.241) into (VI.6.245) we obtain
(VI.6.251) Introducing the new variable
r"
4" - t -(1m l)2
(VI.6.252)
we can rewrite (VI.6.251) as follows:
~[a,b] (s) (J')
.. _1_
res)
[.!!l]2S Joo dyl-l u(a,b] (r,l) 21i
0
(VI. 6. 253a)
1684
u[a,b] (y,T) can be rewritten in a more con-
The integrand function
venient form utilizing the following identity, known as the Jacobian inversion formula:
L
exp [ • lfY(iii2 + 2mx)
mez+v
I ,.
2 lfyX
i
=_e_
IY
If
2
exp[--m +21fh(x+v)J.
(VI. 6. 254)
Y
mel
By use of Sq. (VI.6.254), Sq. (Vl.6.2S3) can be rewritten as follows:
~[a,b] ( )
=
s
(J»
p[a,b]{s,r)
_1_ (1m T )2S p[a,b1 (s T) res) 21i
= L
L
me Z Ii e Z + WI x
Io ¢O
dyy
(VI.6 •2SSa)
'
5-3/2
exp[ - 21fim (Retii+oo2)] x
.2
2
11'
2
exp[ -lfyn (lmT) - -m y
I. (VI.6.255b)
To calculate the limit (VI.6.244) we observe that the function
l/r(s)
has a simple zero at s" 0 1 --,. s
res)
+ h2s
Z+
(VI.6.Z56)
•••
Hence, for any regular function F(s)
which does not vanish at
5
=0
we find
lim 5+ 0
..! ds
(_1_ Fe ») '" res) S
lim F(s) .
(VI.6.257)
5+0
In our case F(s) '"
(1m T )25 p[a,b] (5, T) zlii
so that we obtain
(VI. 6. 258)
1685
-lndet[a] b
~
.. lim
p[a,b](S,T).
(VI.6.259)
s+O
Substituting Eq. (VI.6.25Sb) into (VI.6.259) we get: -R.ndet[a] ll .. lim b
L
p[a,b]eS,T)"
s+O
L
{melt {nez+w1}
exp [ - 21ri m (Re t Ii ~ ~)] 1- (T) n,m
+
mjl 0 1 +
lim S" 0
I
i
{ii eZ of- w } J0
njI 0
dy l-3/2
r
exp .11'1 ti2 (Im t) 2]
1
+
const (VI. 6. 260)
where we have separated the contribution from m" 0 and defined: (VI.6.261)
The integrals of the type (VI.6.261) can be exactly evaluated using the formula
fa
dy Y-3/2
exp [ - (ay +
~) J '" ff exp (- 2 lOA)
(VI.6.262)
we then obtain
n,m (T) = _1 !ml exp( - 21r Imrlml inll
L
(VI.6.263)
We still have to take care of the limit appearing in Eq. (VI.6.260). We set
1686
(VI.6.264)
where
(VI. 6. 265)
If wi" 0 we have IfO(t) " ~ (t),
whicb is finite at t,,·1.
Hence.
using 1 2
f(.-)=.211i"
(VI.6.266)
!. Imt.
(VI.6.267)
we get o{O,t) "
3
In a similar way we get 1 o('2,t) ...
6"'II 1mt.
(VI. 6. 268)
Putting our results together we find -R.ndet[a] fJ" (1b even
a
1T
'2)'3 1mr
+
Let us now consider the various choices of [ :
spin structure
[~J.
J.
In the case of the
for instance, Eq. (VI.6.269) can be written as
1687
-.tn det [011
~ '" !. 1m!
'"
+
3
m
2 L m"l
00
1::lL L m
(qlllll + qlllll)
n"1
,.
" 2
L .!.. 1I.n q -.tn 1 24
I
= -.tn q1/24
Ii
n
(1 + qn) -
n=1
.!.. R.n q 24
(1 + qn J [2
.
tn IT
n=1
(1+ NLeft ,,18,
1
if D" 4
= D + "2 NRight ... NRight = 44, if
26
D" 4
(VI. 7. Sa)
(VI. 7.Sb)
where NLeft is the number of left-moving fermions, carrying internal group indices, and where NRight is the number of right moving fermions. Secondly, one observes what follows. Let us set
(VI. 7. 9a)
ik A ~ - - : 11 (z) a \J (z) 2 z
T(z)
(VI. 7.9b) (VI. 7. ge)
where fA~rr are the completely antisymmetric structure constants of a semisimple group GF,
gA~
is the corresponding Killing metric and
where {i(z)} and {~I(:n} are, respectively, a set of NLeft=dimGF left-moving and a set of NRight right -moving free fermions, obeying the standard OPE's;
A ).l
r
i
gAL
(z)u (w) = - - k z-w
(VI. 7. lOa)
1706 i
oIJ
2
i-w
,,--
(VI. 7. lOb)
By suitably choosing the constant KI , one can easily verifY that Eqs. (VI.7.9) provide a quantum realization of the superconformal algebra (Vl.3.189) with (VI. 7.11a)
(VI. 7.llb)
Hence the authors of Refs. [1,2,3] concluded that by utilizing the three IS-dimensional semisimple groups
(VI. 7. 12a) GF • 5O(S)
* SUeS)
(VI. 7.l2b)
GF ,. 5U(4)
@ SU(2)
(VI. 7.12c)
one could construct four~dimensional superstring models whose modular invariance could be analysed in a simple way since the partition function takes the form (VI.7.4). This programme will be referred to as the fel'lllionic approach to D,. 4 superstrings. Unfortunately the nice features of the fermionic approach with respect to modular invariance are counterbalanced by a serious drawback. Indeed the fermionic string action exists only in the superconformal gauge and cannot be viewed as the gauge fixed form of a worldsheet locally supersymmetric action. Hence the very coupling of the world-sheet fermions to the world-sheet gravitino. which justifies the otherwise essential inclusion of its determinant in the partition function, is dOubtful. Furthermore, the introduction of backgromd fields and the derivation of effective a-models in the later development of the theory are very problematic. On the contrary. such problems are clearly
1707
overcome by the group~manifold geometric approach pursued in Chapters VI.S and VI.S. Therefore we are led to readdress the question of modular invariance in the present tontext. Here the partition function includes, besides the theta functions representing the fermionic functional determinants, also the characters of the Kac~Moody algebras we considered, which can be viewed as the determinants of the corresponding WZW kinetic operators. We will show that in the case of the target group GT " [su (2) }3, already selected by the requirement of massless target fermions, the Kac~Moody characters can be re~expressed in terms of the theta functions, leading us back to the treatable case of fermionic strings. This is the subject of the next section.
VI. 7.2
~lodular
invariance and GNO fermionization
Referring to the programme of modular invariance in the case of group-manifold superstrings, we observe that, in order to carry it through, we should solve the following problems: We ought to define characters not only for untwisted but also for twisted Kac~Moody algebras. This amounts to defining the analogue of a spin structure for group bosons. These bosonic spin structures arrange into orbits under the modular group and playa role in the construction of modular invariants. i)
ii) The Kac-Moody characters are labeled. beSides by the spin structure, by the highest weight of the vacuum, which ranges over the finite number of values allowed at the given level k (see Eq. (VI.5.98b). This label transforms under the modular group simulta~ neously with the spin structure. Therefore we must devise rules for the construction of modular invariants which are a combination of two items: theta functions, labelled by the spin structure, and Kac~Moody characters labelled by both the spin structure and the vacuum weight. We should extend the theory of Kac-Moody characters to higher genus surfaces in order to address the question of higher loop modular invariance. iii)
1708 In this section. since massless target fermions select SU(2) group-manifolds. we focus on the 5U(2) Kac-Moody algebras with symmetric twists. In this case we show that we can indeed solve problem i) and define suitable bosonic spin structures whose modular transformations we can also derive. In this way we can write down the candidate modular invariant partition function as a linear combination of terms whose modular transformations are known. So we can address problem il) which is that of determining the proper coefficients. We solve this problem through fermionization of the Kac-M'oody currents. This enables us to use the rule established in the free fermion approach with, however, a proviso to be explained in the next section. Let us explain in some words what we mean by the above statement. At the quantum level the Kac-Moody currents of a Lie algebra G corresponding to specific levels k can be replaced by bilinears in new fermion fields X(z) (hereafter named fake fermions) transforming in a suitable representation R. In general the stress-energy tensor calX culated as a bilinear for the KM currents is different from the canonical stress-energy tensor of the x-fields. However, Goddard. Nahm and Olive (GNO) [4J have shown that the necessary and sufficient condition for these two forms of the stressenergy tensor to coincide is the existence of a group GF ~ Fr (hereafter named fermionization group) such that
I
GF""r adj
is a symmetric space
GF= adj "r
(VI. 7.13) $
~
Given a level k realization of the GKac-Moody algebra we can easily calculate the dimension of the candidate representation Rx to be utilized for its fermionization. Indeed, since the conformal anomaly of the RX fermions mUSt be equal to the conformal anomaly of the KM' currents, we have
!
2
dim R = ___k___ dim G • X k + Cv
(VI.7.14)
1709
Provided a representation RX with such a dimension exists, we can ask ourse! ves the next question, namely, if a group GF can be fOlmd whose decomposition with respect to Gr fulfills conditions (Vt.7.13). An ....-.... example will suffice. Consider the level k .. 20 realization of SU(2). The dimension of Rx should be (VI. 7.15) which corresponds to an isospin J" 2 representation, namely, to a symmetric 2-index traceless tensor of SO(3) ~ 5U(2). Now the symmetric space SU(3)/50(3) corresponds to the embedding of SO(3) with the following branching rule {S} SO(3{. {3} e {S}
(VI. 7.16)
This is precisely what we need. Hence in our example the fermionization group is Gp " 5U(3) and the GNO symmetric space is SU(3)/SO(3}. Using this method in the last column of Tables VI,S.I, II, III we have identified the ~~O symmetric spaces associated to each of the fifteen group manifolds fOlmd by us in the previous sections. As the reader can see, the allowed group manifolds have been organized according to their fermionization groups. Remarkably, as fermionization groups GF we retrieve the three IS-dimensional groups proposed by Antoniadis et al [2,3j, that is to say. SU{2)6, SU(4) ® SU(2) and SO(S) ® SU(3). The relation between the group-manifold approach and the fermionic approach can be fully appreciated if we consider the supercurrent (VI.3.1S4c) and we replace the KM currents JA(z) with the fermion bilinears
(VI. 7.17) where the matrices RAij are the GT generators in the RX representation. The result takes the following form
1710 (VI. 7.18)
where (VI. 7.19)
are the structure constants of the fel'Dlionbation group, and
(VI.7.20) is the multiplet of free fermions in the adjoint representation of GF• supercurrent (VI. 7.18) is three-linear in the fermions as the supercurrents considered in Eq. (VI. 7 .9a). Thus one might conclude that all we have done so far is just retrieving the fermionic string models and one might decide to jump directly at the results of references [2,3], fOrgetting the group manifold formulation altogether. We think this is too hasty. The
Indeed it has to be stressed that the GNO theorem deals only with local properties of the KM currents. If one wants to extend the fermionization globally to the whole world-sheet, one has to analyse carefully the bomdary conditions. In the case of the twisted KM algebras there is almost always an obstruction to fermionization, simply because, in general, an arbitrary twist on the KM algebra does not match with the one induced by the boundary conditions of the fake fermions:(*) For instance, in Chapter VI.S we showed that the theory corresponding to
GT = SU(2)k=8
* SU(2)k=20 * U(l)
(VI. 7.21)
contains massless fermionic states if and only if we perform the symmetric twist (VI.S.137) on both the SU(2) algebras. (In this case i = J3 n
(~)
n
As remarked in earlier chapters, the normal ordering ambiguity discovered by Bouwknegt and Ceresole, seems to imply that full fermionization can be achieved in a few more cases of those listed in Tables VI.S.I-III.
1711 ·(l J1.2) The prob lem is to find a suitable moding of the an d In+~'' n+~' X-fermions such that the H-generators be integer moded and the Kgenerators be half·integer moded. It happens that there is none of the sort.
Furthel'lilOre it should be that the coboundary of the Virasoro algebra has the same value whether we use the KM current picture or the fermionic picture for its computation. Once more this is not true in general. Indeed a counter-example is provided by the model (VI.7.21). on one hand, we know that the mass-shift is zero if we choose a q" 1/2 twist of the I<M algebra which has no fel1llionic realization. On the other hand. from Ref. [3} we know that by utilizing non-half-integer twists of the fake fermions Sitting in the SO(5) ~ SU(3) adjoint representation we reach the same result.(·) There is a remarkable exception to this state of affairs. namely, the SU(2) Kac-Moody algebra of level k" 4 can be fel'Dlionized as the diagonal SU(2) subgroup of GF" SU(2) e 5U(2), and the representation RX is consistent with the twist defined in Eqs. (VI.5.131) and (VI.S.133), associated to the symmetric space 50(3)/SO(2). Such a construction is the basis of all fermionizable solutions listed in Table I. They correspond to the free fermion theories associated to GF • SU(2}6. We call quasi fermionizable those solutions where the boundary conditions on the I<M currents cannot be reproduced by the xfermions. One example has been quoted. The complete list is given in Table VI.S.II. As we see, the associated fermionization groups are SU(4) $ 5U(2) and 50(5) & SU(3), and correspond to free-fermion models with non-real boundary conditions. Because of the above obstruction, no one of these models can be formulated as a string propagation on a group manifold; hence no claSSical superconformal action seems to exist for such theories. Finally the non-fel'Dlionizahle solutions of Table VI.S.III
(*)
It appears that choosing a different normal ordering prescription the fermionic twists of Ref. [3] can be reconciled with a group manifold interpretation. This is again due to a different resolution of the Bouwknegt-Ceresole ambiguity. In any case one just adds a few more solutions to the fermioni:tation problem.
1712 correspond to Kac-Moody algebras which are part of a larger Kac-Moody algebra including further U(l) factors. Actually tions are fermionizable in lower dimensions (D" 3 when dim D=2 when dim GF '" 24) •
fermionizable these soluGF;; 21 and
Summarizing this discussion. we can say that if we want both a supersymmetric classical action. whose necessity has already been stressed, and the advantages of a fermionic picture, which seems very important to implementing modular invariance. then we must restrict our attention to the four solutions of Table VI.S.! corresponding to (*) ~"8U(2)6. Let us then focus on these models. This means that throughout the rest of this section the target space will be 3-p
n (SU(2).)
Mtarget
" M4 @
i=1
1
(VI. 7.22)
B
where all the generators B 9 Y " 6 1 @ Y2 @ '13 ' Y1 • '(2 0 Y3) of the homotopy group fulfill Eq. (VI.S.134), namely, they are such that
Y~
6
Z[SU(2)j
(VI. 7.23)
Furthermore p is an integer in the range 0 5 p ~ 3. The case p;; 3 corresponds to a fully toroidal compactification of the D" 10 heterotic superstring. On the other hand, the case p" 0 is the opposite extremum where there are no toroidal subspaces. It is the only one compatible with N=1 target SUSY since we will show that we need precisely three SU(2) to construct a GSO operator which projects out 3 of the 4 candidate massless gravitinos. Let us then define the partition fUnction, which is the main object of our study, for the case p'" O. Let t be the modulus of the world(*)
As it is obvious from the previous footnotes. this conclusion is changed by the discovery of the Bouwknegt-Ceresole ambiguity. There are also, using a different normal ordering prescription, a couple of lIlOdels associated to the SU(3) e 50(5) and SU(2) e SU(4) fermionization groups.
1113 sheet torus associated to the one-loop amplitude and let us set, as usual (see Eq. (VI.6.18)), (VI. 7.24)
q = exp(2i'llT) •
Then the one-loop contribution to the cosmological constant reads as follows (see Eqs. (VI.6.2S) and (VI.6.17»: II "
I
tori
exp [- classical actionI
I
dq dq
i ,c, - Tr (q boundary qq !n qq bc
"
11
i = all fields Lo -(1
_
V(';.) 1
Lo-o.
q
) ,
(VI. 7.25)
conditions: be where LO and Lo are the nought generators of the complete Virasoro algebra including the ghost and superghost contributions, and (l and a are the intercepts. The functional integral involves also a summation over the boundary conditions be we must assign to every twodimensional field in order to calculate the Fock space trace of the
· operat or q LO-aq _Lo·a. "-d 1 . . dea1S Wlt . h t he evo 1utlon ~~ u ar lnvarlance weights Cbc to be assigned to each set of boundary conditions bc. Recalling Eqs. (VI.6.294), (VI.6.301) and (VI.6.96) the integral over moduli space displayed in Eq. (VI.7.2S) can be Written in the following form: (VI.7.26a)
WeT) =
i
(b.c.)
where, in our case, the conformal ghosts collectively denoted non-trivial boundary
Z
(b.c.)
(1')
(VI. 7. 26b)
D" 4, and the contribution of the XU field and of has been separated from all the other contributions, by Wbc(t). The reason of this separation is that conditions are a feature displayed only by the
1714 fermions and the group bosons. X~ and the conformal ghosts have always the same boundary conditions and do not contribute any potential global anomaly. The modular invariance of
(VI. 7. 27)
was already shown in Chapter VI.6 (see Eqs. (VI.6.96) and following 2
ones). Since the Weil-Petersson measure ~
is also modular
(ImT)2
invariant we must impose modular invariance on the partition function WeT). To this effect we begin by discussing the structure of a typical addend ZeD.C.) in the case under study (p'" 0). We have
(VI. 7.28)
where the left and the right partition functions are specified below
e[(ll - ai]eOl T) [Il.]1
3
B'1_ _ _WI _-_
i=1
n(r)
n
BJ .
(r)
(VI. 7. 29a)
1 Bj
at:: ](01 1) p
Let us explain the various factors.
1/2 (V1.7.29b)
1715
In Eq. (VI. 7.29)
[ :,] (IU,
1J,l'
e ZZ) is the spin structure of
~. i.e., the world-sheet fermions with space-time indices.
The
structure of the world-sheet supercurrent implies that the same spin structure must be assigned to the superghosts and to three of the groupfermions .\~. Let us explain why. In the case under consideration the supercurrent reads as follows:
G(z) ..
12 ei11/4 : [P
J.I
...IJ (z) + L 3 (A ABC )] : (z)1jI J. {Z)A.A(z) + -212 \AiA'£ABC i=1 1 1 3 1 (VI. 7.30)
where the index i
enumerates the three SU(2) groups, and the level
k =4 Kac-Moody algebra is encoded in the following OPE expansion: A B J i (z)J. (1'1)
26AB i 12 ABC =--_ + £
(z _ 1'1) 2
1
z- W
J~(W)
+
reg. tetlllS.
(VI. 7.31)
1
Tbe available space-like boundary conditions on these currents are dictated by the choice (VI.7.22) of the homotopy group, which leads to
the symmetrically twisted Kac-Moody algebra (VI.S.ll7). Conventionally we can identify the untwisted currents with J~ and the twisted ones with
J~,2(z). 1
Hence We can set
1
i11a1' 1 1 2'11 J.(ze 1) .. e J.(z} 3 2i11 3 J. (ze )" J. (z) 1
1
tJ:(
i ze
2i11)
1'11 b. Indeed. one has
A B 2al) b - 2b I) a ;: 2a' b -
I L a.' b.
i=l j=l
A
+ 8
I
1
+
J
B
X,
i,l=1 j ,k=l
8i
' a1 • b .• bk (mod 16)
)
(VI. 7. 127a)
1745
r
8 2 22 _22 2 b + c + 2al> c = a + b + 2al> b - Jl + bj + j=l
A
+
8.
8
r .r
a1 • aR. • bj • bk (mod 16)
1,1=1 J ,k=l
(VI. 7. 127b)
with a + b + c" I. Using these properties it is now easy to verify that the coefficients cabc satisfy the same symmetries (VI.7.97) as the corresponding even
z[:].
provided that the quadruple product is always
a· b • c· d :: 0 (mod 2) •
(VI. 7.128)
As mentioned in the footnote, when this condition is not fulfilled the gamma matrices introduced in (VI.7.90) lead to extra phases in Eqs. (VI.7.97), and the general solution becomes more involved (9]. VI.7.4
An
example in 0=10:
the 50(32) superstring
In this section we apply the general theory discussed in the previous one to the derivation of specific models. We begin with the ten-dimensional case, which is the simplest. Here the boundary vectors are of the fonn a = (w~w;vl""'V32)
(VI. 7.129)
8-times where w is the boundary condition of the space-time fermions, equal for all the eight transverse components. and v.1 is the boundary condition of the i-til heterotic fermion. If all Vi obey the Sallie boundary conditions then the heterotic fermions carry an 50(32) symmetry. EXAMPLE 1: The 50(32) heterotic superstring in D=10 In this example the group 3 is generated by two elements: b0 '" 1 = (1, ... , 1 ; 1, ... t 1)
40-times
(V1. 7. 130a)
1746
--
bI .. S .. (1 ..... 1; 0 ••• .,0)
(VI. 7.130b)
8-times 32-times These generators fulfill Bqs. (VI.7.80), (VI.7.84) and (VI.7.88)
12 .. -24 .. 0 mod 8 52 .. 8 ..
a mod
8
(VI.7.l3l)
1·S .. 8=Omod4 and hence all the four elements of _. that is. O. I, S and
S ..
5 ... 1 .. (0 ..... 0; 1..... 1)
fulfill the
S8/118
conditions.
(VI. 7. 132)
For instance we can check that:
-2
5 .. -32 .. 0 mod 8
(VI. 7. 133a)
lI. • 5 .. -32 .. 0 mod 4
(VI. 1. 133b}
S • S .. 0 .. 0 mod 4
(VI. 7. ISle)
1 • S• 5
" 1:. {I • S... S • 5 - (5 + I) • S} .. 2
" 1.2 {-32 ... 32}
(VI.7.1S3d)
.. 0 " 0 mod 2 •
Since 1=1=4. according to Eq. (VI. 7. lOS) there are five one-loop invariants.
Hence. recalling Eqs. (VI.7.l03) and (VI.t.ll0) we obtain W SO (32)(T) ..
41 {c001
50S SSl ZOOI'" e Zsos'" c ZSSI
SSt lll ... c ZSSl ... c Z111 }
+
(VI. 7.134)
1747 where the summation is extended to all the independent triplets a, b, c such that the constraint (VI.7.100) is verified. According to the discussion preceding Eq. (Vl.7.122) we can arbitrarily choose the signs associated to any pair of group generators. In our case we can set (VI. 7.135)
with (VI. 7. 136)
Then, using formula (VI.7.125) we can calculate We write
COOl,
cSOS and cSSl .
(VI. 7.137)
which together with Eq. (VI.7.132) suffices to express all the boundary vectors involved in terms of the chosen generators (VI.7.130). We find c001
=
°
0 60
exp [.11l(02 + 02 + 201> 0)/8 ] (c111 ) 4
(VI.7.138)
and since (VI. 7. 139)
recalling (VI.7.112) and (VI.7.136) we conclude: cOOl = 1 •
(V!. 7.140)
Similarly we have (VI.7.141)
1748
Since S I> 0 = S • 1 '" S· lL
= 16
we can write (VI.7.142) where we have utilized the information
Os = -1
J
00
= 1,
SIS
c
= csst = n .
(VI. 7. 143)
Finally we have 1:. 1 SS1 111 SIS 155 (VI. 7. 144) cSSn. = og6S exp [.111(5-2 '" 5-2 '" 2;,1> 5)/8 c c c c
and since (VI. 7.145)
we conclude cSSt
(VI. 7.146)
" os
Substituting these results in Eq. (VI.7.134) we obtain W SO (32)(i)
i {ZOOl '" ZSoS
=
+
n ZSSl
+
ne
ZSSI
+ £
~R1} (VI. 7.147)
The superstring described by the partition function (VI.7.147) has four sectors, two bosoni c {0 and S} and two fermioni c {S and I} • The GSO·projection operators associated to each of the four sectors can be read off Eq. (VI.7.147) by substituting the explicit form of the We have invariants Zab' c
1749 (VI. 7. 148a)
(VI. 7. 148b)
Zsos = -Z[ :] +
ZSSI
+
Z[ ~] - Z[
!] -z[ ~ 1
z[~] - z[~]
= Z[ ~]
Zssn = z[~]
+
(VI. 7.148c)
~1~ z[ ~ ]
(VI. 7.148d)
-z[!] ~ z[;J
(VI. 7. 148e)
+
Z[
and, hence, in each sector we obtain the following combinations of the partition functions: i)
Sector 0:
(VI. 7. 149a)
which corresponds to the insertion in the Fock space trace of the following GSO projector:
(VI. 7. 149b)
ii)
Sector
s: (VI. 7 . 150a)
which corresponds to the insertion in the Fock space trace of the following GSO projector:
1750 (VI. 7. 150b)
iii)
Sector S:
(VI. 7.1S0c)
the associated GSa projector being: (VI. 7.1S0d)
and finally,
iv)
Sector 1:
(VI. 7. ISla)
which yields (VI.7.1S1b)
As the reader may notice the overall sign of the partition fWlction is
positive for the bosonic sectors 0 and
S,
and negative for the
fermionic sectors 5 and 1.
