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<-(zl)
the form (VI.2.IS) of the metric is preserved.
(VI.2.16a)
(VI.2.16b)
Indeed we find
and by redefining
we reduce (VI.2.17) to the form (VI.2.1S). The transformations (VI.2.16) are named conformal transformations and are the elements of a group since the composition of two analytic functions is still analytic. The Lie algebra of this infinite-dimensional conformal group is the classical Virasoro algebra. We easily derive its structure by considering the infinitesimal form of the transformation (VI.2.16) Zl • f(z) • z • E(Z)
(VI.2 .19)
and expanding the small analytic function E(Z) in a Laurent series about the origin z=O:
1397 +<»
e:(z)
n+l
L
III
e:_n Z
n=-""
(VI. 2.20)
The coefficients e: -n are the infinitely many parameters of the Vita-+ sora algebra. Indeed the tangent vector te: t.'hich generates the transformation (VI. 2.19) -+
t
E:
-+
d
"e:(~)-
(VI. 2.21)
dz
can be rewritten as the following linear combinations: -+
+co
\
L
n=-co
e:
-n
L n
(V1.2.22a)
..
n+1 d
z
(VI.2.Z2b)
dz -+
where the tangent vectors Ln are iDlll\ediately seen to satisfy the following commutation relations: (VI. 2.23)
Equation (VI.2.Z3) is the classical form of the Virasoro algebra in which the so-called central charge is equal to zero. In Chapter V1.4 we shall be concerned with the quantum version of the same algebra whose characteristic feature is the presence of the central charge. (See Eq. (V1.4.38». The previous discussion leads to the important concept of conformal class or complex structure. To introduce this notion let us first summarize the have just presented in slightly different words.
re~ults
we
We can say that given an atlas A={U} covering 1: and an arbitrary metric g~)(~) there always exi~ts a new complex atlas A(l) such that in eveT'{ chart U(l) e A(1) the metric gel) takes the fom (VI.2.lS). Given now a sec~nd metric g~)(~). a second complex atlas A(Z) can be found where also g(2) takes the form (VI.2.1S).
1398 The first question which arises is the following. When two different metrics gel) and g(2) take simultaneously the form (VI.2.1S) in the same complex atlas? The answer was already given: it happens when ~) and g~~) are related to each other by a !'Ieyl transformation, that is, when (VI. 2. 24)
where
A(~)
is any itmction on k.
If g~~) and g~) (~) are not related to each other by (VI.2.24) then the complex atlases where they take the form (VI.2.IS) are different. This means that the complex coordinate z (1) adapted to the metric gel) is not an analytic function of the complex coordinate adapted to the metric g(2):
m
(VI.2.25) At this point one might naively define the space of conformal classes (that is, of complex structures) as the space of metrics gaS(n divided by the \'Ieyl group, i.e. the space of equivalence classes with respect to the relation (VI.2.24). This definition would still lead to an infinitedimensional space and it is not the right one. lndeed we have still to divide the space of metrics by the diffeomorphism group. TI>"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 <2 then 1 is elliptic
If
Itr TI
If
Itr TI" la+dl >2 then T is hyperbolic
=la+dl =2
then T is parabolic
(V1.2.78)
Clearly elliptic elements have fixed points on H (Im
~ > 0).
Indeed
z "i is a fixed point for all the MObius transformations of type (VI.2.75); so if T is similar to (VI.2.75) via a matrix A then it has a fixed point in the A-image of z '" i.
It follows that elliptic
elements cannot be used to represent the generators of the homotopy group.
For a similar reason the parabolic elements are also excluded.
Indeed if we consider a Riemann surface
1: = H!G where G c SL(2.C} is a discrete subgroup of the automorphism group containing parabolic elements, then one can show that 1: is not compact since it has as many punctures as there are parabolic elements in G. A surface L with punctures is obtained from a compact surface
1: by removing a finite number of points (the punctures) in such a way that the region around each puncture is conformally equivalent to the pointed open disk
{z, 0 < Izi < l}. Furthermore, identifying the puncture
with the point z =0, every sequence of points {zn} with zn'" 0 must be discrete on the surface 1:. Consequently, since we do not want 'If} (I g) fOr g ~ 2 must be realhed by means of 2g hyperbolic elements Ti (i" 1, •.. ,2g) of SL(2.:R). The hyPerbolic elements pose no problem since their fixed points always fall
punctures the hOlllOtopy group
on the real axis :R and hence have a fixed point free action on H.
1416
In order to be a representation of 111 (Eg) the 2g hyperbolic matrices Ti must fulfil the fundamental relation (VI.2.3S) which reads (VI. 2 • 79)
The finite subgroups of SL(2,R) generated by hyperbolic elements fulfilling Eq. (VI.2.79) are named Fuchsian groups. It follows that. for g ~2. Teichmiiller space, Le. the space of deformations of the metric (0'1' of the complex structure) inequivalent under local diffeomorphisms. is the space of Fuchsian groups inequivalent under SL(2,lR) conjugation. We Can easily count the dimensionality of this space as a real manifold. Every Ti depends on three real parameters so that we have 6g real parameters. Three of them are eliminated by the constraint (VI.2.79) and other three can be fixed by an SL(2,lR) transformation. In this way, we are left with 6g - 6 real parameters. What is not obvious from the above counting is the complex structure of Teichmuller space which will be inherited by the moduli space after division by the mapping class group. To See this let us proceed to the next section.
VJ.2.4 Deformation of the metric, quadratic differentials and the cO~lex structure of Teichmiiller space. It is convenient to revert for a moment to a deSCription of our 2.dimensional surface in teTms of real coordinates ~1, ~2. We consider the space of Metrics Met (E g) already introduced in Section VI.2 and we fOcus on its tangent space T (Met (tg Indeed, since the moduli space M is given by Eq. (VI.2.27) it follows that g its tangent space, that is the Teichmiiller space Teich (t), is given g by the following quotient
».
1417 Teich (Eg)
= T (Met
(Eg»)/T(Oiff) e TeWeyl) •
The tangent space T(gaB) (Met (Eg»
(VI. 2.80)
at a given metric gaB m is
spanned by the infinitesimal deformations of
gaB(~):
(VI.2.81)
where ogaB(~) is an infinitesimal two index symmetric tensor. The linear space T( ) (Met (E» is endowed with a natural scalar progaB g duct. Given two deformations og(l) and og(2) we define their scalar product as follows
where the integration is extended to the Whole compact surface. The tangent space to the Diffeomorphism group T(Oiff) is its Lie Algebra which is spanned by the tangent vectors to the surface. Indeed tie have T(Oiff (E g » = T(E g )
(VI.2.S3)
every tangent vector ta(~) being the generator of an infinitesimal coordinate transformation (VI.2.84)
Also T(Oi£f) is endowed with a natural scalar product
1418 < t 1,t 2 > ..
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
I iF..rgg(f;fl,6gaB6~(\to+"JO\MZyOVot) . IiF..rg
>=
=
(Vot ... Vot - 2Vot)o, .. 0 •
Next we calculate the adjoint of the operator PI
(VI. 2. 94)
defined by (VI. 2.95)
and we get
(VI.2.%)
Hence we can write the following orthogonal decomposition T(Met) '" TCWeyl)
$
H
(VI. 2.97)
1420
where H is the space of traceless metries. On H we define the which maps H into the space of vectors, action of the operator i.e. into T(Diff)
pi
pi:
H'"
T(Diff)
The operator PI and its adjoint decomposition of H:
(VI.2.98)
pi
induce the following orthogonal
t H '" 1m PIe ker P1
(VI. 2. 99)
Then we introduce the following nomenclature: ker
pi '" space of quadratic differentials
ker PI = space of conformal Killing vectors
.1
pi
The reason for the name attributed to ker is still mysterious but why ker PI is named as above is almost obvious. Indeed if ta(~) e ker PI we have that the metric deformation induced by t a (VI.2.l00) is just a Weyl transformation. It is easy to verify that the conformal Killing vectors form a Lie subalgebra of T(Diff). Indeed if t~ and t~ are two conformal Killing vectors then also their commutator (VI. 2 .101)
fulfills Eq. (VI.2.100) and is a conformal Killing vector. Hence ker PI is a Lie Algebra: the Lie Algebra of the conformal automorphism group of the metric gaB:
1421 (VI.2.102)
It contains the subalgebra of the proper Killing vectors fulfilling the more stringent equation (VI. 2.103)
These Killing vectors generate the Isometry group of the metric gaa' have seen in the previous section in the case of the three Simply connected Riemann surfaces the isometry group is always a subgroup of the Automorphism group As ~~
(VI. 2. 104)
t
the other hand the elements of ker PI are the sYJlllletric traceless and divergenceless tensors: On
haa " hSa t
haS e ker PI
aa _ haag .- 0
(VI. 2.105)
naB" 0 Why we call such objects quadratic differentials becomes evident when we go back to complex coordinates. In full generality, if we consider the line element written in real coordinates: (VI. 2.106)
and we introduce the complex ones through the identification (VI. 2.107)
we obtain the equality
1422 (VI. 2. 108)
where (VI. 2. 109a)
(VI. 2.109b)
(VI. 2.109c)
In complex coordinates the infinitesimal variation of the metric is parametrized in a similar way: (VI. 2.110a)
(VI. 2 . HOb)
og-zz
= (ogzz )"
(VI.2.110c)
and we have
The complex basis expressions of the operators PI and easi ly worked out.
[Pl(tl]
zz
[p1(t)J zz-
pi
are also
We find
= 2~ zt z
-
g {g~B~ to}
zz
= ~ zt-z + ~-t zz-
a
~
zz_{g~B~~tel
g
(VI.2.112a)
(VI.2.112c)
1423
t, ] [Plug
Z
"~2
(g
zz, '9 ug 1.
ZZ
fE V~6g-
+g
1.
1.1.
zi '9-6g
+g
Z
1.1.
1.1IJ 6g- )
+g
Z
ZZ
(VI. 2 .113a)
(VI.2.113b)
If we choose the always available set of coordinates where the metric takes the form (V1.2.15) (g zz .. g-_" zz gZZ '" gzi =0) > formulae (VI •2• 111) > (VI.2.l12) and (VI.2.113) simplify considerably and we get: a8
-$
g ogCI B .. 2 e 6g1.1. [Pit
11.Z ..
(VI. 2. 114a)
2'11Zt Z
(VI. 2. 114b)
[P1t]-.. 2V-tz:r. ZZ
(VI.2 .114c)
[Plt] 1.1.- " 0
(VI. 2.114d)
(VI.2.114e)
(VI. 2 .114 f)
From these formulae we realize that a traceless, symmetric, divergenceless tensor (i.e. an element of ker pi) is a holomorphic covariant tensor Il zz '
Indeed if ogzi" 0 Eq. (VI. 2.1l4e) yields
o = gzzV-il ~ o-il Z 12. Z Z1
,,0
(VI. 2.115)
This is the justification for the name quadratic differentials imposed t
on the elements of ker Pl'
1424
On any Riemann surface Eg one considers the
q~di£ferentials
(VI.2.1l6) q
i.e., the analytic (q,O) tensors, and calls H(q)(r) their complex g linear space (that they form a linear space is obvious). What
l>'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" (
zz
d*w=
zz
-iCc-w z z +dw.)dz"dz. zz
(VI.2.143a)
(VI.2.143b)
1430 6.2.2 Definition.
Following a standard nomenclature
III
is named
respect! vely closed or coc1osed depending on whether dw .. 0 or d*IIl" O. If
III
is both closed and coclosed then it is named a harmonic l·form.
Furthermore. a I-form !AI"
"df, f(z,z)
Iil
is respectively exact or coexact if wadf or
being some O-form defined all over the surface.
As it is well known from the general Hodge theory. which applies
to any smooth manifold M (see Chapter 1.6) harmonic p-forms can be taken as representatives of p-cohomology classes, namely they are in one-to-one correspondence with the generators of the H{P) (M) group.
Indeed any closed p-form w(p)
form h (P)
homology
can be written as a harmonic
plus an exact form (VI.2.144)
Furthermore, recalling that the dimension of the p-th homology group
H(P)(M) is the p-th Betti number of the manifold M: (VL2.14S)
and gi yen a set of generators can always choose
B(p)
S~p)
(i" 1, •••• B(»
harmonic p-fonns hjP)
of H(P) (M),
~j ,,1 ..... B(p)}
one in such
a way that (VI.2.146)
In the case of a genus
g Riemann surface the Betti number Bl (1:g )
is
(V 1. 2 .147)
This result is easi ly understood in te:rms of the normal picture of 1:g as a 4g-polygon with identified sides (see (VI.2.32) and (VI.2.SS». Indeed the reader can easily verify that the closed curves b •.•• ,b I
g
a 1,.··.ag ,
are homologically non trivial (they are not boundaries of
1431
any region drawn on the surface) and generate the whole homology group. This means that any other nontrivial cycle can be written as a combination of the cycles 8 1, .•. ,8g, b1, ... ,bg• Hence on the Riemann surface Lg we can construct 2g harmonic I-forms ~l""'~g' ~l,···.Sg: (VI. 2.148)
i " 1, ... ,g (VI. 2 .149)
which satisfy the following relations:
Ja.
a. j
t
°ij
1
Ja.
1
aj .. 0
(VI. 2. 150)
1
S.
jb. S.J =
0
J
°ij
(VI. 2 .151)
1
The harmonic I-forms are closely related to the holomorphic differentials. Let us introduce the general concept of meromorphic q-differentials which was already mentioned in the previous section while describing Teichmuller deformations. 6.2.3 Definition. By a (meromorphic) q-differential p(g) on Lg~ mean an assignment of a meromorphic function 11 (z) in each local ~~~~~~~~~~~~~~~~~~z ... z Foordinate frame on k so that g
/q) (z) " p
z... Z
(z) (dz)q
q
is invariantly defined.
(VI. 2.152)
1432 In the language of Section VI.2.S a meromorphic q-differential is a (q,O)-tensor whose components are meromorphic functions of z. The case q '" 1 corresponds to the case of analytic I-forms. The meromorphic I-differentials are also called abelian differentials. One considers the singularity structure of the q-differentials.
At every point peE z(P)
introducing a coordinate system such that
° we define the order of
=
g
)lq by means or the relation
°
ord 'lJ (q) '" ord 1.1 P z... z
(VIo2.ISS)
If we write (VI. 2. 154)
with g holomorphic and non-zero at
z=O, then ordOlJ z•.• Z =n, As for meromorphic func-
where the number n is a relative integer.
°
tions (which are the particular case q .. of q-differentials) the set {p e 4g ; ordp Uq # a} is discrete and finite.
of points
At any point
tensor)l
z••• Z
P e Eg •
if we set
N ::ordp u(q)
then the (q,O)-
admits a Laurent series expansion 00
)l
'" Z ... 2:
L a zn n=N n
and we can define the residue of the differential
(VI. 2 .155)
jJ (q)
at
P through
the following identification: (VI.2.1S6)
When ordp lJ (q) < 0. u (q) has a pole at P. When ord p 1J (q) > 0, P is a zero for lJ (q) • The points for which ordp lJ (q) :: are the regular points of the differential.
°
1433 6.2.4 Definition.
A meromojhiC q-differential
if it has no poles
(ord p lJ (q
6.2.5 Definition.
HololDOrphic I-differentials are called Abelian
~0
for all
P
)J (q)
is holomorphic
e Eg)'
differentials of the first kind. 6.2.6 Definition.
Meromorphic I-differentials with poles but zero
residues are called abelian differentials of the second kind. 6.2.7 Definition.
Meromorphic I-differentials with residues are
called abelian differentials of the third kind. Let us now go back to
Eqs. (VI.2.I43) and consider their implica-
tions for a hannonic differential. If w is harmonic we conclude that (VI.2.1S7a)
azw-z = 0
(VI. 2 .IS7b)
.
z is an analytic function while
z
Hence
W
tion.
It follows that we can define two analytic I-differentials
W
is an antianalytic func-
(VI.2.1S8a) (VI. 2 .lS8b)
such that the harmonic differential can be written as follows (VI.2.1S9) Furthermore since w has no singularity WI and 11)2 are holomorphic. So we have shown that any harmonic differential can be written in terms of two holomorphic differentials through the relation (VI.2.1S9).
1434
In particular a real haTmonic differential is twice the real part of a holomorphic differential u"ZRew.
(VI. 2. ]60)
We can now prove also the converse theorem. 6.2.8 Theorem. A differential I-foTm w is holomo!phic if and only if w" ex + i *u for some harmonic di fferential a. If a
Proof:
is harmonic then (VI. 2 .161)
with wi and w2 holomorphic. Then using Eq. (Vr.2.142) we get (VI.Z .162)
so that (VI.2.163)
is holomorphic. Conversely, if w is holomorphic then it is also harmonic, and so is its conjugate w. Indeed dw'" d'w = 0 .
(VI. 2.164)
Hence we can introduce the harmonic differential 1
-
(VI.2.16S)
(w + iii)
(VI. 2.166)
ex" 2" (0) - w) and we obtain *a " -
which implies
!.2
1435 w .. IX
i-
i *11
(VI.Z .167)
as we wanted to show. interesting corollary of this theorem provides a simple criterion for the holomorphicity of a differential. M
6.2.9 Corollary: w is a holomorphic I-differential if and on1r if it is closed (dw: 0) and if it is antiselfdual (*w", -iw).
The space of I-foms on the Riemann surface I: can be given a g Hilbert space structure introducing the folloWing hermitian scalar product (VI.. Z.168)
where the integration is extended to the whole surface. In particular given the canonical homology basis identified by the sides of the 4gpolygon and the associated canonical cohomology basis defined by Eqs. (VI.2.148) the scalar product (VI.2.168) can be utilized to define the 2g x 2g intersection mat rix J: (Il1.•
*a.)) J (VI.2.169)
(81'
*aj )
The entries of J can be easily calculated using a lemma which holds true for closed forms.
(VI. 2 .170)
1436 ~:
Consider the fundamental 4g~polygon (VI.2.32). hereafter named , . Since f1 is simply connected the closed form 61 can be written as an exact form on 9 (VI. 2.171)
However it has the property that its values on the identified sides are equal. Using (VI.2.171) we can write (VI. 2 .172)
Using Stokes' theorem in (VI.2.172) we get
(VI. 2.173)
Now by hypothesis
to
is a primitive of the differential 81, Hence if ~ and z and Zl are t~~ equivalent
f
is an interior point of .
pOlnts on ai
-1.
and ai
l>'e
can Wrlte
(VI. 2 .174)
Consider then the following picture
b'l A_----
Zo
1437
By inspection of (VI.2.17S) we see that
: ~ fa. fb. 1
1
8182
=- f
b.
1
81
Ja.
62 • (VI.2.176)
1
Repeating the argument for the other two terms of Eq. (VI.2.173) we obtain the proof of the lemma.
Utilizing lemma (VI.2.170) the
inter~
section matrix J is immediately worked out and we find that J
is
equal to the canonical symplectic metric in 2g dimensions:
(VI. 2 .177)
The reason for the name intersection matrix given to J should now be explained. Geometrically, if w is the closed (harmonic) form associated c1 to a cycle
c l and w
z
C
is the closed harmonic form associated to a
cycle cz, then the antisymmetric scalar product (VI.2.178}
has a simple interpretation:
c10cZ is the number of times the curve c1 intersects the curve c Z. Given a direction on c1 and c2 ' the intersections are cotmted positive if c2 "arrives" from the right of
c1 and negative if c2 arrives from the left of Ct. has
In general one
(VI. 2•179a)
1438 (VI. 2 • 179b)
(VI.2.l80a)
(VI. 2. 180b)
and using lemma (VI.Z.l7D) the intersection number (VI.2.l7S) is found to be: (VI.Z.181)
1m example might be clarifying.
Consider c1 =2a1 and
C
z =bl •
We
find (VI. z.182)
Looking at the following picture we see that c1 has indeed two positive intersections with c2
(VI. 2 .183)
6.2.11 Definition. A basis of generators Cr (1 =1, .... 2g) of the first homology group HI (ig) is canonical when the intersection matrix is the canoni cal one (VI. 2. 177)
1439 (VI. 2.184) As we have seen the cycles
al"); .ag • b l , .... bg associated to the form a canonical homology basis. This. however.
4g~polygon
sides of the
is not the only one.
Indeed any set of 2g linear combinations with
integer coefficients
(Aij • Bij , Cij • Dij
e Z): (VI. 2 • 185a) (VI. 2. 18Sb)
bit = C.ja. + D.jb. 1 J 1 J
such that a!'b!
" °ij
(VI. 2. 186a)
b! 'b! 1 J
" 0
(VI. 2 .186b)
provides another canonical basis.
Conditions (VI.2.186) can be written
0 J "
a! 'a! 1
b! 'a! 1 J
"
1
- O1J••
in matrix form as follows.
)
Defining the 2g x 2g matrix
(VI.2.181)
whose entries are integer numbers we must have AT J A", J co- A e Sp(2g.I) .
6.2.12 Definition.
The discrete group Sp(2g,Z)
(V1.2.188)
is called the sym-
plectic modular group. Any canonical basis is related to another one by a suitable Sp(2g.I) transformation.
1440
We construct now the holomorphic differentials associated to a canonical homology basis. We begin by introducing a collective name for the a., B. l-foms defined by Eq. (VI.2.149): 1 1 I:;i, ... ,2g
(VI. 2 .189)
and we recall that (VI.2.190)
We consider then the matrix
rrJ
defined by (VI. 2 .191)
where the *-operation was defined in Eq. (VI.2.142). 6.2.13 Lemma.
r IJ is a
s~~etric
matrix.
Proof: Indeed we know that (,) is a hermitian scalar product in the Hilbert space of I-forms. Furthermore Yr and YJ are real 1forms. Hence the lemma follows. 6.2.14 Lemma. The matrix rrJ is positive definite. Proof: If we consider any harmonic I-form along the basis Y1:
Since
e
e
it can be decomposed
has necessarily positive norm
o < 1e12 "' J e ~ *-a = {.i. il. I J
the lemma follows.
f,
I -J f,
r1J
(VI. 2.192)
1441
We can now write the 2g x 2g matrix r in g x g blocks:
(VI.2 .193)
and we have the conditions
~ = ~T > 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 <0, zi is a pole ,,'hile if (1. > 0, z. is a zero. It is also cleaT that if 1
Rl
1
and R:2 are two rational functions then the divisor of the product
is the product of the divisors defined in the following way Cl1 an _Bl _8m (R1R2) " (R1)(Rz) "zl .•. zn %1 .•• zm •
(VI.2.269)
where (VI. 2 . 270a)
(VI. 2 . 270b) Generali~ing
the concept to any Riemann surface we can introduce the
following definition.
1460 6.2.17 Definition.
A divisor
a
0.
Ijt
on Ig is a fomal symbol.
~
41:p 1 p 2"' Fk 1 2
(VI. 2.211)
with F. E E and a. e Z~ -3-g JWe can also write the divisor
~
as follows
(VI.2.272)
with a(P) e Z and ClCP) ~ 0 for only finitely many P e I . The g divisors on E form a group Div (I ): it is the free commutative g
g
group (written multiplicatively) on the points of I g • Thus if 41 is given by (VI.2.272) and ~
=
JI
peep)
(Vl.2.213)
PeEg then by definition we have: Ijt,~
=
JI
peI
pClCP) + B( P)
(VI.2.274)
p-a(P)
(VI.2.275)
g
and q,.-l
=
II
PEI g
Next we can introduce the concept of the degree of a divisor.
We define
(VI.2.276)
Clearly the operation deg establishes a homomorphism
deg :
Di~(I
)
g
+
I
(V1.2.277)
1461 from the multiplicative group of divisors onto the additive group of integers. If fez) is a non-identically vanishing meromorphic function on the Riemann surface Lg • then fez) determines a divisor (f) defined as follows (VI.2.278)
i
q ) is a meromorphic q-differential. recalling Eq. Similarly if (VI:2.151) we can define its divisor (pq) by means of the equation:
(VI.2.279)
The divisors associated to llIeromorphic functions have a special property. known from elementary complex analysis: they have zero degree deg (f)
=0 .
(VI. 2.280)
Indeed Eq. (VI.2.280) is nothing else but the familiar statement that the SU1II of the orders of poles and zeros vanishes for a function fez) if it is meromorphic. To be more precise we point out that the meromorphic functions on E form a field I·ler (l:) and therefore an g g abelian group under mUltiplication. The operation () defined by Eq. (VI.2.278) establishes a homomorphism (VI. 2.281)
from the multiplicative group of meramorphic functions into the subgroup of divisors of degree zero. That these latter form a subgroup is selfevident from the definition of the degree operation (VI.2.276). A divisor in the image of the homomorphism (VI.2.281) is called a principal divisor:
1462 'Ii' .. princi pal ~ fiJ e 1m (Mer 0: ) g
.
(VI. Z. 282)
One may wonder whether any divisor of degree zero is principal, that is. whether we can always find a meromorphic function which has poles and zeros at arbitrarily prescribed locations with arbitrary orders, just sU/lUDi-ng up to zero. The answer is no. A deep theorem due to Abel gives a criterion to establish when a degree zero divisor is principal. Before stating it we have to introduce still more concepts and definitions. Abel theorem will be given in the next section.
6.2.18 Definition. The group of divisors modulo principal divisors is known as the divisor class group. Indeed the normal subgroup of principal divisors introduces an equivalence relation on Div Ci:). Two divisors Ii' and fA are equi".,.-1.IS prlnClpa . g. I : va1,ent provl'ded ...~~ o/I"vf) .,.
3£ e Mer (E )/<&,~-1,. g
(VI. 2 .283)
(f) .
On the group of divisors one can introduce also an order relation.
We
begin with the following definition of integral divisors: 6.2.19 Definition. A di visor iii! is integral (in symbols 0/1 > 1) provided (X(?) ~ 0 for all P e Eg• Then we state the following. 6.2.20 Definition. A divisor 0/1 is bigger than the divisor symbols 0/1 > f)) if the di visor 1i'~-1 is integral. This definition has the following meaning. and !J suitable differentials or functions
fflll
a (in
If we associate to and
f.~
~
then we have
fill > ~ if the zeros of fflll are stronger than the zeros of ff) and the
poles of f
f~.
This corresponds
1463 qualitatively to the concept fot/fa "holOl1lOrphic function.
Given
these preliminaries we can introduce the ingredients of the
RieJllann~
Roch Theorem. Ingredients of the Riemann·Roch Theorem: 1)
r(t'} " dimension of the space of meromorphic functions whose divisor is bigger than the divisor 1&'.
Let
be the divisor under consideration. We set (VI. 2.284)
r(en " dim L(!W) and fez)
e
L(~)
In other words,
(VI.2.28S)
... (f) > Ii' .
f belongs to
L(~
if it is holomorphic at all points
which are not p1"",Pk and if at these points it has poles ~~aker or equal and zeros stronger Or equal than those of .... 2)
i(~)
" dimension of the space of abelian differentials
(l~
differentials) whose divisor is bigger than the divisor ~. We set (V1.2.286)
i (...) " dim Q(~
and III " III
z
(z)dz e Q(1i')
CO>
(!II) > 4&' •
The Riemann·Roch theorem can now be stated.
(VI. 2. 287)
1464 6.2.21 Theorem. For a Riemann surface E of genus g and for any g divisor iii! e Div (1:) the following- relation holds , g r(q[-l) _ i(1JI!) " deg t' + 1 - g •
(VI. 2. 288)
For the proof We refer the reader to the mathematical literature (see for instance Ref. [51). We shall rather concentrate on some of its most relevant applications. We consider the abelian differentials, that is, the meromorphic analytic I-forms on the surface (see Eq. (VI.2.lS0) and definitions 6.2.5-7). Each of them defines a divisor, but all these divisors lie in the same class. Indeed if wl and (,)2 are two abelian differentials (VI.2,289)
then their ratio is a meromorphic function
(VI. 2.290)
so that (w1)(2)-1 w
. . Id"IVlsor "prlnclpa
~
(1)~(2) w w • 'V
(VI.2.291)
6.2.22 Definition.
The universal divisor class of the abelian differentials is called the canonical class and it is denoted by Z. Next we prove the following lemma.
Lemma. For an,y the divisor class of 6JJ.
6.2.23
e 01 v
and i(i?I) depend only on Furthermore if w is any abe lian differential
fiJI
(1: ), r(liI!) g
then (VI. 2.292)
1465
~: Let f'l '" t'r Then f'l *'21 is a principal divisor namely tie can find a merolllOrphic function f 12 such that o1§'l.q{;l,. (f 12 ).
Let
h 6 L(1W2) we have f12h II L(~l)' Indeed (f12h) =(h){c§'l ,q[;l) > ~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(
Obviously the roles of L(ti'Z) and L(
The two vector spaces are then isomorphic and consequently we have (VI.2.296)
as we wanted to show. Consider next an abelian differential ,II Q~) whose divisor (,) is bigger than ~. If w is any other abelian differential then ,/00 is a merolllOrphic function f,/oo whose divisor is bigger than ti'w- 1
(')(00)
-1
,.
(f~/w) > ~
-1
w
(VI.2.297)
Hence the mapping (VI.2.298)
establishes a linear isomorphism between the vector spaces L(4'(oo)-l). Hence they have the same dimension
Q(~}
and
1466
(VI. 2. 299) Since we have already shown that (VI. 2. 300) it follows from (VI.2.299) that
this concludes the proof of the lemma. Combining the above lemma with the Riemann-Roch theorem we can now prove the following.
6.2.24 Theorem. The degree of the canonical class Z is always given by: deg Z ,. 2g - 2 .
(VI.2.302)
We know that the holomorphic differentials span a g-dimensional complex vector space. This vector space is n(l) since the divisor of a hOlomorphic differential is by definition integral «00) > 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
""
tho thot, fwcti""
a
e [:b ~
[~l
with ,h,ntt"i"i,
the closed form expression for the following ratio: det
where
•
1" const
[!] ~ (VI. 2.342)
det 0
is the chiral Dirac operator on r g and 0 is the Laplacian
on the same surface. Once established Eq. (VI.2.342) implies that the partition function of every fermion field Z(fermion)
= det
~
(VI. 2.343)
can be expressed in terms of theta functions and, as such, has known transformation properties under Sp (2g ,1) . How the thetas transform under the symplectic modular group will be seen in this section. In Chapter VI.5, Eq. (VI.2.342) will be demonstrated for the case g" 1 by an explicit calculation. For higher genuses, it has been justified by sophisticated arguments in algebraic geometry [61 relying on the factorization properties of the string amplitudes. In principle, using the methods of Ref. [8] it should be possible to retrieve Eq. (V1.2.342) directly in the operatorial approach for any genus.
We will not touch such a complicated problem here.
1476
In any case. Eq. (VI.2.342) is almost an obligato~y identity in view of the relation between the theta functions and the spin bundles we now try to illustrate. We begin by describing the general concept of line bundles over Lg and its relation to the notion of divisors introduced in the previous section. To this purpose we also need the concepts of sheaf and sheaf cohomology. A sheaf over a topological space M (in OUr case M=l: g is the Riemann surface) is a family of groups F(U) (usually abelian) associated to each open subset U c M. The elements cr e F(U) of the group FeU) are called the sections of the sheaf over the open chart U. Given a subspace U eVe M of a subspace V c M there is a map
rv , U
: F(V) ......
(VI.2.344)
F(U)
between the two associated groups. The map rv U is called the > restriction map. It must satisfy the following axioms: a) Given three nested open subspaces (or charts) U eVe have
I~
we must
(VI.2.345)
We denote by crl u the restriction to U of our element cr e F(V b) (J
e
:l
U).
Gi ven any two submanifolds U and V and given any two sections and 1" Ii F(V) such that
F (U)
01 UnV '" 'I UnV there exists a section p e F(U
(VI.2.346)
u
V) such that (VI. 2. 347)
1477 If
c)
(1
e F(U u V)
and o/U
=0lv = 0
then
0=0.
The prototype of a sheaf is the sheaf of C""·flUlctions.
Indeed C'" (u)
is clearly an abelian group lUlder addition
= hex)
f(x) + g(x)
(VI.2.348)
and the restriction operation is the one which obviously falfi 11s axioms a).
Axiom b) is the principle of analytic continuation. and axiom c)
is obvious.
IJ
1)
Sheaves of interest to us in the present context include:
= the
sheaf of holomorphic functions (form a group under
addition)
IJ*
2)
= the
sheaf of holomorphic functions without zeros (form a
group under mUltiplication) 3)
Q(p)
= the
sheaf of holomorphic p-forms.
Next we introduce sheaf-cohomology (Cech cohomology). As in any other cohomology construction
~~
define the vector space
cP of p-cochains and a coboundary operator
(VI. 2 . 349) with the property (VI.2.350) Let
(F. M)
be a sheaf of abelian groups (typically functions) over the
manifold M.
Let
!!. = {'i'a 1
be a finite atlas covering
Then we have the groups
M.
F(U l ), F(UZ) •...• F(Un ). For instance if
n =4 as in the pi cture (VI. Z. 3S 1) we have just four groups F(UZ)'
reUS)'
F(U4)·
F(Ul ) ,
1478
(VI.Z.351)
The space of O-cochains
CO(Q, F)
is defined by (VI.
z. 352)
In other words, a O-cochain is a collection of m-functions (calling functions the elements of the groups
FeUa)): (VI. 2. 353)
defined over UI , Uz,···, Un' On the other hand a 1-cochain is a collection of
functions
(gaB" -gSa
In
(n - 1)
in additive notation): (VI. 2.354)
defined on all the possible intersections collection of .!:.n(n-l)(n-2) 6
2 C
= (h I23 ,
functions:
h 124 ,· .. , h(n-2)(n-l)n)
defined over all the possible intersections and so on.
Ua n US' A 2-cochain is a
(VI. 2. 355) Ua n Upo n Uy (a # 8 of y)
1479 The definition of the coboundary operator is the following.
Using the additive notation for the F-group operation we set:
(VI.2.356)
where (VI. 2. 357)
For instance let n" 4 and let {VI. 2. 358)
be a l-cochain.
The boundary of C1 (VI. 2. 359)
in additive notation is given by
I - f 13IUl23
g123 "f23 U 123 g 124
"f! 24 U124 - f 14 IU124
+ f 12
IU123
(VI. 2. 360a)
+ f
I
(VI. 2. 360b)
12 Ul24
while in multiplicative notation it reads as follows
(VI.2.361) As usual a p-cocyle
C(p)
fulfils the condition
1480 (VI.2.362)
while a p-coboundary b(P) is given by (VI. 2. 363)
for some (P-l)-cochains lp-l) 6 C(p-l). The p-th cohOlllology group is as usual the space of p-cocycles modulo p-coboundaries: p-cocycles
If (M • F) =- - - - -
(VI. 2. 364)
p-coboundaries
To illustrate the concept we consider a very simple example.
Let
PI be the projective line
o +co
(VI.2.365)
which is parametrized by two charts: i)
UO'
the chart around zero with coordinate x,
Ii)
U.. ,
the chart arotmd infinity with coordinate y.
In the intersection
UO "" U.. ,
the transition function is
1
(VI.2.366)
Y"x' Let us then consider the sheaf of polynomial differentials: Q
= P(x) dx
which form a group Jllldel' addition.
(VI. 2. 367)
A typical O-cochain is given by: (VI .2. 368)
1481
where Po", (x) is a polynomial defined in the intersection Uo () U"",
We have
oco ~ ([Po(x)
1 + --2
x
1 1dx) P (-) "" x
(VI. 2.369)
A O-cocycle should have 1
1
PO(x) " - - P H x2 co x
(VI. 2. 370)
which is absurd since both Po and PO) are polynomials. Hence (VI. 2. 371)
1
The cohomology group H (palyn. P1), however is nontrivial since not all the i-chains are in the image of o. For instance ;le have
(VI.2.372)
so that (VI. 2.373)
is a non-trivial l-cocycle. Note that oC l ;: 0 follows from Eq. {VI.2.356) for any C1 since we have just two charts. Equipped with the notion of sheaf cohomology we can introduce the concept of linebundles. A line bundle y(t g ,C) is a bundle having the Riemann surface Lg as base-space and the complex plane ( as fiber-space. It is conveniently described in terms of cross-sections and transition functions.
1482 {'t.J be an analytic at las covering 1: and let z be ... g a the local coordinate in the chart U. Since the atlas is analytic, in Let
a
the intersection Ua n Us we have that (VL2.374)
is a holomorphic function of zB' A meromorphic cross-section (J e .!I' (kg,C)
is an assignment of a
C-valued meromorphic function (Ja(za) of the local coordinate za to each local chart Ua: (VL2.375) In the intersection
and aB(zS) are related by a transition ftmction haB which is required to be holomorphic and nonUa nUB' aa(za)
vanishing in Ua nUB: (V1.2.376)
where we have siIDply called z the local variable z"Za
[in this way
aB(z) =06(z6(Z»]' It follows from Eq. (VI.2.376) that the transition ftmction hBa is related to h
Ba
haa
as follows:
.. _1_
haB
(VI. 2.377)
Furthermore, the follOWing cocycle condition (Vl.2.378)
is necessary in order for the cross-sections (] e !f (1: g• C) to be consistently defined. Recalling the definitions given a few lines above we see that the set of trans! don fUnctions {haB} is a l-cocycle for the sheaf Oi< of nonvanishing holomorphic functions.
1483 From the definition it follows that the set of inequivalent line bundles coincides with the first cohomology group HI (E ,0*). Indeed, if two sets of transition functions coboundary we have
{haS} and
{h~B}
g
differ by a
(VI.2.379) (VI. 2. 380)
and the coboundary can be eliminated by a redefinition of the crosssection (VI .·2. 381)
The simplest example of line bundles is provided by the canonical bundle K, whose cross-sections are the meromorphic I-forms (the abelian differentials) : 10
=wzdz
In this case we have cr
(VI. 2. 382)
ex
=W
z
and the transition functions are given by
(VI.2.383)
By definition the q-th power;eq of a line bundle is that line-bundle whose transition functions are the q-th power of those of :I: (VI. 2. 384)
The holomorphic cross-sections of Kq are the holomorphic q-differentials discussed in the previous section. A
bundle
spin-bundle S is defined as a square-root of the canonical
1484 (VI. 2. 385) It is introduced in the following way.
Consider the line-element (VI.2.1S) in the Ua and U8 coordinates: (VI. 2. 386)
where for later convenience we have redefined we can introduce the vielbein I-forms
~ +
2..
In both patches
(VI.2.387a) e - • e0 - i e 1 • e ~(z.z)d-~
(VI. 2. 387b)
and we can rewrite Eq. (VI.2.386) as follows (VI.2.388)
From the definition (Vl.2.386) of the conformal factoT we deduce its transformation property
(VI.2.389) and we conclude that the transition function for the vielbein 1-forms
is a U(l)-element. the phase of the Jacobian dzaldzS' (VI .2. 390a)
exp[i6aa]
dz
= dZ;
The phase-factors {e
I I
dzS dZa .
ieaa
1
(VI. 2. 390b)
can be regarded as the transition functions
of the canonical bundle· K in the anholonomic description.
1485 In a spin bundle S the transition functions {S s} are given i9 a by square~roots of the transition functions fe as}. Indeed for a spinor. that is for a cross-section l/I e S of the spin bundle we must have (VI. 2.391)
where (VI. 2. 392)
Hence we can set (VI.2.393)
where Aas = 0, 1 mod 2 is a 7.2 element.
In order to be consistently
defined the spin bundle must have transition functions fulfilling the cocycle condition: (VI.2.394)
which is true if and only if the following identity is verified iC6Q+6S +9 }»ill(A o +A S +A +mod2}. 2 ajj Y yo. a... Y ya
(VI.2.39S)
The cocycle condition fulfilled by the transition functions of the canonical bundle implies
eC1B +
9SY + 9ycr.
= 211
(VI. 2. 396)
kapy Q
So we can write i (9 S+9 +6 ) -2 SY yo. 01
where
= ill(wcr.0.., ... ,
mod 2)
(VI. 2.397)
1486 if katly " even (VI. 2. 398)
if
k~ "odd
The proper way of interpreting Eq. (VI.2.397) is in terms of sheaf cohomology. We have constructed a lz·sheaf over the Riemann surface and waBy is a 2·cochain of this sheaf.
Actually it is easy to prove that it is a 2.cocycle. (VI.2.396) it follows
Indeed from
and, hence
oWaByo " wBy6 - Wayo
+
waSo . W aBy " 0 mod 2
(VI. 2.400)
Equation (VI.Z.39l) can now be interpreted in the following way: the 2.cocycle w(2) should be equal to a coboundary 071(1): w(2) "
oJ. (1)
•
(VI.2.40l)
If the homology group H(Z) (Lg'ZZ) were nontrivial then we might find an obstruction to the solution of Eq. (VI. 2.401) and hence to the existence of spin bundles. This is never the case with Riemann surfaces. since one can prove that H(2) (Lg'ZZ) (the second Sitefel-Whitney class) is always zero. Indeed this latter is the lz·reduction of the Euler class which for a Riemann surface, is always even being given by the well known formula for the Euler characteristic:
x" 2 -
2g •
(VI.2.40Z)
When we have a solution of Eq. (VI.2.40l) we can generate others by adding to A(1) a solution of the homogeneous equation:
1487
(VI. 2. 403a) (VI.2.403b) 1 T he di menslonallty . , The Sh1' f t n(1)..lS a 1 ~cocyc e. of the space of spin bundles is then given by H(l) (Lg,lz)' The dimensionality of this homology group is (VI. 2.404) and so we conclude that on a genus g Riemann surface there are 22g. inequivalent spin bundles. To see that this is the case let us cut the Riemann surface L g along the cycles of a canonical homology basis. The result is the usual 4g-sided polygon (VI.2.3Z). The cocycle n~~) can then be viewed as the set of transition functions across the cuts. We consider a section S of the spin bundle. If Sa is the section along a.1 we identify i -1Ii 4>i 1 it with e times S 1 (the section along a~) and similarly we 1 a. 1 1Ii6i (the identify Sb. (the section along bi) with e times 1
. along b~l). The phases ~i an d 6i sectlon 1
a.1 ~. 1
:0
0,1 mod2
= 0,1
mod2
(VI. 2. 405a)
(VI. 2. 405b)
Since we have two possibilities for each of the 2g homology generators, the total number of casas is 22g as claimed. Now to get a better understanding of line bundles we consider their alternative description in terms of divisor classes.
1488 6.2,25 Proposition. The line bundles are in one-to-one correspondence with the divisor classes. Actually they form a group isomorphic to the divisor class group. Proof. Let 11 be a line bundle and let 0 € L be one of its meromorphic sections. a has zeros {P.} and poles {Q.} and hence it 1 1 defines a divisor (VI. 2. 406)
where ni(roi ) is the order of the zero (pole) of a at Pi(Qi)' Any other section 0' e 1t is obtained from the section {J via multiplication by a meromorphic function 0' "
f e Mer (I: ) g
fo
(VI. 2.407)
so that div(a')
~
(VI. 2,408)
div(cr)
the two divisors differing by a principal divisor. Hence a line-bundle singles out a divisor class. Conversely let (D) be a divisor class and choose a representative divisor. In every local chart Ua we can find a meromorphic function fa whose divisor (fa) coincides with the U -restriction of D: ex
(VI. 2. 409)
In the intersection regions UC! functions with the same divisor
n
US' we have
t~o
meramorphic
(VI. 2.410)
Then the function (VI. 2.411)
1489 is holOlllOrphic and nowhere vanishing in Ua. I'l US' It defines a l-cocycle in the sheaf 0" and hence a line bundle. (The cocycle condition is obvious from the definition (VI.2.41l»). The product operation on linebundles is given by the product of sections; hence it is mapped into the product of divisors and we have the claimed isomorphism. In view of this isomorphism each line-bundle is characterized by a numerical invariant, the degree of the corresponding divisor class. Recalling Eq. (VI.2.302) we see that the degree of the canonical line bundle is 2g - 2. deg K = deg Z
= 2g - 2
(VI. 2.412)
Since the spin bundles are square-roots of the canonical bundle it follows that the square of their divisors is in Z.
Hence their degree
is g -1: deg S
II
g -1 •
(VI. 2.413)
The relation between spin-bundles and theta-functions can be seen at the level of divisors and of their image in a special flat variety associated to the Riemann surface. which is now time to introduce: the Jacobian variety. i" Given a Riemann surface Ig and its period matrix Il J Sq. (Vl.2.208». let us consider the complex linear space
!
E
t8
(see
c8.
A point
is a set of g complex numbers
+
z
= (2; 1••••• zg)
.
In Cg we can construct the following lattice L(Lg)
(VI. 2.414) <:.
Cg which depends
on the choice of the Riemann surface through its period matrix: (VI.2.41S)
1494
Using only the definitions (see Ref. [7}) we obtain
a[~r (DIn')
(VI.2.429~431).
= e:(A)e- i1T
after some manipulations
en](O[II)
(VI. 2.433)
where (VI. 2. 434a)
4>(a, b)
t +j = 4'1!-7a' (At C) 'a...... + b' (B D)'b
-
1[-; t ' " ... t + t t 2a'(C B)·b+(a·A -b·B )'(DC)
- -
4
1
(VI. 2 .434b)
d
(M)d denoting the vector whose components are the diagonal entries of the matrix M and e:(A) being an eighth-root of unity depending on the symplectic modular matrix A= (;
(VI. 2. 435)
:)
but not on the characteristic
[~ ] .
....
[i
If we could relate the 22g theta characteristics 1 to the 2g 2 _spin bundles, then Eq. (VI.2.434) would tell us how the spin bundles are mapped one into the other by the action of the mapping class group on the homology basis. The desired relation between the theta characteristics and the spin structures is based on a deep theorem due to Riemann, called the Riemann vanishing theorem.
6.2.28 Theorem. Consider a Riemann surface Eg, a base point and a point t e Jac(E ). Define the function
PO~g
g
(V1.2.436)
1495 where
...
~P
(P) is the Jacobian image of P (see Eg. (Vl.2.419)) and TI
o
is the period matrix of
~.
g
Then either f( p )(P) vanishes identiz, 0
.'
cally, or fCz,po) has exactly g-zeros PI,P2"" ,Pg• FUrthermore, the zeros {Pi} fulfil the following condition (VI. 2.437) ~
where np e Jac(E) depends on the base point Po but not on
o
~p
o
g
...
z.
is called the vector of the Riemann constants.
The proof is as usual omitted and the reader is referred to the mathematical literature [s]. Let us, rather, see how the Riemann vanishing theorem can be utilized in the spin bundle problem. We name the theta-divisor and denote by 0 c Jac(£) the set of g points of the Jacobian variety where the a-function without characteristic vanishes
...e e e ~ SCe+1 TI) For each -; e
e,
(VI. 2.438)
'" 0
the Riemann vanishing theorem implies that we can find
.
...
g-1 points Pl .... 'P g- 1 such that, chooslng l"'e, we have ...
e
+
1 P.
L Jl...~ =...~
g-
(VI.2.439)
i=1 Po
Indeed by hypothesis,
f(e,poJ '" a (;
+
J: o ~11I) = O
6(;11T)
=0
(VI. 2.440)
so that one of the g zeros is given by the base point Po itself. Eq. (VI.2.439) on the other hand can be re-interpreted in the following way.
1496
...e e e
belonging to the theta divisor, we can always find a degree g-1 divisor
Gi yen a point
(VI. 2. 441a)
...
deg DCe,P o)
" g-1
(VI.2.441b)
such that the following degree zero divisor
(VI. 2.442a)
_ .... (VI. 2. 442b )
deg D(e,PO) " 0
...
...
has Ap - e as Jacobian image
o
g-1
=L
(VI.2.443)
i=1
In view of the Abel theorem we conclude that the divisor D(e,P O)
is ..... e in one-to-one
defined up to the principal divisors, so that 15 correspondence with a divisor class of degree (g-I). This already suggests the relation with spin bundles.
To see this relation more explicitly let us consider the RiemannRoch theorem in the case of a divisor class (Oa) whose degree is g-l. Recalling Eqs. (VI.2.288) and (VI.2.292) we have deg 0a
Let write
~a
=g - 1
.. r(Oa-1 )
= r(DaZ-1 ).
(VI,2.444)
be the line bundle whose divisor class is Da' Then we can
1497 dim
HO(Eg'~a) = r(D~l)
(VI.2.44Sa)
dim HO(Eg.K(O ~~1) .. r(DcF 1)
(VI.2.44Sb)
where HO (Eg'~) denotes the vector space of holoJllOl'phic sections in the bundle ~. eqUation (VI.2.444) states that whenever ~ has a holoCL morphic section then so does K(O ~-1. ~
Let ~CL have a holomorphic section. In this case a representative of the [oJ class of the form
~~
can choose
(VI. 2. 446) Since K® ~~1 also has a holoJllOrphic section, we can choose a representati ve of the ZO~l class of the form ZD- l
a
= Q1
(VL 2 .447)
where Q1 ... Q 1 is a suitable set of g-I points of t. Correspond~ g ingly we have a representative of the canonical class given by (VI. 2. 448)
Given this observation, let Sa be the divisor associated to a spin boodle (deg (S ) .. g-I (S )2" Z) and consider the following set of a a points in the Jacobian variety: (VI. 2. 449a) (VI.2 . 449b)
{Pi} being any (g-I)-tuplet of points. It is easy to show that the set A the origin.
Let
ii eA.a
a
is symmetric with respect to
This means that ~ is the Jacobian map of
1498 -1
PIP2 '"
Pg- 1 Sa
where Pl'"
Pg- 1 is some (g-l)-tuple of points.
Equation (VI.2.448) guarantees that there exists another dual (g-l)-tuple of points Ql.···'Qg-l such that Pl'" is the canonical class.
Pg- 1Ql ... Qg-l
Consider the element ; e Jac(L) defined below: g
(V!. 2 .450)
Clearly it fits the definition (VI.2.449) and hence it is a point of A(%. On
the other hand. since
(Sa)2 =Z we have
(Pl'" Pg- 1) (8 a)-1 (Ql ... Qg-l) (5(%)-1; Z (Sa)-2
=1 (VI. 2.451)
and hence ;
=-~.
We have shown that (VI. 2.452)
The elements form
t
e Aa can also be written as a set of elements of the
(VI. 2. 453) Then in view of Eqs. (VI.2.443). as the (g-l)-tuple of points vary we generate all the theta-divisor and we Can write A
€X
= -0
..,. a
(VI. 2.454)
+ E
+
where Ea is given by +
E
a
+
= D.p0
I-g
~
-!J (8 PO)
a
(VI. 2. 455)
Since e(!I~) is an even function. the theta divisor is symmetric with respect to the origin:
1499 0= - 0 •
(VI. 2. 456)
Also Aa was shown to be symmetT
a
(VI.2.457)
CI.
This can be reconciled with Eq. (VI.2.454) only if (VI. 2. 458)
where L(k) is the lattice (VI.2.415) defining the Jacobian variety. g Hence the spin bundles are in one-to-one correspondence with the points of order two of the Jacobian variety. This provides the connection with the theta characteristics previously introduced. Indeed Eq. (Vr.2.458) is solved by setting ...
E
CI.
= -1'" b 2
+
1'" 2
-1!a
(VI.2.459)
... ...
where a. b are vectors whose components take
v~ues
in 12'
Therefore, given a theta-characteristic [~1 the divisor class of the corresponding spin bundle is the solution of the following equation: 1'"
1na " '..' 't::.'
- b +2 2
Po
... -.p (S
[:j
l-g )
p 0
(VI.2.460)
This concludes our discussion of spin structures and the long mathematical chapter. We are now ready to turn to Physics.
1500
References quoted in Chapter VI. 2
[1)
B. Riemann, Uber die Hypothesen, welche der Geometrie zu Grunde liegen, Abhand K. des Ges. Wiss. Gottingen 13, 133 (1868).
[2]
Arbarello, Lectures given at the C.I.M.E. course on "Global geometry in mathematical physics", July 1988, Montecatini, ed. M. Francaviglia (Springer Verlag).
[3]
J. Birman, Links, braids and the mapping class group, Princeton University Press, 1974.
[4}
J.P. Serre, A course in arithmetic, Graduate texts in mathematics, Springer.
[sJ
H.M. Farkas and I. Kra, Riemann surfaces, Graduate texts in mathematics, Springer (1980).
[6]
L. Alvarez Gaume and P. Nelson, Riemann surfaces and String Theories, in: Supersymmetry, Supergravity, Superstrings '86, Proceedings, eds. B. de Wit, P. Fayet, M.T. Grisaru - Singapore, World Scientific (1986).
[7]
J. Fay, "Theta Functions on Riemann Surfaces", Lecture Notes in Mathematics No. 352, Springer Verlag (1973).
18J
P. di Vecchia, F. Pezzella, M. Frau, R. Hornfeck, A. Lerda, S. Sciuto, Nucl. Phys. B322 (1989) 317. F. Pezzella, Phys. Lett. B~ (1989) 544.
1501 OIAPTER VI. 3
The Classical Action of the Heterotic Superstrings and their Canonical Quantization
Pill fe con/orme, e pero plu Ie place
Dante, Paradiso VII, 73
VI.3.1 Introduction The intuitive idea of a closed string moving in a target manifold ~lt
arge t
is realized by assigning an embedding function: (VI. 3.1)
where dim Mt
{xu} t)
are the coordinates of a point P e Mt arge t (U=1,2, ... , and {O','r} ={~a} are the coordinates labeling a point
arge p 6 W5, WS denoting the moves:
world~sheet
sweeped by the closed string as it
(VI. 3. 2)
1502 In Eq. (VI.3.l) 1 denotes the proper-time in the rest frame of the string's center of mass while 0 labels the different points on the string. Since the string is closed (in this book we consider only closedstring theories), X~(a,l) fulfills a periodic boundary conditions in a: (VI. 3. 3)
Equation (VI.3.3) states that when we move along the string (i.e. we increase the coordinate a), after a certain length we come back to the original point. As mentioned in the previous chapter (see Eqs. (VI.2.8») the relation between the world-sheet geometry and the analytic geometry of Riemann surfaces is obtained by performing a Wick rotation (1 .... i I) and then introducing the complex coordinates
= expel + ia)
(VI. 3. 4a)
z "" expel - ia)
(VI. 3. 4b)
z
In this way the coordinates XIJ of the target manifold become functions of z and i (VI.3.S)
and one can also introduce the momentum fields defined by (VI. 3.6a) (Vl. 3. 6b)
For a general embedding function pIJ and piJ have no special property. but if X~(z,i) fulfils the Laplacian equation (VI. 3.7)
1503
then p~ and p~ are respectively analytic and antianalytic: (VI. 3. Sa)
pU
= pll(i)
•
(VI.3.8b)
Furthermore Eq. (VI.3.3) implies that pll(z) and pll(z) are monodromic: p\.l (Zl1Ti)
'"
p].l ez)
(I V .3. 9) a
Referring to Chapter VI.2 and in particular to Section VI.2.S, we see that the momentum fields are cross-sections of a suitable line bundle defined on the Riemann surface. To be more specific, the momentum field is a tangent vector to the target space 50 that it defines a bundle: (VI. 3. 10)
having the Riemann surface as base-space and T(Mt arge t) as fiber. Clearly the transition functions of the bundle are SO(dim Mtarget) matrices, and the general form of equations (VI.3.9) is (VI. 3. 11 a) (VI. 3.llb)
A and
A being
suitable orthogonal or pseudo-orthogonal matrices.
Equations (VI.3.l1) correspond to a more general set of boundary conditions on the embedding functions than those given in Eq. (VI.3.3). This latter is the right definition of a closed string only if Mtarget is simply connected, namely if its fundamental group 1Tl(Mtarget) is trivial. In the general case we can write
1504
M
target
~arget =- - -) (
(VI. 3.12)
\arget
'1f t
where Mtarget is the universal covering space. Let now X be a continuous map from ~ g to M t arge t' t g being the universal covering space of the Riemann surface: X'•
'<'
~g
....
(VI. 3. 13)
"
Mtarget'
Let us see under which conditions X defines a map from I:
g
to
Mtarget' COnsider two points
PI
and P2 of 1:g which are equivalent
under an element be '!fI(I: g): (VI.3.l4 )
Since they are equivalent, they must be mapped by X into equivalent points of ~1target:
(VI. 3.1S)
where B(b) e 1!l(Mtal'get)' Hence B(b)
defines a homomorphism (VI. 3.16)
and for each choice of the homomorphism (VI. 3.16) we have a different set of boundary conditions (VI.3.IS) on the embedding functions. The homomorphism B has an image in the tangent space since the canonical tangent boodle induces a hOlllOmorphism, (Vl.3.17)
1505
and this defines the A and
it
matrices mentioned above.
Let us now come back to the observation that P and P are j.! 11 respectively analytic and antianalytic if Sq. (VI.3.7) is fulfilled. The reason why this is relevant is the following. As shown in more detail in later chapters, in particular in Chapter VI.S, the analytic fields with support on open subspaces U c r of a Riemann a g surface correspond to quantum field theories. called "conformal theories" that are often completely sol vahle. The analytic structure is indeed tied up with the algebraic structure of the Virasoro and Kac-Moody algebras (see Chapter VI.S) and this often allows for a determination of the n-point Green functions by means of algebraic techniques. utilizing the representation theory of infinite-dimensional Lie Algebras. The Green functions of the 2-dimensional confonnal field theory are then utilized to construct the string amplitudes. The latter are the scattering amplitudes of the target space particles. which are in oneto-one correspondence with the states in the Hilbert space of the 2dimensional quantum field theory. To appreciate such a statement. the reader should just recall that the basic idea underlying string theories is the following. All the observed particles are nothing else but the vibrational modes of the quantum string. In particular the light particles, including the graviton, the photon, the gluons. the wt, the ZOo the quarks. the leptons and eventually the Higgses are the zero modes of the string spectrum. In view of these remarks the current line of thought is the following: «every consistent 2-dimensional conformal field theory corresponds to a possible string vacuum and it is a suitable starting point fOT the string perturbation theory. » We shall be a little bit more particular in this respect, since we want to stick to the geometrical picture of the closed string as a
one-dimensional loop moving in a smooth target manifold. Hence we shall regard as possible string vacua only those consistent conformal theories which are generated by embedding functions (VI.3.5) from a world-sheet ws to a target space
1506 X: \'IS .... Mtarget
(VI.3.l8)
Mtarget can be a smooth manifold or a singular variety. The general idea having been introduced. there are still many important details to be clarified in order to make it perspicuous. First of all, we mentioned consistent conformal theories but we did not specify what we meant by this. Let us do it now. The general idea having been introduced, there are still many important details to be dari fied in order to make it perspicuous. First of all, we mentioned consistent conformal theories but we did not specify what "'8 meant by this. Let us do it now. From the 2-dimensional point of view. the embedding functions (VI.3.5) must be regarded as scalar fields: these scalar fields are coupled to the 2-dimensional gravitational field (the metric g~(~)) in such a way that the classical action is invariant both ~ith respect to diffeomorphisms (VI. 3.19)
and to Weyl transformations defined by Eq. (VI.2.24). The typical 2dimensional action was mentioned in Eq. (VI.2.28) where the collective name {fi(~)} denoted both the embedding functions X~(~) and, in the case of superstrings. a convenient set of left-handed and right-handed 2D-fermions 1jh~). Relying on the lore discussed in Chapter VI.2 and utilizing the classical invariances of the action (Vr.2.28) we can always choose a gauge where, in every coordinate patch U. the 2D-metric is given by a Eq. (VI.2.37), i.e., g _
zz
#
exp[2~(z.z)1
(VI. 3.20)
(as in the last section of Chapter VI. 2, we redefined • + 2+ for later convenience) •
1507
6.3.1 Definition. We state that the action (VI.2.28) describes a classical conformal field theory if in the gauge (VI.3.20) the field equations of the 2-dimensional fields ~i(~), generally given by
oS (g,'I') ---=0 o'1i (~)
(VI.3.21)
can be cast into the form:
az -i J (Z,i) ,
= 0
(VI. 3. 22a)
(VI. 3. 22b) where
-i -1 J (z,z) = J (
(VI.3.23b)
are suitable local functionals of the original fields
on the classical shell the fields
J
i
-i and J
{
analytic and antianalytic fields: (VI. 3. 24a) (VI. 3. 24b) and can be interpreted as cross-sections of suitable line bundles on E (see Section VI.2.8). g
In addition to Eg. (VI.3.19), the action (VI.2.28) leads to the variational equation: (VI. 3. 25)
1508
where TaS(~) is the classical 2-dimensional stress-energy tensor. In a 2-dimensional world the Einstein equations reduce to the statement that TaS (;) should vanish. This corresponds to the observation made in Section (VI.2.1) that there is no nontrivial action for the gravitational field. In the gauge (VI.3.20) the classical Weyl invariance of the action guarantees that the only non-identically vanishing components of TaS are
T '" r(Class)
(VI,3.26a)
T-- .. :r(class) zz
(VI,3.26b)
zz
zz zz being identically zero. Furthermore, the invariance with respect to local diffeomorphisms yields the usual conservation law:
T - =T-
(VI. 3. 27)
which. in the conformal gauge (VI.3.20). becomes (VI. 3. 28a) (VI.3.28b)
Hence, the two independent components of the energy-momentum tenSOr are respectively an analytic and an antianalytic field on Lg• From a canonical point of view. the Einstein equations T(class)(z) ::: 0
(VI. 3.29a)
f(class)(i) ::: 0
(VI. 3. 29b)
are a set of first-class constraints fulfilling the following algebra:
1509
This is nothing else but the classical Virasoro algebra discussed in Eqs. (VI.2.19 - VI.2.23). Indeed, if we define (VI. 3. 31)
we see that Eq. (VI.3.30) yields
Ie -) 1Lmclass, Lclass n r(Dirac)
= (m _n) L (class) n+m
(VI. 3. 32)
which is the canonical realization of the commutation relations (VI. 2. 23). The result is not surprising since the energy-momentum tensor is the canonical generator of the infinitesimal diffeomorphisms. Indeed, for each field ~i(z.z) appearing in the lagrangian (VI.Z.28) we have {T(z).I,hw,w)}H = __ 8_ Z ~i(w,w) Dirac (z -w)
+
_1_ 3\./Cw,w) (VI.3.33a) z -w
where {8,~} are two numbers characteristic of tiez,i) called the conformal weights of the corresponding field. As long as the action (VI.2.28) is Weyl and reparametrization invariant, all the equations from (VI.3.26) to (VI.3.3S) hold true at the classical level, irrespectively of whether or not the classical equations of motion can be cast into the form (Vl.3.22). The problems arise at the quantum level. If the action (VI.2.28) does not define a classical conformal theory, then, for a general target manifold Mtarget' the algebra (VI.3.30) is not respected after quantization; indeed in the short distance expansion of the operator product T(z)T(w) we find terms which are not of the form (VI.3.30) and sometimes not even analytic. Their coefficients are complicated functionals of the metric &Uv(X) and the other geometrical data on Mtarget' These functionals can be calculated only perturbatively deforming a space for
1510
which the classical Virasoro algebra (VI.3.30) is respected at the quantum level. Such starting point spaces can be found among those which yield a classical conformal theory. Indeed, although the classical Virasoro algebra is in general violated also in these cases, when the theory is conformal the violation has, nonetheless, a particularly nice character, since it simply corresponds to a central extension of the algebra (VI.3.32). In fact, after quantization Eq. (VI.3.32) gets replaced by: [Lm,t.n]
=
(m
*n)L~n ,.. ~
+ 1c2 (m3 -m)1SJUT~n ,0 + 2bJUom+n,o
(VI.3.34)
where c and b are two real numbers, respectively named the central charge and the Virasoro coboundary characte'cistic of the type of conformal field considered. They are positive fOr physical fields but negative for the ghost fields we are forced to introduce when we gaugefix the classical Weyl and reparametrization invariances. as well as, in the case of superstrings, the local supersymmetry invariance (see Chapter VI. 4) • Therefore, we can find classical conformal theories which maintain the classical Virasoro algebra also at the quantum level by choosing the field content in such a way that all the ci and bit corresponding to the various fields in the theory. sum up to zero. These quantum conformal theories, given by well-defined choices of the target space, are available string vacua. One should note that 2dimensional interacting field theories that are not classically conformal, according to the definition (VI.3.22) may. nonetheless. lead to quantum conformal theories. This happens through a non-perturbative mechanism. Indeed the short distance operator product expansion of the exact quantum stress energy tensor T(~)T(w) can still fulfill the Virasoro algebra (VI.3.34) (with an appropriate central extension c) if all the perturbative violations of the algebra (S-functions) sum up to zero. An example of this phenomenon seems to be the supersymmetric 0model on the Calabi-Yau 3-folds. Although such a theory is not classi· cally conformal. it appears that its exact quantum version is a super-
1511
conformal field theory. It remains to be seen if in these cases the theory cannot be reformulated as the canonical quantization of a model that is conformal already at the classical level. We make the choice of considering only those string vacua that can be obtained as canonical quantization of classical conformal theories. Clearly they can be classified in t~~ steps. First, one can list the classical conformal theories. Secondly, we can select among these the ones which, upon quantization. yield a vanishing central charge c and a vanishing coboundary b. In the present chapter we deal with step one. Actually. since we are heading for superstrings, rather than bosonic strings, we shall construct classical conformal theories possessing local world-sheet supersymmetry in addition to reparametrization invariance. as well as, a fermionic analogue of Weyl invariance. Furthermore, since we are interested in heterotic superstrings, we will choose an N = 1 world-sheet supersymmetry generated by a Majorana-WeyI d:: 2 supercharge. The construction of such classical theories requires a preliminary study of N,,1, 0:: 2 conformal supergravi ty •
VI. 3.2 N:: 1. 0" 2 conformal supergravity and the heterotic superspace
geolJletry stated above the structure tmdel"lying heterotic superstrings is that of N = 1 conformal supergravity in d" 2. By N ,,1 we mean that the gravitino is a Majorana-Weyl spinor. As
The proper formalism to describe superconformal symmetry in d" 2 is fully analogous to the fonnalism [1] utilized in d" 4. We rush through the necessarY steps beginning with Our conventions. The flat metric and the Levi-Civita symbol are given by: nab" nab:: (:
_°1 )
(VI.3.3Sa)
1512
Eab ::: Eab
(0
1)
(VI.3.35b)
0
~l
where a,b =0,1. The gamma matrices are chosen as follows
l.(: -:) " . (: :)
(VI. 3. 36a)
(VI. 3. 36b)
The charge conjugation matrix is defined by (VI. 3. 37)
with (VI. 3.38)
Finally, we have ab
Y
=
E
ab
Ys
(VI. 3. 39)
where
Y3'" (1o 0)
(VI. 3. 40)
~l
It is also convenient to introduce the notation + 1 0 1 0+ Y- '" - (y .. y ) "" y Y2
A Majorana~Weyl spinor
~
= -21
(1 +Y~) ~
satisfies the conditions
(VI.3.41)
1513 C ~T
= I{I
(VI. 3.42)
which are uniquely solved by setting
'!'
= ew!f( 1;) o
1;*
= I;
(VI. 3.43)
•
Similarly a MajoranawantiWeyl spinoI' • satisfies the conditions (VI. 3.44)
which imply
=e
c)
w¥(O)
x'"
(VI. 3.45)
=X •
X
Utilizing these notations we can immediately write down the curva w tures of the Zwdimensional superconfomal group. (VI. 3.46a) k
a
=DKa
a
i 2
- 1'1 .. V - - • " y
p
= D'¥ + 12 W,.
t:J
=at - 1Z w" ••
a
c)
'¥ - iya. ,. V
a
iy&.v " Ka
(V1.3.46b) (VI. 3.46c) (VI.3.46d) (VI. 3. 46e) (VI. 3. 46 f)
where va, 1'1, '1', t, Ka, w are all I·forms and are the gauge fields associated to the translations, the dilatations, the Q-supers)I1IIIII8tries, the conformal boosts, the S-supersymmetries and the Lorentz rotations respectively. Furthermore, va, 1'1, Ka , wah are bosonic, while '!' and c) are fermionic, and obey conditions (VI.3.43) and (VI.3.44) respectively.
1514 It is convenient to introduce a different basis by defining (VI. 3. 47a) (VI.3.47b) (VI. 3. 47c)
rl"
W:!:
(VI.3.47d)
III
Then with obvious notations Eqs. (VI.3.46) become: (VI.3.48a) (VI.3.48b)
(VI.3.48c) -
L =dk
-
+ -00 ~k
i --X"X
(VI.3.48d)
2
(VI.3.48e)
c"
a+ "
1 + dX - -00
2
dw+ + 2i~
A
X - 2~
A
k
-
(VI.3.48f)
"X
(VI. 3.48g) (VI.3.48h)
Equations (VI.3.4S) encode our description of the N" 1 supercon-
formal algebra. The associated superconfol'lllsi transformations should be non-
linearly realized in heterotic superspace. 01-
with two bosonic coordinates z" x Weyl fermionic coordinate e.
+x
By this 0
1
, z" x - x
~~
mean a superspace
and a single Majorana
1515
Its geometry is described by a supervielbein (e+e-~) and an SO(I,I) connection w.
The set of l·forms (e+e-~) provides a basis for the cotangent space. This means that every p-form can be expanded along these fundamental one-forms. For example: (VI. 3. 49a)
n(2) ~ n;:)e+
A
e- + n;:)e+
A
~
+
n~~)e-
A
,
+ n~:),
A
,
•
(VI.3.49b)
Gi ven (e +e-1,;) we can write down the torsion and the curvature 2-fol'llls (VI.3.S0a) (VI. 3. SOb)
(VI.3.S0c) R" dw.
(VI. 3.50d)
The problem of the nonlinear realization of supe:rc:onfol'lllai S)'lIIlIIetry can now be formulated through the following requirements: we want to identifY the supervielbein multiplet (e+e-,) with the corresponding I-forms sitting in the superconfol1llal algebra i}
(VI. 3.48); +
..
the remaining I-forms w·, X, k- appearing in (VI. 3.48) must be viewed as defined over the heterotic superspace and should be expanded according to Eq. (VI.3.49a); furthermore their components should be expressed in terms of the superspace geometry, Le. in terms of components of the curvature and torsion (VI. 3. SO) ; ii)
iii) the resulting parametrizations of both the superspace torsion and curvature (VI. 3. SO) and of the supereonformal curvatures (VI. 3. 48) must be consistent with the corresponding Bianchi identities.
1516 We fulfill the above requirements by writing, on the one band, the
following "rheonomic" parametrization (VI.3.51a) (VI.3.51b) (VI.3.S1c) (VI. 3.S1d)
where the torsion superfield t(z,z,6) superfield !I1(z,i,e) via 211.
= ~ ·t,
is related to the curvature and by setting, on the other
hand, +
III
=
-Ill
= fIJ
1
X = "2 Te
-
•
(VI.3.S2)
The corresponding rheonomic parametrization of the superconformal curvatures read +
r =0
(VI.3.53a) (VI.3.S3b)
p = 0
1
o = -2
(VI.3.S3c) ~ te +
+
1 "e - -4 til; " e
(VI. 3. 53d)
(VI. 3. S3e)
(VI. 3. 53f)
Finally utilizing these parametrization, from the Lie derivative rule we obtain the transformation of the supervielbein under superconformal transformations:
1517 (VI. 3.54a)
{VI. 3. 54b)
(VI. 3. 54c)
",ilere ~+ :: parameter of left translations ~w
€
..
parameter of right translations
=parameter of Q supersymmetry
(anti commuting)
A: parameter of Lorentz'rotations • :: parameter of Weyl transformations ~
= parameter of S supersymmetry
(anticommuting).
These transformations preserve the constraints (VI.3.S1) and allow the choice of a superconformal gauge where (VI.3.SSa) (VI.3.SSb) (VI.3.SSc)
where .(z .!) is an arbitrary real functions and n(z) is an analytic anticommuting spinor. In this gauge we find (VIo3.56a)
1518
(VI. 3.56b) In particular, one can choose $=0 and n .. O which correspond to a flat and torsionless superspace, i.e. to the special superconformal gauge where (VI. 3. 57a) (VI.3.S7b) 1; '"
de .
(VI. 3.57c)
Equations (VI.3.55) are the supersymmetric analogues of Eqs. (VI.2.382). As seen above, the right-vielbein e- is identical in the bosonic and in the supersymmetric case. It is only the left vielbein e~ which feels the presence of the fermionic coordinate e (compare Eq. (VI.3.SSa) with Eq. (VI.2.382a)). This is also evident from the transformation rules (Vr.3.S4) where we see that e is unaffected by a Q-supersymmetry transformation of parameter E. This structure of the heterotic superspace implies that the worldsheet supercharge acts only on the analytic fields (left-movers), the antianalytic ones (right-movers) being supersymetry singlets. Finally observe that Eq. (VI.3.S6a) is the supersymmetric analogue of Eq. (VI. 2.47). Recalling the redefinition (4) .... 24» of the conformal factor, the reader can check that the first component of the curvature superfield is the ordinary curvature (VI.2.47) of the Riemann surface.
VI.3.3 Classical superconformal theories and the WZW-action In the introduction to the present chapter (Section VI.3.l) we defined the classical conformal field theories. The generalization of this definition to the superconformal case is straightforward.
1519 6.3.2 Definition.
A classical superconformal theory is described by
an action
s = II d2~
det
e(~)~(e!(;)'~n(~),~i(~»)
(VI.3.58)
g
where e!(;) is the vielbein,
~n(~) is the iravitino and .,i m is
a convenient set of mattel' fields fitting into supermultiplets. The matter fields are coupled to the 5upergl'avitational ones in such a way that all the local transformati ons (VI. 3. 54) are symmet ries of the action (VI.3.S8). Furthe:rmore, in the superconformal gauge given by the a = 0 slice of £9s. (VI.3.SS)
(VI.3.S9a) e-
= e~(zl!)d-z
(VI.3.S9b) (VI.3.59c)
(oS/o"i .. 0) can all be cast into the form (VI.3.22) USing suitable local functionals i(op,aop).
the e9uations of motion of the matter fields
In the present section lo.'e construct an example of such a superconformal theory. utilizing as matter fields opi (~) the components of a superfield g(z,i,e) which describes the injection (VI. 3.60)
g: SNS + G
of the superworld-sheet into a siqlle group manifold G. is called the Wesl-Zumino-Witten model
Such a theory
(WZN).
By letting the G structure constants {fABCl go to zero, we obtain from it another classical superconfo1'lllal theory whose matter fields describe the injection of the superworld-sheet SWS into a flat manifold with either the topology of Rd or that of ad-torus Td.
1520 In the next section we will consider the general case of an action (VI.3.S8) ~here the matter supermultiplet {~i(~)} describes an injec-
tion «1:
SWS
+
Mt arget
(VI. 3.61)
of the superworld-sheet into an arbitrary target manifold Mtarget We shall conclude that such an action corresponds to a classical superconformal theory if and only if Mt arge t has the following structure M
target
,. M ® M d compact
(VI.3.62)
where Md is the d-dimensional Minkowski space, and Mcompact" Gr/B is a variety whose covering space is a group-manifold Gr , The homotopy group B is a discrete subgroup B·c Gr ® GT, Finally the structure of GT (called the target group) is (VI.S.63)
G.1 being Simple groups or U(l) factors. At the quantum level the possible choices of restricted by requiring
Gr
will be
conformal anomaly cancellation (the total Virasoro central charge should be zero, when the ghost contributions are included), i)
ii)
existence of target space massless fermions.
These two conditions will be analysed in Chapter VI.S. In any case, the results of the next section will imply that the classical action describing the propagation of a heterotic superstring in a vacuum state can be cast into the fOllowing form (d)
S = S(Mink)
n
+
.r
1,,1
(G. )
S(k~) 1
+
S(het)
(VI. S.64)
1521
00
p
where SCMink) contains, as dynamical fields, the X -coordinates and
(G.)
their superpartner ~~ while S(k~) contains the WZW elementary 1
field g e Gi together with its superpartner AA. which is a Gi-Lie algebra valued tlro-dimensional fermion. Finally S(het) contains a nUlllber Nhet (fixed by the conformal anomaly cancellation) of heterotic fermions which have opposite handedness with respect to ~~ and AA and which do not couple to the 2-dimensional gravitino. This latter, (d)
(G i )
appearing both in S(Mlnk) and in all the SCk i )' defines, through its own variation, the appropriate supercurrent of the model. The supercurrent as we are going to see is the superpartner of the energy-momentum tensor T(z) and leads to the superextension of the Virasoro algebra (VI.3.34). The integer number k.1 associated to each group G.1 is the normalization constant in front of a typical cubic teI'ID appearing in the WZW-action, called the Wess-Zumino term. upon quantization k becomes the value of the central charge in the Kac-Moody algebra generated by the currents ~(z) appearing in the form (Vl.3.22) of the field equations. As announced in the present section, by utilizing the rneonomy approach we construct the locally supersymmetric action of the wess-Zumino-Witten model associated to a simple group G; later by letting fASC"O we retrieve the action associated to an abelian group G. If G is the d-dimensional translation group, we get
s~!Ink)' while
if
G=U(1)p then our result corresponds to the non-
semisimple part of Gr. In all cases, whether Mcompatt is GT Or Gr'B is irrelevant at the level of the action functional since the division modulo B corresponds to imposing a convenient set of boundary conditions on the superfield g(z,z,6). This is a step to be taken at a later stage. Hence from the construction of the present section 1>'6 will be able to retrieve the whole classical action (VI.3.64) of a heterotic superstring. Let us turn to such a construction. We start from the results derived in the previous section and we write the definition and the parametrization of the curvature and torsion of the heterotic superspace:
1522 +
T
def + + i .. de +w"e =r~
.. t
(VI. 3. 65a) (VI.3.65b)
• def
T = d,
+
1 rW""
+
= reA e
def J2 + -. R ,. dw":7l e "e - 1t
,
"e
(VI.3.65c) -
(VI.3.65d)
The above equations encompass the constraints on superspace geomet 1')" imposed by the requi rement of superconfomal s)'lJll1letry.
The super-
field r(z,z.a) is the field strength of the two-dimensional gravitino and provides a complete description of our heterotic geometry (the curvature ~ equals twice the spinor derivative of T).
Given Eqs. (VI. 3.6S)
the intrinsic covariant derivatives !j+, ~_.~. satisfy the following algebra: (VI.3.66a) (V1.3.66b)
..
[~ ~J
.. iST
(VI.3.66c) (VI. 3. 66d)
where s is the spin of the field acted on by the derivatives for the scalars. s .. 1/2 for left-handed femions.
(s" 0
s;: -1/2 for the
right-handed fermions). On the background of the above described heterotic supergeometry we intoduce the WZW-model.
We consider the superfield g(z,i,S) e G
describing the map (Vl.3.60).
In terms of g we can construct the left-
invariant and right-invariant I-forms Sl ,. g-1 dg
(VI. 3. 67a)
1523 ..
Q = dgg
-1
(VI.3.67b)
which can be decomposed along a b~is t A of the Lie algebra G: (VI.3.68a) (VI. 3.68b)
rl
and
rI'
fulfill the Maurer-Cartan equations (VI. 3.69a) (VI. 3.69b)
where fABC aTe the structure constants of G defined by (VI. 3. 70)
Our first step is to decompose the l·forms oA and QA along the complete basis e+, e-.~. In full generality we can write
(V!. 3. 71a)
(V1.3.71b)
where XAz),A(z,z.B) is an anticommuting spinor superfield whose first component
A~B=O)
is the superpartner of the wz.W field g(SaO)'
Using the identity 51 '" {I Qg the O-forms
(Q~,~. ~A) are
easily retrieved from the corresponding untilded ones. trate hereafter
onl~
Hence we concen-
on Eqs. (VI.3.71).
In full generality the covariant differential of ')..A: (VI.3.n)
1524 can be decomposed along the complete basis e + J e - J
1;;
as follows: (VI. 3. 73)
FA;:!I).A is an auxiliary bosonic superfield. Inserting Eqs. (VI.3.71) and (VI.3.73) into (VI.3.69a) and using (VI.3.65-66) ft~ determine FA:
Eqs. (VI.3.69) give also the spinar derivatives of
!W• rf+
= - !W '}..A + tBC AB ~
Q~ and ~:
IF+
(VI. 3. 75a)
and the "Bianchi identity": (VI. 3. 76)
Equations (VI.3.71, 73, 75) provide a consistent (compatible with 2 d =OJ rneollOlDic parametrization of the I-forms QA and !W AA. From this parametrization ft'e obtain the transformation rules under supers}'lllllletry of the matter fields g and ')..A. The transformation rules of the supergravity fields e+, e-, l;; are fixed by Eqs. (VI.3.6S). First of ail, instead of 6g, we utilize the tangent variation
-1
A
6y ;: g 6g = 8y t A
(VI. 3. 77)
which is related to the variation of the I-form QA by (VI. 3. 78)
1525
Then we recall that a supers}'lllllletry variation is nothing but a diffeomorphism in superspace along a fermionic tangent vector e=e:a, that is. a tangent vector such that (VI. 3. 79a) (VI. 3. 79b) So we get: (VI. 3. 80)
where !£ is the Lie·derivative. Utilizing Eqs. (VI.3.71a) and (VI.3.69a) in (VI.3.80) we obtain (VI. 3. 81) which yields (VI. 3. 82)
Simi larly. from (VI. 3. 83)
and by use of Eqs. (VI.3.73) and (VI.3.74) we find (VI. 3.84)
where we: is the parameter of a field-dependent 2·dimensional Lorentz transformation and is given by (VI. 3. 85)
1526 Finally, computing the Lie derivatives R.e;e+. 1"e- and 1~:, we obtain also the 6e;e+. 6E: e - and 6E: ~ rules. SUDDDarizing. the world-sheet SUSY rules of all the fields are given by (VI.3.86a) (VI.3.86b) (VI. 3. 86c) (VI. 3. 86d) (VI. 3. 86e)
The field-dependent Lorentz transfonnation we;' being separately
a symmetry of the action. can also be dropped and in this way Eqs. (VI. 3. 86) reduce to the customary set of rules. The next problem is that of writing an action functional from which the parametrizations (VI.3.71a) and (VI.3.7,'S,74) should follow as fermionic (i.e. outer) projections of the variational equations. This will guarantee the invariance of the considered action against the infinitesimal supersymmetry transformations (VI. 3.86).
Before proceeding it is convenient
to stress that the 2-dimensional field theory under consideration is a special case of a locally supersYlllllletric non-linear a-model.
Hence it
is necessary to introduce a target space spin connection wAB ,. - wBA
(VI. 3. 87)
in addition to the target space metric, which in this case is the Killing metric: gAB
~
fARS fBRS •
(VI. 3. 88)
1527 This spin~connection appears in the covariant differential of the 2-dimensional fermion AA: (VI. 3.89)
Using the structure constants fABC we can introduce a oneparameter family of spin connections (VI. 3. 90)
Regarding
r"c
as the vielbein of the group-manifold G. the torsion
and curvature 2-forms associated to the a-connection are
AB "B _ TABe ....8 TA def ._A (a) .. ~r - .. w(a) ",. - (a).'
"C
A"
(VI. 3. 91a)
(VI. 3. 91b)
where TADe (a)
= - (a + '2I)
fABC
(VI. 3.92a)
(VI. 3. 92b)
There are three critical values of i) if)
iii)
11:
a = - !. 2
CL
=0
a = -1 •
The first corresponds to the choice of the metric connection for which the torsion is zero. The second and the third correspond to the choice of a parallelizing connection for which the curvature is zero.
As we
1528 sha.ll see, for each value of action functional.
there is a corresponding supersymmetric
(X
The case a .. 0,
however, will be special since it
involves an extension of the local symmetries: in addition to local supersymmetry (actually to local superconfonnal invariance) at «=0 We gain local chira! G ® G invariance (Kac-Moody symmetry).
It is
this invariance which leads to the form (VI.3.22) of the equations of 1IIOtion). The supersymrnetric action functional is the following:
~(a)
..
! (J~[(st-AA~) •
AA V(a}]"A
+ 21
_!
3
i
e
+
AA_A iI-
A
,
+
-
(!2 +a) fABC AAABAC l; .. e+ +'i+ QA- e+ e-] -
+ 1 (1 + 2et)
6"
+
A
(~e+ -~ e-)
A
A
JM t-.ABC_A -
If.. QB .. QCJ
(VI.3.93)
•
3
let us discuss its structure.
The dynamical variables are
The supergravity background fields
i)
e+, e-, l;
The WZW field g (contained in $l .. g-ldg)
ii)
and its super-
A
partner A iii)
The auxiliary O-forms
algebraic and plays a double role:
gA, ~t
QA whose field equation is
yields
n~"tr(g-la:tgtA)
and it
enforces the rheonomic parametrization. The last addend in (VI.3.93) is the Wess-Zumino term which is written as the integral over a 3-Simplex M3 (of ~~ich the ~~rld-sheet SW Z is the boundary) of the closed (but not exact) 3-form fABC rf "QB ,,$lC. This term might be a non-Single-valued functional of g.
Single valuedness
requires its coefficient to be an integer (in proper units).
This is
the origin of the quantization of the parameter k to be discussed in Chapter Vr.S.
1529 ~~
observe that both the Wess-Zumino term and the three-linear lambda-tem disappear when (1= -1/2, namely. when we choose the metric connection. Alternatively, when a .. O, VI];O =~ coincides with the world-sheet covariant derivative and we have the emergence of a new local symmetry which will be manifest in the superconformal gauge (Kac-Mbody symmetry). We postpone its discussion and we consider the field equations associated to (VI. 3. 93). The larger symmetry will manifest itself also in the
equations of motion where it leads to the existence of conserved chiral currents and hence to a classical superconfomal theory according to the definition 6.3.2. The oyA variation of (VI.3.93) yields
+
• Motl..NRS M. R S 41Q f fn-A A
A
e+
=0
•
(VI. 3.94a)
while the oAA.variation of the same fUnctional gives
+
8
(21 + a)
the one-form IT
A
f
ABCBC
A A I;;
+
A
e .. 0
(VI. 3. 94b)
rrA being defined by
_A + = 1,e +
A-
- Q e- •
(VI. 3. 95)
-
Equations (VI. 3.94) have the same st l'U<;t ute. They state that a 2-form, defined over heterotic supetspace, should vanish. Hence they split into four sectors corresponding to the four independent wedgeproducts:
1530 i) il)
e+ " e e+
" 1;
-
iii) e " iv)
,
I; " ,
which provide an intrinsic basis for the vector space of 2-forms. The coefficients of the structures ii), iii), iv) cancel automaticallY when the rheonomic parametrizations (VI.3.71a, 73, 74) are inserted into (VI. 3. 94). Actually the coefficients of the action were precisely fixed through the requirement that this shOUld happen (i.e. supersymmetry invariance). The cancellation of the e+ "e- coefficients instead is the same thing as the following differential equations:
(VI. 3. 96a) g) )"A +
-
a. rASe ABrF
-" 0 •
(VI. 3. 96b)
The meaning of the parameter a and the difference between the supersymmetric and bosonic WZW model are manifest in Eqs. (VI.3.96). In order to discuss it we choose the superconformal gauge and we consider the critical values of a (ep: 0, a" -1). If the ;..A were zero, namely, if we considered a bosonic WZW model both critical values would lead to a conserved chiral current. Indeed, in the bosonic case a." 0 implies ~Q =9(g-1~g)=O, I.;hile a=-l implies ~Q =~(g-lfjg)=O. -+
-
+
+-
+
-
In both cases, utilizing also the right-invariant one-form n, we arrive at the conclusion that there are a conserved left-moving and a conserved right-moving current: (VI.3.97a) (VL3.97b)
1531 Furthermore. in the bosonic case there are no fermions which couple to the spin connection (VI. 3. 90) and the two choices
cx .. 0 or a,,-1
simply conespond to flipping the sign of the Wess-Zwnino term.
The
Here, if we
situation is quite different in the supersymmetric case.
want a complete decoupling of the fermionic and bosonic degrees of freedom we must choose a .. O. fermion
In this case (and only in this case) the
AA is a free field obeying the equation (VI. 3. 98)
while
~
'"
becomes a conserved chiral current (VI. 3.99)
The sign of the Wess-Zumino term is no longer arbitrary, rather it
is fixed by our initial choice of the left·invariant one-fom flA as If
the ftmdamental object in terms of which we constructed our action. We had used
n rather than
fl we would have fomld the reversed result.
Relying on this link we decide to call left-moving all the fields which, like the left-invariant one-foTID
9l .. O. +
rf. +
satisfy 9." O. -
•
Right-
moving fields
i
will be zero.
Under this condition Eq. (VI.l.93) is the action of a
obey 'instead
classical superconfomal theory:
From now on the value of a
the super WZW-model.
VI.l.4 Heterotic a-model on a general target space and the choice of ...e.;. r; . COR;;. ;.; .; ;.fo;. . ;l'IlIl1. ; ; ; ;. ; ;.I. .:t;.;,.h_eo...!1"",' -Mt arget_-'l..:;e-"'ad.;;.;i...;:;n""g'-t.;..;o;......;.ca_c""l;.;;a_ss-"'i;....;c...;.a..;;,l-'s_up In this section we reconsider the results of the previous one from a more general stand-point.
As already pointed out. the action (VI.3.93)
of the Wess-ZUmino-Witten model is just an eXaJlq>le of a locally supersymmetric two-dimensional o-model.
Indeed. the bosonic kinetic part of
the Lagrangian (VI.l.93) .z,p-k'
ln
_A = ll"""
("A _A -) •• e+ - u·· e
'"
-
+
"A "A e+ .-~ e•••• +
-
(VI. 3. 100)
1532 upon transition to
second~order
formalism can be rewritten as (VI. 3.101)
where
xlJet;;)
are the coordinates of the group manifold G, ~v(X) is
the Ge G invariant metric on that manifold and ~B (1;) is the worldsheet metric. In view of this observation we are interested in the general form of a heterotic locally supersymmetric a-model where the bosonic kinetic term is given by (VI.3.101).
~v(x)
being an arbitrary metric on an
arbitrary target manifold "large M t' In this section we construct such a model and then we investigate for which choice of Mtarge t the corresponding field equations take the form (VI. 3.22). We conclude that this happens only in two cases. namely when M t has for universal d urge covering space either R or a semisimple group manifold Gr' The model on Gr is a superposition of WZW-models, one for each simple group Gi
contained in
0-
Gr·
The a-model on a general target manifold is not a classical superconformal theory but nonetheless it can describe a consistent quantum conformal theory if all its beta functions vanish. The beta f1.D1ctions are differential expressions in the deviation h g~(~)
G
from a group manifold one g IJV(~):
IJV
of the metric
(VI. 3.102) and also in the deviation of the torsion from the group manifold parallelizing torsion. Setting these beta functions to zero we obtain the perturbative expansion of the string field equations around the vacuum state corresponding to the group-manifold GT, The a-model described in this section will be used in Chapter VI.9 to provide the starting point for the analysis of the string propagation on background field configuration. Let us now der! ve the a-model we have been discussing.
1533 The signature of the target manifold Mtarget is chosen as follows (-. +. +, .... +)
(VI. 3. 103)
and its dimension D is given by D = rl + d "loUnk compact
(VI. 3.104)
where, referring to Eq. (VI.3.62), ~ink is the dimension of the Minkowski space and dcompac t is the number of compactified dimensions. i.e. the dimension of Mcompact' The geometry of Mt t is described by a vielbein Va, a spin arge connection wab =_wba • and a 2-form B defined up to gauge trans formations (VI. 3.10S)
A being
a I-form.
We define the torsion, the curvature and a 3-form field strength for B, named .ft: (VI. 3 .106a)
AlIab
;n
_=
dwab +wac
cb
"W
= Rabmn
.JD
V
.J!
"v
(VI. 3.106b) (VI. 3. I06c)
The coefficients Hab' c being the components of a 3-form, build up a tensor which is necessarily completely antisymmetric. r abc , on the other hand, is not a priori antisymmetric in all its indices: it is just antisymmetric in b ...... c. However, as we are going to see in a moment, the requirement of local supersymmetry invariance of the o-rnodel action leads to an identification
1534
(VI. 3. 107)
Tabc " - 3Habc •
Hence Tabc is also fully antisymmetrlc. Furthel'lllOre in Chapter VI.9 we will see that Eq. (VI.S.I07) is only the classical limit of a much more complicated quantum equation relating the torsion to H-field strength (see Table VI.9.V and Eq. (VI.9.238). In addition to va, wab and B, we introduce also a I-form A'*' on Mtarget with values in the Lie algebra SO(Nhet ). where \et is the number of heterotic fermions to be fixed through quantum considerations in Chapter VI.S. The I-fol1ll Ao£, = _ABa is a given by the following 2-form
gauge
field and has
a
field strength
(VI.3.108)
When the world-sheet is mapped into Mtarget by the assignment of an embedding function XP(z.z,a) then the vielbein y& can be decomposed along the basis e+,e,1; of the heterotic superspace (VI.S.I09) the components V~ can be identified with the 2-dimensional fermions l/Ja completing the embedding supermultiplet v~
"
(VI. 3. 110)
l/J& •
Inserting (VI.S.I09) and (VI.3.1IO) into (VI.3.106) we obtain
(VI.3.111a) (VI. 3.111b)
1535 (VI. 3.lllc) which are the generalizations to ail arbitrary manifold of Eqs. (VI. 3. 73) and (VI. 3. 75). In addition to the fermions
a W
we introdu~e the heterotic fer-
mions ~a (ex = 1, ••• ,Nhet ) and we define their covariant differential (VI.3.112)
From the Bianchi identity (VI.S.llS)
inserting the following rheonomic par3llletrization (VI.3.114)
we obtain the condition
(VI. 3. 115)
Given these ingzedients the action of the heterotic a-model is
s ..
2-IJ ~
~
(y8 ~1jIaf;)
;, (nae+ _nae·) + 2i
!pa~!p8
;, e+ +
+-
+
IPaVa ...
+
2H(l~~B
I; +
~ i rabc,l!pb"h ;, e+ ... n~ae+ ,.. eS
"e- + 4
+
FaBf;a~B1jImfe+ nln
~
"e- +
+
I A'l M
(VI. 3. 116)
,5
which is supersymme1;ric invariant for any choice of the target manifold provided the relation (VI.S.I07) between the torsion and the 2-form field strength is verified. Furthermore ~ is the boundary of 143:
~"IlM3'
1536
The action (VI.3.116) is determined via the usual rh~onomic construction by requiring that the parametrizations (VI.3.109-110), (VI.3.111) and (VI.3.114) should follow from the associated variational equations. In order to write the latter we must vary S in O$a, in on! and also in the coordinate (OXP) implicitly contained in the vielb;in.
Rather than oX~, it is convenient to utilize anholonomized variations (tangent vectors to Mtarge t) oX a whose generator is the Lie derivative. So we write (VI. 3.117)
where +
a .... pa
oX "oX
(VI. 3. 11 Sa)
(VI. 3.llSb)
and
we obtain
w~~ being the parameter of a field-dependent SO(D) local transformation, defined by
wab ..
-Iwab
oX .§lJ
(VI. 3.120)
By the same token we obtain (VI. 3. 121)
(VI.3.122)
1537 (VI. 3.123)
where ,,~ is the parameter of a field~dependent SO(Nhet) local transromation, defined by (VI. 3. 124)
in full analogy with Eq. (VI.3.l20). Equation (VI. S.122) follows from (VI. 3. 12S)
which is true as a consequence of d Ii '" O. Utilizing these formulae the variation of the action (VI.3.116) in o~a, 6~:, oXa and o~a yields the following results: The rheonomic parametri zations determined from the Bianchi identities are indeed consistent with the equations of motion projected in the outer directions i)
ii) The auxiliary O-forms projections of the vielbein:
n:
are identified with the inner
(VI. 3.126)
iii) The following inner field equations hold true in the special superconformal gauge t(z,i,8) =0 (VI. 3. 127a)
(VI. 3.127b)
(VI. 3.127c)
1538 Using Eq. (VI.3.106a) which yields (VI. 3.128)
Eq. (Vr.3.127c) can also be rewritten as (VI-3.lZ!)
By inspection of Eqs. (VI.3.127a,b) and (VI.S.129) we see that, in a and general, neither W are left-moving fields, nor ~a is a right-moving one.
v!
Indeed the three field equations can be rewritten as follows (VI.3.130a) (VI. 3. 130b )
(VI. 3. 130c)
From Eqs. (VI.3.130) we easily see that the necessary and sufficient condition to obtain a classical superconformal theory is the vanishing of the spin and gauge connections (VI.3.131) In this case Eqs. (VI.3.130) reduce to the desired equations: (VI. 3. 132)
There are just two cases where the spin connection wab can be taken as equal to zero everywhere: i) the first case is that of a flat space characterized by a vielbein whose exterior derivative is zero
1539 (VI.3.133)
the second case correspbnds to a simple Lie group manifold whose vielbein fulfills the Maurer-Cartan equations ii)
(VI. 3.134)
In the first case we have also a vanishing torsion, while in the second case the spin connection vanishes but the torsion is given by Eq. (VI.3.92a) with a=O.
Hence the most general solution for a target manifold leading to
a classical superconformal theory is obtained by an arbitrary combination of these two items. This corresponds to the choice (VI.3.62,63) of the target space and with such a choice the action (Vl.3.116) reduces to the sum (VI-3.M). To see that this is the case it suffices to split the index a of the vielbein Va according to the following pattern: ~
runs from
1 to
~nk
i-I
from
(d~fink + R,~ 1 dim G9.
+
1)
(Vl.3.13S)
a"
to
(d~link
i +
L
R,=l
dim G9. )
Furthermore, recalling Eqs. (VI.3.63) and (VI.3.104) we have n
L dim G.1
i=l
"d
mp ct co a
(VI.3.136)
We set (VI. 3.137a)
1540
(VI.3.137b)
(VI.3.137c)
(VI. 3. 137d)
A
where X~ is the coordinate of the Minkowski space. g i invariant l-fol'lll of the i-th group, and ki
is the left-
is the integer number (k. )
appearing in front of the corresponding WZI'l action S (G~) • 1
Furthel'lllOte, the torsion field is given by (VI. 3. 138a)
Ai BiCi
T
,,-
f{;ki
1
ABC
(VI.3.138b)
-2 f(l')
f~:~ being the structure constants of the i-th group. All the other components vanish and Babe is given by Eq. (VI.3.107) in terms of
r abe . With these subst! tutions and setting moreover AaB =O. the action (VI.3.116) takes the form (VI.3.64) as claimed.
In this way we have succeeded in finding the general form of the classical action for a heterotic superstring propagating in a vacuum that corresponds to the quantization of a classical superconformal field theory.
The freedoms we are left with are the following: i)
ii)
iii)
choice of the space-time dimension dMink , choice of the simple groups Gi
and of their "levels"
choice of the number of heterotic fermions Nh
iv) choice of the homotopy groups B c
et
ki ,
'
~ ® ~.
All these freedoms will be fixed in Chapter VI.S by the quantum consistency conditions.
1541 VI. 3. 5 Canonical quantization of the heterotic WZW·model and the s~erconformal
algebra
In Sections VI.3.l and VI.l.4 we considered the variations of the actions (VI.l.93) and (VI. 3. 116) with respect to the matter fields but not with respect to the supergravi ty background fields e +, e· " •
As it was stressed in the introduction to this chapter and also in Chapter VI.2, the vielbe!n and the gravitino are not dynamical fields in d .. 2. Rather, they can be viewed as the Lagrangian multipliers associated to the primary constraints of the theory, which are nothing but the stress-energy tensor and the supercurrent.
In the classical theory these
latter are "weakly" zero, while in the quantum theory they are utilized to construct the BRST -charge. Let S be the complete action (VI.3.116) of our theory, where the substitutions (VI.l.IS7) and (VI.l.l3S) have been made.
We define
the superstress-energy tensor and the supercurrent through the following variation: (VI. 3.139)
The l·forms fT+, 9",., and ff., which should not be confused with the 2-forms, T·, T- and T· of Eqs. (VI.3.6S), constitute the super stress-energy tensor. The statement that they are "weakly" zero is the 2-dimensional analogue of the Einstein and Rarita Schwinger equations; indeed it follows from the stationarity of S with respect to oe+, oe - and 01;variations. In general the l·forms ff.,
ff..
and .1'. decompose as follows: (VI. 3. 140a) (VI. 3.140b)
1542 (VI. 3. 140c)
However, if the action S is not only supersymmetric but also super Weyl invariant then we have additional constraints on the components of ff,., ff_ and 9",. We have shown in Section VI.3.2 that when the constraints (VI.3.65) hold the heterotic superspace geometry is superconformal; in addition to the superdiffeolDorphisms and the 50(1,1) Lorena transformations it admits super Weyl transformations (that is, dilatations and S-supersymmetries). In particular we have the following transformation rules for the vielbein and the gravitino: (VI.3.141a) (VI. 3.141b)
0,
= -1
2
(M
+
ow) I; + 2011 e
+
(VI. 3.141c)
where oA, OW, and 011 are the infinitesimal parameters associated to a dilatation, a Lorentz transformation and an S-supersymmetry respectively. Requiring invariance of the action
oS/oA = oS/ow = OS/o11
=0
(VI.3.142)
we obtain the following conditions: ff
+
Y+
A
+ e +ff
A
e
- -9"
-
=0
(VI.3.143a)
I,; '" 0
(VI. 3.143b)
A
e -+2:..9" 2 •
~ 1;
A
e -+2:..9" 2 •
A
(VI. 3. 143c)
from which we deduce
1543 ~+-
.. ff-+
=§_.
=S"•• =0
(VI. 3. 144) (VI. 3.145)
Hence in a superconformal invariant theory then are just thne independent and non-vanishing components of the super stress-energy, ff,ff' ++
and ~.'
--
For later convenience we parametrize them as follows:
ff
.. T(class)
(VI. 3.146)
!T
= _ r(class)
(VI. 3.147)
9"
.. 2!T
++
ill +.
'" __1_ e'4 G(class)
.+
(VI. 3. 148)
fl
In classical canonical formalism T(class) ~ 0, t(class) ~ 0, G(class) ~ 0 are primary constraints and, as we will see, they generate the classical algebra of superconformal transformations. If the total action S is the sum of several actions associated to various matter supermultiplets. then the stress-energy tensor is the sum of the stress-energy tensors of the different supermultiplets. Similarly for the supercurrent. In our case the action has the structure (VI.3.64) and, provided all the addends are superconf01'lllal (namely I Eqs. (VI. 3.61) hold true for each component subsystem). we can write
(VJ.3.149a) i(class) _ T(class) - (Mink) G(c1ass) '"
G(elass) (Mink)
n
I
+
i=l
+
i=1
r
r(claSS) (Gi.k i ) G(class) (Gi'ki )
+
T(clas5) (het)
(VI. 3. 149b)
(VI. 3. 149c)
1544
where we have anticipated that the heterotic fermions do not contribute to either r{class) or G(class) since they do not couple to e+ Or to
1;.
As already stated several times, si~nk) is simply a particular case of the general WZW action, where for the group G we take the abelian translation group G" TCd) • It suffices to study the WZW-model for a general group G, namely, to concentrate on the action functional (2.33). Applying the general definition (VI.3.139) to this specific case and inserting, after the variation has been taken, the rheonomic parametrizations (VI.3.71a, VI.3.73, VI.3.74) we easily see that Eqs. (VI.3.144) are indeed satis~ fied while for ff'++, ,"-_,
~.,
~+
and
we obtain the result (VI. 3. 150a)
(VI. 3.150b)
(VI. 3. lSOc)
(VI. 3. 150d)
Hence we see that the theory (VI.3.93) is indeed superconformal invariant, at the classical level, for all values of the parameter a. However, as already stated, we are just interested in the case a= 0 which is the only one where the effective degrees of freedom correspond to those of a classical superconformal theory and where the local symmetries extend to include Kac-Moody symmetry (see below). Anyhow since we have ascertained that our dynamical system is superconformal invariant it is now our privilege to choose the superconformal gauge or even, if we like, the special superconformal gauge where
.'
1545
e+ l,;
'" dz +
= de
!.2 e de
(VI. 3.15 la)
•
(VI. 3.1S1b)
Furthermore, since we have ascertained that the action (VI.3.93) is rheonomic we can, without loss of information, take its restriction to the d =2 space-time s lice of the heterotic superspace, namely. we can set e =O. With these gauge fixings and after substitution of the firstorder equation QA+ =tr(g -1 c+g t A), the action functional (VI. 3.93) with (l." 0 becomes
s~~j = - 8~ +
HdZ di(Q~Q~ + 2iAAazl) + A
t J fABe rzA "UB " Qe]
(VI. 3. 152)
where (VI.3.lS3a)
Q~ t = QA t = g~l oz-g z A - A
(VI. 3. 153b)
Furthermore, we have inserted an i factor which takes into account the Wick rotation implicit in the holomorphic notation z and z for the left-moving and right-moving coordinates respectively. Our i-prescription is very simply stated in terms of the 2-dirnensional integration measures (VI. 3.154)
As we see, with this prescription the integration measure over the worldsheet goes over into the integration measure defined over the associated Riemann surface. Given the form (VI.3.lS2) of the action functional we derive the correct canonical quantization of the fields g(z.z) and AA(z,z).
1546 Since the two field theories are completely decoupled we can treat them separately. Let us begin with the WZW field g(z,i).
In the gauge we have
chosen. the equation of motion (V1.3.99) reduces to
a.rt z z .. 0 .. fI!' .. rt(z)
•
(VI.3.155a)
Multiplying the above by g(z,i) on the left and by g-l(z,i) on the right we can also deduce that ",A azSf'; z ..
",A-A
0 .. 1l"" '" {2 (z) .
(V1.3.1SSb)
Hence out of an "on-shell" WZW-field g(z,z) we can construct twoanalytic currents:
k ."A JA( z) .. - 1. .
2 +
-A -
k ",A
. k "A = - 1. -k tr (-1 A gag t )
(VI.3.156a)
(VI.3.156b)
.. 1 - . .
z
2
z
• k -A
k
-I A t)
2
J (z) .. -i-w= 1-.0-= -i-tr(3.gg 2 2 z 2 z
whose normalization has been chosen for later convenience. want to show that
Indeed, we
i) our dynamical system is a constrained hamiltonian, characterized by second-order constraints which naturally lead to the use of Di rae brackets,
A ·A ii} the Dirac brackets of the dynamical variables J (z), J (z) are (VI. 3. 157a)
A
-B
(J (z). J (w)
lD'lrac = 0
(VI. 3. 157b)
(VI. 3. lS7c)
1541 If true, Eqs. (VI.3.157) would show that the degrees of freedom of the WZW field g(z,i) can be naturally subdivided into two independent sets of left-moving and right-moving IDOdeS. It suffices to perform a Laurent series expansion of both t{z) and JA(z):
(V!. 3.1SSa)
(V1.3.1S8b)
Using these expansions Eqs. (VI.3.157) become the following: (VI. 3. 159a)
(VI. 3. 159b)
(VI. 3. 159c)
J!. j!
The "modes" form a complete set of dynamical variables describing the WZW field g(z,z).
3!
Canonical quantization is obtained by tuming J~. into operators and replacing the canonical Dirac bracket in Eqs. (VI.3.l59) with the quantum commutator: {I a • JillB} n
Dirac
quanthation [A BJ • I n, J m .. (VI. 3. 160)
and similarly for the other cases. Note that no i-factor is needed in the above fomulation of the correspondence principle since we have already performed the Wick rotation on the classical action (see Eq. (VI.3.154). Alternatively we can say that no i-factor is needed since
1548
in our canonical commutations time-ordering has been replaced by radial ordering of the complex variables z and w. With the above quantization procedure, Eq. (VI.3.160) states that the left~ and right-moving modes of the WZW field satisfy two identical and commuting Kac-Moody algebras G where the value of the central charge (that is, the level k) is identified with the constant k originally written in front of the action (VI.3.93). lmitarity of the -2 Kac-Moody representations requires k/6 e I, where 1\ is the longest root of the G Lie Algebra (see Olapter VI.S). Equivalently, the canonical quanthation can be formulated directly on the currents l(z). jB(f) turning them into operator-valued distributions and identifYing the r.h.s. of Eqs. (VI.3.IS7) with the Singular part of their short-distance operator product expansion (OPE): A B k J (z) J (w) " 2
(V1.3.161a)
k
(VI. 3.161b)
In this way we see that once Eqs. (VI.3.IS7) are established the canonical quantization of the WZW-field is also established. Let us then prove these crucial equations. First of all, using the tangent variations
= oyA tA
(VI. 3. 162a)
6y~ .. g oy g-1 • '" ogg-l-A "oy t A
(VI. 3.162b)
6y = g
-1
6g
we notice that the variation of the action functional (VI.3.152) can be written as follows:
oS = i !. 411"
I
dz
~
di
az-rtz 0/ " i J!.. 411
I
dz ;. di
aZrtz oyA (VI. 3.163)
1549 Next, we recall the structure of Dirac brackets in a theory whose Lagrangian L is linear in the velocities. The general structure of such an L is n L"
I
.
A. (q)
i=l
q1
(VI. 3.164)
1
where Ai (q) are functions of the dynamical variables qi and where, for simplicity, we have considered a finite number of them. The Euler-Lagrange equations are obtained from the variation of the action S = di L(q,q) and read
I
(VI. 3. 165)
the matrix Cij(q) being defined by the following equation dS "
f dr (oqi clJ.. qj)
.
(VI. 3.166)
Using (VI.3.164) the explicit form of Cij is given by:
a
a
, , - A).(q) - - . A.(q) • cqi cqJ 1
(VI. 3. 167)
Now the very fact that L is linear in the velocities implies the existence of second~class constraints. Calculating the canonical momenta Pi" aL/cqi, we find the constraints
w.1 "p. - A. 11
::: 0 •
(VI. 3. 168)
They are of the second class since their Poisson brackets are not weakly zero, rather they are equal to the matrix C1) .. : l~· ,~·l 1
J Poisson
-;
0 .
(VI. 3.169)
1550 This observation has the important implication that for the class of theories tmder consideration, once the matrix Cii has been identified, we can immediately calculate the Dirac bracket of any two dynamical variables X(q)
and Y(q): (VI. 3.170)
This procedure can be immediately applied to the WZW-model. us consider Eqs. (VI.3.163) and take Z.. T as the time variable. identifying the oq~variation with
Let Then,
or:
6'1~A .. ogg -1~" .. "'1
(VI. 3.171)
from consistency it follows that ~ is the velocity ~
ll;'; ..
z
a-gg Z
- 1 . -1
.. gg
Oq • .. - .. q • OT
(VI. 3.172)
which explicitly identifies the operator Cij with integration over the variable z and
d';B .. (L)
in the case of left-
For the latter, sUllDBation on the index i
moving fields.
ik 411' .
~~
is substituted
can write
oAB a •
{VI. 3. 174)
z
~
A
1:.
.A nA
Al ternatl vely, if we choose T" z, uy .. uq, y
""z
we find the
operator CR appropriate to rignt-moving fields: (VI.3.175)
1551
Having identified CL• the derivation of Eqs. (VI.3.157) becomes straightforward. We just need the inverse of (VI.3.174) which can be represented as an integral operator: (VI. 3. 176a) -1 41fi CAB(Z - w) .. 10g(z - w) 0AB • k
(VI.3.176b)
Using this kernel in the definition (VI. 3.170) of the Dirac bracket we find:
(VI.3.l77)
which is easily calculated using the identity 6JA(Z) .. &yM(u)
!!.
tr(g{z)tAf 1 (u)t M)
2
(u - z)2
(VI. 3. 178a)
which follows from (VI.3.178b)
The result of the calculation is the Eq. (Vl.3.1S7a) above and in a perfectly analogous way one derives Eqs. (VI.3.1S7b) and (VI.3.157c). The quantization of the fermions is much simpler and easily done.
Their contribution to the action is read off from Eq. (VI.3.1S2) (G) k S(fermions) '" 411"
I dz
-
A
A A dz A el zA.
(VI.3.179)
which immediately leads to the following second-class constraint: (VI.3.IS0)
1552
where we have denoted by ~A the canonical momentum conjugate to )..A. (Remember that both )..A and l£A are anti commuting fields). Calculating the Poisson bracket of $A(z) with ~B(w) we find (VI.3.181) which defines the matrix C1) .• of Eq. (VI.3.169) in the case of free fermions. Since 1/2~i (z - w) is the identity operator (in the distri-1 i -1 bution sense) we get C '" - k (z - w) and can write i {jAB k z-w
(VI. 3.182)
(the index :!: appended to the Dirac bracket tells us whether it is fermionic or bosonic, symmetric or antisymmetric). This explains also the not ation of Section VI. 3.1. The canonical quantization of the field )..A(z) is then obtained by turning it into an operator and identifying the r.h.s. of Eq. (Vr.3.182) with the singUlar part of the OPE: A
B
i
oAB
A (z) A (w) "' - - - - .. reg. k z-w
(VI.3.183)
Recalling Eqs. (VI.3.148, 149, 156) we can easily write the explicit expressions for the classical stress-energy tensor and the supercurrent in terms of fields whose canonical Dirac brackets are known: (VI. 3. l84a)
(VI. 3. 184b)
(VI. 3. 184c)
1553
In the formulae above, the suffices (G,k) recall that the stressenergy tensors and the supercurrent are completely determined by the choice of the group and the constant k. Using (VI.S.184), (VI.3.182) and (VI.S.lS7) we find Ir(claSS)(Z), TCclass) (W)}H '" (Dirac) 2T(class)(W) aT(class)(w) ..
(VI. 3.18Sa)
+ - - - - " -....
(z_w)2
z-w
I
T(ClaSS)(Z), G(ClaSS)(W)JC-) : (Dirac)
=1
G(class)(w) + oG(Class)(w) 2 (~ _ w)2 z •w
\ G(class) (z), G(class) (w)
I
(+)
= 2T(c1ass) (w)
(Dirac) - ~(class) - }e-) T (w) 1 T~(ClaSS) (z),
(Dirac)
(VI. S.
l8Sh)
(VI. 3.18Sc)
z -w '"
2T(class)(w) (i - 14)2
+
or(class)(w) i -W
•
(VI,3.18Sd) while all the other brackets vanish. Equations (VI.3.18S) define the classical superconformal algebra of primary constraints. As already pointed out, the two-dimensional vielbein and gravitino are the Lagrangian multipliers of the hamiltonian constraints: T(class)(z) '" 0 G(class) (z) ::: 0
t(c1ass) (i) '" 0 (VI. 3.186)
which are primary precisely because their Dirac brackets are weakly zero, namely, proportional to the constraints themselves (Eq. (VI.3.18S)).
1554 Equation (VI.3.185a) coincides with Eq. (VI.3.30) and leads to the classical Virasoro algebra. Expanding in modes both T(z) and G(z} from Eqs. (VI.3.l8Sa,b,c) we obtain the supersymmetric extension of the classical Virasoro algebra. Its geometric interpretation is very simple. As the Virasoro algebra generates the diffeomorphisms maintaining the conformal gauge (VI.3.20), in the same way the super Virasoro algebra generates the superdiffeomorphisms maintaining the superconformal gauge (VI.3.SS). Upon quantization Eqs. (VI.3.18S) are replaced by the Corresponding operator product expansions. The case of the composite fields T(z). fez) and G(z) is however different from the case of the elementary fields JA(z), ~(i), AA(z). In the latter, declaring that the Singular part of the OPE coincided with the classical Dirac bracket was our privilege and embodied our quantization prescription. In the former, the OPE of the superconformal generators, which are composite fields, is no longer a matter of definition, rather it is something we can calculate and compare with the r.h.s. of the classical Dirac bracket. The two answers are usually different but most of the difference can be reabsorbed in multiplicative renormalizations of the composite fields. The part which cannot be reabsorbed is the conformal anomaly. It manifests itself in the OPE as a higher singUlar term whose coefficient is a C-number. That C-number is the central charge of the quantum superconformal algebra. In the case of the WZW-model we consider the OPE's of the superconformal generators defined by Eqs. (VI.3.184). A straightforward calculation shows that in order for the simple and double pole terms to have the same form as in the classical Dirac bracket (Eq. (VI.3.18S» we just need two multiplicative and finite renormalizations. Indeed, if we define the renormalized quantum generators by TCG
,
1 A A k){Z) • ------ : J (z)J (z) k + C
v
T(G k)(z) '" -1- : -AC-)-A(-) J zJ z ,
k + Cv
ik 2
(VI.3.187a) (VI.3.187b)
1555
(VI.3.187c) where the second Casimir Cv is the number defined below
(VI. 3. 187d) then we obtain the fOllowing OPEI S T(z)T(w) .. £. __ 1_.. 2T(w) .. aT(w) .. reg 2 (z _ w)4 (z ~ w)2 z- w T(z)G(w) ..
!
~
2
(z-w)2
..
aG(w) ... reg
(VI. 3. 189b)
7.-W
G(2:)G(W) .. ~ c __ 1 _ + 2T(w) .. reg 3 (z _w)3 z· W T(f)T(w) ..
(VI.3.189C)
§. __1_.. ii'(w) + 3'i'(iIi) .. reg 2 (i _ w)4
(z _ w)2
i-if
where the values of the central charges c and c .. c(G.k) '"
~ dim k .. -V
G+
c .. c(G.k) .. _k_ dim G •
k
+
Cv
(VI. 3.189a)
!
2
dim G
c
(VI.3.189d)
are given by: (VI.3.190a)
(VI. 3. 19Gb)
In Eq. (VI. 3.190a) the first addend is the contribution of the KacMoody currents. while the second is the contribution of free left-moving fermions ~A(z). This explains why in the right-moving sector (Eq. (VI.3.190b» we have only one contribution. This concludes our treatment of the WZW-model, which, as shown in Section VI.3.4, is the only other building block, besides the heterotic fermions, of the heterotic superstring vacua corresponding to classical superconfozmal theories.
1556 APPENDIX TO CHAPTER VI.3
Rules for the Wick rotation of spinots Before Wick rotation (in Minkowskian signature) e+ and e
are
real I-forms and 1; is real as a consequence of the Majorana condition. Under the Wick rotation the action must be multiplied by a factor i: S+
is .
(VI.3.Al)
This corresponds to the transformation t .. it
Furthermore, ,/
+
dz. e - ... eli but we must also keep t rack of the
Majorana condition.
Originally we had
(VI. 3.A2)
so that we wrote
(VI. 3.A3)
After Wick rotation we must replace yO tion (VI.3.A2) becomes
+ -
i yO.
so that condi-
(VI.3.A4)
Maintaining the fOrm (VI.3.A3) (VI. 3.AS)
1557 we get
(VI. 3.A6)
wilich implies (VI. 3.1\7)
If we set
(VI.3.A8) OUT
equation is satisfied.
hi
-'4
In conclusion. after Wick rotation a Majorana spinor is e times a real anticammuting field.
References for Chapter VI.!
[1]
L. Castellani, P. Fre, P. van Nieuwenhuizen. Ann. of Phys. 136 (1981) 398.
1558
CHAPTER V!. 4
The DRST Charge and the Ghost Fields
VI.4.1
Introduction
In this chapter we give a brief account of BRST covariant quanti. tat ion. Its characteristic feature is a global invariance first dis· cussed by Becchi. Rouet and Stors [lJ and Tyupin [2} in the context of gauge theories. The phase-space description of a physical system with local
invariances includes a set of (first-class) constraints .A (qi 'Pj) ,,0. A:l ..... m. i.j =1 ••••• n. satisfying the Poisson bracket relations: (VI.4.la) (VI.4.1b) where ~ means that the equality holds on the constraint surface 'A (q.p) ,,0. These constraints generate the local invariances of the theory (see Refs. 3-6 for reviews).
1559 To quantize this system, we lllay proceed in two ways: we introduce m additi~al constraints XI\ (q,p) '" 0 (gauge choices) such that $1\ and XA together become second class, Le. i)
(VI.4.2) These constraints can be solved to obtain an unconstrained theory defined on a 2 (n-mJ-dimensional phase space. Then the usual canonical quanti:l.ation can be applied. ii) We quantize the theory in the 2n-dimensional phase-space (qi,Pj)'
and project out the physical states by requiring (VI.4.3)
The hat denotes quantum operators.
We have to check whether the quantum
version of Eqs. (VI.4.1) still holds.
If it does not bold, we have
anomalies in the quantum algebra of constraints. In field theory, Eq. (VI.4.3) may be too strong a reqUirement, and one imposes the weaker condition (VI.4.4) The states Ix:>" 4>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 >
=
=0
(VI. 4. 7)
i.e. iff it does not affect observable quantities
(lIP> 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 G
But
X. Q takes the form (VI.4.31)
and the condition QX =0 is equivalent to (VI.4.32) Indeed nA cannot annihilate a state X satisfying (VI.4.29.30). We see therefore that the BRST-invariant states X are G-invariant states that do not contain ghosts. and these states we call physical states. The condition QX= 0 projects out these states. when X has ghost number zero. In the case of infinite-dimensional Lie algebras, there are two caveats. Because of normal ordering in Q (necessary, for example, to ensure its hermiticity), the nilpotency condition Q2 = 0 may develop an anomaly.
Also. normal ordering in U will alter the condition U.. 0
for physical states.
As we will see in Sect. V"I.4.6. for the string
the physical states have U= -1/2.
VI.4.4 The BRST invariant hamiltonian and the Fradkin-Vilkovski theorem Let us suppose that Eq. (VI.4.1b) takes the form
with constant V/
and HO = classical hamiltonian.
BRST-invariant hamiltonian H as follows:
One can construct a
1566 since
[H.Q] " 0 •
(VI.4.35)
Moreover, by introducing a gauge fixing X with ghost number" -1, one can construct a BRST invariant gauge fixed hamiltonian HX: (VI. 4. 36) (VI.4.37)
For example, if we choose X::: lISXB. conditions of (VI.4.2). we have
l
being the gauge fixing
(VI. 4. 38)
Consider now the path-integral
tz
'f tldt (q· ip.· An lIA - (H - [1 x.Q) +
1
Zx
= J Dq DpDn D!!
1
e
1)
Theorem:
Zx
2)
~:
(f'radkin-Vilkovisky); ZX::: ZX"
is ERST-invariant.
The proof of these theorems is not difficult and can be found in Ref. [8]. Theorem 2) states that the transition amplitudes are independ~ of the particular gauge fixing function X. as they shOUld, being physical quantities.
VI.4.5 BRST quantization of string theories VI.4.S.1 The bosonic string The action for the bosonic string is {see, also, Eq. (Vr.6.61))
1567
J
S :: _1_ do dt r-g gaB 0 Xll 21fa'
Cl
(loX P
(VI.4.39)
II
where X~(O.t)
0;
D-dimensional Minkowski coordinates
gaS(a,i) :: metric of the world-sheet
Idet gaB'
g ::
a' :: "Regge 5 lope". a'
has the dimensions of an area.
The variational equations are oS - - : : 0 ....
ogaB
aCtxiJ
1 doX - p
II
2
g~o up
yo g
aYxII a..,xjJ ::
(VI. 4. 39a)
0
\J
(VI. 4. 39b)
Eq. (VI.4.39b) is the propagation equation for XiJ(a.i), whereas Eq. (VI.4.39a) is a constraint, since the derivatives of gaB do not appear in the action (VI.4.39). We can solve Eq. (VI.4.39a) by taking ll 0SX ' the square root of its detel'lllinant. Defining GaB:: lJ G:: Idet GaB I, we have
ari
GaB:: G ::
i
t
gaB gYO
g
(gYo
ayx~ ~oXjJ
allJ coX)
and substituting into Eq. (VI.4.39) yields the Nambu action:
(with
iJ..i::.!. XlJ x'jJ::.!. XiJ ) aT'
spanned by the string.
00
which is the area of the world-sheet
Consider now the Lagrangian: (VI.4.40)
The corresponding primary (first-class) constraints are given by:
(VI.4.41)
I'.'here p "0 L/ oXlJ • The hamiltonian II
HO ::
I
PlJ ill da - !! = 0
(VI. 4. 42)
is itself a primary (first-class) constraint generating T-reparametriz2tions. The canonical variables Xll, Pv satisfy the (equal time) Poisson brackets: (VI.4.43)
From (VI.4.41). one arrives at the constraint algebra (VI.4.44a)
(VI.4.44b) Following the prescription of Sect. VI.4.3. we construct now the BRST operator for the bosonic string:
=:
=:
Jda[L I'}+ +L n-] _ 1 fdadal [IT (0) +IT (0 1 )]0 0(0 -o')n+(a)rtco') +
12 Jdada ' [n
2
w
(a) H w
= JdoCL+O+ + L_o-)
-
+
(0 1 )]3 Q(a 0
+
...
a
-O')T]-(O)O-CO')
Jda[1T+(Oan+)n+
-11' Joan-)n-]
(VI. 4. 45)
1569 ~
where the ghosts n and antighosts
{n(a;r), ~((1I,T)}
satisfY
= tHo-a') .
(VI.4.46)
We now choose the gauge fixing ("conformal gauge"):
(VI. 4.47)
so as to obtain the BRST -invariant gauge fixed hamiltonian (see (VI. 4.36)) :
HX
=
HO
+
{X,Q}+ =
=' /dO(L+ + L)
+
/dO[1T+(}on+
-~ _a()"n-]
(VI. 4. 48)
Notice that the ghost Lagrangian is then 9'(ghost} = I(1f+(}Tn+ +~_a·ll-)do =
J(~+(}()"n+ -If_
I[
r. +
(VI. 4. 49)
i.e. the same one would obtain from ordinary path-integral-quantization (cf. Ref. [9]). In the conformal gauge, the equations of motion are
xP = {xP,
HX} '" np~ .. "X-x"=OX=O
pI! = {pIJ, HX} ...
(VI. 4. SOa)
~ (x~)"
= {+ n, HX } =
;; a n+ -
=0
(VI. 4. SOb)
(VI.4. SOc)
(VI. 4. SOd)
1570
all + -
=0
a• L+ = a+L =0
(VI.4.50e)
•
(VI. 4. 50f)
For closed strings, Eqs. (VJ.4.S0) imply the mode expansions:
co
n+ .. tL cn e -in(T+o) -co co
-
n "
-In(T-o)
'"L b
-in(T+o)
-00
1T" +
1T
-
•
~
t
L
cn e
n e
-CIQ
= '"I
.in(T-.,)
b~
n -co
e
co
L .. +
L =
-
•
I
L e-1n(T+O)
-00
n
-GO
n
rL
e-in(T-a)
The commutations (VI.4.43) and (VI.4.46) give: (VI. 4.51a) (VI.4. SIb)
{cn'
b} m + .. 0n+m. o·
(VI. 4. SIc)
After expressing Ln in terms of x~ modes (cf. Eq. (VI.4.41»:
r Xllnom Xlim
GO
L .. -
n
.!.2
..,.
(VI. 4. 52)
1571
we find the Virasoro algebra (VI. 4 . 53)
To quantize the theory, we now replace the Poisson brackets {}, { 1+ by commutators and anti commutators respectively. Moreover, it is necessary to introduce normal ordering in the infinite sums on modes (Vr.4.52). The quantum algebra of L becomes then n
ILn ,
L J
m
= (m - n) Lm+n + .-£. n (n 2 - 1) <5 12 m+n, 0
(VI. 4. 54)
with c" D (dimension of target space-time). In terms of the Fourier modes, the normal ordered, BRST operator (VI.4.47) becomes co
Q"
l -co
L
c -m m
~ l
(m-n)
-co
c c
b
-m -n m+n
(VI. 4. 55)
with a = normal ordering constant. The ghost-number operator is co
u"
L
-co
(VI. 4. 56)
c b
-m m
For closed strings we should add the left-moving contributions to Q and U. Now, we know that Q2=O holds classically, i.e. when using Poisson brackets and considering b, c as being grassmann numbers. However, when one replaces the Poisson brackets by (anti)commutators and introduces normal ordering in the sums on modes, the nilpotency of Q is not ensured any more. In fact, we find (VI. 4. 57)
so that Q2 " 0 ~ D " 26,
a = 1 •
(VI. 4. 58)
1572 Let us now discuss the physical states of the bosonic string. nomal ordering of U that ensures its hem ticity is
The
co
U=
I (Coho -boco) nfl (c.nbn +
(VI.4.S9)
-b_ncn) •
The ghost and antighost zero modes cO' bO commute with the hamiltonian (VI. 4.48). The gl'Otmd state is therefore a representation space for the operators
cO' boo satisfying (VI.4.60)
The irreducible representation of the algebra defined by (VI. 4.60) has dimension 2. and we denote its basis vectors by
1t>
and
1~>.
We can
choose these as follows: (VI.4.61) so that we must have
(VI.4.62) The ghost numbers of I~>o
1+> are
ul*> .. - .!2 1*> ult>
= .!.2 It>
(V1.4.63)
since
n 1-1-> .. cn [.r> .. bn1+>
b
i.e.
(I~>.
It»
= cit> n
'" ()
Cn > O}
(VI.4.64)
is a ground state annihilated by the destruction
operators b. c .(n >0).
n
n
Physical states do not contain ghost excitations and we expect 0
therefore the corresponding Fock space to decompose as
1573
Ix> 1-1->
(VI.4.6Sa)
Ix> It>
{Vr.4.65b}
or
i.e. as a tensor product on which the (anti)ghost operators act on the second factor. Only the first of these choices, with ghost number U .. -1/2, is really what we call a physical state. Indeed the action of Q on (VI.4.6S) is given by (VI.4.66a)
Qlx> It> ..
[ln>O
C
LJ -n n
Ix> It> •
(VI.4.66b)
Thus only in the first case the condition Qlx> /-1-> =0 reproduces all the physical state conditions (LO-1) Iphys >.. 0, Ln>olphys >.. O. We conclude that the physical states of the bosonic string are BRST cohomology classes of ghost number -1/2.
VI.4.5.2
The BRST charge in complex variables
In the conformal gauge (VI.4.47), the bosonic string becomes a conformal theory. Going over to complex coordinates z
= exp(t i- iO')
z = exp(t - iO')
(VI.4.67)
we see that Eqs. (VI.4.50f) imply analyticity of the stress energy tensor L(z), renamed T(z) in the following.