We can now analyse the spectrum of massless states. Recalling Eqs. (VI.7.77) we have (VI. 7. 152a)
o" ~
1
~
1 + - nR + N • 16
(VI. 7. 152b)
1751 i) In the O-sector, we have nL" nR" 0 and hence the candidate massless states are characterized by N= 1/2 and N= 1. The states fulfilling these conditions are (VI. 7.l53a)
(VI, 7. 153b)
The eigenvalues of S·P and I'F on the above states are easily calculated: (VI. 7. 154a)
(VI. 7.154b)
SoFlllIJ > .. IUIJ >
(VI. 7.154c)
(VI. 7.1S4d)
It can be seen that both
1Jl'J
> and
correspond to the eigenvalue one of Po' that is, they are physical massless-states. IlJV > decomposes into a symmetric traceless part (3S-polarizations corresponding to a graviton gjl\l)' an antisymmetric part (28-polarizations corresponding to a 2-index photon BlJV)' and a trace-part (I-polarization corresponding to the dilaton D): III IJ >
(VI.7.15S) The state III IJ > .. -IIlJI > describes, instead, the gauge field of the 50(32) group:
IlllJ > = AIJ II
e 50(32)
,. 8 -
$
496 •
-
(VI. 7.156)
1752 ii) In the sector 5, we have nL.. 8, nR.. 0 and hence the candidate massless states are chancterized by N .. 0 and N" 1.
There are two possibilitIes: (VI. 7• 157a)
(VI. 7.157b)
The reason why we have written the additional index a in the righthand side of Eq. (VI.7.1S7) is that the Is> vacuum must support the 50(8) Clifford algebra of the 8 Ramond zero modes I/J~" rl!. Hence Is > is an SO(8) spinor. The action of the Fermion number operators on the above states is given by (VI. 7. ISBa)
(VI. 7.158b) where r\) .. r1r2 ••• r8 is the ninth gamma mattix. Hence the GSO projector (Vr.7.150b) reduces on these states to the chirality projection operator (VI. 7.159)
The meaning of n becomes clear from (VI. 7. 159) • Oloosing n" 1 we admit in the spectrum all the right handed spinors while we project out the left-handed ones. Oloos!ng n" -1 we do the reverse. The state la.}! > decomposes in a gamma tr~celess part corresponding to the 56polarizations or an on-shell Majorana-Weyl gravitino plus a gamma trace part corresponding to the 8 polarizations of an on-shell spin 1/2 Majorana-Weyl particle (gravitello o~ dilatino):
1753
!a.,~ > = ~J.l ~ Xa. =S6 + S • ~
(VI. 7.160)
The state 1~,IJ >, instead, has the !e496 polarizations of the SO(32) gaugino, Le., of the supersymmetric partner of the gauge field (VI. 7.156) :
la,IJ
>
= ~IJ e 50(32) = 8 @ 496
.
(VI. 7.161)
If we now look at the mass-equations (VI.7.1S2) in the case of the sectors 5 and I, we find in the first case (VI. 7.162) and in the second (VI. 7. 163) In both cases Eq. (VI.7.1S2b) cannot be solved since we should set N= -1 which is absurd, N being a positive-valued number operator. We conclude that the sectors S and 1 do not contain massless states. ~
~
Our results are summarized in Table VI. 7.[1, where the numbers refer to the SO(8) e 50(32) representation aSSignments TABLE VI. 7. II Massless States of the 50(32) Heterotic Superstrin&
~
SUGRA MULTIPLET
=(35 , 1) --
2
g
3/2
!/I
1
B
1/2
X=(8 , 1)
0
D=(!.,!>
)JV
lJ
50(32) GAUGE MULTIPLET
= (56, 1) --
IlV
=(28, 1) -s
A1J =(SV ,496) J.I
--
), IJ = (!s • 496)
1754 VI. 7.5 A second example in 0=10:
The 1: S • E~ and 50(16).80(16)
heterotic strings To construct these superstrings we need a larger E group with an additional generator. To bO and bl , defined by Eqs. (VI. 7. 130), we adjoin b2 .. m = (0, •••• 0 i 0•••.• 0 i 1•••• ,1) 8·times
16-times
(VI. 7.164)
16-times
{lI., s , m}
in this way creating a set of lbo' bi • b 2} ::
which fulfills
the conditions (VI.7.80). (VI.7.84) and (VI.7.8S). The new group E has 8-elements:
E = {O. s, 1, S. Ill. iii. v.~}
v =s
+
m ;
= m+ lI. ; v = v + 1
iii
(VI. 7.165a)
•
(VI. 7. 165b)
The number of independent one-loop invariants is M:
(8+1)(8+2)
=15.
(VI. 7.166)
6
In addition to the five invariants listed in Eqs. (VI.7.148) we have
ZDDDl
= zl:]
r: J
Ziiilll : Z
ZVOV
= -z[~]
zwl . z[:J
- z[~l· zf:J Z[ :] +
+
+
+
z[; J
z[~] - Z(~] - z[~J + z[~J - z[~J
z[~] + z[~J
(VI. 7. 167b)
(VI. 7.16 7c)
(VI.7.167d)
(VI. 7. 167e)
1755 Zwl ..
Z[~] . z[:] ~ z[~]
(VI. 7. 167f)
Zmsv" -Z[:] . Z[~] zt:] - Z[:l -z[:] +
zsvm" Z[~] + Z[~]
+
Z[!l
Z[:1
z[:] + Z[~] + Z[~]
+
(VI.7.167g)
(VI.
7. 167h)
-Z[:l z[!] - Z[~]
(VI.7.167i)
Zmsv" -Z[:l Z[!J - Z[:l- Z[!] + Zl~l- Z[:l
(VI.7.167j)
Zvd" -Z[:l
+
Z[~] - z[~]
+
+
+
For the free coefficients we choose cssl ,. n. c111 = E as in equation (VI. 7.135) and
IRSV
•
C
(VI. 7.168)
.. Ul
where 2
2
(VI. 7.169)
6=w=1.
. . The coefflclents
c
001
50s • C
and c
sst
have the saDIe values as before.
Calculating the remaining coefficients we obtain an invariant depending on four si gns:
+
Zmom
+
en£ Z--l VV
+
6 ZmmI + Ul
t
£6 Ziml
ZIRS"-
+
+
Zvav
nw Zs\lm-
+ a~
ZVVI
+
+
(VI.7.170) Combining Eq. (VI.7.170) with Eqs. (VI. 7. 167) we obtain the explicit form of the GSO projectors in the eight sectors corresponding to the
1756 eight elements of = (see Sq. (VI.7.16Sa». In particular we are interested in the sectors which contain massless states. Recalling Eqs. (VI.7.1S2) we see that these are i) ii)
o ~ (n L .. s
~
DR .. 0)
(n L .. 8 , nR .. 0)
iii) m ~ (n L .. 0 , DR .. 16) iv) iii .. (n L .. 8 , nR " 16) v)
'J ..
vi) \i
=0
(n L .. 8 , nR .. 16)
(n L .. 0 , DR .. 16)
Note that 0, m, and \i are bosonic while s, iii and 'J are fermionic. The candidate massless states in the sectors 0 and s are obviously the same as in the previous example (see Sqs. (VI.7.1S3) and (VI.7.1S7». The GSO projectors. however. are different. Indeed we have
(VI. 7. 171 a)
(VI. 7.171b)
The eigenvalue of the fermion number moF on the states by Eq. (Vl.7.1S3a) is zero
(moF)llJv> .. 0
I~v >
defined
(VI. 7.1n)
so that the graviton. the antisymmetric tensor and the dilaton are not affected by the modification introduced in Po by the new generator m. They continue to exist as massless physical states (see Eq. (VI. 7. 155». On the other hand the states III IJ > defined by Eq. (VI. 7.1S3b) split into three sets with different moF eigenvalues. To see it we subw
1757 divide the range of the heterotic index I into two subranges of dimension sixteen: i
I
a
I
= i'
• i
a
1 ••••• 16
(VI. 7.173a)
• 1=17 ••..• 32
(VI. 7.173b)
and we obtain (VI. 7. 174a)
(VI. 7. 174b)
(VI. 7. 174c}
The states I\l ij >, I\l i' j I > associated to the gauge fields of the SO(l6h 50(16) regular subgroup of the original 50(32) group survive the new G50 projection. while the states III ij' > corresponding to the gauge fields in the 50(32)/80(16) @SO(16) coset directions are projected out of the physical mass-spectrum. Considering now the fer· mionic states la,p > and la,IJ > belonging to the s sector we find (VI. 7.175a)
moFla,ij > = 0
(VI. 7. 17Sb)
moFla,i'j' > = .2Ia.i ' j' moFla,ij'
>
= -Ia,ij'
~
>
(VI. 7.17Sc)
(VI. 7.17Sd)
Hence we have two possibilities: i) If Wa 1 we have Na1 target space supersymmetry in 0=10. Indeed the gravitino and the dilatino, contained in la.\! > survive the GSO
1758 projection and so do the 50(16) e50(l6) gauginos The fermions
Ill,ij > atld
la.i'jl >.
la,ij I > are projected out from the physical mass
spectrum. ii)
1f w=
~1
the target space
supers~try
is N.. O.
Indeed the
dilatino, the gravitino and the SO(16) eSO(16) gauginos are annihilated while the fermions Ill,ij I > in the .!!,® 16 representation of the gauge group survive the GSO projection. Let us next consider the by IPs'
new sectors m and v. Here the candidate massless states are (VI. 7.176)
in the m-sectol', and Iv > "
la, A'
v~sector.
in the
(VI. 7.177)
>
The index A'
taking 256 values spans the 50(16)
spinor representation associated with the choice of Ramond boundary conditions for 16 of the 32 heterotic fermions. The GSO projectors are
(VI.7.178)
and
1 ~ 6n{-l)V'F) (1 ~ 6nH)1·P ) (1 + ewe ~l)m'F ) = ( (VI.7.179)
P
2
\I
2
2
The actions of the fermion numbers on the states (VI. 7. 176) and (VI.7.177) are
(~1)
s'P III
A'
.> ..
~11I A' >
(VI. 7.180a)
(VI. 7.180b)
1759 (VI. 7.l80e)
(VI. 7. 180d) (VI. 7.l80e)
(VI.7.180f)
In the non supel'S)'IIDDetric case (Ill = -1), we see fI'Olll Eq. (VI. 7.180) that all the bosonic states IJ.I A' > are projected out so that the sector m contains no additional zero modes. With the same choice 1Il=
the action of the GSa projector II\) on the zero modes contained
-I,
in the v-sector is (VI.7.181)
It follows that we have 128 spin 1/2 massless particles transforming in the chiral spin representation of 50(16). The 50(16) and the space-time chiralities are left or right depending on the values of n and e. On the other hand. in the supersymmetric case (tI)= 1) both Pm and Pv projectors reduce to chirality operators
t
er 17)
lPm
=
lP\)
1 =4 (1 - enfgr17 )(1 + er l7 )
Hence the
(1 +
Gsa projection
(VI. 7.182a)
is survived by 128 gauge bosons
(VI. 7. 182b)
!J.I A' > and
la,A' > in the Majorana-Weyl spin representation of the group 50(16)'. This representation together with the adjoint of SO{16)' completes the adjoint of an ES group. Indeed we have 128 gauginos
ad; E8
= 248 -
~ 120 +
50(16)
E! .
(VI. 7. 183)
1760 If we analyse the sectors v and m we find a completely identical situation for w>= 1: the massless states are the missing 128 vector multiplets which promote the other 50(16) to a second ES gauge group. In the non~supersymmetric case w= ·1, instead, the gauge group 50(16) is not enlarged and we simply get 128 spin one·half particles. In conclusion, with the group E of Sq. (VI. 7. 165) we can realize two consistent heterotic strings. The first is target space supersymmetric and has Eg l6I as the gauge group; the second has N=O target SU5Y and 50(16) 161 50(16) as gauge group. The field content of their massless sectors is displayed in Tables VI.7.II and VI.7.III, where the numbers refer to 50(8) ® (f sl6I ES' ) or 50(8) 161 (50(16) 161 50(16» representation assignments.
fa
VI.7.6 Examples in 0=4 explained at the beginning of Section VI. 7.3 in 0=4 the boundary vectors have the form (VI.7.72) and we can choose freely only the boundary conditions of the X fermions. those of the A fermions being determined in terms of the previous ones by world-sheet supersymmetry. When a basis of generators fulfilling these conditions and the conditions (VI.7.80), (VI.7.S4) and (VI.7.8S) has been found, the construction of the modular invariants proceeds along the same lines. The simplest example of D=4 superstring is provided by the perfect analogue of the 50(32) model in 0=10. The group : contains the four elements 1,0,S,5 " s ... .D., where s is defined by Eq. (VI. 7.76). The five oneloop invariants have the same form (VI.7.148) as in the D=10 case and the complete partition function is once more given by (VI.7.147). As before the explicit form of the GSO projectors is As
(VI. 7. 184a)
(VI. 7. 184b)
1761 TABLE VI. 7. III Massless States of the Eg ®
I~
Eg Heterotic Superstring Eg E'8 GAUGE MULTIPLET
5UGRA MULTIPLET
2
gU\)
3/2
;p
1
B
1/2
X" (§.
0
D = (!:.' !
!l
IN
(w = 1)
1. ' .!)
= (35,
= (56, 1 , 1) - .. (2g. 1 , 1)
-
5
-
A
-
lJ
v • 248 • 1) = (B--
$
(BV
, 1 • 248) ---
A = (§.s • 248 ,.!) 61 (!s . 1.,
.1.,'!)
248)
. .!)
TABLE VI. 7. IV ~Iassless
States of the 50(16)050(16) Heterotic String w"-l
~ 2
MATTER FIELDS
GRAVITY
FIELD g
.. (35, 1 • 1)
B
= (28.
IN
-
-
-
3/2 1
J,!v
!. !)
,1 iI
1/2
A = (8 v , 120 , 1) jl
A = (§.s,~, 16)
s
(§. 0
D"
(!, !, .!)
$
---
.!. 128)
(8 v , 1 , 120) ---
$
(§.S .128 •.!)
4)
1762 (VI. 7.184c)
(VI. 7. 184d)
The massless states of the above theory correspond to the field content of an N=4 Supergravity coupled to the N;:4 vector multiplet of an SO(44) gauge group. In the O-sector we find the following massless states: "1/2 XVI! - 0
:>
(VI. 7. 185a)
"gVv e BeD pv (I " 1,2,3)
;: A~
(i = l,2 •...• 6)
(VI. 7.185b)
(I " 1,2,3)
" ~i ,Al:
J
(VI. 7.l8Se)
(VI.7.185d)
1763
where we have adopted the following index convention: the index i, running on 6 values, enumerates the six femdons
that is, the third components of the fake and true fermions are associated with the three SU(2) target groups. The index A, running on 44 values. enumerates the 3S heterotic fermions plUS the 3 x 3 =9 fake A1 which realize the right-moving [sU(2)1 3 Kat-Moody algebra. fermions X The states (VI.7.l85a) and (VI.7.18Sb) correspond to the bosonic field content of the N=4 supergravity multiplet, encompassing one graviton, six graviphotons, one scalar and one pseudoscalar. The latter is represented by the antisymmetric tensor BJ,lV' whose field strength is dual to the derivative of a pseudoscalar field n(x): (VI. 7.186)
On the other hand, the states (VI.7.18Sc) and (VI.7.18Sd) correspond to
the field content of the N=4 vector multiplet for the group 50(44), encompassing dim 50(44) gauge bosons and 6 ® dim 50(44) scalars. A little work reveals that the superpartners of tbese bosonic massless states do indeed appear in tbe s~sector and survive the GSO projection. In order to reduce the theory to a chiral N=l superstring, one has to introduce two further boundary vectors, for example: (VI. 7.187a)
(VI. 7. 187b)
It is easy to verify that they satisfy all the constraints and that the associated projectors eliminate all but one gravitino. The possibility of adding these further boundary vectors has a simple geometrical interpretation. In the case of s, the untwisted SU(2) currents were Jl
1764
for all SU(2) groups. In the case of bl , they are J2 for the first 3 1 . SU(2), J for the second and J for the third. In the case of bz' they are J~, J~ and J~ respectively. Thus we see that in the models with s, bl and b2 boundary vectors each SU(2) group is twisted in two different ways. Recalling Eqs. (VI.7.3S) and (VI.7.36) this means that the group Be SU(2) is 12 @ 12 for all the three SU(2). In this section we have analyzed only the string corresponding to Gr '" SU (2) 3. One could also treat the other fermionizable cases listed in Table I simply by adding extra space-time dimensions and reducing the number of SU(2) groups. It follows for instance that in the case GT= U(1)2 ® SU(2)2 the sector s is described by Eq. (V!. 7.76) with T" 1, •.•• 4 and i = 1,2. However it is possible to construct only one independent boundary vector of the type (VI.7.187). Thus we have N=2 SUSY at least. Similarly in the case ~ = U(1)4 ® SU(2), there exists? sector s with T= 1, .•. ,6 and i= 1, but it is impossible to introduce a consistent boundary vector of the kind (VI.7.187), and we are thus stuck with N=4 target supersymmetry. Let us ourselves to of heterotic feasible and
(*)
conclude by pointing out that, although we restricted a list of only a few examples, the complete classification superstrings compactified on SU(2)3~groupfolds appears actually is in progress at the time of writing [10].(*)
At the time of correcting the proofs the classification mentioned above was completed. It is reported in the chapter VI.lO added in proofs.
1765 References fOr Chapter VI.7
[11
H. Kawai, D. Lewellen and S. Tye. Phys. Rev. Lett. 57 (1986) 1832; Nucl. Phys. 8288 (1987), Phys. Lett. 191B (1987) 63. [2] 1. Antoniadis, C. Bachas and K. Kounnas, Nuel. Phys. 8289 (1987) 87. [3] I. Antoniadis and C. Bachas, Nuel. Phys. B298 (1988) 586. [4] P. Goddard, W. Nahm and D. Olive, Phys. Lett. 160B (1985) 111. [5] I. Antoniadis, C. Bachas, C. Kounnas and P. Windey, Phys. Lett. I7lB (1986) 51. [6] V.G. Ka~ and D.H. Peterson, Advances in Math. S3 (1984) 125. [71 D. Gepner and E. Witten, Huel. Phys. B278 (1986) 493. 181 L. Dixon, V. Kaplunorsky, C. Vafa, Nuel. pnys. B294 (1987) 43. [9] R. Bluhm, L. Dolan, P. Goddard, DAMP! preprint 88/9 (1988). [10] L. Castellani, P. Fre, F. Gliozzi, M. Rego Monteiro, work in progress.
1766
CHAPTER VI.S
QUANTUM CONFORMAL FIELD THEORIES, VERTEX OPERATORS AND STRING TREE AMPLITUDES
VI.S.l Introduction In the present chapter we shall exhibit the relation which exists between the 2-dimensional quantum field theory defined over the worldsheet and the quantum theory spanned by the local fields, living on the target space, that are associated with the string vibrational modes. In particular we shall focus on the field theory of the massless modes which, as emphasized many times previously, corresponds to the field content of suitable supergravity models. The relation we alluded to can be described in general terms at the level of scattering amplitudes. Let the spectrum of the superstring model under consideration be composed of the following states: lon-Shell state>
= Ik.~,n
>
(VI.S.I)
1767
where kll is the on-shell momentum of the particle (k 2 .. i), !; (k) is its polari1ation tensor or spinor:tensor and ~ is a short-hand notation for the set of its internal quantlml numbers. To each of' these particle states one associates a suitable 2-dimensional quantum field V(k./;,llj :'.,z)
(VI.8.t)
named the emission vertex for that particle type. Every emission vertex can be constructed in terms of the basic fields introduced by the superstring Lagrangian (VI.3.116) and is. therefore. a local operator in the quantum field theory defined by that action. In particular one can evaluate the N-point Green functions. (or correlation functions):
(VI. 8. 3) where T denotes radial ordering IZll S IZ21 ~ IZ31 ... ~ I:'.NI. The dependence of G(1.2 ••••• N) on the arguments (ki'~i'lli) has been separated from that on Zit ii because, from the 2-dimensional point of' view. the choice of the momenta and the polarization tenSOrs is simply a speCification of the particular quantum fields whose correlation we want to consider. The parameters :'. 1.• ~.1 are. instead. the coordinates of our 2-dimensional space-time points. Given the correlation functions (VI.8.S). the integral S(g)(kll;11l1·····kNtNllN) ..
. Jd\.l(g)
(zl ..... zN)G[k 1l;iIl'1
; ... ; kN!;N1fN](zl"l'· .. ·zN!N) (VI.8.4)
1768 where d~(g)(Zl' ••• 'ZN) is a suitable measure on the g~handled world~ sheet. is interpreted as the order g contribution to the loop expansion of the N~point scattering amplitude
(VI.8.S)
for the corresponding particles. For genuses g ~ 1. the measure d~g depends also on the moduli of the Riemann surface and we are supposed to integrate also on these (see Chapters VI.2 and VI.6). In the present chapter. we focus only on the case g = 0 which corresponds to the calculation of tree amplitudes. Equations (VI.8.4) and (VI.8.S) can be taken as axioms to define the scattering matrix within the context of the superstring theory. In the particular from (VI.8.4 - VI.8.S) one can calculate the scattering amplitudes for any number of massless particles. The result can be compared with what one would obtain from a Lagrangian field theory in d-dimension. The Lagrangian, which order by order in perturbation theory yields the same results for the scattering amplitudes of the massless particles as formulae (VI.8.4~S), is named the effective Lagrangian of the superstring theory. In the present chapter we will show how N=l anomaly-free supergravity (discussed at length in the next chapter) emerges as the effec~ tive Lagrangian of D=10 heterotic superstrings. In order to carry through such a programme we have to construct the emission vertices for all the massless particles. Such a construction, on the other hand, is most conveniently discussed within the context of an axiomatic approach to quantum conformal field theories which we shall introduce in the next section (Vl.8.2). We also need various quantum equivalence relations between
1769
conformal fields which are generalizations of the fermionization discussed in the previous chapter. Furthermore, we need an appropriate discussion of the ghostsuperghost conformal field theory and its vacuum, which is essential in two instances: to determine the correct measure utilhed in Eq. (VI.S.4) and to construct the fermion emission vertices. The analysis of these issues, respectively performed in Sections VI.8.3 and VI.8.4, naturally leads to a concept a~ich will prove to be unifying and very powerful: covariant lattices. Relying on this and the naturally associated notion of spin fields, we shall be able to construct all emission vertices and compute the tree
level amplitudes, fixing the structure of the effective Lagrangian. This will be done in Sections VI.8.S and VI.8.6.
VI.8.2 Quantum Conformal Field Theories and Emission Vertices Our starting point in this section is the quantum realization of the superconformal algebra, which is most conveniently encoded in the operator product expansions (OPE's) of the stress-energy tensors T(z) and r(z) and of the supercurrent G(z). These OPE's were given in Eqs. (VI.3.l89) and are repeated here for the reader's convenience: T(z)T(w)
c
= -2 3
1
2T(w)
aT(w)
(z-w)2
(;r;-w)
--- + --- + - - +
(z_w)4
G(w)
aG{w)
(z_w)2
(z-w)
reg. terms
T(z)G(w)
=-2
G(z)G(w)
= - c - - - + - - + reg. terms
--- + --- +
2
1
2T(w)
3
(z _ w) 3
(z - 1'1)
C 2
(VI. S.6b)
reg. terms
1
2T(w)
aT(w)
(i _ w)4
(z _ w)2
(z - w)
- - - + - - - + -.--_- +
(VI. 8. 6a)
(VI. 8.6c)
reg. terms. (VI.8.6d)
1770
In Eqs. (VI.8.6) the numbers c and c are the central charges. Their actual values will be varying throughout the chapter depending on which (super) conformal field theory we are considering.
In the previous chapters we saw that. given a classical supetconformal field theory, the stress~energy tensor and the supercurrent can be defined as variations of the action with respect to either the viel+ beins (oe-) or ~he gravitino (o~). Upon quantization of the elementary fields, out of which T(z), and G(z) are constructed, these operators are seen to fulfill the OPE (VI.8.o) with values of c and C characteristic of the model under consideration. Actually, as we have discussed at length, BRST invariance requires the cancellation of the central charges associated with the matter fields against that associated with the ghost-superghost system.
rei)
Here we take a more abstract stand-point. We define a quantum superconformal field theory directly in terms of Eqs. (VI.8.6). The operators T(z). r(z) and G(z) are supposed to exist and fulfill (VI.8.6), independent of any particular Lagrangian model. A conformal field ~(6.b)(z.z) of conformal weights is defined to be any local quantum operator having the following OPE with the stress-energy tensors:
+
reg. terms
(VI.8.7a)
+
reg. terms •
(VI. 8. 7b)
The conformal weithts (6.6) are characteristic of the conformal field under consideration.