We recall from (VI.3.30) that the real commutation relations (VI.4.44a) are equivalent to the Dirac brackets:
1574 {T(z), T(w)}
D
=.1!.1.!L + dT(w) (z_w)2
•
z-w
(VI.4.68)
One retrieves the Virasoro commutations (VI.4.S3). after defining L n
f
= ~ zn+l C 21Ti
T(z)
(VI.4.69)
where the closed curve C goes around the origin once. All the results of the previous section can be recast in this language. In particular. the ghost action corresponding to (VI.4.49) becomes (see also Eq. (VI.8.131) and Sect. VI.8.4 where b-c systems are treated in general)
S(ghost )
= - l..I[-2bdC+Cdb] 21r
Adz
(VI.4.70)
yielding the equations of motion:
a_z c .. a_z b
.. 0
(VI.4.71)
implying that b and c are analytic fields. Hence the ghost system is a conformal theory. The canonical commutations (VI.4.46) are replaced by {b(z). c(wH D '" _1_ •
z-w
(V1.4.72)
The ghost stress energy tensor, obtained by varying (VI.4.70) with respect to the metric. is T(gh)(Z) .. c db - 2(ac) b •
(VI.4.73)
Using the basic anticommutator (VI.4.72), we find [ T( h) (z). T( h) (W)] ,,_2_ T( h) (z) + _1_ aTC h) (z) • g g D (z _ w)2 g z -w g
(VI.4.74)
1575
Quantization in this formalism means to replace Dirac brackets with operator products. The commutations (VI.4.68) and (VI.4.74) become:
0 1 ~-2 (_w)4 z
I{z ) T(w);;; -
T( h)(z) g
Te gh)(w)
2T(w)
aT(w)
regular tel'llls
+ --- + -- + (~_)2 z-w ~ 'II
-13
2T(gh)(w)
,. - - - . . .
(z_w)4
(z-w)2
(VI.4.75)
aT(gh) ('II) +
z~w
...
regular terms (VI. 4.76)
where the coefficients of the most singular terms in the r.h.s. are equal to half the anomaly of the corresponding system. Using (VI.4.69), it is straightforward to retrieve the quantum Virasoro algebras (VI.4.S4) corresponding to the matter system (c(matt er )" D) and the ghost. system (c{gh) ,,-26) respectively. We see from Eqs. (VI.4.75, 76) that the total anomaly of the matter ghost systems, c(matter) when D,.26.
+ C (gh)'
cancels
The BRST charge QBRST is defined as follows: (VI .4.77) with j BRST(z) "
c (Tmatter(Z)
" - 1. c axax ... 2
+
iT (gh) (z) ) "
bc{
(VI. 4. 78)
The reader can verify that (VI.4.77) indeed coincides with the BRST charge constructed in the real fOl'lllalism of the previous section. The action of QSRST on a field fez)
is defined by (VI. 4. 79)
and one finds, by using operator products to evaluate the contour integrals,
[QBRST'
x~l
= c axP
1576
= T(matter)
+T
(VI. 4. 80)
(gh)
Q~RST=O is equivalent to TCmatter) +T(gh) closing on the confol:roal algebra (V1.4.SS) without anomaly. Hence Q2 =0 * 0 =26. BRST
VI.4.S.S The NSR superstring
In this subsection we discuss the ghosts and the BRSI charge for the NSR superstring. We recall the quantum super-Virasoro algebra, in terms of O.P.E. (VI.S.IB),
T(z) Tew) __
~
3
T(z) G(w)
Gez) G(w)
1
2T(w)
G(w) (z _w)2
aG(w) z- w
aT(w)
--- + --- + -- + (4 _w)4 (z _w)2 ~- w
2
--- + -- +
2
reg.
reg.
1 2T(w) = -2 c - - + - - + reg. 3
(z _ w)3
(VI. 4.81)
z- w
where the stress-energy tensor and the supercurrent are given by (cf. (VI. 3.184»)
i
T(z) = axax - - 1/1 a 1/1 2
G(z) ,.
1/1
ax .
(VI. 4.82)
o
3
The conformal anomaly c(matter) is c( matter ) '" Cx+ C1/1 ,. D + -2 = -D 2' P the bosonic matter x contributing D and the fermions 1/1 contributing !o (cf. (VI. 3. 18B». 2
Expanding T(z) and G(z) in Laurent modes: I;'
T(z) ,. l. Ln
Z
-n
1577
G(z) = ~ G
t. n+~-(w/2)
z-(n+2-(w/2»
(VI. 4. 83)
the O.P.E. (VI.4.81) yield the commutation relations below 1)
W"
(NS superconfol'lllal algebra):
0
[L • L] ... (m -n)L
m n
{G
m+~'
ti)
n+m
G
n+~
+
..£.. 12
(m 3 -m)6
m+n, 0
2 +m)6 n+m+l +£(m 3 m+n+l,O
}=2L
(VI.4.84)
w" 1 (R superconformal algebra):
[L , L ] .. (11\ - n) L
n+m
m n
+
..£.. 12
(m 3 - m)6
m+n, 0
= (-12 m -n)Gm+n (VI. 4. 85)
The corresponding BRST charge is
. -
~
[12 n . !.2
~ L
.e
(, n,S
n,s
w. e -s-~(l-w) en+s+~(l-w) cn ..
(1- )] . -
• -s-~(l-w)
e
n+s+~(l-w)
c • - CleW) c
n•
0
(VI.4.86)
where the ghost fields, having the same moding as the stress-energy tensor, satisfy
c
[cm' n1+ 6 - n+m,O
1578 (VI. 4. 87)
and the superghost5, baving the same moding as the supercurrent, satisfy the commutations
rem~(l-w)'
en+~(l-w) J '" 6n+m+l-w.O .
(VI.4.88)
In order to construct the BRST current for the superconformal algebra (VI.4.81), we need the stress-energy tensor of the superconformal ghosts: T(gh) .. -
~yal3
+
t Bay
+ cab + 2b(ac)
(VI. 4. 89)
where Y. 13. e, b have the Laurent expansions
~
e(z) .. L em z
-m+l
bez) .. L~ e- z -m-2 m
The ghost supercurrent is
G(gh) .. - caB
1
+ -
2
3 yb - -2 (acJB
(VI.4.90)
The reader can verify that -
1
L(gh)n ;: 21ri _ 1.
G(gh)n ;: 2ni
f I
n+1 z T(gh) (z)
n+1
j z
( )
G(gh) Z
(VI. 4. 91)
(VI.4.92)
1579 do indeed coincide with the ghost part of tbe "improved" generators .tn' lin defined by (VI. 4. 93) (VI. 4. 94)
with QBRST given in terms of mode operators in (Vr.4.86)*. current is
jBRST" c[Tmatter(Z) T
+
tT(gh)(Z)]
Y[Gmatter(z) +
t
The BRST
+
(VI.4.9S)
G(gh)l
and the QSRST charge is given by:
(VI. 4.96)
Again the reader should verify that QBRST given in (VI.4.96) is equal to (VI. 4. 86). The nilpotency of QSRST implies that the total stress.energy tensor T .. T (matter) + T (gh) and the total supercurrent G" G(matter) T G(gb) close on the superconformal algebra (VI.4.81) without anomaly. Let us compute the algebra of T(gh) .. T(be) + T (By)
and G(gh) "
G(be) + G(By):
* This is actually the way we have derived T(gh) find
L(gh)n' G(gh)n
and G(gh){Z)
and G(gh): we first
from (VI.4.93, 94). and then construct T(gh)(z)
whose Laurent moments are
L(gh)n
and G(gh)n'
1580 (-26 TIl) 1 2T (gh) (w) aT (gh) (w) T) 2 --- + + + reg. (gh) (z 1(gn) (w) ... (z-w)4
(z-w)2
z-w
(VI.4.97a)
3 G(&h) (1)1)
T(gh) (z) G(gh) (w) ..
'2
(z _ w)2
oG(gh){w) z _W
T
g
1
2T(gh)(w)
3
(z-w)3
z-w
The ghost superconfomal anomaly c(gh)
c egh ) = c(bc)
+
c(Sy)
a
(VI. 4. 97b)
reg.
2
G( h) (z) G( h) (w) .. - (-26 + 11) - - +
g
+
+
reg.
(VI. 4.97c)
is therefore
(VI. 4.98)
-26 + 11 = -IS .
The algebra of the total stress-energy tensor and supercurrent closes as in (VI.4.81) without central charge if the total superconformal anomaly
c" c(matter) vanishes. 1. e.
fOT
+
c(gh) • (D
+
tD)
+
(-26
+
11)
(VI. 4.99)
D=10.
Finally. from (VI.4.89). one deduces the ghost action: S (gh)
=-
~
f(-
2b dc - c db) " e+
- .!.. I(- ~2 Bd'V + !2 ydB) '" 21r
eT
(VI. 4. 100)
As already stated the ghost b-c system and the superghost S-y system correspond to two special cases of conformal field theories of the b-ctype described, in general terms, in Section VI.S.4. The reader is referred to that section for further informations.
1581
References quoted in Chapter VI.4
[1] [2] [3] [41 [5] [6] [7] [S]
[9}
C. Beechi, A. Rouet and R. Stora, Phys. Lett. 52B (1974) 344; Ann. Phys. 98 (1976) 287. I.V. Tyupin, Lebedev preprint FIAN No. 39 (in Russian, unpublished). P.A.M. Dirac, Lectures on quantum mechanics, Yeshiva Lectures (1964). A. Hanson, T. Regge and C. Teitelboim, Constrained Hamiltonian Systems, Contrib. Centro Linceo Scienze Mat. N. 22, Rome (1976). E.e.G. Sudarshan and M. Mukunda, "Classical Oynamics - a modern perspective", Wiley, New York (1974) pp. 78-107. L. Castellani, Ann. of Phys. 143 (1982) 357. M. Henneaux, Phys. Rep. 126 (1985) 1. E.S. Fradkin and G.A. Vilkovlski, Phys. Lett. 55B (1975) 224; L.A. Batalin and G.A. Vilkoviski, Phys. Lett. 69B (1977) 309; E.S. Fradkin and T.E. Fradkina, Phys. Lett. 72B (1978) 343; T. Kugo and I. Ojima, Suppl. Progr. Theor. Phys. 66 (1979) 1. M. Kato and K. Ogawa, Nucl. Phys. B212 (1983) 443; S. Hwang, Phys. Rev. 028 (1983) 2614.
1582 OlAPTER VI. 5
Quantum Determination of the Target Manifold and Kac-Moody Algebras
VI.S.l
Introduction
In Chapter VI.3 the classical action of heterotic superstrings was determined and it was found to be given by Eq. (VI.3.1l6). Under the condition that the target manifold Mt arget is of the type (VI.3.62, 63) and that we compactify the superstring on a classical superconformal theory one has the identifications (VI.3.l37), and the torsion field is given by Eqs. (VI.3.138). With these choices, the action S reduces to the sum (VI.3.64) corresponding to a linear superposition of different classical superconformal theories which can be independently quantized. The freedoms in the choice of Murge t left by our classical considerations were listed at the end of Section VI.3.4. For the reader's convenience let us rewrite this list: i) ii)
choice of the space-time dimension DMink • choice of the simple groups Gi
and of their "levels" ki .
1583 iii) choice of the number of heterotic fermions Nhet , iv)
choice of the homotopy group B c G.r
0
Gr-
The purpose of the present chapter is to show how these classical freedoms are fixed by quantum consistency conditions: The first three items in the above list are related to the con~ formal anomaly, i.e. to the central charge of the Virasoro algebra. Requiring the vanishing of the BRST charge 2
QBRST ;;; 0
(VI.5.1)
discussed at length in the previous chapter, we will find that there is an upper bound on the dimensions of Minkowski space: 0Mink S 10 •
(VI.5.2)
Furthermore, for each choice of DMink in the range (VI. 5. 3) there is a finite number of available choices of Gr (see Eq. (VI.3.63» given by suitable products of simple groups G. with suitable levels 1 ki . For each chOice of Gr one has an appropriate choice of the integer number Nhet • The fourth item in the list of freedoms, that is, the appropriate choice of the homotopy group B c ~ ® GT (see Section VI.3.l for a preliminary discussion of this topic) is related to a different consistency requirement: the existence of target space fermions and, eventually. of target space supersymmetry. Mathematically the choice of a nontrivial S corresponds to a twist of the Kac-Moody algebra generated by the currents (VI.3.156). Henceforth we will study the representation theorY of twisted Kat-Moody algebras and the dependence of the Virasoro coboundary (see Eq. (VI.3.34)) on the twist. ~fassless target fermions exist for few of
1584
the available target groups suitablY twisted. Eventually, if we require chiral representations for such massless fermions, then the choice of Uy is unique in every even dimension DMink S 10. We find the following answer: * D~fink " 10
Gr"
DMink " 8
Gr = SU(2)
(VI.S.4b)
D:.l ink " 6
GT .. SU(2) xSU(2)
(VI.S.4c)
Gr " SU(2) "SU(2) XSU(2)
(VI. 5.4d)
Gr"
(Vr.S.4e)
DMink
=4
DMink " 2
(n.5.4a)
1
SU(2) "SU(2) xSU(2) "SU(2)
Furthermore for each SU(2)i appearing in 8i can be chosen in two ways:
Gr
the first homotopy group
(VI.S.S)
leading to a certain number of possibilities for the full homotopy group (VI.S.6)
The classification of the chiral superstring models is then reduced to the classification of the homomorphisms (VI.3.1o) between the fundamental group of the Riemann surface and the fundamental group of the * Actually as already stated in Chapter VI.l a recent result by Bouwknegt and Ceresole shows that changing the normal ordering prescription a few more solutions exist be~ides those displayed in Eqs. (VI.S.4).
1585
target space. This is what is done by means of the study of modular invariance, addressed in Chapter VI.7. In the present chapter we deal with the derivation of Eqs. (VI.S.4) and (VI.5.S). VI.S.2 The BRST-charge: cancellation of the conformal anomaly, boundary conditions and intercepts. As stated by Eq. (VI.3.64) the heterotic string theory is the direct sum of several independent superconformal field theories which interact with each other only through the 2-dimensional background fields. Hence, in the superconformal gauge these theories appear fully decoupled. Correspondingly the complete quantum stress-energy tensors and the complete quantum supercurrent are given by the quantum analogues of Eqs. (VI. 3.149) : n
T(z)
= T(Mink)(Z)
+
.r 1 T(G.
1=
k.}(Z)
(VI. 5. 6a)
l' 1
(VI,5.6b) n
G(z) ; G(Mink)(Z)
+
.Il G(G.,k.}(Z) 1; 1 1
(VI.S.6c)
The addends with the suffix (G.,k.) were defined in Chapter VI.3. As 1 1 stated many times, the addends with the suffix (Mink) can be easily obtained from the above ones by taking the limit where the group G becomes abelian (fABC ~ 0). Let us see the details of this construction which was just sketched in Eq. (VI.3.137a). for Simplicity we set DMink
=D
(VI.S.7)
and as group G we choose T(D). where reD) is the D-dimensional group
1586 of translations in D-dimensional Mlnkowski space; its commuting genera. tors are called tlJ' (U- O,l •••• ,l)...l). The signature is (-,1",+,+, ... ,+).
An element g(z,z) e reD) can be written as follows in terms of the target space coordinates: g(z,z) .. exp(tu XIJ(z,z})
(VI. 5. 8)
and we find
g-1 dg" dgg -1 .. dXIJt
(VI. 5.9)
I.!
The I-form dXll is the analogue of both Eqs. (V1.3.67).
nA
and QA defined in
Its superpartner is a fermion field f(z,z)
and the
analogues of Eqs. (VI.3.71. 72. 74) read as follows: (VI. 5.10a) DXI.! .. D
,IJe+ + D- -fe - - .!.2 a+xll I;
(VI.5.l0b)
+
In the case of an abelian group the Wess-Zumino term is zero and
Hence. the value of the normalization constant k is irrelevant (the factor k/(k +Cv)" 1 for all values of k).
So for later convenience
we arbitrarily choose k - 2 and we iJDJDediately obtain setting fINP - 0 in Sq. (VI. 3. 93). The result is
seD)
(Mink)
=
~ J[(dXU - ~~)(nUet + nl.!e-) 411 ++ 2iw\J~ll
where we have used
A
e+
+
fdxlJ
A
S~~nk)
by
+
~ + n~TI~e+
A
e-]
(VI. 5.11)
n~ in place of nCo
The analogues of the Kac-Moody currents are the momentum fields: (VI.5.12a)
1587 ik
~"
P'-(z) .. - -
2
nl.l .. - i a xlJ -
-
(VI.S.12b)
which, upon quantization, obey the abelian limit of the OPE (VI.3.161), that is, \.1\1 pJl(z)P"(w) .. _n__ (z _w)2
-t
reg.
(VI. 5.13a)
(VI.5.13b)
p\.l(i)P" (w)
1J"V .. _11_-2 +
(z
The OPE of the
-w)
reg.
(VI.5.13C)
1Jf fermions is gi ven by the analogue of Eq.
(VI. 3.183) (with k:: 2) :
(VI. 5. 14)
Finally the stress-energy tensors and the supercurrent have identical expressions in the classical and quantum case, due to k/k + ~ "I, and are given by: (VI.S.lS)
a I
(-)
(Mink) z ..
2'1 p-\.I (z-)-pPc-) z
r7< i1r /4 _.u P G(M1nk)(:z.) .. r 2e 'IT (z)P (:z.)
(VI. 5 .16)
(VI.S.l7)
They fulfill the OPE (Vl.3.189) with the following values of c: c(M1nk) .. D + c(Mink) .. D •
1
3
2 D =2 D
(VI.S.18) (VI.S.19)
1588 Finally we have the addends with the suffix (het). These are the contributions of the Nbet-heterotic fermions, which, by definition, are right-handed Majorana-weyl spinors. Recalling Eq. (VI.3.116) and setting the gauge field AaB to zero we see that the heterotic fermion action is given by
f
1 2i ~;:»'~ Phi P "e S(het) "4;
(VI.S.20)
where. to avoid confusions with later uses of the Greek letters, we have replaced a + p. Recalling also Eqs. (VI.3.1l4) and (VI.3.lIS). in the absence of background gauge fie Ids, we have (VI.S.2I),
The OPE of the corresponding quantum fields is (VI.S.22)
Calculating the stress·energy tensors one finds (VI.5.23) (VI.S.24)
satisfy the superconformal algebra (VI.3.189) with the following values of the central charge:
~nich
(VI.S.2S) ~het_!N
c
- 2 het
(VI. S. 26)
1589
Having defined the building blocks of our theory the next question is: which are the allowed combinations leading to a consistent superstring II\Odel? In other words we would like to know, in every dimension 4 S D S 10, which choices of groups G1 and levels ki are permitted and how many heterotic fermions are required. As anticipated in the introduction to this chapter, the anSlo,'er is provided by the nilpotency of the BRST-charge QBRST which requires cancellation between the conformal anomaly (= central charge) of the ghost-superghost system and that of the matter system (see Chapter VI.4). Indeed, one finds the following implication: Q;RST = 0 ~ _cghost _ csghost
= cMink
+
n
I
= 26
- 11 =
c(G.,k.)
i"l
1
(VI.S.27a)
1
.... _cghost = 26
= cMink
n
+
~ c(G. ,k.) + chet i=1 1 1
(VI.S.27b)
Inserting Eqs. (VI.3.190), (VI.S.18,19) and (VI.5.2S,26) into (VI.S.27) and recalling that single-valuedness of the Wess-Zumino term (or equivalently unitarity of the Kac-~body representations) implies k./~~ e Z (w.1 being the longest root of G.), we see that Eqs. 1 1 1 (VI.S.27) are a system of Diophantine equations: IS
3
= -2 0 + In
26 = 0
(k.
1
'-1 k·1 1-
+
n (
I
+
Cvi
2
(VI .5. 28a)
G. 1
k.) dim G. 1
. 1 k· + Cv 1 i
1=
1) dim
+-
1
+
1 -N h 2
e
t
(VI. 5.28b)
whose complete solution for the case d =4 is displayed in the first two columns of Tables VI.S.I, II, III. In this table much more information is supplied which the reader is requested to disregard for the
1590
time being. Furthermore the normalizations of Cv and k are those explained in the next section. The point we want to streSS here is that the quantum consistency condition Q~RST=O implies more constraints besides Eqs. (VI.5.27). Indeed it fixes the values of the intercepts a and a which appear in the mass-shell equations: (LO - a) Iphys > ,. 0
(VI. S. 29a)
(LO - u) Iphys > = 0
(VI. S. 29b)
LO and Lo appearing in Eqs. (VI.5.29) are defined as the zero modes in the Laurent expansion of the full stress-energy tensors of the matter system: T(z) =
In 2 -zn+2
+'"
(VI. 5. 30a)
-00
~ _
T(z) =
+00
In
L --n+2
-co
(VI. S. 30b)
z
Since the mass-spectrum of the string is read off from Eqs. (VI.S.29), the values of a and a are clearly crucial in determining how many and which type of massless excitations we might construct. Which of them do actually appear in the string spectrum and are not projected out by generalized GSO projectors is decided later by modular invariance of the I-loop amplitudes (see Chapter VI.7). It is therefore of primary importance to make a preliminary study of the int~rcepts and decide for which of the fifteen solutions found in Tables I, II and IIr massless target space ferrnions can exist. The restriction of our attention to such particular solutions can be justified in two ways: i) Common experience tells us that fermions do indeed exist in our low energy world. ii) All indications support the conjecture that target space supersymmetry is necessary to make
1591
superstring theory fully consistent and in particular, finite as a quantum theory. On the other hand, space-time SUSY requires massless gravitinos which are fermion states. Thus motivated towards the study of the intercepts we remark the following. While the conformal anomaly c is a local property of the 2-dimensional field theory so that it depends only on the number and type of fields we consider, the intercepts, which are associated with the trace anomaly, are a manifestation of a global property. In other words, they depend not only on the type of fields we consider but also on their boundary conditions on the world-sheet. This has a series of implications: i) In order to discuss boundary conditions we can no longer restrict our attention to the covering space of the compactified manifold Mconpact' namely, to the target group GT, but we must also specifY the homotopy group ~l(Mcompact)' Indeed, if Mat contains camp c non contractible loops. the string can wrap around them and create new soliton states which, in particular, can be massless fermions. In the case of superstrings propagating on Mcompact whose covering space is a group-manifold GT, it appears that the massless fermionic states are precisely solitons of this type. More specifically we have (VI.S.31)
where B c ~ & GT is some discrete. abelian subgroup of G.r & Gr that will be determined by boundary conditions together with modular invarianCe, We assume in this section that the world-sheet of the closed string is diffeomorphic to a cylinder, represented by the complex plane z with the origin removed. In this case the homology basis is formed by a circle wrapping the origin and the most general boundary condition one can impose on a WZW field is the following (VI.5.32)
1592
where (b,b) e B. In other words the actual target manifold Nt arge t = c;,JB is the set of equivalence classes of GT group elements under the relation: g1'" iz iff g1" bg2 b. The elements of the group B are pairs (b,b) of elements of <>r. If B is such that either b Or b is always the identity element, then Mtarget is an ordinary coset manifold of <>r with respect to a discrete subgroup B c~. As such, it is a smooth manifold. If, on the other hand, both b and b are non trivial elements, then Mtarget is a singular variety. Indeed in this case, the action of B on ~ is not free from fixed points. Now let (y,y) e B be a generator of the homotopy group. From Eqs. (VI.5.32) and (VI.3.67) we obtain (VI.5.33a) (VI. 5. 33b)
so that, recalling Eqs. (VI.3.156), we find (VI.S.34a) (VI.S.34b) where rBA(cr-l)BA) is the matrix representing y(y-l) in the adjoint representation of <>r:
B 'Y -1 t Y "
rBA t A •
(VI. S.3S)
The matrix r decides the type of moding of the ~ac-Moody generators (see Eqs. (VI.l.IS8» and will playa very important role in determining the fermionic mass-spectrum. Here we note that the non-abelian structure of the group ~ induces a natural asymmetry between the left-moving and the right-moving Kac-Moody currents. The asymmetry corresponds to the possibility of choosing y and y independently. Each generator 'Y singles out a cyclic subgroup:
1593 (VI. 5. 36)
where NeZ is the first power of y for which N Y
=1
(VLS.37)
•
The shift in the Kac-Moody current moding will not be lIN, however. Indeed, let N5 N be the first power of y for which yN belongs to the center of GT: y
N e Z def = center of G1
(VI. 5.38)
Then, since all of Z goes into the identity element of the adjoint representation, we find (f)N
=1
,
(VLS.39)
which implies that the eigenvalues of f
N-l N
are
exp(2~ifiw),
with
(VI. 5. 40)
These are the frequency shifts which will appear in the Kat-Moody current moding. They will enter in the calculation of the intercepts and hence play a fundamental role in selecting among the fifteen distinct solutions of Tables I, II and III those which admit massless space-time fermions in their spectrum. To sholq this we must first resume our general discussion of the BRST charge and of the relation between boundary conditions and intercepts. Generally speaking, while considering operator product expansions we do not feel the boundary conditions since the latter affect only the regular parts we disregard. To probe them we must revert to the mode expansions. For instance, from the OPE of Eq. (VI.3.189), which contains only one C-number term (the central charge c), by inserting Eqs. (VI.5.30) we obtain the Virasoro algebra:
1594
[L,L]"'(m~n)L m n n+m +lc2(m3~m)o m+n, 0+2mMn+m, 0
(VI. 5. 41)
which involves two C-numbers: the two-cocycle c and the coboundary b. The reason why b is called a coboundary is that it can be eliminated through a redefinition of the generator LO:
c, instead, is a cocycle because no such redefinition exists by means of which we can eliminate it. Eq. (VI.5.41) is identical with Eq. (VI.3.34) where the notion of coboundary was anticipated. Nevertheless, the coboundary b is physically significant. Indeed, the definition of LO is uniquely fixed for every conformal field by normal ordering and cannot be changed at will. With this understanding each type of conformal field has its own proper value of the cocycle c and of the coboundary b. The coboundary b depends on the boundary conditions, while the cocycle c does not. The ghosts and superghosts make no exception to the rule. Then the nilpotency of the BRST-charge is achieved if and only if a) the cocycle of the ghost and superghosts cancels that of the matter fields (see Eqs. (VI.5.22)) b) the intercepts a and ~ appearing in Eqs. (VI.4.19) are given by
a '"
~ bghost _ bsghost _ bMink -
Q = ~ bghost _ bMink _
r
i .. l
I beG.,k.) ,
i,,1
beG.,k.) • 1
1
(VI. 5.42a)
1
(VI. 5. 42b)
1
In order to discuss Eq. (VI.S.42) we begin by observing that the operator product expansions (VI.3.189) correspond to two different superconformal algebras, the Neveu-Schwarz (NS) and the Ramond (R) algebra, depending on which boundary conditions are imposed on the supercurrent G(z). Introducing a number III whose values are III "
{
0, for NS fermions 1, for R fermions
(VI. 5.43)
1595 we can write the boundary conditions on
G(z)
in the following way: (VI. 5.44)
Eq. (VI.5.44) leads to the mode-expansion below: '" G
I
G(Z)"
n~-w/2.
(VI. 5.45)
n" .... zn+2-{fJ/2
Inserting Eq. (VI.5.4S) into Eq. (VI.3.l89) we find the two superconformal algebras: NS superconformal Algebra
A)
[Lm, Lnl .. (m-n)Ln+m +
~2
(Ill;: 0)
(m 3 -m)°m+n,o +
+ 2mb
(Vl.S.46a)
6
NS n+m,O
ILm' {G
Gn+ltl ;;
m+~'
(t m- n - t)Gm+n+~
G} ;; 2 L n+~
n+m+l
+
.£.3
(VI.5.46b)
(m2 + m)o
m+n+l,O
+
(VI. S.46c)
B)
R superconformal
Algebra Cw" 1)
+
[L • G ] " (-21m- n)G
m n
D\+n
2mb 0
R n+m,O
(vr .5.47a) (VI.S.47b) (VI.5.47c)
1596
The ghost fields have the same moding as the stress-energy tensor. 50 they are always integer moded independently of the boundary conditions:
{cm, cn}
= 0n.m, 0'
{cm• cn} = {cm, cn} = 0 .
(VI. 5.48)
On the other hand the superghosts have the same moding as the supercurrent so that they are integer or half-integer moded depending on the boundary condition: (VLS.49a)
(VI.S.49b) In these conventions the BRST-charges take the following form (see Chapter VI.4, Eq. (VI.4.86»):
Q
BR5T
=
(c
i' L
n=-oo
- -1 2
I r,s
-n Ln
e
G
-n-~(l-w) n+~(l-w)
(r - 5): C c c : -r-s r s
rl n n:s lZ
+ i'
+
+
.!.2
(1- w)]:
) -
+
e-s-~(l-w) en+s+~(l-w) cn'.
- l:
-
(VI. S. 50 a)
n,s
(r-s):c
cc -r-s r s
(VI. 5. SOb)
and they are nilpotent if and only if Eqs. (VI.S.27) and (Vr.5.42) hold true. The cocycles and coboundaries of the ghost and superghost Virasoro algebras have the following values: cghost
- -1, = -26 , bghost -
(VLS .Sla)
1597
cghost = -26 csghost ;; II
. ". ,
b~ghost
-1
bsghost"
1 _~ . 2
(VI.S.Slb) 8
(Vl.S.Slc)
as one can easily check by explicit verification of (VI.S.41) using the definitions ghost Lm
=I
~ghost
= l.\'
Lm
n
-
(m - n)·• cm+ncon'•
(m - n):
n
sghost
Lm
= {.\' [12" m n
(VI.S.S2a)
c-n+mc-n : 1
1-
n - '2 (1 - fIl) ; em+n+~(l_fIl)e -n-~(l-fIl): (VI.S.S2b)
In this way the calculation of the intercepts is entirely reduced to the calculation of the coboundaries b for the matter fields. These numbers depend on the boundary conditions imposed on all the fields entering our theory, namely, P~(z), pP(z), ~~(z). JAez). jA ei ). AA(z) and ;p(z). World-sheet supersymmetry gives constraints on the available choices for these conditions, which must be consistent with Eq. (VI.S.44). Since the translation group is abelian, all its discrete subgroups are normal and we necessarily have (see Eqs. (VI.S.33» (VI.S.S3a) (VI. S.S3b) Then in order that the Minkowski contribution eVI.S.1Sc) to the full supercurrent obeys the boundary condition (Vl.S.44) we must have _LU
!jT
2ri
(z e
11111)"
)" e
WI' (z)
•
(VI.S.S4)
This shows that the world-sheet fermions with space-time indices have always the same boundary conditions as the superghosts. Eqs. (V1.5.S3)
1598 and (VI.5.54) give rise to the following mode expansions: (VI.S.SSaJ (V1.S.55b)
(VI.5.55c) where the modes obey the canonical commutations [ XiJ
i.,)1 nl1\1m6n+m,O
m' n •
(VI. 5.56a) (VI.S.S6b) (VI. 5. 56c)
If we normal order the Minkowski contribution to the stress-energy tensors.with respect to these modes, we obtain the definition of L~ink and I~lnk. We can then easily calculate the coboundaries of the corresponding Virasoro algebra. They are bMink "
bMink
..!!. D 16
" 0•
(VI.5.S7a) (VI.S.S7b)
-het The derivation of b is equally Simple. We divide the set of heterotic fermions into two subsets according to the following partition of the integer Nhet : (VI. 5.58) The first NM fermions are Majorana and admit either NS or R boundary conditions:
1599
• (VI. S. 59 a)
where v
is either one or zero:
p
,/1 (R)
v p
=
(VI. S. 59b)
~O (NS)
The remaining 2ND real fermions combine to form nO complex fermions:
which admit the following more general boundary conditions:
(VI.5.61a)
.
",p( -z e21Tl) ::
-211i9
=e
P ::=p ez) - •
(VI.S.61b)
where E is the'" complex conjugate of _ and Sp is a real number in the range [0, 1:
-r
(VI. 5.62)
The boundary values 9 =0 or 9 = l correspond to ~Iajorana fermions. P
P
2
The above boundary conditions lead to the following mode expansion
(VI. 5.63a)
(VI.5.63b)
1600
(VI. 5.63c)
where the modes obey the canonical commutation rules: (VI. 5 .64a)
(VI.S.64b)
Inserting Eqs. (VI.5.63) into Eq. (VI.S.20b) and normal ordering, we get the definition of i~et. These operators obey the Virasoro algebra (Vl.S.41) with the already utilized value (c= tNhet) of the twococycle and with the following value of the coboundary:
(VI, 5. 65)
To complete the determination of the intercepts we still need to compute the values of the coboundaries b(G.,k.) and b(G.,k.) as functions of 1 1 1 1 the matrix rAB introduced in Eq. (VI.S.34). This calculation is more complicated and involves the notion of twisted Kac-Moody algebra: it will be performed in the next section.
VI.5.S Twisted Kac-Moody Algebras and Massless Target Fermions In this section, we want to determine the dependence of the Virasora algebra coboundaries on the generator y of the homotopy group. The best way to approach this problem is through the use of root formalism for the Lie Algebra GT of the target group. Let G be any of the simple factors in the direct product (Vl.3.63) and let r be its rank. Choose a Cartan basis for the Lie Algebra and let a, B, ... be the root vectors belonging to the root system 4l.
1601
The currents JA(z) are subdivided in the following way: r of them correspond to the Cartan subalgebra generators and are renamed Hi(t) :
(VLS.66) The remaining dim G - rank G correspond to the roots. To each pair of roots (a., -a). where a. > 0, we can associate a pair of real currents JCi.(z), J-a(z), in terms of which we can define the following complex currents: &a (z) :; - 1 [a J (z) - 1. J -a (z) ]
(VI.5.67a)
C-a (t):; - 1 [a J (4)
(VI,5.67b)
12
fi
i J -u] (z)
+
.
Using these variables the OPE (VI.3.161) becomes (VI. 5.68a) . ai a Hl (z) ,f
z-w
+
reg.
(VI. 5.68b)
z-w Ca(z) Cl\w) "reg. ,a(z) .,-a(w)
= _
ai
z-w
(VI.5.68c) (if a + B is not a root)
.
k
1
Hl (w) + - - - + reg. 2 (z _ w) 2
where the normalization constants N(a, B) N(a.
S) :;
-N(B,
al :;
(VI. 5. 68c' J
(VI.5.68d)
fulfill the usual properties:
-N( -a, -8) " N(-u-B , (l)
which suffice to determine them (given the root system).
(VI. 5.69)
1602 It is convenient to perfor. the same change of basis also on the group fermions. Hence we set (VI. 5. 70s)
(VI.5.70b) ~a
'" (z) .. -
1
12
r ex
w(l]
LA (z) + i A (z)
•
(VI. S. 70c)
With these redefinitions, the energy-momentum tensor (Vr.3.178a) and the supercurrent (VI.3.178c) take the following form (VI.5.71a)
(VI.5.71I»
G() z ..
- - e i1rj4 ~ k+ Cy
: {l/hz) 1hz) +
I a>O
(",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
and by a.1 (i .. 1••••• rank G) the simple roots of G. The weight-lattice A is defined as the set of vectors ). which fulfill the condition w 2 ).. (1 e Z (12
Va e ~ .
(VI.S.87)
In particular. the flDldamental weights ). i (1" 1, .... rank G) are dual to the simple roots
Ai. ~ i 2--=6. (1.2 3
(VI.5.88)
j
The irreducible lDlitary representations of G are in one-to-one Correspondence with the highest weights. These are the weights p e).w whose components along the fwdamental weights are non-negative integers: (VI.5,89)
Finally the highest root of the algebra (that is the highest weight of the adjoint representation) is denoted bye. Extending G+ G we gain two more generators which commute among themselves and with the Cartan generators H~. These are the central charge k/2 and the LO operator of the associated Virasoro algebra. This explains why the roots of the Kac-Moody algebras are vectors with two more components with respect to the roots of the corresponding finite Lie Algebra.
1608
We follow Goddard and Olive notations [2J. The step operators era. are associated with the roots (a.,O,n) while (n 1 0) are associate~ with the roots (O,O,n) =n~, g being by definition the following vector:
H!
6 " (0,0,1)
•
(VI.5.90)
The rank of the Kac~Moody algebra G is r + 1 (1''' Tank G) and the r + 1 simple roots are the following:
'"0.0 = (-6,0,1)
(VI.5.91a) (VLS.91b)
They live in a (2 + rank G) -dimensional vector space V on which the following Lorentzian scalar product has been defined:
Vv = (v,q,m) e V
Vu " (u,p,n) e V " U·
v"
= U·
v + pm + nq •
(Vl.S.92)
" have the following form The fundamental weights of G "0
1 2
A ,,(0, '26 ,0)
(VI. 5. 9Sa) (VI. 5. 93b)
where the integers mi are the dual Coxeter numbers defined below: rank G . _6_" lel 2
L
i=l
m1
",i ...
(VI. 5. 94)
lex.IZ 1
In the same way as for finite Lie Algebras, the unitary irreducible representations'of Kac-Moody algebras are in one-to-one correspondence With the highest-weights U whose components along the fundamental weights (VI.S.93) are non-negative integers:
1609
(VI. 5.95) The structure of the vector V defined by Eq. (VI.5.9S) is the following: (VI.S.96) where (VLS.97) is a highest weight of the finite Lie Algebra G and k/2 is the central charge of the Kac-Moody algebra, whose relation with the WessZumino term in the action (VI.3.93) has been extensively discussed. Eqs. (VI.S.9S) are equivalent to the following condition: k = (no + nimi) "jSj2
E
Z
(VI.5.98a)
~::: a·lJ
(VLS.98b)
whose interpretation is very simple. Indeed, the highest weight lJ identifies the irreducible representation of G spanned by the multiplet of vacuum states and Eq". (VI.5.98b) tells us that there is a finite number of choices for this representation at each level k. To be more precise the multiplet of vacuum states is defined by the following conditions:
(VI. 5.99a)
(VI. 5.99b)
gf'!v;(lJ, lk,O) > n
2
,,0
(n > 0)
(VI.5.99c)
1610 (VI.S.99d) where Cg is the quadratic Casimir of G in the irreducible representation {lJ} of highest weight lJ (VI.S.lOO) and
is any weight vector belonging to {lJ}
\I
\lehJ}.
(VI.S.IOl)
Clearly the whole multiplet of vacuum states is generated by the action of loa. (a.> 0) on the highest weight state
IlJ;(ll, tk,O) > which ful-
fills the additional condition (VI.5.102)
Bq. (VI.5.98a) gives the spectrum of permissible k levels and Eq. (VI.5.98b) specifies which vacuum multiplet is allowed for each choice of k.
On the other hand the full tmitary representation of the
Ka~
Moody algebra is identified once lJ is given. In the following we shall be specifically interested in the SU(2) Kac-Moody algebra.
In this case we have r'" 1 and a single positive
(and obviously simple) root which we normalize as follows: a= 12. The fundamental weight is A'" 1/12
and the vector p is given by p'" 12/2.
The weight and root lattices correspond to points and circled points, respectively, in the following picture:
o . A
0
. 0 '-0-' 0
. 0
.
@
(VI. 5.103)
highest weight lJ has then the following form
IJ '" ZJ A '" 2J _1_ '" J
12
12 '"
Ja
(2J
e Z)
J being the isospin (a conventional name for J).
(V1.5.104)
1611
From Eq. (VLS.98) (since the allowed values of J are J
='k4 '
J '"
k
1
'4 - "2 '
J
k
e.. 0:)
= '4 -
we conclude that at level k
1 , ••. , J = 0
(VI. 5.105)
t,
in particular at level k'" 4 we have J =: 1, O. In terms of J the Casimir is Cg ;;; 2J (J 'I- 1) and Eq. (VI. S. 33) can be rewritten as: (VL5.106) Indeed Cv.. 4 for SU(2) as can be easily checked from Eq. (VL5.100). More generally, the normalization of the Casimirs and the root lengths we used in solving Eq. (VI.S.28) are given in TABLE IV.S.IV. Having established these notations and conventions, we come back to the calculation of the coboundary bt • From the commutation relation (VI.5.107)
it follows that if [0 > is a vacuum state, i.e. a highest weight of the Virasoro algebra, characterized by a Virasoro weight h: (VI. S. 108a)
LjO>=O. n
(v!. 5.108b)
then the value of b is immediately given by the following relation: (VI,S .109)
the operators L
n
being defined by the following contour integrals (VI.5.110)
in terms of the normal ordered stress-energY tensor (VI.s.7l),
1612
°
In the case of the untwisted Kac-Moody algebras t" the highest-weight states Iv; ell, ik,O) > 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 <1>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)
is given by
Its meaning is the same as that of Casimir of the subgroup H c: G. Indeed P1 is identified with the p-vector of the H-subgroup and
)Jt
is identified with a highest weight of the H-algebra. Explicit calculation of the nom IIL.llo> 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)<9SU(2)
ISO(S)k=2] <9 U(l)
37
No
SO(6} ® 50(3) 50(5) 50(2)
[5U(3)k:2) <9 [U(1))2
36
No
SU(4} <9 50(3) 5\](3) <9 U(l) 50(2)
[SlI(2)k:i e [U(1))2
34
Yes
50(6) e 50(3) 50(4) <950(2) 50(2)
[5U(2\=12 J 2 eU(lJ
33
Yes
5tJ(4) <9 SO(3l 50(4) 50(2)
Fermionization Group GF • SOCS} fI SU(3) [SU(Z}k=/ fI U(l)
38
No
SU(3j SO(5) fI SU(2) e U(1) SO(4)
(SU(2}k=/ <9 [SU(2)k=20)
37
No
50(5) fI SUCS) 50(4) 50(3)
lSU(2)k=Z] fI [SU(2)k=a]fI U(1)2
34
No
50(5) SU(3) <9 SO(3) fI 50(2) 5U(2) fI U(1}
33
Yes
50(5) <9 SU(3) SO(3) <9 50(2) 50(3)
J
[SU(2)k=SJ e [5U(2)k=20 fI U(I)
1628 TABLE VI.S.!II Non FeTmionizable Group-Manifolds for D=4 Superstrings (dilQ
Target Group = fir
Ii"
> IS)
l\et
Massless i'eTmions
GNO Symmetric Space
[5U(3)l<=2J -ISU(2)k=4]
37
No
_ SUeZ) _ SU(2) SU(4) SU(3)etJ(l) StI(2)
[50(5)k=4]
36
No
(SU(2}k..
SO(7)
5O(S) • 50(2)
SU(3) SU(2) • tI(l)
[SU(2)k=2] • II
for 0< 4
41- [sU(21 k=sl
1SU(J)k=lOJ
35
No
34
No
It
50(3) It 50(2) Sp(6)
sueS] .U(IJ
TASLE VI.S. IV
NQmaliution of Roots and Casimir Invariants GROUl'
CASIMIR
G
Cv
LONGeST JIQOr Ll:1/GfIf 82
SU(n}
2n
2
5p(2n)
4n.4
4
50(2n + 1)
4n·2
2
50(2n)
24
2
E6
24
2
£7
36
2
"8
60
Z
G2
12
.3
•
5O(S)
• 50(4) 50(3)
1629 CHAPTER IV. 6
The Polyakov Path Integral and the Partition Function of String l40dels
VI.6.1 Introduction As pointed out in Chapter VI.3 (Section VI.3.1) the basic idea underlying string theories is that the observed particles are the vibrational modes of the quantum string. Therefore the most iJllt)ortant aspect of a string model is gi ven by the spectrum of its two-dimensional hamiltonian which is to be regarded as a candidate for the mass-spectrum of the world. A convenient tool to study the spectrum is suggested by statistical mechanics and is given by the partition function defined below: Z(S)
1:
Tr exp(- SH)
(VI.6.1)
where S = l/kT is the inverse temperature and H is the quantum hamiltonian of the system. The trace is taken on the Hilbert space of
1630
physical states and the anti-Laplace transform of Z(8) yields the the microcanonical density of such states at a given energy level: +i OO
aCE)
= J . dB eBE -1""
(VI.6.2)
Z(B) •
In string theories, as we shall see, the role of energy is played by the mass squared m2 of the particle associated to a given vibrational mode. Recalling Eqs. (VI.2.B) one sees that the time variable t on the two~ dimensional world-sheet can be identified with the following combination: (*)
t
~
t (ig z
+
~g z)
(VI. 6. 3)
Recalling on the other hand Eq. (VI.2.22b) one realizes that Lo and Lo are the translation generators in the variables ~g z and tg z, respectively. Hence the hamiltonian H, i.e. the time translation generator, is given by (VI. 6. 4) where L and L are the modes of the left-moving and right-moving n n stress-energy tensors of the complete system (see Eq. (VI.S.6)) and where c1 is some suitable constant, representing the vacuum energy. By the same token, the difference (VI.6.5)
is the generator of translations in on the world-sheet.
(*)
0,
that is, in the space-variable
The reason we changed notation and we named t the time variable rather than T is to avoid confusion with the modulus of the torus.
1631
For a closed stTing model the partition function is then given by Z(cl07ed)(8) ..
strlng
rr(cl0 7ed)eXP[*13 strIng
(LO+Lo-c 1)]
(VI. 6.6)
states where we were careful to specify that the trace must be taken on closed string states. This is very crucial for the following reason. In Chapter VI.3 we associated a vacuum of the superstring with a classical superconformal theory which, upon quantization, leads to two Fock spaces: the Fock space ffL of left-moving modes and the Fock space ffR of right-moving modes. Any state in the spectrum can be written as Istate > '" 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
where (VI.6.27) is the mapping class group invariant measure on the moduli-space M g and the partition fUnction ~(i)(tl" .. ,'t 3g _3) is given by (VI.6.28)
the result of the functional integral (Vl.6.2Sb) being a function of both the moduli and the boundary conditions. The full partition function ~(i) (t 1" .. ;1" 3g-3) must be invariant under the transformations of the mapping class group, which on the boundary conditions related to the homology basis acts as Sp(2g,l). This (*)
For simplicity we have numbered the modUli from one to 3g-3 which is correct in all cases except for g=l and g=O as we know from Chapter VI. 2.
1638 requirement yields a set of constraints on the coefficient.s ci[bC] which must be adjoined to a ftmdamental factorization property. To tmderstand this property consider the usual picture (VI.2.S3) of a genus g surface and go to that region of moduli space where (VI.2.3S) degenerates into the following surface
..... ..... ~ .
~
(VI. 6.29) We can arrange the 3g~3 moduli in such a way that the first g of these are the moduli or the g~tori of picture (VI.6.29). while the remaining 2g-3 describe the shape of the necks. There are limiting values r.1 (i=g+l, .... 3g~3) of these moduli for which the surface CVI.6.29) degenerates into the disconnected sum of g tori. the necks becoming so thin that eventually they disappear. In this limit the partition function (VI.6.28) is in general singular. and the residue of the singularity has to be the product of the partition functions of the g tori: (VI. 6.30)
Indeed Eq. (VI.6.30) is the fundamental factorization property of scattering amplitudes. Since the coefficients ci[bc] depend only on boundary conditions but not on the moduli, Eq. (VI.6.30) can be true only if these coefficients are factorized in the follOWing way: g
c.[bcJ'" n c.[(bc)J 1 r=l 1 where (be)
r
are the boundary conditions on the r·th torus.
(V!. 6. 31)
1639
Combining Eq. (VI.6.3l) with the requirements imposed by the Sp(2g.Z) modular invariance we ob~ain the possible sets of coefficients ci[bc] on an arbitrary torus; through Propositions 6.6.1 and 6.6.3 each of them defines a GSO projector and hence a superstring model
(VI. 6. 32) This discussion elucidates the relevance and the implications of Propositions 6.6.1, 6.6.2 and of Theorem 6.6.3. We are left with the task of convincing the reader that they are true. We shall comply with this task in various steps beginning with a heuristic proof of Proposition 6.6.1.
Consider a single scalar field ~(x) of mass Lagrangian :
m2
and its kinetic
(VI. 6. 33)
At the quantum level f will contribute to the effective cosmological constant in the following way (VI.6.34)
On performing the gaussian integration, Eq. (Vl.6.34) can be rewritten as follows: (VI.6.35) To evaluate the functional determinant appearing in (VI.6.35) we utilize the t-fUnction regularization method.
1640 6.6.4 Definition. Given an operator A, the logarithm of its determinant is defined by the following formula
i. !;(A) (5)
lin det A = - lim
(VI.6.36)
s'" 0 ds
where the i;-function of the operator A is defined below. [ 1 I;(A)(s) =1- [ dtt s-1 Trexp(-tA).
res)
(VI.6.37)
0
If {)..}
The argument underlying the above definition is as follows. are the eigenvalues of the operator A, we should set det A = IT )..
(VI. 6.38)
{)..}
which is in general divergent. However if we define the i;-like function i;(A)(s)
=
I:' l
-s
-s
A = Tr(A )
(VI.6.39)
{)..}
we obtain (VI. 6. 40) and hence
I;~(O) = -
L
lin
A= - lin
{A}
II A = - lin det A . fA}
(VI.6.41)
This shows that Eq. (VI.6.36) makes sense with the definition (VI.6.39) of the i;(A)(S) function. On the other hand, this definition coincides with the previous one (VI.6.37) as shown by a Simple change of variables, _l_JOOdttS-lTre-tA=_l_[dttS-l
res)
0
res)
0
L {A}
e- tA
= L _l_f"'dttS-le- tA {A}
res)
0
(V1. 6. 42)
1641
Applying this regularization scheme to Eq. (VI.6.35) we obtain
Aff = lim ..!{_l_ fdttS-l e
s+O ds res)
JdDpexp[-
tel + m2)]J
(VI.6.43)
where p~ is the momentum in the D-dimensional space-time. The function l/r(s) has a simple zero in s =O. Hence if we are willing to ignore the divergence of the t-integral in the extremum t =0, Eq. (VI.6.43) can be rewritten as follows:
idtJD d pexp [-
Aeff '" JOt
2 t(p2 + m)]
(VI.6.44)
The t = 0 divergence of the integral (VI.6.44) is simply a sympton of the ultraviolet divergences plaguing ordinary field theories and requiring regularization. We will see in a moment that the geometric reinterpretation of the integral (VI.6.44) inherent to string theory provides its natural regularization by means of selecting the proper integration domain. To this effect consider now a theory ~nich. instead of a simple bosonic field ~(x) of mass m2 , contains an infinite number of fields 2 9i(x). both bosonic and fermionic with masses ~. Define (VI. 6.45) where dB(m~) and dF(m~) are, respectively the number of bosonic and 1 1 2 fermionic states at mass level m.. 1
Since each field in the spectrum gives a contribution (VI.6.44) to the cosmological constant, counted positive if the field is a boson, and negative if a fermion, the overall one-loop cosmological constant can be wTitten as follows
A ff e
ex
[ -dtJ() dP 0 t
L
00
N=O
2 2 d(~) exp[- t(p
+
2 mN)]
(V1.6.46)
1642 Let us now assume that the spectrum utilized in Eq. (VI.6.46) is that of a superstring. In this case the mass-shell equations (VI.6.47a)
(LO - allen-shell> " 0
(VI.6.47b) imply (-
-1 DN2 + 2
( - -1 m..2 2 N
+
n Cg I -+ Ni=1 k + C
v
(l
)
I - - + N~ - a)
Cg i=1 k + Cv
lon-shell>" 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
..
·0
1 2 _ L(Mink)
-Zp·O
1 2
+Zm
(VI. 6. 51)
1643
while for the group parts J~J~ is the Casimir operator. Using Eqs. (VI.6.48) the mass ~ appearing ~n Sq. (VI.6.46) can be regarded as the eigenvalue of the following mass operator: (VI. 6. 52)
the degeneracy of the eigenvalue being precisely d(~) if we take into account only the closed string states belonging to a consistent Hilbert subspace fic ~ (see Eq. (VI.6.11)). Using the delta function (VI.6.14) to select the closed string states and inserting the GSO projector 1Pi to project onto .#'1' the effective cosmological constant (VI.6.46) takes the form A
eff
a:
i dt Jot
f-
-co
dx 21r
Tr(§"){pi exp[- t(LO + Lo -Il- (X) + ix(LO-
LO - a+ ii)j) (VI. 6.53)
J
where the integration dDp over the eigenvalues of the O-mode oscillator P~ has been included in the symbol Tr (~) denoting the trace over all states in the big Fock space (VI.6.8). Comparing Eq. (VI.6.53) with Eqs. (Vl.6.15) and (VI.6.16a). we see that we can identify the dummy parameter t with the inverse temperature B and hence with the imaginary part of t: t ::
(VI. 6.54)
B :: - 211 Imt
which allows the rewriting of (VI.6.S3) in the following manner: A
eff
::
i
J0
dImt 1111 t
I+ co dRet _1_ ~.(t) _CD
1m t
1
the partition function .'t. (t) being defined by Eq. (VI. 6. 17) • 1
(VI. 6. 55)
1644 In this way we have obtained the desired interpretation of the integral of 1L. (1") on the entire upper complex plane; it is the onel loop contribution to the cosmological constant as stated in Proposition 6.6.1 (see Eq. (VI.6.2D). The integral (Vl.6.S5) is divergent as are the individual integrals (VI.6.44) of which it is the infinite sum. The reason behind the divergence. however, becomes clear only in Eq. (VI.6.55) which naturally suggests the proper regularization. Suppose that the partition function !t.1 (t) happens to be PSL(2,1)-invariant, i.e. it fulfills conditions (VI.6.23). In this case, since the Weil-Petersson integration measure is also PSL(2,1)invariant, the integral (VI.6.5S) diverges for a simple reason: by letting t vary on the whole upper complex plane we take infinitely
many copies of the same finite integral [~i (t) d(Weil-Petersson), evaluated on the fundamental domain ~ of the modular group (see Eqs. (VI.2.246)). Hence, if this is the case the regularization is obvious: we must restrict the integration region of 1" to the fundamental domain a. Why should we expect that sri(t) is PSL(2,1)~invariant? There would be no ground for such an idea if the complex parameter T had no geometrical meaning. Actually T is the very modulus of a world-sheet torus and because of this Eqs. (VI.6.23) are a mandatory consistency requirement stating the invariance of a physical amplitude under the mapping class group. The above assertion follows from Proposition 6.6.2. Let us close this section by giving a heuristic proof of this proposition. Reconsider the trace (VI.6.17) and the definitions (VI.6.4) and (VI.6.S). Note that the operator exp [- 21T 1m T H]
1645 propagates a closed string through imaginary time - 2u 1m 1.
A matrix
element < fiexp(- 21flmt H)ii > 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 <
1,0[1,0 > = < 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
(VI. 6.156)
A key point now is that the norms utilized in defining the gaussian integration measures (VI.6.1S1) are invariant under diffeomorphisms, although they are not invariant under Weyl transformations. Hence we have (VI. 6. 157a)
(VI.6.157b) Now the decomposition (VI.6.148) inserted into the measure (VI.6.151e) yields
1665
J [dt~] g d a exp( -2' II~lt I ) exp( -2' aras Hrs(P» k
• J(l)
1
2
1
(VI.6.158) where we have set
k = dim ker Pl = dim [Aut (Eg)] H (P)
rs
=
f it Ii'
(VI. 6. 159a)
L .. II AS • -aprs
(VI. 6. 159b)"--
Using (VI.6.158) and repeating the argument for the calculation of the Jacobian defined by Eq. (VI.6.152) we can write 1
= J(21f)-k/2 (detH(p»1/2
Id(01:) d(li.)- [dtJ- [dCoX)]-dka' g
g
g
1 ~~~r N21g a 2 'exp(-'2!1liX+t.aX+a Ar·axli -'2i!V't+oq,g+ot. Cl1:g11 :J(2lT)·k/2 (detH(p»1/2 jd(OT)d(OI/l)_ g
[dtJ-g
>=
[d(OXJ]_dkaexp(--zl Q) g
(VI.6.160)
Let us calculate the gaussian integral. We define two new objects:
x(r)
as
a - --g I - -yo a as 2 as g -at g r yo
(VI.6.161a)
=-g ilt r
(VI.6.1610) and we obtain
III"/{a t S} +
o$iaS
+
Ot~ a~ iaSII2 = 1101/1' g 112 r
+
!Iohll 2 (VI.6.162)
1666 where (VI.6.163a) (VI. 6. 163b) the vector bs being defined by the equation which characterizes the conformal Killing vectors: (VI.6.164)
and the operator PI being defined, as usual, by Sq. (VI.2.92). Hence, using the following 5-row vector notation
(VI. 6.165)
and introducing the matrix 1 l'rVa.,Sr' 0, bs 0, , 0, 0, 0 A"
0,
0 , 1 , 0, 0
0, 0(1!' 0,
o,
(VI. 6.166)
1, 0
0 , 0, 0, 1
whose determinant is one det A '" 1
(VI.6.167)
we have the following expression for the quadratic form Q appearing in the exponential of Eq. (VI.6.160):
1667 (VI. 6.168a)
v'
= Av
(VI. 6. 168b)
where the matrix M has the following block diagonal form
M '"
(
1, 0, 0) 0,
if,
02
0,
0 ,
W
(VI.6.169a)
with
Pl1
J ..
('
X~rll
PI Xes)
t •
)
(VI. 6. 169b)
X~r) X(s) JJ
w2 '"
( 1
w~)
"['J t]J
)
(VI.6.169c)
JJ
W(r) tIl(s)
In Eqs. (VI.6.169), X~~) is to be regarded as an operator which acting on dr converts it into the traceless metric deformation: s (VI.6.170)
t The operator Xes) is determined from the scalar product
<
t s h • Xes) dr (s) > '" X(s)h dT ... (VI. 6 .171)
1668 Hence we have
(VI.6.172) In a completely similar way w~(~) is the operator which acting on as transforms it into an embedding variation 6XU(~). We have (VI.6.172a) (VI.6.172b) Inserting Eqs. (VI.6.168) and (VI.6.169) into (VI.6.160) we obtain the expression of the Jacobian J
det U2 det \'12
=
(VI.6.173)
det Hep)
where the numerical factors have been dropped. To calculate det u2 we diagonaUze the matrix u2 • The traceless metric deformation was written as in Eq. (Vl.6.163) but it also admits the decomposition (VI.6.174) where ~~) is a basis for the space of quadratic differentials. that t is, for ker Pl' Comparing Eq. (VI.6.174) and (VI.6.163) we obtain ljIr 6T , = P1(t-t') +
r
l6-r
r
t and, introducing the projection operat?r on ker Pl'
(VI. 6.175)
1669 (VI. 6.176)
Defining the matrix (VI. 6. 177)
we <:an write
(VI. 6. 178)
where (VI.6.179) Obvi ous 1 y we find
det
~
= det Tsr •
(VI, 6. 180)
Now we observe that calling
(VI.6.181)
we have
(VI.6.182)
and since
1670
I r,,1'1 ~l we
(VI. 6. 183)
can write {VI.6.184}
Now
I~p
>
H~l pq
<
~q I is the identity operator in the space ker ptl
while Pl(P!Pl)-lPi is the identity operator in (ker Pill. we have (VI. 6. 185)
from which we obtain (VI.6.186)
where, as usual det I denotes the determinant where the O-mode contribution has been omitted. This result, inserted into Sq. (VI.6.173), allows the following transcription of the original path integral (VI .6.104):
where .9"(T) ..
I I lg dX
detWe-S(i,X] ..
. f [dxl g detWexp[-}< X,flgX "I·
(VI. 6. 188)
A very similar integral was already defined in Eq. (VI.6.118) and calculated in Eqs. (VI.6.120, 121. 122). The only difference resides in
1671 the restriction to the slice X transverse to the action of Aut (kg) and in the presence of the determtnant of the matrix W. In the Appendix to this chapter we show that (VI.6.189) where v(.) is the volUlIle of the finite group Aut (kg):
Vet)
=I
~
1=1
da. 1
= Vol
(Aut
(E
g
»
(VI.6.190)
so that utilizing the result (VI.6.122) we can finally write
fI =
I
n det < ljiSI Xl' > 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+
I'" (V1.6.270)
n",l
Comparing Eq. (VI.6.270) with Eq. (VI.6.236c) we conclude that
~
det
=
(VI. 6.271)
[6] In a similar way, we show from Eq. (Vl.6.269) that for the other even spin structures the following is true:
e[:](OIT)
12
(VI. 6. 272)
n(T)
Case B: Odd SEin Structure In the case of the odd spin structu~e the calculation of the ~ function (VI.6.228) is performed in a manner identical to the previous case except for one particular. In order to use the Jacobi inversion formula (VI.6.254) it is necessary that all the summations go over all integer numbers.
Hence before applying the identity (VI.6.254) the integrand function u[l,l] is rewritten as follows:
1688
2
+00
l
x
m exp[ .1fy(m -2mnReT)]
+
·00
2
+00
Lm exp(-1fym].
+
(VI. 6. 273)
-00
The first term in (VI.6.273) contributes to
[1
~~'
II' (s)
an addend
_1_(Imr)2S p[l,l] (S,T)
res) zliT which can be treated as in the previous case. The second term gives an additional addend:
1 (IffiT)2S[ dyys-l \'
Q(s;r) '" - - - -
res)
l.
211T
[ 2]
exp -l1ym
(VI.6.274)
m/O
0
By a change of the integration variable we obtain
Q(S,l)
= L ( -ImT)2S
2-5 1 (m) -
res)
m/O 211T
r
dww5-1 exp(-w)
= 2 [ -Clm't)2]S ; , ; - l;(2s}
(VI.6.275)
Then we calculate the limit
Sl!~ d~
2[ cr::)z ]\(2S) = • tn(Im e)2
+
= Un [ (1::)2
]~(O)
const .
= (VI. 6. 276)
USing this result we can write
fi]
- ...nn det'
IJI P
= - ",n (1 me)2 •
+
sl';mo p[l,I] (5,"') '
(VI.6.277)
1689'
where in full analogy to the previous calculations we have lim s+ 0
l
+
l
exp{ - 2tri mRu - 2n
Imt Imllnl J ..
( mezt fnelt m~
..
w
01 ln~ 01
Ron InCr)!
4
•
(Vl.6.278)
Combining Eq. (VI.6.27S) with Eqs. (VI.6.277) and (VI.6.2S0) we finally get (VI.6.279)
which is the promised proof of Eq. (VI.6.220). Using this identity the results we have obtained for the Dirac determinant on the torus can also be rewritten as:
det il
l:1
(&[:1(0\10)1/2 (
Ii detlmnY/4
2
(VI.6.280)
(det I A) 1/4
where II .. 't is the period matrix. Various arguments in algebraic geometry together with the discussion of Section VI.2.S suggest that Eq. (VI.6.280) should be the correct answer for det
~[bl
not only on
the torus but also on higher genus Riemann surfaces. Equation (VI,6.280) is very important for the following reason: as it appears from the explicit formulae we obtained for the case of the torus, the function
1690
(VI.6.281)
is analytic in the moduli. Recalling the structure (VI.6.230) of the Dirac operator we can rewrite Sq. (VI.6.280) as follows: (VI.6.282) This is a case of a general phenomenon known as holoDlOrphic factorization. Given an operator A which maps a nolomorphic bundle B on the Riemann surface into another bundle 8' and given its adjoint (VI. 6. 283a)
A : B .... 8'
(VI. 6. 283b)
we find that (VI. 6.284)
where geT) is an analytic function of the moduli. When this happens we are entitied to define the chiral determinant of the operator A by setting (VI. 6. 285)
det A " get)
In our case we can write
det
(-2i VA) '"
[;1
z
a]
ef (oln) ( _ _ _.....J...,b'-''--_ _ __
(J Ii det 1m n) -1/2 (det' A) 1/2
)1/2 (VI. 6. 286)
1691
which on the torus reduces to
(-2i ()_) =
det [ :]
Z
z[ a] (T) '~
:] (olt) (VI. 6. 287a)
neT)
b
:] (olt) (V!. 6. 28Th)
n*(t)
The formulae (VI.6.287) admit an operatorial verification. Let A(z) be a Majorana-Weyl world-sheet spinol'. Its space-like boundary conditions are specified by a number a e Z2: (VI. 6. 288)
the mode expansion of A(z) depends on the value of a (which decides whether A(z) is a Ramona or a Neveu-Schwarz spinol' (see Chapter VI.S» and the corresponding normal ordered L~ operator reads
r
1.... 1 I a =Lo [n+-+-(1-a)1'A A 2 n=-oo 2 2 • ~n-~(l-a) n+~(l-a)
(VI.6.289)
Apart from L~ one can also define the fermion number operator a 1 +00 F =- ~'II
(VI. 6. 290)
A
2 n;-~' -n-~(l-a) n+~(l-a)
In terms of this operator one finds
Tr exp(21Ji
[(L~-..!..+ a2 )t+ ~Fa]} .. 48
16
2
(e[:](0It)jI2 net)
(VI.6.291)
where the parameter b is also a z2-element (0,1). To show Eq. (VI.6.291) it suffices to recall the infinite product representations (VI.6.236) of the thetas. Eq. (VI.6.291) shows the identity between the
1692
operatorial and the functional definition of the partition function also at the level of the fermion contributions. Furthermore we see that, while the boundary conditions on the a-cycles correspond to the choice of either Ramond or Neveu-Schwarz moding, those on the b-cycles corresFa pond to the insertion of either a factor 1 or a factor (-) in the operatorial trace. Combinations of the following type (VI.6.292)
are projection operators in the Foet space. This is the logical link connecting GSO projection operators and the summation on different spin structures required by multiloop modular invarience. VI.6.7 The gravitino ghost In the case of the heterotic superstrings the cosmological constant can be written as follows:
Aa
I d(Weil-Petersson) ~(tl,···,tn)
(VI.6.293)
where the partition function is a product of several terms: n !'-Z(D).Z(D). II Z - (X) (~) ial (Gil
·z
(het)
·z
(gravltino)
(VI.6.294)
Let us discuss the various factors: Z(G.) is the partition function associated to the supersymmetric 1
WZW-model of the group G1.. It will be discussed at length in the following chapter. z~X) was defined in Eq. (VI.6.192b) and, on the torus, it takes the explicit form (VI.6.213).
zi~~ is the determinant of the chiral Dirac-operator acting on ~, the 2-dimensional fermions endowed with space-time indices.
view of the results explained in the previous section we can write
In
t693
D/2 (VI.6.295)
(J Ig det 1m IT) -1/2 (det' Il) 1/2
where [;, J is the spin structure of the 1jJJl-fermions. Similarly Z(het), which is the partition function of the heterotic fermions, can be written as follows: *'1/2
(f Ii det 1m IT) -1/2 (det' Il) 1/2 where [::]
}
is the spin structure of the i-th fermion field
('1.6. 296)
~i.