1771 For instance, if we recall Eqs. (V1.S.B2) we see that a left~ moving Kac-Moody current l(z) is a conformal field of weights (ll.'= 1. A=0) while a right-moving;·current is a conformal field of weights (ll.= 0, a= 1). Considering next a set of NF free left-moving fermions AA(z) characterized by the OPE:
(A'= 1,2, •••• NF) ,
A
1:
i61\1:
A (z» .. (w) .. - - - - + reg. terms k z-w
(VI. 8. 8)
(where. for the sake of the present argument, k is a normalization constant) and the associated stress-energy tensor: T(z) " ~
ik T : AA(%}az" A(z)
(VI.8.9a)
(VI.B.9b) we can easily check that each of the AA(z) is a conformal field of weights (A =1/2. ~"O):
The given examples suggest that in most cases a conformal field ~(A,!)(z.i)
can be constructed as a product (VI. S.ll)
where 'PA (z) is an analytical (A.O) conformal field and 'P,6(z) is an antianalytical (O.A) conformal field. The factorization (VI.S.II) reflects the commutativity of the left algebra with the right one and allows an independent discussion of the
tm left and ~ight conformal field theories. Prom now on we shall just choose to work with the left~algebra. OPt:, (z)
If we assign boundary conditions to both the conformal field and the associated stress~energy tensor ~n(ze
2ni
) =e
Zwi(!:,+t) () ~A z
(VI.8.12a)
(VI. 8.12b)
where the numbe't" t. taking values in the range [0.1}. is nameli the twist. we can write the mode expansions: +co L T(z) = _n_ . n= .... zn+2
l
(VI.8.13a)
(VI. 8.13b)
and from Eq. (VI.8.7) we obtain the commutation relation:
(VI.8.14) which is also equivalent to (VI.8.15) Equation (VI.8.IS) is the infinitesimal form of the transformation (VI. 8.16)
where (VI. 8.17)
is an element of the Virasoro algebra and z' = fez) is the corresponding analytic transformation on the variable z. If we restrict A to lie in the SL(2,1R) subalgebra spanned by LQ• Ll , L-1: (VI. 8.18)
we obtain A
e "a(z)e
-A
= (cz+d) -2Il
<pa{ZI)
(VI.8.19)
where (cz+d)
-2 = dz'dz
(VI. 8.20)
is the Jacobian of the corresponding MObius transformation: az T b
z,
(VI. 8. 21a)
cz .. d
with
(V1.8.21b)
having defined the 2)( 2 matrix representation of the generators Ll , L_!
LO'
as I
La "
Ll "
Q
1/2
( 0
Cl
I
-1/2 )
:1
(VI. 8. 22a)
(VI. 8. 22b)
(VI.8.2Zc)
1774 Equation (VI.8.19) leads to an important conclusion. If the vacuum of the conformal field theory is Mobius invariant, i.e., (VI. 8.23)
then the correlation function (VI. 8. 24)
transforms under (VI.8.21) as follows M
n (cz.+d)
i,,1
211
G(zl •.. ·,zM)
(VI. 8. 25)
1
Furthermore, since we have
z!
1
z!
J
(VI. 8. 26)
(CZi + d) (CZj + d)
if we introduce the following differential (N-3)-form
(VI. 8.27)
we can easily verify that it is
~obius
invariant (VI. 8.28)
if and only if 6" 1.
This result demonstrates that conformal fields with weights IJ. =6 '" 1 have special properties under projective transformations. We will see in a moment that the emission vertices must have precisely these left and right weights.
1775
Furthermore the holomorphic (N-3)-form (VI. 8. 29) is just the holomorphic square-root of the integration measure d~(zl, ... ,zN) appearing in Eq. (VI.8.4):
(VI. 8.30)
In order to justify these statements we proceed as follows. We observe that in view of Eq. (VI.8.16), a conformal field with weights (A,~) transforms under an analytic diffeomorphism z =fez) as shown below: I
(VI. 8.31) Hence, a (l,l)-conformal field can be viewed as the coefficient of a (1,IJ-form defined over the world-sheet. which we can identify with a viable (conformal invariant) interaction Lagrangian whose integral can be added to the free action So of Eq. (VI.3.58): (VI, 8. 32a)
Lint(z,i) = $(l,I)(Z.f)dZ"dz
So ..... So
+
JLint(Z,z) ,
(VI. 8, 32b)
In this way we introduce the interaction of the superstring with external background fields corresponding precisely to the fields of its particle spectrum. To see this let us assume that the emission vertices for bosonic and fermionic particles have the following general structure: 8 V1,··U (8) V (k.r;.1t; z,z) " r; n(k) W
Ill' .. ].In
(If;
z,z) exp(ik Xll(z.z)) II
(VI, 8, 33a)
1776
l
(k,Il,1f : z,i) =
(VI. 8. 33b)
where
(VI. 8. 34)
is the Minkowski coordinate field from whose derivatives we obtain the conformal fields P (z) and P (i) (see Eqs. (VI.l.6) and (VI.S.12). JJ
\l
(8)
W
(If; z,i)
is a conformal field transforming as an
Ill' .. )In
indexed Lorentz tensor while
Wa
(1\' ;
Ill" ·Iln
transforming as an n·indexed spinor tensor.
Z ,z)
is a conformal field
As above,
1\'
denotes the
(BJ
internal quantum numbers.
The conformal weights of
up with those of exp(iklll(Z,z»,
to make
(1,1).
n~
(11)
Finally
~
JJ l ·· ·Iln
W and
(F)
W sum
which turns out to be {ll" ~" k2/2) , -a
(k) (Il,. ,,(k) is the polarization .. 1··· ..n tensor (spinor) of the corresponding tensor (spinor) particle. If the emission vertices have this structure the most general interaction Lagrangian is obtained by taking a linear combination of these vertices, that is. by summing up on the free parameter kJJ with independent coefficients (polarization tensors):
1m
This formula has an illuminating interpretation. The integrals (VI. 8. lOa)
(VI.8.lOb)
are the Fourier representation of a background tensor (spinor) field T(X) (~(X)) which is pulled back from the target space to the worldsheet through the embedding function (VI. 8. 37) Hence the conformal fields
(B) W Vl···]Jn(z,z)
and
(F)c£ WlJ ,,(t,i) 1" '''n
appearing in the definition (VI.8.33) of the emission vertices have a natural interpretation. They are the coupling vertices of the string fields to the background fields in the a-model Lagrangian:
(VI. 8. 38) We shall see further on that the BRST invariance of the vertices, necessary to make them emission vertices of physical particles, imposes transversality conditions on the polarization tensors (spinors). For instance, for massless fields we have kV~ k2 = 0
1JVl" ,vn _1
= 0
(VI. 8. 39a)
wd { \.11, .. \1
U a
n(kv)Q Yv B= 0
(VI. 8. 39b)
Equations (VI. 8. 39) are ordinary field equations in the momentum space. This means that the Lagrangian (VI.S.38) is BRST invariant and hence respects conformal invarisnce also at the quantum level. if and only if the backgrolDld fields fulfill the appropriate differential equations which we interpret as their equations of motion. This is the point of view underlying the a-model approach to string effective theory. Instead of calculating the scattering amplitudes of the massless modes we couple them to the string as backgrolDld fields and then impose quantum conformal invariance on the resulting nonlinear a-model. In this way we obtain a set of equations of motion for the background fields which are the same as one would obtain by varying the effective Lagrangian. In this chapter we do not pursue this point of view any further. We, rather, turn to a heuristic derivation of Eq. (VI. 8.30) which gives the integration measure to be used in the calculation of the scattering amplitudes (VI.8.4). We begin by noting that the conformal fields are in one-ta-one correspondence with the highest-~eight states Ih > of the Virasoro algebra defined by the following: (VI.8.40a) (VI. 8. 40b)
Indeed denoting by (VI. 8. 41)
the vacuum of the conformal field theory (which is a product of the left vacuum with the right vacuum) we can easily prove that (VI. 8. 42)
1779
is a highest.weight state for both the left-moving and the right-moving subalgebras of (VI.8.6). Indeed from Eqs. (VI.8.7) and (VI.S.13) it follows that +~
r
n"~Qo
1 - 2 L '(A
z.R+
n
A) (0.0) 10,0 ,
A >" -.(A A)(O,O)IO,O > +
z2
•
(VI.8.43) where BR(w,w) denote the infinite new fields corresponding to the regular terms omitted in Eqs. (VI.8.7). Equating term by term the two Laurent series appearing on left hand side and right hand side of Eq. (VI.8.43) we obtain
th~
(VI. 8. 44a) (VI. 8. 44b) (VI.8.44c) £qs. (VI.8.44a,b) prove our statement that the state (VI.8.42) is a highest-weight one characterized by h =11, while Eq. (VI. 8. 44(;) provides the definition of the new fields Bn(t,Z) called the descendants of the primary field .(A.b) (z,i). Let us now recall the mass-shell equations (Vl.S.29) characteriz.lng physical states (VI.8.4Sa) ~(matter)1
LO
phys >
= Q~I phys
L(matter)lphys > = 0 n
>
(VI.8.4Sb)
(n > 0)
(VI. 8. 45c)
f780
r(matter)/phys > = 0 n
where L(matter) n
(n> 0)
(VI. B.45d)
are the Virasoro generators associated with the stress-
energy tensor of the matter fields which excludes the contribution from the ghost-superghost system. Equations (VI.8.4S) state that a physical state is a highest-weight state created from the vacuum of the matter system by a conformal operator V(a,a_)(z,i) of weights equal to the intercepts (a,a): Iphys>
= V(ma:ter)Co 0)10 (a,a) ,
>
matter
•
(VI. 8.46)
If we recall Eqs. (VI.S.123) we see that in general the intercepts are less than one so that V(C ma: t) er) (z,i) is not a (1,1) conformal field. a,a ( tt ) In view of our previous discussion v(~~a)er (z,t) cannot be identified with the emission vertex operator. It should however be closely related to it. To find the precise relation of v«ma~t)er) with the correct a,a vertex, let us firstly rewrite the intercepts in the fOllowing way:
a ;;
L
1 -
b (field)
(VI. 8. 47a)
b(field)
(VI. 8.47b)
(fie Ids # ghost)
L
1 -
(fields # ghost) where b(field)
is the coboundary of the Virasoro algebra associated
with each field. Secondly let us anticipate a result from the following sections. The coboundaries b of the matter fields are nothing else but the conformal weights of certain additional conformal fields not appearing in the Lagrangian and named the spin fields. The spin fields S(z) are associated with those matter fields which admit two different boundary conditions (typically the fermions) and have a conformal weight equal to the difference of the Virasoro coboundaries corresponding to the two situations. Usually for one choice of the boundary conditions, we have b = 0 and the corresponding vacuum 10 >, which is annihilated by L±l and LO is SL(2, R)
1781
invariant.
For this reason let us call it the
10 >SL
vacuum. The
2
other vacuum 10 >, corresponding to the second choice of boundary conditions and a non-vanishing value of the coboundary b, is obtained from 10 >SL by application of the spin field 5(0): 2 (VI. 8.48)
This suggests that we include the spin fields in the definition of the vertex by the replacement
and that we let the new operator v(n+b,a+b) (O,O) vacuum to create the physical state.
act on the
10
>SL 2
The would-be vertex V(~+b,a+b) has conformal weights closer to one than does V( Ct.,a~)' However we are not yet there, since in Eq. (VI.8.47) there is also a contribution from the coboundary of the superghost system. This contribution is different in the Neveu-Schwarz and in the Ramond sector. Bosonizing the superghosts, as we do in the next sections, we shall discover two operators similar to the spin fields, which when applied to the 10 >SL vacuum yield either the NeveU-Schwarz vacuum 2
or the Ramond vacuum. Their conformal weights are respectively 1/2 and 3/8. Let these conformal fields be named LNS(Z) and LR(Z). If we make the replacement (VI. 8. 50)
we can conclude that every physical state is a highest-weight state created from the 10 >SL vacuum of all the fields (with the exception 2
of the diffeomorphism ghosts) by the action of a vertex which is a (l,l)-conformal field. Hence we can write
1782 Iphys
>
= VeO,O)IO,O
LOV(O,O) 10,0 >SL
2
>5L
(VI.8.SIa)
2
= Lov(o,oJlo,o
>SL 2
= V(O,O) 10.0
>SL 2 (V!. 8. SIb)
where
La. La La
=
are the following series:
I
(fields; ghost)
L~ield , LO ::
I
i.~field)
(fields; ghost) (VI. 8.52)
The rationale for treating the ghost, antighosts on a different footing from the superghosts and superantighosts is that, while constructing vertices and amplitudes, what matters is the conformal, rather than the superconformal, symmetry. Indeed, the integration is over the bosonic world-sheet coordinates zl •••.• zN. Hence we regard the superghost system as an additional conformal system contributing to the stressenergy tensor r(z) that couples to the conformal ghost c(z). In due time we shall also demonstrate how the factor
appearing in the measure (VI.8.30) can be traced back to a ghost correlation function. For the time being, we exclude the ghosts from the construction of the emission vertices which, in this way, always have conformal weights l\ =A=1 with respect to the stress-energy tensor of all the other fields.
Let us now point out that, recalling the relation (VI.3.4) between the holomorphic coordinate z and the string intrinsic time T, the point %:: ~ = 0 of the world-sheet corresponds to the very remote past T :: - lim V(k,(;.z,z)jO.O >
z"'O i"'O
(VI.8.53)
1783 is the analogue of an asymptotic "in" state of an ordinary quantum field theory. In complete analogy, an "out" state can be created fl'om the conjugate vacuum < 0,01 applying a vertex operator evaluated in the limit z,z ... ., which corresponds to the remote future. Explicitly we set
= V(O,O)IO,O
>SL
(VI. 8.S1a) 2
(VI. a.SIb)
where LO'
La ..
Lo
are the following series:
I
L~ield ,
(fields:f ghost)
Lo ..
I
(fields:f ghost)
L(field) o (VI. 8.52)
The rationale for treating the ghost, antighosts on a different footing from the superghosts and superantighosts is that, while constructing vertices and amplitudes, what matters is the conformal, rather than the superconformal, symmetry. Indeed, the integration is over the bosonic world-sheet coordinates zl"",zN' Hence we regard the superghost system as an additional conformal system contributing to the stressenergy tensor T(z) that couples to the conformal ghost c(z). In due time we shall also demonstrate how the factor
appearing in the measure (VI.a.30) can be traced back to a ghost correlation function. For the time being, we exclude the ghosts from the construction of the emission vertices which, in this way, always have conformal weights !J. .. 6=1 with respect to the stress·energy tensor of all the other fields. Let us now point out that, recalling the relation (VI.3.4) between the holomorphic coordinate z and the string intrinsic time i. the point z .. f .. a of the world-sheet corresponds to the very remote past T =~CO so that the physical state
Ik.1;;
> lim V(k,l;.Z:.z) 10,0 > z ... 0 £+0
(VI.8.53)
1783
is the analogue of an asymptotic "in" state of an ordinary quantum field theory.
In complete analogy, an "out" state can be created from the conjugate vacuum < 0,01 applying a vertex operator evaluated in the limit :L ,i -+ which corresponds to the remote future. <Xl
Explicitly we set
',w) = - - O(A,W) + reg.
(VI. 8. 99)
z-w
Comparing Eqs. (VI.S.75) and (VI.8.99) with Eqs. (Vl.S.68a) and (VI.5.68b) respectively, we are led to wonder whether under suitable restrictions, the operators O(n,z) cannot be identified with the ladder currents 10(z) of a non-ab~lian current algebra. Indeed if we do so and identify the derivative currents (VI.8.74) with those belonging to the Cartan subalgebra, then Eqs. (VI.S.75) and (VI.8.99) are the first two Correct OPEl s corresponding to a Kac-Moody algebra of level k
=2 .
(VI. 8.100)
Consider then the case of a simply laced Lie algebra a; of rank and let the system of its roots be denoted by .(~). As we know, in this case, all the root vectors
same length, which can be taken to be equal to 2.
Ij
=r
-+
have the In this way we obtain oe 4>(6)
(VI. 8.101)
and from Eqs. (VI.S.IOI) we also deduce that
.~
-+
V Cl.B e 4>(Ii)
(VI. 8.102)
we identify i\z) =O(Cl,Z) and utilize Eqs. (VI. 8.101-102) in (VI.S.92b) we get
If
1796
-+
.... .... f - -1- + = exp[-in·Mo.] (z_w)2
. ... " exp[.il! .aM8] a
a·Jf(w) + reg. } (z-w)
t - 1 a+....8(w) z- w
5
I (z) I (w) "reg.
&
(VI. 8. 103a)
........5) 2 = 2 (VI.8.103b) + reg. } if (a+
........ 2 if (0.+6) ,,2.
(VI. 8.l03c)
Eqs. (VI.8.103) together with (VI.8.99) and (VI.8.7S) reproduce the OPE's of a level k/e 2 .. 1 Kac-Moody algebra (Vl.S.68) provided that i) the system of roots ~(6) contains all the vectors of the root lattice whose length is equal to two, and
ii) the matrix M is chosen in such a way that the la(z) are bosonic (i.e .• commuting) fields.
(VI.8.104a) .... +
...
-+
V 0. B 60) exp[-i'lfo.· loiS] " N(a,8) •
(VI.8.104b)
A particularly important and illuminating example corresponds to the case of an SO(2r)" D current algebra. In this case, the 2r(r - 1) roots r are gi ven by the vectors of the form (VI. 8.105)
corresponding to the length .. 2 points of the r-dimensional cubic lattice. The ladder currents are given by fe . .t e.
I
1
J ez) .. O(te i ± e j ,z) " : O(:tepz)
oC±ej ,z)
(VI. 8.106)
1791
which together with the identity (VI. 8.107) reproduces the fermionic representation of the Kac-Moody algebra discussed in Chapter VI.S. Comparing Eqs. (VI.8.1.06) with Eq. (VI.8.96) we can now appreciate the general rule underlying the above constructions. Let us consider the set of vertex operators (VI.8.9l) where the + momentum vector A is constrained to belong to a lattice A,
A= {A =
r 71 r n.A In. e zI i=1 1
(VI.8.l09)
1
spanned by the basis vector 71 ).. •
.,..
If A1 are identified with the fundamental weights of a simply laced Lie algebra ; of rank r: (VI.8.1l0)
then the lattice A is the weight lattice of I: (VI.8.Ill)
In this case the operators O()..,z) can be grouped into multiplets. {O(A,Z} Ae R(~)}, each multiplet containing the operators associated with the ",-eights of an irreducible representation R(~) of highest weight t In particular, one always has the adjoint multiplet associated with the lattice points of length 2, that is with the roots. The construction (VI.8.96) of the free fermions shows that these latter are associated with the weights of the vector representation in the case of the algebra SO(2r).
1798
This makes a lot of sense since in the Neveu-Schwarz-Ramond formulation of superstrings the free fermions carry a vector index and transform in the vector representation of the tangent group SO(Mt l. arget In the general case of an SO(r) algebra we can write the following OPE: A
E
i AE 1 ---2 z-w
A (Z)A (w) = - - 0
_ i I:f
+
reg.
(6AfJEACw) _ 6f~JEf(w)
z-w
(VI. 8. 112a)
+
oEAJAr Cw ) _
(VI. 8. 112b)
(VI. S.l12e) (VI. 8. 112d) where the vector index A has been defined in Eq. (VI.S.S4), the free Fermi field AA(z) are given by (VI.S.SS), and the currents JAL(z) , in the vector notation, are identified with those in the Cartan-Weyl notation by the following relations:
_ s. J 2i - 1,2j _ i J 2i - 1,2j-l) J
(VI. 8. 113a)
(VI. 8. 113b)
In the equations above, 5 i denotes the sign in front of ei left hand side of the equation.
in the
1799
This suggests that in even space-time dimensions d .. 2r
(VI. 8.114)
and at the quantum level we can replace the d-free fermions ~(z) with r free bosons .,i(z) relying on the correspondence (VI.8.93). For instance, in d:l0 we can set ±i e( feZ) : e
1(z) +- W 6(z) I c(±e l ) .. ein/4rW
(V 1. 8. 115a)
±i e2' ,,(z) : c(±e2) .. ei~/4[~2(z) +- , 7(z) ] : e
(VI. 8. 115b)
±i e3" .p(z) : e(±e3) .. eiW/4[~3(z) e ±ie4' ,,(z) e
+ ,8(z)]
(VI. 8. USe )
c(±e4) .. ei~/4[~4(z) + ,9(z)]
(VI. 8. 115d)
±i e • .p{z) : e 5 : c(±eS} .. ei1l'/4 [~S(z) +- ~10 (z) ]
(VI. 8. l1Se)
Once the bosonization has been achieved, in addition to the free feTmions we started from, we can construct all the conformal fields corresponding. to all the weights of the 5O(2r) weight lattice. Among them we have the -~ spin fields S(z). They correspond to the case where A is a spinor weight. We write (VI. 8.116) r-components having set s1 .. ±1 for all i .. 1, ••• ,1'. This implies (VI.8.117)
With these notations, if we define
1800 a
-+-
5 (z) .. 0(5,Z)
(VI. 8. 118)
where a is a spinor index which enumerates the spinor weights -+s, then utilizing Eqs. (VI.8.SS) and (VI.8.92) we obtain a d/16 a T(z)S (w) ,. - - - 5 (w) (z _ w)2
1
+--
(z • w)
a
as
(w) +
reg.
(VI. a.IISa)
(VI. 8. 119b)
(VI. 8.1I9c)
Where rA are the usual gamma matrices in d=2r dimensions (see Chapter 11.7) .
To see that this is the correct result we introduce a gamma matrix baSis adapted to the Cartan-Weyl basis. We realize the Clifford algebra (VI. 8.120)
setting (VI.8.12Ia)
(VI. 8.121b) where (VI. 8. 122a) (VI.8.122b)
1801
A convenient representation of the r
:teo 1
matrices is the following: (VI. 8. 123)
1 (1 . The matnces . h were a3> a± = '2 a:t i (J2) are ord'lnary Pau l'1 matnces. (VI.8.I23) act on the spinoT space ~nich is the direct product of r 2dimensional spinors. The basis
(VI. 8. 124a)
( 1/2 \) ~
\°I
rl2 ) \°I zr { O\
S
=
, 1/2 )
(1/2 ) ~ ... e ( ° ) \ 0
® (
1/2 \
(VI. 8. 124b)
1/2
~
... ( 0 \
0)
\ 1/2 )
(0\ ® ... ®
)
1/2
~
(112 ) \
0
(VI. 8. 124c)
(VI. 8. l24d)
I
is naturally associated with the weights through the correspondence (VI. 8. lZSa) (VI. 8.12Sb)
(VI.8.12Sc)
"1802
and provides one possible way of defining the spinoI' index fl. basis and with the definition (VI. 8. 123) we see that the OPE
In this
. 1
O(±ei,z) 0(5) .. (z - w)
+ 1 -5.2' 1
2 ej,w) = .
exp{ilr M.. 53} O(±e. 1)
1
1 j + - S
2
eJ. ,w) +
reg.
(VI. 8.126)
can be rewritten as follows
Indeed, from the sign of sign s. in
(VI. 8. 126) we see that there is a singular term only when ei in the first operator is opposite to the corresponding the second operator. This guarantees that the operator
appearing
the coefficient of a singular term is again of the form
1
llS
where the signs {sjl} are the same as the previous signs {sj} except for the sign in the i-th position which is flipped. Flipping the i-th spin is precisely the result of applying the matrix r±ei and hence we are allowed to perform the covariant transcription (VI.8.127) of Eq. (VI.8.126). Actually, this is the place where we can fix the yet undetermined signs of the cocycle matrix Mij in such a way as to match completely the definition (VI.8.12S) of the gamma matrices. Combining now Eqs. (VI.8.121) with the inverted form of Eq. (VI. 8.93):
,2i-l( zJ .. 2e 1 -ilr/4{O( e,z i )
A
+
O( -e,z i )}
(VI.8.128a)
1803 1 e-i'lr/4{ -O(ei ,z) A2i (z). 2
+
} O(-ei,z)
(VI. 8. 128b)
one obtains the desired result (VI.8.119b). The result (VI.8.119c) follows next by means of a second OPE. Eqs. (Vl.8.119) encode all the properties of the spin fields. They are conformal fields with confomal weight (ll
= -d • -A. 16
0)
and because of the OPE (VI.8.119b) which introduces a branch cut change the boundary condition of the corresponding femion field lA(t). For this reason, as anticipated in the previous section, given a set of free femions we can identify the Neveu-Schwarz vacuum with the true SL2invariant vacuum. and define the Ramond vacuum by means of Eq. (VI.8.48), where S(z) is the spin field of the appropriate SO(2r) algebra iC1,R > '" lim S(X(z) !NS
z+O
>SL • 2
(VI. 8. 129)
The shift in the Virasoro coboundary from bNS '" 0 to bR... d/16 is accounted for by the conformal weight of the spin field. Different from the femions XA(z), the spin fields are not free fields, since, relying on Eq. (VI.8.92). we can write the following OPE
(VI. 8.130)
where cfl6 is the charge conjugation matrix (see Chapter II.7) and aI' a2 are suitable coefficients. (In addition to a c-number singular term we have an operatorial one.) Notwithstanding this fact the spin-field correlators
1804 can be computed explicitly by either utilizing their vertex operator representation or solving suitable differential equations imposed on the correlator by the OPE (VI. 8. 119c).
We do not touch this point.
We recall now that the spin fields are just one of the two items we were lacking in order to construct all the emission vertices. The other item is the bosonization of the superghost system. This topic is discussed in the next section.
VI.8.4 b-c Systems, Superghost Bosonization and the Background Charge
In this section we consider a dynamical system described by the following action:
s = - 1. J[- Abdc 211"
+
e:(A-l)cdb]
~
e
+
(VI.8.13!)
where e: -=!l is a parameter which decides the statistics of the fields b(z,i), c(z,i):
= - e: c(w,w)b(z,z)
(VI. 8. 132a)
b(z,i)b{w,w) " - e: b(w,w)b(z,i)
(VI. 8. 132b)
c(z,i)c(w,w) " - e: c(w,w)c(z,z) .
(VI.8.132c)
b{z,z}c(W,w)
With such a convention,
€"
1 corresponds to Fermi statistics while
e: =-1 corresponds to Bose statistics.
Furthermore, A is a parameter
lIhlch corresponds to the conformal weight of the field b, while the field c bas conformal weight (I-A). Let us see how this happens. We observe that the equations of motion obtained by varying the action (VI. 8.131)
a
b
=0
(VI. 8. 133a)
a
c
=0
(VI.8.133b)
1805
imply that b = b(z), c =c(z) are both analytical fields and that the presence of a second-class constraint leads to the following Dirac brackets: {c(z),b(w)}
= __1__
(VI. 8. 134a)
=__E:__
(VI. 8. 134b)
z-w
E:
{b(z),c(w)}
z- W
13:
{c(z},c(w)} = {b(z),b(w)}
=0
(VI. 8. 134c)
which, upon quantization, translate into the following OPE's: c(z)b(w)
=-1- +
reg.
(VI. 8. 13Sa)
b(z)c(w)
= _E_ + reg.
(VI. 8. 135b )
c(z)c{w)
= reg.
(VI. 8. 135c)
b (z)b (w)
= reg.
(VI. 8.13Sd)
z-w
z-w
Calculating the stress-energy tensor:
(VI. 8.136)
one finds 1"
+
= T++e+
+ T
+--
e
(VI. 8. 137)
where (VI.8.138a)
1806 T+~
=0
(VI. 8.13Sb)
and by direct evaluation one obtains {T(z) .b(w)} .. _A_ b(w) + _1_ ab{wl (2: - 1'1)2 2: - W I-A
{T(I1.),c(w) }
II
---
(z-w)2
(VI. 8. 139a)
1 + - - Clc(w)
c(w)
(VI. 8. 139b)
z-w
which prove our statement that A and (I-A) are respectively the conformal weights of b(z) and c(z). Upon quantization Bqs. (VI.8.139) become the following OPE: T(z)b{w) .. _A._ b(w)
+
T(z)c(w) .. -i-A - - c(w)
+ -
(z _ w)2
(z _ 1'1)2
_1_ ab(w) z- w 1
Z-
1'1
ac{w)
+
reg.
(VI. 8. 140a)
+
reg.
(VI. 8. 140b)
Calculating. next. the OPE of the stress-energy tensor with itself, we find that (VI. 8.141)
fulfills the algebra (VI.8.6a) with the following value of the central charge (VI.8.I42)
where the number Q .. (1 - 2 A)
(VI.8.143)
is named the backgromd charge, for reasons that will be apparent below. Here we point out the deep geometrical significance of Q, recalling
1807 the formulation (VI. 2.329) of the Riemann-Roch theorem which can be restated as follows: #
zero modes of b{z) - " zero modes of c(z)
= - Q(g-l) (VI. 8.144)
where g is the genus of the surface and Q is the background charge (VI. 8. 143). Indeed. by zero-mode we mean a classical field b.(z) (or ci (z) which when multiplied by the differential (dz)A (or l(dz)l-A) defines a A (or (I-A) holomorphic differential:
p~A) = b. (Z)(dz)A
(VI. 8. 145a)
(I-A) Pi = ci (z)(dz) I-A •
(VI. 8. 145b)
1
1
With this understanding (VI.8.144) is precisely the same as the RiemannRoch theorem: dim HCA ) _ dim H(l-A) H(q)
= - Q(g-l)
(VI. 8.146)
denoting the vector space of the holomorphic q-differentials.
The most important values of the parameter ;. and listed in Table VI.8.1.
TABLE VI.S.I b - c Systems
A
2 3/2
I-A ~l
-1/2
E
Q
c
1
-3 -2
-26
-1
11
1
0
1
-1
-2
1/2
1/2
1
0
1
E
are those
1808
In the given order the cases enumerated in the table correspond to the system of reparametrization ghosts (J.. .. 2). supersymmetry ghosts (J,. .. 3/2), gauge ghosts (J,. = 1). and finally (J,. .. 1/2) to a pair of physical spin 1/2 fermions. The last column of Table VI.8.1 allows a straightforward calculation of the critical dimensions for the various string theories. As an exemplification we can evaluate the critical dimensions of the N=2 superstring. The N=2 supetmultiplet is made of two scalars x).I,.p).l and two fe1'lllions ~ (A= 1.2) corresponding to the Xli (z,6) superfield: (VI.8.147)
Hence each space-time dimension gives a contribution 3 to the central charge:
2
1 2
t
t
t
t
Xll
~
",IJ 2
(z) we need an additional conformal system of central charge c" -2. This is provided by an auxiliary b-c system with A" 1 and e:" 1 (see Table VI.8.1). We designate the corresponding fields n(z) and ;(z) and obtain from Eq. (VI.8.l41) (VI. 8.187)
T(n~) ez) " - nCz)
J,I'
emits
~
l-
-
(VI.S.255c)
J,I
emits
.. ).aA
(k;z,z)
(VI. 8. 255a)
(VI. 8. 25Sb)
~ AA
(k;z,z)
(q)aA Vu
g.lJv· BJ.IV' D
~
(q)a
(q)A
..
(VI.
s. 2SSd)
emits In the above formulae, the index A runs on the adjoint representation of the gauge group (either 50(32) or ES ® ES). The emission vertices must fulfill many requirements.
have conformal weights t:.. A = 1 as I\"e thermore, they must be BRST invariant:
They must have stressed many times. Fur-
(q)
[QIlRST'
V
(z,z)}
-= 0
(V1.8.2S6)
in order to create physical states. be
chosen in such
a
Finally. their normalization must way that the corresponding state (VI. 8.257)
has unit norm
~
1.
(VI.8.258)
1840 Naively one might think that we can just choose the canonical picture and Set q =1 for the bosonic vertices and q =1/2 ror the fermionic ones, forgetting altogether the other pictures. This is not true since in all amplitude calculations the correlation functions (ql)
SLz
= --------- x
x e
(VI. 8. 276c)
i~(zl)
iipCz3) <e
e
1 >=--
z3 - zl
(VI. 8. 276d)
(VI. 8.276e)
(VI. 8. 276£)
1847
Eqs. (VI.8.276) fOllow the colinearity conditions (VI.8.269) and the assumed transversality of the vectors f~i): (VI. 8. 277) The explicit form of the correlators which we have given above can be obtained by utilizing a general formula that holds true for the expo· nentials of free fields. We call free any Set of conformal fields A.(t) that fulfill the 1 following relation: (VI.8.2?S) where < Ai (Z)Aj{W) > is the two·point function. can always write the identity
lIIexp(l ortant point to exhibit the relation between the arbitrariness existing in the field equations and the one we are facing here. A particular way of selecting the possible interactions all~~d by supersymmetry is obtained by demanding that the CUrvatures should take the new minimal form (i.e., Eqs. (VI.9.434-436» also in the presence of Chern-Simons. This essentially means that the auxiliary field S is set to zero by authority. Whether this is a wise choice and whether string dynamics really chooses this possibility is not yet clear, at least to the authors of this book. It is anyhow a very interesting possibility, as in this case all the formulae simplify dramatically and one is able to write all the SUSY transformation rules in terms of the same objects as for the 10D case, namely, the torsion field t. which becomes the only carrier of arbitrariness. a Let us then specialize our results to the new minimal supergravity. In t becomes very siDq>le this case the structure of the corrections W. W a since it reduces to purely fermionic terms. Indeed, as it is apparent from Eq. (VI. 9.434-436) the choice of the new minimal model S:; ~ EC =0
2021
and the field redefinition implied by the choice K 2 " -I, 1i(f)
+ ipi'(z)
... -i8~Zl(z,.i)
(V I.I0.45a)
= pi(i) + ipi'(E) = -i8.. Zh(z,z)
(V 1.10.(56)
'Pi(:) = piC:)
where Zl/R(z,z) are two triplets of complex coordinates describing two different T8 tori (the left and the right ones). In general we cannot identify Z1 with Z~ bec:ause the boundary conditions fulfilled by 'Pi(:) and PO( E) are different as a consequence of the non-equality of b with b in the elements (b,6) of the group B. Therefore we conclude that Z1 and Zh are in general chiral free bosons describing what is named an asymmetric orbifold {29c]. In those spec:ial cases where 'Pi(z} and 1>i(E) are given equal boundary conditions we can set Z1 = Zi and we obtain a symmetric orbifold. Let us now consider the table VI.IO.n where we have listed a few examples of type II superstring models. In each case we have written the explicit form of the additional 'Yi generators and the associated element (b,6) of the B group. The corresp,?nding orbifold is immediately read off by comparison of eq.s (VI.I0.41) with (VI.10.26). To see how this works let us first note that in the iO, 'Y+ and 'Y_sec:tors the currents (VI.IO.45) are all periodic:
zi =
'Pi(ze2lri ) == 'Pi( z)
(VI.I0.46a)
1>i(1e-211"1) == pi(Z)
(V 1.10.(66)
the same way as in the O-sec:tor. Therefore if we do not introduce any ii boundary vector (case K ::::: 0), then the orbifold is just an untwisted T6 torus. This is consistent with the fact that the target space SUSY can be chosen to be N=8 by appropriately fixing the e,u: signs. Therefore we just need to analyse the twisting introduced by the additional generators ii. Consider for instance the second case of table VI.IO.II. In the i1 sector we find:
2041
pi(:e2!l'i) = - P(z) (i =1,2) r(ze2"") == p3(Z) pi(Ie-2'lri) = pi(E} (i =1,2,3)
(VI.10.47a) (VI.I0.47b) (VI.I0.47c)
Since pie:) and pi(l) obey different boundary conditions they cannot be regarded as the holomorphic and antiholomorphic derivatives of the same free field Zi(z,I). For this reason we get an asymmetric orbifold. The right-movers are the coordinates of an ordinary untwisted torus The left-movers, illStead, correspond to an orbifold T'/Z2 where the Z2 identification group ads as follows:
re.
Zz: {
Zi
zi
=* =*
-Zl Z!
(i =1,2)
(V1.10.48)
With a similar analysis we see that, in the fourth case of table VI.10.II, we have a Z:a which acts on both the left and the right coordinates according to eq. (VI.I0.48): Z • :I.
{Zlzt =* -Zl =* Z!
Zh Zk
==> - Zit (i =1,2) ==> Zl
(VI.I0.49a)
plus an additional Z~ that acts only on the left coordinates as it follows: (i
=1,3)
(VI.lO.49b)
In this way one can proceed and work out all the correspondences listed in the third column of table VI.lO.lI. The general rule is easily obtained. The boundary vectors 8' and 8" correspond, respectively, to the Zz and Zz' twisting of eq.s (VI.IO.48) and (VI.10.49b). The boundary vectors band b', on the other hand do not introduce additional twistings but modify the zero-mode spectrum: they correspond to a translation in the moduli space of the orbifold. To understand the fourth column of table VI.lO.U we observe that the local world-sheet supercurrent (VI.I0.15a) can be rewritten in terms of the orbifold currenl$ (VI.I0.41) and of the free fermions (VI.l0.43) as follows:
G'Q ...1(Z)
= eit v'2¢x(z)px(z)
(VI.lO.50)
Introducing the SO(6) Kat.Moody currents: (V 1.10.51e)
2042
JXY(z)JZW(w) ::;; _i
12
2
(.; - 111)2
(6XZ6YW _ 6XW 6Y.1)
(6 XZ JYw(w) _ 6xW JYZ(1,O)
+ 6YW JXz(1,O}
_ 6YZ JXw(1,O})
';-1,0
+ reg.
(VI.IO.SIb)
under which ",X(z) transfotms as a vector
and pX(.;) is inert (VI.I0.S3)
we see that Glocfll( z) is just one element of a multiplet of 6 supercurrent5 transforming as a vector under SO(6). These currents are part of a chiral superalgebra that we name n ;; 6: it is not properly a superconformal algebra since it includes generators of weight higher than h == 2, yet it is characterised by 6 generators of weight h". 3/2 and for this reason the name n ::; 6 is justified. If we do not introduce additional generators 1i, all the 6 supercurrents a.re global symmetries of the internal conformal field-theory which is, therefore, properly named it. (6,6) theory. The addition of the boundary vectors 8' and 8" breaks, successively the superconformal symmetry to n == 4 and to n ::; 2. How this happens will now be described in detail for the left-movers. Repeating the same analysis for the rightlIlOVet$ one arrives at the results listed in the fourth column of table VUO.I1. The relation between world·sheet supersymmetries and target supersymmetries is encoded in the gravitino vertex whose construction involves the SO(6) spin field. Given the 8 )( 8 SO(6) gamma ma.tl'ices:
(VI.I0.54) we can introduce the 8-component internal spin-field EP(z) , which is defined by the following OPEs:
",x(z)E.f>(w) =
!e-i~ (fX)PQ 2
JXY(z)E.P(w)
= iV2 (rXyt~ I:~(w) 2
.
T(z}EP(1,O)
EQ(w)! + reg. (z-w)t (; - til)
+ reg.
!. 1 P = (z-w 8 )2 tP(w) + --DE (w) + reg z-w
(V 1.10.500)
(V /.1o.s6b)
(V J.IO.55c)
2043
+ _l_e-ii (rxct~
1/IX(W) 1 + reg. (z - w),: (VI.IO.55d) where C is the charge conjugation matrix and T(z) i3 the stress-energy tensor. As one sees from eq.(VI.IO.55c) the spin-field ,£P(z) has conformal weight equal to 3/8. Taking the 3uperghosts into account, the vacuum state corresponding to the boundary vedor 8, is given by: :EP(z):EQ(w) == -
iv'2
1
C pq
(z - w).
2v'2
(VI.lO.56) where ~·g(z) is the free boson introduced by the superghost bosonization (see Chapter VI.8) and S4(z) is the four-component space-time spin-field that is defined by the following OPEs:
1/I1'(z) So,{w) :; !e-if (")'I')c,,; 2
i·
T(z)SQ(w) ;:::; ( )2 S"'(w) z-w
SP(w) (z - w)i
+ reg.
1· +- &S"'(w) + reg. z-w
(V I.I0.57a)
(V I.IO.57b)
(V I.IO.57c) being the space· time gamma matrices and C the associated charge conjugation matrix. The values of the coefficients in the expansion (VI.10.57c) are fixed by consistency with (VI.IO.57a) and with the OPE:
"(I'
'l/J1'(z)1/I"(w)
=
1 rJ"" -i---
2z -
(VI.1O,5S)
w
Utilizing the definition of eq.(V1.10.26a), eq.(VI.10.56) suggests that we introduce the following unprojected gravitino vertex in heterotic superstrings: (VI.IO.59) On this massless \'ertex the GSO projectors of eq.(VI.I0.39) that correspond, respectively to N::::4, N=::2 and N=l supersymmetry take the following very simple ga.mma matrix form: (V I.IO.60a)
2044
(V 1.10.60b)
pWs~l)
=
~(1 + 75rd~(I- i15r33.)~{I-
i')'sru*)
(VI.IO.60c)
In order to analyse the action of these projectors on the vertex (Vl.l0.59) it is convenient to introduce the following decompositions of the 50(6) and 50(1,3) spinors into two-component objects:
(VI.1O.61a)
(VI.lO.6Ib) This decomposition corresponds to a splitting of both seta of gamma matrices into 2 x 2 blocks. The basis we utilize for the two Clifford algebras is the following: 50(6) Olifford algebra
r,
~ (.! Z 0'3
T)
0
0 0
il13
0 0
-i0'3
0
0
(J, r. ~ (.! r2
rl -
""
-iO'l
0
0
0
0
0
0 -i112
i 112 0
0
0
fa
=
0=
COl it) o
0 0 1 0 0 o -1 0
(-~,
0 0 0 -()'2
;
0'2
0 0 0
i)
fa-
=
(i~
n
T) T)
0 iO'l
0
0 1
0 1 0 1 0 0
=
a
10'2
C· o
0 -it 0 0 0 0 it 0
rT = oo o
0 1 0 0 -1 0 0
C·
(VI.lO.62a)
(V 1.10.62b)
-~t )
(V I.I0.62c)
jJ
(VI.lO.62d)
2045
SO(l,3)·Cli1ford a.4ebra 7& =-
G~)
C ::
n
j
~1)
0 7;'" ( fI; j
75 ==
-fl") 0•
i
=1,2,3
(-~O"Z i~2)
(V 1.10.634) (VI.lO.63b)
where 0"; denote the standard Pauli matrices. For the space-time spinora we also adopt the standard convention:
(VI.IO.54) Inserting (VI.lO.54) and (VI.IO.63) into (VI.10.60) and applying the projectors to the vertex (VI.10.59), we obtain the following result for the physical gravitino vertices: gravltino vertex in tbe N=4 heterotic theories
(VI.lO.6Se)
(V I.I0.65h) gravitino vertex in the N=2 heterotic theories
(V I.I0.66e)
(V 1.10.66b) gravitino vertex in the N=l heterotic theories (VI.IO.67e)
(V I.10.67b) These vertices have exactly the form of the gravitino vertices discussed by Dixon and Banks [9] from an abstract conformal field theory point of view. Following the line of thought of those authors, the internal spin-field E(z) is the key to understand the relation between target space supersymmetry and the extended world-sheet superconformal symmetries in heterotic superstrings. According to their general argument, one must find the following scheme:
2046
j) In the N=4 case, the world-sheet supersymm.etry should be given by a chiral algebra containip.g six supercurrents which, under the action of an appropriate SO{6) s;:j SU(4) Kae-Moody algebra transform as a 6-veetor: under the same KM algebra the spin-field lJP(z) should transform as a chital four-component spinor. A chiraI algebra. with six supercurrents necessarily contains generators of conformal weight h > 2 so that it is larger than a purely conformal algebra, whose maximal superextension is given by four supercurrents. The explicit structure of this chiral algebra is not particularly relevant, since the same argument wbich implies its existence implies also that the corresponding field theory is free and hence immediately solvable. il) In the N=2 case, the internal conformal field theory should be the direct sum of two theories, namely some representation of the n=4 superconformal algebra with central charge e = 6, plus the representation of the n=2 superconformal algebra with central charge c = 3 that is provided by a complex Cree superfield. Under the SU(2) Kae-Moody currents belonging to the n=4 superalgebra, wbich we denote ji(z) (i == 1,2,3), the internal spin field E"(z) appearing in the gravitino vertex (VI.lO.66) should transform as an isodoublet: ji(z) I:"(w) =
_! (O'i)" ~(w) + .,.eg. (z - w)
2
(V 1.10.6&)
(V I.I0.6Sh) It is v.-orth to point out that under the action of the same SU(2) currents also the four supercurrents should arrange into an isodoublet (see eq.(VI.I0.73e,f». On the other hand, under the U(l) Kae-Moody current contained in the n=2 superalgebra, whlch we denote j{z), the spin field should transform as follows: j(z)~(w)
=
1 I:A(W) 2(z _ 10) 1 ~(w)
j(z)~(w) "" 2(z _ w)
+ reg.
+ reg.
(V 1.10.694)
(V I.10.69b)
Furthermore, always relying on the same general argument the OPE of the internal spin-field with itself must be the following ones: (V1.10.7(4)
(Vl.1O.70b)
'" - v'2 (z
e"·
s
- W)i
+ .,.eg.
(VI.IO.70c)
2047
where tr3(Z) is the lowest component of the complex free superfield which generates the c == 3 representation of the N=2 superalgebra. ill) In the N==l ease the interna.1 degrees of freedom must span an n:;:2 superconformal theory with central charge c =0 9. Under the U(l) current of the n=2 algebra, the spin fields E(z) and Et(z) appearing in the gravitino vertex (VI.IO.67) and identified respectively with 1:1(z) and E2(z). should transform as follows: .
j(z) E(w) :::: j(z) 1:t(w) ==
1 E(w}
-2 (z
_ w)
I 1:t(w)
2(z
_ w)
+ reg.
(V I.lO.7la)
+ reg.
(VI.10.71b)
It is fairly easy to identify these algebraic structures within the explicit conformal theories generated by the :£ermionization of the C!'-model$ on SU(2)3-group folds. The N==4 case is very easily discussed: indeed the 50(6) Kae-Moody algebra. was already identified ill eq.s (VI.I0.50.51). The OPE (VI.l0.55b) verifies the statement concerning the internal spin field. Finally the theory is clearly a. free ollC since it is described by a set of free fermions. Let us then focus on the other two eases and begin by writing our normalizations for the n=2 and n=4 superconformal algebras. T.be n=2 algebra
T(z)T(w)
c
=:
2 (z
1 _ w)4
2T(w) _ w)2
+ (z
8T(w)
(V 1.10.724)
+ (z-w) + 'reg. 3 1 g:!::) T( Z )g:!::() w::: 2 (z _ w)2 {w
T(z)j(w)
_() g+() z g w
+ (zag:f:(w) _ w) + reg.
(VI •10.1_ .... L)
= (z-w 1 )2 jew) + ( 1 /;(w) + reg.(V 1.10.72c) Z-1O =
2
'3 c (z
1 _ w)3
2j(w) _ w)2
+ (z
2T(w) + 8j(w) (z-w ) + reg. .:!: g:t(w) j(z)g (w) = ± ( ) + reg. z - w j(z)j(w) ==
i (z .: w)2 + reg.
+ (VI.lO.72d) (VI.IO.72e)
(VI.10.72 f)
The n=4 algebra c
= 2(z
T(z)T(1II}
1
- 111)4
(8T(1II» z - 111
2T(1II) - 111)3
+ (z
+ 1'eg.
(VI.10.13Il)
+ (OO"(1II} ) + reg. Z - 111
(VI .10.7'.>~) ow
= 2! (Z-1II 1 )2 0"(111) + (00"(111» + reg. Z-1II
(VI.lO.13c)
T (Z )g"() 111 = -23 (Z - 1 111)2 g"() 111 T(z)0"(1II}
+
T(z)ji(1II) = ( 1 )2 ji(1II) Z-1II
ji(z)gtl(1II) == ji(z)04(1II) ==
+ (Z-1II 1 ) ai(w) + reg.(VI.10.73d)
i g6(~111~(')6" + -! ((J;r~O~~) + :)
reg. .
(VI.lO.73e)
reg.