Finally, Z( . . ) is due to the functional integration on gravltlno the gravitino field ~ a (z,i). This contribution we shall discuss verY briefly. As it happens for the metric the gravitino ~ can be gauged a away in any local chart by use of the invariances against local supersymmetries and super-Weyl transformations (see Eqs. (VI.3.54». This operation leads to a Faddeev-Popov determinant which is represented by the integration on the superghost and the antisuperghost fields. The corresponding contribution to the Lo operator is read off from Eq. (VI. 5. 52c)
L~sghost) = I n
[n +
t
(1- w)] :
en+lz(l-w) e -n-~(l-w)
(VI.6.297)
Defining the superghost number operator F(sghost)
=I e n+~(l-w) n
the operatorial trace
E
-n-lz(l-w)
(VI.6.298)
1694
(VI.6.299) is immediately evaluated and it reads W
J
Zgravitino [ 00' (L)
=
IS[:, (OIT) ]-1 J
(VI.6.300)
neT)
one sees the gravitino contribution cancels against the contributions of the two Majorana fermions with the same spin structure [:,]. These are two of the D-components of -f. Indeed, the fermions with space-time indices have necessarily the same spin structure as the gravitino since the coupling term
As
appearing in the action (VI.5.11) must transform as a cross section of the btmdle K- I e K- 1 (it is a 2-form on the Riemann surface). The same conclusion can be reached also by looking at the'supercurrent (VI.S.17) whose spin structure is that of the superghost system to which it couples in the BRST-charge (VI.5.S0). The exact cancellation of Z with Z(longitudinal) gravitino (w)
z
I W1Z(longitudinal) f W] '"
gravitino 1w'
(l/I)
w'
1
(VI. 6. 301)
is what allows the use of a light-cone formalism where only (D-2)X~ and (0-2) ~ are introduced. Eq. (VI.6.301), however. is true only on the torUS. On higher genus surfaces there are additional complications. First of all. in the same way as it happens for the metric, the gravitino can be gauged away only locally. Globally there exists a finite number of i-differentials
1695
which cannot be obtained as the result of a local supersymmetry. These
differentials span a linear space analogous to the Teichmlliler sp~ce: the space of supermoduli. Riemann-Roch theorem illll1\ediately yields its dimensionality. Setting q .. 3/2 in Eq. (VI. 2.329) and recalling Eqs. (VI.2.3IS) and (VI.2.102), we obtain • supermoduli 3
= dim
" (22" - 1) (g - 1)
H(3t 2) •
+
r(Z
1/2
)" 2g - 2 •
(VI. 6. 302)
If ~i is a basis of H3/2, the orbit space of two-dimensional geometries is parametri~ed not only by the metric gaS (t 1 ••.. ,tn> but also by (VI. 6. 303) where ai are anticommutlng parameters. The functional integral reduces to an integral on the moduli '[ 1' •••• T and the supermoduli Ql ••••• a2g- 2• The latter integration, however~ can be explicitly performed and the final partition function reduces to the expression (VI. 6. 294) where (VI. 6. 304)
is a function of the moduli encompassing both the Faddeev-Popov determinant of supersymmetry and the integration on supermoduli. The explicit form of the function (VI.6.304) is still unknown at the time of writing although some progress has been reported [1]. In particular, the spin Structure dependence of the left hand side of Eq. (VI.6.30l) has been clarified. It equals one only on the torus. Let us define
1696
...
...
z [ W J zlongitudinal [ ...OJ gravitino;:;, (!jJ) w'
]
(VI.6.305)
On genus g =1 we have
x[~I=x[~l=x(~]=x[~]
1.
(VI. 6.306)
On higher genuses it is found that the correct generalization of the identity (VI.6.306) is not
x[ ~] but rather
(VI. 6. 307)
1
[l,2J
(VI.6.308)
Eq. (VI.6.3GB), as we will see in the next chapter, ensures the correct relation between spin-statistics and modular invariance.
References for Chapter VI.6
[1] [2]
E.
Verlinde and H. Verlinde, Nucl. Phys. B288 (1987) 357. D. Arnandon, C.P. Bachas, V. Rivasseau, P. Vegreville, "On the vanishing of the cosmological constant in four dimensional superst ring models" (87)
1697 APPENDIX VI.6.A
A Detailed Treatment of the Conformal Killing Vectors We shall discuss the last integral over the embeddings (VI.6.Al)
where the integration is perfomed over a slice of the embedding space
I. which is required to be transversal to the action of the "conformal (0) is such thl:1t a conformal Killing Killing transformations" Gt .
x
transformation never leaves it lUlchanged.
Let us call j e G
the
t
diffeomorphisms of the finite dimensional group: we have
This means that the points of the space of embeddings can be written as (VI.6.A2)
Using the invariance of the classical action under both diffeomorphisms and Weyl transformations we have (VI.6.A3)
where we have established that . - () _ () 0(0) i.g . aB a : gaB r ~
(VI.6.A4)
Now if we integrate over all the conformal Killing group we get
1698
(VI.6.A5)
where dWT is the measure on the conformal Killing group dw
"(
=
k i II da
(VI. 6.A6)
i=l
and vCr) is its volume
vCr) '"
J nk
. dal
•
i=1
Let us now consider the normalization equation
1
= J d[6x]g
exp(-
t lto)(II~)
(VI.6.A7)
and parametrize )( ,. j*:t. . This implies
6x", j*
and the Jacobian of the transformation comes out by substituting
(VI.6.A8)
and using coordinate invariance of the norm we write
(VI.6.A9)
1699 It remains to observe that the quadratic form in the exponent precisely defines the matrix
w2
previously introduced in Eq. (VI.6.169c) so that (VI.6.AlO)
J " det W . We can write
[dx]g = (da) [di] .-1_ )
where j
(VI. 6. All)
det W
g
is the transformation which brings the point x to the slice
i. Instead of Eq. (VI.6.A2). we have (VI.6.A12)
Then we can rewrite Eq. (VI.6.AS) in the following way:
W(T.+) " __1__ J [dx]_ e-S[g.x] . VeT)
(VI.6.A13)
g
To obtain this we just have to show that the measure (VI.6.A14)
is invariant under conformal Killing transformation.
We do this in two
steps: A)
Let
p
n
be the eigenfunctions of the Laplacian (VI.6.A15)
and let A
IIU1
[j1
be the matrix which represents the transformation j
on these eigenfunctions:
1700 (VI.6.A16)
The embedding oxlJ(a) can be expanded as (VI.6.A17)
Then, given the norm (VI.6.A18)
by
coordinate invariance we have
IIcSxll~*g "
J;g
ia(j;l 6xlJ j;l oxll )
= rl AT (i-I) A(j)a
(VI.6.A19)
so that the measure differs by [d(ox)]. _ J g
B)
= (det A) -1 [dox] _
Theorem for j e
g
•
(VI. 6.A20)
Gr
(det A) 2 ,,1.
(VI.6.A21)
Proof of the Theorem. Defining the conformal factor
pea)
associated to j E Gr
by
(VI. 6 .A22)
the scalar product (A,B)
=
Jia,l Me) A(a)B{
(VI. 6.A23)
1701
and
i*A(
(VI.6.A24)
we get (VI. 6. A2S)
Furthermore, from (VI.6.A26)
which can be rewritten (VI. 6.A27)
we get AnAnm (j) =. AnAmn U-11
(no summation on m and n)
(VI.6.A28)
which means (VI. 6. Al9a)
or T
A AA = A ... (det A)
2
=1 .
(Vr.6.A30)
1702 Olapter VI. 7
MOQULAR INVARIANCE, FERMIONIZATION AND THE PARTICLE SPECTRUM OF HETEROTIC SUPERSTRINGS
VI. 7.1 Introduction In this chapter we impose the condition of modular invariance on the multiloop partition function. By so doing we obtain a set of equa· tions which must be satisfied by the coefficients (VI.6.3l) weighing the different sets of boundary conditions. Each solution of these equations corresponds to a viable heterotic superstring vacuum characterized by a GSa projection operator and a well·defined particle spectrum. The massless sector of this spectrum yields the field content of a supergravity theory coupled to a convenient set of matter multiplets. This supergravity plays the role of an effec· tive theory for the superstring model under consideration. The tools to investigate the structure of the effective theory will be discussed in Chapter VI.8 and in Chapter VI.9; here we confine our· selves to the most basic information, that is, to the spectrum. By modular invariance we mean the invariance of the full partition function (see Eqs. (VI.6.26) and (VI.6.28»):
1703 (V1.7.1)
against the action of the symplectic modular group Sp(2g,Z) . We recall that Sp{2g,Z) is the homomorphic image of the upping Class group on the homology basis which is utilized to fix the boundary conditions. Clearly. in order to impose modular invariance, we need the modular transfomation rules of the partition functions ff(bC) (r 1" •• , r 3g-3) calculated by functionally integrating on the fields with fixed boundary conditions (be]. These transformation rules are easily obtained from Eqs. (VI.2.429) and (VI.2.430) if the functions ff(bc)(T i ) depend on the boundary conditions only through Riemann theta functions. This situation is realized in ten-dimensional superstrings where the only matter fields are the Minkowski space coordinates x~, their spinor superpartners ¢P and the 32 heterotic fermions needed to cancel the conformal anomaly. The corresponding one-loop partition function reads as fOllows:
" (1m rf4
In(T) 1-16
(et:. ]
~ (6 b~
a. ]
(OIT»)4 n(r) i=1
(olt) *
)1/2
n*(r)
(VL7.2)
where [:,] is the spin structure of the
f
-fermions, while [
the spin structure of the i-th heterotic fermion.
:~]
is
Introducing the symbol
(VI. 7.3)
1704
the ten-dimensional partition function can be rewritten as a particular case of the general form
(VI. 7.4)
where
o = 10
= 8 = D-2 M = 32 N
(VI. 7.5)
(VI. 7.6)
6.7.1 Definition. We say that a superstring model is fermionic or, more preCisely, that it is fully fermionized, if its one-loop partition function can be written in the form (VI.7.4) for suitable choices of the positive integers D, N and M. As we are going to see in Section VI.7.S, for fermionic strings the problem of modular invariance can be completely solved by reduction to a fairly simple set of algebraic rules determining the coefficients
that weigh each of'the terms (VI.7.4). As we have seen, the tendimensional superstrings are fermionic, but the four-dimensional ones, apparently, are not. This is due to the presence of the factor ZCGi) (see Eqs. (VI.6.224)) associated to the WZW fields.
1705
Actually, in some very interesting papers published recently [1,2,SJ it was observed that a fermionic description of D= 4 superstrings should exist, for which one chooses D" 4
N= 0 - 2
+
18
= 20
(VI. 7.7)
M" 44
The argument underlying Eq. (VI.7.7) is the following. First one notices that Eqs. (VI.S.2S) for the conformal anomaly cancellation can be solved using only fermion fields by setting 15 "
f I D+
NLeft => 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<1].
.. e
J:( )
(VI. 7. 32)
i Z
where ai e Z2'
If we compare Eqs. (VI.7.32) with Eq. (VI.s.34a) we see that eLi .. 0 corresponds to the choice
r"
1 for the i th SU(2) group.
On
the other hand, a." 1 implies that for that particular 5U(2) we have chosen
r .. e 3,
1
where
63
e 50(3)
is the following matrix:
o -1
o
(VI. 7.33)
1716
Clearly, instead of e3 we could equivalently choose r '" ~ 1 or r", eZ' having defined:
o -1
~
o
-1
) ; e
2= (~1 0 ~) 0
0
(VI. 7.34)
-1
Of course, with these choices, the untwisted current is, instead of J3, either Jl or Note that {e 1,eZ,e 3,1} form a ZZ® Zz subgroup of SO(3). In the complete theory the full homotopy group S c SU(2)3 ® SU(Z)3 is generated by a set of elements (y,y) e B where
i.
(VI. 7.35a) (VI. 7. 35b)
e. e SU(2).
e,. e SU(2).
J
Ij
so that the boundary conditions of the l(~ /iTi) J
=
(VI. 7. 3Se)
J
lj
Kac-~loody
currents are
A B
(e·)sJ·(z)
(VI. 7. 36a)
~B j~(z e21ri ) = (e.,. ) AB J.(z)
(VI. 7. 36b)
J
Ij
lj
]
J
the matrices e. being defined by Eqs. (VI.7.33-34). 1
From Eq. (VI.7.30), we see that the three A~(Z), which multiply 1 the three periodic Kat-Moody currents, must have the same spin structure
f:, J
to
as rp)J and the superghosts. This explains the power S/Z assigned
e[; ]/n
in·Eq. (V1.7.29).
corresponding to the superghost
Indeed, 5=4+3-2, the subtraction contr~bution.
1717
Consider now the six A: which multiply those Kat-Moody currents which are possibly twisted. Let a1 be the space-like boundary condition already defined on the pair of twisted currents J~(Z) relative to the i th SU(2). and 8i be the time-like boundary condition on the same currents (which has still to be defined); then the corresponding pair of AiA(z) will have spin structure [11)w' -~] -8i • This explains the three factors
e[:,:~! ]/n
in Eq. (VI. 7.29a).
a
The next item in the same equation is BJ
{B] (T).
By this symbol
we denote the boson partition function associated to the k=4 SU(2) KacMoody algebra with spin structure [;] and vacuum state in the J representation of SU(2). As shown in Chapter VI.S the allowed values of J are 0, 1/2 and 1. We will define the B-functions and their modular transformation properties in a moment. Before doing that we want to complete the scanning of Eqs. (VI.7.29) by considering also the right partition function. rn this sector there are no A-fermions and hence we have only the BJ j
[:~]
factors where the spin structure is
determined by the y generators of Eq. (VI.7.3Sb). Finally, the 35 square-roots of the theta functions
e[~l
are associated with the 35
heterotic fermions. Before imposing modular invariance there are no a priori constraints on their spin structure.
A; ]
Let us now go OYer to the discussion of the B
partition
functions. The proper mathematical concept we need for their description is that of the characters of a Kac-Moody algebra [6,7]. Let A(lJ) be an irreducible module. namely. an irreducible unitary representation associated to a highest weight state lJ (see Eqs. (VI.5.96). (VI.S.97) and (VI. 5.98)) and let p.., tk, - n) =Ae A(p) be a typical weight belonging to the module. The grade n is the energy label, i.e. the eigenvalue of the number operator /oJ ..
La -
~.
Cli+2p)
--:--~ k+C
v
(VI. 7. 37)
1718
In the untwisted Kac-Moody (rank G)-component complex another complex parameter. A{U) is defined to be the
algebras we have n E N. Let now ! be a vector. 1 a complex parameter, and t The character of the irreducible module following complex function of !, T and t:
where JIlUlt A(A) denotes the degeneracy of the weight A. Le., the dimension of the eigensubspace of A belonging to the eigenvalues ~k, • n) of the commuting operators (Ho• ~k, - M).
ex'
IdentifYing the parameter T in Eq. (VI.7.38) with the modulus of the world-sheet torus, it makes sense to study the transformation properties of the characters under the modular group. Indeed Kac and Petersson [6] have shown that the following objects transform as modular forms: (VI. 7.39)
where q is given by Eq. (YI.7.24) and Su is the so-called "modular anomaly": 1- {+Jj. (II .... s)l =+ 2p) k+C V
- -k
24
dim GJ •
(VI. 7.40)
..
The functions BJj(Z,T,t) can be easily rewritten as operatorial traces: -~~
Bu = q
k + Cy
24
.~+ blkt (La 2in.no ) e TrA(II) q e
(VI. 7.41)
From Eq. (VI.7.41) one sees that BIl (0.1.0) is a q-power times the partition function the representation S and T defined B~(t,T.t) are the
of an untwisted Kac-MDody algebra corresponding to ... where the vacuWll has weight U. Under the generators in Eq. (VI.2.244) the transformation rules of the following:
1719
(VI. 7. 42a) (VI. 7.42b)
~'
where the summation range for
is given by the set of the highest
weights allowed at level k. SV~I and T~~, are unitary matrices which have been constructed by Kac and Petersson for several levels of several algebras.
In the level k=4 SU(2) case p is identified with the isospin J which can take the values J =I, 1/2 and O. Then Eq. (VI. 7.41) becomes (VI. 7.43)
where we have identified the Cartan subalgebra generator HO with 12 J~. This corresponds to the choice r = e 3 . Clearly equally good identifications are flO = 12J~ or 110 =12 J~ corresponding to r" e 1 or r" e2 • In this particular case the matrices SJJI and TJJ • are respectively given by SJJI "
A
sin{ ~ (2J + 1)(2J'
(VI. 7. 44a)
+ IJ)
and
. f (2J + 1)2
TJJ1 :: 0JJ' exp {llTL
8
-
1
1}
(VI. 7.44b)
4" j . 1
Ordering the J values in the following sequence ('2,0,1) one can explicitly write
SJJ' '"
t
(
0
12
12
1
-12
-,(2) 1 1
(VI. 7. 45a)
1720
o e -1'rr/8
o
,,:.J
(VI. 7. 4Sb)
Relying on Eqs. (Vr.7.42) we are able to introduce a consistent definition of spin structure for SU(2) group bosons.
We just draw an analogy
between characters and theta-functions and regard the function BJ(z;r,t)
as the analogue of the theta function without characteristic. Then the BJ-function with characteristic
[;]
is naturally given by the ana-
logue of Eqs. (VI. 2.426) and (VI.6.235):
\'lith this definition, the transformation properties of the "partition
functions"
(VI. 7.47)
are given as
(VI. 7. 48a)
(VI.7.48b)
where the action of S and T on T is defined by Eqs. (VI.2.244), while their action on
( ~] p
is the following:
(VI. 7.49a)
1721
(VI. 7.49b)
Eqs. (VI.7.49) hold true if a2 , 82 are either zero or one. As we see the transformation of the bosonic characteristic is very similar to that of the fermionic: ones but there are two important differences. The first is apparent from Sq. (VI.7.49b), which is to be compared with the Ttransformation of a theta-function characteristic, (VI. 7.50) The second difference is related to the minus Sign in Eq. (VI.7.49a). Also in the case of theta functions the outcome of an S-transformation is given by (VI.7.49a). In that case, however, we have
sf ~a ]
(l) "
e[:]
el) ,
(VI. 7 .51)
so that the minus sign becomes irrelevant and we can write (VI. 7.52)
In this way the spin structures are defined over l2' The reduction to l2 elements is possible also for the bosonic spin structures, but instead of (VI.7.S1) we must use the identities (VI.7.53a)
(VI. 7.S3b)
which easily follow from the explicit transcription of the BJ [;] functions as operatorial traces:
(VI. 7.548)
(VI. 7.541:»
[01
m·
(1
BJ 1 = BJ -
1
)
T• O
- i6
.. q
I
L
TrJ q
J3. J3
0 - 0 111 0 I
e
I
(VI. 7.54c)
(VI. 7. 54d)
Incidentally Eqs. (VI.7.54) are the very justification for our definition (VI.7.46) of the bosonic spin structure. Indeed. if we recall .... 1 t Eqs. (VI,5.84) we see that for t" the Lo operator of the twisted algebra is related to LO of the untwisted. one by
m'
(VI. 7.55)
Hence apart from a prefactor which is given by a power of the q-variable, the function BJ [~ J iS the partition function of the twisted algebra. O 0] is the partition function of the untwisted in the same way as algebra.
BA
The miracle which links BRST invariance with modular invariance. i.e., the cancellation of local conformal anomalies with the cancellation of global anomalies, appears at the level of the above-mentioned q-power prefactors. Indeed the partition function was written in the form (VI.7.29) since, by using Eqs. (VI.5.123) for the intercepts, we Io"ere able to cancel all powers of q obtained by replacing the trace of the evolution operators (Trq La) for all the fields with either theta functions or BJ-flttlctions.
1723 Finally we note that. once BJ [ ~ 1 was singled out, the definition of the remaining partition funct~ons was no longer our privilege, but rather was determined by applying modular transformations. In this way we discovered that the role of fermion number is played by the third component of the isospin. Indeed the factor (_lJ F of Eq. (VI.6.29l) _ 1S
_
J3
substltuted here by (-1) 0
At this pOint, by means of the above formulae, we have established the modular transformations of all the items appearing in the partition function (VI.7.28). We should now devise rules to construct modular invariants at one loop to begin with, and at higher loops later. As mentioned above we solve this problem by fermionization, i.e. by reducing both Eqs. (VI.7.29) to a product of Riemann theta functions. In this way we will make contact with the work of Refs. [2,3] and use the rules explained there to write dOl-In modular invariants. On the other hand we shall gain extra constraints on allowed invariants by demanding that one should always be able to go back and rewrite the modular invariant partition function in the language of Eqs. (VI.7.29). The fermionization procedure is based on the realization of the k=4 SU(2) KM~algebra by means of "fake" fermions l(z) transforming in the adjoint representation (A" 1,2,3). The currents fulfilling the OPE (VI. 7.31) can be written as the following normal ordered bilinears: (VI. 7.56) where 0o 00 denotes normal ordering with respect to the modes of the fermionic field XA(z). We can introduce boundary conditions on XA(z) by writing, as usual, (VI. 7.57) where wA e 71.2' Defining next the contraction function
ll(wlz,w)
= - -i - 1
4z-w
\ (l-w)
z+w) ~ 2"zw
+ -W
(VI. 7.58)
1724
we obtain o ABo XA (z)xB (w)" oX (z)X (w) 0
where ~ '" {wI
,i ,w3}
+
hAS (w.... 1z,w)
(VI. 7. 59)
and the 2-point function hAB is
hAB (w.... 1z,w) '" 0ABA(wBlz,w}
(no
sum
on
B) •
(VI. 7.60)
Considering next the I(.\t-currents JA(z), we see that their boundary conditions are fixed by those of the fake fermions. Indeed:
(no sum on A) • (VI.7.61)
Hence if all the wA are equal we have an untwisted Kac-Moody algebra. while if one is different from the other two the algebra is twisted. Referring to Eqs. (VI.5.134-137), let H be the untwisted subgroup and G/H the symmetric space spanned by the twisted generators (If H= Uti) we have an actual twist, while if H:SU(2) there is no twist). With these notations we can write: A
A
xA
A
x
J {z)J (w) " xJ (z)J (w)x
2_1
+ __ (z _ w)2
+
dim H +
1. dim (G/H) z~ I 2
v zw
r
(VI. 7.62)
where : : denotes normal ordering with respect to the moding of the Kac-Moody current. The stress-energy tensor (VI.5.71b) is therefore given by the limit B
1
XA
A
x
T (w) '" - lim xJ (z)J (w)x 8 z .... w
'
(VI. 7. 63)
while the canonical energy-momentum tensor of the fake fermions is given the other limit
by
1725
A little bit of algebra and the use of Wick theorem reveals that
2
+1 t.w) - (tr hew+1 z.w}) - 16(tr h (w
2} = 0, (VI. 7.65)
the trace is taken on the 3-dimensional space and the 2-point function (VI.7.60) is regarded as a matrix. This is the proof of full fermionitation: the energy-momentum tensor of the Kac-Moody syste~ is equal to that of the fake fermions independently of the boundary conditions. We have four different sectors depending on the relative numbers of Ramond and Neveu-Schwarz fake fermions. They will be denoted by (R)3. (R)2(NS). (R)(NS)2 and (NS) 3. (R)3 and (NS)3 correspond to the untwisted algebra, while the other two cases correspond to the twisted one. They decompose into irreducible representations according to the following rules: ~~ere
untwisted algebra twisted algebra
I (R) 3 ... (J = 2)1 (NS)3 ... (J = 0)
I
1
= '2)
(J
$ $
(J = 1)
(R) (NS)2 ...
(J "
!.) 2
$
(J "
!.) 2
(R)2 (NS)
(J "
0)
6)
(J =
I)
+
=}
For instance, in the (R) 3-case the J vacuum is the eli fford vacuum of the three Ramond fermions and all the states built with any number of fermion oscillators belong to the same irreducible module. In the (NS)3 case, instead, the states built with an even number of fermion oscillators belong to the J=O module while those built with an odd number belong to J=l irreducible module.
1726
As a final check of fermionization and as a preparatory step for
A~]
the next section we have verified, numerically, that the B
func-
tions relevant to us can be rewritten as suitable products of theta functions as we show in Table VI.7.!. The same table establishes a valuable one-to-one correspondence between the choice of J, a and e and the choice of a spin structure for the fake fermions. The table must be understood as follows. According to Kac and Peterson [6] the numerical value of the characters for the k;4 SU(2) algebra is given by
(
~
BJ-,T,-U
)
12
1 = L
2J C1J,{1")0(2JI2)(T,Z,U) J'"Q '
(VI. 7.66)
where (VI. 7.67a)
(VI. 7.67b) Co 2 ;: -
1
c
1
1 { n ( -) 1" 2[11(1)]2 2
11(21") =---
- e-in/4
I} 11 ( -1" + -) 2 2
(VI. 7.67c)
(VI.7.67d)
[n(l) J2
are so called "string functions" and the capital theta series is defined as 0(2J,2)(l,Z,U)
=e
-4illU
(VI. 7.68)
Performing a shift in the variables one can also introduce capital-thetas with characteristics:
1727 TABLB VI. 7. I Fermionization of level }<=4 SU(2) Partition FlBlctions ,;
Free Fermion Partition Function
KM Bosonic Partition FlBlction
z[
BA:J Bl/2[~]
a1 a2 a 3 ] b I b 2 b3
_1
z[ 000 III J
12
Buzl ~]
12
1
zIIOO] 000
B1/ 2[ ~]
72
1
Z(111]
B1/ 2[ ~ 1
e· i11 / 4
f2
011
z[ 1 00] 011
BO[ ~]
llz[ 0 00] z[ 0III °°11 2 000
Baf ~]
[011] '211 Zoo 0
Bo[ ~ ]
!{z[OOO] z[0001l 2 OIl 100
Bo[ ~ 1
eitT2/ 4 Z[011] • . [0 11 100 1Z 011
+
+
. [Oll]} 1Z 1 11
+
I
1I2 [OOOJ lOOO]} 000· Z 111
Bl[~l
2
BIl ~ J
21[[011) z 000
BII ~]
1Iz[000] . z[OOOJl 2 011 100
BI [
~1
H
e 3il1/ 4 2
• iZ[
{z(Ol1] 100
+
~ !~ )}
iZ[OllJl 011
1728 0(2J,2)[;] (t,z,u) .. =e
-4niu+ill(<<+4J)S/4 il8 q
\'
q
L
2p2+ap
e
-4nipz
,(VI. 7.69)
pel+~
A: ]:
and write the analogue of Eq. (VI. 7.66) for the B
BA:Hlz ,T, -u) =J1t c~,0(2J,2)[:1(T'Z,U) .
(VI. 7.70)
In Table VI. 7. I we have verified that for z .. U" 0 the numerical values
A: ]
of B
as given by (VI. 7. 70) are equal to the values of certain
suitable combinations of Riemann theta functions. The reason why we restricted our attention to z = U =0 is that we were interested, for the time being, in vacuum to vacuum amplitudes. This, however, introduced some ambiguity, since sometimes we were dealing with the factor
a(: J(Oh)
which, being zero, could not be seen. To resolve the ambiguity we shifted to non-zero values of 4 and we also established the following identities:
(6 n.
[ 1J 0
1/2
[ 1]
° (zit)
(VI. 7. 7la)
B [O](_z ,t,O) ,,_1 (e[~](Olt)~/2 a[ ~}(zlt)
(VI. 7. 71b)
B [O}(_Z 1/2 0
n."
1/2 1
12
BO[~J(.,;
t
0)
= _1
12
(OIT»)
e
net)
net)
J
net)
nCt)
,T,O) - Bl[~](Jr ,T,O) =
=_ilrrz('6[~](OIT)f2 l](t) J
6[!
(zit)
nCt)
(VI. 7. 7lc)
1729 Eqs. (VI. 7.71) are the justification for the way utilized in Table VI. 7. I to write vanishing objects like Sl/2[ ~ (1) or
1
so[ ~] (1) - Bl [!] ('r).
Finally let us explain the notation we have adopted. regarded as functions of i alone (Le.. z =t =0).
BA: J
are Furthermore
corresponds to the partition function of N Majorana fermions with spin structures
[ :~ ], [::].....
[~]
(see Eq. (VI. 7.3») • Looking at
Table VI.7.1 and considering the possible partition functions of three fermions we see that we have apparently missed four combinations, namely,
i)
ell) 100
ii)
(i00) 100
iii)
(Ill) 11 1
iv)
POD) 111
This is not serious since the above combinations are zero being proportional to
a[: }(OIT)/n(l).
Hence, without making any error we can
add them, multiplied by any coefficient. to the expressions for
Bl/2[~]' Bl/2[~1. BI/2[~]
and
Bl/2[~]
respectively. Thisnumeri-
cally harmless addition allows, on the other hand, the rewriting of 81/2 [ :] inserted.
as a fermion Fock space trace with a projection operator
1730
VI.7.S Modular invariance and spin structures In Chapter VI.S we have selected a few string models with massless fermions. All of them, but one, contain U(l) factors in the target group. Then. according to a general argument [81. they correspond to non-chiral theories which are not very interesting from a physical point of view. Thus we are led to consider a tmique superstring action in four dimensions with G.r" SU(2) 3 which is the only one compatible with N-l space-time supersymmetry, as we shall see. \'Ie have shown in the previous section that this model is fermioni~able; this means, in particular, that the physical spectrum can be described in terms of the Fock space associated with a set of free fields containing, besides the bosonic coordinates Xll, 20 left-moving and 44 right-moving world-sheet fermions, as in the free-fermion approach to 4D heterotic superstring. However, there is a difference between the allowed botmdary conditions in the present approach and those considered in the ordinary free fermion description. We devote this section to the study of the constraints on the botmdary conditions. We include also all the constraints arising from modular invariance. \'Ie start by listing the whole set of 64 world-sheet fermions. In the left-moving sector we have. besides the tt.'O fermions (T =1,2) associated to the transverse coordinates, six fermions x~,,,: (A= 1,2.3) for each of the three SU(2) groups (i = 1.2,3). where t's are the fake fermions used to represent the KM currents si(~) having ~:(z) as fermionic partners.
·i.
In the right-moving sector we have nine fake fermions X~(i) whose bilinears represent J~1 and thirty-five heterotic fermi~ns
~p(Z).
A first important constraint comes from the compatibility of the twisted KM algebras with the existence of massless fermions already discussed which forces all the world-sheet fermions, excepting ~p(z). to obey real (i.e. R or NS) boundary conditions, while in the ordinary are reqUired to be real. free-fermion approach only the
l
Another constraint is imposed by the world-sheet supersymmetry and is a generalization of the one identifying the boundary conditions of
1731
the ~'s with those of the gravitino ~. It can be obtained simply by considering the explicit express~.on of the two-dimensional supercurrent G(z) given in Eq. (VI. 7.30). After fermionization the term J. A becomes bilinear in the X's; any arbitrary choice of real boundary conditions CBC) for the X's fixes an allowed twist of the SU(2) current JA(z). Then the boundary conditions of the A's are uniquely fixed by the requirement that J. A must have the same Be as ~ . In other terms, once given the BC for ~, we can freely choose the BC of the nine X's, while the Be of the remaining left-moving fermions are uniquely determined. A similar analysis in the ordinary free-fermion approach, where the supercurrent G has a different expression, shows that there one can choose arbitrarily the boundary conditions of 12 left-moving .fermions. In the right sector the boundary conditions that can be imposed on the fake fermions ~ are those induced by a twist of the SU(2) Kac-Moody algebras (see Eqs. (VI.5.34)). The other constraints we wish now to describe are not specific to the present formulation, but are common to all fermionic strings (here we use the definition eVI.7.1)). It is convenient to represent the BC of the world-sheet fermions of a given string sector by a 64-component boundary vector (VI. 7.72)
where
r"
is a NS fermion;
A
1 if II
(VI. 7.73) is a R fermion •
Using this notation the two constraints mentioned above can be written as follows:
1732
a~ T a A TaB = a C (mod 2) • Xi
Xi
~i
i
= 1,2.3.
t ABe = 1
(VI. 7. 74)
(VI. 7. 75)
Since the boundary vector a is a sequence of 1's and 0' s it can be specified simply by listing the Ramond fields. For instance, using a shorthand notation we can write s = {~T (T= 1,2).
xi,
A~
(i = 1,2,3)}
•
(VI. 7. 76)
to denote the boundary vector associated to the string sector characterized by the above R fields. This boundary vector has an important role in the theory, since it describes a space-time fermionic sector (because Ctl/J =1) whose lowest state (gravitino) is massless, as it can be seen using the mass formulas given by Eqs. (VI.5.127) which can be rewritten in the following form: (VI. 7. 77a)
(VI. 7. 77b)
where N and N are the occupation numbers in the LM and RM sectors, while nL and nR denote the numbers of Ramond fields in the two sectors. The equality of the RHS of the two equations (VI.7.77) is the additional constraint one has to impose on the direct product of open string physical states in order to describe closed string states. This additional constraint acts also on the boundary vectors. Indeed, subtracting Eq. (Vl.7.77b) from Eq. (VI.7.77a) we can write
which implies that nL• nR is a multiple of eight.
1733 It is convenient to introduce a Lorentzian scalar product for boundary vectors as follows
aob
def 20
= I /l.a. i=1
1 1
~
64
1:
a.a.
j=21 J J
(VI. 7. 79)
Then the additional constraint (VI.7.78) implies 2 def _ a = a a = 0 (mod 8) • 0
(VI. 7.80)
As we have already seen in the previous section, one· loop modular invariance requires that we define the sum of boundary vectors: and
ai
if ai are the components of a and b, then !,:he components 'Yi of
c = a + bare gi ven by (VI. 7.81)
The same definition of sum can be obtained by considering the coupling of three strings, characterized by three arbitrary boundary vectors a, b and c;
indeed the Ramond fermions carry a fermionic number in the
target space whose conservation can be expressed precisely by Eq. (VI. 7. 81), hen ce a + b = c. This fact implies in turn tha.t the set of boundary vectors form an abelian group :: where the identity element is the null vector. :: -I K+l
- - 2
Actually we have, for some integer K, (VI. 7.82)
since a+ a= 0 Vas:; K+l is the number of generators bo' bt , •.• , bK which form a basis of ::. Note that in general Eq. (VI.7.80) is not invariant under the sum (VI.7.8l) in the sense that if a and b are two boundary vectors
satisfying Eq. (VI. 7. 80), their sum a + b does not satisfy automatically such a condition. curious relation
However, one can immediately derive the following
1734 (a+b)
222 =a +b
(VI. 7.83)
-2a o b
which yields, when combined with Sq. (VI.7.80), the following constraint a' b :: 0 (mod 4)
(VI. 7.84)
for every pair of allowed boundary vectors. However this constraint is not yet sufficient to guarantee that the sum of the three boW\dary vectors also fulfils Eq. (VI.7.80). The point is that the scalar product defined in Eq. (VI.7.79) is not linear in its arguments; rather, one has a•c
+
b' c - (a + b) • c = 2a' b • c
(VI. 7.85)
where the triple product a' b • c denotes the net nUlllber of Ramond fermions (LM minus RM) common to the three boundary vectors. Similarly one can define a quadruple product a 0 b • C • d as 2a • b • c • d = a' b • c
+
a' b • d - a' b • (c + d) ,
(VI. 7.86)
which gives likewise the number of R fermions common to a. b, c and d and so on. Using this notation one can write (a + b + c)
2
z
222 a + b + c - 2a' b - 2a' - 2b' c + 4a' b • c
C -
(VI. 7.87)
which gives the further constraint a' b • c :: 0 (mod 2) •
(VI. 7. 88)
Such a constraint is sufficient to ensure that the sum of an arbitrary number of boundary vectors satisfies Eq. (VI.7.80), provided that all the addends also obey the same conditions. Indeed the coefficients of
1735
the mUltiple products (with more than three entries), needed to express the square of the sum of more than three vectors, are multiples of eight. There is one further constraint on the boundary vectors following from one-loop modular invariance. In order to discuss such a constraint it is convenient to build explicitly the modular invariant vacuum to vacuum amplitudes (or partition functions) of the model. An ordered pair of boundary vectors a, b defines a one-loop spin structure [:] where a and b give the boundary conditions in a and T respectively. To each spin structure there is associated a contribution to the vacuum amplitude which, according to the previous section, can be rewritten as
(VI. 7. 89)
where the trace is taken over the states of the string sector characterized by the boundary vector a, and b· F", I S. Fi , Fi being the h . 1 fermion number of the it world-sheet fermion AI. More precisely, we have Ni (-1) NS
i
i NNS
(-1/
Ni
(_I)Rf2A~, NiR
I
Ait Ai
r" ~ r r
, for a.1 "
Y Ai\i ,
n"l n n
0 (NS) (VI. 7. 90)
for a.1
1 (R)
A~ is proportional to a gamma matrix, and the operator (~l)b'F ~ S.F i
n.1 (~l)
is well defined only when a prescription is given for the . order to be taken in the product of the A~'S' It is possible to show that there are prescriptions such that 1
1736
(VI. 7.91)
(D) Apart from the factor Z(X)(T) (see Eq. (VI.7.27)) the total one-loop partition function is given by Eq. (VI.7.26b):
Wei)
=k
I
(a,b)
c[aJ b
z[a]b .
(VI. 7.92)
The normalization factor k and the set of the needed spin structures will be specified later. Note that
c[:J
are the one-loop boundary condition dependent
coefficients introduced for the first time in Sq. (VI.6.2S). In the multiloop case the total partition function of the fermionic string has a similar form: (VI. 7.93)
where z~~~ is given by Eq. (VI.6.192b) and
is the multiloop analogue of Eq. (Vl.7.92), the Z symbol in (VI.7.94) being the multiloop analogue of the one-loop Z symbol defined in Eq. (VI. 7.3): 1/2
(f rg det Im II) 1/2 (det
I
fs) 1/2
1737 * 1/2 N+M
II i=N+l
The factor X
(I ,rg det 1m 1T) 1/2 (det
(VI. 7.9S) t
A) 1/2
a:II1 , .•. ,a1/1] g is the contribution of the gravitino ghost 1jJ
1/1
hI' •.. ,bg
and of the two longitudinal 1jJU discussed in Section VI.5.7. By a~. 1/1 + 1 bi we denote components of the boundary vectors t 1 , hi corresponding to D~2 ~-fermions endowed with transverse indices. It is missing in the g=1 case since there it is equal to one as explained in Section VI.6.7. The reader should not confuse them with the double set of labels attributed to the boundary vectors. The upper index enumerate the components of the vector and indicates which fermion we are dealing with; the lower index enumerates the handles of the Riemann surface. Following the arguments of Section VI.6.2 the multiloop coefficients appearing in Eq. (VI.7.94) must have a completely factorized form: (VI. 7.96)
in terms of the our goal.
one~loop
coefficients (VI.7.92), whose determination is
We begin by conSidering the constraints due to invariance. The S and T generators of PSL(2,1) act on
z[aJ-...!.-~eiTla2/8z[ a ] b a+b+lI
one~loop
modular
z[: 1 as follows
(VI.7.97b)
1738
where ]. is the botmdary vector with ing Eq. (VI. 7.97b) to
0:." 1
1 for any i. (*) By apply-
z[ ~ l we see that such a vector is always
contained in the group _. 1 eE•
(VI. 7.98)
This is the further constraint on botmdary vectors contributed by modular invariance. We shall see that it is possible to build modular invariant partition functions out of any group E of botmdary vectors fulfilling Eqs. (Vl.7.80), (VI.7.84), (VI.7.8S) and (VI.7.9S). Starting from an arbitrary spin structure [: ] and applying the S and T operators several times, we can generate an orbit of the modular group. This orbit contains at most six distinct spin structures which are made with all the possible pairs of the three boundary vectors at b and c .. a + b + 1. Provided they satisfy the constraints (VI. 7.80) and (VI.7.84), it is possible to construct the following modular invariant combination:
(VI. 7. 99)
with a+b+c=l.
(*)
(VI. 7.100)
A careful treatment of the ordering of the zero-modes associated with the Ramond fields may lead to extra phases in the transformations (VI. 7.97) unless a' b • C • d:: 0 (mod 2). It is however possible to find a modular invariant solution of (VI.7.92) even when this last condition fails. See Ref. [9}.
1739
Zabc is the only PSL(2.1) invariant containing the above six spin structures; it has simple symmetry properties under arbitrary permutation of its indices: (VI. 7. lOla)
(VI. 7. lOlb) Note that a· b =b • c =c· a as it can be verified using the simple
identity 2 a
+
2 2 2 b + c + 11. .. 2a' b + 21
(VI.7.102)
where c is given by (VI.7.100). If one of the boundary vectors
a. b or c is 1, then in Eq.
(VI.7.99) there are only three distinct spin structures and we divide
Zabl by two. Similarly ~.Il contains the single spin structure and we therefore divide it by six.
[~]
The most general one-loop modular invariant can then be written Wet) :: k
~
cabc Zabc
(VI. 7.103)
{a,be ::l where for consistency cabc must obey the symmetry relations (VI.7.101). Comparing Eq. (VI. 7. 103) with Eq. (Vl.7.92) we can write (VI. 7. 104a)
c[ b] a
= cabc
e
i'!f(a+b)2/8 .
(VI. 7.104b)
It is easy to calculate the number M of independent one-loop modular invariants contributing to (VI.7.103):
1740 f.f"
(1:1 + 1) 0=1 + 2)
(VI.7.lOS)
6
where
131
is the number of elements of -.
The number of free parameters is actually much less than M, because one has still to impose some constraint following from the physical interpretation of Eq. (VI. 7. lOS) , which implies in particular that if one collects all terms associated to the sector a of the string, it must be written as a suitable trace over string states:
(VI. 7.106)
where
oa = (-1)
a l/I
(VI. 7.107)
takes into account the connection between spin and statistics and P a is a generalized GSO projection operator(*) (VI. 7.108) Condition (VI.7.108) is fulfilled if
(VI. 7.109)
which implies again, like the three-string coupling, that : shoUld form a group under the addition (VI.7.al). Moreover choosing the normalization factor
(*)
GSO stands for Gliozzi, Olive, Scherk who originally introduced the projector in the context of the 10-dimensional NSR spinning string (see also the Historical Remarks).
1741 k
= _1_
(VI. 7.110)
l=!
the
~oefficients
C
fa l x] are pure signs: (VI. 7.111)
Indeed from Eq. (VI. 7.109) and choosing x =y= 0 we obtain
c[ ~]
=
~a
(VI. 7.112)
while for x =y;' 0 we deduce (VI. 7. 112a)
Utilizing this extra information we can write the general solution for the
c[:]
coefficients. Prior to that, however, we want to emphasize
that Eq. (VI.7.109) is also the additional constraint needed to promote one-loop modular invariance to multi loop modular invariance. Consider the two-loop Case and the following element of the Sp(4,Z) group
(VI. 7.113)
Acting on the homology basis r produces the following transformation (VI. 7. 114a) (VI. 7.114b)
1742
(VI. 7.114c) (VI. 7. 114d)
which clearly mixes the cycles associated with the two handles. be shown that f 12 , together with
(~ ,{
0
-1
0
0
\0
0
0
T1
1
=
1
(j
0
0
1
0
0
1
0
0
0
1
0
0
;)
0
1
0\
0
0
1
0
o
0
0
0
1
-1
0
0
o
,
T2
=
I . ',. (~ I
1/
i)
;)
It can
(VI. 7. USa)
(VI. 7.l1Sb)
generates the full Sp(4,Z) group. In Eq. (VI. 7.115) , {T. ,S.} are the 1 1 generators of the SL(2,Z) subgroup associated with the i-th handle. Recalling Sq. (VI.7.96) we immediately obtain the additional constraint on the coefficients c[:] implied by the invariance of the partition function (VI.7.94) under the action of f 12 • From Eq. (VI.2.430a) we get
(VI. 7.116)
Furthermore, combining Eq. (VI.2.429) with Eq. (VI.6.30B) we can write the invariance condition associated with r 12 :
(VI. 7.117)
1743 Choosing b2 =a1 in (VI. 7.117) we obtain
(VI.7.118) Then, using Eq. (VI.7.104) we find
(VI. 7.119)
which combined with (Vl.7.118) yields
(VI. 7.120)
Sq. (VI.7.120) coincides with Eq. (Vr.7.109) if we choose the normaliza· tion of our coefficients by setting
(VI.7.121) This completes the proof that two·loop modular invariance coincides with the requireaent of reinterpretability of the partition function (VI.7.92) as a trace of the evolution operator with a GSO projector inserted. Higher loop modular invariance adds no new information since we just have a generator r11+ .. 1 for any two neighbouring handles i and i + 1. Each of them implies the same condition (VI. 7.117). Let us return to the problem of determining the c[:] coefH· cients. Combining Eq. (VI.7.109) with (VI.7.119) we have (VI. 7.122)
1744 In particular. for x = y one has
(VI. 7.123)
where i = a + 1. More generally we can choose arbitrarily the sign cijx associated to any pair of basis elements b.• b.• Eq. (VI.7.122) then 1 ] allows one to extend the assignment of signs to a generic coefficient cabc An explicit solution can now be written. Putting
8
A
a
= I
where ai
i=l
a1 • b,.
and bj
I
(VI. 7.124)
b. j=1 J
are basis vectors. one can !>Tite
(VI.7.12S)
at> b,.
rB
j=l
a' b.
(VI. 7.126)
J
This is a kind of asymmetric scalar product which is linear. by construction. in the second argument, and this property allows to verify at once the factorization properties of Eqs. (VI.7.109) and (VI.7.122). On the other hand, it is more difficult to extend the symmetry propera.bjc, . ties (VI.7.l01) from the basis element c 1 1J to the generic coefficient C8bc because of the asymmetry of a I> 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 :: -<0 so that the physical state Ik,l; > 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 <
k,~1 =
lim (z,z)2 < O,OIV(k,,; z,z)
(VI.8.S4)
Z"''''
z... The factor
00
(z,i)2 appearing in Eq. (VI.S.S4) is easily explained.
Writing the vertex in a factorized form (VI. S. S5)
where, for brevity, we have omitted in the right hand side the polarization ~, and expanding in modes: V (k) r _n_ n=-"" zn+l +00
VL(k,z)
=
(VI. 8. 56a)
(VI. 8. 56b)
we obtain, after comparing (VI.8.S6) with (VI.8.53), (VI. 8.S7a)
(VI. S. 57b)
Hermitean conjugation requires that {VI. 8.S8a}
1784" (VI. 8.58b)
This is precisely what we obtain if we define < k,~! (VI.8.54) •
as in Eq.
The formula (VI.8.4) for the scattering amplitude and the related integration measure (VI.8.30) can now be justified relying on an analogy with the ordinary quantum field theory. Consider a scattering process of N+3 target space particles and the following picture:
I
I I
I I
,
\
Fig. VI. 8. I which describes a closed string world-sheet with the topology of a sphere (g =0). By means of a suitable conformal transformation the sphere can be reduced to the cigar-like shape of Fig. VI.8.1 which helps our intuition in grasping the analogy we want to draw. At time 1" '" -00 corresponding to Z =0 we have an incoming state of momentum ki. This is a closed string in its ground state oscillation mode, that is, a point-like object. At T =co (corresponding to Iz I +00) we have an outgoing momentum state k~, that is, another
1785 point -like object. While propagat ing from t = -co to T =.., the closed string vibrates in all possible ways and behaves as a truely one-dimensional object. In this way it sweeps the world-sheet. At times '2.T4, ••• ,TN+3 corresponding to the world-sheet locations ZZ'%4 ••••• zN+3. the interaction hamiltonian provided by the vertex operator induces the emission of particles of momenta kZ.k4, .••• kN+ 3. One associates naively to such a process the following scattering amplitude:
2
• d
%z
N+3 2 II d z. 1=4
(VI. 8. 59)
1
where T denotes the radial ordering in the moduli
1%.1 : t (VI.8.60)
Since all the world-sheet points are equivalent, we integrate over the locations of the particle emissions. There is, however, a problem which i!lllllediately sholfs up if we do so; the result for the amplitude is infinite. This happens because we have neglected the automorphism group of the genus zero world-sheet (see Eqs. (V[.2.56»: Aut (Eo)
= SL(2,[)
(VI. 8.61)
and pretended to sum independently on the gauge equivalent configurations. To see that this is the case, let us utilize the factorization (VI.8.S3) of the vertices and rewrite (VI.8.59) as follows: (VI. 8.62)
1782 jphys >
= 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 <
k,~1 "
lim (z,i)2 < O,OIV(k,r,;; :L,i)
(VI. 8. 54)
z ... 00
z-+ c:o
The factor (z,z)2 appearing in Eq. (VI.8.54) is easily explained. Writing the vertex in a factorized form (VI. 8. 55)
where, for brevity, we have omitted in the right hand side the zation r,;, and expanding in modes:
VL(k,z)"
+"" V (k) _n_ n=-CIQ zn+1
I
polari~
(VI. 8. 56a)
(VI. 8. 56b)
we obtain, after comparing (VI.8.56) with (VI.8.53), (VI. 8. 57a)
(VI. 8. 57b)
Hermitean conjugation requires that (VI. 8.58a)
1788 Since the Kac-Moody currents are already given in terms of Fermi bilinears, it follows from hosonization that the non-abelian currents are eventually expressible in terms of abelian free bosons. This is the Frenkel-Kac vertex operator representation mentioned above. It works- for level k/S2" 1 simply laced algebras. It wi 11 be manifest in what follows that both the bosonization of fermions and the FrenkelKac representation are just particular cases of a general construction, by means of which we can write the conformal fields transforming in any representation of a simply laced algebra ~ in terms of a set {~i(z)} of r free bosons, r being the rank of ~. The spin fields correspond to the particular case of this construction where ~ is an orthogonal algebra and the representation considered is the spinor representation. The stage being set, let us begin by introducing the abovementioned free bosonic fields (z) (i" 1, •.. ,r). whose OPE is declared to be the following:
i
(VI. 8.68) In terms of oscillator modes Eq. (VI.8.68) corresponds to
(VI. 8.69a)
(VI.8.69b)
(VI. 8. 69c)
Relying on (VI.8.69) we can write the decomposition
(VI. 8.70)
where we have set • (VI.S.71a)
1789 V!
i(z) ,. 1. <
t.\ -1 z~n .p i
n
(VI. 8. 71b)
n
i() ,. qi - lp . i .lvg Z
(VI. 8. 7Ic)
V! 0 Z
The notion of normal ordering is defined as follows. In a normal ordered operator all the creation operators .p (n < 0) sit on the left of all n . absorption operators opn (n> 0). Furthermore, ql sits on the left of . pl. With this definition we get
(VI. 8. 72a)
,. : 'I';cz)
"'~(W)
: - oij tn (1-
;J
(VI. 8. 72b)
yielding
(VI. 8. 73)
which is an eqUivalent form of Eq. (Vl.8.68). derivative currents
If we introduce the
(VI. 8.74)
we can immediately verify that they fulfill the following abelian algebra: oij
2
+
reg.
(z - w)
In addition to Hi(z) we also introduce the vertex operators:
(VI. 8. 75)
1790 U(A,Z) = :
exp{i A • f(Z)} : :: (VI. 8. 76)
where Ai is a constant vector in the r-dimensional space, If ,i(z} are inteTPreted as the coordinates of a torus then Ai is tbe momentum on that torus. Utilizing Eqs, (VI.8.72) it is straightforward to verify that the vertex operators U(A,Z) obey the following relation: U(J.,z) U(u.w)
= (z - 1'1».0).1
:
exp{U. ,
(z) + i).l 0 ,
(w))
(VI.8.m
which implies
U(A,Z)
U(ll,Z) = exp{ilr A·).I} U(ll,W)
U(A,z) •
(VI. 8, 78)
Furthermore, expanding the operator A(z,w)
= : exp{i A• , (z)
+
ill' ., (I'll)
(VI. 8. 79)
in Taylor series about w we get the following OPE:
U(]..,z)
U().I,w)
= (z - 1'1) ]..0).1 U(A + ll. 1'1) x
Equations (VI.8.78) and (VI.S.80) are the key to understand bosonization, the Frenkel-Kac representation and, ultimately, the lattice approach to superstrings. Let us begin with bosonization. Using the vertex operator we show how from r free real bosons we can construct 2r free Majorana fermions, Consider the following 2r vectors:
1791
~\ o
±1
ith row
o
•
(VI. 8. 81)
o From Eqs. (VI.8.78) and (VI.8.79) we obtain - U(±e. ,1'1) U(±e. ,z) • (if i J
U(±e. ,z) U(1e.,w) J
1
=
1
=j)
{
(Vr:8.82a) U(±ej,w) U(±ei,z) , (if i'; j)
U(+e. ,z)
oij
U(~ej ,1'1) '" - - +
z-w
1
reg. terms
U(+e.,z) U(+e.,w) " reg. terms 1
(VI. 8. 82e)
J
U(-e .• z) U(-e.,w) = reg. 1 J
(VI. 8. 82b)
terms.
(VI. 8. 82d)
Equations (VI.8.82) suggest that we define
(VI. 8. 83)
since with such a definition Eqs. (VI.8.82) translate into the canonical OPE's of a set of 2r free fermions: A 1: i A (ZlA (1'1) '" - -
2
<5
/,1:
1
-- +
z-w
reg.
(VI. 8. 84a)
1792 where the index A runs on 2r values:
I
2i - 1,
A"
i,. 1, ... , r (VI. 8. 84b)
2i
, i=l, ... ,r
There is just one problem with this identification which is manifest from an inspection of Eq. (VI.8.82a): while Ai(X) anticommutes with itself and with A-i(Z) it commutes with )..±j(Z) for j#i. Hence the A's are not yet true fermionic fields. This difference is remedied modifying Eq. (VI.8.83) via the insertion of a new c(te.) called 1 the cocycle:
by
c (±e. ) U(±e. ,Z ) • eiW/4(,2i-lCo) /\.. i 1
l' /\ ,2i(z)j
1
.
(v!. 8. 8S)
The cocycle is constructed in such a way that we get
AA(Z)A r (w) ,. - Ar (W)A A(z)
(VI. 8. 86)
without modifying the OPE (VI.8.84). A proper choice for cCte l ) is the following; c (±e.) " exp [.tire1.• MPJ 1 where
p
(VI. 8. 87)
is the momentum operator appearing in the mode expansion
(VI. 8. 69a) :
1 pl. , , _ 21fi
f .
dz HI (z) , , 121f
f
dz
.
a '/ z
(VI. 8. 88)
and M is a r x r matrix whose structure we shall now determine. Extending the definition of the cocycle operator to any rdimensional vector: (VI. 8.89)
1793 and using the Baker-Hausdorff formula AB
B A [A BJ
ee=eee'
A+B
=e
e
f [A,B]
(VI. 8.90)
we easily verify the following equations: C{),.)C(Il}
= c(\I}c(A) = c(J..+ \I)
(VI.8.9Ia) (VI. 8.9Ib)
Hence if we define the improved vertex operators as
OCh,Z)
= c(A)
U(A,z)
(VI. 8. 91)
we obtain the new equations OCh,Z) O(Il,w) " (VI. 8. 92a) +
+
O(h,Z) O(Il,w) = exp{ -ilr UIIJ} (z - w)
X
{1+
)"'IJ
0(.>. + Il, w} x
(z -w)iA' o",(w) + ... }
(VI. 8. 92b)
and can write Eq. (VI.8.SS) as (VI. 8. 93) It is now easy to see that the matrix M can be chosen in such a way as to fulfill our requirements and give rise to true fermions. Indeed, since we have: (VI. 8.94)
1794
a solution to our problem is provided by a lower triangular matrix with zero diagonal elements and non-zero elements ±I. that is, by Mij ,,0
,
(i> j)
Mij .. ±l
Mij " 0
,
(VI. 8.95)
(i < j) .
Choosing the signs in (V1.8.95) amounts to a choice of a gamma matrix basis (see below), Using Eq. (VI.8.95) we obtain (VI. 8. 96a)
1
O(+e.,2.) O(-e.,w) " - - 6.. J
1
(z _ w)
+
1J
(VJ.8.96b)
reg.
(VI.8.96c) (VI. 8.96d)
which, upon insertion of Eq. (VI.8.93) lead to the desired result, i.e. to Eqs. (VI.S.86) and (VI.8.84). The bosonization we have achieved can be checked at the level of conformal weights. Introduce the stress-energy tensor of the free-fields T(2.) "
I'
i
1
i
-rcz)A' (z) " - -2 az ~ az " 2
~
i
i
(z)
(VI.8.97)
we immediately check that for any improved vertex operator tbe following equation holds true; T(z)
i2/2 O(;\.,w) .. - - - O(;\.,w) (z_w)2
I
+ - - aO(;\.,w) +
reg.
(VI.8.98)
(z-w)
+2
~
This proves that O(;\.,w) has conformal weights (1I" A /2, 1I" 0). In particular, O(!ei.z) has conformal weights (1/2,0) as a respectable fermion should have.
1795
Equation (VI.8.9S) follows from the OPE written below, whose verification is also immediate: i Ai ;0 H (z) O(>',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
3" 1
$
!.
(j)
e 1
(VI.8.148)
so that the anomaly cancellation equation reads as follows: 3D - 26
+ reparam. ghosts and has the solution
+
22
+ two supersymmetries ghosts
- 2
" 0
(V['8.149)
t
one gaugc ghost
D.. 2. In Eq. (VI. 8.149) the contribution of the gauge ghost is due to the presence of a U(l) current in the N.. 2 superconformal algebra. Since we are specifically interested in tree amplitudes, we focus on the case g .. O. We find that the following holds true:
1809 #
zero modes of e(z) - " zero modes of b(z) = - Q.
(VI. 8. ISO)
Thus minus the background charge is just equal to the number of c-zero modes minus the number of b zero-modes. In the case of the reparametrization ghosts. Q.. -3 is due to the well-known presence of three conformal Killing vectors (c-zero modes) corresponding to the generators SL(2.E). Similarly in the superghost case. Q= -2 is due to the presence of two conformal KUling spinors. The reason why we naJlled background charge the number Q defined by Eq. (VI.8.143) can now be , given. Consider the ghost CUrrent j(z) .. -
:b(z)e(z):
(VI. 8.151)
which together with the stress-energy tensor. fulfills the following algebra ~~ere c is given by Eq. (VI.8.142): T(z)T(w) =.£ __1 _ + 2T(w) 2 (z-w)2 (z-w)2
+ 3T(w) + (z-w)
• Q Jew) aj(w) T(t») (w) .. - - - + - - - + - -
(z_w)3
j{z)j(w) " -e:
2
(z_w)2
1 + (2: _ w)2
reg.
(z-w)
+
reg.
reg.
(VI. 8. 152a)
(vr.8.1S2b)
(VI.8.IS2c)
The OPE's (VI.8.1S2) differ from those describing the semidirect product of the Virasoro algebra with a Uti) Kac-Moody algebra (see Eqs. (VI.S.68) and (Vr.S.82» in two respects: the presence of the triple pole in Eq. (VI.8.1S2b) and the sign £ in (VI.8.1S2e). which for the superghosts (e:" -1) corresponds to a negative central charge. Expanding in modes . ()
\' -n-l.I
n
(VI. 8.1538)
z-n-2 Ln
(VI. 8.1S3b)
J z "L. z n
T(z) "
I n
1810
we see that Eqs. (VI.8.152b,c) are equivalent to 0 m nJ,. ·nj m+n + gm(m+l)6 2 m+n,
[ L ,j
[ .;
~m'
J'
nJ
= f.2
m il. m+n,O
(VI. 8. 154a)
(VI. 8.154b)
The algebra (VI. 8. 154) is not compatible with the assumption that the ghost number operator jo be hermitean. Indeed, if we set (Vl. 8.155)
for consistency with Eqs. (VI.8.1S4) we must also set
.t
'
3-n = -3 n -
Q~ u
(VI.8.156)
n, 0
From this property it follows that if jq > is an eigenstate of jo
with eigenvalue q:
(VI. 8.157) then its ad; oint
Iq
>t
is a state < -Q - q I fulfilling the condition (VI. 8.158)
Let the vacuum state 10 >q =Iq > have charge q and let us define the q-vacuum expectation value of any operator A by the following equation:
q
= < -Q- ql AIq
> •
Let us, furthermore, say that A has charge q' the condition:
(VI.8.1S9) if it fulfills
(VI.8.160)
1811
A simple manipulation shows that for an operator of charge q' the following equation is true (q I + Q+ q) < A > .. 0 • q
(VI.8.161)
Therefore either A has charge q' =-Q - q or its vacuum expectation value vanishes. This observation has an immediate and very important implication for the string amplitudes: it provides the operatorial justification for the integration measure appearing in Eqs. (VI.8.67). In that equation the state 10 > is the SL(2,t) invariant vacuum for the system composed of the matter fields and superghosts. The reparametritation ghosts are excluded. The reason for this exclusion is that their contribution is already factorized out and is nothing but the integration measure
That this is true follows precisely from the need of cancelling the ghost background charge (VI.8.162)
To see this we begin by expanding the fields bet) and c(z) in modes:
b(z)
=I
b z-n-~
(VI.8.163a)
c () z
= L~ Cn Z-n- (1-~)
(VI. 8.163b)
n n
n
and observe that for any b-c system the SL2 invariant vacuum is characterized by the conditions: (n ~ I-A)
(n
~
11)
(VI. 8. 164a)
(VI. 8. 1Mb)
1812
en
:S 1.-1)
(n S -A)
(VI. 8.164c)
(VI. 8. 164d)
and tlierefore has charge q =O. One could prove that this is the correct choice by calculating the explicit form of LO' L±l in terms of the modes and showing that they annihilate the state characterized by Eqs. (VI.8.164). A more geometrical proof is based on the uniqueness of the SLZ-invariant two-point function. Let us define ~q(z.w) If the state
= < -Q - qlc(z)b(w)!q
iq
> •
(VI.8.165a)
> is SLZ-invariant then we have
(VI. 8.l65b)
for any Zl
~15bius
transformation w'
CtW + B
yw + <5
(VI. 8.166)
Up to a normalization constant the functional equation (VI.8.165b) has the following unique solution:
(VI. 8. 167) Therefore, the correct SLZ-vacuum is one for which the 2-point function is given by (VI.8.167). Inserting the mode expansions (VI.8.163) into the definition: (VI: 8.168)
and using Eqs. (VI. 8.164), we obtain·
1813
., L
z-n~(l-A) wn - A " _1_
n~A
z- w
(VI. 8.169)
which is indeed the SL(2)-invariant 2-point function. Let us now regard the string amplitude integrand as the expectation value of a product of N+3 vertices
(VI.8.170)
in the universal SL(2)-invariant vacuum which includes also the ghost contribution. Eq. (VI.8.161) implies that the operator 0 should have ghost charge q" 3 in order to yield a non-vanishing result. If all the vertices V(z.) were constructed out of the matter fields and the 1 superghost above, then 0 would have charge zero, since each of the vertices has ghost charge zero. We correct this situation by replacing three of the N+3 vertices with V(z.) -. c(z.)c(i.)V(z;.) 1
111
(i
=1,2,3)
•
(VI. 8.171)
In this way the ghost charge of 0 takes the correct value Q" 3. a result of our procedure we find that the correlation function
As
(VI. 8.172)
of the matter-superghost vertices is multiplied by the correlation function of the ghosts:
(VL 8. 173)
The result (VI.8.173) follows from
1814
setting (VI. 8.175) From the above discussion we have not only gained a deeper understanding
of the SL(2)-invariant integration measure. but also learnt that the vertices must cancel the background charge of the b-c systems involved in our construction. The backgrolUld charge Qgh" -3 of the ghosts having been taken care of, it remains to see how we can cope with the background charge Qsgh = -2 of the superghost system. To this effect we turn our attention to the general problem of bosonization for a b-c system. The proper way to address this problem is to consider the OPE's (VI. 8.152) and to look for a Sugawara-like realization of the stressenergy tensor. If the background charge Q were zero. then Sqs. (VI.8.1S2) would describe the semidirect product of an abelian Kac-Moody algebra with the Virasoro algebra and we could identify the stress-energy tensor with a current bilinear. To SCcOlUlt for the triple pole appearing in Sq. (VI.8.152b). we write instead r(z) "
t
(j(z)j(z) - Qaz Hz»
(VI. 8.176)
which reproduces Eqs. (VI.8.1S2a.b) with the following value of the central charge: (VI.8.177) Comparing Eq. (VI.8.l77) with Eq. (VI.8.142). we see that there is a difference between the ghost and the superghost systems. In the ghost case we have
1815
g(1,-3) " £(1.-3) ;: -26
(VI. 8.178)
and if we bosonhe the ghost current by setting j (gh) (z) ;: i £
az 41 (gh) (z)
'" i
az .(gh) (z)
(VI. 8.179)
where ~h(z) is a free scalar field that obeys the following OPE:
" - R.n(z - w)
+
reg.