(VI.lO.13/)
Given these normalizations, the problem of identifying the n==2 and n=4 algebras implied by N=2 target supersymmetry reduces simply to the problem of finding the correct embedding of the corresponding SU(2) ® U(l) Kae-Moody algebra. into the 50(6) current algebra (VI.IO.51). We set
j(z) ==
~J33'(z) :::
4:
Al3 )(z)xi3)(z) :
(VI.IO.74)
and
(VI.IO.75a)
= t/11~(z)t/12'(z) + t/11(z)t/12(z)
(VI.IO.7Sh)
2~ (J 1*1 + J22') == t/11* (z)tP1(z) + t/12(z)t/12* (z)
(V I.10.75c)
2-/2
j2(Z)
(J
+ t/12'(z)t/11(z)
jl(z) = _1_
1' 2
= 2~(il'2'
t(z) =
+ J2'1) + J 12 )
== t/1 1*(z)t/12(z)
2049 which are easily seen to fulfill the OPEs (VI.lO.12f) and (VI.lO.73h), respectively. The specific choice of the SU(2) generators encoded in eq.s (VI.10.75) is motivated by the fact that with respect to this particular subgroup the complex 8-dimensional spinor and the real6-dimellsional vector representations of 80(6) branch as follows: ~ -+
2 (1) (9 4 (1)
(V 1.10.764)
~--+1(9i(92(1)
(V l.l0.76b)
so that both the spin field E"( z) and the following combina.tion of free-fermions:
(VI.lO.77) transform simultaneously as two component SU(2) spinors under the KM currents (VI.10.75). Indeed from eq.s(VI.IO.55b} and (VI.10.51), utilizing the gamma matrix basis (VI.10.62) and the definitions (VI.IO.iS) we obtain both eq.s(VI.1O.65) and :
. j'(z)IP"(w)
IPb(w) = - -21 (')d /1' + reg. (z - w)
(VI.10.78)
Eq. (VI.lO.78) is vital in order to construct a set of supercurrents behaving as an isospin doublet as it is required by the structure of the n=4 superalgebra (VI.IO.73). To complete the identification of our superalgebras we introduce the following complex spinor:
(VI.IO.79) whieb is just the lowest component of the free complex superfield predicted by Dixon's theorem and appearing in the opera.tor expansions (VI.10.iO). Let us complete our programme of identifying the extended 5uperalgebras (VI.IO. 72) and (VI.IO.73). We set
T(n=2)(Z)
= ~ (p3(z) p3(z)
+ pS" (z)p3* (z) )
_ iy.3(z)&P3(z) - iy.3"{z)&p3*(z)
= ~J~}(z)J(1)(z)
-
2i'\~)(z)8'\~)(z)
(VI.lO.80a)
g(;.=2)(z) == eit (y.3(Z)
+ iy.3*(Z») (p3(z)
- ip3"(z») (VI.IO.80b)
g~=2)(z)
- i1jl3*(z») (p3(z)
+ ips'(z») (VI.tO.SOc)
= eii (,p3(z)
T(""'4)(Z) ==
t[~ (pi(z)Pi(z) + pi"(Z)pi*(z») i=1
2050
2
=
~ [~J~(Z}J(1)(z)
''In=4)(Z) ::: eft { (1/1 ' (%) - (,,2(%) _
2iA~)(Z)8At~")(Z)]
-
+ i,pl'(Z») (p2(z)
(V I.10.81a)
- ip2t(z»)
i,,2'(%») (pl(z) + iP)'(;»)}
(VI.1O.8Ib)
"l..=4)(Z) = cit { (1/1 (%) + i1/l2'(z») (p2(z) - iPZ*(z») 2
(V I.lO.81c) gln=.)(Z)
= e-if { (,pl(z)
- (,,2(Z)
+ iP2*(z»)
- i,pl'(%») (p2(Z)
iP1"(z»)}
+ i1/l2'(z») (pl(z) -
0'. . 4)(Z) = e'f { (,,2(Z) - i1/l2'(z» + (1/1 1(%) + itjlP(z») (pI(Z)
-
(p2{Z)
(VI.IO.81d)
+ ;p2"(z»)
iPI'(z»)}
(VI.10.8Ie)
It is tedious but straightforward to verify that with these positions, together with the positions (VI.lO.50) and (VI.lO.51), the algebras (VI.IO.72) and (VI.I0.73) are satisfied with the correct values of the central charges: ct,.=2)
=3
;
ct ..=4)
=6
(V 1.1o.s2)
The internal part of the local supercurrent (VI.lO.IS) can be expressed as a linear combination of the global supercurrents by means of the following identity:
VI.10.5 - The h-map Gepner has pointed out that, given a modular invariant type II superstring model, there is a general procedure that associates with it two modular invariant heterotic superstring models, respectively based on the Kae-Moody algebra of the group SO(26)
2051
and on the Kat-Moody algebra of the group SO{lO) @ Ea. The idea is that of substituting the c '" 13 conformal system composed by the superghosts (c :::: 11) plus the space-time fermions 1/>1'(z) (c = 2)-.with a e ::::: 13 conformal system of matter fields whose character transformations under the modular group are isomorphic to the character transformations of the SO(2) Kac-Moody algebta spanned by the transverse 1/>/-'{ z) fields (p. ::;: 3,4). The reason why SO(26) and SO(10) ® Es are seleeted is twofold. On one hand they have both rank;::;: 13 and therefore, at the lowest level klfP ::= 1, they yield a c ::= 13 conformal system. On the other hand, as we show below, their characters transform isomorphically to the characters of SO(2) and, as a consequence, they are interchangeable with these latter without spoiling modular invariance . Amore geometrical way of understanding these two particular groups was pointed out by Gepner and relies on the interpretation of the d "" 4 superstring models as compactifications of the d "'" 1() model on a suitable 6-dimensional manifold. The argument is that on the same 6-manifold M6 (leading to the same internal conformal field theory with c = 9) we can compactify both the type II superstring and the heterotic superstring. In both cases the left-sector contains the six interBai fermions 1/>X(z) (X ::= 1,2,3,4,5,6) that couple to the background spin-connection wXY(X). The difference resides only in the right-sector where, for the type II case we have the six internal fermions ~X(z), also coupling to the spin-connection, while in the heterotic case we have the 32 heterotic fermions ~P(I) coupling to the gauge field AP1(X) of either SO(32) or SO(16) ® SO(16) (in the second case the group SO(16) ® SO(16) is promoted to Ee @ E8 by the twisted sectors}. In the heterotic case, in order to cancel the q-model anomalies one has to introduce the Lorentz Chern-Simons term into the definition of the axion field-strength 'H and to satisfy the Bian(:hi identity:
d'H = TrRI\R -
Tr~l\~
(VI.IO.84)
A general way to do thls is to set 'H = 0 and to identify the background spinconnection w with the background gauge field A, so that one gets:
TrRI\R =
Tr~/I.~
(VI.10.85)
Thls is the geometrical counterpart of the heterotic h-map from type II superstrings to heterotic ones. Indeed, identifying the background spin connection w with the background ga.uge field means that six of the thirty-two heterotic fermions are assimilated to the six internal fermions ~x (f). The remaining 26 heterotic fermions couple to the gauge field of either SO(26) or 50(1O)@E8' which is the normalizer of SO(6) (the internal holonomy group) in either 50(32) or E3 ® Es. This norma.lizer is the residual massless gauge group in the d :::: 4 model. In the ease of special internal manifolds the holonomy group can be a proper subgroup of SO(6):
1iol(M6}
c
SO(6)
(V 1.10.86)
2052 In these cases the normalizer in SO(32) or Ea 0 Ea is larger and we have an enhanced gauge group. A typical case is provided by Calabi-Yau compactifications, where ?iol(Ms) :::: SU(3) and as a consequence SO(10)0E8 is promoted to Ee 0E•• Other examples are provided by the orbifold compactifications where ?id(M.) becomes a discrete group and the gauge group is even larger. In particular on Zs orbifolds it becomes SU(3) 0 E& 0 E•. What we want to point out is that this geometrical interpretation of the h-map holds true also in the SU(2)'-approach, where the internal manifold is 9-dimensional, rather than 6-dimensional. For a generical 9-manifold which is an exact zero of the beta function (not necessarily the group-manifold SU(2)3 ) the holonomy group is ?iol(M,) :::: SO(9). The gauge group coupling to the heterotic fermions, on the other hand, is either SO(35) or SO(19) 0 SO(16) (possibly promoted to SO(19) 0 E. by the twisted states). After identification of the spin connection IN with the gauge connection A, the massless gauge fields are associated with the normalizer of the holonomy group SO(9) in either SO(35) or SO(19) 0 E8 • This normalizer is once more equal to SO(26) and to SO(10) 0 E. respectively. The actual gauge group is enhanced if the holonomy group ?iol(M,) is a proper subgroup of SO(9) (possibly a discrete subgroup as in the case of the orbifolds). As in. th!, previous case, what we are actually doing by identifying IN == A is to assimilate 9 of the heterotic fermions to the 9 internal fermions 1/Il(z) (1=1,... ,9) of the type II atring. Let us now describe the h-map in technical terms. As we are going to see, in the fermionizable models we have been discussing, the h-map admits a particularly simple and elegant formulation directly in terms of the boundary vectors and the c[:l coefficients. Denoting by: (VI.lO.S?)
[:1,
the partition function of 2n free fermions endowed with the same spin structure the characters of the level one SO(2n) Kat-Moody algebra can be written as follows:
B1'''>( T) = Z(2 ..) [~] (r)
+ Zl2 ..}[~] (1')
(V [.lO.88a)
B~2n)( T) =Z(21l) [~] (T)
- Z(2,,) [~] (T)
(V 1.1O.88b)
B~2n)(1') =Z
where the index i = 0, v, 8, S denotes the singlet, the vector, the spinor and the conjugate splnor representation respectively.
2053
Under the one-loop modular group, the characters transfonn as follows: ~2"){"
+ 1)
B~2")(_~)
:::: Ti~") B~2")(,,)
(V 1.10.894)
s!r> Bj2")(T)
(V 1.10.89b)
=
where
(VI.lO.90a)
(VI.10.90b) The algebraic basis for the SO(26) h-map is contained in the following identity pointed out by Gepner: T.+,g->.} =2Lo +2).]0 + 3{A - 4
Since for any state 11fr}
we find
(V I.10.110a) Therefore in the NS sector:
(VI.lo.nOb) and in the R sector
2060
(V I.IO.ll0e) By definition crural and. antichiral fields 4i are defined respectively by
g+(z) 4ic/';""'(lII) "" reg. g-(.;) t ....tic.\·....'(lII) ;:;: reg. Equivalently, crural and antichiral states are defined by:
G~ t wki""')
=0
G: w,ntie.\;".') "" 0 1
•
±l
Using eq. (VI.IO.ltOa) with A "" we lee that chiral and antichiral states in the NS sector saturate the first inequality of (VI.I0.110b). i.e.:
If the state It) is also primary we have (VI.IO.Ula) G~_t!4i ....tiC'dT"'} = 0;
n;::: 0
(VI.10.l11b)
The conditions (VI.IO.I1la) or (VI.lO.11lh) remove half of the superpartners of the field t. In the superfield language these conditions are equivalent to
where D:I:. are the covariant derivatives of the N=2 supersymmetry (see Sect. II.a.3). Analogous statements hold for the right moving sector. Of interest in the construction of massless multiplets is the case h = corresponding to q "" ± 1. Following Gepner [11, and Dixon, Kaplunovski and. Louis [11], the abstract (1.1)forms are identified with the weight h == ii = ~ primary chiral-dUral fields
t,
(a "" 1.2.......,hl,l) while the abstract (2,1 }-forms are identified with the h "" antichiral fields :
(a ;:;:
it ""
1.2 ........ h2•1 )
! primary chiral-
2061 hI,! and h~,l are the abstract Hodge numbers of the (2,2)... theory and X 2 (hl,l - h'l,l) is its Euler characteristic.
=
a
By applying the right-moving n=2 superalgebra to the tI';: ~1)( z, i} primary fields one obtains: '
:l:(l,t) -
±(hl) - +
1/2 T-(-) Z tI'.. 1, ±l (w,w) "" (i _ tli)2 tI'" 1, ±l (w,w) (z
~ tli) 8,.tI':(1:11)(w,tli) + r-eg. (VI.I0.112a)
~(-) ::1:(1,1) ±l Z tI'" 1, ±1 (w,w) == (I _ tli)
J
:I:(l,l) -
tI'.. I, ±l (w,w)
+ reg. (VI .lO.112b)
(}:I: (z) t;=(l:'ll)(W,tli) "" reg G'" (I) tI'; (1:'11) (w,tli) (}± (z)
-
t;(t~) (w, tli)
(VI.IO.112c)
== (I _1 tli) 41; ,., 8/h
o:!)
+ reg(VI.IO.1l2d)
(j ~ tli) tI';(1:'11)(w,tli»
+ 1'eg (V I.I0.1l2e)
. . (1 1)
G'" (z) 41; ~:O (w, tli) == reg where the new primary fields
(w, tli)
(V I. lO.ll2f)
9; (i:D (w, tli) are the upper components of the short
n=2 multiplets whose lower components are the tl't'Uil){Z,Z). The one-te-one correspondence between these t~ sets of primary fields is the world-sheet counterpart of the natural one-to-one correspondence between the moduli and th~ charged multiplets one obtains in the Kaluza-Klein compactification of D = 10 supergravity on Calabi-Yau complex 3-folds. To complete the list of ingredients entering the construction of massless multiplet vertices we also need the primary fields of 50(10)@E~ current algebra, corresponding to the (0,0)0,13 theory of table I and the primary fields of the conformal field theory on Minkowski space (2,0):;1"'·. The 50(10) ® E; current algebra is generated by the set of 10 $16 free fermions introduced in the previous section and denoted /P'(l) (p :;;: 1,2.... 10) and e(l) (I == 1,2, ''', 16), respectively. Introducing a gamma matrix basis for the 80(10) and 50(16) Clifford algebras:
{rl', r'}:;;: 25"
(V I.I0.113a)
{ rI , rJ } ==
(VI.10.nab)
2S lJ
2062
such that the corresponding charge conjugation matrices have the form
r ll = (~
~1)
;r
(!
=
17
~1)
(Vl.1O.114)
the 50(10) and 50(16) spin fields :
Sl(z) = (S1(z») s (E) SM (z)
(AlA
= 1.........,16)
(V 1.10.115a)
= (;~~;D (M,M = 1, ........,128)
(V1.10.11Sa)
satisfy the OPEs:
P (l) SA (w) ...
~ e-;f (E
_ lUi)1/2 (rp)Ab s,& (Ui)
+ (z
T(,i)SA(Ui);;: (z 5/8_ Sl(w} _ '10)2
+ reg.
_1 Ui) 8t7>Sl(-) '10
(V 1.10.11611)
+ reg. (V 1.10.116b)
SA{,i)SB(Ui)
=
OAB
(z - Ui)i
+(
cons~ 1 (rp)lB P(Ui)
f -
+reg
'10)8
(V 1.10.116c) 1.~ I
1 (l)Mfl!l (z _ Ui)t r s (Ui)
1 M (,i)S (Ui)
e
=2
t
(z) Sit (Ui)
= (z-w 1 -)2 SM (tV)
C- •
+ (_1:-'10 1 _)
+ reg.
8wS M (Ui)
(V l.lO.l16d)
+ reg (V 1.10.116e)
As one sees the 50(16) spin fields have conformal weight equal to one and together with the bilinear currents e(f}e(z), close OD the Era current algebra. On the other hand the SO(10) fields will enter the construction of massless vertices assigned to Ea irreducible representations. Following the notations of chapter VI.S for the space-time spin-fields we are now ready to write the emission vertices for all the massless multiplets contained in the e1fC(tive N=l supergravity. We have the following multiplets: 1) The gravitOtJ multiplet (2,!) containing the graviton k,." and one gravitillO ",p.a
2063
!)
2) The axion multiplet (0+,0-, containing the dilaton " the mon hi 2 supermultiplets, while in group-spa.ce they correspond to the completion of ET representations. In order to find the analogues of eq.s (VI.10.118-124) and (VI.lO.127-128), we still have to discuss the analogue. in the (4,4) theory, of the wral primary field multiplet of the (2,2) theory described by eq.s (VI.lO.1l2). This analogue is given by the short representation of the n = 4 superalgebra provided by the following set of fields (a = 1,2; a= i,2):
1JI[~,t]«(z,.i)' .[~ll]"(z,Z)' n[I,l]"(ZIZ) 2'2
2,0
~,O
which, together with the generators of the right-moving n = 4 algebra, satisfy the following OPE's:
T(E)'i
1 [1 1]"4 [1 1]G4 [1rrI]Qil(1IJ,w)::--L..( __ W)2'i rr (w,w)+(E~_)8tb'i rr (1IJ,w)+reg. 2'2
Z
2'2
1IJ
2'2
(V I.10.138a)
2070
~
_
[12,1
_
1
t,oJa(w,w) + (Z_tli)8Il>II [1t,O]4(w,w) +reg. 2,1
T(z)II ~,OJ4(w,w):: (Z_tii)2 11 ['
_
1
2,1
_
(V1.1G.13&)
(VI.IO.13Se)
(VI.IO.l38h)
(V I.10.138i)
gi(Z) II
1].(wW)=6 a...( I}i[M]~6(w,w) [1ttD' H ) +reg. (f-tli) 2'
ai
(V1.10.138j)
Let US now enumerate the short multiplets whose lowest component is l}ii
[I'1~1]""(.tIZ) 2'2
by an index i taking the values i=l,..., hl,l where hl,l is the abstract Hodge number of the (4,4)(8,6) theory. Indeed if this latter can be geometrically interpreted as
describing a non linear O'-model on a complex 2-foJd of vamshing first Chern class, i.e. on a complex manifold of SU(2) holonoID1, then hl,1 is the number of harmonic (1 ,I)-forms on such a manifold. For this reason the q;j can be named the abstract {l,I)-forms of the (4,4)(G,8) theory.
[ttJ
2071
These abstract (l,l)-forms enter the construction of the emission vertices for the massless hypermultiplets in the 56 representation of E? To each of these families of E7-charged hypermultiplets we can ass?ciate a hypermultiplet of ET singlets, namely the corresponding modulus. This follows from n == 4 world-sheet supersymmetry since the vertices of the moduli hypermultiplets are constructed with the 11) .1] and
[i
II [i'~l associated with each qi 3 ~ 'l Let us discuss the supermultiplets appearing in an N their emission vertices. We have
Ii'll.
t
2.0
= 2 heterotic model and
1) The graviton multiplet (2, 2
y,.c."'(k,z, z) =et~··(z)
sa (:)1 [1::] "'(Z)1 (t:)(z) Pi'(z)e[ilt.X(Z,i)! (y [,1O.140c)
3) Graviphoton ED Axionpboton vertex
Next, if we replace in eq.s (VI.IO,140a-d) the conformal field P"(z) with the Kac-Moody currents jA(z) we obtain, in the given order, the emission vertices for the corresponding gauge bOSOll$, gauginos and gauge scalars completing the vector multipl~t of S0(10) ® E'a. Similarly, if in the same equations we replace PV(z) with either j'(z) or J(z) we obtain the SU(2) ® U(l) group that is needed to promote SO(10) to ET• To see how this happens it suffices to recall the decomposition of the ET adjoint under SU(2) ® U(l) ® SO(10). We have: 133 SU(2)@~~SO(lO) (j
=I, Q=0,1)
ED (j =0, Q= 0,1)
ED (i=O,Q=0,45) ED (j=O,Q=l,lO)
(YI.IO.141) The vertices corresponding to the first three terms in the decomposition (VI.I0.141) have already been identified. Those associa.ted with the last three representations are provided by the spectral ftow in the right sector. For the gauge bosons we have:
2073
A) (1=0. Q=l, 10) gauge boson vertex
V"±P(k,z,z) =2e''fr/4. eW'(a) ,pIL(:) 1 [~:~] 1
(::±;)
(1) UP(z)e[ik.X(.,i)!
(V I.10.142a)
B) (J=h Q=t, 16) gauge boson vertex
VIL~(k, I:,z) = 2e,,,/4 ei."(a) ,p"(z) 1
[::ir (~: D
(1) SA (z) elik.X(a.JI)
{i)l
(V 1.10.1426)
0) (I=!, Q=-t. 16) gauge boson vertex
VIL,;,..t(k, Z,%) == 2 e iw/ 4 ei."(z) ,pl'(,,) 1
[~:
ir
(E)I(:"_\) (1) SA (E) elik,X(z.J}(
(V1.10.142c) and repeating the same right-moving spectral flow on the vertices (VI.10.140b-d) we obtain also the corresponding gauginos and gauge scalars. Let us now discuss the hypermultiplets. We begin with the E7-families in the 56 representation, that are in one-ta-two correspondence with the abstract (1,1)forms of the (4,4k. theory. To understand their structure it suffices to recall the decomposition
-
56
SU(2)~~~SO(10) (j =
1 -
••
!,Q == 1,1) 2
1
$ (j ==
!,Q = -1,1) 2
_.
1-
$(J=Z,Q=O,lO) $ (J=0,Q=-2,16) $ (J=O,Q=2,16) (VI.IO.143) As it is evident from eq. (VI.l0.143) the 56 representation is pseudoreal so that 56"" 56.
The emission vertices for the doublet of complex scalars that sit in each hypermultiplet families are given below == == 0, 10) :
or the 56
(j t, q
(VI.I0.143a)
(j == 0, q = -~, 16) :
2074
(1- ... 0, Q- "" 1-) 2 .16 : (V1.10.143c)
11;m;;'(k z •
I,
!]m;;. (.I i)l (1) 0, 2 elik. X(z,!»)
i) = ei."(~)~. [ 12':1 •
!2'2!
i
'
0I ±l
=l,2, .... kJ ,1
(VI.I0.143d) (V I.1O.143e)
and the left-moving spectral flow can be utilized to complete the N =: 2 hypermultiplet. For instance the spin one-hall partners of the = t, Q... 0 t scalars a.re emitted by the following vertices:
(J
Via "'P(k,z,i) ... eH'·(z) So.
10)
(Z)~i (~: tJ "'(z,i)l (::~)(z) (11'(i)e1iJo •
X (z,ii)]
(VI.1O.144a)
V. a;;"(k,z,z) =: e t .'·(3) sa (z)t'i
[t:]
;;'(.;,i)1
ct~)(Z) IJP(i)elU.X(Z,i»)
(VI.IO.144b) In a similar way one deals with the other cases. Next we consider the hypermultiplets that are E,.-singlets. In order to be neutral under Er the conformal field associated with these states must be chara.cterized by
]=0, Q=O Hence the general structure of the vertex fur the E7-singlet scalars is
(V 1.10.1454) (V I.IO.145b) One possibility corresponds to h2 =: 0, h.. = 1. In this class we retrieve the moduli associated to the (I,l)-forms via the right-moving supersymmetry transformations (VI.10.l38). The remaining singlets in the same class can be interpreted as describing
2075
the deformations of the tangent bundle to the hyperKihler 2-fold tha.t supports the (4,4)6,$ theory. This interpretation, however, is not possible tor the singlets with
h2 ;' o.
r
This fact shows that, in general, we cannot understand the spectrum of an N = 2 heterotic model by regarding it as a compa.c:tification on Ks ® T2 • Indeed the statement tha.t every (4,4)6,6 conformal field theory corresponds to a non linear O'-model on either T4 or K a, these being the only complex 2-folds with van· ishing first Chern class, is correct only if the partition function of the (4,4)... theory is modular invariant by itself. This is not necessarily the caSe in the N =2 heterotic compactifications we consider, since the non-inva.rlance of the (4,4).,6 partition function can be compensated by the other factors appearing in the full partition function, in particular by the factor corresponding to the (2,2)s,3 theory. The models that appear in our classification give an explicit confirmation of these statements.
Finally we have the vector multiplets of the enhancement group. In analogy with equation (2.27), the gauge bosons of GE are emitted by the vertices V,.1J(k, %, i) == 2e'1f/4 e'''''(') 1/I1'(z) 11 [O~,h~ }(f) .1 J
eO,h:)(z) eli A
·X( ••• )]
(V 1.10.146a) (V 1.10.146b)
The currents! (V 1.10.147) close the Kac·Moody algebra of the GB group. The general structure 'of GE is the following. The subset of currents with hJ = 0 and hI == 1 close a subalgebra that corresponds to the affine symmetry of the (4,4).,6 theory. Similarly the currents with hI = O,hl =1 close the sub algebra of the (2,2h.3 affine symmetry. If there are no other currents GE is a direct product (VI.IO.148) If we have also currents with both hI =1= 0 and hJ '# 0 then Gs is not factor· ized and the group G~) ® G~) is promoted to a larger one by the additional mixed genera.tors. This phenomenon occurs in the examples of our classification.
2076
VI.IO.S Embedding ofa (2,2)9,9 into the direct sum (4,4),,8 e (2,2)8,8
=
Eventually we are interested in the N ::; 1 truncation of the N 2 models due to an additional aso projection preserving just one of the two gravitini. For this reason it is convenient to look at the (4,4).,8 + (2,2)3.3 theory £rom a (2,2)9,9 standpoint. Indeed this is the same as decomposing the (N '" 2) ® Br supermultiplet& into (N :0: 1) ® &6 supermultiplets. The n .. 2 algebra with c :: 9 is obtained by setting
j(z)::: 2j3(Z) + J(z) T(C=9)(Z)::; 7(.=6)(Z) +T(=3)(Z} 0("c=9)(Z) ::; g[.=6)(Z} + gt=3)(Z)
(V [.10.1490.) (V [.lO.149b)
G(=9){z) "" gtc:oG)(z) +g(~=3)
(VI.IO.149d)
(V 1.10.149c)
Analogue definitions are introduced in the right.moving sector. Eq. (VI.I0.149) implies that the U(l).charges (q,q) of the (2,2)9,9 theory are given by:
q=2m+Q
(VI.I0.150o.)
q:::2m+Q
(VI.IO.150b)
while the free bosonic fields 'P( z), ~(oi) are identjfied with:
(V 1.10.1510.) (V [.IO.151b)
The available (1,1) and (2,1) forms of the (2,2)9,9 theory are easily traced back to the (l,l)·forms of the {4,4)a,6 and (2,2h.3 theories. Indeed, recalling that for a (2,2)s.3 theory the only chiral primary fields are the spectral flows of the identity, we obtain:
(V 1.10.1520.)
(VI.1O.152b)
20n (V!.l0.IS3) Hence the (2, 2)9,9 theory that can be embedded in the (4,4}'.G + (2,2h.3 theory is necessarily non-chiral since it has x::::: O. Its truncation, however, might be chiral if the GSO projection is devised in such a way that it removes a different number of families and antifamilies. Actually, in the set of theories we have classified, this phenomenon never happens and also the truncation remains non-chiral. Let us remark that equations (VI.IO.IS3) can be group-theoretically understood. Indeed under N :;::: 1 supersymmetry, every N :;::: 2 hypermultiplet branches into two Wess-Zumino multiplets:
(V 1.10.154) while an N multip!et:
= 2 vector multiplet branches into a vector multiplet plus a Wess-Zumino
(
2~1
2[0+1,2[0-1
)
-t
(!)2 e (0+ '~0-)
(VI.I0.155)
At the same time, under U(1) ®Ee the 133 and 56 representations of Ey branch as E.0U(1)
133 ==> (78, Y
) = O)e(l, Y::::: O}e(27, Y == -2)e t¥i \27, Y = 2
(V I.I0.156a)
56B·~(l)(27,y = l)e(27,y = -l}e(l,Y:;::: 3)e(1,Y:;::: -3) (V I .10.1560) where Y denotes the U(l) hypercharge. Hence every hypermultiplet in the 56 of B1 contributes both a 27 family and a 27 anti family. In addition we have an extra antifamiiy coming from the N ::::: 2 vector multiplet of E1 •
2078 VI.10.9 Classification of'the SU(2)S groupfold realizations of the internal conformal field theory
In this section we consider the explicit construction of the internal superconformal field theory by use of the SU(2)* groupfolds. Correspondingly we obtain the classification of the available internal superconformal field theories by means of classifying the available type II superstring vacua. The h-map, whose free fermion formulation is explained in section VI.lO.5, allows the immediate identification of the corresponding heterotic: vacua. We recall that the set of free fermions spanning the (c == g,t :::: 9) internal superconformal field theory is given by the 18 left·moving fermions xt(z), ~t(z)
(A == 1,2,3; i::: 1,2,3) ,
(VI.10.157)
=1,2,3; i = 1,2,3) ,
(V 1.10.158)
plus their right-moving copies (A
where A enumerates the generators within each SU(2) and i labels the three SU(2) factors. The fake fermions xt(z) and xt(z) originate from the fermiomzation of the SU(2)3 -Kae-Moody currents (see eq.s (VI.I0.12), while the true fermions At(Z) and it(!) are the world-sheet superpartners of these currents. The normalization of the OPEs for this set of fermions is given in eqs.(VI.10.13), while the form of the local supercurrent is given in eq.s (VI.IO.15). Here, our primary concern is that of classifying all inequivalent boundary vector groups E that are consistent with multiloop modular invariance and with the specific Corm of the supercurrent that follows from the geometric interpreta.tion of the free fermion sY5tem as describing propagation On an SU(2)' groupfold. The groups 5 are freely generated by a set of generators
hAl :: h'",1'+,1'-,Y;}
(i :: 1,2, •••K)
t
(VI.I0.159)
where 1'0'1'+,1'-, that correspond to the universal N=8 type II superstring model , were defined in (VI.I0.24-28), while the additional generators "Y' are model dependent and have to be found. By means of a computer program we were able to classify all the inequivalent maximal sets of 7i generators that can be adjoined to the universal system {Y",/+,1'-} . The result is displayed in Table VI.lO.III, where the information is codified as we now explain. We begin by observing that the basis of the generators /i can always be chosen so that in each 7. botb the left-moving and right-moving transverse space-time fermions 'ifJT, ~T T = 1,2 have Ramond boundary conditions. Boundary vectors of this sort are named RR. vectors. Then, arranging the left-moving fermions according to the order shown below
2079 Mink
SU(2).
1fJT
x~ 6i-5
0
~
xi
.\'i
xl
6i-4
6i·3 '
6i-2
6i-l
A!• 61
(V1.10.160a)
and the right-moving fermioDs in an analogous order, the structure of a 'Yi boundary vector will be the following
(V 1.IO.160b) where z .. !f;,Zhii,Yl,Zi are 6-component vectors that represent the boundary conditions of the fermions belonging, respeetively, to the first, seeond and third SU(2) group both in the left (zi,lIi,zi)and in the right (ii,y.,Zi) sector. For each SU(2) group!old the structure of the supereurrent with Ra.mond boundary conditions allows a choice of eight possibilities that we enumerate aecotding to Table VI.IO.IV. The reason why the possibilities are only eight is that, once the bound· ary conditions on the fake fermions X1 (z),X2 (z),r(z) have beeen assigned, those on the true fermions ..V (z), A2(Z), >.3(Z) follow. Hence we have the choices
xJ
x~
x~
#
1
1 1 0 0 1
1
0
0
1
1 1 1
0 0 0 0
1
0 0
I
2
0
3 4
1 0 1
0
5 6 7
(V 1.10.161)
xt
where the sequence of Z2 numbers giving the boundary conditions is interpreted as the complementary of the base two trlLllscription of an integer number in the range 0:5 n S 7. Let us also remark that for the SU(2)' supercurrent of the usual tree fermion approach, the Dumber of possibilities would be much larger. It is clear from these considerations that every generator 1'i C8Jl. be codified by a sequence of six digits a;yzzyz , where z == 0,1,2, ...7 identifies the twist state of the first SU(2) in the left sector, !f does the same thing for the second SU(2) and so on; i,y,% give analogous information in the right sector. For instlLllCC, we have 'Y"
t->
000000 ; 1'+ tot 333000 ; 1- ..... 000333 •
(VI.I0.162)
2080 Note, however, that ")'+,;_ are not RR boundary vectors as all the other generators 1'i (i;;:: l,2••.K) are. The boundary vectors (V1.10.162) have to be added to each of the systems of four or five generators displayed in Table VI.lO.II1, in order to get the corresponding complete maximal system. Thus, the boundary vector group Em... generated by each maximal system is usually (Z,)T and occasionally (Z,),. The subset of RR vectors form a group with respect to the following composition rule
aob
= a + b + "),,,
(V1.10.163)
The notations in (V1.10.161) are chosen in sueh a way that eq (VI.IO.163) can be easily implemented in a computer program by use of the logical function
ioj
= iorj -
(VI.IO.l64)
iandj ,
where i,J E {O, 1, ...7} are the digits composing & boundary vector. For instance, in the BI system of Table VI.lO.lII, the composition of the first two generators yields 71 o"'h ;: 1164160553300 =
445116 .
Obviously every maximal system generates all the subsystems one obtains by deleting one or more generators. The list of all the possible systems and subsystems is not yet equal to the number of available type II models (and hen« of internal superc:onformal theories ), since one has still to take into account the various choices of the free signs encoded in the f:AI: tensor defined by eq.s (VI.I0.34-38). The explicit realization of the superconformal theories corresponding to the various boundary-vector groups and , in particular, the identification of the primary fields entering the construction of the massless vertices, will be discussed promptly. However," before considering the details we want to describe the general features of the classification encoded in Table VI.10llI. We begin by explaining the grouping of the maximal systems into the three classes named A,B and C. In every system the first two generators 71 and 1'2 are chosen in such a way as to make a sequential projection N 4 -+ N =: 2 - N ... 1 on the space-time supersymmetry generated by 1'+ • As the reader can check by inspection, the left-moving part of the generators 71 and "'h has a universal form, within each class A,B and C, respectively given by
=
A
=?
{
71 == 636···' 663iyi • B {71 ::: 1l6:i!ji . C {11 == 116.i!Yi 553···' 454···
"'h
=:
~'!Iz
=?
1'2 ==
~lIZ
=?
12 ...
~yz
(VI.lO.165) Looking at tables VI.lO.1I1 and IV we see that the vectors 71 and 1'2 are both octets of left-moving Ramond fermions for class A, while they are one octet and one decuplet for class B, and two decuplets for class C. This fact has some important consequences on the general features of the corresponding superstring vacua.
2081
In the systems of the type A, each projection introduces new massless sectors Dot present in the parent theory. This implies that the effeclive N=l supergravity, corresponding to the system containing both "11 and 12, is not a truncation of the higher N supergravity corresponding to the system without "11 or "Il' In the systems of type B, one of the projections (that generated by the decuplet 11) usually does not introduce new massless sectors, while the other projection (generated by the octet "12) does. This implies that the N=l effective supergravity can, in most cases, be regarded as a truncation af an N=2 supergravity. Finally, in the systems oC type C, both projections usually do not introduce new massless sectors, so that the N",l effective supergravity is most often a truncation of N=4 supergravity and ,sometimes, a truncation of N=2 supergravity. Each system in Table VI.10.m yields a (2, p) internal superconformal field theory, where p can take the values 0,2,4,6 (see Table VI.lO.V), corresponding, in a type II superstring interpretation, to Nrlcht = 0,1,2 or 4 • Nrisht is the number of gravitinos emerging from the right sector. As remarked earlier, in the heterotic interpretation, the same values of p correspond to gauge groups of the type
having defined E5 = $0(10). Here we focus only on (2,2) vacua. We have considered all subsystems of the (2,2) maJCimal systems that are still (2,2) theories. This involves also an appropriate choice (case by case) of the free signseAt. Some of these theories are bosonizable, in the sense that the 18+18 fermions are permanently coupled in pairs having the same boundary conditions and can, because of that, be replaced with 9+9 free bosons. This is always the case for type A systems and for a fraction of the type B systems, while no C system is bosonizable (see Table VI.10.VI). Bosonizable models yield a gauge group of maximal rank rank G
1 =22 ~ rank G B = 9 - '2' ,
(V [.10.166)
and admit a transcription in terms of covariant lattices (see chapter VI.S). They are particularly manageable because, after full bosonization, the analysis of their spectrum and the construction of the primary fields is easily implemented on a. computer. We have analyzed all the (2,2),,9 compactifications generated by the bosonizable boundary vector groups E. Table VI.10.VII shows the type of massless spectra that emerge from these explicit constructions. In the first three columns we list one half of and the Hodge numbers h1 ,1 and 2.1 • The fourth column the Euler characteristic lists the number of deformations of the tangent bundle, abstractly defined by:
h
(V I .10.167) The fifth column lists the enhancement groups. Finally, in the six column we list an example of a model that has the type of massless spectrum in consideration. The
model is completely specified by assigning the boundary vector group E and the choice of free signs. The group E is identified by selecting a subset of a. maximal system of Table VI.lOlI!. For instance, A2(123} means the subsystem of A2 generated by 11012,")'3 • The choice of signs is specified as follows. By convention one always sets f:++
e+i
=f: __ == t+_ ... t:_+ ::: 1 == f:-i ::: -1
(i;:;:; 1,2, ••• K)
(V 1.10.168a) (V I.10.l68b)
The reason of this choice will be motivated later on , while discussing the detal1s of the bosonization. Next, for the signs tij we set
(VI.I0.169) unless a specific component 'ij is listed after the E group. The listed components aze assigned the value -I. For instance. in the model {A2{123}23} we have tii =t12 ;:;:; t:l~'" I, e23 = -1. The remaining components aze determined by the symmetry (V 1.10.170) In all the chiral models, changing the sign of &12 has the e1fect of exchanging the role of the moduli h1,1 ..... h2 ,1 so that the Euler chazacteristic changes sign
x -+ -x .
(VI.IQ.171)
Therefore all our chiral models exhibit the mirror symmetry (VI.lO.In), already observed in a. class of Calabi-Yau :manifolds embedded in weighted projective spaces. The Ilon-bosonizable models, based on left-right paired systems (LRP) can be analyzed in the following way. To each complex left-moving or right-moving fermion we substitute the corresponding lelt-moving or right-moving free boson. To each pair of left-right coupled fermions we substitute instead the primary fields of the critical Ising model. In this wa.y the final superconformaI field theory is spanned by a. certain number of free bosons tensored with a certain number of critical Ising models. As a result, the rank of the gauge group is decreased, and one has
rankGB
=9 -
1 1 -p - -nLl~ ,
2
2
(V 1.10.172)
where nLR is the number of Jeft-right pairs. An explicit example is worked out in detail in Appendix C. As far as the left-right unpaired systems (LRU) are concerned, we have not yet tried to construct the associated superconfonnal theory. Although the one-loop partition function is well defined and one may also calculate the spectrum, yet no definition of the emission vertices and of their correlation functions is available. Another remazk is in order. The existence of LRP models shows that the free fermion constructions aze not a subset of the free boson models and that not all the free fermion theories can be retrieved in the covariant lattice approach.
2083
Finally, we point out that type A systems are common to the SU(2)3 groupfold supereurrent and to SU(2)$ supereurrent of Antoniadis et al. [6,8]. Systems of type B or 0, instead, are compatible only wi~ our supercurrent and not with the SU(2)' supercurrent. ,. VI.10.10 Details ot the SU(2)3 grouptold construction with emphasis on bosonization
With the conventions we ha.ve adopted, the gravitino vertex (VI.lO.USa-b) and its right moving counterpart, that is the emission vertex (VUO.122a-b) for the (16,q::::: 3/2) gauge boson of E., are respectively located in the "Y+ and "Y- sectors. This allows a stralght£orward identification of the ¥I(Z) and ~(z) U(l).bosoDs of the (2,2).,9 theory and of the superalgebra. generators. Let us focus on the left sector, since everything is repeated identic:a.lly in the right sector. The 18 free fermions of eq. (VI.IO.160a.) are naturally split by 7+ in the two disjoint subsets of 6 and 12 fermions respectively, as discussed in section VI.I0.4. The sextet eonwns!
x: ,A:
(i
=1,2,3)
(VI.I0.173)
In the new basis provided by the definitions (VI.lOA1) and (VI.lO.43) the OPEs characterizing the internal superconiormal field theory are given by:
(V 1.10.17441) (Vl.1O.l74b)
(VI.1O.174c) and are clearly invariant under 50(6) linear transformations:
t/lx(z) = AXY t/lY(z) pX(z) ;::: A XY pY(z)
(V 1.10.175)
A E SO(6)
In pameular one c:an consider matrices AXY :::: n XY E ~ C 50(6) that belong to the permutation subgroup of SOC6). H we restrict ourselves to the universal system ho,7+,7-} we have a (6,6) superoonformal field theory and N=4 space-time &upersymmetry. We step down to N=l supersymmetry by a sequential GSO projection -t N 2 -+ N = 1 , going through a (4,4)6,6 Ell (2,2)3,3 theory. Hence the generators of the n ::;: 2, c 9 superalgebra are alwa.ys given by formulae (V1.10.149) where one utilizes, as generators of the n == 4, c == 6 and n =: 2, c == 3 algebras. those defined by eqs. (VI.lO.80) and (VI.lO.81).
=
=
2084
There is however a proviso: the "'X (z) and pX (x) to be used in the quoted formulae are not, in general, those originally defined by eqs. (VI.lO.41) and (V1.10.43)j one should rather use those rotated by a &Uitable permutation matrix XY E $ C 50(6). The choice of XY depends on the choice of the maximal boundary vector system (BVS) and is related to the varioU$ ways one can arrange the Six fermions (VI.10.41) into perma.nent pairs having the same boundary conditions within all the boundary vectors of the system. Indeed, although not all the BVS are fully bosonizable, yet in all of them the six fermions (VI.lG.(1) are permanently coupled in suitable pairs. In type A systems the matrix is the identity matrix and, furthermore, the 18 fermions are always coupled in the same pairs, namely each >.f with the homologous Relying on this property, We can introduce 9 left-moving free bosou:
n
n
n
xt.
(VI.IO.176) defined by the following bosonization formula: e['Fi ,,",t 1- A(4)]
= ei7r/ C Vi
[>.t(z) ± i xt(z)]
(V I .10.177)
where the eoeyele factors have been supressed. An a.nalogous set of 9 right-moving free bosons ~1(i) is provided by the rightmoving analogue of eq.(VI.l 0.177}. In type A systems by comparison of eqs. (VI.IO.177) with (6.2) we get: e['F; 'Ps(:)]
= e;"/4
(+I(z) ± i tf/l'(z»)
(VI.10.178a)
e[ 'Fi .,.8( ')1 == ei "'/4 (+2(z) ± i tf/2' (z})
(VI.I0.178b)
el'Fi ,,9(.)]
(V 1.10.178c)
= ei1t /
4
(+3(z) ± itf/SO(z»)
The permutation n we alluded to few lines above is determined by requiring that eqs. (Vl.lO.178) should hold true (in the permuted basis) for every BVS. Indeed it suffices to keep +2 fixed and find out which of the remaining +x fermions is paired to it: this is the new +1'. Next one repeats the argument with the remaining fermions and determines n. For instance in all the systems of the type B one finds that the left n is given by the matrix U shown below:
1
2 3
u= 12" 3"
1 0 1 0 0
2 0 0 0 0 0 1 0 0
3 P 2" 3" 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
0
(V LI0.178)
2085
while the right n can be either U or the identity. In any case we rea1ize that the splitting of the (=9 theory into a c=6 E9 c;::3 theory is permanent in all the B-type systems. Indeed the property ot the matrix U is that of keeping the directions 3,3" fixed, just making the following permutation: 1 -+2 1*-+1
(VI.IO.179)
Type C systems are characterized by the appearance of more general permutations. Having fixed the conventions (VI.lO.178), the SO(6) spin field EP(:) can be written as follows:
~ = e[iC:I: 'P,:I:.".:I: 'P.>j
p = 1, ... ,8
(VI.IO.180)
Comparing equations (VI.10.1l7) and (VI.10.l40b-c) with the form of the gravitino vertex ~ven in eqs. (VI.10.67) and (VI.lO.66), we conclude that:
(V [.10.181a.) (V I.lO.181h)
(V I.lO.18Ic)
(VI.IO.181d) where the values &=1,2 of the spinor index correspond to III == -1/2, 1/2 ,and similarly for the dotted case. Equations identical to (VI.lO.181) hold in the right sector. From these identifications we learn that T(Z) and O'(z) are linear combinations of 'Ps, 'P6 , 'P9 and we also verify eqs. (VI.I0.151), yet we cannot specify which combination is which field, this being a matter of convention. Indeed, doing this is the same thing as specifying which of the 8 combinations of ± signs in (VIolO.ISO) corresponds to the first spinor component, which to the second and so on. Our conventions are fixed by the choice (VI.IO.16S) of the f+A and £_/\ signs which we now motivate. Let us first discuss the form of the OSO projection operator in bosonized form. Let A = o o,A 7A1 (1./\ E Z2, be any element of the h-mapped E group; the GSO projector on the corresponding heterotic sector is given by :
E:.
2086
(VI.I0.182) where 6~ is the statistics of b and where the coefficients c[.!]Aet are determined in terms of the e~D tensor, by first computing their type II counterparts employing the modular invariance rules (see eqs. (VI.IO.sa) ) and then by applying the h-map encoded in eq.(VI.lO.I04). Consider next the bosonization of the space-time fermions and of the 10+16 heterotic fermions. This is done by setting:
= e(itt/4) exp [Ti rpf+i(1}] = e(itt/4) exp [=Fi y>D(:)]
[,,7'=1(:) ± i ",T=:2(:)] [g2H(z) ± i 02i{f)]
(VI .10.183'3
=
= Ai ;:; --12
(V 1.10.190a)
= ).9 ;:; -21
(V I.IO.190b)
).$
Recalling eqs. (VUO.181) we conclude that the U(l) and SUeZ} free bosons related to the spectral How phenomenon are given by:
(V 1.10.1914) (VI.1O.191b)
(V 1.10.191c) The choice (5.12) of the E-A signs implies identical identifications for the right moving fields '9; m= H>'3+Ae);
(VI.10.193a) (Vl.l O.193b) (VI.I0.193c) (VI.IO.193d)
Q = >'9
m' =
H.\, - ).6);
-
The numbers m' ILlld in' corresponding to the momenta in the -r',T' dire3, f{'6, 1(>1,1(>8, 'P9, both in the left and in the right sect}>r. The remaning 8+8 fermions can be left· right paired, namely we can distribute them into pairs composed of a left and a right fermion that have the same boundary conditions in aU boundary vectors. These left-right pairs are the following ones corresponding to as many critical Ising models: 11 .... (1,3)
12 ..... (2,4)
13
1, -+ (4, iO) I, ...... (8, i)
....
(3,9)
15 .... (7,2) IT -+ (9,8)
(VI.I0.C8)
I, .... (10,7)
Ea.ch Ii (i::=1,...8) Ising model contains the corresponding left and right Majorana fermions wi(z),ciii(z) identified by eqs. (VI.I0.C8) and in addition the weight h = h:::: f& twist fields ~~(z,z) obeying:
.'
w'(z)~~(w, w)
::::
50j
.
1
(z-w)' 6ii
.'
cii'(z)~~(w,w) =
~~(W, w)
+reg.
.
1
'E~(w,w)+reg.
(z -w)' . ' 5ij [ 1 . I:±(z,z)~~(w'W)::;;---l (z-w).w'(w)
Iz-wlt
+ (z-w).w'(w) l'
]
trego
(VI.10.C9) In terms of the 10+10 free bOiona plus the 8 Ising model conformal fields, we can write aU the vertices of our model. The appropriate representation of the fermion number operator in the LRP systems is given by: (V I.I0.GIO) where >.[IIJ is the momentum on the reduced lattice spanned by the free bosons associated with the permanent left-left and right-right pairs of fermions, while denotes the reduced boundary vector restricted to the left-right paired fermions. As we see eq. (VI.10.ClO) is the generalization of eq. (VI.lO.188). In our case ).!b] has 1+5 left and 1+5 right components while hClli 8+8 components. For the left-right paired fermions we have:
"Ii
"If
ntK
nLK
i=1
i=1
(_lpi. F = II (_lpiFi-'iV'. = II (_lpi(F;-i'o)
(V I.10.Gll)
where nLR denotes the number of these pairs, namely of critical Ising theories. The action of the fermion number (-Il' -1\ is defined as follows:
2100
(-lli-I'i"l(z) = -w'(z)(-1t·- t , (_1)F,-l'i Wi (Z)
=-wi(z)(-ll'-I';
(V I.IO.Cl2)
(_l)Fi-i';E~(z,i) = ±E~(z,l)( -It;-FI In this way we obtain the obvious generalization of eq. (VI.I0.199) expressing the GSa operator in the LRP systems. In the model under consideration, the spectrum of the N=l theory can be calculated in this formalism and yields the result presented in Table VI.I0JUV. The gauge group 01'1=1 is the non regularly embedded subgroup or the 01'1=2 gauge group, given by eq. (VI.10.C4) :
where: 90(3)1090(3)11 C 90(6)
(VI.lO.CI4a)
U(I)! C 9U(2)
(VI.I0.C14b)
U(l)l1 ®~ C ET
(V I.10.C14c)
Note that the 90(2)3 factor has been suppressed altogether. This accounts for the reduction or three units in the rank of GE. The additional reduction of one unit is accounted for by the non regular maximal embedding (VI.10.Cl4a). Since, as we have claimed several times, the massless sector of the N::::1 theory is a truncation of the N=2 theory, it follows that the massless states we found are a subset of the N=2 massless states. Indeed in Table VI.l~.XlV we show which N=l multiplet comes from which N=2 multiplet. This has the implication that the combination of critical Ising fields appearing in our massless vertices can also be rewritten in terms of exponentials of the free bosons associated with the deleted rank
(i.e. f(JhV'2'V",V'5,fPh ~2' rp",fPs).
4,
For example in the sector (see Table VI.10.XIII) we have a WZ multiplet in the (2,2) representation of 90(3) 0 90(3) whose emission vertex is: V( k, z, i)
=eilc.x( _,I) eW'( r) e(i/2l( "3(~)+v>.(~)+'1;;1(l)+ •• 2:i:~'(!)1 l:;(%,i)l:~(Zt i}
(VI.1O.C17)
=~p/2)I.,..(.c)+;l.(2)1
The other combinations of the Ising twist fields can be obtained by considering the other vertices of this sector and in the sector 423. Perhap!l it is worthwhile to mention that we can see explicitly the obstruction to bosomzation by considering the massive sector al' In this case what we get is only half of the bosonizable combination (VI.10.C17) of the twist Ising fields, for example Ei without its partner E~, and this Ising twist field alone can not be rewritten in terms of the bosons like in (VI.10.C17).
2102 BIBLIOGRAPHICAL NOTE
As we stated in the Historical Remarks we have not attempted the compilation of an exaustive superstring bibliography. In the Sanle spirit here we just provide a very limited list of references that should help the reader to find his own way through the literature relative to the topics touched upon in this cbapter.
References
(2.2)·compactiJications aDd .h-map II] D. Gepner, Nucl. Phys. B 296 (1988) 757; Phys. Lett. B 199 (1987) 380, Trieste lectures at Superstring school 1989 12] F. Englert, H. Nicolai 8.Ild A. N. Sehellekens, Nucl. Phys. B214 (1986) 315 [31 W. Lerche, D. Luest and A.N. Schellekens, Nucl. Phys. B 281 (1987) 477 and Phys. Lett. B 181 (1987) 45
7Ype H superstrings on free fermion systems or on groupfolds
[4&1 S. Ferrara. and O. Kounnas LPTENS.89/4, CERN·TH.~8, UCLA·89·TEP·14 [4b] S. Ferrara and P. Fre', Int. Joorn of Mod. Phys. AS (1990) 989 I4c] L. CastellaDi. P. &e', F. Gliozzi IIlld M. R. Monteiro, DFTT 6/90 to appear in Int. Journ. of Mod. Phys. A (1990). SU(2}3 groupfolcl formulation
151 P. Fre' &lid F. GJiozzi, Phys. Lett. B 208 (1988) 203 ; Nucl. Phys. B236 (1989) 411•
.Free Fermion construction of heterotic superstrings
16] I. Antoniadis, C. Ba.ehas IIlld C. Koonnas, Nucl. Ph),s. B 289 (1987) 87 17] M. Kawai, D.C. Lewellen IIlld S.H. Tye, Phys. Rev. Lett. 57 (1986) 1832; Nucl. Phys. B 288 (1981) 1
181 1. Antoniadis, C. Dachas, C. Kounna.s &lid P. Windey, Phys. Lett. B 111 (1986) 51. Relation between world·sheet SUSY aDd target SUSY
[91 A. Sen, NucI. Phys. B 218 (1986) 289; Nucl. Phys. B 284 (1981) 423; T. Banks, t.1. Dixon, D. Friedan and E. Martinec, NucI. Phys. B 299 (1988) 613; L.J. Dixon, Lectures at Trieste Spring Sehooll981; T. Banks and L.J. Dixon, Nucl. Phys. B 301 (1988) 93.