(VI. 8.180)
then the stress-energy tensor of the ~h-system (VI. 8.181)
is equivalent to the stress-energy tensor of the original b-c system as both have the same central charge c;: ~26. The complete bosonitation of the ghost system is obtained by the replacement: (VI. 8. 182a)
b(z) ;: exp[-i ~(gh)(~)]
(VI. 8. 182b)
In the superghost case we have instead f(-1,-2) ;: 11 '" g(-I.-2) - 2
(VI. 8. 183)
and, although we can still bosonize the current by setting (VI. 8.184)
where 4l(sgh)(z) is a free scalar with the unphysical sign of the propagator
1816
q,(sgh) (z)q,(sgh) (w) " _ E: 2n(z - w)
+
reg.
" !n(z - w) + reg.
(VI.8.185)
the stress-energy tensor of the cp(sgh)(zJ-field
(VI. 8.186)
is not equivalent to the stress-energy tensor of the original B-y system as it has central charge c =13 rather than 11. This is the signal that the superghosts e(z) and y(z) cannot be bosonized solely in terms of 4>(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)
Therefore, the correct bosonization formulae for the superghosts are
B(z) " e y(z) " e
-H) (sgh) ez)
i
;}z i;(z) n(z)
(VI. 8. 188a)
(VI. 8. 188b)
which substituted in Eq. (Vr.8.141) yield the desired result: (VI.8.l89)
The superghost bosonization introduces the building block we were still lacking in order to construct correct emission vertices having a conformal weight equal to one and cancelling all the background charges. This building block is the superghost-sector vertex operator
1817
(VI. 8.190) For any b-c system, if
~(z)
is defined by
Hz) .. ie: a (l{z)
z
(VI.8.191)
the vertex operator U(q.z) '"
exp[i Q9(z)1
(VI.8.192)
has the following properties whose verification is immediate: a)
It is a conformal field fez) U(q,w) .. _ h _ U(q.w) + _1_ 3 U(q,w) + reg. (z _ w)2 z- w w
(VI. 8.193)
with conformal weight h .. -e: q(q + Q) • 2
(VI.8.194)
b) It is an operator with charge q . () i q 9(w) J z e .. -q- ei q (l{w) + reg. z-w
(VI. 8.195)
c) It obeys the operator algebra: U(p,z) U(q,w) ..
.. (z - w)€pq U(p+q,w){l + (it - w)pj(w) + ••• }
(VI. 8.196)
Using formula (VI.8.194) one easily retrieves the values of the superghost contributions to the intercepts, (see Eqs. (VI.5.Sl» and
1818
discovers the explicit form of the fields L(NS)(Z) ER(z) announced in Eq. (VI.8.S0). Indeed if we set (VI. 8. 197a)
(VI. 8. 197b)
it follows from (VI.8.194) that I(NS)(z) has conformal weight h" 1/2 and IR(z) has conformal weight h .. 3/8. The complete Ramond~vacuum of any superstring can now be viewed as the result of applying to the universal SL(2)-invariant vacuum the product of the w~-spin field S(z) whose conformal dimension is d/16 (see Eq. (VI.8.119)) and the ghost field kR(Z) whose conformal weight is 3/8 as stated above: (VI.8.198)
In d=lO we have
.!Q.+~=l 16
8
(VI. 8. ISS)
so that the operator (VI.8.200)
has a weight equal to one and is a correct form of the fermion emission vertex. In d < 10 the operator v(1/2) (z) has weight fl < 1 since it must be completed by additional spin field factors carrying the internal quantum numbers of the emitted fenuion. These observations take us into the topics of the next section.
1819
VI.8.S The Covariant Lattice for D=lO Superstrings In the present section we fo~s once more on ten-dimensional heterotic superstrings which we us'e as laboratories to develop the covariant lattice construction of the emission vertices.
In particular we want to show how to each solution of the modular invariance problem we can associate a lattice which encompasses all the properties of the string model. The main idea of the lattice approach can be summarized in a few words. Using the techniques discussed in the previous sections. one bosonizes all the fermions and then treats the reSUlting bosons on the same footing as the free scalar ~sgh(Z) related to the superghosts. By means of such a procedure the emission vertex for any state can be written as a vertex operator with the momentum vector lying on a suitable Lorentzian lattice. times the exponential factor exp(ik~X~(z,i», and the derivatives of the scalar fields (both XP(z,z) and the scalars obtained from bosonization). The D=lO models contain the follOWing matter fields: i) the 10 coordinate scalars XP(z,z), ii) the 10 free fermions ,p(z), and iii)
the 32 heterotic fermions ~p(Z) (p ... 1, .•.• 32).
Bosonizing the ten space-time fermions WU(z), we obtain 5 scalars ./(z) (i= 1 ..... 5) fulfilling the OPE .pi (z) .pj (w) -= _ oij in (z - 1-1) +
reg.
(VI. 8. 201)
and spanning the rank of 50(1,9). Bosonizing the 32-heterotic fermions we obtain 16 scalars ~ (1* .. 1, .... 16) fulfilling the OPE ~I* (i) ~J* (w) = _ ol*J" R.n(z - w) + reg.
1*
(~)
(VI. 8. 202)
1820 and spanning the rank of either 50(32) or E8 @ ESp or of some other rank-16 Lie Algebra. The superghost scalar field whose OPE is given by Eq. (VI.S.lSS) can be ajoined to ~i(z) and be regarded as the O-th component of a six-dimensional vector in a space with Lorentzian metric (VI.8.203)
Indeed setting opsgh(z)" ,lC'l.) and I" 0.1,2,3,4,5 we can write I
J
'I (z) 'I (w) " -
nIJ R.n(z - w)
+
reg.
(VI.8.204)
Using these scalar fields as primary building blocks, we consider the conformal fields of the form (q)
V (z,i) ; exp(ikPX (z,i» ]J
O(oo,z)
O(w,z)
(VI. S.205)
where (VI.S.206)
00 "
.:;:
-1
-16
W= A = (A , ..•• A )
(VI.S.207)
are momentum vectors for the left- and right-moving sectors respectively. The number q being the O-th component of the left vector is the superghost charge. All emission vertices for massless and massive states are either of the form (VI.8.205) or of the form (VI.8.205) multiplied by derivatives of the free scalars '1 I (z) (or ~I·(z», with momenta 00, W which are vectors in a suitable lattice. To show this we begin by calculating the conformal weights of a field of the form (Vl.8.20S) multiplied by N derivatives of the left-moving scalars and Nderivatives of the right-moving ones. \'Ie
find
1821 (VI. 8. 208a) ~ 1 2 1 +2 ~ ~=-k+-A+N
2
2
(VI. 8. 20Sb)
2
2
Since k .. -m, where m is the mass of the emitted state, requiring that ~ = =1 we obtain the mass-shell equations;
a
1 2 1 1 +2 - m .. - q(q - 2) + - A + N - 1
(VI. 8. 209a)
222
(VI. 8. 20gb)
which must be compared with Eqs. (VI.7.77). Next we utilize the information obtained from the analysis of modular invariance to decide the .; structure of the momentum lattice to which belong A and A. For this purpose we must pause for a moment and digress to lattices and Lie algebra lattices.
-
DIGRESSION ON LATTICES LIE ALGEBRA LATTICES
~~D
A lattice 1\ is defined as a set of points in a vector space V of dimension N of the form
(VI. 8.210)
e
where 1 (i .. 1, .•. ,N) is a set of basis vectors for V. We shall consider real vector spaces V= IRN or V= lRlIl-p,p the first being enclidean and the second having the Lorentzian signature (-, .... -
+,+, ... ,+)
1822 The covariant metric of the lattice is defined by the scalar products of the basis vectors: (VI. S.211)
The dual lattice A* is defined by introducing a dual basis e~ 1
(i" 1 ••••• N)
such that (VI. 8. 212)
The points of A* are vectors of the fom ni +* ei : (VI. S. 213)
A
1)
lattice
A
is called
unimodular if
lidet gij I = 1 2)
integral if ... +
+ -+
V v. w e A
3)
(VI. S. 214)
v' we Z
(VI.S.2IS)
even if A is integral and if
VveA
+2
v .. Omod2
4)
odd if A is integral but not even
5)
self dual if
A" A* •
(VI. S. 216)
(VI.S.217)
1823
Obviously A is integral if and only if A c A*. Since a lattice A is a group under vector addition, it is natural to consider sub lattices As c A and decompose A into cosets.; In general, if As has the same dimension as A we have a finite number of coset representatives: (a." 1,. .. ,P)
(VI. 8. 218a)
{VI. 8. 218b}
and we can write P
A"
2
0.=1
-+
(8 + A ) a. s
(VI. 8. 219)
We shall be concerned mostly with simply laced Lie algebra lattices. By this we mean sublattices of the weight lattice of a semisimple simply laced Lie algebra, which is a direct sum of An' On' E6, E7, E8 addends. Such algebras have the property that all their roots have the same length. Choosing their normalization so that 0.2 =2 for all the roots, we obtain that the weight lattice Aw is the dual of the root lattice hR' which is itself a sub lattice of the weight lattice. The root lattice is defined by (VI. 8. 220)
where
....
~i
II ;: .'W
where
11
are the simple roots, while the weight lattice is defined by
tn.1
.....
~?
In.1 e I}
are the simple weights:
(VI. 8. 221)
1824 ....i .... A • ~ ....i ... >=2---=.>. .a.j (a.. )2 J
(VI.8.222)
The weight lattice Aw can be decomposed into cosets with respect to the root lattice. In this case the cosets are called conjugacy classes and are the elements of a finite group, the center of the Lie algebra under consideration. As we are going to see, there is a close relationship between this finite group and the group = spanned by the boundary vectors (see Chapter VI.7). The lattice most relevant to us is that associated with the Lie algebra Dr' that is. with the orthogonal algebra in an even dimension: 50(2r).
we shall now discuss in some detail its properties. The simple roots are: (VI. 8. 223a)
r
....
~
...
= e2
...
(VI. 8. 22 3b)
- e3 .. (0.1.-1.0 ••..• 0)
(VI. 8. 223c)
(VI. 8. 223d)
....
....
l ...
Hence a root lattice vector a e AR * a .. n at has the following general fotm:
(VI. 8. 224)
r
I
i=1
k. .. 0 mod 2 1
1825
as the reader can easily prove utilizing Eqs. (VI.8.22S). The simple weights are instead given by Al " (1,0, ... ,0)
(VI. 8. 225a}
1. 2 " (1,1, ... ,0)
(VI. 8. 225b)
(VI. 8. 22Sc)
(VI. 8. 225d)
so that, as the diligent reader can once more easily check, the weight ... lattice is composed of the vectors a" (k i , ... ,kr) whose components k i are all integers or all half-integers:
(VI. 8.226)
7L+l2 The group
1~/AR
is composed of four conjugacy classes: (VI.8.227a)
{v} " AR
+
(VI. 8. 227b)
(0,0,0, ... ,1)
(VI. 8. 227c)
(VI.8.227d) whose multiplication table under addition is two:
eithe~
one of the following
1826
1)
2)
Case of ro: 2.t-SO(4R.): 0
s
v
5
0
0
s
v
S
s
s
0
S
v
v
v
S
0
s
S
S
v
s
0
(VI. 8. 228a)
Case of r= 2R.+1-SO(4R.+2):
0
s
v
S
0
0
s
v
S
s
s
v
5
0
v
v
S
0
s
S
S
0
s
v
(VI. 8. 228b)
As one sees, in the case ro: even, lli/AR is isomorphic to 12 e 1 2, while in the case r= odd it is isomorphic to 1 4 , The irreducible representations of the semisimple Lie algebras are characterized by their weight vectors, which, for a given representation, all belong to the same conjugacY class of the lattice. Hence we can specify conjugacy classes by glving the representations that belong to them. This is the rationale for the names assigned to the four conjugacy classes (VI.8.227). The class {a} contains all even-rank tensor products of the vector representation (which includes the singlet and the adjoint). The class {v} contains all odd-rank tensors (which includes the vector representation). The classes isl and {5} contain all the left-handed and all the right-handed spinor representations respectively. An important property of conjugacy classes for simply laced algebras is the following.
1827
The inner products modulo integers depend only on the class and not on the individual vectors. Indeed, if BI , 82 e AW and aI' a2 e AR we have ~
~
~
~
(VI. 8. 229)
Using this property one can show that the norms of two vectors in the same conjugacy class are always. equal modulo 2. The essential lore about Lie algebra lattices can be condensed in the following table which closes this digression. TABLE VI. 8. II Simply Laced Lie Algebra Lattices Lie Algebra
conjugacy class
smallest representation
norm of the weight mod 2
{OJ
singlet
0
{v}
vector
1
{s}
spinor
r/4
0
r
\s}
conjugate spinor rank p-tensors
A
r
E6
E7
fS
rj4
O~p~r
p(r+l-p) ri- 1
to}
singlet
0
{II
(27)
4/3
{2}
(t7)
4/3
{oj
Singlet
{I}
·56
{OJ
singlet
(p)
0 3/2
0
1828 We can now resume our discussion of the vertices for the D=10 models. In the previous section the canonical Neveu-Schwarz and Ramond vacua were assigned superghost charges qNS =1 and qR" 1/2 respectively. Then the superghost vertex operators which create such states from the SLZ invariant vacuum were included in the definition of the vertices in such a way as to make the total conformal weight equal to one. The restriction of our attention to only such charges is not correct since the OPE
+ .""
(VI. 8.230)
implies that for closure of the operator product algebra we need also vertex operators with different superghost charges other than the canonical ones, qNS", 1. qR", 1/2. NS R The correct way of proceeding is to regard q ,,1 and q =1/2 as representatives of the equivalence classes of integer and halfinteger numbers. The Neveu-Schwarz sector can be defined as the sector where q € I, while the Ramond sector is the one where q € I + 1/2. Within each sector we have an infinity of equivalent vertices for each of the physical states the string can emit. Equivalent vertices V(q) (z,i) integer
and V(q')(z,z) have superghost charges differing by an
q = q' mod l
.
In every ghost sector there is a copy of the canonical vertex operator (corresponding to the choices qNS =1 and qR= 1/2): the different copies are also called pictures. Actually there is an operation which maps a given vertex with superghost charge q into one with q-l: (q-l)
-
V (z,z)
f dw
= -. 2111
pew)
(q)V (z,z) .
(VI. 8. 231a)
1829 The operator P(z), named the picture changing operator, is nothing but the supercurrent Gez) multiplied by exp{ _ i $0 ( ... )}:
(VI. 8. 231b) In this way we see the existence of infinitely many different pictures as the manifestation of world-sheet supersymmetry. Choosing the canonical picture corresponds to fixing a gauge for this symmetry. Considering next the OPE
........
O(w,z) O(w' ,w) " (z - w) A'A'-qq' O(w + w' ,w)
+ •••
(VI. 8. 232a)
We see that it is natural to define the scalar product of the vectors w and w' by the formula w' w' " ~qq' +
........ X' AI
(VI. 8. 232b)
•
It follows that the w's can be regarded as elements of a lattice D~Wi endowed with the Lorentzian metric (VI.8.203) and composed of vector~ of either one of the following two forms: Ai
el
(NS sector)
1 2 3 4 5
w" (q!A ,X ,X ,A ,X) where
(VI. 8. 232c)
Xi e
l +
21
(Ramond sector)
The lattice DiWi is named the covariant weight lattice and has a , structure somewhat reminiscent of a 06 weight lattice. Due to the Lorentzian signature, however, its vectors are not aSSOCiated with the representations of a Lie algebra. Nevertheless, we can identify a sub~ lattice
D~~i
c
Di~i similar to the root lattice and define the space
or conjugacy classes, We set
(W)
(R)
D(5.1)/0(5.1)'
1830
q, Ai e Z (R)
{VI. 8. 232d}
00 e 0(5,1) ~
5
q
+
L
i=1
.
Al
= 0 mod
2
and in full analogy to Eqs. (VI.8.227) we find the following conjugacy classes: {o}5,1
= DS(R),1
{v}5,1 :: { }
(R)
DS,l (R)
s 5,1 :: 0S,1
+
+
°(R) _ (R) w(v) - °5,1
00(0) :: 5,1
_
+ w(s) -
(VI, 8. 233a)
I
(VI. 8. 233b)
(1 1,0,0,0,0)
+
(I 0,0,0,0,0)
+
(2 2'2'2'2'2'2) (VI. 8. 233c)
1 11
(R)
DS,1
I
+
1 1 1 1
1
(VI. 8, 233d)
Under addition the multiplication table of these classes is the following:
{ols,1
{S}S,l
{v}S,1
{S}S,1
{0}S,1
{O}S,l
{s}s,}
{V}S,1
{s}S,I
{S}S,I
{S}S,I
{ols,l
{s}S.I
{v}S,l
{v}S,l
{v}S,l
{S}S,I
{O}S,l
{S}S,1
{s}S,1
{siS,l
{vIs , 1
{s}S,l
{ols,1
(VI. 8. 234)
Comparing with Eqs. (VI.8.228), the reader will notice that D~W~/D~R~ is isomorphic to Z2 ® 12 and hence to D~I~) ID~R), The origin;1 ' lattice Ds' corresponding to the odd case, was instead isomorphic to 14' This is not a coincidence, rather, it is that which allows the introduction of a iattice analogue of the light~cone formalism.
1831
we
know that states (or vertices) with tmphysical values of the
superghost charge are related to th:ir images in the canonical pictures by means of the picture changing operator. this may be seen as follows. ing:
From the lattice's viewpoint
Any lattice vector admits a unique splitt·
W '" wlight-cone + 6
(VI.8.23S)
with
wlight.cone ..
weO) +
rl
w(v) +
bl
wr + il
(VI. 8.236)
We!!) + w(s)
where
{j)'t
is a transverse "root" vectol': (VI. 8. 237a)
4
ni e Z. and
a 6
X n.
• 0 mod 2
(VI. 8. 237b)
1=1 1
is a light-like root' vector
= (plo.o,o,O.q)
(p+q '" 0 mod 2)
(VI. 8.238)
Recalling the structure of the ten-dimensional supercurrent G(7.) '"
12 e h / 4
p
jJ.
(z)~(z)
(VI. 8. 239)
and introducing bosonization. we see that the picture changing operation on lattice vectors corresponds essentially to the addition of light-like root vectors
a.
This provides an intuitive relationship between states
in the unphysical ghost sectors and physical states. More explicitly, W
what we have achieved is a decomposition of the weight-lattice DS• 1:
1832 (VI. S.240) where the transverse coset D:/D! is isomorphic to
D~.l/O~.l'
Using this formalism the states and bence the vertices of the two ten~dimensional superstrings can be associated with tbe conjugacy classes of the following lattices: A)
Covariant la.ttice of the 50(32) superstring
. . _
(W)
(If)
rS , 1'16 - DS 1 e D16
B)
(VI. S. 240a)
Covariant lattice of the EseES superstring (VI. 8. 240b)
There is indeed a simple relationship between the language of boundary vecto.s uti lized in Chapter Vi. 7 and the language of lattice conjugacy classes adopted here which we want to clarify by discussing the Simplest of the examples treated in Chapter VI.7, namely, the 50(32) superstring.
VI.8.6 Conjugacy classes and GSO projectors: The 50(32) example in 0=10 As stated above, in this section we utilize the 50(32) D-10 theory as a laboratory to clarify the relation between boundary vectors. GSO projectors and conjugacy classes of the lattice. Recalling the results of Chapter VI.7, in the example under con~ sideration, the group = is composed of the following boundary vectors:
0= {O •...• O O•... ,O} 8
32
(VI.8.24Ia)
1833 5 = {l •••• ,l ; O••••• O}
8
s ..
32
{O, ... ,O; 1.... ,l}
= {l, .•• ,l
(VI. 8. 241c)
32
8
1
(Vl. 8. 241h)
; l ••.. ,l}
8
(VI. 8. 241d)
32
As we see. the transverse space-time fermions and the 32 heterotic fermions constitute two subsets that have always the same boundary conditions. This is the reason why we have stated that the lattice corresponding to this superstring is the one given by Sq. (VI.8.240a) •. In the lattice language, NS boundary condition correspond to the conjugacy classes to} and {v}, while Ramond bOlBldary conditions correspond to the conjugacy classes is} and Is}. Therefore. the sectors [0]. (5]. [5] and [IJ correspond to the following sets of conjugacy classes:
(o}
[5]
[51
=
{Ols,!
®
{0}16
{ols,l
®
{vl 16
{v}S,l
$
{o}16
{v}S,l
®
{vl I6
{s} 5,1
® { O116
(s}S,l
®
{S}S.I
® {O}I6
{S}S,l
® {V}16
{O}S,l
®
lvl l6
(VI. 8. 242a)
(VI. 8. 242b)
{s}16
{ols.1 e {si I6 MS,l
®
{s}16
{v}S.l
@
{5}16
(VI. 8. 242c)
1834
{S}S,1 ~ {s}16 [11 ..
{s}S,l ~ {s}16
{s}S,I ~ lS}16
(VI. 8. 242d)
{s}S,1 ~ {s}16 altogether summing up to the 4 x 4 .. 16 possibilities of combining the four conjugacy classes of DS,1 with the four conjugacy classes of D16 • The effect of the GSO projection is to save one combination for a sector, discarding the remaining three. Indeed one can prove that if we fix, for instance e: .. I') =1 then the conjugacy classes surviving the projections (VI. 7. 149b} , (VI.7.150b,d) and (VI.7.1S1b) are the following:
[oj ~ {O}S,l
& {O}l6
(VI. 8. 243a)
[51 ~ {S}S,l ~ {O}16
(VI. 8. 243b)
[5] ~ {vIs,l ~ {s]16
(VI. 8. 243c)
fl] ~ {s}S,1 ~ {s}16
(VI. 8. 243d)
To prove the result it suffices to write the fermion number operator F (see Eq. (VI.6.290» in bosonized form. For any set of 2r fermions, >hz) (A= 1.... ,2.), fulfilling the OPE (VI.8.84a). the fermion number operator is defined by the property (VI. 8.244)
and is represented by the following integral: (VI. 8. 245)
1835 SUbstituting the mode expansion of ).A(z)
into Eq. (VI.8.24S) one
retrieves the formula (VI.6.290) expressing F in terms of the ferOn the other nand, re~alling that
mionic oscillators.
: ).hw}>hw) : .. lim {>hz»hw) z+w
i
2:r 1
-r 2 z-w
+ -
{VI. 8.246)
and utilizing Eq. (VI.8.93) we obtain
+
O(e. ,w) O(-e. ,z) 1
1
1+ -i2
r
i -1:} 2 w-z
-- + z~w
(VI. 8. 247)
so that, upon use of Eqs. (Vl.8.92) and (VI.8.9S), we finally get
F ..
-!i ..rfdW+ -. e. 1 2111 1
r... ,,- L ei
-+ 0
H (w) ..
-+
(VI. 8. 248)
' p
i=l
Equation (VI.8.248) shows that the fermion number operator is, essentially, the momentum operator on the lattice.
It follows that given a
1A > which is created from the SL2invariant vacuum by the action of a vertex operator O().,z), the operator (_l)boF acts on I). > as follows:
boundary vector b and a state
(VI. 8. 249) Sin~e
b COUllts the Ramond fermions, from the conjugacy classes' point
}b
of view, is the representative of a spinor conjugacy class {s}, i.e., it is the vector 00(5) for some D(W) sublattice of the total p
lattice. Hence, the "elementary" projector
1836 (VI. 8, 250)
acts as follows on the lattice:
(VI. 8, 251)
where Q .. (QS,!,?i) denotes the lattice vector. In particular, extending the notion of fermion number also to the superghosts (for the superghosts it is nothing but the charge q), the four GSQ projectors (VI.7.149b), (VI.7.1S0b), (VI.7.1S0d) and (VI.7.151b) admit the fOllowing bosonized transcription:
In >
(Vl.8.252a)
(VI. 8. 252b)
1(
s
4
x ( 1-
16
2ni<W (S),Q»
JP _I n > ,,- 1 + En e
x
21fi [< Ws 1(5) + 00 16 (5) ,n >] ) En e '
In >
(VI. 8, 2S2c)
(VI.8.252d)
1837
To work out the result (VI.S.243) at this point, it suffices to note that for both the Lorentzian lattice DS,l and any enclidean D4v lattice the following inner product relations hold true:
ez
(VI. 8. 2S3a)
< (0) , (i) > e l.
(VI. 8. 253b)
< (i), (i) >
.!.2
< (v) , (5) >
e l.
+
< (v) • (5) >
eZ
+-
(VI.a.253d)
Z +.!. .
(VI. S. 2S3e)
< (5) • (il > e
1
2
2
(VI. 8. 253c)
Changing the sign of the parameter E has the effect of replacing the spinor representations {S}16 of the gauge group 50(32) with their chiral conjugates is} 16' Similarly. flipping the sign of n replaces the representations {s1 S,l of the Lorentz group with those of opposite chirality {s}S l' I
In full analogy one can treat the other examples given in Chapter VI. 7.
For the ES ES case, the lattice is given by Eq. (VI.S.240b) so that we have 43 .. 64 possible combinations of conjugacy classes of which only a few survive the GSO projection. Utilizing the fact that the union of the root lattice D~R) with its spinor conjugacy class {sIs is equal to the root lattice of E8, one realizes that the G50 projected lattice is composed of conjugacy classes of the lattice (VI. 8. 254)
Later on we shall spend few words on the covariant lattice associated with ~4 theories. Anyhow, the idea of how the lattice is associated
with the solution of the modular invariance problem should be clear from the e~amples. It is now time to focus on the massless emission vertices and the low energy effective theory.
VI.8.? Massless emission vertices and the effective theory of D=10 superstrings The final goal of heterotic superstring theory is the classification of the N=l, 0=4 models and the derivation of the associated effective supergravity theories. As we have stressed several times, such a classification and the investigation of the D=4 effective Lagrangians is the subject of current research. Hence, for pedagogical purposes we have chosen to work with the D=10 models that are already classified and whose structure is much Simpler. From the conceptual point of view, this choice does not imply any limitation. Indeed, all the features of the D=4 theories are already present in the ten-dimensional models. In this section, therefore, we list all the emission vertices for massless particles in N=l, D=10 heterotic superstring theory. Then, utilizing these vertices we derive the three point functions that fix the structure of the low-energy effective supergravity. Our main goal is to determine the relation between the dimensionful parameters entering the effective Lagrangian and the string tension al • Furthermore, we want to show the emergence within string theory of the Lorentz Chern-Simons form modifying the definition of the axion field strength. This is the essential motivation for the study of Anomaly Free Supergravity which is the subject of the next chapter. Indeed, the Lorentz Chern-Simons form is what, on one side, makes the theory non-anomalous (Green-Schwarz mechanism) and on the other side causes its highly nonlinear structure. Let us begin our study of the massless emission vertices. Recalling Tables VI.?II and VI.?III, we realize that we just need four
1839 emission vertices of which the first is associated to the emission of the graviton, the axion and the dilaton, the second to the emission of the gravitino and the dilatino (also called gravitello), the third to the emission of the gauge bosons and the fourth to the emission of the gaugino: (q)
VJ,lv(k;z,z)
emits
Vj.I (k;z,z)
\
)
",a
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 <
state J state >
~
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<
21
V (2: 1,zl)
(VI.8.259)
are well defined and non-zero if and only if the sum of the supetghost charges cancels the background charge: N
L
q.
1=1 1
=2
(VI. 8. 260)
(see Section VI.8.4). Such a situation cannot be obtained if we utilize only the canonical picture. For instance we cannot write three-boson or four-boson amplitudes. Equation (VI.8.260), on the other hand, can always be satisfied if we have two pictures in our stock: the canonical picture and the O-th picture characterized by q =0
in the NS sector (VI. 8. 261)
q
= -1/2
in the R sector
Hence, in order to be able to write down all the scattering amplitudes, we just need eight vertices, that is, the four vertices (VI.8.25S) in the canonical picture and their copies in the O-th picture. Let us begin with the canonical picture. We write
(VI. 8. 262a)
1841 i1r
(~~A(k;Z,i) = 2 ell ei~{Z)W~(Z)jA(i)eik'X(Z.Z)
(VI. S. 262b)
(VI. S. 262c)
(VI. 8. 262d) where Pv(z) and Wu(Z) have the OPE given by Eqs. (VI.5.13c) and (VI.5.14) and admit the mode expansions (Vl.S.SSb) and (Vr.S.SSe), X~(z,z) is given by the mode-expansion (VI.6.71), Sa(z) is the spinfield of the $0(1,9) algebra fulfilling the OPE (VI.8.119) with W~(z) and JUVCz), and jA(i) is the Kac-Moody current constructed with the 32 heterotic fermions and (in the Eg® Eg case) their spin fields and obey the k=2 algebra
oAB
+---
(VI. 8. 263)
Ci - w)2 ~BC being the structure constants of the gauge group fS
(SO(32) or
Eg).
The reader can check that the bosonic vertices are correctly normalized in order to create unit norm states:
(VI.8.264a)
(VI. 8. 264b)
lilA>
=
lim
z,z'" 0
(1) A V (z,z)IO >SL
U
2
(VI. 8. 264e)
(VI.8.264dJ
1842 The normalization of the fermionic vertices (parameter a, b) is left to the reader as exercise. If we recall the explicit form (VI.S.lS) of the supercurrent, the picture changing operator can be written as r:; i1l/4 pew) = 12 e e -iA.{w) 'I' rp (w)P II (1'1)
(VI. 8. 265)
\l
Inserting (VI.8.26S) and (Vl.8.262) into Eq. (VI.8.231) we obtain·the explicit form of the vertices in the O-th picture, We confine ourselves ~o the bosonic vertices (the only ones we need in the sequel):
(~)
(k;z,i) =
~
(~~ A(k;z.z)
12 [p
= 12
II
(z) +
[pll(z)
2ikA~\ (z).p A
II
(z)] ji (f)eik'X(z.z) v (VI. 8. 266a)
+ 2ikA1/I), (z)l/Il-I(z)
I JA(z)e ike X(z, i)
.
(VI. 8. 266b)
Given the above vertices we calculate the following three-point functions: 1)
Three "graviton" vertex
(1)
I
Vll1V1 (k 1;zl,i 1) 0 > (VI. 8.267a)
1843 2)
Two "gluon"-one "graviton" vertex
(1) A.
_
I
Vu1 (k 1;Zl'Zl) 0
>
(VI. 8. 26 7b)
3)
Three "gluon lt vertex
where p~)
is the polarization tensor of an on-shell tensor particle or 0 depending on whether it is symmetric traceless. antisymmetric or proportional to nllV ), and £ti) is the polarization vector of a gauge boson AA, G is the string coupling constant which \.I is dimensionless. For real momenta the three amplitudes (Vr.8.267) vanish since they are proportional to a conservation delta function (g~v' B~,
1844
which, together with the shell conditions, (VI. 8. 268) yields the colinearity conditions: (VI. 8. 269)
k·k=k·k=k·k",O 1 2 2 .3 .3 1
whose only solution on the real axis is kl = k2" k3" O. By analyticity we can however extend our amplitude to complex momenta, and in this way we obtain a non-vanishing result which can be compared with the trilinear vertices of the low energy effective Lagrangian. Since our amplitudes are on-shell to Eqs. (VI.8.268-269) we must adjoin the transversality conditions: (VI. 8. 270)
From the 2-dimensional point of view, Eqs. (VI.8.270) are implied by the requirement of BRST invariance: (VI. 8. 271a)
(VI.8.271b) Utilizing Eqs. (VI.8.268. 269, 270) the amplitudes (VI.8.267) can be explicitly calculated and one obtains the following results:
lJ\I
.. _ ~ G P .3 .3
;;r
3
lJ\I f} 2 2
2
lJ\I
p lIt
1
[
(k) t lJ 3lJ 2lJ 1
\)3\)2\)1
\I
\I
\I]
2
1
3
(k) + k .3 k 2 k 1
(VI. 8. 272a)
1845
(VI.8.272b)
(VI.8.272c)
where the tensor t
1J 311ZlJ 1
(k) is defined as
(VI. 8.273)
The derivation of Eqs. (VI.8.272) is quite simple and provides an
excellent illustration of how the 2-dimensional conformal field theory is utilized to derive string amplitudes. Recalling Eq. (VI.6.71) the exponential factor exp(ik·X(z,z») can be rewritten as follows:
·k' X( Z,Z-) exp (l
= e ik·X(z)
eikoX(z)
(VI.8.Z74)
where we have defined xJ.l (z)
+
ipll tg z - i
.!. Xll z-n n#O n n
(VI. 8. 27Sa)
= .!.qIJ
+
iplltgz _ i
I .!.j(ll z-n n,lO n n
(VI. 8. 27Sb)
2
XIJ (2)
I
= .!. qlJ 2
1846
Upon use of Eq. (VI.8.274) the vertices (VI.8.262) and (VI.8.266) factorize into the product of a left-moving vertex times a right-moving one. In this way the amplitudes (VI.8.266) are also factorized into the product of a function of zi I times a function of zi' The building blocks in the calculation of these functions are provided by the following correlation functions (and their analogues with z +>- z): (VI. 8. 276a)
(VI. 8. 276b)
1
ik1'X(Zl}
> = --------- 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
For such fields we
(VI. 8.279)
Since all the conformal fields appearing in the vertices are either free fields or exponentials of free fields, Sq. (VI.8.279) suffices to calculate any correlator. in particular those of Eqs. (VI.8.276). Inserting the explicit forms (VI.8.262) and (VI.8.266) of the vertices into the definitions (VI.8.267) of the three·point amplitudes and utilizing the correlators (VI.8.276) the result (VI.8.2?2) follows iJlllllediately. Equations (VI.8.272) are sufficient to work out the essential features of the low energy effective action. As we know since Chapter VI.?, the massless states of the heterotic superstring correspond to the field content of N=l, D=lO supergravity coupled to the gauge multiplet of either the SO(32} or the Eg® Eg group.
1848
Since supersymmetry has to be preserved the effective Lagrangian must be the Lagrangian of ten·dimensional N=l supergravity. ·which is studied at length in the next chapter. There it is shown that there are actually two versions of the theory corresponding to the gauging of two inequivalent free differential algebras. Both algebras are encoded in Eq. (VI.9.23) and are obtained from a specialization of the parameters (l3;y). If we set B" y =0 we deal with pure supergravity without the Yang.Mills multiplet. Since the superstring contains also the gauge degrees of freedom, B has to be different from zero. Fixing the normalization of All from the canonical choice of its kinetic Lagrangian. the parameter B gets fixed by supersymmetry and takes the value (see Eq. (VI.9.301)):
B=- 4
.
(VI.8.2S0)
On the other hand, as far as supersymmetry is concerned the parameter y remains free. If we insist on y " 0, we obtain the so-called Chapline·~lanton Lagrangian (CM) which is limited to second-order deriva· tlves. This theory is anomalous for any choice of the gauge group. On the other hand, if we allow non·zero values of y.
the theory becomes highly non· linear (actually it contains an unlimited number of derivatives). and for a special value of y in the case the gauge group is either SO(32) or E8 e E8, all field theory anomalies cancel. This value is y"
1 32
(VI. 8. 281)
The theory characterized by the values (VI.8.Z80) and (VI.S.l8I) of the parameters is called Anomaly Free Supergravity (AFS). Since the superstring is not only supersymmetric but also nonanomalous. its effective Lagrangian in the point-like limit a' + 0 should be the AFS Lagrangian. This is what we shall presently' verify by comparing the three· point functions obtained from the cubic interaction terms of the AFS
1849 Lagrangian with the stringy three-point functions (VI.8.272). Through this comparison, besides retrieving Eqs. (VI.8.280) and (VI.8.281) we also obtain the relation between the string parameters a' and G and the field theory parameters, i.e., the Newton constant gauge coupling constant g.
K
and
To this effect we refer to the bosonic part of the action, as it is given in the Chapline-Manton normalizations (see Eqs. (VI.9.316)): L(Bosonic)
A
(det V)
[
A
2
_1.. ~-2 16
- -1"R(w) - -3 ~-3/2" ~ H H d
~d ~
~]J
4
+
~vP ~vP
! ~-3/4 FA FA ] . 4
J.IV llV
(V 1. 8. 282)
The hatted fields are related to those used in the geometrical formulation of Table VI.9.V! by the transformation (VI.9.30S). These latter are related to those utilized in the solution of Bianchi identities for AFS (see Table VI.9.lI!) by the Weyl transformation (VI.9.290). Eq. (VI.8.292) describes the Chapline-Manton theory and hence does not contain the parameter y. Since we limit ourselves to cubic terms with at most four derivatives, it suffices to introduce into Eq. (VI.8.282) the modification (VI.9.12a) of terms originating from the coupling do not contribute In addition to introducing
the axion field strength. All the other supersymmetry completion of this additional to the cubic vertices we shall be considering. y through the shift (VI.9.12a), we perform
a rescaling of the fields which reinstates the dimensionful coupling constants: (VI.8.283a) 2
A1.1\1 ".L f2 K
B"v ..
(VI. 8, 283b)
(VI. 8. 28Se)
(VI. 8. 283d)
1850 The dimensions of the gauge and of the gravitational coupling constants
are respectively given by
(VI. 8. 284a) (VI. 8. 284b)
After this transformation the bosonic part of the AFS Lagrangian becomes
!I '" det V ( __1_ R(w) _ 1 e-teD/f'l pA pA 2K2
-
4
t
).IV IN
~2 e- KD/1"2 HJ.IVP).IVP H - 13 03 D) + 211lJ
•
••
(VI. 8. 285)
where
H '" j.IVP
a[II -Bvp]
+ -1 8K(F
- - -2 g -A -A A - ) A lJ\I P 3 -~.~ P
8
I K -ab -ab 2 -ab -ac -cb - - y - (R-!.II - -!.II !.II !.II ) •
2
g2W P
In (VI.8.28S). the fields If we write
3 lJ
"
P
{VI. 8.286)
- - -D have canonical kinetic terms. B\.I\I'~'
(VI. 8.287)
also hllv has a canonical kinetic term. Expanding up to the cubic order in the fields we obtain the following interaction Lagrangians
(VI.8.288a) y(S} OM
= _K_ Da llA"C3 2/2
A _ 3 A) ).1\1
\Ill
(VI. 8. 288b)
1851 (VI.8.28Se)
(VI. 8. 288d)
3 !eBh(3h) .. Y ~ g2
(aIl aahb"II a~bp ~
a a h. ;hap) )(
1l a--bv
x (ailgVP + aVB PIl + aPBIlV)
!l (3) .. _ ! (3 A AAA 2 ll'v
_
(V1.8.288e)
avAIl ) (A·P·v A )
(VI. 8. 288f)
Which in momentum space lead to the following interaction vertices:
(VI. 8. 289a)
(VI. 8. 289b)
(VI. 8. 28ge)
(VI. 8. 28ge)
+ ( kl - k3 )
1l2
\.13\.11 T)
( + k3" k2
)lJ 1
1l31l 1] T}
(VI. 8. 289f)
where sVV(k) is the symmetric polarization tensor of the graviton, aUV(k) is the antisymmetric polarization tensor of the axion and D(k) ,
1852
which we omit is the singlet polarization tensor of the dilatonj finally ~~ is the polarization vector of the gluon. The field-theory vertices (VI.S.29B) match the stringy ones, given in Eqs. (VI.8.272) if we identify
(VI. 8. 290a) (VI.8.290b) and if we enforce the following relations - -
8
f2
- -
8
12
K
G " - (2a')
2
(VI. S. 291a)
2
G =<
-
2y
K3
~
(2a')
2
(VI. S. 291b)
g2
(VI. S. 291c)
__S_G
= _2_g
f2
f2
(2a.)3/2
(VI. 8. 291d)
The normalizations (VI.8.290) follow from a comparison of the vertices AhOD ' AOSS and ADAA · The relations (VI.S.29I) correspond, respectively, to the identification of the vertices AhOO ' AShh ' ASAA and AAAA of model with their analogues in field theory. The factors inserted in such a way as to make G dimensionless. The which was set equal to one for convenience throughout the reinstated in this way.
the string (2a') are string tension chapter is
Eqs. (VI. S. 291) imply B:: -4 and y" - 1/32 and yield furthermore the relation
(VI. 8. 292)
1853
As can be seen, the gauge coupling constant and the Ne.~on constant are not independent: they are both expressed in terms of the string tension. This is the signal of the unification of the gravitational interactions with the gauge interactions provided by the underlying string theory. If the gauge constant is of order one then the string tension is of the order of the Planck mass. The massive modes are therefore really massive and can be safely neglected in most considerations. The values of S and y guarantee that the low energy theory is anomaly-free supergravity in D=10. Similar procedures can be utilized to investigate the effective theory of the f):4 models. Here however we have further complications due to the variety of available theories and the presence of the scalar field geometry. In the next chapter we shall study the general structure of anomaly-free supergravities both in 0=10 and in D=4.
1854
CHAPTER VI.9
EFFECTIVE SUPERGRAVITY THEORIES AND THE COUPLING OF THE LORENTZ CHERN-SIMONS TERM
VI.9.l Introduction The D=10 theory is a very useful laboratory to investigate general properties of string theory at the various stages. We already utilized it in Chapter VI.7 to illustrate the solution of the modular invariance constraints and in Chapter VI.S to illustrate the relation between string amplitudes and the effective theory. We saw that this latter is in any case an (anomaly-free) supergravity theory. There are two methods to construct the effective supergravity of a given string model. One method is based on the calculation of string tree amplitudes for the massless modes. One tries to retrieve a Lagrangian which reproduces them in a field theory framework. An example of tbis procedure was given in Chapter VI.S. Another method is based on the a-model. One considers the propagation of the superstring on a classical bosonic background described by the massless fields (i.e. the metric, axion and dilaton in 0=10; in D=4 there are also the scalar fields). Quantum consistency of the 0model requires vanishing B-functions. This yields differential equations
1855
on the background fields which can be reinterpreted as equations of motion of the effective Lagrangian in the tree approximation. In either approach one finds that the effective theory contains arbitrarily high derivative interactions. This is the counterpart, in the field theoretic language, of the extended nature of the string. Hence, it is essential to determine the general structure of the higher derivative supergravity theory corresponding to the chosen number of dimensions and of supersYllllnetry generators. So far we have limited ourselves to second order derivative supergravities. This is not necessary. The expansion of a nonlinear supergravity in the number of derivatives can be. viewed as a power series expansion in the dimensionful parameter 1'2(ii. The relation between Newton r s constant and (l' depends on the model and for the D=10 heterotic case was given by Eq. (VI. 8.292).
This expansion, as we are going to see, is due to the simultaneous presence of two mechanisms. First one has infinitely many higher curvature invariants that one can supersymmetrize and add to the second derivative Lagrangian. Their coefficients are arbitrary and can be fixed only by comparison with the superstring amplitudes or with the a-model B functions. For the D-10 case we will argue that this type of higher derivative structure is fully controlled by a single spinor tensor ~~ich appears in the superspace parametritation of the ax ion field strength, precisely in the (O,3)·sector (see below). The explicit expression of the spinor-tensor in terms of the fields of the theory is what supersymmetry cannot predict but must be worked out by comparison with the string amplitudes. Secondly one has another source of higher derivatives coming from the transition from first to second order formalism in presence of the Lorentz Chern-Simons terms. This type of non-locality can be generated by the solution o~ a differential equation. namely, the torsion equation. This equation is just one of the field equations in the first order approach to supergravity, which is a distinctive feature of the rheono~ framework. To understand how this can happen let us consider the following toy example in 0=4.
Let
1856 S..
d J (RabC "v ... V- €abcd
+
abcd ahabcd R "R )
(VI.9.1)
be the (nonwsupersymmetric) action containing the coupling of a scalar field ~ to a term of order a in the Lorentz curvature (without the scalar field in front, the term Eabcd Rab ... Rcd would be a total deriw vative). By variation of the Va,. and wah fields we obtain the equations of motion: ab
R
c
"v Eabcd
(VI.9.2)
=0
ab cd a Eabcd R " R
=0
c d cd T "V - 2a d4> .... R )
where TC is the torsion
(VI. 9.3)
E:abcd
=0
r C =f!) yC.
(VI. 9.4)
new feature wi th respect to usual gravity theories is the fact that the torsion equation (VI.9.4) ab is differential rather than algebraic in the spin connection II) • By means of standard manipulations Eqs. (VI.9.2 w4) become A
(VI.9. Sa)
(VI.9.Sb)
[c
d]
T [pq 0 m]
+
cd 2a a[p~ R qm j(w)
=0
.
(YI.9.Sc)
Contracting the indices d and m in Eq. (Vl.9.Sc) and using R' c (Ul) .. 0, as it follows from (VI.9.Sa), one obtains 'q
(VI. 9. 6)
This is a first order differential equat~on for the spin connection wab~. We can solve it perturbatively in a by setting
1857
(VI .9. 7)
oab
where
is the usual metric connection.
IJ
&I
At a= 0 we find
(~) =0, and Eqs. (VI.9.S) become the usual Einstein equations. pq first order in a we find RC
(1) c f 4 -
PI [V ~ v]
W
me
0
9 a am~ R IJV (W)
+ -
RmclJV being the usual Riemann tensor. in solving Eq. (1.2.42) one finds (1)
w ablJ
=
t a Va IA Vb Iv
3p
=0
At
(VI. 9. 8)
By the same method already used
~ [RPAlpv(£)
+
RPlJlvA(~)]
RPvIAUC£) -
(V1.9.9) oab
(1) b IJ
Inserting w + a w a J,I
into the equations (VI.9.Sa, b) one obtains
the equations of motion of the physical fields . 1y: a and a -1 respective
glJv and
~
at order
010
Rab (w) -
= -
2" nab R(w) =
coR. 9"4{ a 9 a (34)) Rcb(w) - 9 (ac~)
- aC4>
~R.
Reb[aR. +
2 ~i
~j
Rij
(~)
co c 0 R b[aR.(w) + a 4>~b Rac(w) -
+
2ai~ ~
Rij
(£)}
+ 0(a 2)
(VI. 9.10) o
0
R(w) R. j (Ul) ~
i j
a Ij)
1
+
Rij
lu
Rm
r,;,
-
2
1J
a .. =
jlkR. ~i mV
~.
R.,
jk
R
i
~ ak~
" .. ~i R - °i v ~ ij
ij
. 2R +
1
a ..
2 tV
R. '1'. ~ a ~ J.. III
m
R
1
R
+
R.. _ 2 Rij 3 4> 9 1R Rt. _ 9 Ij) Rab Icd 9 Rt + 1J t ljm t a bed
+
2 aR.4> Rij
+
9 1 (R2 _ 4 Rij R., ij [Itt ) (0 16 1J + R Rij Iu. + 0 a ) .
a
J
+
(VI. 9.10b)
1858 We see that higher derivative interactions are generated by the order a solution of the differential equation for the torsion field ~bc' Proceeding in the same way for higher orders in the a parameter, it is clear that one generates terms with an arbitrary number of derivatives (or curvatures). As we discuss later on, precisely the same type of mechanism is present in the effective theory of the D=10, N=l heterotic superstring, and in general, in any effective supergravity theory. In both D=lO and D=4 effective supergravity theories, this mechanism, pro~ ducing non~locality through a differential equation, is due to the presence of the Lorentz-Chern-Simons form (VI.9.l1)
This term was introduced by Green and Schwarz in order to cancel the gauge and Lorentz anomalies. The field strength .It of the 2~form B must be modified as follows: o
0
K
jf+.It".fI' - y -
l
{lew)
(VI. 9. 12a)
where
(VI. 9.12b) is the field strength of the Chapline~Manton formulation [Ref. 6J. is the Yang-Mills Chern-Simons form
{leA)
!'I(A) " tr(A"dA •
'23 gLLA)
and K and g are respectively the gravitational and pling constants in ten dimensions.
(VI. 9.13) Yang~Mills
cou-
The parameter B is fixed by supersymmetry once the gauge field is canonically normalized. We will show in later sections that its value is B=-4. The same value was calculated in the previous chapter
1859
from superstring tree amplitudes. The parameter y is free as far as supersymmetry is concerned and gets fixed by the requirement of anomaly cancellation (y" -1/32). This value was also calculated directly from the tree amplitudes. The addition of new) to} in the ChaplineManton Lagrangian given in Eqs. (VI.9.316) breaks supersymmetry. The supersymmetric completion of Q(w) is thus the first problem to be solved in order to construct an anomaly-free effective theory of the 0=10 heterotic string. As we see in Sections Vl.9.8-l0, an analogous problem arises in the D=4, N=l case. Also there the Lorentz ChernSimons form appears in the axion field strength. While the D~lO axion is part of the gravitational multiplet, the D=4 axion 8 is part of \.IV the linear multiplet, containing also the dilaton ~ and the dilatino X. The spin 1/2 field X has the vacuum quantum numbers. As already stressed, besides the supersymmetric completion of the Lorentz Chern-Simons form, there are infinitely many more higher curvature interactions needed to cancel the 2D superconformal anomaly. They correspond to independent supersymmetric invariants. The lowest of these invariants, discovered by Grisaru, Zanon and van de Vent corresponds to a R4 term in the action with a transcendental coefficient ~(3). The spinal' tensor appearing in the parametri~ation of the axion field strength we mentioned before is responsible for the ~(3) invariant and all the higher ones. To study the structure of higher derivative supergravities, in the following sections we utilize Bianchi identities in superspace. In D=10. these close only on shell, and hence imply the field equations. This is not the case for D=4, where the field equations are retrieved only after specifYing the dependence of the auxiliary fields on the physical ones.
VI.9.2 The algebraic basis of D=10, N=l matter-coupled
super~ravity
The theory contains two supermu1tiplets: the graviton multiplet and the gauge multiplet. They have, respectively, 64(964 and (86)8)®496 on-shell degrees of freedom which are distributed as follows among the massless fields of Table VI. 7. I: 35 fOr the graviton v~, 28 for the
1860 axion B , 1 for the dilaton ., 56 for the gravitino ~, 8 for the PV P dilatino (or gravitello) ., S®496 for the gauge boson \, and Se496 for the gaugino AA. Let us introduce the Free Differential Algebra (FDA) underlying our field theory. According to the procedure explained in Chapter 1II.6 we start fl'Oill the D,,10, N=l super Poincare Lie algebra ~ah
!Jr-
ab - f.lIa ... wcb ,,0 c
"dw
(VI. 9.14a)
(VI.9.14b) (VI.9.14c) and look for non-trivial cocycles. Taking into account the Fierz identity (VI. 9.15)
which immediately follo~s from Table II.S.XIV, we find the following non-trivial cocycle Q: (VI. 9.16)
a being an arbitrary constant. Hence we can introduce a 2-fotm B and adjoin to Eqs. (VI.9.14) the following: (VI. 9. 17)
where we have set <:4" -1/2 by convention. Equations (VI.9.14) and (VI.9.17) are a possible starting point, but not the most general one, for the construction of pure 0=10 supergravity theory. There is indeed a further non-tr~vial extension of the previous FDA; it can be easily found by recalling the Chevalley-Eilenberg second theorem of Sect. III.6.3. In fact the 3-form:
1861
(VI. 9.18) is a nonwtrivial cocycle of the SO(1,9)wLie algebra. Consequent"ly the definition (VI.9.17) can be extended in the following way: vP-
... :: dB
w
i-IjI 2
-
A
raljI ... Va
1 - -
3
yewab .. w.0 c
c
A
(jl
a
)
..
0
(VI.9.19)
y being a constant. At this point we could construct the supergravity theories corresponding to the FDA's (VI.9.l4, 17) or (VI.9.14, 19). However, ~~ know tqat the effective theory of the tenwdimensional heterotic string contains also a gauge field A (and its superpartner, the gaugino A). It is therefore convenient to consider the differential algebra of the coupled supergravity super YangwMills system (SUGRA 61 SYM in the following). The dynamics of the (A,A) multiplet has been studied in the rigid supersymmetry case in Sect. 11.9.3. The gauge group MaurerwCartan equations are obtained by stating that the field strength , defined in (II. 9.34) * is zero:
(VI. 9.20) Adding (VI.9.20) to the supergravity FDA one gets a direct sum of FDA's and apparently no coupling is obtained, even in the gauged case (that is, when all the "curvatures" are taken different from zero). However the presence of the gauge field A makes it possible to find a new cocycle for the combined systems (VI. 9.14, 19). (VI. 9.20). Indeed, the presence of the YangwMills potential allows the extension of the previous FDA by means of the new cocycle: o
Q(A) =
w
3'1 Tr{A" A .... A)
(VI.9.21)
where Tr denotes a trace over the adjoint representation of the gauge group G. Therefore Eq. (VI.9.19) can be further extended as follows: * Since we have denoted by B the 2-form defined in (VI.9.17), the gauge field denoted by B in Sect. Il.9 will be denoted by A, in agreement with standard notations.
1862 i- a l l abc A': dB - -21/1"r IPAVa - -3 B Tr(A"A"A) - -yew ... 00 c ... 00 a ) 3 b
=0
(VI.9.22)
S being some constant. In this way one gets a coupling between the supergravity and the Yang-Mills sectors. Let us now gauge the FDA (VI.9.14), (VI.9.l9, 20). Out of the vacuum we define gpab
..
db>ab
-
II)
a ab c"W
(VI.9.23a)
(VI. 9. 23b)
(VI.9.23c) 4 .If .. dB -
"2i e'3° ~" ra~"Va + B Q(A)
-
y Q(w)
(VI. 9.23d)
(VI. 9.23e)
where neAl and !l(w) are the Chern-Simons forms of the gauge group G and the Lorentz group SO(l,9) respectively: Q(A) .. Tr(A.. § -
1
"3 A.. A" A) = Tr(A
0
A
§ + OrA))
(VI.9.24a)
note that we are using dimensionless § and .If- fields; that is, the physical fields B, A and their field strengths defined in (VI.9.l7, 20) have been rescaled as follows: We
2
gA ... A
LB+ B K
1863 Let us coment on the modifications of the left hand side of Eq. (VI.9.23d) with respect to its definition in the vacuum, namely Eq. (VI.9.22). First of all we note the appearance of a dimensionless scalar field in front of the 3- form ~ r 1/1" Va which we have arbitrarily a normalized to e4 / 3cr (x). As we are going to see in the sequel, this field is needed in oreier to find a consistent solution of the ; - Bianchi identity. The field 0' (x) is identified with the dilaton field ($(x) ;; e4/3 a(x», the missing bosonic degree of freedom besides Va A
II
By definition, we have a(x) ;; 0 in the vacuum, The gravi. tello X is then defined as the spinor component of dO' along ~:
and
Hll"P'
(VI. 9 .25) o
On the other hand, the replacement of the Lie algebra cocycles Q(A)
o
and Q(w) with the Chern-Simons forms Q(A) and Q(w) is justified by the requirement of gauge invariance of the equations of motion. Indeed, let us write the Bianchi identities that follow from the exterior differentiation of Eqs. (VI.9.23). We obtain
fi §lab .. 0
(VI. 9. 26a) (VI.9.26b)
~
~ p +
'41 r abl/l
i
+'2
A
fJ
ab
.. 0
(VI. 9. 26c)
4/3O'-..a ab lIr .. ral/iAi -6Tr(§ .. §')+y{a A"ab)=O
e
(VI. 9. 26d) V§EdF+A"§'-:i"A,,O
(VI.9.26e)
where ~ is the G-covariant derivative and we have used the well-known relations
1864 d Il(A) .. Tr (.1'" §,)
d
ab
new) .. - ~ . .
def
~ ab
:: Q(§')
def ::
- Q(~)
(VI. 9. 27a) (VI.9.27b)
and Q(Il) being. respectively> the G and Lorenu gauge invariant 4-fOrms. They are known as the second characteristic classes of the G and of the 50(1,9) groups respectively. Since Bianchi identities imply the equations of motion, we see that the properties (VI.9.27) are essential to get a gauge invariant theory. There are no other 3- forms, o 0 besides n(A) and neoo) , that reduce to n(A) and new) in the vacuum and that would give rise by di,fferentiaUon to gauge invariant 4-forms. Therefore Eq. (VI.9.23d) is justified. Q(")
Let us also note that the modifications 8+ g we just conSidered are in line with the point of view adopted in Sect. 111.8.4 where we regarded a gauged FDA and its Bianchi identities as a new FDA in which the generalized curvatures play the role of contractible generators. Finally we remark that while the Jf-curvature defined by (VI.9.23d) is SO(I,9)- and G-gauge invariant this is not so for the 2-form B. It varies non-trivially under G and 50(1,9) in order to cancel the variation of the Chern-SilllOns teX'lllS and make Jf invariant. Furthermore the noninvariant nature of Il(A) and Qew) implies that the characteristic classes Q(jr) and Q(~) defined by Eqs. (VI.9.27) are closed but not exact, since they are obtained through differentiation of non-gauge invariant 3-forms. We conclude this section by writing the parametrization of the dilatino field X appearing in the spinor derivative of cr, Eq. (VI.9.2S). Quite generally we can write (VI. 9. 28)
being a matrix in the spinor space. From the do parametrization (Vr.9.28) and considering the tlBianchi}dentitytl d20'=0 in the 21/1sector we find (j
1865 (Vr. 9. 29)
whose solution is '"~ '" 21'"°aO ra + rijk
zijk + rij
2'ij
+
rijH
Z"ijk~
(VI.9.30)
the Z's being generic superfields. Taking into account the opposite chi tali ty of X and 1/1 we must set Z!." Z'! 'k" ,,0. Therefore> the 1J 1J '" O-forms a and X admit the following parametritations: (VI. 9. 3Ial ~
. .a a X" f» X v + (2i () (] r a a
+
r ijk
Z. 'k)1/! • 1J
(V1.9.3Ib)
The related Bianchi identities are (VI.9.32a)
2
1 4 a
§PX+-rb~
ab
X .. O
(VI.9.32b)
The antisymmetric tensor Zijk will be determined in terms of the physical fields in Sect. VI.9.4. VI.9.3 The general solution of the D=10 super Poincare Bianchi identities The aim of the present and the following sections is to work out the most general solution of the Bianchi identities (VI.9.26). The key observation is that the first three Bianchi's (VI.9.26a,b,c), associated to the generators of the super Poincare group, can be given a general rheonomic solution independently of the Jf- and jr -parametrizations. Let us start from the ansatz: (VI. 9. 33a)
1866 (VI.9.33b)
'k2 r abe 11'i A r a'l' 'ft 'k d, rabijk'b Tijk + 1 3'1' A 'I'
+ 1
(VI. 9. 33e)
where k1, k2 and kS are three coefficients, to be fixed by the Bianchi's. The ansatz (VI.9.33) defines a rheonomic parametrization of the curvatures: besides the tensor-spinor eab c, determined as usual by the Bianchis in terms of Pab (the space-time components of p), all the other nonzero components of the super Poincare curvatures are expressed in terms of the torsion space-time components r abc • The torsion TOOc is assumed to be completely antisymmetric in a, b, c. From the scaling behaviour (V1.9.34a)
(VI.9.34b) (VI.9.34c)
it follows that:
(VI. 9. 35a) (VI.9.35b)
[Rab
ccJ
= [w- 2]
(VI ,9. 3Sc)
[pool = [w- 3/ 2]
(VI.9.3Sd)
and we recognize that the ansatz (VI.9.S3) exhibits, besides Lorentz invariance, also homogeneous scaling behaviour according to the rules of Sect. 111.3.12.
1867 As we are going to see, the so far underdetermined coefficients k1, kZ' k3 appearing in (VI.9.S3) are uniquely fixed by the Bianchis (VI.9.26a,b,c). Therefore it is important to find out whether the given ansatz is the most general one, or if we might have more general para~ metrizations of Rab , r a and p. w(p)
Let us first introduce a convenient nomenclature. Each p-form can be written in the anholonomic basis of superspace provided by
the "supervielbein" (VI. 9. 36)
Hence we have ( )
w P =w
(p)
A
1'"
Ai
A E
" ... "E
Ap
•
(VI.9.S7)
P
Separating the summation on the indices into subsets we rewrite Eq. (VI.9.S7) as follows: (VI. 9. 38)
where ( )
w
(p-q,q)
The forms
w(p).
w(
=wp
p-q,q
a1 .. ·ap-q Cli···aq
a1
V
a Cl 1 CI " ••• " V p-q IjI " ••• " 1/1 q (VI. 9. 39)
) will be named the (p-q,q)-sectors of the form
In particular the equation (VI. 9. 40)
implies w(p-q,q) -- 0
(VI.9.41)
1868 a
a
due to the linear independence of the monomials VI ....... V p-q al a W ... $ q. Equation (VI.9.41) is named the (p-q,q)-sector of Eq. (VI.9.40). Using this formalism, the ansatz (Vl.9.33) implies the following rheonomic constraints: (VI. 9.42a)
a
(VI.9.42b)
T(1.1) ",0
P(0,2) '" 0
(VI.9.43a)
Tijk I .~ kr r P(l,l) = I m ijk !P"v
(VI.9.43b)
(VI. 9. 44a)
(VI.9.44b)
On the other hand, the (2,O)-sectors of
r a , P. !!tab just define the
space-time field-strengths: abc V V (2,0) -- T b" c
Ta
(VI.9.4Sa)
(Vr. 9.45b)
ab fJ (2,0)
ab
= R cd
c d V .. V •
(VI. 9.4Sc)
Let us now discuss the constraints (VI.9.42-44). At first sight Eqs. (Vl.9.42-44) seem to be rather restrictive. Consider for instance the first three constraints (Vr.9.42, 43a). It is easy to give a non-zero value to T(0,2)' T(l,l) and P(O,2)' by using the extra components of the SUGRA-SYM multiplet {Habc ' X, cr, Fab' A} utilizing, when needed.
1869 the dimensional constant la' =64 g2/K ([a l ] =[L2]). For instance, since [xl'" [w·l/2] (see Eq. (VI.9.25)) we could write (VI.9.46) (VI. 9. 47a) (VI.9.47b) where the dots denote other possibilities. IIDwever, it must be kept in mind that, in any case, a particular set of constraints is not invariant under field· redefinitions. For instance let (VI.9.48) be a Weyl redefinition of the field 1/1 in terms of an arbitrary super· field L. Then "
p
1 ab " L • - III "r bl/! '" e (p + dL ~ 1/1)
= dl/l A
a
4
Calling
p
(VI. 9.49)
the (O,I)-component of dL: (VI. 9.50)
we find a nonzero (O.2)·sector for "
p(O,Z)
=- e
p:
L • ljI"l/IU '" L -
= - e [1/1" r
a
•
1/1\.1 +
1/1" r
a 1• .. a5
1/111]
(VI.9.S1)
In view of this ambiguity the important question is whether field redefinitions might bring the constraints to the form (VI.9.42·44).
1870 The iUlswer is the following. Starting from the most general rheonOJDic ansatz for r a, p, Rab, it is always possible to reduce it to the fom (VI.9.42-44) by means of field redefinitions, except for Eq. (VI.9.42a), namely TCO ,2)" O. Actually (VI.9.42a) can also be considerably weakened. Let us expand TCO .2) in the SO(1,9) irreducible (0.2)-sectors: a
alb -
TCO 2) :; P •
1/1
ala l ···as
A
fbifi + P
!/I" f
a1· .. as
1/1
(VI. 9. 52)
the P tensors being generic fields of weight zero. By means of field redefinitions one finds 1050
Pa/b -- 0 • paIa ···a ~ p---S a l ..• aS l a
1050 ,,'here L - is the al· .. aS
~
a/'1 ••• aS part of L • Therefore, the only
a
essential constraint in (VI.9.42-44) is (VI.9.S3)
We do not give the proofs of the above statements. They can be found in Ref. 1 of this chapter. As a very simple example let us show that we can always retrieve the particular f-matrix combination appearing in P(I,I)' namely,
by means of a redefinition of the spin connection. more generally
Indeed, let us set
1871
ab
III
~
Aab =wab
-rUj
.,.~b
(VI. 9. 56)
+t.>W
It follows that (VI. 9.57) Therefore ,..
1
4"
P(l,l) " P(l,I) -
r
ij
• "W
ij
. .m ,I, m v A'f
•
(VI.9.58)
Hence setting (V1.9.59)
we retrieve the particular combination (VI.9.52) by choosing 6= kl (3~ 4nJ. We note that the field redefinition (VI.9.S6, 59) transforms the (2,0) (space-time)-sector of r a as follows:
abc
= (1 + B) T
Vb AVe •
(VI. 9.60)
Therefore the chOice 6 = -1 implies Ta ", O. This result clarifies the meaning of the field r abc appearing in the rheono~ic parametrization (VI.9.42-44). ~~ile it is possible and convenient for physical reasons to identify Tabc with the antisymmetric part of the space-time torsion, this is not necessary. If we put Ta = 0, through the above redefinition Tabc appears to be an "auxiliary field" in the rheonomic parametrization (VI.9.33). (It will be shown below, (Eq. (VI.9.114». that on Tahc the Bianchis just imply the constraint !»ffi Trnab" 0). Indeed, the value of Tabc will be determined by coupling the Bianchis (VI.9.26a, b, c) to the £- and ff - Bianchis. Summarizing, provided Eq. (VI.9.53) is satisfied, the constraints on the super Poincare curvatures can always be given the form (Vl.9.42-44) where Tabc is an
1872 antisymmetric tensor and eab is a spinor-tensor whose explicit form c will be given later. What about the essential 1(0,2)" 0 constraint? A nonzero Ta{O') implies the introduction of a dimensionful constant 1050 in the corresponding ansatz for P-- . We do not have at the a1" .as a 1050 moment any reason to exclude a priori a nonzero P-- . However, as al" .as a far as we know, it seems very problematic to get a consistent Bianchi solution in the presence of T(0,2) ~ O. For this reason we stick to Eq. (VI.S.S3). The constraints (VI.9.42-46) being justified, we proceed to work out the explicit solution of the super~Poincare Bianchis.
I·
To this purpose it is convenient to decompose the exterior derivative d according to the (r,s) sectors. The operator d maps a p-form w(p) into a (p+l)-form dw(P). Since both w(p) and dw(P) can be decomposed into (r,s) sectors (p) W
"
dw
(p)
w(p,O)
+
wCp-1,l)
" dwCp+1,0)
+ ••• +
....
uw(p,l)
+
+ ...
w(O,p)
(VI.9.61a)
d t
w(O,p+l)
(VI.9.61b)
a(r,s )(r + S " 1)
the operator d can be viewed as the sum of operators which map a (p,q) form into a (ptr,qts)-form:
Furthermore, each of the operators OCr,s) two operators: d
(r,s) " V(r,s)
+ ~
can be viewed as the sum of
(VI. 9.63)
(r,s)
where by definition V( r,s ) acts on the anhoionomic components of a p-form and ~( r,S ) acts on the vielbein basis:
V
(r,s)
w(p)-
-
" VCr s) (WA
,
Al A EA •••
1'" P
A
A E p)
Al -
def
(V(r s)wA
A)E
1''' P
A .......
I'.
E P
1873 (VI.9.64)
~bile ~(r,s) is a Lorentz covariant algebraic operator, VCr,s) is the Lorentz covariant derivative acting on O-forms. Hence it exists only for (r,s) = (1,0) or for (r,s)'" (0,1).
V(I,O) and V(O,I) are explicitly defined below: (VI.9.65a) (VI.9.65b)
where LA
A is a spinor or a r-matrix valued boson depending on 1'" P
whether wA A is a boson or a spinor, respectively. In the follow1''' p A lng V(O.1) is called the spinor derivative, and the 0- forms LA 1'"
are called the spinor derivative components. write
P
(Note that we may also
(VI. 9. 66)
Da being the tangent vector dual to lji: ljia(DB) ='/'-B)' On the other hand the action of the p( 1",S ) operators on the supervielbein (Va.~) can be determined explicitly by using the superspace constraints (VI.9.4Z-44). Indeed, from the definitions (VI.9.Z3b, c) and the constraints (VI.9.42-44), we obtain
(VI.9.67) (VI. 9.68) Equation (VI.9.67) shows that ~ acting on yS, (the (I,O)-sector by abc definition) generates a term in the (2,O)-sector, T Vb" Vc' and a
1874 tetlll in the (0,2) -sector. namely
a '2i l/I" r l/I.
part ~(1.0) creating another Va and no destroying the vi and creating two I/I's:
Therefore
W.
~
contains a
and a part UC -1,2)
(VI. 9.69)
(VI. 9. 70a)
(VI. 9. 70b)
In an analogous way. from (VI.9.68). (Vr.9.43b) and (VI.9.4Sb) one deduces (VI. 9. 71)
(VI. 9. 72a)
(VI. 9. 72b)
furthermore, one obviously has (VI.9.73)
In addition, it is straightfo1'\'lard to check that the closure of the 2 exterior derivative operator, d = O. reflects into the following set of identities: 2
~(-1,2)
2
= ~(2,.1)
=0
(VI. 9. 74a)
(VI. 9. 74b)
(VI. 9. 74c)
1875
(VI.9.74d) (VI. 9. 74e) where the brackets denote anticommutators. We stress that Eq. (VI.9.74a) implies that ~(-1.2) and ~(2,~1) are cohomological operators. In the following table we summarize the d-decomposition {we set k1 " 1/36 anticipating the result (VI.9.S?)). Coming back to the super
Poincar~
Bianchis let us now
deco~se
the torsion and gravitino Bianchis into the different (p,q)-sectors,
using the previous decompositions for d and the constraints (VI.9.4244). We obtain Torsion Bianchi identity: (O.3l-sector: (VI. 9. 75)
0" 0
(1.2) -sector:
Ta
~(-1,2) (2,0) +
""ab ::II
V
.j
(0,2)" b - IIp,.
ra
P (1,1) =
0
(VI. 9. 76)
(2.1) -sector: (VI. 9. 77)
(3,0) -sector: (VI. 9. 78)
Gravitino Bianchi: (O.3)~sector:
1876 1
11(_1,2) P(1,l) +
'4 f ab$ ...
ab fJ (0,2) "
°
(VI. 9.79)
(1,2) -sector:
1
V(-l,2) P(2,O) + 'V (0,1) P(l,l) +
ab
(VI. 9.80)
4" fab W a (I,l) A
(2,1) -sector: 1
V(o,l) P(2,O) + (V(l,O) + \l(I,O»
P(l,l) +
ab
4" f ab1jJfJ (2,0)
:=
0
(VI. 9.81)
(3,O)-sector: (VI .9. 82)
Considering first Eqs. (VI.9.76) and (VI.9.79), which are purely algebraic, we can easily fix the coefficients k1, k2' k3• Using the constraints (VI.9.42-44) and Table VI.9.r we find from (VI.9.76)
(VI. 9.83)
corresponding, respectively, to the vanishing of the (l,2)-subsectors d'1'.. rad,'f'... Vb and ;i.'I' ral '" as,,,'f'_V .. In of (VI.9.76). From (VI.9.79) one finds
-3k
I
k2 9 + - - - k ,,0 4 2 3
(VI. 9.84)
corresponding to the vanishing of the irreducible (O,3)-sector 144 ::a "l/IA~Ara1/l
(VI, 9. S5)
1877 TABLE VI. 9. I DECOMPOSITION OF THE EXTERIOR DERIVATIVE TN
a(-1.2)
N:l SUPERSPACE
D~lO.
= Jl (-1.2)
d(2._1) " Jl(2,_1)
On the component
'if
W
(1,0) AI" .Ap
V(O l)w J
i .. o
AI""
wA A of a p-form: 1'" p
= 9#a (wAI"' .A
A." P
)Va
p
iii
L
Ai'"
A
P
implies Eqs. (VI.9.74).
(see Eq. (VI.9.66))
In particular
Jl~_IJ2)=Jl~2,_l)=O.
1818 (see Sect. 11.8.8 for notations). In working out (VI.9.79) one must use the relation (VI. 9. 86)
Which can be easily deduced from Table II.S.IV. From £qs. (VI.9.83, 84) one finds (VI. 9. 87)
Let us now consider Eqs. (VI.9.77) and (VI.9.80). tensor nabc by
Defining the spinor-
(VI. 9. 88)
Eq. (VI.9.77) becomes (VI. 9. 89)
while from Sq. (VI.9.SO) we obtain
(VI.9.90b)
- a b ~ a1···aS corresponding to the two (1,2)-subsectors '" r .. '¥ ... V , 'A r WA., respectively. Eqs. (VI.9.89, SO) are a coupled system of equation for the two unknown spinor-tensors ea\ and rAabc• From (VI.9.89) one finds aabc in tel'lllS of Qabc (and !lab) with the usual manipulations: (VI. 9.91)
1879
Substituting (VI.9.91) into (VI.9.90) we obtain two relations between which DlUst be cons~stent. To analyze the cOl\sistency it is usefulato decompose the spi~~r-tensors ~~c. asb and P b into C a SO(l,9}-representations. These decompositions are supplied in Table
nah<:. and P b
II. S.XIII. The conventions adopted there are modified in the case of the P b . 560 144 16 a fleld-strength. The irreducible parts Pii). t>;- and r will be denoted by 0ab' aa and 0' respectively. We set (VI. 9.92)
One easily finds that the representations 1200 of Q b and 3696, 1200, ab -ac ---720 of e are equated to zero by Eqs. (VI.9.89, 90). In the 560 and -c -~ sectors the two equations (VI.9.90) give consistent results, namely, 560
(VI. 9.93a)
~,. - 6 Cab
(VI.9.93b)
Instead, for the 16 sector one finds incompatible relations between ~ and 0' so that one obtains the constraint (VI.9.94)
Recalling the coefficients appearing in the decompoSitions of Table II.8.XIII, the final solution is 1
Qabc
"
2" f[a
560
S'lbcJ -
3
56 f[ab
144 Q~"
•
9 3 f[a abc] - 4" f[ab O'e] (VI.9.9S)
or. using the full Pab field-strength. (VI. 9. 96)
1880 Substituting in (VI.9.9l) one also finds
(VI.9.97) As a last step we analyze the (2,l).sector of the gravitino Bianchi, Eq. eVI.9.8l), which determines the spinor derivative (D Pab) (O,l}' so far unkl\own. Using the constraint (VI.9.43) and defining the two matrices /.lab' and lola through the decompositions (VI. 9. 98a)
(VI. 9. 98b)
we find that Eq. (VI.9.8l) can be explicitly written as follows:
1
r r .. k [ a -r·kn·[)§lbJ ·-e 36 1J J 1 a
. jk +
(VI. 9. 99)
To solve this equation we again decompose each term of Eq. (VI.9.99) into 50(1,9) irreducible representations. In order to do this we need the decomposition into irrepses of !ia Tijk , Tijk Tpqr and Rab cd ' and of the spinor derivatives Mab and Na defined by Eqs. (VI.9.98). We set _
(945)
~a Tijk - Lijk
a
~
(210)
+
Laijk
+
.
(45)
8 °a[i LjkJ
(VI,9.100a)
1881 firA!
mab
.. L(45)
(VI.9.100b)
ab
(VI. 9.101a)
(VI.9.101b)
(VI.9.101c) (VI. 9.l0ld)
(VI. 9.101e)
corresponding to the decompositions: ,
(VI. 9. 102a)
T" = 10®lZO = 2101&945$45 all k - - - - -
--
;; 41251& 1050+ 1&1050· iii 770 + 54 + 210 + 1
(VI. 9. 10Zb)
----
-
The tensors L and K are implicitly defined by Eqs. (VI.9.100, 101). For the Lorentz tensor Rab cd we set ab
R ij"
(770) Rab ij
+
1 (54) 2"°[a ~]
[i
j]
R(210) +
abij
+
!
&(945) . 2 jab i
(VI. 9.103)
1882
corresponding to the decomposition (45 ® 45) = 770$ 945 ED 210 e 54 6145 $1 • - -symm. - - -
(VI. 9. 104)
However, not all of the irreducible components in (VI.9.103) are independent fields. Indeed, a straightforward analysis of the (3,O)-sector of the Lorentz Bianchi (VI.9.26a) gives, upon use of the previous decompositions, R(210) _ L(210) abij - abij R(945 J
jab i
+ 2K(bZ~~)
a
1)
:: L(945) jab
(VI. 9. 105a) (VI. 9. 105b)
i
(VI.9.10Sc) Finally we decompose the matrices Mab and Na into irreducible components: (VI. 9.106a) N = 160144" 1050-6)945ED210$54$45
a----------
(VI. 9. 106b)
and, using the following properties of Alab and Na' (VI.9.107) which are write
~nherited
from the irreducibility of crab and 0a' we can
1883
(VI.9.l09)
Using now Eqs. (VI.9.l0S, 109), (VI.9.l0S) and (VI.9.94) and the con· straints (VI.9.42·44), we can work out Eq. (VI.9.99) in the various irreducible sectors. The f·matrix algebra is rather lengthy but straightforward. In particular, when rearranging the terms, one has to remember the cyclic symmetry property of the Young tableaux;
1884 the sum being extended to the cyclic permutations of the set bj,al •.. an • bj being any of the indices in the second column. One obtains the follOWing system of equations for the irreducible M- and N-tensors:
a
M(S940) '"
(VI. 9 • ll1a)
!M(lOSO)- '" _ ~ K(1050)2 72
_!. N(lOSO)
= _l
2
_!
K(lOSO) of-
_!
(VI.9.111e)
216
= _ .!. L(945)
M(94S) _ ! N(945) 4
9
(VI.9.l11b)
M(210) +
6
.!. 1'1(210)
'" _5_ K(210) + .!. L(210) 216 72
24
S6
(VI. 9 • 111f)
= _ 1.. L(4S)
(VI. 9.ll1g)
32
M(77D) '"
1.. K(770) 12
_1_ K(210) 24 216
96
_
1:. R(770)
(VI. 9.112a)
4
M(945) _ ~ 1'1(945) '" ! L(945) 4 6
M(llO)
+
~N(llO) = _ ~
4
(VI. 9.llle)
_.!.
_1_1'1(210) _ _ 1_"PIO) '" (24)2 24·30
_ .!. 14(45) + 1.. 1'1(45)
(VI.9.111d)
36
K{l10)
12
(VI. 9 .1l2b)
_! 6
L(lIO)
(VI-9.l12e)
(VI. 9. l1ld)
_! 8
M(4S)
+
1. 1'1(45) = _ 1.. L(45) 32
48
.'
(VI.9.ll2e)
(VI.9.l13a)
1885 ~_l_K+_l_R=O 270 180
(VI.9.113b)
Eqs. (VI.9.l11), (VI.9.112) and (VI.9.113) are respectively associated to the cancellation of r
b 1b2b 3b4
blb2
, r a n d lL terms.
In most sectors, the system (VI.9.111-113) poses no problem and yields an immediate solution. In the sector 210 we have three equations for two unknowns (M{210) and N(210») Whi~re however compatible so that no constraint is implied.
In the sector 45, instead, we have three incompatible equations for two unknowns which imply the following constraint
rnab = a
L(45) :: ~ m
(VI. 9.114)
and hence (VI.9.11S) Finally Eq. (VI.9.113b) can be read as a constraint on the ture scalar:
curva~
(VI. 9.116)
The final solution of the system in terms of the surviving
tions is: N(S4)
= !. R(S4) 2
N(210)
=~
~
.!.. K(54) 18
~
18
K(210) ~
'
.!.. L(210) , 12
representa~
1886 M(S940) .. 0 • M(1050)~ .. _ ~ ,(10S0) 36
•
N(lOSO)+ .. __1__ K(1050)+ 108 M(945) .. ~ L(945) 16 • N(945) .. ~ L (945)
(VI. 9. 117)
36
In this way we have completed the solution of the torsion and gravitino Bianchis. In principle one should also consider the Lorentz curvature Bianchi, Eq. (VI.9.26a), and check if it gives consistent results. This is however not necessary; indeed a theorem by Dragon [2] assures that !iii Rab =0 is automatically satisfied if the torsion and the gravitino Bianchis are satisfied. In Table VI.9.II we summarize the results obtained for the super Poincare curvatures. We see that, besides the parametrization of the Rab , Ta and p-
curvatures we have obtained a set of space-time constraints: 45 m m Rab " 0 - R amb .. R bma
(VI.9.1I8)
1 = Rmn _ 2 Tijk Jr mn - '3
(VI.9.119)
T
ijk
(VI.9.120) (VI.9.12l)
The first constraint expresses the synunetry of the Ricci tensor, while the last three are the field equations of the dilaton a. the gravitello X and the 2-index tensor BJ.I\I respectively. This will be evident when we couple in the next section the previous general solution
1887 TABLE V1.9.II
D"lO, N=l Paramet,rization of the S~er Poincare Curvatures
_ a b P - 0ab V AV
'I-
1 r ..;n ij k 36"rm ijk1/l" T
§tab '" Rab .. Vi Avj 1)
'I-
~ Sab A..;n m
+
i ~ Tabc~ Af 1jI 6 c
where
Sable" - 4i f[a Pb]c - 3i f[ab rm pc]m fj
m
firnn n
TIDab .. 0
ron
"~Tijk T
3
ijk
of the Poincare Bianchis to the H-Bianchi identity. In particular, from the dilaton equation (VI.9.119), we can eXtract a very important information concerning possible string compactifications on a classical four-dimensional background. Let us consider a classical solution of the complete D=10, N=l SUGRA-SYM theory, the effective theory of the heterotic string. Let such solution represent a cornpactified background M4 0 M6, where M4 is the 4-dimensional Minkowski space and M6 is a compact 6-dimensional space. On this background the dilaton equation (VI.9.119) becomes
1888 (VI. 9. 122}
where we denote by a, B. y the flat indices of the internal manifold M6• If one separates the spin connection into a metric and a torsion part
Eq. (VI.9.122) becomes (VI.9.123)
Now in our convention the right hand side of Eq. {VI.9.I2S} is a negative definite quantity
due to the negative signs in the 100 metric("): (+, -, - , .•• ,-). Hence, M6 must have a negative definite scalar curvature in order for the dilaton equation to be satisfied. This simple observation rules out the possibility of compactifications on group or coset manifolds since (in our convention) they have a metric scalar curvature which is strictly positive. Note that we have arrived at this conclusion by just looking at the super Poincare sector of the theory without worrying about the implications of the remaining Bianchi identities. In particular this conclusion is independent of the actual form of H, that is, of the presence or absence of the Chern-Simons forms in its definition. Indeed, the absence of such compactifications was first reali~ed in the minimal 0=10 SUGRA-SYM theory (see Ref. 3). What we have shown is that the same conclusion applies to the most general effective theory of the 0=10 heterotic string. Note that this conclusion does not mean that we cannot have a consistent string theory, on a manifold which is the product of the four-
(*)
Notice that in Chapters VI.2-8 we used the opposite convention.
1889
dimensional space~time with a group (or coset) manifold. Indeed the 4D strings previously discussed in Chapters VI.9.3~7 propagate on such a manifold. (**) It only means that such models may admit a multiple geometric interpretation as a~models on different target spaces. When seen as compactifications from D=lO. the target space is not a coset manifold. We conclude this section by deriving the solution of the YangMills supermultip1et Bianchis on the supergravity background described in Table VI.9.II. In Sect. 11.8.9 we found the rheonomic parametrization of (jr.~) (~.ab = Ta = p = 0) (see Eqs. (11.9.13). Actually, one can easily verifY that the same rheonomic parametrization holds also in the curved background of Table VI.9.II. Only the equations of motion implied by the Bianchis undergo a change due to the non~trivial background. Let us write down the parametrization found in Sect. 11.8.9 in the flat case: in the rigid supergravity background
a b • a !F = Fab V ,,1/ ~ 21 A r a'II" V
(VI. 9.125a)
(VI. 9. 125b)
where VA denotes the gauge and Lorentz covariant derivative. Let us assume a non~trivial background described by the curvatures of Table VI.9.!l. If the constraint (VI.9.126) which was implicitly assumed in the in the non~trivial background, then defines the supersymmetric partner change. The VA-parametrization. on (+*)
flat case, is supposed to hold also the parametrization (VI.9.125) just A of A and therefore does not the other hand, is fixed by the
More precisely the manifold is a group manifold madded out by a discrete group B.
1890
Bianchi
V~
=0
in the {I,2)-sector. Since the super Poincare curvatures
do not appear in this sector it is obvious that Eq. (VI.9.126) is true
.also on a non-trivial supergravity background. Let us now derive the equations of motion for " and Fab' We use the Bianehi (VI.9.26e) for §" and that derived from (VI.9.12Sb) for A: (VI. 9. 127) The gaugino equation of motion can be derived from the {O,2)-sector of (VI.9.127). It is therefore necessary to compute first the spinor derivative V(O,l)Fab appearing when one differentiates (Vl.9.125a). This can be derived from the (2,l)-sector of V§" = O. Setting (VI.9.I28) one finds
which, with respect to the flat case. involves also the supergravity field T.1)'k' Using this result in the (D.2)-sector of (VI.9.127) one finds the following constraint: (VI.9.129b) that is, the gaugino equation of motion. Furthermore, taking the spinor deri vati ve of the last equation one obtains the space-time equation for the Yang-Mills field:
vmFma
+
fermion bilinears
=0
•
(VI.9.130)
The above results are true if the constraint §"(O,2) =0 is assumed to hold as in the flat case. Are we justified in taking this simplifying assumption? We remark that the most general form of '(O,2) is:
1891
(VI. 9.131) The field redefinition A-+ A + C Va allows us to put C ,,0, so that a a the only non-trivial part of jr(0,2) is given in terms of a 126 50(1,9)representation. Observing that (VI. 9.132) we see that, without introducing dimensionful parameters, the C a1... as tensor should be constructed in terms of gaugino bilinears or of the torsion field Tab' c This is forbidden by Lorentz covariance (~r abcA being the only non-vanishing current). There is not, however, an a priori reason for discarding a nonzero jr(O,2) if one allows for the presence of a dimensionful constant. The situation is thus very similar to that already discussed in the case of the supergravity constraint
T~O,2) ,,0. In the following we shall assume .1'(0,2)" 0, the reason being the same as for the T(O,2)-constraint, namely, it seems very unlikely that the Bianchis (Vl.9.26e) and (VI.9.127) should admit a consistent solution if we allow .1'(0,2) t- o.
VI.9.4
H-Bianchi identity in the (0,4)- and (l,3)-sectors: determination of the H-parametrization
~
While the solution of the Yang-Mills Bianchi identity is straightforward as we just saw, that of the H - Bianchi is more laborious and is based on cohomological considerations. Let us write Eq. (VI.9.26d) d (Jf + Y) "
f3 Q(§')
- y
Q(&1')
(VI.9.133)
where Y is the following gauge invariant 3-form: (VI. 9.134)
1892 As it has already been stressed, the 4-forms Q("-) and Q(Bt), defined in Eqs. (VI.9.l27), are closed, but not exact. The right hand side of Eq. (VI.9.133) is analogous to the electromagnetic current in the right hand side of the inhomogeneous Maxwell equation for the 2~form F • The closure of the 4-form ~ Q(.f) - y Q(dI) is the counterpart of jlV current conservation in electromagnetism. It is clear that the most general solution of Eq. (VI.9.133) can be obtained by adding the general solution of the homogeneous equation d (.1'+ Y)
=0
(V1.9.13S)
to a partiCUlar solution of the inhomogeneous one. Eq. (VI.9.13S) is nothing else but the .r! - Bianchi at ~ '" y = O. Let us discuss Eq. (VI.9.l3S). It is a cohomological equation in superspace and we must find its rheonomic solution. This means that we have to write an ansatz for the 3-form Jf parametrized by the spacetime components {Rab cd ' Tabe , Pab , Habe' a, xl. In Eq' (VI.9.I3S) the form d Y. explicitly given by
(VI,9.136) is known in terms of the p, Ta and a parametrizations (see Table VI.9.Il and Eq. (VI.9.2S»). Let us now remark that ~lO. N=l Supergravity does not allow off-shell formulations with auxiliary fIelds, so that any solution of the complete set of Bianchi identities necessarily implies a set of equations of motion. This will be explicitly verified by our calculations. Indeed, once the most genel'al parametrization or , has been established from the (0,4)- and (l,3)-sectors of the JIf'- Bianchi, it follows from the remaining sectors that certain differential constraints have to be fulfilled by the phYSical field strengths. The first part of
1893 the programme is carried through in the present section. The second in the next. Before proceeding to the actual calculations note that Eq. (VI.9.13S) must be solved in terms of a form Jf + Y that is closed yet not exact.
Indeed 1f + Y +.if + Y + de
corresponds to a redefinition of
the 2-form B. To begin with, we recall some Fierz identities that can be derived
from Table II.S.XIV: (VI.9.137a)
(VI.9.137b)
(VI. 9 • 137c)
(VI. 9. 137d)
(VI. 9 .137e)
(VI. 9.137f)
1894 144
Equation (VI.9.137a) simply expresses the irreducibility of E--- (see a Table II.8.XIV). Equation (VI.9.137b) is the same as (11.8.87). Equation (Vr.9.137c) can be obtained from (VI.9.137b) by using the selfduality of iPA r 1/1, CEq. (IL8.83b)). Equations (VI.9.130d, e), ar··aS
stating that the left hand side are pure 54 and 10S0+ irrepses are simple consequences of the previous ones; indeed Eq. (VI.9.137d) implies (VI. 9.138a) (VI. 9. 138b) These equations are satisfied because of (VI.9.137a) and of the commutati vity of the currents iP" r a1/1. Analogously, irreducibility of the left hand side of (VI.9.137e) requires
(VI. 9.139a) (VI. 9. 139b) which are obvious consequences of (VI.9.137b,c). Finally (VI.9.137f) is deduced as follows. The 44-sector, (w"tjI"w"l/i)aBYo' has dimen. 19,18-17-16 " 3876 . .In a, B,y, 1:u. Its 50(1,9) S10n , b' eing symmetrIC 4-3'2 content is (VI. 9.140) The two irrepses 54 and 1050+ are defined by Eqs. (VI.9.137d,e). Hence (VI.9.137) follows. The two coefficients in front of the 1050' and 54 fragments can be calculated from the identity
(VI.9.141)
1895
that follows from the
1/1 ~ ~
decomposition of Table II. 8.XlV.
After these preliminaries, let us write the (0,4)- and (1,3)sectors of (VI.9.13S). Using the decomposition of the d-operator given in Table VI.9.l and the further decompositions (VI. 9. 142a)
(VI. 9. 142b)
we
find
~(-1,2) H{l,2)
+
V(O,l) H(0,3)
o
(VI.9.143a)
(VI. 9. 143b)
where we have used the relations
~(-l,2) Y(1.2) = 0
(VI.9.144)
which follows from Table VI.9.I and Eqs. (VI.9.137a), and furthermore,
(VI.9.14S)
-i xl/!
being, by definition, the spinor derivative of cr (see Eq.
(VI.9.31a»).
(r + s =3)
On the other hand, the most general expression of H(r,s)
is given by the following ansatz:
H(3,0)
abc
= Habc V ~ V ~ V
(VI. 9 • 146a)
1896 (VI.9.146b)
(VI. 9. 146c)
(VI. 9. 146d)
the two expressions in (VI.9.146d) being equivalent upon use of Table II.8.XIV. The tensors and spinor-tensors appearing in the ansatz (VI.9.146) have the following 50(1,9) content (we are using the convention of Sect. II.8.8 for the indices of irreducible tensors):
(VI. 9.14 7a)
(VI.9.147b)
560 F;ab .. ~;b
+
144 f[a ~b]
16
+
fab
r
(VI.9.147c)
(VI.9.147d)
(VI.9.147e) Actually we can slightly reduce the number of independent irreducible 50(1,9) fragments by adding an exact form de to H. As already emphasized, this is just a redefinition of B. Indeed setting:
1897 •.a h a - a C=CabV "V-+l/InA"V + l/I . . r l/IF + a a (VI. 9.148)
where All is and
~
vecto-r-spinor (144 (16), and Fa and F are 10 al"' .as irreducible tensors respectively. 11
In the (O,3)-sector we find
(VI.9.149)
where IT {= 144 QI ill and II (= 1200 III ille J&) are the spinor a a l " .as derivatives of F and F • respectively. We see that we can a a a1,··aS 144 choose A on such a way as to cancel . - in (VI. 9 • 146d) . Further672 a more we can require that 41should not be the spinor derivative al"' .as of a 126 tensor, since in this case it might be cancelled by IT • Ill' .. as
In an analogous way for the sector (1,2) one has:
(VI,9.150)
Sa defining the spinor derivative of A. Again, we can a use Cab to cancel the 45 antisymmetric part of Lalb (and, further-
the matrix
more, to require that the L-tensors should not be spatial derivatives of the lower rank tensors in
F, F
a
a1... a5).
Finally, the singlet contained
Lalb can be suppressed by a field redefinition of o.
Therefore.
in the following we shall assume L4b5 a
= LI=
.144 a
= O.
(VI.9.1S1)
1898
Let us now try to solve (VI.9.143a). Using (VI.9.145c. 147a,b) and Table VI.9.I, we find
is a matrix in spinor space defined by the spinor
where A
al' "as
derivative of 16
-
Ii
41672
a1,,·a5• Corresponding to the decomposition
672 ---~'" 2772
-
50(1.9)
--
$
6930
--
$
1050+
--
(VI.9.1S3)
we have (VI. 9.154a)
(VI. 9.154b)
The field
6930 does not contribute to Eq. (V1.9.152) since a l ,,·a5
f(--
210
bl'"b5 ~"rb b b 1/1=0. Furthermore, l 23
L a- a
1'" 4
also gives a vanishing cantri-
bution due to the Fierz identity (VI.9.139a). Therefore, in order to satisfy Eq. (Vl.9.152) it is necessary to set 54 = 0; a
L-
b
1050 10S0 = A-a1" .as a1" .as
I;--
b
b
(Vl.9.1SS)
1899 (VI. 9, 156)
;:
Equations (VI,9,15S) determine the tensor L
Ib
in terms of the 210 spinor derivative of'-:-,except for its 210-component, L , al ···aS a1,··a4 wbich remains free at this level. On the other hand, Eq. (VI.9.1S6) is 672 a strong restriction on the spinor derivative of a generic .-a a' aI' .• as
672
r"
S
We do not know a priori if it can be satisfied at all and, in case it . 6n can, we do not know the general form of a superfie1d v.:-. This a 1... as
672
is not the only restriction on
~
a1,· .as
since the
(l.S)~sector
of
(VI.9.143) provides a further restriction, Let us consider the (1.3)sector of Sq. (VI.9.143b). A~plYing the P(_1;2)~operator and recalling its cohomological property 11(_1.2) = 0, we obtaln (VI. 9,157)
where Q(I,3)
= V(O,I)
H(1.2)
+
(V(l.O)
+
11(1,0» H(o.S) = 0 (VI.9.1S8)
having used 11(-1,2) Y(l,2)=O which follows from (VI.9.137a). Therefore. gel.S) must be a cocycle of the coboundary operator P(-l,2)' This fact strongly limits the possible SO(l,9) content of G(l,3)' Indeed all the representations which are not projected out by the cobol.Uldary operator 11(_1,2) must vanish. Actually one finds that the most general ~(1.3) satisfying (VI.9.IS7) can contain only the 1440, ~ and ~ representations, so that
(VI.9.1S9)
1900 To prove (VI.9.1S9). we write down the most general form of nel •S) (VI. 9. 160)
is defined in Table II.S.XIV. The SO(1,9) content of ~alb
Qa
a
1'" 5
is given by
Ib " ill ~
10
= g§!!. elliQ.
(VI.9.161a)
(VI.9.161b) ~672
Applying the P(_1,2)-operator and using the explicit form of -&1" .a s one obtains
(V1.9.162)
Recalling (VI.9.137d,e) and using the explicit form of ;::,~ll.
-aI' .. as
one
a1 ,,·aS " ~ rb1/l and 1/1 ~" r&1jI " ~ f 1/1 have the following SO(1,9)-content: g.§.2. and 720 144 respectively. Using the orthogonality of irreducible representations one concludes that
easily finds that
5280
g-a1... aS b
Eo 672
720
A
A
A
144
=n-"sz-·o a a
(VI. 9.163)
b
so that Eq. (VI.9.1S9) is proved. Inserting (VI.9.lS9) into (VI.9.143b) one finds the following identifications: 144
~=o;
16
4 1 30
16
r=6"e X+r:
S60
S60
~=~
(VI.9.164)
1901 (VI. 9. 165)
Equations (VI.9.164) just determine the tensor-spinor ;ab in terms of the gravitello X and the 9-tensors defined in (Vr.9.1S8-1S9). Equation (VI.9.16S), on the other hand, is a new constraint which we do not know a priori if it can be solved or not. Recalling the definition of 9(1,3)' Eq. (VI.9.1S8), and the identification (VI.9.1SS), the constraint (VI.9.16S) can be rewritten as follows: 1440
9-
a1,··a4
::
,,1440 n--
a1 ···a4
+
['i/
210
(0,1)
]1440
L-a1••• a4
contains the following
" 0
(VI.9.166)
a1•• .a4
~
structures;
J1440 means that we must project the components a1 ···a4 of the spinor derivative inside the bracket onto 1440.) We see that ,,1440 672~ is constructed out of 4iand its spatial divergence ar .. a4 al· .. aS
(The notation [
and double spinor derivative (recall that A10SO is contained in al ·• .as V
(0,1)
672 nQ m , - ) . Since L was left free by the (O,4)-sector, a1' •. aS a l ••• a4
we conclude that Eq. (VI.9.l66) is a new constraint on the original . 672 fleld ~ , implying that ~d440 ~,-8 1 ,,·aS a1... a4 should be expressible as 8 pure spinor derivative of the so-far arbitrary field L210 • Note
that L210
L~
8 1 · .. a 4
a1' .. a4
aJ ••• a4
is fixed except for the possibility of adding to
a field M~ satisfying the constraint a1···a4 (VI. 9.167)
1902
The presence of such
tensor
210
does not change Sq. (VI. 9. 166). m 126 (Tensors M""'"'"'" gi ven by a pure divergence fJ P would be a1",a4 ma1,··a4 trivial since Eq. (VI.9.1S0) shows that it could be reabsorbed in the redefinition ;r~;r + de). Let us summarize the results obtained so far. With the definitions (VI.9.154) and (VI.9.lS8, 159) the general solution of the homogeneous Bianchi (VI.9.IlS) restricted to the (0,4)· and (l.3)·sectors is given by the following H·parametrization: &
210
~
&1" '&4
(0) H{2,1) "
(VI.9.168a)
(0)
"(1,2) "
(VI.9.168c)
provided the fields ..v 672
• ,.210 and M210 w satisfy the a1 ,,·a4 a1' .. a4
al'"aS
constraints (VI.9.156, 166. 167). The suffix (0) on the outer compo· nents of H reminds the reader that (VI.9.170) is the most general solution of the homogeneous Bianchi (VI.9.13S). that is in absence of the Chern-Simons forms (8 .. y:: 0). Noti ce that since (VI. 9. 169)
does not enter in the (0,4)· and (1.3)-sectors, "abc is for the moment unconstrained. Hence Eq. (VI.9.169) is essentially a definition of
1903
Habc ' Restrictions on Habc will be found in the (2,2)-sector of the Bianchi (see next section), and this fact will restrict Habc to the . on-shell configurations only. Let us now discuss Eqs. (VI.9.1S6, 166, 167). It is obvious that a very simple (but very restrictive) solution of these equations is obtained by setting: (VI. 9.170)
(VI. 9.171)
~
12.
which imply also Qab = Q takes the very simple form:
H(3,0)
def
=
~
0.
abc Habc V "V "V
In this case the solution (VI.9.168)
(VI. 9.ma)
(VI. 9. 172b) (0)
H(l,2)
0
= H(O,3) = 0
(VI. 9.172c)
In the following, Eqs. (VI.9.172) will be referred to as the minimal parametrization. Likewise Eqs. (VI.9.l71) will be called minimality constraints. Supergravity theories obtained with the restriction (VI, 9. 171) will be named "minimal S!lPergravities" in both the presence and absence of the Chern-Simons forms. Indeed, we shall see that the possibility of imposing the restriction (VI.9.171) is quite independent of the source term a Q(Y) -y Q(~) (see Eq. (VI. 9. 133». Instead, if non-trivial solution of Eqs. (VI.9.156, 166) do exist, the corresponding theory will be named non-minimal (or i~roved) supergravity.
Do non-trivial solutions of Eqs. (VI.9.156, 166, 167) exist?
1904
We can argue that they should exist from the appearance already mentioned above, in the superstring effective theory, of higher curvature terms with transcendental coefficients. The argument goes as follows. We will see that the presence of the 4-form Q( bf) in the Bianchi identity (VI.9.26d) (see also definition (VI.9.27b)) reflects into a modification of the equation obeyed by the spin connection (torsion equation), which becomes differential rather than algebraic. Solving such an equation by iteration and replacing the solution in the Lagrangian leads to an infinity of higher derivative terms. However, it should be noted ~hat such a procedure does not introduce new transcendental numbers besides those already contained in the first-order Lagrangian (where w~b is an independent field, and is not replaced by the solution of the torsion equation). Since the coefficient of the Lorentz Chern-Simons form (y'" -1/32) is an algebraic number, it follows that any term with a transcendental coefficient cannot be traced back to the supersymmetrization of the yQ(w) term; it necessarily corresponds to an additional SUSY invariant. On the other hand, since the only freedom one encounters in the solution of the Bianchi identities is located in the general solution of Eqs. (VI.9.1S6, 166, 167), it follows that such equations must admit solutions with arbitrary parameters. In the next section we shall show that any solution of (VI.9.156, 166, 167) contributes to the torsion equation in a differential fashion and in this way generates additional higher curvature terms besides those generated by the Lorentz Chern-Simons. Therefore the transcendental coefficients appearing in the Lagrangian must be accounted for by fixing the arbitrary parameters in the general solution of (VI.9.156, 166, 167). In Ref. 4 it was argued that non-trivial solutions of the equations under consideration can be generated in the following way. Consider a scalar superfield S and its n-th order antisymmetrized spinor derivatives:
From the 8-expansion of a generic scalar superfield 5, we find that the SO(1,9) content of the derivative~ Sen) is given by the following table:
1905 TABLE Vr.9.III
Representation Content of a Scalar Superfield in D=10 n 8
50(1,9) -content 8085 • 4125 • 660
9
8800 • 2640
10
4312 • 3696
11
3696 • 672
12
1050 • 770
13
560
14
120
15
16
16
1
The case n and 16-n have the same SO(I,9) content, except that all the representations are replaced by their complex conjugates. In flat superspace Rab .. T a,. p .. 0 we can satisfy the constraints (VI. 9.156, . M210 ' ~ . h t he 166 , 167) bY settlng ,. 0 an d 1'dentl' fY1ng Wlt a1···a4 a1···aS irrepses appearing at the level n=11 of the table. Of course, if we now take a curved superspace, we shall find that the constraints are violated. However, one can conjecture* that the violation can be compensated by the lower-order superfields SCm) m
Then the corresponding Bianchi d (J'f .. Y)
= B Tr (.1' ~ .1')
* Actually the conjecture has been proved in Ref,
(VI.9.173)
113J.
1906 is actually identical to the homogeneous Bianchi (VI.9.I3S) in the (0,4)and (l,3)-sectors.
Indeed, from the parametrization (VI.9.l2S) we have
F(0,2) =O. We therefore get (V1.9.174) It follows that nothing is changed with respect to the homogeneous case.
Even in presence of Yang-Mills Chern-Simons Sqs. (VI.9.170b-d) and (VI.9.172b-d) are the correct Jf" -parametrizations in the non-minimal and minimal cases respectively. Actually, the Yang-Mills coupling term B Q("-) shows up only in the lower sectors of (VI.9.l73), namely, (2,2), (3,1), (4,0).
There it
modifies the equations of motion of the supergravity fields.
In parti-
cular! the Habc field strength at 13 " 0 (Yang-Mills coupling) and at f3 =0 (pure supergravity) satisfy different equations. The derivation of the equations of motion from the analysis of the (2.2)- and (3,1)sectors of the minimal case Bianchi is given in the next section. The resulting theory will be referred to as the linear D=10, N=I supergravity coupled to the Yang-Mills multiplet or, concisely, the linear D=10, N=l SUGRA-SYM system.
Indeed, when yeO and the constraints (VI.9.171) are
imposed, no higher derivative coupling appears, so that one obtains an ordinary field theory with second-order field equations.
y"
ii) Let us now consider the most general case B" 0, 0, so that both the characteristic classes B Q(Y) and y Q(a) are present. In this case the (0,4)- and (I,3l-sectors of the general JrBianchi d (Jf" .. Y) ..
f3 Q(ff) - y Q(a)
(VI. 9. 17S)
get non-trivial modifications with respect to the homogeneous case. Since Rfo,2) " 0, R~t 1) I 0, we also have Q(O,4) (91 H o. Q(l,3) (91) ,,0. We do not know a priori whether there exist solutions of the inhomogeneous equations:
1907
(VI.9.176a) (VI.9.176b)
However by a clever trick even in this case Eqs. (VI.9.176) can be reduced to the same form as for the homogeneous case. Following an idea originally introduced by Bonora, Pasti and Tonin (Ref. [41) we show that in case the 4~form Q(R) admits the decomposition described below the H -Bianchi identity can be sol ved by the same tokens as before at the level of the (0,4)- and (1.3)-sectors. Furthel'lllOre II. theorem to be proved in the next section and due to these same authors, (BPT-theorem). guarantees that once the Bianchi identity is satisfied in the (0.4)- and (1.3l-sectors it is also satisfied in all the other sectors, since it simply implies consistent additions to the space-time field equations with respect to the homogeneous case. The decomposition one needs to solve the Ii -Bianchi in the (0,4)and (l,3)-sectors is the following: Q(gt) = dX + K
(V1.9.l77)
where dX is exact. i.e., it is the differential of a (Lorentz) gauge invariant 3-form X.
a)
b)
K is a closed 4-form with vanishing
(O,4)~
and (l,3)-sectors: (VI.9.178a)
dK .. 0
•
(VI.9.178b)
(Note that the closure of K. dK .. 0, is a consequence of the closure of Q(gt), dQ(gt) =0). In this case the Bianchi (VI.9.17S) can be rewritten as follows:
1908 (VI. 9.179)
d(Jf· + Y + yX) .. yK
and therefore d (Jf + yX + Y) (0.4) " 0
(Vr.9.1SOa)
=0
(VI. 9. 18Gb)
d (JI" + yX + Y) (1,3)
o It follows that in terms of JI" :: JI" + yX the inhomogeneous Bianchi
(VI.9.17S) has the same structure in the (O,4)· and (1,3)-sectors as the corresponding homogeneous one in terms of JI" ; therefore Eqs. (VI.9.180) can be solved. We notice that the outer parametrization of the 3-form H at y ~ 0 is actually different from that of the homogeneous case (which
also works for f3 ;! O. y .. 0),
and is given by (VI.9.181a)
H(2.1) .. ft(2,1) . y XC2 ,l) H(I.2) ..
eo •3)
H
~(l,2)
(VI. 9.181b)
. Y X(I.2)
eo •
(V1.9.18Ic)
.. fi(O.3) - y X 3)
o Her. s) (5 ~ 0) are given by Eqs. (VI.9.168a-c) or (VI.9.172b-c) in the non-minimal or the minimal case respectively. The proof that the
\Ot'here
decomposition (VI.9.177) with the properties aJ and b) actually exists, can be given by actual computation. Let us decompose Eq. (VI.9.177J along the (0,4)- and (1.3)-sectors of the superspace. We get (VI. 9. 182)
Q(l,3)
= ~(-1.2) +
X(2,l)
(V(l.O)
+
+
VeO,I) X(I.2)
U(I.O» XeO • 3) •
+
(VI.9.l83)
1909 In order to solve Sq. (VI.9.182) we begin by writing the explicit form of Q(O,4)' utilizing Sq. (VI.9.141) and the parametrization of Table VI.9.II we obtain:
(VI. 9.184)
Equation (Vl.9.182) is then solved by setting (VI. 9. 185a)
where W is an arbitrary scalar and
'§
a1' .. a4
is an arbitrary four·
index anti symmetric tensor. The reason why these functions are arbitrary is that they respectively multiply the two (2,1)-cocycles of the operator P(-1,2)' Indeed we have (VI. 9. 186a)
(VI. 9. 186b)
according to Eqs. (VI.9.138a. 139a).
1910 The terms contained in power w=
f§
and '#a
1 .. • a4
must have scaling power
hence the most general expressions we can write are the
~2.
following: (VI. 9.187a)
(VI.9.187b)
where hI' h2, h3 are numerical coefficients. However, we can set hI =0, since the term 'I can be reabsorbed in a redefini don of the dilaton a: a'" a'. by setting
as it is evident from the definition of the 3-£01'111 Y in Eq. (VI.9.134). Ne also notice that (VI.9.185) is a particular solution of (VI.9.182). Other solutions with X(O,3) 10 can be reabsorbed in a redefini tion of the field
6
Let us now consider the (l,3)Msector of (VI.9.183).
(VI~9.177).
namely Eq.
Quite generally we can set (VI.9.188)
where
tab
is a spinor tensor, antisymmetric in its indices and of
scaling power
II"
-5/2.
Hence in the construction of ~ab we can use two types of teI'lllS! either Pab times the torsion Tijk or the derivatives of Pab . Now, using the panmetrization of ~ab given in Table VI.9.II, we obtain
1911
(VI.9.l89) where aab c is given by Eq. (VI.9.9?). Furthermore using (VI.9.185b) and the property of the JJ(.l,2) operator, we can write
(VI. 9. 190)
(VI. 9.191) Using the 3~ irreducible basis of Table II.B.XIV we can split Eq. (Vl.9.183) into the 672 sector and the 144 sector. The 672 sector is --~2 . . ery easy to work out. Collecting the coefficients of E-a from a 1'" 5
(VI.9.189-191) we obtain: (VI. 9.192a)
(VI. 9. 192b)
(VI. 9 • 192c)
We see that there are two independent structures to be cancelled separately. leading to the two equations
1912
(VI. 9.193a)
-~ ~ 2h
+ 8h ,. 0 923
(VI. 9.193b)
whose solution is h
2
.,!
(VI.9.194)
9
In this way the 3-form XC! 2) is completely fixed. On the other hand, , ab the analysis of the 144 sector of Eq. (VI. 9.183) shows that ~ , and hence X(2 1) is also fixed uniquely. The algebra needed to arrive at the explicit determination of ~ab does not present any conceptual difficulty, but is very cumbersome and lengthy. The interested reader can find the details in Ref. 5. Here we just quote the final result
for X(2,1) together with X(l,2) and X{O,3) already determined above: X(2,1) _ {3
-
-
20 -
2 -
ijk
ITabmwom-TwrrOslbrs+g1/JrijaObkT
+
+
32 2 5 S-lj;raOijTbij - '3 TijaHbiOj + 12 1PrijOaTbij +
+ ..!...,i.rabijokr ..
12
'I'
13k
+! ,i.rijkaobT .. 4
'I'
1Jk
_ ~ ,j,rabiojkT .. } V V 9 'I' 1J k " a" b
(VI.9.195) X(1,2)
=
(VI. 9.196)
1913
(VI. 9.197)
xeD,S) " D •
In this way the outer parametrization of :.11' is fixed also in presence of the Lorentz Chern-Simons form. To complete the decomposition (VI.9.177) also in the lower sectors (2,2), (3,1) and (4,0), we write do~n the corresponding decompositions for (VI.9.177):
(VI.9.198)
+
Q(4,0)
~(2,-1) X(1,2)
(V' (1,0) + ].l(1,0»
+
K(3,1)
(VI,9.199)
X(3,O) + \J(2,_I) X(2,l) + K(4,O}
(VI. 9.200) From Eq. (VI.9.198) we can compute the quantity K(2,2) +1l(_l,2) \3,0)' as all the rest is known. The separation between K(2,2) and \l{-1,2) X(3,O) depends on the particular choice of the non-trivial cohomology element K in Eq. (VI.9.177). We shall fix this ambiguity later on (see Eqs. (VI.9.225, 226». K(3,1) and K(4,0) are then fixed by the last two equations (VI.9.199, 200), all the other terms being known. Thus our statement about the solubility of the H- Bianchi is proved. We shall see in the next section that in the (2,2)- and (3,1)sectors the inhomogeneous Bianchi identities modify the y =0 result in a consistent way, that is, they just add new possible interaction terms to the space-time equations of motion of the B=y =0 theory. Of particular importance is the modification that occurs in the (2,2)sector. Indeed, as we have already anticipated for the y/O case, the torsion equation for wab~ becomes differential, quite independently of the minimality constraints. Therefore, we arrive at the conclusion that,
1914 in presence of thel Lorentz Clem-Simons form
(y'; 0), the theory contains
higher derivatives interactions, even in the minimal case. We shall refer to the minimal theory in presence of Lorentz (and Yang-Mills) Chern-Simons fol'lll$ as the 0=10, N=l minimal anomaly-free supergravity (MAFS in the following). In this case, the Lorentz and Yang-Mills anomalies, which are absent in the heterotic string theory, also cancel at the effective field theory level. If we remove the minimality constraints (VI.9.171) (that is, if we allow the most general solution of Eqs. (VI.9.1S6. 166, 167) while keeping y
(and B) different from
zero), we obtain the non-lIIinimal or improved anomaly-free supergravity. which is by definition the complete effective theory of the heterotic superstring.
Summarizing the previous discussion, we have essentially
four types of 0=10, N=l SUGRA-SYM theories which are listed in the following table. In the first coltDllll of Table VI. 9. IV we have anomalous theories.
In Sect. VI.9.6 we construct the Lagrangian for the linear
SUGRA s'SYM theory.
Except for the linear SUGRA Ii SYM system, the other
TABLE VI.9.IV 0=10, N=l Supergravities Coupled to
the Yang-Mills Multiplet of Either
,gg.!.E g or
y=o
MINIMAL THEORIES
NON MINIMAL THEORIES
50(32)
y"
-1/32
Field theory with gauge and Lorentz anomalies
Field theory where all gauge and gravitational anomaly cancel
Linear SUGRA. SYM Theory (OlaplineManton theory)
MINIMAL ANOMALY-FREE SUPERGRAVITY (MAFS) (no 1;(3) terms and similar ones)
Nonlinear SUGRA. SYM theory
NON-MINIMAL ANOMALY-FREE SUPERGRAVITY (NMAFS) (1;(3) and all the other additional SUSy invariants are present) :: Effective theory of the heterotic string
(Higher derivatives but still anomalous)
1915
three cases are higher derivative theories corresponding to the separate or simUltaneous appearance of the Chem-SimQns form and the ~ , La210 a
a1" .as
and M210 fields. As already anticipated, the super1'" 5 a1 • .. aS symmetric completion of these terms, realized through the solution of the Bianchis, indicates that they are totally independent.
In the next section we discuss the structure of the equations of motion of these theories, and derive their explicit expressions for the minimal case.
VI.9.S The (2,2)- and (3,1)-sectors of the Y- Bianchi identity and the equations of motion So far we have analyzed the content of the H- Bianchi identity in the (0,4)- and (l,:;)-sectors in both the homogeneous (S"'y .. O) and inhomogeneous (6;' 0, Y f 0) cases. We have seen that in these sectors the Bianchis determine completely the Jf-parametrization. Now we shall analyze the lower sectors (2;2), (3,1) and (4,0). As already stated, for these sectors we do not find any new independent condition on the already determined Jr-parametrization (VI.9.168b-d), (or (VI.9.172b-c) in the minimal case). Rather, we find algebraic and differential constraints on the physical field strengths which are, by definition, the equations of motion of the theory. Let us prove this statement. USing the decompositions of Eq. (VI.9.142) and Table (VI.9.I!), the inhomogeneous Bianchi identity (VI.9.133) takes for the (2,2), (3,1) and (4,0)sectors the following form: Sector (2,2):
1916 Sector (3,1):
(VI. 9.202)
Sector (4,0):
(VI. 9.203)
Eq. (VI.9.203) does not need further illustration being an ordinary Bianchi identity on space-time. In the minimal case, using (VI.9.172b),
we explicitly have
= B F[ab
Fed]
+
tm tm Y R [ab R cd] •
(VI. 9.204)
Let us concentrate on the (2,2)- and (3,ll-seetors. We first show that in the (2,2)·sector we just get an equation relating irreducible 120 tensors (i.e., 3-index anti symmetric tensors). This statement is the first part of the BPT theorem stated in the previous section. For the proof we proceed as for the case of the (l,S)-sector. Applying ~(-l,2) to Eq. (VI.9.201) we get
~(-1,2) A(2,2) where
=0
(VI.9.20S)
1917
+
lJ(2._1) H(O.!) - 8 Q(2,;O(ff)
+
'Y Q(2.2)(~) • (VI.9.206)
All the irrepses in A(2,2) which are not projected out by the cobounequation (VI.9.aOS) must vanish identically. Now, the most general expression of a {2,2)-form is the following: dary
(VI. 9.207)
Hence,
Jl(_1.2) A(2,2)
=i
a - a - b c S bcl/J"r I/I"t/I"r 1jJ .... V +
(VI.9.208) a1.. ·aS The SO{I.9) content of the tensor sabC and S ab is the following: (VI. 9.209a)
(VI. 9. 209b)
On the other hand, using (VI.9.137d,e) the 4-1/1 terms appearing in (VI.9.208) are seen to be pure S4 and 1050+ representations. Hence the tensor products
(VI.9.210a) (VI.9.210b)
1918
lIlUSt not contain the ~ or the 1050+ representations, respectively, if
Sq. (VI.9.20S) is to be satisfied. It can be easily verified that this happens only for the llQ. representation. Therefore, we have BQ.;; 10 .. 3696;;
m= m .. 0
and the statement is proved.
The meaning of the {2,2)-sector just examined is the following. It establishes a relation between the so far tmdetermined 120 tensor, Tabc appearing in the parametrization of the super Poincare CUl'Vatures
of Table VI.9.I and the l1Q. tensor Habc defining the field strength
of Bpv' Taking into aecotmt the constraint (VI.9.114), this relation implies also the equation of motion of Habe' Let us now make explicit the content of (VI.9.20l) in the minimal case, that is, when the constraints (VI.9.171) hold true. In the first stage of the calculation it is convenient to set also y .. O. In this case what we are actually doing is the computation of the (2.2)·sector for the linear SUGRA-SYM system. The modifications at y # 0 will be treated at the end of the y .. 0 computation. Using the parametrization (VI.9.172) in Sq. (VI.9.20l) we readily obtain
4 i 3'0 e 1/1 36
• -
- BTr(F
A
rarmr.130kl/l Tij k
" F)(2,2) .. 0 .
. .m .08 A
V "V -
(VI. 9. 211)
In the following we set B" - 4. as determined in Sect. VI. 8. 8. This value Can also be retrieved by requiring supersyaetry invariance of the linear action (see Sect. VI.9.6). The explicit value of Tr(Jr" Jr}(2.2) is immediately computed from the Jr·parametrization (Eq. (VI.9.125)) •. We thus find
1919
(VI,9.212) The second term in (VI.9.21l) can be evaluated by using the spinor derivatives V(O.l)O and V(O,l)~ given by the parametrizations (VI.9.3l). One finds
(VI. 9. 213)
Using
(VI.9.214a)
~
-i6 Ar b A =0 a c
•
(VI.9.214b)
These two equations correspond to the vanishing of the two independent (2,2)-forms i~Ar[a1jJ~VbAVC] and i~ArabcpqlJl ... vPAvq respectively. The two equations (VI.9.214) can be solved for the so far undetermined fields Tabcand b. We obtain a Ze (VI.9.21S)
192D
(VI. 9. 216)
Eq. (VI.9.216) determines the gravitello transformation law. Eq. (VI.S.21S), on the other hand, determines the torsion field entering the parametrization of the super Poincare curvatures in terms of Habe , cr and A. Recalling Eq. (VI.9.I21), we also obtain
(VI.9.217) which is the equation of motion of the B~,,~field in the y= 0 case, that is, in the linear SUGRA s SYM theory. We can now explain what we have anti cipated in the previous section, that is, the appearance of higher derivative interactions (even in the present y =0 case) when we consider the non-minimal version of the theory. The point is that the additional terms present in the non-minimal parametrization (VI.9.168) transform the torsion equation (VI.9.2IS), originally algebraic in wab , into a differential equation in wabu' That it must be so can be se~n by the fOllowing argument. Let us write in shorthand notation T, R, P for the field strengths T b ' Rab d and Pab • Then, considering the field 672 a c c +:-= we see that it can be written as a sum of terms of the type a1· .. as (VI, 9.218)
672
a ' being the dimensionful parameter. Notice that, since <11-
ar .. aS
is a spinor, at least one of the spinor fields Pab , X must enter the functional f a ... a . l
s
Consider now the term 17(0,1) H(2.1) appearing in Eq. (VI.9.201). Schematically we have 560 V(O,I) H(2,l) c V(O,l) Q -
"'"
(VI. 9. 219a)
1921
(VI. 9. 219b) (VI.9.2l9c) 672
We need three spinor derivatives of
to construct the al'"aS term V(O,I) H(2.1)· On the spinor fields Pab and X we have ~-
successivelY*
V(O,I) ~
X V(0,1)
p
~
V(0,1) V(0,1)
Z ., T R+
+
• p + more
more
(VI. 9. 220)
R+~+~T
,
V(0,1)
more
(}R + TR +
'07(0,1)
more
~
3p
+
Tp
+
more (VI. 9. 221)
Hence VeO,I) H(2.1) will in any case contain terms constructed with the field strength Rab cd' This wi·ll make the torsion equation differential in the spinor connection wab~. Recalling the discussion in the previous section, we see that ~~oving the restriction "(0,3) = 0 and introducing the non-trivial ~--, we obtain a higher derivative 8 1.. ·aS theory. We remark that this phenomenon is quite independent of the inhomogeneous source term t3 Q($") - y Q(a). After this digression we come back to the study of the (2,2)sector and analyze the modifications implied by y 1- O. Recalling the decomposition (VI.9.177-179) we see that when Y1- 0 the (2,2)-sector of the H-Bianchi can be written as follows:
* The structures appearing in the successive V(O.l)-derivatives in Eqs. (VI.9.220-22l) are easily retrieved by looking at the formulae (VI.9.2I7). (Vl.9.216), (VI.9.88, 96), (VI.9.9B, 117) and (VI.9.98, 117), (VI.9.29, 33c). (VI.9.98. 117) respectively.
1922
(VI.9.222)
where X(2,l) and X(1.2) are given by Eqs. (VI.9.19S, 196) and we have used (VI.9.197). Utilizing Eqs. (VI.9.lSl) we can write Eq. (VI.9.222) in the following form
Since our general theorem (see Eq. (VI.9.20S) and the following ones) guarantees that only llQ. tensors can appear in Eq. (VI.9.20l). we can set def. P(-1,2) X(3.0) + K(2.2) + .... (5} ;j, A 1 "abc'!'
..
r abcmn,,,'1''' Vm... Vn
1
(1) abc Wabc lfi ... r 1/1 ... V "V + (VI. 9. 224)
We notice that since the first term on the right hand side is the (2,2)sector of the exact form ~ d(W~~~ ya . . vb "Ve ). we have the identifications: X(3,O) ..
2 (I) all
'3 Nabc V
c "V- ... V
(VI. 9. 225)
(VI.9.226) In fact this is just a particular choice of the cohomology class K appearing in the decomposition (VI.9.177). This fixes the ambiguity mentioned after Eq. (VI.9.200).
1923
We conclude that the effect of switching on a nonzero value for y is the presence of the (2,2}-form (VI.9.224) on the left hand side of (VI.9.223). In particular, in the minimal case, the only change in the previous equations is the addition to the left hand side of the equations (VI.9.2l4) of the two terms y w!~~ and y w!~~ respectively. Eqs. (VI.9.214) then become 4 3 - H 2
abc
+
'3 0 1 e {- T
3 abc
- i
+ Z
abc
Xr abcA + y 1'1(1) abc
- -
i
288
Xf
abc
X}-
(V1.9.227)
0
4
"3° {-T 1 e
36 abc
i 1 i (5) +~Xr X--Z }--Ar A+YW =0 64-27 abc 6 abc 6 abc abc (VI. 9.228)
Solving for Tabc and Zabc we find 4
--0
Tac b
=e 3
[- 3 Hac b + 4i
~r ac b A
- 2y Wac b
I
(VI. 9.229)
where we have set
w
abc
::;
(VI. 9. 231)
W(l) + 6 1'1(5)
abc
abc
Using (VI.9.121) we find the equation of motion of Habc:
(9 rn - i3
amaH - 3 Hrnab
+ 4i
~r ac b A
- 2y \II b ) ac
=0
•
(VI. 9. 232)
Comparing Eqs. (V1.9.229, 230, 232) with those for the Y'" 0 case, Eqs. (VI.9.21S, 216, 217), we find the relations:
1924
(VI.9.233) 4
= Zab Za b c e (Y" 0) + 6y e
- 3"0
(5) Wabc
(VI. 9. 234)
where Habc(Y= 0) and Zabc(Y= 0) satisfy Eqs. (VI.9.21S, 217) and (VI.9.216) respectively. Eqs. (VI.9.229, 230) give the general transformation laws of the gravitino and gravitello fields in minimal anomaly-free supergravity. Let us now discuss the eXplicit computation By comparison of Sq. (VI.9.224) with of the tensors w~~~ and webS)' a c (VI.9.19B). we can write
; Q(2,2) - ~(O,l) X(2,1) - (~(l.O)
+
~(1,0» X(1,2)'
(VI.9.23S)
Except for the Q(2.2) term the computation of the right hand side of Eq. (VI.9.23S) is very cumbersome and lengthy, especially for the spinor derivative V(O.l) X(2,1)' due to the complicated form of X(2,1) giyen by Eq. (VI.9.19S). The details of this computation are given in Ref. [ScJ. We just mention that one can take advantage of the fact that each term on the right hand side must contribute to the ~ tensor only, so that one can project each term onto the 120 irrep, disregarding all the others. This simplifies considerably the ~ornput~tion of V(O,l) X(2,1)' Let us quote the results: the tensors W~b~' W~b~ and their linear combination Wabc appearing in the torsion equation (VI.9.227) are the following:
1925 +
1 T~
rn
3 'am 'bn
ttocR. . 1.. T.. Tijk T S4 13k abc
_ ~
3
ro (ab T.. Tij e] 1J m
+
• mec ec 5 1 T1jk T 1 2 T 3 4 +-10" 3 . • rs r +--e: 0" + 3S! abcijkc 1•.• c4 m 4 abc rs
+
35·r
-6 4 i orb a c 0"r
+~TR.
9
am
,.m tt bn
4S . -
• -
cR.
16 1
r
(VI.9.236)
o[a b IJc]
__l_T ..
TijkT
108 IJk
2 . •r rb IJ] - - 11 1. .0 r - -9 l O'La r c r 384
abc
rab
c IJr
1 .. r 32 1 O[a b (1 c] +
• -
(VI. 9.237)
- 2. Tijk T 27
ijk
T abc
+
!. ors r 2
abc °rs
+
~ • 8 1
?l r
0"
abc r
_
39 . 0 8
1
r
[a b O"c]
(VI. 9. 238)
Eq. (VI.9.229). together with (VI.9.238). is the explicit form in the minimal case of the differential equation satisfied by Tabc' or equivalently by
wab~. which
we anticipated several times earlier.
1926
Even in the purely bosonic case it looks very complicated and can According to the discussion of Sect. VI.9.I, it implies an infinite set of higher derivative interactions for the equations of motion in second-order formalism (see Sect. VI.9.1). we stress once more that the higher derivative structure generated by the expression (VI.9.238) is due to the presence of the Lorentz characteristic class y Q(~) and is completely independent of the one discussed before, which is due to the removing of the minimality constraints (Vl.9.170-171). In the effective theory of the heterotic string they are both present. be solved only iteratively in the y-parameter.
To complete the study of the H -Bianchi we finally analyze the (3,1)-sector, namely, Eq. (VI.9.202). The most general form of the (3,1)-sector equation is
(VI.9.239) where Aabe is a tensor-spinor antisymmetric in a, b, c. Equation (Vl.9.239) obviously implies (V1.9.240)
A b :: 0
a c
and, taking into account the SO(1,9) decomposition, (VI.9.241) we see that (VI.9.240) implies an equation for each of the irreducible fragments of (VI.9.202), namely, 1200 560 144 16 "-b- ,,1I:: 11- '" 11-:: 0 . a c ab a
(V1.9.242)
As we show below, Sq. (VI.9.240) involves the gravitino and
t
gravitello field strengths /lab:: 0ab - r[a 0bj and ~aX respectively. Hence it must be an identity in the llQQ and W sectors, since
1927 the gravitinQ equation lOUst be a statement on its lli·part, naQlely, on " free. Liki ' 1 10 0a -ra = Pab , W1t'h 0ab .§.@. =Pab re:a~nlng ew se, t he grav1te equation must be a statement on r .~ X, that is, on the 16 part of the a dilatino (gravitello) field strength,
A!:O ;: 0
To show that A!~~O" are identically satisfied. we proceed as follows. First we verify this property at 'Y" 0 by an explicit calculation.
Then We prove that the same property holds also at y ~ 0
by using the closure of the 4-form K introduced in Eq. (VI. 9. 177).