Retrieving the superconformal algebras
2103 in tbe groupfold approach [IO} R. D'Auria, P. Fre', F. Glicnzi ~~ A. Pasquinucci DFTT/89 (Torino preprint) to appear in Nucl. Phys. B
(2,2)-moduli a:t1d effective Lagrangi.aDs
Ill] L. Dixon, V.S. Kaplunovski and J. Louis, Nucl. Phys. B329 (1990) 27. Effective Lagrangia:t18 of N::::2 truncations
112} S. Ferrara, C. Kounnas, L. Girardello, M. Porrati, Phys. Lett. B194 (1987) 358. Calabi· Yau spaces and compacti1ications
113} M. B. Green, J. H. Schwarz and E. Witten, "Superstring theory", Cambridge University Press, 1987.
Moduli spaces of (2,2) a:t1d (4,4) theories
[14a] N. Seiberg, Nucl. Phys. B303 (1988) 286 [14b] S. Cecotti, S. Ferrara and L. Girardello, Int. Jour. Mod. Pbys. A 10 (1989) 2475
.
Representations of tbe n=4 world·sheet supez-algebra
[15aJ T. Eguchi, A. Taormina, Phys. Lett. B196 (1987) 75; B200 (1988) 315; B210 (1988) 125. [1ShJ M. Yu, Nucl. Phys. B294 (1987) 890 IISeJ T. Eguchi, H. Ooguri, A. Taonnina. and S. K. Yang, Nuel. Phys. B315 (1989) 193.
[1Sd] A. Taormina, CERN·TH 5409/89, to appear in: Proceedings of the acl regional conference in Mathematical Physics, Islamabad, Pakistan, 1989.
Oovariant Lattice approach and bosonized GSO projector [16} for a review of the covariant lattice approach aee: W. Lerthe, A.N. Sc:hellekens, N.P. Warn~, Phys. Rep. 17'1(1989)1. [17} R. Bluhm, L. Dolan and P. Goddard, Nuel. Phys. B309 (1988) 330. OompactiJicatioll$ on weighted P" spaces [lS} P. Candelas, M. Lynker and R. Schimmrigk, UTTG-37·89 and NSF·ITP·89·164 preprints, (1989).
Effective Lagrangians of free fermion superstrings
2104
[19] S. Ferrara, C. Kounnas, L. Girardello and M. Porrati, Phys. Lett. B192 (1987) 368. 1201 1. Antoniadis, J. Ellis, E. Floratos, D. V. Nanopoulos and T. Tomaras, Phys. Lett. Bl9I (1987) 96.
Special Kabler manifolds, vector mu1tipJels MId Galabi- Yau moduli spaces [21a] S. Ferrara and A. Strominger CERN-TH 5291/89- UCLA/89/TEP6, Proceedings of the Texas A.M. String Workshop (1989), World Scientific (1990) [21b] B. de Wit, P.G. Lowers, R. Philippe, S. Q. SU and A. Van Proeyen, Phys. Lett. B134 (1984) 37; B. de Wit and A. Van Proeyen, Nue!. Phys. B245 (1984); B. de Wit, P. G. Lowers and A. Van Proeyen, Nuel. Phys. B255 (1985) 569; J . P. Derendinger, S. Ferrara, A. Masiero and A. Van Proeyen, Nuel. Phys. BUO (1984) 307. [21c] E. Cremmer, C. Kounnas, A. Van Proeyen, J. P. Derendinger, S. Ferrara, B. de Wit and L. Girardello, Nucl. Phys. B250 (1985) 385. l21d] S. Ferrara and A. Strominger, CERN·TH·5291/89, UCLA.89·'1'EP/6, (1989) [2Ie] V. PeriwaI and A. Strominger, UCSB preprint NSF.ITP·89.144, (1989) [21£] J. Bagger and E. Witten, Phys. Lett. Bll!) (1982) 202. [21g1 J. Bagger and E. Witten, Nuc!. Phys. B222 (1983) 1. [21h] L. Castellani, R. D' Auria and S. Ferrara, Phys. Lett. B, (1990), to a.ppear; Int. Jour. Mod. Phys. A, (1990), to appear.
Massless spectra in (2,2) theories from minimal model tensor products and also Kazama-SuzuId coset constructions
1221 M. Lynker and R. Schimmrigk, Phys. Lett.B208 (1988) 216, Ibid B215 (1988) 681, preprint UTTG·42·89 (1989).
[23] C.A. Liitken and G.G. Ross, Phys. Lett.B213 (1988) 152. [24] P. Zoglin, Phys. Lett. B218 (1989) 444. {25J Y. Kazama. and H. Suzuki, University of Tokyo preprints UT-Komaba 88·8, 8812. \26) A. Font, L. E. Ibanez and F. Quevedo, CERN·TH.5327/89 and LAPp·TH242/89. [27J P. Candelas, G.T. Horowitz, A.Strominger and E. Witten, Nne!. Phys. B 258 (1985) 46
Effective Lagrangians and Calabi· Yau spaces [28a] P. Candelas, P. S. Green and T. Hubsch, UTTG·17·89 (Austin report) (1989). 128b] A. Strominger, Pbys. Rev. Lett. 55 (1985) 2547; A. Strominger and E. Witten, Comm. Math. Phys. 101 (1985) 341; A. Strominger, in Proceedings of the Santa Barbara Workshop, "Unified String Theory", World Scientific (edited by M. Green and D. Gross), (1985).
2105
128e] P. Candelas, Nucl. Phys. B298 (1988) 458. \28dj R. Dijkgraaf, E. Verlinde and H. Verlinde, preprint THU·8Y/30 (Princeton reo port); V. P. Nair, A. Shapere, A. Strominger and F. Wilctek, Nucl. Phys. 13287 (1987) 402; A. Shapere and F. Wilczek, Nne!. Phys. B320 (1989) 669; A. Giveon, E. Rabinovid and G. Veneziano, Nuc!. Phys. B322 (1989) 167; S. Ferrara, D. Lust, A. Shapere and S. Theisen, Phys. Lett. 225B (1989) 363; J. Lauer, J. Mas, H. P. Nilles, Phys. Lett. B226 (1989) 251; W. Lerche, D. Lust and N. P. Warner, Phys. Lett. B231 (1989) 417; E. J. Chun, J. Mas, J. Lauer, and H. P. Nilles, Phys. Lett B233 (1989) 141; C. Vafa, Harvard preprint HUTP-89/ A021(1989); S. Ferrara, D. Lust, and S. Theisen, Phys. Lett. B233 (1989) 147; B. Greene, A. Shapere, C. Vafa and S.T.Yau, Harvard Preprint HUTP-89/A047 and IASSNS· HEP-89/47j J. H. Schwarz, Caltecl! preprint CALT.68-1581/19. Toroidal and orbifold compactincations [29a! K. S. Narain, Phys. Lett. 169 B(1986) 61 !29b] L. Castellani, Phys. Lett. B166 (1986)54; L. Castellani, R. D'Auria, F. Gliozzi and S. Sduto, Phys. Lett. 168 B (1986) 47. [2ge] K. S. Narain, M. H. Sarmadi and E. Witten, Nuc!. Phys. B 279 (1987) 368. 129dJ L. Dixon, V. Kapiunovski and C. Vafa, Nuc!. Phys. B 294(1987) 43. [2gel L. Dixon, J.Harvey, C. Vafa and E. Witten, Nud. Phys. 261 (1985) 678 and B 274 (1986) 285 L. E.lbanez, H. P. Nilles and F. Quevedo, Phys. Lett. B 187 (1987) 25 and B 192 (1987) 332 A. Font, L. Ibanez, H.P. Nilles and F. Quevedo, CERN-TH 4969/88(1988}
2106 Table VI.lO.I
InterlJ&! Supercon/otmaJ fidd Theories from k-mllp Theory
left-light tonrormnl decomposition I:"g(p, ~)"l
15011 =\,£,
(2, 2)9,8$(0, O)!.~ln)/i>B~
ISCI/: I ,1I,
(2,4).,.61(2, 2)a,QIB(O, O}~gilO)&Z:
ISCP =I,B,
(2,6)9,9$(0, ()}:,?J'O)&S;
ISC!f:2.B.
(4,1)6,0$(2, 2)a,al1l(0, O):,~~IO)&£:
ISCP=2.B,
(4,4}o.e$(Z, 2)3.sl!i{O, O)~~~IOI&B:
ISCP=2.Z,
(4,0).,0111(2, O).,oEfJ(O, 6)o,oEfJ(O, O):.~~IO)&£;
ISCP="z.
(6, 2)909$(0, O):.~f'OJ.B;
15CIl"',B.
(6, a).,oe(O, 4)••aEfJ(a, 2)0,3$(0, O)~,~'~)&E',
15CI'="£'
(6, 6)9,961(0, OI:g~tO).8; Table VI.lO.n Exampl.. ofmodeJs with K
1'1
1'l
[.' It] -
=0,1,2
C()uesponding O,buold
World-she.t and target S'USY
-rB
(6,6) => N:::S
(f,)r. o {-rB}R
(4,6) => N::;6
:c z.
(4,4) => N",,4
(i0 etIlH" ea0 ea®e.)
[$'I;'J(e3®is 0 el,
e.0 e30ed
[1'1;1-
!l/lb') ":
(ia 0ia €lei' ea0 es®·tl
(e,®i\ 01,
["Ib] -
(i ~Ht €Ie" "30 •• 0 41)
/I
(:/,4) => N::3
•• 00\ 0'8)
["'IV] -
(i\ ~Hl €Ii,
(z.tz.) L €I (-rB) R
(2.6);;;;:} N::::5
48 ®e\ 0/q 342 differential k-form 45 dilatation subgroup of SL(2, R) 1786
=
2136
dilatations 1409, 1513, 1542 dila.tino 1752, 1839, 1859 dilaton 1756, 1839, 1854 dimensional reduction 1148 on a. I-torus 1134 dimensional reduction, trivial 1148 Dirac brackets 1547, 1549, 1513, 1805 Dirac determma.nts 1389 Dirac equation 433 .. of a massless spin 1/2 field 189 Dirac gamma ma.trices 519 Dirac operator D 1154 Dirac singleton 442, 451 dista.nce function on cosets 224 div(E,) 1460 divisor 1458 class group 1462, 1488 classes 1481 dual basis 11 dual Coxeter numbers 1608 dual formulation of Lie algebra 105 dual formulation of the superalgebras 307 dual lattice 1822 dual space V*(n/m) 342 dual vector space 10 duality group USP(28,28) 111S duality group in six dimensions 1131 duality relation sa2 duality rotation 1019 duality transformation 924, 1984 dualiza.tion formula 891 for r-ma.trices 835 Dynkin basis 1292 Dynkin labeling 1293 8-dimensional spinors 834 Er 1118 E1(-7)/SU(8) coset ma.nifold 1118 E7, real form of 1118 Es@ E~ 1760 eft'ective cosmological constant 1643 eft'ective Lagrangian 1376 of the superstring theory 1769 eft'ective supergra.vity theories 1389, 1838 eft'eclive theory 1102, 1854 eigenfunctions of the Laplacian opentor 1236, 1656, 1677
2137 eigenfunctions of the Lichnerowicz operator 1235 eigenvalue of the Lichnerowicz opera.tor 1237 eigenvalue spectrum of the invariant operators 1259 Einstein-Cartan action 141, 165 Einstein equa.tion, linearized 434 elliptic transformation 1414 embedding supermultiplet 1534 emission vertex. 1768 emission vertices, massless 1838 energy operator 284 energy-momentum tensor of the field 622 Englert's solution 1193, 1250, 1343 enlarged Fock space 1564 Euler characteristic 162, 290, 1399 even spin structures 1682 even subspace 310 exceptional algebras D(2,l, ), G(3) and F(4) 324 exceptional selies EXl, EX2, EX3, EX4 in Mpqr mass spectra 1326 exceptional superalgebra D(2, 1, a), G(3), F(4) 331 extended action principle 661 extension mapping 651 exterior algebra of forms on Vn 22 exterior algebra on Mn 46 exterior differentiation 47 exterior forms on vector space 10 exterior (or wedge) product 17,21 F(4) ex.ceptional group 332 factorization property of scattering amplitudes 1638 Faddeev-Popov determinant 1693 fake fermions 1723, 1730 fermion determinants 1646 fermion emission vertex 1818 fermionic approach to D = 4 superstrings 1706 fermionic coordinates 339 fermionic mass spectrum, longitudinal 1302 fermionic shifts 1080, 1081 fermionic strings 1704 fermionization 1620, 1632, 1723 group 1708, 1709 fermion number 1752, 1834, 1836 fiber bundle 120,585,586, 644, 645 fiber bundle P P(M4,{i) 585 field equation of a massive spin 3/2 particle 612 field equation of a massless spin 3/2 particle 612 field of I-forms 44
=
2138
field of k·forms 46 field redefinition 1869, 1944, }945 field redefinition in D == 10 supergravity 1947 field redefinition, nonlocal 1984 Fierz identities 305, 308, 367, 536, 545 Fietz reazrangement 485 finite theory of gravitation 1107 finite transformations 204 first·clus 1508, 1558 first homotopy group 1584 first order formalism 143, 172, 186, 509 for ga.uge .fields 693 supersymmetry transformations 628 first-order Lagra.ngian 1904 first order transcription of the 2nd order action 517 fixed point free subgroups 1401 fiat and torsionless superspace 1518 Hat directions 1383 flat or intrinsic indices 54 flat superspace 302 Fock space 1280, 1572, 1631 Fock vacuum 1287 Fourier representation of a background tensor (spin or) field 1777 free differential algebra 305, 609, 795, 1860 of D == 11 supergravity 863 of D == 6 supergravity 841 rree differential algebra, extending the D == 11 Poiacue supergroup 867 free differential algebra, iterative constrllction 798 free differential algebra, maximally extended extension 780, 908 free-fermion approach 1730 free fermion constructions 1384 Frenkel·Kac vertex operator representation 1787 Freund-Rubin solution 1155, 1193 Freund-Rubin, solutions of D == 11 supergravity 1192 Freund-Rubin type compactifications 1346 Fubini-Study metric 942 fuchsian groups 1416 functional determinant 1389, 1677 function ring 38 functions on a manifold 37 functions on the circle 1175 fundamental domaill 1644 fundamental group 255,1401 of a genus g lIurface Ef 1401 1I'1(M 'arget) 1503
2139
fundamental region 1635 for the modular group PSL(2,1) 1453 fundamental Wl!ights 1607,1608, 162.1, 1797 G(3) 332 g = 1 mapping class group 1451 g 1 moduli space Ml 1412 G-covariant derivative 775, 798, 1233 G-ga.uge transforma.tioD 646 GIB vielbei.n 1177 G-index 1365 G-iueps 1179 G-left invariant metric on GIB 214 G-Lie algebra. valued }.form 584 GNO fermioniza.tion 1707 GNO symmetric spa.ce 1709 GNO theorem 1710 GSO projection 1834 9perator 1702, 1740, 1748 GSO pIojectors, generalized 1590 gamma.-ma.trices 891 gamma matrix algebra 304 gauge and gravitational coupling constants 1850 gauge bosons 424 gauge comp~nents 700 gauge coupling constant 727, 1151, 1853 gauge field of supersymmetry 611 gauge bed hamiltonian 1566 gauge fixings 434, 1566 gauge ghosts 1808 gauge group G 1377 gauge groups EsxEs or SO(32) 1155 gauge hierarchy 999 gauge invariance 616 gauge mUltiplet 1859 gauge operator B 1561 gauge subalgebra 689 gauge supersymmetry transformation 628, 646 ga.uge symmetry 1150 gauge transformation 88 gauge transforma.tion, infinitesimal 126 gange translation 145 gauge variation of a non-abelian vector field 1188 gauging 593 gauging of the D = 11 F.D.A. 870 of the vector fields 1039
=
2140
gaugino 568, 1753 gaussian functional integrals 1655 gaussian integral 1652 Gelfand~Zetlin labeling 1292 general coordinate transformation, linearization 634 general coordinate transformations in the extra dimensions 1163 general graded Lie algebra GL(m/N) 326 generator 648 genus of the surface 1380, 1393 geodesics 222 geometrical actions based on F.D.A. 812 geometrical coupling of a massless spin one field 185 geometrical lagrangian 159, 171 geometricity 692 geometric symmetries 1147 geometric theory with auxiliary fields 812 gbost and antighost zero modes 1572 ghost and superghost Virasoro algebras 1596 ghost antighosts 1782 ghost current 1809, 1815 ghost fields 441, 1388, 1561, 1594 ghost number 1564, 1573 operator 1571, 1810 ghost states 1559 ghost stress energy tensor 1574 ghost supercunent 1578 ghos~8uperghost conformal field theory 1769 Gliozzi, Olive and Scherk projectors 1632 Goldstone scalar 1166 Goldstone vector 1166 graded antisymmetric tensor fields 801 graded antisymmetry 22 graded extension of the anti de Sitter group 710 graded matrices 301, 341 graded structure constants 314,364 graded vector space V(n/m) 341 grading 313 Grassmann algebra 302, 301, 333 Grassmann algebras, generators 333 Grassmann algebra, Z2~grading of 335 gravitational coupling constant 642 gravitational multiplet 1239 gravitello 568 gravitino 306, 386, 614, 1066, 1732, 1839 gravitino l~form 386
2141
gravitino ghost 1692, 1737 determinant 1389 gra.vitino mass 998 matrix 1078,1081, 1126 graviton 306,424, 1756, 1839 multiplet 1859 Green-Schwarz action 1961 Green-Schwarz formulation of the D 10 superstring 1960 Green-Schwarz Lagrangian in a Don-trivial sl1pergra.vity and Yang-Mills background 1962, 1965 Green-Schwarz mechanism 1838 group fermions 1715 group manifold approach 661 group manifold potential 1233
=
H(q)(I:,) 1424 hamiltonian constraints 1~3 harmonic I-form 1430 harmonic dilferential 1433 harmonic expansion of D-dimensional fermions 1153 harmonic expansion of the SO(7) spinor 1303 harmonic fOrms 1214 harmonic one-forms 1231 harmonics 1306, 1307 harmonics (C)pl 1306 harmonics (C)p2 1306 harmonics, longitudinal 1322 harmonics on G/H 1183 H-compensator 210 H-connection 212 H-covariant deriva.tive 690 H-covariant Lie derivative 221 hermitean almost complex manifold 935 heterotic fermions 1380, 1534, 1535, 1583, 1588, 1103, 1819, 1964 heterotic string 359 heterotic superspace 1514 heterotic superstring 1107, 1377, 1378, 1511 heterotic Wess-Zumino-Witten model 1388 H-gaugeinvariance 691,695 H-gauge transformations 702 H-harmonics on G/H 1180 H-horizontality condition 126 Higgsinos 583, 983 Higgs particles 416,983,998 Higgs phenomenon 920 higher curvature interactions in D 4, N 1 supergravity 1982
=
=
2142 higher-dimensiOllal theories 1141 highest root 1601, 1621 highest weight 1291, 1607, 1608, 1701 highest-weight state of twisted algebra. 1612 highest-weight st&tes 1778 of Virasoro algebra 1612 Hilbert space 1631 H-irreducible fragments 1183 H-inep 1178 Hodge-de Rabm operator 1204, 1213 Hodge dual 690,1428 Hodge duality 737, 1053 mapping 57 operator 26,57,664 holomogy and cohomology 251, 268, 1428 holomolphic coordinate transformations, infinitesimal 984 holomorpbic diferential 1807 holomorphic factoriza.tion 1690 holomorphic Killing VectOlS 984 holomorphic 1-diferentials 1433, 1412 holonomic basis (IJ/8?J·,8/(j,"') 497 holOllomic indices 55 homogeneous scaling law 697 homogeneous spaces 190 homology basis 1439, 1474, 1631 homotopy and (co)homology of coset spaces 262 homotopy exact sequence 262 homotopy group 1401,1520,1583,1712 horizontality 120 constraint 658 H2 hyperboloid 201 hyperbolic transformation 1415 hypermultiplet 416,583 ideal 323 improved genera.tors 1562, 1579 improved vertex operators 1793 second order formalism 726 torsion, in the n&tural frame 92 index conventions 194 index of iD 1364 index theorem 1364 induced mapping 41 induced representations 392 infinite-dimensional Lie algebras 1505 infinitesimal generator of the difeomorphisms 61
2143
infinitesimal transformations 210 inner components 319,657,700 inuer derivatives 495 inner direction 495 inner product or contraction 25 inner sector 509 inner space 689 lnonu-Wigner contraction 136, 301, 322, 764 integra.bility conditions of the rheonomic constraints 656,651,672 integral divisom 1462 illtegration measure 1655 interaction Lagrangian 1775 intercepts 1590,1591, 1597, 1600, 1713, 1780 internal manifold 1382 illternal space, size of 1150 internal symmetries 609, 1147 intersection matrix 1435, 1437 intrinsic covariant derivatives 1522 invariant measure 225 invariant operator on G/H 1181 inversion formula. 530 invisible sector 999 involutory automorphism 1618 irreducibility constraints 432 irreducibility-transvemality 434 of the H == SO(l,3)ol?IO(N) group 538 irreducible representation m(I') highest weight II 1797 irreducible massless representation of N = 2 supersymmetry 731 irreducible representations 1826 irreducible supennultiplet 402 irreducible transverse vector-spinol 1216 ISO(2) 1407 isometries or extra compactified dimensions 1154 isometries of MK G/H 1187 isometry 74 isometry group 1408 isometry group, global 1150 isospin 1610 isotropy irreducible subspace 233 isotropy subgroup 191 for structure constants 101
=
Jacobian 1663, 1668 Jacobian variety 1489 points of order two 1499 lacobiidentity 62,100,311
2144 Jacobi inversion formula 1684, 1681 Jacobi map 1490 Jordan algebras 1133 Jordan structure 1280 Kae-Moody algebras 1548 Kae-Moody characters 1389, 1632 Kae Moody, level 1548 Kae-Moody extension of the Poincare algebra 1165 Kae-Moody symmetzy 1528 Kibler connection 941 Kanler coset manifolds 940 Kibler form 1348 Kahler manifold 937, 1404, 2002 Kibler manifold, restricted lllO, 1112 Kihler metric 936, 2003 Kihler potential 1376, 1404, 2002 Kibler Riemannian curvature 938 Kibler transforma.tion 937 Kihler 2-form 948, 2003 Kaluza-Klein mechanism 692, 1150 Kaluza-Klein miracle 1150 Kaluza-Klein supergra.vity 1318 Kaluza-Klcin theories 782, 1141 Ie-anomaly 19797 Ie-anomaly. cancellation of 1961 It-symmetry 1390, 1960,1971 Ie-transformations 1911 Killing equation 984 Killing form 114 KiDing metric 113, 1263, 1526, 1705 Killing multiplet 1239 KiDing spinor 1082, 1083, 1198, 1232, 1344 in D = 7 1233 Killing vector 14, 210, 301, 381, 927, 1020, 1190, 1215, 1231, 1342 on G/H 1186 kinetic operator for a field of spin [~J in D == dim G/H 1184 Klein-Gordon equation 184 ladder currents 1795, 1796 ladder operators Ea 1291 lagrangian, building rules 681,825 lagrangian multiplier, Siegel method 844 lagrangian multiplier o-form. 859 lagrangian of N == 1, D == 10 supergravity coupled to Yang-Mills supermultiplet 1948 Laplace-Beltrami operators 1180
2145
laplacian 429 lattice L(E,) 1489 lattice approach 1389 lattice, cubic r-dimensional 1796 lattice, even 1822 lattice, integral 1822 lattice, odd 1822 lattice, points of length 2 1797 lattice, self-dual 1822 lattice, unimodlllar 1822 Laurent modes 1576 left-invariant I-forms 104, 362, 1522 left-movers 1384, 1513, 1518, 1631 left-moving fermions 1705 left rig~t invariant vector fields 100 left transla.tion 98, 381 leptons 416, 983, 998 Lichnerowicz operator 1204, 1215 Lie algebra 62, 100 Lie algebra., lattices 1821 Lie algebra of vector fields 62 Lie algebra, reductive and symmetric 130 Lie algebra valued matrix of I-forms 111 Lie bracket 310 Lie derivative 219,306,102, 1525 of wah, va 131 of I-forms 66 on superspa.ce 609 operator 490 Lie derivative, cova.ria.ht 219, 489, 1180 Lie group 98 Lie super algebra. 1219 light-cone formalism 1694 light-cone frame 392 light-cone gauge 1972 light-like root vector 1831 line bundles 1481 linear differential operators 928 linear mUltiplet 1859, 1983 linear opera.tors, on the tangent space 929 linearized action 634 little group GO 474 local gauge translations 613 local supersymmetry 607, 704 transformations 611 local translation invariance 607
2146
local translations 613 locally supersymmetric Lagrangian 704 locally supersymmetric two-dimensional q-model 1531 longest root 1548 longitudinal representation 1210, 1230, 1267 Lorentz Chern·Simons form 1838, 1858, 1904 Lorentz Chern·Simons terms 1389,1855 Lorentz and Yang-Mills anomalies 1859, 1914 Lorentz covariant derivative 88, 429 Lorentz transformation, field dependent 140 Lorentzian lattice 1819 low energy el£ective action 1847 lowest energy quantum numbers 1274 M.XSI Maxwell theory
1166
MtXSl space-time 1166 magnetic potential 1040 Majorana condition 1266 Majorana Killing spinor 1233 Majorana spinor 311, 393, 526 MajoranarWeyl spinor 527,568, 1154, 1512, 1588 manifold, orientable 32 mapping class group 1380, 1412, 1448, 1474, 1636, 1644, 1703 mapping moduli space, invariant measure 1637, 1671 mappings, between manifolds 38 mass 426 in anti de Sitter space 425 massive and massless representations 392 massive modes 1148 massive multiplets 411, 457 with and without central charges 395 massive spins, infinite tower 1173 massive supersymmetries 1225, 1226 mass matrix 119, 1066, 1185 mass operator 1154 mass spectrum of a KaluzarKlein theory 1185 mass-shell equations 1780 mass sum rule 1240 massless gravitinos 1591 massless higher spin representations 457 massless modes 1854 massless multiplets 395, 416, 1239 massless particles 428 massless supersymmetries 1226, 1235 massless target fermions 1388 matter coupled supergravities 1151