At
and in the minimal case, Eq. (VI.9.202) becOIIIes
y .. 0,
(0)
V(0,1) H(S.O) (y- 0)
+ (V (1,0) +
)l(I.O»
H(2,l)
+
(0) +
)l(2,-1) (H(l,Z) + Y(1.2»
+ 4 Tr (ff ,. 9")(3,1) ,,0
(VI.9.Z43)
wnere ~(1',5 ) \~:~~ are given by the parametrization (VI.9.l72b-cl. which. we recall, is also valid in the presence of the Yang-Mills source term Q(Y). Let us evaluate sepal'ately the four terms in the left hand side of (VI.9.Z02). The spinor derivative V(O,I) H(3,O) is evaluated by using the "1= 0 torsion equation (VI.9.215) and the spinor derivatives '1(0,1) Tabe , '1(0.1)0. V{O.l)A given in Eqs. (VI.9.8S. 95). (VI.9.S1a) and (VI.9.12Sb) respectively. We obtain V(O.I) "(S.O)(Y- 0) = 4 1 "3° 4 ) . .a b c .. V(0,1) [ - '3 Tabc e + '3 i abc>' v '" V "V
Ar
- II. 9' X Tabc
.. 1jI" (.
=
4 +
"3° ira 0bc + '4 i r ab \1 c)e 3
+
(VI.9.244)
The second term in (VI.9.202) gives (using the properties of of Table VI.9.II):
~(l.O)
1928
[V(l,O)
+
P(I,O)] B(2,1)
=
4
'"
~A 1- t rab ~c(xe30") + 1
4 3 ;6 e r:t f ijk rabcXTijk -
!r:t fmaXTmbc }Va"Vb..c V
- '3e
(VI. 9. 245)
•
The third term is immediately evaluated using (VI.9.172c), (VI.9.134)
and the Pe2,-I) definition (Eq. (VI.9.72-73»; o P(2,-I) (H(1,2)
4
'30" + Y{l,2»" - e
4
. '3° -
= - le
1/1 ..
ljiAr a
..a
1
abc Pbc V "V "V '" c
b
(r [a r:tbe1 - 4" rab 0" c) v .. V V
(VI.9.246)
A
where we have utilized the decomposition eVI.9.92, 94). (VI.9.12S) the fourth term gives
Finally, using
(VI.9.247)
Collecting the coefficients of $" Va" Vb VC in (VI.9.244-247) we get the y'" 0 value of Aabc defined in (VI. 9. 239). 1200 560 Let us now verify that Aabc '" J\.- '" O. First of all we see that the two terms containing the 560 representations O"ab appearing in (VI.9.244) and (VI.9.246) cancel identically. To verify the cancellations of the other l£Q[ and 560 irrepses contained in the terms X Tabc and A Fab , we use the decompositions: A
T
X abc
( T) 1200
= X abc
+
1
r
_1_
144 bCf - 563 r [ab (T) X c}
(T) 560
'2 [a X
r
(1)16
- 720 abc X
(VI. 9. 248)
1929 (VI. 9.249)
(VI. 9. 250)
~ X = (~ )144 a Xa
+
1r (~ )12 10 a X
(Vl. 9. 251)
where we have used the conventions of Table II.S.XIII. We can then . 1200 560 560 easl1y verlfy that the terms (X Tabcl--, (X T)ab and (A Flab cancel identically in Aab . Eq. (VI.9.240) therefore splits into its c 144 1§. 144 and 16 S0(1,9) parts, A;-" 0; A ,,0. After some manipulations we find (VI. 9. 252) ",-here (0)' 1 J "-6
a
12
. 'k
(r"1J ka XTIJ
-
. 'k
r 1J
3
2
XT .. )'20i(~ X+-3UX) + lJa a 3 a
- ia
1m3 - -60i r.1a(!iI i X + -43 ai ox) - -4 (8 rBl F + F. F)A e 5 rna .. ma
(VI. 9.253)
is the suparcurrent. at y"O.)
(The index
(0)
reminds us that we are working
We notice that the left hand side of (V1.9.2S2) is a pure
144 while the supercurrent contains both a 144 and a 16 part.
Indeed
a using the constraint (VI.9.94), r 0a" O. found in the super Poincare sector. one has
which is the gravitello equation of motion. ~X
=
Explicitly we have:
(0)
J
(VI.9.254a)
1930 (0)
J
4 2 -abe 4 c . ij - Sa .. - - r b XT- - - a a r X - 8Ir .• F Ae 9 a c 3 c 13
(VI. 9. 254b) Thus ~~ have justified our previous assertion that the constraint ~ aa = p12 = 0 is the gravitello equation (see Sect. VI.9.3). Equations (VI.9.252-254) are the fermionic equations of motion in the supergravity sector for the linear SUGRA-S'iM theory (Chap line-Manton theory). To complete the set of equations of motion for the linear SUGRA-SYM system we still need the two missing equations of motion, namely, the Einstein and the dilaton equations. They can be fOlBld by taking the spinor derivative of (VI.9.252) that is, by a supersymmetry transformation. The computation is cumbersome but straightforward, using the spinol' derivatives of all the fields derived so far. Here we just quote the results obtained by setting to zero all the fermion bilinears:(*) m R n
1m 21.m m 81n m --6 R=-(-Ooo -aaa)i'-(-3aaao-
2
3
n
10
n
9 10
n
n
n
4
_ m _ .!. ID Tijk _ - Sa ! ab m _ ..mil aoano) 15 0 n Tijk e (SF Fabo n 8t' Fnr) (Vl,9.255) (VI. 9. 256)
4
Do
i'
!
3
aaa
a0 = e a
--0
3
2 F"_ FR.m MIl
1.,Jj k 6
T •
(VI. 9. 257)
i3k
where the dilaton equation is derived from the trace of the Einstein equation by utilizing the constraint (VI.9.n9), R = T2. fOlBld in the super Poincare sector. We are therefore justified in calling the
i
* We do not write the complete equations including the fermions bilinears, since in the next section we write out the action for the linear SOORA-SYM system, and the complete form of the equations (VI.9.254. 256) including fermion bilinears can be derived in a much easier way from the variational principle. (However, we shall redefine the fields in that case (see Sect. VI.9.6)).
1931
constraint (VI.9.IIS) the dilaton equation of motion. Eq. (Vl.9.256), on the other hand, is the constraint (VI.9.I2l) which insures the s)'llIIIIetry of the Ricci tensor '\m' Let us now consider the (3.1) -sector in the case Y Fo. We proceed as for the (2,2)-sector. Using the decomposition (VI.9.l77) and the relations (VI.S.ISl) and (VI.S.233), Sq. (VI.9.IS3) can be rewritten as follows:
+
o P(2.-1) (H(1,2) + Y(l,2»
- 8 Tr(;§'" .1')(3.1)
+
+ Y[IC(3,l) + I]{O,l) (X{3.0) - tWabc va"yh ",Ve )} " O.
(VI. 9. 258)
Furthermore. using the identifications (VI.9.225. 226) and the definition (VI.9.23l). the expression proportional to y in (VI.9.2SS) takes the simpler form:
(VI. 9. 259)
Therefore Sq. (VI.9.2S8) can be written as follows: (VI.9.260) where Aabc{Y" 0) is the left hand side of the (S.l)-sector equation at y" O. previously analyzed, and t.-e have defined (VI. 9. 261a)
(VI. 9. 261b)
1932
In order to shoW that the presence of the y Q(91J) 4-form gives consistent lIIodifications of the fermionic equations of IROtion, we must prove that the spinol' tensor 4 eabc - Tabc has vanishing 1200 and ~ sectors. This can be shown without the explicit knowledge of 6abc and Tabc' Indeed, from the closure of the 4-form K, which is a consequence of (VI.9.177J. we have. for the (2,3)-sector (dK)(2,3)
=~(-1.2)
K(3.1)
+
VeO,l) K(2,2) = 0
(VI. 9. 262)
where we have made use of Bq. (VI.9.17Ba). Using the identification (VI.9.226) and the definition (VI.9.261), Eq. (VI.9.262) becomes a o = '23.11/1 ... r 1/1 ... ~ +
-
abcmn
il/l" r
b
e
Tabc V "V
-
1/1 ... 1/1
eabc
+
Vm Vn A
=0
(VI.9.263)
Decomposing Tabc and 6abc into irreducible parts (with the conventions of Table II.B.XIII) and using the irreducible 31/1 basis of Table II.B.XIVone easily finds 1200
1200
a~= T~= 0
(VI. 9. 264a)
560 560 a-;- t-;:b =0
(VI. 9. 264b)
4
This proves that the (3,ll-sector of the ;r~Biancbi give a consistent solution even in presence of the Lorentz characteristic class y Q(~). This statell\ent is the second part of the BPT-theorem. To find the modifications to the y =0 fermionic equations of IROtion, (Eqs. (VI.9.2S2-254» we notice further that in the 144 and M. sectors we also get from (VI. 9. 263) 144
r-= a
3
144 ea
(VI.9.265a) (VI.9.26Sb)
1933 We can then easily find the fermionic equations at y ~ 0;
(VI. 9. 266)
(VI. 9.267) o
0
where J a and J are defined by Eqs. (VI.9.2S3. 254a). Notice that the terms proportional to y on the left hand side of (VI.9.266, 267) can be consistently interpreted as the supersymmetric completion of the supercurrent
(3a .3)
at y~O.
Eqs. (VI.9.266, 267), (VI.9.232) and
the torsion equation (VI.9.229) are the equations of motion of the D=10 minimal anomaly-free supergravity for the fields 111. x. B and T b J.l IN ac respectively. The Einstein and dilaton equations can be found by adding to the right hand side of the y =0 equations the spinor deri vati ve of the terms proportional to y in Eq. (VI.9.266, 267). Neglecting as before the fermions bilinears we get m 1 m R n (w) - - 0 (ro) '" 2 n
rnn - -127y .. 7... (W
3
4 -- 0 e 3
T ij
ij(a b)
i
-i L
ab
..
.
..!... 0
10 ab
Do ..
"I
'3" a "aO
•
= T - lye
-!o
c
1 0 ijk l - 10 ab T Wijk ) J (VI. 9. 268a)
4
4 "a
Lc
m abc (L m - i Wabc T )
(VI,9.268b)
T mn and T are the right hand sides of Eqs. (VT.9.255, 257) m and L n is the ra-coefficient of the spinor derivative of the spinortensor 9abc defined in (VI.9.261b): where
(Vr.9.269) We remark however, that the spin connection wab entering the Lorentz ah curvature R cd is different from the corresponding one in the y =0 case since wah J.l is now a solution of the differential equation (VI.9.229). Thus, if we want to express the left hand side of Eq. (VI.9.268. 269) in terms of the Riemann tensor
RabCd(~)' we must solve
1934
Eq. (VI.9.229) iteratively in the y·parameter. Therefore, in the same way as in the toy model of Sect. VI.9.l in the metric fomalism we get an infinite set of terms of higher orders in the Riemann tensor (and its contractions). Other higher order terms are found of course from the explicit evaluation of the spinar derivatives V(O.l) e!44 and 'O'CO 1) 616 appearing in Eq. (VI.9.269). The explicit farID of the eq~tions of motion for MAPS requires the computation of 6;44. 61ft and their spinor derivatives. Although straightforward, they are very cumbersome computations and. as far as we know, no explicit evaluation has so far been attempted. Let us just conclude this section with some observations. A first observation concerns the appearance in the left hand side of the Einstein equation (VI.9.268) of terms which do not vanish at T. ok" O. Indeed, let us select among the various terms contributing to 1J 144 ° ° 2 (5) '0'(0,1) ea--. which.ls proportlonal to '0'(0,1) Wabc by Eq. (VI.9.261b), those of the form ~O,l) ClTabc (see Eq. (VI.9.237». One can then easily check that
'" (terms vanishing at Tabc"O) '" fermion bilinears.
(VI.9.270)
Thus (in the minimal case) the Einstein equation is satisfied on a torsionless Ricci flat background. Secondly. we observe that the Yang·Mills equations of motion, given by Eqs. (VI.9.129, 130), are left formally unchanged by the solution ~f the Jf·Bianchi given in the last two sections. Indeed, the Yang·Mills equations feel the presence of the ;F·field only through the presence of the torsion Tabc ' which is related to Habc and the other SlJGRA·SYM fields through the torsion Eq. (VI.9.229). Since.this equation does not contain the field strength Fab , it would seem that it is impossible in ~FS to generate terms of higher orders in the Fab field strength, which are known to be present in the effective theory of the heterotic string. We can however readily show that such tel'DlS can be
1935 generated in the second-order formulation of the theory. Indeed. considering the iterative solution o~ the torsion equation (VI.9.229) we have the following phenomenon. At the zeroth order in r we have 4
T(O)
abc
= _3H
abc
e
--0
3
+
more
(v!. 9. 271)
At the first order we get (v!. 9.272)
So that at the second order we find a term T(b2)
a c
g:
O· H. [ b Ri] + many more terms 1 a c
(VI. 9 .273)
where Ri is the Ricci tensor. Taking the derivative of the Bianchi c identity (VI. 9. 274) and substituting for the Ricci tensor the zeroth order of the energy
JllQmentum tensor (VI. 9. 275)
we get T(2)
abc
I I J J ~ P Fpi Fbc Fim Fam + more.
(VI. 9. 276)
This shows that by substituting (VI. 9. 277)
in the field equation of the Yang-Mills field (VI.9.130)
1936 (VI.9.278) higher order terms in the F curvature are generated. As a final point we want to discuss briefly the possibility of
solving the Bianchi identities of the lO-dimensional supergravity in terms of a dual formulation. By this we mean that we replace the twoindex field B" by a 6-index field B "so that the correspond~ l.lv
Ill" '''6
ing space-time field strengths are related by a duality transformation. Let us consider the two Maxwell-like equations obtained for the field strength Hab' c namely, Eqs. (VI.9.204) and (VI.9.232). It is convenient to rewrite these in terms of Tabc ' Eq. (VI.9.232) was obtained from Eq. (VI.9.114), f).m Tmab"'O' using the torsion equation (VI.9.229). Using this latter equation also in (VI.9.204), the two Maxwell-like equations in terms of T b have the following structure: a c (VI.9.279a)
d *T .. 0
dT
+
(VI. 9. 279b)
(more) '" J
where we have set T "Tabc Va vb VC and J is the current, that is, the 4-form 13 Q(§) - y QUI) (restricted to space-time), appearing on the right hand side of (VI.9.204). In the absence of external matter, we have J" 0 so that Eqs. (VI. 9. 279) simply reduce to d 'r =d T + (more) =o. A
A
In this case, it is our privilege to choose which of them we want to identify with the Bianchi identity, and which with the field equation. If we choose (VI.9.279b) to be the Bianchi identity, then r is the field strength of a 2-form B whose equation of motion is IN (VI.9.279a). If we do the reverse, the 7-form ·*1 is the field strength of a 6-form B whose field equation is given by J.l 1 ,,·J.!6 (VI. 9. 279b). Now, precisely as in Maxwell's ~heory, the above ambiguity is resolved by the coupling of external 'turrents. Of the two Maxwell equations, the true Bianchi identity is the one which remains homogeneous
1937 also in presence of external currents. The inhomogeneous one is the field equation. From this point of view, we see that Eq. (VI.9.279b) is to be interpreted as the dynamical field equation while Eq. (VI.9.279a) is the true Bianchi identity. Consequently, the 7-form *T is the field strength of what is worthy the name of true "electric" potential, namely an appropriate 6-form B • Pc .116
In other words, we conclude that what we originally named the Bianchi identity of the 3-form ;r(3) (that is, Eq. (VI.9.26d» is rather to be regarded as the inhomogeneous Maxwell's equation associated to a 6-form. The above observation can be strengthened by the following simple computation. Let us try to solve the Bianchi identities in the dual formulation by introducing a 7-form ;f(7) defined as follows: • (VI. 9.280) where a is the dilaton field, v is a number to be determined and B(6) is the aforementioned 6-form which plays the role of true electric potential of our system. The Bianchi identity associated to Eq. (VI.9.280) reads as follows
Using the universal parametrization of the super Poincare curvatures given in Table VI.9.II we try to solve Eq. (VI.9.281), namely, to find a rheonomic parametrization for ;f(7). We arrive at the following conclusions.
1938
i) No solution exists unless (VI. 9.282)
v .. 0 •
This fOllows from the (5.S) projection of Eq. (VI.9.281). ii) If given by
V"
O.
then there is a tmique solution of (VI.9.281)
(VI.9.283) iii) The solution (VI.9.28S) is "off-shell" in the sense that. when inserted into Eq. (VI.9.281). it satisfies it identically without implying any further constraint on the Poincare curvatures besides ~ m ab =0, which was already implied by the super Poincare Bianchi identities themselves.
ro
This very simple remark tells us that if we formulate anomalyfree supergravity in terms of the 6-form B(6) and consider only the corresponding superspace Bianchi identities (VI.9.26a.b,c) and (VI.9.281), we arrive at an "off-shell solution" for the entire system which is given by Table VI.9.II and Eq. (VI.9.28S). No information on the field equations can be extracted from these equations whose solution does not depend on the interactions considered. The dynamics is then introduced by supplementing the Bianchi identities with Eq. (VI.9.26d) which is now regarded as a true dynamical equation in superspace, no longer as the Bianchi identity of a 2~form. In this interpretation. the field strength of the physical field is Tabc (defined by Eq. (VI.9.283)) and "abc is an auxiliary quantity solved in terms of the physical field strength by Sq. (VI.9.229). equation (VI.9.204) yields the dynamical field equation of the 6-form B • The explicit form of this equation depends on the choice of UI .. ·!J6
the current 4·form J in two ways.
First, because J a
a = [6 Q(.n -
1'" 4
Y Q(91')j
al" .a4
appears in the right hand side of Eq. (VI.9.204),
secondly because the relation between the auxiliary quantity Habc and
1939
Tabc is also J-dependent as one sees from Eq. (VI.9.229). We conclude that the superspace formulation of anomaly-free supergravity based on the 2-form B(2) and the 6-form :](6) are entirely asymmetrical. The 6-form is the electric potential while the 2-form is the "magnetic" one. Correspondingly, the Bianchi identity of the magnetic potential B(2) is the field equation of the electric one, namely, the inhomogeneous Maxwell equation which furnishes all the needed information on the interactions. The Bianchi identity of the electric potential, on the other hand, being the homogeneous Maxwell equation, is a mere identity and yields no information on the dynamics. In Table VI.9.V we have summarized the most important results of the last two sections for the minimal supergravity e super Yang-Mills system.
TABLE VI. 9. V CURVATURES fif
ab
ab a cb "dw - w c "w
Y=dA+LA
PARAMETRIZATION OF THE CURVATURES arab" Ra\j vi" vj + . 1
+
~(- 4i
fa Pbc - 3i r[ab rill Pc]m) " VC
~ ,j, r ,I, Tabc + i;;, rabijk,/, T 6 ,'I' ~ c'f 36 'f" 'I' ij k
+
1940
Table VI.9.V (cont.)
va
P .. Dab
Vb
h
+
1 rm rijk't'.,."" Tijk 36 A
4
1 30' a b V ... V - - e X r bl/! V V 6 a
abc
.#' .. H b V
h
a c
A
X(2,1) - Y X(1,2)
- y
§
A
a
.. F b V
a
A
b . a V - 2l )..r 1/1 A V •
a
Parametrization of the O-forms X, a and ).. (gravitello, dilaton and gaugino)
da=coVa
a
1--X1P 4
FIELD EQUATIONS Torsion equation:
-1<)
Tac b
=e
.
[- 3 Hac b
+
4i ~r ac b A - 2y Wac b ]
1941
Table VI.9.V (cont.) Einstein equation: am 1 8 2 1 a R bm - 2" 0 b R = (10 00 0 b -
a
a
3"
abo) +
4 0 12 -y e- '3
[
7
. i c -lLb .. -obL a 10 8 C
Dilaton equation:
~3 Tijk
R"
4
-_ e- 3'0
~
T ijk
Do
+
± a80 3
d
0 =
a
4
F~ F~m _ -1 r ijk T. 'k
2
m
. -3 0 (Laa-1. W Tabc ) -lye abc
1)
Axion equation: fjj
m
Tmab " 0
:0
(9)
m 4 m - -3
a 0)(- 3 Hmab
+
-
4i Ar abcA - 2y Wabc ) " 0
Gravitino equation:
rm
Pam
= ~ cr 162
ijka X
Tijk _
-±cr 144 6. 3lY e ea
+ 7
rij X TiJa. ) -
1942 Table VI.9.V (cont.)
Dilatino (gravitello) equation:
Yang~Mills
V
Fm!
m "
equation:
= fermions
bilinears
Gaugino equation:
144 where'. w(5), abc' Ifabc'• Lab''eabc :-; ~ 1. 56 rab ac ~ J.. 6! rabc ~ are def"lned by Eqs. (VI.9.237); (VI.9.238); (VI.9.269) and (VI.9.26lb) respectively.
V1.9.6 The Lagrangian of N=l. 0=10 matter coupled supergravity at y= 0 In the present section we address the problem of constructing the Lagrangian of the linear SUGRA 1& SYM system. In the literature the abovementioned Lagrangian is known as the Chapline-Manton Lagrangian. Its standard form will match our previous results for the equations of motion after appropriate field redefinitions. It is not difficult to see that the purely bosonic Lagrangian leading to Eqs. (VI.9.255. 257) and (VI.9.130) is
1943
4 (} ( _1_ e3 T Tabc .. 1080 abc
+.lr 90
ab
pab).
(V!. 9. 284)
We observe the following features of (VI.9.284): Ca) The Einstein term is noncanonical. exponential of the dilaton.) (b)
(It is multipled by an
The explicit kinetic term of the dilaton is not present.
(c) The.Jf kinetic terros and !T kinetic terro are canonical; they have no dilaton factor in front. This is needed in order to produce, upon variation in the B field. the constraint (VI. 9.285)
which is the two-index photon field equation as expected.
Cd) The third term appearing in Eq. (VI.9.284) can be interpreted as a Lagrangian mUltiplier. With the variation oEm we obtain the constraint
ofna
,,0
(VI,9.286)
1944 where f1 are the components of the torsion 2-fotlll. On the other 1'5 hand, the variation OfJ.)ab yields for ~s the follOWing result
f11'5 = Tnrs
+
On[ r (as0
-
t 5 I)
(VI. 9.287)
so that combining (VI.9.286) and (VI.9.287) we get the identification of Es with the dilatoD derivative: (VI. 9. 288) Finally, we observe that the variations of (VI.9.284) with respect to the dilaton 60 and to 0 Tahc yield, respectively, R- ~T
- 3 ijk
Habc
1
Tijk
(VI.9.289a)
4
3'0
= - -3 e
Tabc •
(VI. 9. 289b)
Equation (VI.9.289a) is the constraint following from the Bianchi identities and is maintained also at the nonlinear level. Equation (VI.9.289b). on the other hand, is the linear (y= 0) bosonic part of Eq. (VI, 9 .229). The features a) - d) are quite different from those of the standard Chapline-Manton Lagrangian. Since the theory is the same, the two formulations must obviously be related by a field redefinition. This relation can be easily found by considering the supersymmetry transfotmation laws of the present formulation. The fields must be redefined so as to yield a new parametrization of the curvatures, corresponding to the supersymmetry transfotmations of the Chapline-Manton theory. The field redefinition turns out to be the "following:
a= "a
(VI. 9. 290a)
(1
V "exp
[16" o]Va
(VI. 9 • 290b)
1945 A
1/J ..
exp [1] - 0 (1/1 12
~
[1] 0 (I/I
IjI .. exp -
12
+ -
rax va)
(VI. 9. 290c)
1 .v.a-X r ) a
(VI. 9. 290d)
i
24
- 24
(VI.9.290e)
(VI. 9. 290g) "-
A .. A
(VI,9.290h)
"X ,. exp (- 1 - 0) X
(VI. 9. 290i)
" exp(- -alA 3 A"
(VI. 9. 290j)
12
4
(VI, 9. 290k)
the "hatted fields" are the new ones and the "unhatted" fields are those of the previous formulation. We see that using this redefiniH b and the parametrization of tion, we trade the O-form Tabcfor ae all the physical fields becomes (omitting the "hats")
~ilere
(VI. 9. 291a)
__ i_ (rabcm + 5 rab ocm)lJ;" V j(r X2048 m abc
(VI.9.291b)
1946 {tab" Rab
" "yf1
+
other terms not explicitly given, (VI. 9. 291c)
mn
(VI.9.291d)
(VI. 9. 291e)
Furthermore the derivatives of the ..a
da=30v
a
1 ~ --XI/I 4
O~forms
are given by
(VI. 9. 292aJ
(VI. 9. 292b)
(VI. 9. 292c)
The construction of the N=l, 0",10 linear Lagrangian in terms of the new fields obeying the transformation laws derived from Eqs. (VI.9.29l. 292) will be performed through the well-known procedure illustrated in Parts III and IV. Let us recall the various steps involved.
i) We construct a geometrical action for the free differential algebra (VI.9.23) with y=O: (VI.9.29S)
such that it is a polynomial of degree one in the curvatures. which is Lorentz invariant and homogeneous with weight W" 8 in the scale transformation (8' is the weight of the Einstein term), and such that the field equations
1947 I) y(geom)
oVa
6 y(geom)
oy(geom)
",.::....::..--=0
oij,
oS
(VI. 9.294)
vanish identically at curvatures equal to zero, i.e •• they are linear in the curvatures. In this way we take into account the requirements A, B, C, D of Sect. 11I.6.7 (or 111.3.9).
to
ii) We complete the Lagrangian for pure supergravity by adding .1(geom) a further term lly(SUGRA). lly{Sugra) is homogeneous with weight w= 8, as y(geom), and is
Lorentz invariant; it contains the first-order kinetic terms of B (this requires a O-form Habc ' 0, X. and all the other possible terms of correct weight one can write using X. do and the curvatures ra,;r, p, ~ab. This int roduces a set of unknown coefficients a1" .• am' and takes care of the kinetic terms for the O-forms and the 3-form .If. iii) To describe the Yang-Mills multiplet and its coupling to supergravity. we introduce two further terms, y(YM) and Il~(mixed) . .1 (YM) is the rheonornic Lagrangian of Sect. 11. 8.9 in flat superspace with some minor modifications explained below. 1l5f(mixed) is Lorentk invariant. homogeneous of weight w; 8 and is constructed by multiplying the supergravity curvatures ra,Jf, p and ~ab by the Yang-Mills
fields.
Thus, we introduce a further set of coefficients, b1,···,b. ~jP(mixed) describes the interaction between the Yang~MillS and them supergravity multiplet. The total Lagrangian is therefore 1'SUGRA;-SYM " !l (georn) ... 1l1'(SUGRA) ... ,y(YM) ... t\y(mixed) (VI. 9. 295)
Next, we determine the coefficients a1, ..• ,am; b1, ...• bm by inserting the rheonomic parametrization of the curvatures, CEq. (VI.9.28l, 292), in the field equations derived from the variational principle and requiring identical cancellation of all the terms except those along the maximum number of zehnbein ya, which correspond to the x-space propagation equation. (For convenient normalization, one requires also
1948
the space-time torsion T~ to be zero.) Such a choice corresponds in second-order formalism to ~he use of a metric spin connection w~ These steps correspond to implementing the requirement E, F of Sect. III.6.7, that is, rheonorny and consistency. Let us first consider the construction of ~(geom). Using the Pierz identities discussed in Eqs. (VI.9.137-139) we see that there is no Lorentz invariant constructed out of four ~ and six Va as required by scale invariance. Hence there is no A(l/I, V) term and yajom) is homogeneous in the curvatures. We write
where the normalization of the Einstein term has been arbitrarily fixed 1
.
ab
a
~
to - 8 for convenience. Varying (VI.9.296) In 6w • 6V , 6~ and 6B and requiring identical cancellation at .'If .. p .. §tab" Ta .. IJ .. 0, we obtain (VI. 9.297) Next we
write
Ll.t' (Sugra) .. (l)
1949
+ (a4 e
-0
abc Hb H +
a c
-0
+ a6 e
A
al
_
Jf" Xr a
a c1 as Ear)v
r · .a6"'" V,.. •••
•••
A
cto V
e;
,,+
c1......10
a6 "V
+
(VI.9.298)
and we recall
.tH~)
which was constructed in Chapter II.9:
1950 (VI. 9. 299)
Comparing Eq. (VI.9.299) with Eq. (11.9.38) we notice the following differences, which are now explained: -0/2)0 (a) !I and Fab have been consistently multiplied b y e. These factors become one when the dilaton is zero (0 =0) and cannot be seen in the rigid theory. Ho~ever, they are needed when one considers the coupling to supergravity (namely, when one varies ~YM in o~ and oVa as well). (b)
The term - 84i Tr (09' "A 1
-
+
+-A~AhA)1/IAr
3
a1
l/I"V
a1 ···aS
as
A
...
"V
(VI. 9. 300)
in Eq. (II.9.38) corresponding to a coupling of the gauge Chern-Simons form to a bilinear gravitino current has apparently disappeared from (VI.9.299). Actually (VI.9.300) has been reabsorbed in the definition of .If and in this way transferred to the geometrical action (VI.9.296). Comparing (VI.9.23d) with (VI.9.296, 297), we see that this reabsorption fixes the 6-coefficient of (VI.9.23d) to the value
8 =- 4 .
(VI. 9. 301)
This is exactly the same value for B that we have deduced in Chapter VI.8, Eqs. (Vr.8.290) and (VI.8.301), from string amplitude calculations. (A third way to fix B would be to reqUire gauge anomaly cancellation.) It is worthwhile to mention that the modification (VI. 9. 302)
1951
introduced by Chapline and Manton and leading to the Pauli-like term could readily be inferred from the earlier rigid supersymmetry results of Chapter 11.9. (c) The term
being proportional to lO-zehnbeins Va, has no w-projection and its coefficient f can be fixed by requiring supersymmetry invariance of the action (£1 d Y '" 0). Finally, let us write the most general form of ~y(mixed):
A '"
uoZ
(mixed) _
(1)
+ b6
-
-
)..ra
a a AX- r tP" Vp "Vq "V
1 2 3
pq
a4 " ••• ValO A
£
a l •·• alO
I
•
(VI.9.304)
We are now ready to insert the rheonomic parametrization (VI.9.29l, 292) into the field equations obtained by varying the Lagrangian (VI.9.295). As claimed. this fixes all the coefficients except the 4-fermions bilinears, which are proportional to ten vielbeins. They can be fixed either by requiring invariance of the action under supersymmetry variations (these can be read off Eqs. (VI.9.291». or by considering the trilinear terms in the gaugino equation of motion derived from the
1952 Lagrangian (VI. 9.295) and comparing these with the analogous terms in Eq. (VI.9.129) (after using the transformation (VI.9.290». The final result is given in the following table, which summarizes the D=10 supergravity coupled to 0=10 super Yang-Mills without Lorentz ChernSimons fOrlll. TABLE VI. 9. VI
N=l. 0=10 Supergravity Coupled to N=l, 0=10 super Yang-Mills without Lorentz Chern-Simons form. Definition of the curvatures:
.1f
_
i
cr-
a
= dB - 4n(A) - -2 e 1/1" r 1/1 ... Va
do :: do !F::dA+A"A
Rheonomic. Bianchi consistent parametrization of the curvatures:
1953 TABLE VI.9.VI (cont.)
a
d(j"EV
a
1--xl/! 4
_ i _1_ 2048 - i.
(r abem + srab oCDl),I." V x-r x-
1..8 (raDem + sr ab
VA " 'i/ A.V a
a
___1_ 1536
A;=
.a
+ 21l(T "V
a
~r
ocm) 1/1" V
b A-
mae
1 1 ab - 2(J - - r b1/l F e 4 a
Complete Action:
m abc
'I'
Xr sbcd r
+
3 64
3 - ab Ar Xr b 128 a
(-AX" -
)1/1
X abed
f!l
+.1fe
-(j-
)"I/I"r b
b ljI"V 1'" 5
b1
b5
" ... "v
-
1954
TABLE VI.9.VI (cont.)
1955
TABLE VI.9.V! (cont.)
The Lagrangian of Table VI. 9. VI is the Chapline-~lanton Lagrangian in first-orde~ formalism. In the following we give, for the reader's benefit, the translation from our conventions to those used in Ref. 6, and rewrite the Lagrangian of Table VI.9.VI in components (and in second-order formalism) utilizing the Chapline-Manton conventions. Let the unhatted quantities be those utilized in this paper and the hatted ones the corresponding quantities in the normalizations of Ref. 6. Then we have
vaIl = VaII W Il
1
= 1:2
(vielbein) A
$1.1
(gravitino)
(spin! in the graviton multiplet 2
(VI.9. 30Sa)
(VI. 9. 30Sb)
= gravitello) {VI. 9. :SOSe}
1956
AI' " _1_ Xl
412
B IN
::_l_A 12 jJv
AI " j,J
!2jJ iI
(gaugino)
(VI. 9. 30Sd)
(2£)
(VI. 9. 305e)
- orm
(gauge boson)
(VI. 9. 305 f)
1.0
e3
$
(dilaton)
wab ,,~ab Il
J.l
(VI. 9 • 305 g)
(spin connection) .
(VI. 9. 305h)
If we rewrite the action of Table VI.9.IV in second-order formalism, the spin connection is the solution of the torsion equation a
T ,,0 ~
~a
~ab
Ab
i
The definition of the field strength ;..
J'f j.lVp
a A
~
a[jJ vv1 - W[II vvJ - -4 1/J[ j.l r
"f2
"O! J.l Avp ]
A
'"
f JlVp 2
1 ::
1/1v
(VI. 9. 306)
0 •
reads explicitly A
h
A
1)
- - 2 (F[ jJV A] P - -3 gAl J.l Av AP
+
(VI. 9.307)
and from the rheonomic parametrization of Table VI.9.VI we get the 0form field H , a 1a2a3 (VI. 9.308)
Similarly we have
(VI. 9.309)
1957
and the O-form I:a reads
aj.l
(1
=
3 I f2 a].Ia .. aaa VIIa = -4'" ~ a• + j.l 4 A
A
0"
>"1/1
J.i
•
(VI.9.310)
Finally the field strength of the gauge field is (V!. 9. 311)
(Vl.9.312)
Furthe1'lllOre. we recall that the Riemann curvature utilized in this book being defined by ""ab
:n
...,..ab = uw
BC
- III
cb
(VI.9.313)
"W
differs in normalization from the standard one: 2!J ab
IIV
=!J"ab
IIV
..
ab ab aw -aw II v v ].I
+ ...
(VI.9.314)
Using this vocabulary, the space-time second-order projection of the action of Table VI. 9, IV becomes the one given below, where we ha.ve consistently factored out a common coefficient (-7! 2) and dropped all the hats. The quantities appearing in Eqs. (Vl.9.316) however. are always the hatted ones, and the spin connection is the one solving Eq. (VI.9.306). This allows us to drop all the torsion terms. We write y= Y(Un) + .YCPauli} + YC4-Fermi)
where
(V!. 9, 315)
1958 !fk m ,
:f
=~
1 i ~ -2 det V 91 (00) - -2 t/I
pr
,= i 12 Paull 16
.-3/4
(det V) Jif
~vp
~vp
D 1/1
vp (det V)
~ r CIINP131j1 ex
13
~
+
(VI. 9, 316b)
+
1 -I
I-J pvp J
384 X fJ.IVPX X r
X (det V)
(VI. 9.316<:)
1959
We can also write the supersymmetry transformation rules as (VI. 9. 317a)
os .. ]J\I
i {2 2
~3/4 iP
[lJ
r \I
e _ ! ¢3/4
J
4
Xr
E: _ j.I\1
i {2 ¢3/8 -I 2 X
r)lEAI\I (VI.9.317b)
o~
{2-
=- -
3
'PAe:
(VI. 9. 317c)
(VI. 9. 317d)
i - -256 (r,APOlJ + sr,liPg0lJ )EX-I rApf1XI
-
(VI. 9. 317e)
(VI. 9. 317f)
VI.9.7 Retrieving the superspace constraints from the K symmetry of the Green-Schwarz string formulation In Chapter VI.3 we have discussed the construction of the classical action of the heterotic superstring in the so-called Neveu-SchwarzRamond (NSR) formalism, characterized by world-sheet superconformal
1960
invariance. For heterotic a-models describing superstrings propagating on a general bosonic target space, we have shown that superconformal invariance requires T b =-3 H b (see Eq. (VI. 3.107) ), whi ch is the a c a c a .. y .. 0 approximation (with a .. 0) of the torsion equation (VI.9.229) calculated in 0=10, N=l supergravity. We did not investigate the tonditlons under which superconformal invariance is preserved at the quantum level. However, it is by now a well established fact that the vanishing of the superconformal anomaly, i.e., of the generalized a-fwctions, is equivalent to the field equations derived from the effective actions. In particular. in first-order formalism we would expect to retrieve the full torsion equation of the effective theory, Eq. (VI.9.229) being its minimal version. However, the NSR formulation and the effective actions deri ved in this framework lack manifest supersymmetry in the target space-time. Indeed Mtarget is by definition a purely bosonic manifold. There exists, however, another formulation of 0=10 superstrings, given by Green and Schwarz. This formulation is, in a sense, dual to the NSR formulation: the fields entering the Green-Schwarz a-model are superfields in the target space and purely bosonic fields on the world-sheet. As a result the 20 action has only 2D conformal (and not superconformal) invariance. The role of superconformal invariance is played here by a fetmionic symmetry called Siegel symmetry (or ~-syrnmetry). The effect of this symmetry is to decouple half of the fermionic degrees of freedom of the superstring, so that we retrieve the necessary matching between bosonic and fermionic degrees of freedom. Therefore K~Symmetry must be preserved in the quantum theory. If the Green-Schwarz superstring is embedded in a non-trivial background. K-symmetry gives restrictions on Mtarget in the same way as it happens for the superconforrnal symmetry in the NSR formulation. Since Mt arget is now 0=10, N=l superspace, these restrictions take the form of rheonomic constraints on the superspace curvatures. Indeed we show in this section that classical Ksymmetry implies the supergravity and Yang-Mills constraints of the S =y .. 0 theory (that is pure supergravity and pure Yang-~1ills theories without Chern-Simons forms). This result is the counterpart of the = -3 b we have fowd in the NSR a-model at the classirelation Tabc aHe cal level.
1961
The preservation of K-symmetry at the quantum level, that is, the cancellation of the K-anomaly, implies the superspace constraints we have used in the construction of the 0=10 heterotic string effective theory. This is the counterpart of the equations of motion for the bosonic fields one finds in the NSR approach. (Recall that superspace constraints imply the equations of motion through the Bianchi identities). We shall not give the explicit proof of the results one obtains at the quantum level (see Ref. 7 of this chapter), but we only sketch the main line of reasoning. The Green-Schwarz approach is quite convenient for comparing the results obtained by direct construction of the 0.10, N.l, SUGRA-SYM system with those obtained from a-model computations. Let us now construct the Green-Schwarz action. We proceed as in the case of NSR superconformal models, trading world sheet supersymmetry for target space supersymmetry. We first construct a 20 conformal theory utilizing as matter fields, ~i(~), the components of a target space superfield. More precisely, our fields $i(~) are the embedding functions for the injection ~i(~J: WS + Mtarget where WS is the bosonic world sheet described by the closed string propagating on Mtarget' a superspace. As in Sect. VI.S.3 we first identify Mtarget with a supergroup manifold G. In the superconformal case the bosonic group G was unspecified at the classical level. Here the supergroup G is fixed a priori to be the N=l, 0=10 supertranslation group, that is, N=l, 0=10 rigid superspace. Therefore, the Green-Schwarz model can be considered as an ordinary (bosonic) Wess-Zumino-Witten model, the bosonic group G being replaced by the supertranslation group. Consider now the superalgebra element
A" Xa pa
dio =
0
0
xa
0
0
0
and the associated group element
J
- -oi2 sr a 0
/
(VI.9.318)
1962 g(X,6) = exp (Xa Pa
+
QS)
=
(VI. 9. 319)
Then the left-invariant superform o
= g-1
a dg; V Pa
-
+ ~
(VI. 9.320)
where (VI. 9. 321a)
(VI. 9. 321b)
1j! '" de
satisfies the Maurer-Cartan equations (VI. 9. 322a)
d1j! :: 0
(VI. 9. 322b)
which are the analogues of Eqs. (VI.3.69). The pull-back of va, 2D world sheet is given by
1/1 '" 1j!+ e
+
+
1/1. e
-
Won
(VI.9.323b)
e+, e- being the "tweibein" defined in (VI. 3, 47a). The Maurer-Cartan equations are (VI .9. 324a)
1963 llljl-
-
-
(VI. 9. 324b)
+
The WZW a-model action can be written as follows: (VI. 9. 325)
where the trace is on the indices of the matrix representation given in Eq. (VI.9.3I9), *0 is the 2-dimensional Hodge dual of 0: (VI. 9. 326)
and c is a constant. Usjng the trace formulae (VI. 9. 327a)
Tr (Q(). P) a
= Tr (Q0. Qa) ; p
0
(VI. 9. 327b) (VI. 9. 327c)
and the algebra (VI. 9. 328a) (VI. 9. 328b)
one finds (VI. 9. 329a) (VI. 9. 329b)
We notice that the 3-form (VI.9.329b) is the cocycle of the super Poincare algebra, which we introduced in (VI.9.16, 17) to enlarge the
1964
Lie algebra (VI.9.14) to a free differential algebra. Therefore, on the trivial background we are now considering, we can write (VI. 9. 330)
The kinetic term (VI.9.329a) will be rewritten in first-order form to avoid the presence of the 2D Hodge duality operator according to the a the rules of the geometric approach. Introducing the O-forms n:!:. first trace in (VI.9.325) is substituted by [Va ,,(ITb+ e+ - ITb_ e- ) + n+a ITb_ e+ "e -] nab
(VI.9.331)
which has exactly the same form as the bosonic kinetic term in the NSR case (see Eq. (VI.3.100)). To complete the a-model Lagrangian we also introduce 32 heterotic fermions whose action is 2i Shet = 4i
f
s
~ (d~)
s e-
(VI. 9.332)
A
Mz where we use the same notations as in the NSR case: the indices r,s, .•. run from 1 to 32 and belong to the fundamental representation of 50(32) (or of 50(16) x 50(16) for the Ea x ES gauge group}. Moreover, since. we are on flat background, we set A=wab == 0, and V+ d. The complete 0=10, N=l action of the Green-Schwarz heterotic amodel on a flat supergravity background is given by
s == 1.. 411"
f M2
{(Va ...
(nb +
e+ -
nb -
+ c B + 2i ~s d ~s "e-} •
e-) + na nb e+ A e-)n b + +
-
a
(V1.9.333)
The coefficient c will be fixed later by requiring K-S)'Dlllletry. We can now immediately extend the action (VI.9.333) to the case where nontrivial background fields are present. In this case the background potentials ya.~. B and A have nQnvanishing curvatures defined by Eqs. (VI.9.23). There are just two modifications which are needed on a
1965 nontrivial Mtarget' The first is the replacement d+V in the kinetic term for the heterotic fermions: (VI. 9. 334) since now Frs # O. Secondly, non-trivial lOD backgrounds imply, as it is evident from the definition (VI.9.23d) of the B-field strength, a non~ero value for the dilaton field o. It is clear that a functional of the dimensionless field 0 may appear in front of the three terms of the action (Vl.9.333). It turns out that the correct choice is given by the following replacement:
4
3'(1
... e
a
(V" (II
b + e
- Ii
b -
e) +
b + na+ II)e '" e -
11 b
a
(VI.9.335)
that is, a simple rescaling of the kinetic term for the coordinate fields (Xa , eQ ). (The ~ in e4/30 is necessary in order to agree with our previous convention for D=10, N~l supergravity). Therefore, the Green-Schwarz Lagrangian in a non-trivial supergravity and YangMills background is given by
where the background fields Va, 1/1, B, F, 0, X now satisfy the relations (VI.9.2S) and (VI.9.3l) and the Bianchi identities, (Vr.9.26) and (VI.9.32). Comparing the action (Vl.9.336) with that corresponding to the NSR o-model (~q. (VI.3.116)), we see that the former can be obtained from the latter by killing the 2D-supersymmetry cl:: 0) and replacing the structural equations of the bosonic target space (Eqs. (VI.3.106, 108)) with those defining the supertarget, namely, (VI.9.23). Further~ more, one has to perform the rescaling (VI.9.33S). It is obvious that
1966
the action (VI.9.336) has 2D reparametrization, Lorentz and Weyl invariances. The gauge fixing of these invariances has already been discussed in Sects. VI. 3.1, 2. Setting +(z,z) == wab =0 (a,b =0,1) we can reduce the 2D zweibein to the fOrm (VI. 9. 337a) (VL9.337b) (cf. Eqs. (VI.3.57) restricted to the conformal subgroup of the superconformal transformations). We now observe that the use of the supel'conformal gauge transfOrmations in the NSR case enables us to fix the gauge where only the transverse degrees of freedom of Xa and Aa do propagate. In the present case Xa has 8 and aa has 16 on-shell degrees of freedom. Therefore, we need to kill 8 fermionic degrees of freedom in order to have a consistent supersymmetric theory on target space. This can be done since the action (VI.9.336) has a further fermionic symmetry which we shall now discuss. Let us consider the general variation of S under a variation of all the 2D fields, namely, ei , xa, aa and nt. This enables us to find both the equations of motion and the new syrmnetry. Actually. instead of varying the coordinate fields (Xa,aa), it is more convenient to vary the two I-forms (Va ••) (this is the same tangent variation we used in Chapter VI.S). (Notice that also B and a undergo a variation, being functionals of Xa and aa). The general variation is given by
cSS
=
J
1.0 e
3
142
f
a _b
. . . .ab
+ cSe ,. (v
b
+
-
ab~
+
h - b + on a- (-V-" e + II + e "e )11 ab
-I
IT+- 1 II 'n e)nab + oe ,.. I (-V -
.rr}
+ 21 ~ 'V ~
+
+
1cSII + (V- .. e + n~ e .. e )Tlab
+
3+aa ,,(n+ e - n- e)
flV
4
ab+ n1+'nne )n 3 b +-
ab
+ -3 oo(V
a A
...
+
(nb e+ - nb e) +
+
~
J doB.
rra b + ] e + 2i ~ rrss~ OA ~"e) + c nne", e )n b + (4i cS~ 'V ~ +
-
a
A
143
(VI. 9.338)
1967
The equations of motion for the fields
n±
±
and e
are
Expanding Va as in (VI.9.323a) we find the identification (VI. 9.340)
establishing the transition from first- to second-order formalism for V:. From the zweibein variation we have . .a b
(v
n.. • n..a I lb- e- )11 ab
65 ab <se. '" 0 '* l'Jab (- V n..
+
" 0
ab+ ,..l'r IT.. TI _ e ) + 2i I; A'VE; ,,0.
(VI. 9. 341a)
(VI .9. 341b)
Equations (VI.9.341) give the explicit form of the Virasoro constraints, Eqs. (VI.3.29).
Substituting Eqs. (VI.9.340) into Eqs. (VI.9.341) we find (VI. 9. 342a)
(VI. 9. 342b)
(VI. 9. 342c)
To obtain the equations of motion for the coordinate fields xa and Sa we proceed as in Sect. VI.3.4 by considering the Lie derivatives of the background fields B, 0 and A(XS with respect to vectors tangent to the superspace manifold. (Notice that the fermionic background fields $, X do not enter explicitly in the action (VI.9.336). As in ... the NSR model, we have a bosonic tangent vector oX a p ;: oX. In addition .... a we also have a fermionic tangent vector oe'J. Da :: E:. These vectors satisfy
va,
1968 (VI. 9. 343a) (VI .9. 343b)
We must therefore consider Lie derivatives of the background fields along the bosonic vector oX, and also along the fermionic vector t these latter corresponding to supersymmetry variations of
ya.
B. A and
1.1.
From the Lie derivative formula
(VI. 9. 344)
+
setting t,. 1+
oX+
+ +
and t" & we get
ya " iepa", + 1.J Ta
-
(II wab)Vb
(VI. 9. 345a)
&
(VI.9.34Sb)
(VI. 9. 346a)
(VI. 9. 346b) R.+ Al'S " .!Jj'"l'S + v(~] Ars )
e:
(VI. 9. 347a)
(VI. 9. 347b)
(V!. 9. 34Ba)
(VI.9.348b)
1969 (See for CO$parison Eqs. (VI.3.117-125) applying to the NSR case). In the right hand side of Eqs. (VI.9.34S) and (VI.9.347) there appear fielddependent Lorentz and Yang-Mills gauge transformations with parameters wab and .1J Ars respectively. We can forget these as far as the classical action is concerned, since they correspond to exact symmetries of (VI.9.336). At the quantum level, however, the effective action will have gauge anomalies so that the presence of these extra transformations cannot be overlooked. We shall come back to this point later.
11
Substituting the Lie derivatives (VI.9.34S-348) into Eq. (VI.9.338) and equating to zero the coeffi cients of 61" and Ell:: 6iia , we find the equations of motion for the coordinate fields:
(VI. 9. 349a)
4 4 30" b +- -3 e ~ dO" V+ Vb!- e ... e ,,0
(VI.9.349b)
where we have set (VI. 9. 350a)
1970 (VI. 9. 35Gb)
We
we
notice that in evaluating the term bilinear in the heterotic fermions have performed the gauge rotations (VI. 9. 351a)
(VI.9.3S1b) in order to compensate for the field dependent gauge variation of An given in Eq. (VI.9.347). Finally. the equations of motion for ~r is given by (VI. 9.352)
Let us now investigate the fermionic symmetry of the action (VI.9.336). It is convenient to consider first the case of a trivial background: Rab =~ =p =H=a '" ,Frs .. O. In this case the variation of the action can be written as follows:
(VI. 9. 353)
where we have used Eqs. (VI.9.340) and the left hand side of Eqs. (VI.9.349. 350) in the flat case. We now set
1971
(VI. 9. 354a) (VI. 9• 3S4b)
K+ being a 16-dimensional Majorana-Weyl spinoT and a 2-dimensional vector and 0 is a constant. We try to detel'l!line 6e + and 6e- so that identically oSflat" O. This happens for (VI, 9. 355)
6e- = 0
(VI. 9.356)
y " 1
(VI.9.357)
K .. 0 +
(V1.9.358)
Notice that the last condition means that K= K e- is a 2-dimensional self-dual vector. Hence we find that the 2D action (VI.9.333) is invariant under the following transformations (K-symmetry): (VI. 9. 359a) oe + ,,- 2i
K_
w_ e-
(VI. 9. 359b)
(VI. 9. 359c)
The transformations (VI.9.359) can be interpreted as a restricted supersymmetry in the laD flat superspace, the restriction being given by the particular form of the oe-parameter. Notice that the action is invariant only if we perform the simultaneous notation of the 2D-zweibein described by Eq. (VI.9.359a, b). The K-transformations do actually close an algebra only on the onshell configurations of the physical fields (Xa,e). This can be easily + verified by computing the cQmmutator of two K-transformations on e and ea. One obtains a new K-transformation, a 2D general coordinate
1972
transformation and a Virasoro constraint (which vanishes on-shell). The conformal and K-symmetries are fixed through the following light-cone gauge conditions: +
+
X =x
+
+
(VI. 9.360)
Pt
(VI. 9. 361) where we have set 1;0:? and pll is the a Moreover, for any 100 vector u we have should not be confused with the 2D ones). is the so-called light-cone gauge for the only the 8 transverse coordinates survive
100 momentum of the string. + 0 9 set u- =u ± u (there ±
The gauge condition (VI.9.360) bosonic string; in this gauge as physical fields,
Equation (VI.9.361) extends the light-cone gauge to the fermionic coordinates Sa of the heterotic string and reduces the 16-dimensional e spinor to an 8-dimensional one, matching the eight bosonic degrees of freedom. Using (VI.9.360, 361), the gauge-fixed action takes the following fom:, (VI. 9.362)
.r:i? S
where we have set S =
and used the following identity
= '21 -der ar+ r- e = 0,
(except when a= -).
(VI. 9. 363)
The action (VI.9.362) describes a free theory and has the following global supersymmetry invariance: (VI. 9.364) (VI. 9.365)
1973 (VI. 9.366)
For the quantization and the spectrum of the corresponding theory we refer the reader to the literature. Let us now investigate the extension of K·Symmetry (Vr.9.359) in the presence of a non-trivial background. In this way, we shall retrieve the rheonomic constraints on the supergravity and Yang·Mills curvatures. If we keep fqs. (VI.9.359) unchanged, taking into account Eqs. (VI.9.338, 349). we get oS
"' (v..a+a+ = JM ioe" V I )e '" ",66-a (-
4 a '" "e-}e'30' 2ir t/J V)e a-+
2
~!...a + - 40' MI '" oa-a{ ~ T (V Ie", V I e)e '" ~ If a+ a+ 2i~r ~5rs~s
e+ Ae-}
+ 2i~rl7(oe~£J
4
~
b
} +
-3 a do" V+ V- Ib e
A
+ 4iOr.rl7~r
•
Ars)F,s
e-
(VI. 9. 367)
The terms inside the first brackets do not contain the SUGRA-SYM curvatures and therefore, as in the flat case, they vanish upon the use of Eqs. (VI.9.359). An integration by parts shows that the last two terms cancel each other if or.r is given the transformation law (V1.9.3S1b). The cancellation of the residual terms containing the background curvatures can be achieved by imposing suitable constraints on the curvatures, as we shall now show. Let us set (VI. 9. 368a)
~I
~F
rs
_I
=~
rs
(~(1,1) +
.... rs oT
(0,2»)
(VI. 9. 368b)
(VI. 9. 368c)
(VI. 9. 368d)
1974 On the torsion and Yang-Mills curvatures we impose the following
constraints: i)
(VI. 9.369)
§
rs (0,2) .. 0
(VI. 9. 370a) (V!. 9. 370b)
ii)
On the 3--form .If defined in (VI. 9. 23d) we require the absence of
the Chern-Simons forms (Le. B" Y= 0) and the minimal parametrization given by Eqs. (VI.9.172), namely (VI.9.371a) (VI. 9. 371b)
(VI.9.371c)
In Sect. VI.9.S we proved that the actual parametrization of
"-(1.1)
and H(2 .I) is fixed by the § - and Jf-Bianchis once the constraints (VI.9.370a) and (VI.9.371a. b) have been imposed.
Therefore. the con-
straints (VI.9.37Gb) and (VI.9.371c) follow from (VI.9.369. (VI.9.371) respectively.
370a) and
Furthermore. we also recall from Sect. VI.9.2
that the constraints (VI.9.369) fix the parametrization of p and (modulo field redefinitions) in the super Poinca~ sector.
~ab
It is now
straightforward to verify that. using the constraints (VI.9.369-371) and the form of
69
given by (VI.9.359c). all the terms in the second
brackets of (VI.9.367) cancel identically except for the residual terms
(VI.9.372)
1975 However this expression is proportional to the Virasoro constnint
v! Va/
+
~d can therefore be cancelled by changing the transformation as follows: ).
~e
llUi of
In this way we find
oS .. 0
In conclusion, provided the
identically.
supergravity and super Yang-Mills curvatures satisfY the constraints (VI.9.369-S71) the action (VI.9.336) describing the Green·Schwarz heterotic string propagating in the N-I, ,n..10 background is invariant under the following Siegel transformations: i)
for the 2D-fields:
6 e .. Krbvb I(
-
61( e+
ii)
(VI.9.374b)
'"
= - 21 K.I/!.
+
3 (i K}bXV~ aio
f or t he background fields
1.0
R 1,1.,.
~
B 1. e3·e: r l/I '" .~ v a ..
+
2i ii;),rs~r~S)e-
(VI.9.374c)
(e: =- c\:6):
4\1 I ! - ab - - e e:r XV Vb 6 a A
(VI. 9. 37Sa) (VI.9.375b)
R
1,1
K
1 Ca·-ex
(VI.9.375c)
4
(VI. 9. 375d) (VI. 9. 375e)
1976
where we have used Eqs. (VI.9.345*348) and the constraints (VI.9.369, 371) (Eq. (VI.9.375f) is added for completeness even if the gravitino ~ does not appear in the action (VI.9.336». At this point we recall that the coupling between supergravity and super Yang-Mills is enforced by the presence of the Yang-Mills Chern-Simons term (8; 0). Hence the constraints (VI. 9. 370, 371) (which include (6" 0) imply that the 100 background is that of the decoupled (minimal) SUGRA. SYM system. Actually, if the constraints (VI.9.369. 371) are introduced into the Bianchis (VI.9.26), we shall find the equations of motion of the decoupled SUGRAeSYM system (they are obtained from Eqs. (VI.9.129b. 130, 217. 252-254, 255-257) by removing the coupling terlDs (a= 0)). It is worthwhile to note that we have obtained a constrained superspace, and in particular. the equations of motion of the background fields, by the requirement of K-Symmetry invariance at the classical level. In the NSR model the background field equations are obtained only at the quantum level (n-loop a-model computations of the relevant a.-functions). On the other hand, a fully consistent treatment of the GS heterotic string in an arbitrary background must of course imply a careful study of the conditions, under which the classical symmetries of the GS a-model are preserved at the quantum level. In particular, we must expect that the reqUirement of cancellation of the K-anomaly should modify the constraints (VI. 9.368-372) and the condition i3 .. y ,,0 obtained at the classical level. These modifications in turn should reproduce the constraints obtained in Sect. VI.9.4 for the most general effective theory of the D.,10, N=l SUGRAeSYM system, namely, the non~minimal constraints of the inhomogeneous case. This indeed happens as recently shown in a series of papers (see Ref. 7). We shall content ourselves to give only a rough sketch of the main line of reasoning. Let us suppose that the GS string model has been quantized and renotmalized consistently. (That this is indeed possible has been recently shown in Refs. 7, 8.) We call
r " S + h r (1)
+ •••
(VI. 9. 376}
19n the renormalized effective action. At first-order in Jf is defined by
or = c5S .. ftst
~
the anomaly
(VI. 9 .377)
where 6 is the nilpotent Slavnov operator, which on functionals of fields and ghosts can be identified with the BRST variation, and (VI.9.378)
The anomaly cancellation condition is (VI. 9. 379)
and, taking into account the nilpotency condition 0 2 :: 0, we get the Wess-Zumino conSistency condition: (VI.9.380)
6Jf '" 0 •
The anomalies we have to consider are those related to the conformal and K-invariances of the classical action; the invariance under 2D diffeomorphisms need not be considered since there exist invariant regularizations (for example dimensional regularization) which preserve diffeomorphisms. To treat the K-anomaly let us consider its general structure. On the basis of general considerations
[7j one finds that st has
the same general structure as 6S given by Eq. (VI.9.367). MOre precisely, one can set
.. .£J Jf A
r
+
r s
2i ~ ;
.£J §" TS A
+
-}
e,.. e
(VI.9.381)
where a and JTS are 2-forms and'; is a 3-form. Substituting Eq. (VI.9.379) and using Eq. (VI.9.367) one obtains
1978
+
S!l
h
(f + fl .
+ 2i ~rt;S ~ (ffrs
+
11 jrs)e+ "e~}. (VI. 9. 382)
Aa
~rs
A
fII' and flJf' are speci~ "a and ff"rs , and in the fied only in the sectors (0,2) and (1,1) for T " From Eq. (VI.9.383) we see that sectors (0,3), (1,2) and (2,1) for f. the anomaly can be cancelled if .£! ffa. S!l §rs and .£1 Jf contain terms Ars "'of order fl which can compensate .£1 ff ,i!I § and.ill Jf respecti ve~ Aa ""'rs "1y. Since T, 1" and Jf' are curvatures, this can happen only if a, jrs and ;. satisfy the relevant Bianchi identities in superspace. The important point is that the consistency condition (VI.9.380) implies that the Bianchi identities (VI.9.26) are satisfied by the new forms a "a rs "rs T '\" fl T ,ff + flff and Jf + flJf'. Indeed, let US for example conArs "a sider the case where ff = T ; 0 so that the anomaly has the form We note that the quantllJll corrections
h T "
~a
r
A
.!I
K
a "oe ~.1f M
=J
(VI.9.383)
2
The consistency condition (VI.9.380) gives
(V1.9.384)
We note that
.£J
~ d; has only the (0,4), (1,3) and (2,2) sectors.
We first require (VI,9.38S)
1979
...
that is, the vanishing of dJf in the sectors (0,4) and (1,3). This is equivalent to l'equiring that t~e 3~form; satisfies the Bianchi identity in the (0,4) ~d (1.3) s~ctors. At this point we do not know the explicit fol'lD of Jf. However, we recall that a K-tl'ansformation induces field-dependent gauge and Lorentz transformations on the background fields (see Eqs. (VI.9.345, 347) with eo' =0Keo, given by (VI.9.374b). If the 2D heterotic fermions are integrated out in the functional integral. one gets an effective action whose gauge variation is anomalous [9]. That it must be so is clear from the- fact that the heterotic fermions ;r have a chiral gauge coupling in the action (VI. 9. 336). As a consequence, the K-symmetry, which involves the gauge transfOl'lDation (VI.9.374d). is also anomalous. In Ref. 9 it is shown that both the K- and gauge-anomalies can be cancelled through the redefinition: (V1.9.386) with 13 .. - 4. In an analogous way one can also show that the effective action possesses a gauge Lorentz anomaly which induces a K-anomaly through the wab -field dependent K-transformation. The further redefinition ."..(13 =y= 0)
+
B 03{A)
~
.;f(B =y;; 0)
+
B 03(A)
- Y Q3(w)
(VI. 9. 387)
with y .. -1/32 cancels the 10D Lorentz anomaly as well as the related 2D K-anomaly. Therefore. in order to solve (VI.9.385) we set ....
.Jf = B n3(A)
~
y n3(w)
+
X
(VI. 9. 388)
and try to determine X in such a way that (VI.9.38S) is satisfied. Fl'OIII the previous analysis of the JI"-Bianchis in the sectors (0,4) and (1,3) given in Sect. VI.9.2, we know that the solution of the consistency condition (VI.9.385) exists; its explicit form is determined by
1980 the parametrization (VI.9.181a, b) with X given by Eqs. (VI.9.19S-197). In the sector (2,2) the condition (VI.9.385) cannot be fulfilled because Eq. (VI.9.388) implies "
(VI.9.389)
(d·fI")(2,2)" y 1(2,2 - 8 Tr (y" §)(2,2)
However, the theorem proved in Sect. VI.9.4 tells us that Eq. (VI.9.389) contains only the SO(1,9) irrepses 120 so that it has the general form ... . "(1) abc (dJf)(2.2) " 1 Nabc ljI"r 1/1 V "V + + i
webS) ~,. r abcmn•
ac
A
yo" va
(YI.9.390)
and webS) are actually given by the tensors N(bl ) - !4' ~r b~ and ac ac a N(Sb) +! Xrab A according to Eqs. (VI.9.211, 212, 223, 224). Thereac 6 c fore, introducing the expression (Vr.9.390) into (VI.9.384) we get
ebl) (Wac
J2 15 aa 15 a8 ~ m(d ;)2 2 " =f eral) a vb V + i ;;(5) 15 erabcron 15 e v M abc abc m M
K
K
(i W(!) I)
,
C
A
K
A
K
K
K
V ) ::
n
2
"JMz (i
W(l) i< r rar
abc - p
K
q-
yP yQ yh VC +
+
+
+
-
+ i ;;;(5) i< r rabcJIIllr K vP vq vm Vn)e+ "eabc - p q- + + + -
(VI.9.391)
Using r-matrix algebra ~~ see that the first term vanishes identically, while the second term is proportional to the Virasoro constraint (VI.9.342a), yo+ V+ Im:: O. Ne conclude that the consistency condition (VI.9.387) is satisfied if we use the new definition of H given by .Jf:: JI" (8 " y ,,0) + 11; with ; g1 ven by (VI. 9.388), and if its parametrization is the standard one discussed in Sect. VI.9.4 (Eqs. (VI.9.168) or (VI.9.172b, c), if the .:;tronger constraint H(O,3) '" 0 is imposed). In an analogous way, one proves that for the other 2-terms
1981
in the general expression (VI.9.384) the consistency condition implies which are satisfied if §' " a set of conditions on ;- and ~ ta -B Aa ,. + 1\ ff and T :: T + fl T obey the constraints
ra.,
§(0,2)
'a
T (O,2)
(VI. 9. 392a)
=0 =0
(VI. 9. 392b)
•
Eqs. (VI.9.392) are indeed the constraints that we have assumed to hold in solving the § and Ta Bianchis; * they therefore imply the parametrizations (VL9.l25) for § and VA. and the parametrizations of Table VI.9.II for the super Poincare curvatures. In conclusion, we see that the quantWll consistency of the GS heterotic string implies that the amended supercurvatures of the SUGRA$ SYM system satisfy the Bianchi identities implied by the constraints discussed in the previous sections. We know that this set of constraints also implies the equations of motion for the background fields. Therefore, quantum consistency implies the equations of motion of the most general SUGRA 6l SYM background in exactly the same way as in the N5R approach. In principle one should also prove the cancellation of the conformal anomaly. However, we know that the conformal anomaly is proportional to the K-anomaly, since the vanishing of both implies the field equations. In this section, we have discussed only the form of the K-anoma1y and shown that the Wess-Zumino consistency conditions are essentially equivalent to the superspace Bianchi identities. The actual calculation of the K-anomaly would determine the explicit form of ;f(0.3) , but is probably as difficult as the calculation of the conformal anomaly in the NSR cr-model.
*
[}126
°
[ , a] 1050
Actually only ';-(0,2) -;; and T (O,2) =0 are essential since the other 50(1,9) fragments are coboundaries of the BRSToperator. This agrees with the discussion given in Sect. VI.9.2.