2147
matter multiplets 306, 608, 1239 Maurer-Cartan equations 105, 124, 228, 360,687 for the Poincare group 111 '. Maurer-Cartan equations, generalized 795 Maurer-Carta.n equations, of the gauged E7·7 1124 maximal compact subsuperalgebra. 1280 maximal subgroup 196 Maxwell equation, self-interaction term in D = 5 787 measure on the g-handled world-sheet 1768 meromorphic q-differentia.ls 1431 metric 45, 113, 1854 connection 1527, 1857 on Vn 15 metric, constant curvature 1426 metric, hermitean 935 metric tensor 83 microcanonical density 1630 microscopic quantum theory 1376 minimal algebra 796 minimal coupling 172, 1003 of a spin 1/2 field 188 minimal generators 869 minimal grading of ISO(l,4) 761 minimal grading of SO{2,4) 758 minimality constraints 1913 niinhnal supergravities 1903 Minkowskian metric 315 Minkowski N-extended superfields 371 mirror fermions 1361 Mobius transformation 1408, 1414, 1773, 1812 mode expansions 1772 modular anomalies 1380, 1718 modular forms 1718 modular group, transformation properties of the characters 1718 modular invariance 1474, 1380, 1590, 1702, 1722 constraints 1854 moduli 1383, 1448 space 1383, 1393, 1413 modulus of the world-sheet torus 1635 momentum fields 1502 momentum lattice 1787 momentum \lectors 1820 monopole-like configurations 1366 moving frame 55, 77 Mpqr solutions 1156, 1302 Mpqr spa.ces 264, 633, 1149, 1251
2148
Mpqr spaces, geometry 249 multiloop modular invariance 1393, 1741 multiplet shortening 455, 465 N = 1 anomaly-free supergravity in D :::: 10 1769
N == 1 chiral supergravity 1376 N :::: 1 massive multiplet 404 N :::: 1 matter coupled supergravity in D
= 10
1982
N = 1 supercon{ormal algebra 1514 N = 1 supergravity in component formalism 609 N = 1 supergravity in D :::: 4 358 N:::: 1 supersymmetric Yang-Mills theory 309 N = 1 supersymmetty 614 N:::: 1, D == 2 conformal supergravity 1511
N = 2 couplings in D :::: 4 1109 N = 2 massless multiplet 726 N = 2 simple Ilupergravity 726 N = 2 superstring 1808 N :::: 3 supergravity 1033 N :: 3 vector multiplet 1034 N :::: 4 supergravity multiplet 1763 N = 8 graviton multiplet 1115 N = 8 supergravity 1148 N = 8 supergravity, non compact ga.ugings 1120 N = 8, SO(8) gauged D == 4 supergravity 1358 Nambu action 1567 natural basis 78 nega.tive norm states 1559 Neveu-Schwarz (NS) and Ramond (R) algebra 1594 Neveu-Schwarz sector 1829 Neveu-Schwarz vacuum 1803, 1829 Neveu-Schwarz superconformal algebra 1595 new minimal in superspace supergravity 806 new minimal set of auxiliary fields 1993 Newton constant 273, 1853, 1855 N-extended Minkowski superspace 370 N-extended Poinca.re superalgebras 323 N-extended anti de Sitter superspace 370 N-extended superconformal algebra. in D = 4 330 N-extended supergravities 424 N-extended supermultiplets 304 N-extended supersymmetrk versions of Ya.ng-Mills theory N'oether coupling method 615, 1108 Noether method 609 non-abelian current algebra 1795 nonchiral (1, 1) N = 2 superalgehra 1378
424
2149
Doncompad G/H 201 Don compact symmetry 1016 nOD contractible loops 1591 nonheterotic 8upeutrings 1378 non-linear #-model 1526 non-minimal supergravity 1903 non-standard coupling 278 non zero torsion 274, 756 norm 337 normal ordering 1562, 1723, 1724 normalization of the Cashnirs 1611 normalizer 1253 ofHinG 235 N-point Green functions 1768 null eiJenspinors 1197 number of on-shell degrees of freedom for fields of spin running from two to zero 159 I-form 10, 303 I-form on a manifold 42 odd subspace 310 off-shell multiplet 477 off-shell representation 411 off-shell rheonomy 812 off-shell vector multiplet 1987 old minimal set 1986, 1993 one-loop cosmological constant 1641 one-loop modular invariance 1734, 1737 one-parameter groups of transforma.tions 60 on-sheU Bose and Fermi degrees of freedom 476 on sheU spin 3/2 particle 1227 on-sheU states 476 on-shell supersymmetry 677. 720 open algebras 1563 operator product expansion (O.P.E.) 1552 operator products 1575 operator-valued distributions 1548 order of pq 1432 orienta.tion 17 of a. manifold 46 oriented volume 17 orthogonal algebra. in even dimension: 80(2r) 1824 orthogonality and completeness rela.tions 1176 orthogonal roots 1612 orthosymplec'ic algebras Osp(2p/N) 324,326, 351, 1278 orthosymplectic algebras Osp(4/N) 301
2150
Olthosymplectic metric 349 Osp(4/1) 802 Osp( 4/1 )-cova.riant derivative 715 Osp(4/1) curvatures 710,958 Osp(4/2) 726 Osp(4/2) curvatures 129 Osp(4/2) theory,lagrangian 744 Osp(4/N) Maurer-Cartan equations 1088,1202 OsP(4/N) superalgebra 352, 364,426 Osp(4/N) superrnultiplets 308 outer automorphisms 398 oute! components 379, 653, 700 outer derivatives 495 outer direction 495 outer sectors 509 outer space 689 out state 1783 parabolic transformation 1414 para11e1izable manifold 119 para11elizing connection 1527 para11elizing torsion 1251 parallel transport 227 parity conservation 708 partial supersymmetry brea.ldng 308, 1077, 1107 partition function 1388, 1400, 1473, 1629, 1630, 1702, lil2 of the fe!mionic string 1736 path.integral quantization 1392, 1589 p-cohomology classes 1430 period matrix 1443, 1448 p-forms 16, 343 on manifolds 45 physical fields 476 physical operator 1561 physical states 1560 picture changing operator 1829, 1842 pictures 1829 Planck mass 1853 P(n) and Q(n) algebras 324 Poincare-Bianchi identities 136 Poincare gauge transformation 145 Poincare group curvatures 136 Poincare Lie algebra. 136, 316 Poincare Lie algebra-valued curva.ture 2·form 143 Poincare polynomials 268 Poincare supermultiplets 307
2151
PolliOn bracket 1549, 1558, 1561 polarization tensor 1168, 1776 Polonyi model 1014 '. Polyakov path integral 1388, 1632, 1647 Pontriagyn number 162 potentials for the Osp(4/8) superalgebra 1115 (p,'l) lIuperalgebra 1378 prepotential of the Killing vectors 981 primuy constraints 1541, 1568 principal divisor 1461 problem of chiral fermions U53 projection operator 163 projective coordinates 205 pseudo-Majorana condition 763 pseudo-Majorana spinor I·forms A 555 pseudoscala.r particle 405 p-th Betti number 1214, 1430 pullback of forms 24, 50 punctures 1415 pure supergravities 608 q-differentials 1424 Q(n) algebra. 331 Q-supersymmetries 1513 q-th power of a line bundle 1483 quadratic Gasimir 428, 1610 quadratic differential 1420, 1455, 1661, 1673 quadratic holomorphic differentials 1424 quadruple product a·b·c·d 1734 quantum bosonization of fermions 1787 quantum genera.ting functional 1399 quantum realization of the superconformal algebra. 1769 quantum superconformal theories 1389 qUilks 416, 983, 998 qUilks and leptons 1361 quasi fermionizable 1711 quasi-massltss multiplets 1239 quasi-massless scala.r multiplet 1240 quasi-massless supermultiplet 1275 quaternionic: manifold 1109, 1112, 1136 radial ordering 1768 Ramond sector 1829 Ramond vacuum 1818 rank of G/R 1181 rank of G 1788
2152
Rarita.-Schwinger equation 432 Rarita-Schwinger field, Lagrangian 615 Ranta-Schwinger Lagrangian 617 Rarita.-Schwinger, Lagrangian, local gauge invariance 634 Ranta-Schwinger operator 1154, 1204, 1206 real projective Sp&Ce 36 reality 337 reductive algebra 323 redllctive G/H 196 Regge slope 1567 ofthe old dual models 1654 relative cohomology of a Lie algebra G with respect to a subalgebra H 803 renormalized elective action 1977 reparametrization ghosts 1808, 1811 representation of the global $upersymmetry algebra 614 rescalings 231, 1263 Iheonomic action principle 508 Iheonomic conditions 306, 495 Iheonomic constraints 654, 1868 rheonomic extension mapping 492, 652 rheonomic parametrization 1516, 1867 rheonomic theory 654 rheonorny 299, 308, 654, 700 and Bianchi identities 716 and supersymrnetly invariance of the action 610 for a superfield action 509 rheonomy method 307 rheonomy principle 649 Ricci tensor 91,228,939 Ricci two-form 1348 Riemann curvature tensor 93, 1404 Riemannian connection 85, 227 Riemannian manifold 45, 80, 85 Riemann-Roth theorem 1425, 1463, 1471, 1695, 1807 Riemann surface 1379, 1386, 1393 Riemann theta functions 1389,1472, 1492, 1646, 1703, 1728 Riemann vanishing theorem 1494 right action of a group G 235 right isometry group 236 righ~movers 1394, 1513, 1518, 1631 right moving fermioDS 1705 right translation 99, 101, 381 rigid superspace 644 rigid supersymmetry transformation 633 Robertson-Walker metric 295 root diagram of SO{S) 240
2153
root formalism 1600 roots 1292, 1603 loot lattice 1796, 1823 ofEs 1837 root space 1292 root system 1600, 1795 loot vectors 1600, 1795 round S1 1250 R-symmetry 2007 7-dimensional compact homogeneous Einstein spaces 1194 S2 191,206 S% structural equations of 288 S2-zweibein 286 S1 23JJ, 1156 scalar culva.ture, manifolds with vanishing 1153 scalar density 57 scalar field potential 960, 966, 1064 scalar harmonic 1216, 1263 scalar multiplet 583,919 scalar particle 405, 429 scalar potential 172, 1065,1128 scalu product {or boundary vectors 1133 scalar superlield in D = 10 1905 scale invariance, rigid 158,691 scale, of supersymmetry breaking 999 scale weight 517 of the Einstein term 691 scaling beha.viour 1866 scattering amplitudes 1766 Schottky problem 1446 Schur's lemma 523 SchwalzschUd radius 294 Schwarzschild solution 283, 294 Schwinger terms 1562 second chuacteristic class 1864 second-class constraint 1549, 1805 second order Casimir 426 second-order constraints 1547 second order formalism 147 sections of the sheaf 1476 self-conjugacy 1110 selfdual and anti-selfdual tensors 838 seU-interaction term of the spin I-field 1132 semi-simple group 115 semisimple Lie algebra 323
2154
sheaf 1476 cohomology 1476 sheaf of holomorphic-functions 1477 sheaf of polynomial differentials 1480 short distance expansion 1509 short-distance operator product expansion 1548 short massive representations 451 short multiplets 1287 shortening of the representation 441 Siegel supper plane 1445 Siegel symmetry 1959, 1960 Siegel transformations 1975 .,-model 1854 Lagrangian 1777 signature of metric 26 simple roots 1292, 1608, 1823, 1824 simple weights 1823, 1825 simply connected Biemann surfaces 1407 simply laced algebra 1788, 1795 simply laced Lie algebra lattices 1823 singleton multiplet 1295 singleton supermultiplet 1297 Sitter or anti-de Sitter group in D dimensions 131 SL(2,C) 1407 SL(2)-inva.rla.nt vacuum 1811, 1818 SL2-vacuum 1812 Sla.vnovoperator 1977 sleptoJls 58, 983 smooth manifold 31 So 191,199,216,229 SO(l,3)-Cactorization 667 SO(I,3) gauge invariance 158 80(1,9) irreducible representations 1880 80(1, 10) irreducible representations 563 80(1, n-1) covariant derviative 89 SO(2)-charge 727 SO(3)-Maurer Cartan equations 287 SO(3,2) unitary irreducible representations (UIR's) 1288 SO(8)-covariant derivatives 1225 SO(8)-covariantIy constant spinor 1226 SO(8)-field strength 1123 SO(8} irreps 1289 SO(10) Es current algebra 1383 80(16~SO(16) 1760 SO(44) gauge group 1762 soft I-forms, non-left invariant 122
2155 soft P.D.A. equations 812 soft group manifold 119, 122, 645 softly broken symmetry 999 soft Poincare group manifold 121 soft superspa.ce 644 Sohnius-West model 806 Sp(2g, l) modular invariance 1639 Sp(2g, I) transformation properties of the partiton functions 1474 Sp(4, 1) group 1741 space of COll$tant curvature metria 1427 space of holomorphic q-differentials 1468 space of spin bundles 1487 space of supermoduli 1695 space of tnceless metria 1420 space-time action, noninvariance of 855 space-time supersymmetry transformations, geometrical interpretation of 615 special superconformal ga.uge 1537, 1544 spin 1/2 O-modes 1342 spin 1/2 and spin 3/2 fields, canonical field equations 1227 spin 1/2 shift 1126 spin 3/2 O-mode 1342 spin bundle 1485 spin connection 79, 84, 227, 633, 1856 spin fields 1769, 1780, 1787 spin fields, vertex operators 1389 spin one kinetic term 737 spin structure 1312, 1474, 1649, 1619, 1693, 1703, 1707, 1735 for SU(2) group bosoll$ 1720 spin structure, odd 1682 spin-bundle S 1483 spin-field correlators 1803 spin-sta.tistics and modulaz invariance 1696 spinor bundles 1474 spinor conjugacy cla.ss is}, 1837 spinor derivative 1524, 1873 spinor ields on a. manifold 55 spinor harmonics 1264 spinorial derivative 879 spinorial generators Q 317 spinar representaUoll$ 1826 spinor-tensors 543, 1879 spinor weight 1799 spinors, chiral 527 spinors, Ramond or Neveu-Schwatz 1691 spontaneous compactifi.cation 305, 308, 1107, 1193 to M4$Mk 1148
2156
spontaneous supeIsymmetry breaking 305,583 squarks 583, 983, 1256 squashed seven-sphere h 1255 S-supersymmetries 1513, 1542 stability bound (Breitenlohner-Freedman) 1241 stable vacuum 180 step operators 1608 stereographic coordinates 193, 1262
stereographic projection 34 stress energy tensor 1392, 1508, 1541, 1573, 1588, 1724 stretched solutions of D 11 supergravity 1347 string amplitudes 1854 string tension 1838, 1853 string tree amplitudes 1854 string vibrational modes 1766 strongly geometrical theory 692, 756 strong geometricity 159 structure c.onstant5 100, 194 structure constants, boosted 1046, 1085, 1124
=
structure constants, generalized 795
structure equations 83,85 structure functions 124, 614 SU(I. 1) 1407 SU(l. 1) coset representative 1089 SUp, 1) Fnbini-Study Kahler potential 1016 SU(l. 1), symmetry 1016 SU(l, 1)/U(I) 1088 SU(2) 1407 SU(2) doublets 1301 SU(2) mass-shift 1616, 1620 SU(2) root diagram 1617 SU(2,2/1) 766 SU{2,2/1) algebra 758 SU(2.2/N) 329. 330 SU(3) transformations 1035 SU(3)xSU(2)x U(l) 1360 SU(3)xSU(2)x U(l) representations 1306 SU(3.n)-invariant metric 1040 SU(8) compensators 1119 SU(8) curvature 1123 SU(N) symmetry, global subalgebra 312
1080
sublattices 1124 Sugawara-like realization of the stress-energy tensor 1603. 1814 Sullivan's fundamental theorem 797
2157
superalgebras P(n) and Q(n) 331 supetalgebra U(m/n) 1278 superconformal algebra 1578, 1595, 1706 of primary constraints 1553 superconformal anomaly 1530 superconformal calculus 1109, 1110 supetconformal gauge 1517, 1544, 1554, 1535 supetconfc>Imal ghosts 1578 superconformal map 1994 superconformal theory, classical 1538 supetcoset manifold 370 supetcurrent 1521, 1541, 1588 supercurvatures 385 supetdeteIminant 348 superdifeomorphisms 1554 superlield 303, 304, 339 superghost antisuperghost fields 1594, 1693, 1783 superghost number operator 1693 superghost vertex operator 1816 supergra.vity, linea.rized theory 633 supergroup 333, 345, 346 manifold 303 • sllpethelicity 420 sllperhermitean 349 super-hermitian ba.sis 1279 super Higgs phenomenon 920, 921, 999, 1001, 1152 super Lie algebra 302,310,687 supermanifold 302, 338 supermultiplets 304,390, 614 in anti de Sitter space 307 ofOsp(4/N) 1232 supermultiplet, self-conjugate 420 superOSClllators 1217, 1285 super Poincare algebra 315 with a chiral charge 810 super Poincare group 302 snperpotenteJ 1004 superspace 301, 668 constraints 1390 superspace, geometry 642 superspace structural equations 643 superstring generated supergravities 567 superstring tree amplitudes 1389 sllpersymmetnc anti de Sitter background 960 sllpersymmetric critical points 1083
2158
supersymmetric field theories 391, 476 sllpersymmetric: Minkowski ba.c:kground 960 supersymmetric models 584 supersymmetric V&Cll11m 1194 supersymmetry algebra automorphism group 1108 supersymmetry algebra, off-shell closure of the 812, 1984 supersymmetry algebra. on-shell or of-shell closure of 610 supersymmetry breaking 1082 supersymmetry ghosts 1809 supersymmetry, order parameter of 1003 supersymmetry shifts 1085 supersymmetry trlUlsformation 624,627.655, 656, 702, 1235 supersymmetry transformation, on-shell first-order 720 supersymmetry transformations, of-shell dosed algebra of 822 sllpertorsion 385 super-torsion 2-form 619 supertrace 348, 1000 superunitary algebras SU(m/N) 324 superunitary algebras SU{p,q/N} 328 superunitary group 351 supervielbein 385,643. 1198, 1515 super Weyl invariant 1542 super Yang-Mills theories 582 symmetric G/H 197 symmetric Lie bracket 313 symmetric rescalings 239 symmetric space 689 symmetric twists 1618 symplectic modular group 1632, 1439 symplectic modular group Sp(2g, l} 1703
lO-dimensional Lorentz group SO(I,9) 567 lO-dimensional supergravity dual formula.tion 1936 3-form A catalyst for the spontaneous compactification 1149 3-index antisymmetric tensors 834 2-component complex Weyl spinor 393 2·djmensional quantum field theory 1766 2-dimensional superconformal group 1513 2D superconformallUlomaly 1859 2·£orm B 1533 (2,2) superconformal field theory 1383 T (met (v,») 1416 tangent space 39 to p/q 341 to the difeomorphism group 1417
2159 tangent space, isotropy illeducible 1263 tangent vector 39, 361, 688 on the (soft) group manifold 647; target manifold 1377, 1382, 1391, 1582 target space fermions 1583 target space gravitino 1379 target space massless fermions 1521 target space spin connection 1526 TeichmiilIer parameters 1661 Teichmiiller space 1413, 1416, 1424, 1454 ten-dimensional supercutrent 1831 tensor multiplets 1136 tensor, self-dual or antiself-dual 834, 838 tensors 12 on manifolds 44 theta characteristics 1679 theta-constants 1680 theta-divisor 1475 theta-functions 1720 theta series 1726 thetas with characteristic 1492 three-glnon vertex 1843 three-graviton vertex 1842 T-identities 1125 topological manifold 30,31 topological term 1018 Torelli group 1450 toroidal compactification 1712 torsion-constraint 1988 torsion equation 1855 torsion field 1582 torsion of the almost complex structure 934 torsion on space-time 779 torsion 2-form 79, 82, 227 transition from first- to second-order formalism 1053 transition functions 31 of the canonical bundle 1484 transitive action 190 translation gauge transformation 647 translations 1513 transversality condition 616, 1777 transversality constraint 432 transverse 3-form 1216 transverse harmonics 1211 transverse representations 1211, 1230, 1267
2160 transverse loot vector 1831 transverse symmetric traceless tensor 1216 transverse two-form 1216 transverse vector harmonic 1216 tree amplitudes 1769 triangular gauge 1158 triple product 1734 trivial principal bundle 120 twist 1772 twisted Kac-Moody algebra 1388, 1583, 1600, 1606, 1707, 1710 twisted vector 1603 two gluon-one graviton vertex 1843 two-cocycle 1600 two-dimensional hamiltonian 1629 two-dimensional supercurrent G(z) 1731 U(l)-bundles over Kahler 6-manifolds 1343 U(l)-current in the N = 2 superconformal algebra 1808 U(l)-gauge invaria.nce 1163 U(l)-harmonics 1159, 1167, 1111 U(l}-superspace 1987 U(2,/4) Sp(4,m.)xO(8) 1284 _ U(N)-automorphism group 341 ultraviolet divergences 1641 uncontracted Osp(I/4) group 710 uniformization theorem 1400 unitary irreducible representation 304, 390 of 80(2, 3) 435 of supersymmetty 397 unitary representations of SO(2,3) 426 Usp(Z8,28) Lie algebra 1117 vacuum 159, 164, 698 vacuum configuration 177 in the presence of the scalar matter 118 vacuum existence 698 vacuum in the Osp(4/1)-case 712 vacuum of the conformal field theory 1774 vacuum state 959 van del Waelden notations 393 vector field 38, 41 kinetic term 1073 vector harmonics 1263 vector multiplet 583, 919, 1239 vector of the Riemann constants 1495 vertex operators 1189
2161 vielbein 212, 1158 field 82, 612 frame 55,77 viel hein, linearized 633 vielhein, of the scalar manifold 1122 vierbein 386 Virasoro algebra 1165,1571, 1773 Virasoro and Kac-Moody algebras 1505 Virasoro coboundary 1510, 1583 Virasoro constraints 1967 visible sector 998 volume element 46 volume form 17,56 volume of coset spaces 228 WZW sigma-model action 1963 Ward identity 1078 warped solutions of D = 11 supergravity 1352 warp factor f(y) 1343 weak hypercharge group 1305 weak isospin group 1305 weight 1800 weight lattice 1603, 1624, 1823 weight space 1292 weight vectors 1603 weights of the vector representation 1197 Weil-Petersson integration measure 1644, 1113 Weil-Petersson metric 1413 well-adapted frame 930 Wess-Zumino consistency condition 1977 Wess-Zumino model 307,308,477 Wess-Zumino multiplets 405, 457, 943, 1376, 2004 Wess-Zllmino term 1528, 1586 Wess-Zumino-Witten model 1519, 1961 Weyl condition 834, 1266 Weyl gravitino 835 Weyl group 1658 Weyl holonomy group 1197, 1251 Weyl spinol' algebra in D 6 834 Weyl spinors 537 Weyl transformation 1398, 1506 Weyl transformation, infinitesimal 1418 \Veyl unitary trick 201 Wick rotation 1502, 1545, 1556 Wick theorem 1725 world-indices 55
=
2162
world-sheet 1377, 1391 fermion. 1379, 1597, 1715 supetsymmetry 1377 SUSY lilIes 1526 wrap factor 1192 Yang-Mills action 144 Yang-Mills Chern-Simons form 1858 Yang-Mills cocycJe 1861 Yang-Mills propaga.tion equation 585 Yang-Mills theories 307 and supergravity theories, differences 642 Young supertableault 1285 Young tableau 563, 1264, 1308 labeling 1292 for O(N) tensor 544 Yukawa couplings 1361 IO>SL2 vacuum 1181 zero-forms 692 zero-mode action in D = 5 KaluzarKlein gravity 1172 zero mode of the Hodge-de Rahm operator 1241 zero-modes of the antighost and ghost fields 1472 zero modes of the Lichnerowicz and RaritarSchwinger operators 1232 zero nonn states 441, 1559 zeta-function regularization method 1389,1639, 1617 zero-th picture 1840