1982
VI.9.S Bianchi identities and off-shell formulations of Nal. 0=4 s~ergravity revisited The effective theories of four-dimensional heterotic superstrings with N=l target supersymmetry are N=l supergravities. As in the tendimensional case these effective supergravities are not limited to second-order derivatives but must include higher curvature interactions and incorporate Chern-Simons foms. In view of this, we reconsider the structure of D=4, N=l supergravity, discussed in Part lV, in order to generalize it to the higher derivative case. Standard N=l matter coupled supergravity was constructed assuming the following: i) There is a Kibler a-model structure underlying the Wess-Zumino
multiplets. The Kibler connection Q, which is a U(l) gauge field, is thus a function of the coordinates zi, zi* describing the bosonic degrees of freedom of the Wess-Zumino multiplets (zi.i): Q= 1m dG(z,z}, G being the tihler potential. As a consequence, the auxiliary field Am appearing in the parametrization (VI.3.52b, c) is identified with a bilinear in the matter fermion fields (see Eqs. (VI.3.25b), (VI.3.6la) and (VI. 3.125) • ii) The auxiliary field S is a fUnction of the zi. ii· coordinates (see Sq. (VI.3.61b». The Bianchis then fix 5 in terms of the Kahler potential (Eq. (VI.3.84)). Moreover, the other auxiliary fields At and X are set equal to zero. a It is clear that the assumptions i) and ii) are very restrictive. Indeed, they are so strong as to imply the standard second-order theory examined in Part IV, where any higher derivative structure is absent. In principle. one could think that the various auxiliary fields depend in an arbitrary way not only on the matter multiplets, but also on their higher derivatives. The only restrictions are the relations implied by the Bianchis on the auxiliary field functionals. Obviously the dependence of the auxiliary fields on higher derivatives implies the introduction of a dimensional constant which we can naturally identify with
1983 the string tension (a t )-1/2. This is the way the higher derivative structure enters into the 4-dimensi~nal supergravity theories. The previous discussion makes clear that it is very difficult to extract information from only the Bianchi identities. Different ansatte for the auxiliary fields correspond to different higher derivative structures. To decide the correct ansau we have to resort to either string amplitude or a-model B-function calculations. Nevertheless. the study of the Bianchi identities gives us information on the general form of the theory. There are strong analogies between the 0=10 and the D=4 cases. The massless spectrum of heterotic superstrings in 0=10 has been extensively discussed. In particular, one has the follOWing massless fields: (VI.9.393a) (VI. 9. 393b)
The same states are present also in 4-dimensional superstrings. Thus in D=4 we also have, besides the graviton and the gravitino, the axion B~v' the dilaton a and the dilatino X. While in 0=10 the states (VI.9.393) belong to an irreducible multiplet, in D~4 they fall into two different irreducible multiplets:
o = 10"
D =4
(VI.9.394)
The matter multiplet (B~v. X. 9) containing the 2-index gauge field, tRe dilaton • and the spinor X (dilatino) is named the linear multi~let. From the point of view of physical states the linear multiplet (B~. x. +) is equivalent to a chiral multiplet (~ .. A+iB. Xl. i.e. it deSCribes a scalar, a pseudoscalar and a spin 1/2 particle. The relation between the two fie ld representations. however. is given by a
1984
field redefinition (a duality transformation), which implies a substantial difference at the level of the interaction and quantum properties. In particular a chiral multiplet contains no gauge fields and therefore gives rise to no anomalies. The Green-Schwarz anomaly cancellation mechanism corresponding to the substitution non~local
H
PVP
...
H ;; ~VP
H _y pVp
nell
pvp
(VI.9.395)
(~'here H = a B is the B field strength and nell is the Lorentz IJVP ~ 'JP ~ ~P Chern-Simons form) takes place in the D=4 as well as in the 0=10 superstring field theory limits. Hence it is important to discuss the structure of supergravity theories ~'here the invariant field strength of the antisymmetric tensor is given by formula (VI.9.395J, namely. where BIJ'J transforms non-trivially under the local Lorentz group. In the tendimensional case, where the supersymmetry algebra can satisfy the Jacobi identities only on-shell, the modification (VI.9.39S) of the H field lJVP strength can be reconciled with supersymmetry in an almost unique way; then from the closure of the SUSY algebra we obtain the field equations of all the physical fields. we consider the same problem in the D=4 case, the major difference we have to face is the off-shell closure of the supersymmetry algebra. This implies that the solution of the super Poincare Bianchi identities is no longer unique; rather, we have a few different solutions, each involving a different set of auxiliary fields. The auxiliary fields appear in the parametrization of the super Poincare curvatures, and hence in the SUSY transformation rules of the fields va, ~ and wab • Thus in our approach the choice of an auxiliary field~setlJcorres\.l ponds to the choice of a certain form for the SUS! rules of the Poincare fields. ~ben
There are two sets of auxiliary fields one can use to describe matter coupled supergravity: 16 e 16 and the li eli. The two descriptions turn out ta be equivalent in the linear case, but inequivalent in the higher derivative case! In particular. if we use the 1£.12 set, the fI-Bianchi identities are not modified in the (O,4) and (1,3)
1986
sectors when both the Yang-Mills and Lorentz Chern-Simons forms are inserted. (In D=10 this happens only for the Yang-ltills Chern-Simons). Therefore, with this choice of auxiliary fieldS t the higher derivative structure due to the Chern-Simons terms becomes particularly simple. In the 1218 12 formulation, another similarity with the 10-dimensional case is that the whole higher derivative arbitrariness is located in the choice of M(O,S)' In 0=10, the most general solution for R(O,S) is given in terms of an irreducible spinor tensor $(672) • with a a1.. ·8S constraint on its spinor derivatives. Here the analogue is an irreduwith constrained spinor derivatives. cible spinar tensor
.!:J
There are two off-shell solutions of the N=l. D=4 Bianchi identities.
The first is given by Eqs. (VI.3.S2) and contains the auxiliary fields Aa' A~, 5:: ~ + i Y". 1;. Once the Wess-Zumino multiplets are introduced we set A' =l; :: 0 so that we are apparently left with a system a a of 12 G) 11. off-shell degrees of freedom. (Remember that VIJ and l/Ill have 6 and 12 off-shell degrees of freedom respectively). However, the description of the coupling of the Wess-Zumino multiplets to supergravity by means of a Kahler potential really implies that the true set of auxiliary fields contains 4 iii 4 additional degrees of freedom. Indeed, the covariant derivative defining the gravitino curvature in Eq. (VI.9.27c) also contains a Uri} gauge field, which was a priori identified with the Kahler connection Q. The Uri) gauge field can be regarded as an auxiliary field. Indeed the covariant derivative p" 'ill/; contains a term L.1/I. which, when brought to the right hand side of Eq. (VI.3.S2b), appears in the transformation law of the gravitino. Since the I-form A is a gauge field, it carries 3 off·shell bosonic degrees of freedom. Successive spinor derivatives of the S field provide an additional spinor and scalar field respectively, so that the off·shell multiplet has.!! G) 16 degrees of freedom. On the other hand, a U(l)gauge auxiliary field Ap is also present in the "new minimal" offshell solution of the super Poincare ~ Uri} Bianchi identities given in Sect. III.6.6. This solution can be retrieved as a truncation of the 11111'16 solution. A different truncation of the latter can also be obtained by killing the auxiliary gauge field Ap; it corresponds to
1986
the old minimal set*. The situation is best appreciated by looking at Table VJ.9.VII TABLE VI.9. VII
16
Bosons Va jl
$
16
Fermions $\.1
OLD MINIMAL Bosons Fermions Va jl
ta
ta
1PjJ
NEW MINIMAL Fermions Bosons Va II
!/Ijl
ta
(Data" 0)
s+
s+
s-
sA
A
II
lJ
~
C
To unify the notations we have renamed A in the solution (VI.3.52), and f \.I a vector t can be naturally identified a torsion
t the auxiliary field named a in the solution (I1I.6.72). The with the dual of the space-time
(VI. 9. 396)
* It must be stressed that this is not the same as the definition of auxiliary field set utilized in the superconformal approach. There one defines the auxiliary field set by means of a choice of the compensating multiplet. For instance, one identifies the old minimal formulation with the case where the compensating multiplet is chiral, and the new minimal formulation with the case where the compensating multiplet is linear. The transformation rules of the physical fields follow from these choices of the compensating multiplet, but their explicit form depends on the matter multiplets which have been coupled and do not take a fixed form. In our Bianchi identity approach the auxiliary field formulation is defined by the tensor structure of the curvature parametrization and corresponds in one-toone fashion to the superconformal definition only in the absence of matter fields.
1987
Furthermore, the U(l) gauge field is named A~ in both cases (it was called ~ in Sect. IV.3.2). Fina~ly we set S=S- +i5+. rather than 5=I1+i:l' as in Sect. IV.3.2. C "and ~ are the extra auxiliary fields one finds in the spinol' derivatives of S as mentioned before. One observes that (A~.~. C) correspond to the field content of an off-shell vector multiplet. Hence it is conceivable that we could suppress them altogether without spoiling off-shell supersymmetry. This is indeed true and the result of this operation is the old minimal parametrization which is totally deprived of any U(1) symmetry. The new minimal formulation corresponds instead to a more subtle truncation. + One sets ~ = C =0 and imposes S- =0 in the ol/l-transformation rules. Under these conditions the supersymmetry algebra closes if and only if t is divergenceless: Dat =O. This means that the torsion field a a . T L is a closed 3-form and, at least locally, can be viewed as the ave ( curl of an auxiliary 2-form B aux) W
After these preliminari~s, we discuss in detail the 161&'!§' offshell solution and its truncations. The material presented here will be necessary in Sects. VI.9.9, VI.9.10 in order to treat the coupling of the linear multiplet in the presence of Lorentz (and U(l») Chern-Simons forms. The superspace we consider is the so-called U(l) superspace where one gauges, by means of an auxiliary connection, the U(l) automorphism group of the N=l supersymmetry algebra. The corresponding curvatures are
(VI, 9. 397a) (VI, 9. 397b) (VI. 9. 397c)
(VI.9.397d) where wah and A are the 50(1,3) and U(1) connections respectively, va and f are the vielbein and the gravitino I-forms spanning a local
1988
reference frame in superspace (supervielbein). Furthermore ~ Va and ~W denote the Lorentz covariant differentials •.a
~v
a
a
b
(VI.9.398a)
=dV -wb"V
!!J l/I = dl/I - !4 wab '" YabW
(VI. 9. 398b)
while V~ is the 80(1,3)0 U(l) covariant differential defined by Sq. (VI.9.397c). The curvatures obey the following Bianchi identities: (VI.9.399a) ab
r..a
:P
T + R
"Vb -
.-
a
1 1jI" Y P
=0
(VI. 9. 399b)
(VI.9.399c) dR~ = 0
•
(VI. 9. 399d)
U(l} superspace was introduced in the treatment of the new minimal model in Sect. III.6.S. The extra auxiliary 2-form B appearing there will J.lV be traded here for a divergenceless vector. In the following we shall utilize the decomposition of superforms in (r.s)~sectors explained in Sect. VI.9.3. The decomposition of the d-operator will be given after the solution of the Bianchis has been completed. In solving the Bianchis (VI.9.399) we assume the following torsion-constraint:
where
K2
is some numerical parameter. Then
t(l,l)
= 1(0,2) = 0
(VI.9.401)
T(l, 1) = 0 can always be imposed by a suitable redefinition of the supergravity fields, while this is not true for T(0,2-)" As in the 0=10 case,
1989
we easily recognise that any reasonable ansatz for T{O,2) in terms of supergravity or matter fields would imply the presence of a dimensionfUl constant. We do not know if the Bianchis ha.ve a solution in this case. In the fOllowing we shall maintain the constraints (VI.9.401). On the other hand, the presence of a non-vanishing antisymmetric torsion field t b "E b J: t f on space-time does not correspond to anything essential a c a CJ, ab since we can always redefine the spin connection w in such a way that ~ = O. flowever we find useful to set the torsion constraint in the form (VI.9.400) since, in any case, the field t f will appear in the parametrization of the other curvatures. Hence it is conveniently identified with the torsion field. Furthermore, the use of the constraint (VI.9.4oo) makes the analogy between 100 anomaly-free supergravity and the present theory stronger. Using (Vr.9.400) the general solution of the Bianchi identities is given by the following superspace parametrization of the curvatures: (VI. 9. 402) P " Pab
Va ... Vb
+ (5- YSY a +
1- K2
i
S+ Ya +
i KiYsta -
a
(VI.9.403)
- -2-YsYabtb)l/IAV
(VI.9.404) ab .. Rab cd Vc Vd
R
A
+
;toab c IjIQ C V A
- (. ab -1 5 Y5Y
+ 1/1 A
+ ab 1 + K2 b d - S Y - i - - ea c t yd)1/I 2 c
(VI.9.40S)
where Pab , Fab and Rab cd denote the space-time (intrinsic) components of the 2-forms p, Re and Rab respectively. Here and in the following we use the irreducible components 0ab' 0a and 0 of the gravitino field strength Pab defined by the following: (VI.9.406a)
1990
a
_ a
yob-yo;;
a
0
a
(VI. 9. 406b)
•
Using the decompositions (VI.9.406). the spinors a~ and ~a are given by ~
a ;; -2(1-1C 1)0a
+
6(1+IC 1)Y a+2{1+2Klh I;
a
a
(VI.9.407a)
(VI.9.407b)
where the auxiliary spinoT I; is defined by the spinor derivative of S'+:
(VI.9.408)
Notice that (S+, 5-) is U(l) charged with Weyl weight 1: (VI.9.409)
l , K2 reflects the freedom we have of redefining the spinot connection mab and the U(l) connection The presence of the parameters
K
A: ab' -_ IIIab
III
+
B ~abcf <.0
A' ;; A + a t a V
a
t
f
Va
(VI.9.410a) (VI.9.410b)
Their values will be fixed later on in such a way that the solution of the linear multiplet Bianchi identities in the presence of Chern-Simons forms should assume the Simplest expression. We also need the spinor derivatives of tfie auxiliary fields S-. t a , 1;. Defining (VI. 9.411)
1991 (VI.9.412)
(VI.9.413)
we get (VI. 9. 414a)
wa '" YSO a ~ 3ysyaa ~ 2ySYat i a U '" C 1 - -- (fP t 4
1
a
(VI. 9.414b)
. 1 + 1- Kl - 21(- - 'i/ S + - - S 4 a 2 _
1- Kl
+
t}y a
a
a
- 21 (- - 'i/ S + - - S t)y y 4 a 2 a 5
(VI. 9.414c)
The spinor
tives of Pab • Recalling Eq. (VI.9.400) and setting (VI.9.41Sa) (VI.9.41Sb)
(v!. 9. 41Sc)
VIe find
(VI. 9. 416a)
1992 1 (9i') 2 1 III II =- R + (1 ~ K )(t tb ~ -6 bt t hb + aza 2a 4a m b +
t (~[a tb]· t6ab~mtm)Y5Yb
+ .
(VI.9.416b)
1
+
i{· 1-
+
"4 VaS
1(1
+ -2- 5
i'
1- Kl _ a 1 _ + - 2 - S ta)Y + (- (VaS -
tahSY
- 5·5-) 1 +
1:. ~a 8
a
t
1 _ K2 1 2 III + + + (- 24 R· -8- t tm+S 5 -
a
'YS
(VI,9.416c)
where +
N
a
-
= -43 VaS+ + -23 3
(1 + K1)S
-
t
·3+
N = - V S - - (1 + K)S a 4 a 2 1
t
(VI. 9.417a)
a
(VI.9.41Th)
a
and where we have decomposed the Lorentz curvature as follows:
- !.4 6mq !=IPt P)opq cd
+
K
6 [
g
1
2ea bdmn
1C 2 1 ab - -4 E:ab <:d ~Pt P + -6 0cd R •
!II t
mn
-
(VI.9.418)
1993 +
R~O}. R(9) and R being the Werl tensor, the traceless Ricci cd +
&
tensor and the scalar curvature, belonging to the irrepses !Q., of SO(I,3) respectively,
~
+
and.!
According to the notations introduced in Sect.
11.8.8, the indices of the SO(I,3) irrepses are written using the Young tableaux patterns with rows and columns interchanged. As already anticipated, the 16-.16 multiplet is reducible in two different ways. suppressing the
We can truncate it to a 12.12 multiplet either by would~be
"U(l) gauge multiplet"
(\.~,
C)
sup~
or by
pressing S .. S- +is+, C and ~ simultaneously implementing the con~ straint ~a ta .. O.
Let us see how it Boes in the two cases.
In the
first case the U(I) superspace reduces to the ordinary superspace cribed by the curvatures (VI.9.402) where A .. R" :0.
des~
The surviving
auxiliary fields s+. S~. t a span the well-known "old minimal set". In • the second case, we begin by setting S+ =S- ., 0, which implies ~ .. U = O. Then Eqs. (VI.9.418) implies, besides C =0, ~a t
a
=0
•
also the constraint (VI. 9.419)
In this way, we are left with the "new minimal set" of auxiliary fields
(A , t) where A is a gauge field and t satisfies Eq. (VI.9.419). ].1 a \l a The constraint (VI.9.419) can be actually solved by introducing an l: Baux ln " superspace. Indee, d def"lnlng . aUXl"I'lary 2-.orm (VI. 9. 420a) .. .aux at!
.~ a i ~ a =l1/1"Yp"V --l/I"y~"t a 2 a
(VI. 9. 420b J
one finds (VI.9.421a)
(Vr.9.421b)
1994
It is in terms of the auxiliary superforms A and B(aux) that the U(l) superspace was first introduced in Sect. III.6.S. A different and more subtle way ot obtaining the new minimal 12 e 12 formulation from the 16 Q) 16 mult.i.plet is via a suitable superconformal transformation. We consider the following superconformal map: (VI. 9. 422a) (VI.9.422b) (VI.9.422C) (where tVR=YSYR, t/lL :·YS"'L) and we demand that after the transformation the new value S of the field 5 should be zero. In Sq. (VI.9.422) w and t are a scalar and a spinor parameter respectively, to be deter· mined by the above requirement. The transformations (VI.9.422) imply also a change in the SO(1,3) and U(l) connections:
A = A + oA
(VI.9.423a) (VI.9.423b)
The shift in the spin connection can be computed by expressing the new torsion 2·form i a in terms of the old one. We obtain
owab
=
K{.
abcf w/2 -
-
) c
tf - tf V -
(e
-
abcf
+ 'iYabtVL • 'RYrL E
a[aWv.b] + tRYabljlR Vf
+
(VI. 9.424)
and the constraint (V1.9.42S)
where TR,L is the spinor derivative of w:
1995
(VI. 9.426) Next we apply the mapping (VI.9.422) 'on the p-curvature. We have
~
+ S
- ~ w/4 ifiL... v .. e
(V(Ul,A) +
' '4l dw + IOA
+
(VI.9.427) Now we set (VI. 9. 428a)
(YI.9.428b) (VI. 9. 428c)
where 1rR• L denote the spinoT derivatives of CA, and La and Z are a real vector and a complex scalar which enter in the parametrization of 1ft as it follows from d2w'" 0 in Eq. (VI.9.426). Using Eqs. (VI. 9. 397) and (Vl.9.403) and the definitions (VI.9.428) we find that Eq. (VI.9.427) has the following meanings. In the {O,2)-sector one finds ..
.. 3 .
"R(L} .. -
'21
(VI.9.429)
TR(L)
In the (1,1) sector we have three equations corresponding to the three
va.
different (I,ll-forms YaifiR . . YabWR"yb and 1/IR"Va ; they yield the required transformation law of the auxiliary fields in terms of the complex scalar Z and the vector La defined in (VI.9.428b). Setting s .. S- + is'' we obtain (VI.9.430a)
1996
s* .. e-w/ 2 (5* + i2 z - !4 t L• L) ~
ta
=e
-w/2
= Aa +
(ta - La
(VI. 9.430b)
1
+
(VI.9.431)
'4 1RYal'L)
(2K 1 + l)(t a - ew/ 2 t a:)
(VI .9. 432)
The (2,O)-sector. giving the relation between Pab and Pab' is not relevant to us here. Therefore, the transition to the new minimal supergravity can be accomplished by choosing w and 1;" such that
it
i i S--Z+-tT =0 2 4 RR
(VI. 9. 433a)
i i 5 * +-Z--'tT .. 0
(VI. 9.433b)
2
4
L L
and Z being defined in (VI.9.426) and (VI.9.428b). As explained in the next section Eqs. (VI.9.433) can always be fulfilled by choosing for the parameters w and T, the first two components of a linear multiplet. T
We conclude this section by rewriting the parametrization (VI.9.402.40S) and the other relevant formulae in the new minimal supergravity. In the following equations we have set 1<2 =-1, 1<1 =1 since in this case the Chern-Simons corrections will take their simplest form. We have (VI.9.434a) ~ a b.a R .. Fab V "V + 1 l/r(S4!a" V
(VI.9.434b) (VI.9.434c)
1997
(VI.9.435a)
4>
a
= 12 Yaa
(VI.9.43Sb)
=0
~
c ..
(VI. 9. 43Sc)
0 .
(VI. 9, 43Sd)
The corresponding values of Mab , Na and P are given by M
ab
~
-
cd 41 R(IO) ab y +
( 2' .. rs tabrs + 1 Ys vab
cd
1
1
. (ll Frs Na
1
= 2"
(9+)
IRa
Yb
b
-
+
) €rsc[a Yb]c •
l'
4
+
2" C~[a tb] - '4 °ab ~ tm)YSYb -
9 r t s)
i
~ Eabcd C} Fed
(VI. 9. 436a)
1
+
~c tdJrb -
m
t (i
Fab + 9[a tbjhSyb (VI.9.436b)
p
VI.9.9
= - ..!.. eabed 48
F y ab cd
-..!.. R• 24
(VI. 9. 436c)
Chiral multiplets, the linear multiplet and the geometrical interpretation of R-symmetry
As we have remarked at the beginning of the previous section, the effective low-energy supergravity of an N=l, D=4 superstring theory necessarily contains a linear multiplet. The on-shell degrees of freedom of the linear multiplet (B~v, X, 4» are the same as those of a Wess-Zumino multiplet (A + iB, X). Indeed, the dual field strength of B is a vector, h, which can be thought of as the gradient of llV l! a pseudosealar field h: h .. a h. Because of this on-shell equivalence, l! 1.1 it is clear that, given a set of n Wess-Zumino multiplets, the linear
1998
multiplet must be an appropriate functional of these. In geometrical terms, we must embed the linear multiplet into the Kahler manifold Mchiral' In a Lagrangian approach this is usually done by means of a Legendre transformation on the action. From our point of view, we rather solve the embedding problem by means of a duality transformation on the solution of the linear multiplet Bianchi identities. Let us introduce the physical linear multiplet (., X. Bpv )' BVV being the space-time components of the axlon, and let us solve the Bianchi identities of this multiplet in the supergravity background described by the 16 $ 16 solution of (VI. 9.402-405) . We define the curvature of the 2-form B as*: (Vl. 9.437)
where (VI. 9. 438a)
(VI. 9.438b) are the Chern-Simons forms of the SO(I,3) and U(l) group respectively, and 'VI' Y2 are dimensionful parameters of dimension [L2]. Notice that the dilaton • appears in the definition (VI.9.437); its spinor partner X (dilatino) is defined by the expansion (VI.9.439)
The Bianchi identity of H is
(VI.9.440)
* Here and in the following we do not introduce the Chern-Simons forms of the gauge vector multiplets: they are not essential to our conside rations.
1999
where we ha\>e set
Setting (VI.9.441)
Eq. (VI.9.440) can be rewritten as (VI.9.442) where ab
Q(R) .. It
(VI. 9. 443a)
"Rab
(VI.9.443b) Comparing it with Eq. (Vl.9.133) we see that the 10D and 40 problems are completely equivalent. Ou~ 4D ease can therefore be solved in exactly the same way as for the 100 case. In particular. in analysing the (0,4) and (l,3)-sectors of Eqs. (VI.9.442), one finds that the most general solution for the H-parametrization corresponds to the existence of an H{G,S) f. 0 satisfying suitable constraints, in complete analogy to the non-minimal solutions considered in Sect. VI.9.4. We shall analyze this "non-minimal" solution of the H-Bianchi identity in the next section. There we shall assume from the beginning the following "minimal" ansatz for .1f:
.11'" -
in analogy (VI.9.442) subject of Y1 =Y2 '" 0, the sector
1
f
abe V
'3 Eabef h V "V
A
+
I). a b k e X Yabl/l" V "V
(VI.9.444)
to Eqs. (VI.9.172). Let us consider the solution of Eq. when Y1" Y2 .. O. the solution at y 1 f. O. Y2' 0 being the the next section. Inserting (VI.9.444) into (VI.9.440) at one finds that the sector (0,4) is trivially satisfied while (1,3) just enforces (VI.9.445)
2000
We now project the sector (2,2) on the following 4-forms: (VI. 9. 446a)
(VI. 9. 446b) .11/1 ...
c
b
y 1/1" V
A
d V Eabcd
(VI. 9.446c)
corresponding, respectively, to the 1T, 1-, !- irrepses of the SO(1,3) group. We then obtain (5 .. S- + iST, H.. H+ + iH-) (VI. 9. 447a)
e~ (-is *
1 2
+ -
H - -41 X-Y ) .. 0 K"R
(VI. 9. 447b)
(VI. 9.448)
where H and M appear in the parametrization of 9X as it follows a from d2~", 0:
9X R(L)
= 9aXR(L}
va
+
(t aa~ y& + Ma Y5ya}l/IL(R) + H1/IR(L) (VI. 9.449)
Let us now compare Eqs. (VI.9.448) with Eqs. (VI.9.433): we see that Eqs. (VI. 9.433) are satisfied if the parameters w and 1;" belong to a linear multiplet together with a 2-form E~v defined in a way analogous to Eq. (VI.9.437). but without Chern-Simons forms (y 1" Y2 .. 0) • Furthermore, working at y 1'" Y2" 0 we can actually identify the multiplet of parameters (w, T. E~v) with the physical linear multiplet
-}t
(VI, 9.450)
2001
In this case if we apply the Weyl transformation (VI.9.422) on the 3form .ff in terms of the physical fields
2002 1 XLX - ) + Yl W" 0 e~ (is - '12 H* • 4" L
e~(-iS* ~
1
+
t 1
H . iXR~) 1 -
+
(VI. 9. 455a)
(VI.9.455b)
Y1 W* " 0 1
e (-t - -M - - X Y X ) - -h 2a 2a ZRaL 2a
+ y
If 1a
=0 •
(VI.9.4S6)
Solving for S, S* and introducing the result into Eqs. (VI.9.430) one finds that the parameters of the Weyl transformation allowing the transition to the new minimal mo4el cannot be expressed (in a local way) in terms of the physical fields ~ and X. This obstruction implies that we cannot perform local field redefinitions such that ." 0 as in (VI.9.4S2). In other words, ~; 0 and R(aux} 'Iff. Similarly. comparing Sq. (VI.9.431) and (VI.9.456). we conclude that the identification does not hold any more, i.e. B(aux) ~ B. This being clarified. we try now to embed the linear multiplet (!P, X. Bp..) into the Kiihler manifold Mchiral of an arbitrary number of Wess-Zumino multiplets. Since we are going to utilize the results obtained in Part IV. Sects. IV.3.4, we shall now give a short resume of these. The standard N=l, 0=4 matter coupled supergravity can be naturally formulated by identifYing the U(l) gauge potential with the U(l) connection associated to the spinor Kibler transformations. These latter act on the spinor fields in the chiral formalism (see Sect. rV.3). More precisely one sets (Vr.9.457)
G(zi,zi*) being the Kibler potential of a manifold whose complex coor· dinates zi (i = 1, ••• ,n) correspond to n Wess-Zumino multiplets. Notations are as follows: if (Ai, Bi , Ai) are the fields describing the (O+, 0·, content of the i-th Wess·Zumino multiplet, we shall define the complex fields by
t)
(VI.9.458a)
2003 (VI .9.4SSb)
and the chiral projections by i 1+Y5 i X =--A 2
(VI. 9 .459a)
i* l-yS i* X ... _-1\
(VI.9.459b)
2
Notice that the identification (VI.9.4S7) implies that the curvature R& .. dA is the Kihler 2-fol'lll; & i -j * R .. i gij* dz "dz
(VI.9.460)
gij* being the Kiihler metric; gij*,"i\aj*G. The superspace parametrization of the "curvatures" dz;i and
vl
is given by: (VJ.9.461) (VI.9.462)
where Hi is the auxiliary field of the chiral multiplet, and VXi is a derivative covariant with respect to Lorentz and U(l) transfol'lllations, and also with respect to zi-diffeomorphisms. i.e. (V1.9.463)
ilk} being the Christoffel connection in the Kahler manifold. Note that all the fields with i.j •••• indices are Kanler world vectors (except zi themselves). The corresponding formulae for the di i * and VXi * curvatures are obtained by the substitutions Zi ~ .i* z
,
i i* X"X • (VI.9.464)
2004
The U(l) gauge sYJIIIOOtry acts on A and on the fetmions as follows. Corresponding to the Kibler transformation (VI. 9. 465)
f(zi) being an arbitrary analytic function of zi • we have the follow· ing U(l) transformation: A ... A + d(Im fez)) .
i
t
Xl = X e
iIm f
(VI.9.466a)
; ~R
t
=W Re ljIL
Urn f
= WL
(VI.9.466b)
. 1 iIm f e 2
(VI. 9. 466c)
Notice also that the auxiliary fields S = S- + is+ and Hi are U(l). charged: (VI. 9.467a) (VI.9.467b)
Finally we give the on-shell values of the auxiliary fields S, t a , Hi in the 16 e 16 formulation, in terms of the chiral multiplets (determined from matter coupled supergravity Bianchi identities): (VI.9.468a) (VI. 9. 468b)
(VI.9.468c)
The above formulae refer to the case where no gauge multiplets are present; we restrict our analysis to this case for simplicity. (Recall that in presence of gauge multiplets S, t and Hi acquire terms a
2005
~nich are bilinear in the gaugino fields Ai). In the above formulae e is a dimensionful parameter which distinguishes between theories with and without superpotential. If e = 0, then S = Hi = 0 and the theory is automatically in the new minimal form for any choice of the Kahler potential G. On the other hand. if e f. 0 then we have a superpotential and the theory can be brought to the new minimal form only when the K'ahler potential G is "R·symmetric". This will be shown in the following, together with the explicit form of the auxiliary fields A, ta and Hi in the new minimal formulation. Let us now come back to our embedding problem. We set (VI. 9. 469)
and by differentiation we find (VI. 9.470)
Comparison with (VI.9.439) gives (VI. 9.4 7la) (VI. 9. 471b)
We shall now compute the V-derivative of both sides and equate the (0.1)sectors. USing the definitions (VI.9.449) and (VI.9.462). We obtain the following equations:
(vr. 9 • 472a)
(VI.9.472b)
(VI.9.473)
2006
Substituting the left hand side of these equations into Eqs. (VI.9.447) and (VI.9.448). we find the duality transformation of the Bianchi identities of the linear multiplet:
. s*
-1
-
1 a Hi - -41 (V . a.1/I -2.f 1 1 J
1 • 1 - - h + e (- t
2 a
2 a
1 - - lm
2
+
(a.1*1/1
a., a••) x-i.j x- " 0 1 J
(VI.9.474a)
i* a
Z )(VI. 9.475)
Eqs. (VI.9.474) admit a very simple solution. Substituting Eqs. (VI.9.468a. c) into (VI.9.474) we get (VI. 9. 476a)
(VI. 9. 476b) and their complex conjugates. The geometrical meaning of these equations becomes transparent by means of the following redefinition i -i* • " log D(z.z ).
(VI.9.477)
Eqs. (VI.9.476) then become (VI. 9. 478a) (VI. 9.478b)
The geometrical interpretation of the new equations is given below (see Sect. IV.4).
2007 i)
D(Zi.ii*) is the prepotential of a Killing vector
ki .. i gij" aj" D(z.Z i -1*}
(VI. 9. 479)
ii) The Kahler potential G is invariant under U(l) transformations generated by ki. Therefore, the embedding problem is solved by considering a Uel} invariant Kibler potential and identifying the first component of the linear multiplet with the logarithm of the Killing prepotential as shown in Eq. (VI.9.477). The Killing vector ki deter~ mines the direction along which the spinor partner X is aligned since from Eqs. (VI.9.439, 470, 471) we find -1 i
j*
X .. - iD (k X gij"
+
(VI. 9. 480)
c. c.)
Recalling the discussion given in Section VI.9.S. we conclude that. when 'VI" 'V2" 0, the transition to the 12 $12 new minimal set for a theory with superpotential can be done only provided we have a Uel) invariant Kahler potential. Essentially this result was already known in the superconformal approach, the U(l) symmetry being the so-called R-symmetry. Here we have a better geometrical insight into its meaning. Let us now consider Eq. (VI.9.47S). Using the further Killing condition
v[.1 a.*]+ J
+
3.$ 1
a.J
ft. . .
(VI.9.481)
0
and the explicit form of t, Eq. (VI. 9. 468b), which is valid at a Yl = Y2" 0, we can rewrite (VI.9.47S) as follows: •
i
.1o
h .. D(Zl, ~l ) (1m (a. In D Z a
1
a
) + t )
a
(VI.9.482)
which is the duality transformation linking the 2-form field strength to the imaginary part of a suitably chosen chiral scalar gradient. We notice that formulae (VI.9.47l-473) can be used to find the expliCit form of the auxiliary fields A. ta and Hi in the new minimal formulations at 'Vl" O. Indeed, since at y 1" 0 the identification
200S
(VI.9.4S0) is valid, we can substitute the duality transformations (VI.9.471-473) into Eqs. (VI.9.431, 432) after the identification - H, Z=
Z*
=_
H+, La = - Ma' TR,L:- XR,L
(VI.9.483)
and impose S=O. In this way one finds the value of the auxiliary fields t a and A in the new minimal formulation at y 1 = 0:
(VI. 9. 484)
A,:, 1m dG - (2Kl + l)[Im {(a. In DlZi) + 1
a
(VI.9.485)
where for simplicity we omit the tilde notation for the transformed quantities. Moreover one can easily verify that under the same superconformal transformation (VI. 9.422) Hi undergoes the transformation
(VI. 9. 486)
Hence, in the new minimal model we find (again omitting the tilde)
(VI.9.487)
2()09
Interesting for superstring derived supergravities is the case where the Kahler manifold ~n is the product .In
=.111 flJ.Jfn- 1
(VL9.488)
the one dimensional Kahler manifold ~l containing the field zn:: S = e41 + iA associated with the dilaton • and the axion A emerging from the string spectrum. The axion A is related to H
jJVP
by the duality transformation
The manifold .Jfn_l contains all the other Ness-Zumino multiplets zi (i,. 1••..• n-l). According to Eq. (VI.9.488) the Kiihler potential is (VI. 9.489)
With this choice Eq. (Vr.9.476b) can be rewritten as
asD
asasY
-=-D asY
(VI. 9. 490)
This equation implies 0 = asY. and moreover that y and D depend only on S + S. (Peccei-Quinn synunetry).
In particular from dimensional reduction of D=lO N=ll Supergravity or directly from calculations of string amplitudes one concludes that the space ."1 parametrized by the field S is the coset SU(l,l)!U(l) characterized by yeS +
5) = 1
= In(S
+
+
5)
(V1.9.491a) (VI.9.491b)
0=-S +S
41
In(S
S)
(VI.9.491c) (VI. 9.491d)
2010
A simple way to recover this result is as follows. Consider the D,,10 N.. l Lagrangian given by Table VI.9.VI; restricting our attention to the terms containing the dilaton 0 and the 3-form Jr, after dimensional reduction on a torus T6 we have
(VI.9.492)
where the 6-volume of the internal torus has bee3 factored out. Setting eO " ReS; do .. e-o d(ReS)
(VI. 9. 493a) (VI.9.493b)
.ff
A
(VI. 9.493c)
V e: b mac a em
V = d (ImS) V '" Vb
A
the Lagrangian (VI.9.492) becomes
1
- (-- ReS ReS 12 m m
3
-4
cl
[mS ImS )e:e V m m l"'C4
A
(VI.9.494a) which after elimination of the first order field Sm" ReSm+ i ImSm yields: (VI.9.494b) corresponding to the a-model on the SU(l,l)/U(l) Kahler coset manifold.
2011
Let us now consider the same embedding problem in the presence of Chern-Simons terms. Eqs. (VI.9.474) and (VI.9.475) now become
~.
e (1S ~
1
i*
'2 ai*~ H
1
4"
-
(Vi .. aj*~
+
i* j*
ai*~ aj*~) X X )
+
'VI W .. 0
(VI. 9. 495b)
(VI. 9.496)
where we have again assumed that the dilaton $ is a local function of ti, zi, and W. W*, Wa are the corrections defined in Eq. (VI:9 .455, 456) to be computed in the following section. They are no longer local . .* functions of Zl, Zl. In the new minimal formulation we still obtain S =0 and if we insist that ~ .. In 0 then Eq. (VI. 9.495. 496) reduce to
- -21 c.I D Hi
+
Y1 W..
=0
(VI.9.497a) (VI. 9. 497b)
1
i"-
- -2 (h a - Im{a .• lnD Z )) + D(S,S)t + Y1 W .. 0 1 a a a
(V1.9.498)
Hence we come to the conclusion that in presence of Chern-Simons Hi is no longer given by Eq. (VI.9.487) but by the following expression: . 1/2 "* G/2' . c.D . w* HI .. 2e 0glJ o. G e _ XIX) _J_ + 2y 61 , n _ J* D 1 " 0 "i
(V1.9.499)
2012
In particular, using Eqs. (VI.9.491) we can write
(VI.9.500)
VI.9.10 0=4 Chern-Simons cohomology and the linear multiplet As pointed out in Section VI.9.S the complete solution of the linear multiplet Bianchi identities involves certain "corrections" which were named W, W* and Wa (Yl'Y2)' They arise from the presence of the Chern-Simons forms which, in string theory, play an essential role for the anomalies cancellation a la Green and Schwarz. The explicit solution can be found in closed form utilizing the same procedure employed in 0:10, N=l anomaly-free supergravity (see Sect. VI.9.4) where the corresponding problem was also completely solved. Following the same steps as in the 100 case, we first reduce our problem to a well defined cohomological one in superspace. Let us decompose the characteristic class Q=Rab " Rab + a R& R& as follows: A
Q = dX
+
K
(VI.9.s01)
where K is a closed 4-form (dK" 0), X is gauge invariant and a: -Y/Yl" Then the analogue of the theorem proved in Sect. VI.9.4 (Eq. (VI.9.177» states that the solution of the Bianchi identities (VI.9.440) exists provided the following condition are satisfied:
=0
(VI.9.502a)
K(1.3) " 0
(VI. 9. 502b)
K(O,4)
Indeed, defining
H' " H + Y1 X
(VI. 9.503)
2013 it is clear that ferr the sectors (0,4) and (1,3) the Bianchi identity
written in terms of H' has the same form as the corresponding identity written in terms of H at y 1'" Y2'" 0; hence it can be satisfied. In the sector (2,2), K(2,2) f: 0 implies a modification of the structure of the Bianchi identities with respect to the linear case (y 1 =Y2;: 0). In order that the correction Yl K(2,2) be compatible with the previously stated Eqs. (VI.9.447, 448), it must be expressible in terms of the 4~forms (VI.9.446), in other words, the 4-forms K(2,2) must con~ tain only the irrepses 1+, 1- and 4; of SO(I,3). In this case, the corrections previously named 1'1, 11" and Wa just modify the right hand side of Eqs. (VI.9.447, 448). We show that this is indeed the case by analyzing the closure condition of K: dK = 0
(VI. 9. 504)
•
At this point it is useful to introduce the decomposition of the exterior derivative operator d as given in Sect. VI.9.3 (see also Table VI.9.!): d ;: ~(~1.2)
+
VeO,l)
+
V(I,O)
+
~(1,0)
+
~(2,-1)
(VI.9.50S)
where the operators V(~l,2)' V(O,!)' ~(l,O)' ~(2,-1) satisfy the relations (Vr.9.74). In particular, ~(-l.2) and ~(2.-1) are cohomological operators: (VI. 9. 506)
The action of the v-operators on the (super)vielbein basis is given by '(VI.9.507a)
(VI. 9. 507b)
2014
(VI.9.507c) (VI.9.507d)
lJ(_1,2)1/i .. 0
11(2.~1) ya ..
(VI.9.507e)
0
lJ(2.~I)1/I = Pab
(VI. 9.507f)
..,.".; •
These relations are the 4D analogues of the lOD ones given in Eqs. (VI.9.70. 72, 73). Then in the sectors (0,5) and (1,4) the equation dK= 0 yields (VI.9.S0Sa) 11(_1,2) K(2,2) ... VeO,I) K(l,3) ... (VCI,O) .. P(l,O}) K(O,4} .. 0 (VI.9.S08b) which, using the conditions K(O.4) =K(l, 3) "0, reduce to the simple conomological equation (VI.9.509)
To find the nontrivial cocycles of Eq. (VI.9.S09) we write down the most general decomposition of K(2,2) along the supervielbein basis: (VI.9.5l0) and exploit the irreducible representation contents of the tensors Ka!bC and Kab!cd'
In terms of irrepses their decompositions are (V1.9.S11a) +
Kabled .. .!Q. + ~
-
+ ~
1'-
+.2. ... 1
+
1
(VI.9.Sl1b)
2015
This result is immediately retrieved by considering Ka /bd as a tensor product of a vector and an antisymmetric tensor, and so on. Then, applying ~(-1.2) to (VI.9.S10) We get ~H,2)
·K:T· a b c K (2,2)=l a lbc 'i''''Yl/I .. l/I"YI/I .. V + +
~ ab - c d iKabI cdl/l . . y 1/1 .... 1/1 .. y 1/1" V •
(VI.9.S12)
The two basis (1,4)-forms of (VI.9.S12) are in the h 4+ and 1604+ v v representations respectively. So we obtain
-
-
1/1 '" y a"'" W YbW Vc e9 e-4 v = 16-+-16'- +v 4 A
A
(VI. 9.5l3a) (VI.9.513b)
Now if we impose the cocycle condition (VI.9.S09), using the orthogona-
1;
lity of the irrepses we find the result that Kable is a pure and Kab Icd contains only a and a 1, so that our assertion is proved. Notice that this theorem is the analogue of the theorem in VI.9.S, proving that the (2,2)-seetor of 100 Bianchis contains only the 120 representation of 50(1,9). Let us now proceed to the actual calculation of the cocycle. We assume the following form for X:
t
x = X(1.2)
+
X(2,1)
(VI.9.514)
where X(0.3) and X(3,O) are set to zero. ~~ validity of setting X{O,S) to zero with no loss of generality is discussed later; X(3,O) is set to zero as we do not want any modification to the.:tf space-time field strength. We begin with sector (0,4) where Eq. (VI.9.S01) can be rewritten as follows (VI.9.51S)
which yields the explicit form of X(1.2):
2016
(VI.9.516) where the C's are the ~(_1,2)-coboundaries in the sector (1,2). They belong to the representations 1+,~, i+ and !- respectively. We can set Cab and C to zero since this is equivalent to a redefinition of the 2-form B and the dilaton $. In the sector (1,3), Eq. (VI.9.S01) defines X(2,1): j.l
H,2)
X
(Z,l)
=Q (1,3)
-v (0,1)
X
(VI. 9. 517)
(1,2)
Since the most general form of X(2,1) is (VI. 9. 518)
this sector determines ~ab
= 8(1 + K2h[a
~ab'
which turns out to be
Z + Zo.(l- KZ) JO'[a tb] O'bJ f t f + 2 l (1- )(2) 1
+ 4(1 + Kzhf[a
O'b] t
f
+
Z
2(1 - KZ)yabO'f t
f
+
-
- 2[(1+K 2)(5+91<2) + 6O;(1+K 1)(3+K 1)jy[a tbf + +
2 2 (1 + K2)(1 - 3K2)yabfO' t f - 4 [ (3 +l
+
(VI. 9.519)
Finally we come to the most interesting sector, i.e. the (2,2)-sector. Eq. (VI.9.S01) enables us to calculate K(2,Z)'
the corrections to the
2017
supersymmetry transformation rule of the dilatino, and the complete form of the field strength ha • Indeed, Eq. (VI.9.501) can be rewritten as
Following the previous discussion we can write the most general expression for K(2,2) as
(VI. 9.521)
where W, It =(1'1 + 1'1*)/2 and \'1- =(\'I - W*)/2i correspond to the 4-, 1+ a - and 1- non-trivial cocycles of Eq. (VI.9.504) respectively. After' lengthy calculations the \'I-tensors turn out to be
2
+ 4 (K l + 2Kl - 1) t
a
+ +C + 45 'i/ S - 4S 'i/ S -
a
a
(VI. 9. 522)
2018
(VI. 9.523)
(VI. 9. 524)
In order to check our algebra we have considered the cancellation of the other irrepses (.!2.-.!Q., ~+, ~-, §.. .1.+) which was found to occur as . required by the BPT theorem previously stated. In the 10D case, further analysis of the Bianchi identities in the (3,1)-sector leads to higher order corrections of the fermionic equations, namely, the gravitino and the spin equations. In our case, we do not expect any information from this sector since the theory is essentially off-shell. Indeed a straightforward analysis of the (3.1)-sector shows that it is identically s~tisfied both in absence and in presence of the corrections w+, W· and Wa' so that no equation of motion can be retrieved for the fermions. Finally in the (4.0)-sector one finds
t
a
cp •
2 ~ah - 6e Xa
+
ab ab Y1 (R cd R ~ - ~ Fed Ftm)Ecdlm
=0 • (VI. 9.525)
2019
Let us briefly illustrate the meaning of
OUT
results.
i) The 4--current of the (2~2)-sector modifies the relation between the 2-fol'lll field strength 'h and the torsion t already a a introduced in Eq. (VI.9.448) in the absence of Chern-Simons forms: 1
- -h 2a
+
• 1. 1 1 e (- -4 XRYaXL - -2Ma-2 - a t )
+
Yl a If .. 0
(VI.9.526)
Equation (VI.9.526) is the analogue of the equation relating Habc and Tabc in the 100 case (see Sq. (VI.9.229». The difference here is that, in addition to the torsion t, we have also scalar auxiliary a fields S and C and their superpartner ~. ii) The 1+ and 1- currents of the (2,2)-sector yield a differential equation for the complex field S .. S- + is+: IPI
e
(2' H -
." 1 +S - '2 X~R) + Y1 (11 + 2i W ) .. 0 •
1
(VI.9.527)
We note that Eq. (VI. 9.527) contains also H.. H+ + i H-
(VI.9.449). that, owing (VI.9.472). the Bianchi
defined in Eq. In order to determine the auxiliary field S, we recall to the relation between H and Hi given by Eqs. ~~ must couple the differential equation (VI.9.527) with identities associated with Eq. (VI.9.462):
_2 i
\IX ..
1
b
4" Ra
" 'Yabl
1"
I Rfll•
+
. *. k Rm!j!k dzm "dzJ X .. 0 (VI.9.528)
where Rm!j Ik is the Riemann tensor of the K"ahler manifold. The (0,2)sector of (VI. 9. 528) yields 9(0,1)
Hi
= 2i
i
..
X S •
(VI. 9. 529)
In the absence of Chern-Simons, if one assumes that Hi is simply a function of the chiral scalars ti, one obtains a differential equation (see Eq. (VI.3.95))
(VI.9.530)
2020
yielding the solution (VI.9.468c) for fli(t,l). This exhibits the arbitrariness intrinsically contained in an N=l theory. Indeed, assuming that Hi is a function of z alone (and not of its first and higher derivatives. or even of more general objects like curvatures) corresponds to restricting one's attention to actions which are of the second order in the derivatives. This is an arbitrary choice which is justified by the absence of a fundamental dimensionful parameter such as u'. In the macroscopic theory of a superstring, this is not allOWed and one should consider more general ansitze for Hi. introducing arbitrary functions for its dependence on the higher derivatives. Once this is done, Eqs. (VI.9.SZ7) and (VI.9.S30) become a system of differential equations for such functions and the auxiliary field 5, in rerfect analogy with the previous case. Alternatively. one can try anslitze for the auxiliary fields S and dete"'llline Hi. This teaches us the following lessons. Differently from the 100 case, even if ./e set Heo •S) =0, we still have arbitrary choices in the solution of the differential equations for the torsion and auxiliary fields. This corresponds to the existence of arbitrary supersymmetric terms which can be introduced into the Lagrangian, and it would be a very iDq>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, 1<1" 1 yields a;, 2) =ReO, 2) ",o. Therefore the conditions (VI.9.S01) and (VI.9.502) are automatically satisfied by setting X;; 0 •
(VI.9.S31)
Hence the sectors (0,4) and (1,3) of the H-Bianchis are left untouched by the presence of the Chern-Simons forms and only the sector (2,2) is affected. However, we have (VI.9.S32)
*
ab and using the expressions of R(I,I) and R(I,l) given in Eqs. (VI.9.434), one finds (VI. 9.533a)
(VI.9.S33b) (VI. 9. S33c) The same results can be retrieved from fqs. (VI.9.522-S24) by setting on the right hand side (VI.9.534)
In particular, if we use the new minimal equations (VI.9.533) in the duality transformed equations (VI.9.49S, 496), we retrieve the results obtained by the Lagrangian approach in Ref. 10, where the higher order interaction has been fixed by a particular choice of the quadratic curvatUl"J terms (square of the Weyl tensor). It is clear that the relations (VI.9.495, 496) (or correspondingly (VI.9.497-498) in the minimal framework) depend in an essential way on the ansatz (VI.9.477). If we allowed ~ to depend also on higher derivatives of the Wess-Zumino chiral multiplets, then the duality relations would be modified in a
2022
non-local way. This arbitrariness has its counterpart in the Lagrangian approach in the choice of a different higher derivative interaction. As a final point in our discussion we shall now study what is the most general parametrization of .~ in the outer (0,4)- and (l,3)-sectors in the same way as we did in Sect. VI.9.4 for the 100 case. The most general parametrization of .!f contains the (0,3)- and (1,2)-sectors, which were absent in the ansatz (VI.9.444). It is sufficient to add to the old Jf-parametrization a further 3-form Y so that the Bianchi identity dJf :: d(Jf{old)
+ Y)
=0
(VI.9.535)
is satisfied in the (0,4)- and (l,3)-sectors. There we have denoted by "Hold" the parametrization (VI. 9.144) with k =1/2, according to Eq. (VI.9.145). Cd
.!fold
Since by construction we have
) (0,4)
= (dJf
old
) (1,3) " 0
(VI. 9.536)
our problem is reduced to solving the two cohomological equations (dY) (0,4) " 0
(VI. 9. 537a)
(dY) (1,3) ,. 0 .
(VI. 9. 537b)
If a solution of Eqs. (VI.9.537) exists, then in the sector (2,2) we find modifications to the equations (VI.9.45S, 456) relating the auxiliary fields. (In the (3,l}-sector we must again find 0" 0 as in the minimal case, because the theory is off-shell). Let us then investigate whether solutions of Eqs. (VI.9.537) exist. With no loss of generality we can write (VI. 9. 538a)
2023
t" YablP " ve -
a __b
~
Y{2,l) .. 1/1 ~ab'" V
(VI.9.S38b)
(VI.9.S38e)
"v- .
All the other possible representations can be eliminated by adding an exact 2-form Z. that is. by redefining (VI.9. S39)
Y .... y+dZ.
This can be proved in the same way as we did in the 100 case (see the discussion after Eq. (VI.9.147e». Now the first condition, Eq. (VI.9.537a). can be rewritten as (VI. 9.540) Inserting Eqs. (VI.9.538a. b) into (VI.9.540) we get -
- (8)
-(l/I"YabljlAl/I)
(8) i ~(9)'V(O.l) $ab - '2 K~ l/I"YaWA 1/J"Ybl/l-
i -(16)1/I"Yabl/l,,1/J"Yclj/aO.
(VI. 9. 541)
-'2 Kab c Since
(8) 'V(O.l) $ab
(10)
. (16) Yed + lMab Yc ed c
= Nab
(6)
+ O[a
Mb]d cd -
1
1
+
(16)
'2 Eabcd Ned
'2 Eabed
YSYb +
b
(6)
(6)
Med YS + Mab 15 (VI. 9.542)
we obtain the constraint (V
~(8»(lO)
(0.1) 'Yah
= .,(10) - 0 - '~ab cd
(VI.9.S43)
2024
and the conditions (VI. 9 • S44a)
'M(16) _ K-(16) lab - ab
c
(VI. 9. 544b)
c ~4+
-4-
while the representations Ka , Ka identities
remain free because of the Fierz-
(VI. 9. 545a)
(VI. 9. 545b)
Next we work out Eq. (VI.9.S37b) and rewrite it as follows:
~(-1,2) Y(2,l)
+
~(O,l) Y(l,2)
+
(~(l,O)
+
Pel,O» Y(O.3)
=0
(VI. 9. 546)
Applying the operator P(-1.2) to the left hand side of Eq. (V1.9.546), we derive the equation
(VI.9.547)
which is the analogue of Eqs. (VI.9.157-1S8).
From Eq. (VI.9.547) one
derives that the most general form of Q(1.3) is (VI. 9. 548)
The above result can be retrieved as follows. Q(1,3)
as
We write the most general
2025
(VI.9.S49) and analyze the irrepses contained in the $'5, getting (VI. 9. SSOa) (VI. 9. 5S0b)
abl c ,,20... -12 •
Now we impose Eq. (VI.9.547): i " " (8) ablc 1l(_1.2) 51(1,3) " - 2' (1/1" YahI/!" ljI) $ ,,1/1 YeW -
i-alb -
(VI,9.S51)
"ljIyblP
-2'W"YaW
and, using the fact that ~"Yatp"~"Yb1/JA1/I and ~aby"ljI"~"Yct/l"ljJ are pure 24 and 20 representations respectively, we immediately obtain Eq. (VI.9.S48). Inserting Sq. (VI.9.S38c) and Eq. (VI.9.548) into Eq. (VI.9.S46). we get the constraint $ (12) a
=0
.
(VI. 9. 552)
The results (VI.9.S48) and (VI.9.S52) are the counterpart of Eqs. (VI.9.IS9) and (VI.9.16S) in the 10D case. As in (VI.9.166), we can rewrite the constraint (VI.9.S52) as -I'
$(12) " $(12) + [17
a
where ~~12)
a
(0,1)
(j(4
a
-
... j(4 )]
. a
(12)
a
=0
(VI. 9. 553)
contains the following 12-structures: 16 (12)
['101 ~l
(VI. 9. 554)
c
Hence Eq. (VI,9.553) is a new constraint on ~~) (since M!~ is part (8) of V(O,I)
2026
~±
K4±-structures. Therefore
a
fields
:J:
M! (V
are fixed modulo the addition of new
satisfying
(0,1
± ) M4 ) (12) ,,0
•
a
(VI. 9.555)
This parallels the discussion leading to Eq. (V1.9.167). Thus we
may conclude that a non-trivial solution of Eqs. (VI.9.537) exists provided: i) the field
+!:)
entering in the definition (VI.9.538a) satis-
fies the constraints (VI.9.543) and (VI.9.553); ii} there exist solutions of the constraint (VI.9.555).
10D case, there are solutions of (VI.9.543) which are Indeed, if
As in the
trivial.
.(8) ..
ab
[V
(0,1)
f
ab
1(8)
with fab antisymmetric in
(VI.9.S56) (a,b), than the constraint (VI.9.543) is
automatically satisfied. However, such a solution is trivial since the corresponding contribution to YeO,3) is the (0,3) part of the exact form dZ With (VI.9.557) and thus amounts to a redefinition of the fields appearing in Y.
If
non-trivial solutions to exist, we have an additional arbitrariness in the field equations due to the non-trivial cohomology (VI.9.S37).
If
the string dynamics "chooses" the new minimal form. then all the higher curvature interactions can be produced only by Y(0,3) and
M!.±
2027 References for Chapter VI.9
[11
I2l I3J
[41 [Sa] [Sbj [5c] [6} [7]
[8J
[9] [10]
[111 [I2} [13]
L. Bonora, M. BregoIa, K. Lecliner, P. Pasti and M. Tonin, Int. Journ. Mod. Phys. AS (1990) 461. N. Dra2on, Z. Phys. C2 (1979) 29. D.Z. Freedman, G.N. Gibbons and P.C. West, Phys. Lett. B124 (1983) 491. L. Sonora, P. Pasti and M. Tonin, Phys. Lett. B188 (1987) 335. R. D'Auria and P. Fre, Phys. Lett. B200 (1988) 63. L. Bonora, M. BregoIa, K. Lechner, P. Pasti and M. Tonin, Nucl. Phys. B296 (1988) 877. R. D'Auria, P. Pre, M. Raciti and F. Riva, Int. Journ. Mod. Phys. A3 (1988) 953. G.P. Chapline and N. Manton, Phys. Lett. S120 (1983) 105. M. Tonin, J. Mod. Phys. A3 (1988) 1519; Int. Journ. Mod. Phys. A4 (1989) 1983. M.T. Grisaru, H. Nishino and D. Zanon, Phys. Lett. 206B (1988) 625. R. Kallosh and M. Rahamanov, Phys. Lett. B209 (1988) 233 and Phys. Lett. B214 (1988) 549. P. Pasti and M. Tonin, Int. J. Mod. Phys. A4 (1989) 2959. J.J. Atick, A. Dhar and B. Ratra, Phys. Lett. 169B (1986) 54. S. Cecotti, S. Ferrara, L. Girardello and P. Villasante, Int. Journ. Mod. Phys. A2 (1987) 1839. R. D'Auria, P. Fre, G. De Matteis and I. Pesanolo, Int. Journ. Mod. Phys. A4 (1989) 3577. K. Lechner, P. Pasti and M. Tonin, Mod. Phys. Lett. A2 (1987) 929. K. Lechner and P. Pasti, Mod. Phys. Lett. A4 (1989) 1721.
2028
CHAPTER VI.10 (2,2) SUPERCONFORMAL FIELD THEORIES AND THE CLASSIFICATION OF N=l,D=4 HETEROTIC SUPERSTRING VACUA
VI.lO.I Introduction As anticipated in the introduction to PART SIX, the present is a chapter added in proofs, when the rest of the book was already in press. Although we have attempted to be self-contained, in some instances the reader is referred to papers listed in the bibliographical note. Here we try to solder the lore discussed in chapters Vl.l - VI.9 with the most . recent perspective on superstring compactifications, based on abstract superconformal field theories. Rather than focusing on the internal compact manifold, one jumps directly to the 2-dimenaional superconformal quantum theory that describes its degrees of freedom. The key argument, in this respect, is a powerful relation that exists between the number of space-time supersyrometry generators (N) and the number of globally conserved world-sheet supercurrents in the left sector (11.). The number of globally conserved supercurrents in the right sector (ii) is instead related to the chiral gauge group (E5+v (y =: 0, 1,2,3), Es;: SO(10». Let us denote by
a superconformal field theory where the number of left-moving and right-moving conserved supercurrents is respectively 11. and n and where the central charges are, respectively, c and C. With these nota.tions the internal superconformal field-theory (ISe) associated with an N=1,D=:4 heterotic superstring has necessarily the following structure: T
IseN :
1
= (2,p)g,co .
+ 1: (O,pi).,.; i=1
(VI.10.1a)
2029 r
(VI.lO.1b)
CO+L Ci=22
'=1 while in the case of a vacuum with N=2 target SUSY we have necessarily: r
ISO N =2
""
(4,p~,.o
+ (2,P')3";' +
L (0';;)0":1
(VI.10.2a)
';=1 r
CO+C~+ECi=22
(VI.10.2b)
1=1
For general compactifications these results were derived by Dixon and Banks The conformal field e~+"S"E appearing in the D=4 gravitino vertex (see VI.lO.67) defines, when integrated on the world-sheet, the spacetime supercharge Q. Requiring {Q,Q} i:::j P implies that the OPE of E(.;) with Et(lII) contains a U(l) current j(z) that does not commute with the local world-sheet supercurrent Gi
191. We sketch the line of reasoning in the N=l case.
= =
=
= =
=
[SON=l ... (2 , 2)9,9
+ (0 , O)sO(lO)eE~ _,n
SO{lO~B'
(VI.IO.3)
In eq.(Vl.I0.3) the symbol (0,0)0.13 • denotes the current algebra of the group S0(10) ® E;, whose characters have the same modular transformations as the characters of the SO(2) current algebra spanned by the world-sheet right-moving fermions with space-time transverse indices ~T (T '" 1,2) that appear in a type II superstring. In this w~ we can directly associate a modular invariant heterotic superstring vacuum to each modular invariant type II superstring vacuum. With the choice (VI.10.3), each (2,2)9,$ superconformal field-theory describing an admissible type II superstring with N=2 target supersymmetry, yields also an admissible heterotic superstring with N=l target SUSY and gauge group
G,AUge
::=
GE
@
Et @ . (
(VI.I0.4a)
rankGE :::; 8
(VI.lOAb)
where the model dependent factor GE is named the enhancement group. Applying the approach of the previous chapters also to type II superstrings, in order to obtain massless gravitinoll both in the left and in the right-moving sectors, we focus on SU(2)*.groupfolds. Generally speaking by a GT·groupfold we denote the quotient GT/ B where B C GT @ GT is a. discrete subgroup of the isometry group of the f'0up.manifold GT. In other words, denoting by (b,b) the elements of B (where b, b E GT), one identifies the points of GT through the equivalence relation:
(VI.IO.S) IUhe generators of B are all of the form (l,b) or all of the form (b,l), then GT/B is an ordinary left (respectively right) coset manifold and, as such, it is a smooth manifold. In all the other cases the groupfold GT/B is a singular variety since the action of B on Gx has fixed points. On a groupfold GT/ B the WZW field g(s, z) has as many twisted sectors as there are elements (b,b) in the homotopy group B of the compact space Mcompact '" GT/B. In each of these sectors the WZW field g(z,z) obeys the following boundary conditions: (VI.I0.6)
These reliect into twisted boundary conditions for the Rae· Moody currents of the target group GT = SU(2)$ *: J(z) :: -2ig- 1(z,f)8z g(z,i) = Jt(z)t~
(VI.IO.7a)
= -2i8!g(z,f)g-1(z,i) = jt(z)t~
(VI.I0.7b)
j(l)
Locally, these currents satisfy the SU(2)3 affine Lie algebra, encoded in the following OPEs:
Jt(z)Jf(w) ::: ~ij (
26AB
-( )2
z-w
+ iJ2e z-w
ABC
If(w)
)
+reg. tenns
(V 1.10.8)
corresponding to level k/e'l ::::: 2. Globally, the same currents are subject, in each sector, to the following twisted boundary conditions:
(VI.10.9)
* t~ are the generators of SU(2)S, A=1,2,3 enumerate the generators within each SU(2) and i=1,2,3 enumerate the SU(2) factors.
2031
which eventually break the symmetry to U(1)3 and give rise to massless fermion states. The rdation between the choice or the homotopy group B and the twisting or the Kac-Moody currents was already explained in chapter VI.5 and we do not insist on this point. It was also pointed out tli'at the choice or the generators (6, b) is related to the boundary vectors one considers in the construction or IIlOdular invariant free fermion theories. This was explained in chapter VL7. Here we will make this relation more precise. Once the groupfold theory is fully fermionized and, as such, is described by a set of 20 + 20 component boundary vectors, then the h-lJlIq) can be recast in a very simple set or rules yielding the 20 + 44 component boundary vectors of the corresponding heterotic model. The free signs appearing in the GSO projectors are fixed by those appearing in tbe type II projectors. Following this method we will show that, on SU (2)3 -groupfolds, one can construct heterotic superstring vacua. where the internal superconformal field-theory is of one of the types shown in Thble VI.IO.I, where, in the various cases, ISCN=~,EH' yields a theory with N=x target SUSY and ga.uge group
(V 1.10.10)
GE being some appropriate enhancement group. The corresponding type II superstring has target SUSY: (VI.10.n) We will present a complete classification of the super&tring vacua based on the SU (2)3 groupfolds with level k/9 2 = 2 and discuss their genera.! properties. Furthermore, we shall establish a precise relation between the compactification approa.ch based on a.bstract conformal field-theories and the constructive framework based on free fermions. This mea.ns that, in the case or (2,2)9" compa.ctifications, by adopting the philosophy introduced by Gepner, we can identify the abstract (1,1) and (2,1) forms a.nd assigD Hooge DUmbers to the groupIold compactifications. We obtain theories for which the Euler characteristic X is a multiple of 8. This suggests that many or these vacua may correspond to special points in the moduli spa.c:e of certain CaIabi· Ya.u 3-folds. It appears on the other hand that this class of Calabj· Yau 3-folds is almost disjoint from the dass one describes with tensor products of minimal models or twisted versions of the sa.m.e. In our constructions the enha.nced symmetry plus the information that the final theory is obtained through two GSO projections from a parent N;::::4 theory provides a very useful set of constraints on the effective Lagrangia.n apt to determine it in full. Of special interest in this respect are those (2, 2)8,9 compaetifieations that can be viewed as pure truncations or an N=2 heterotic model based on a (4,4),,8 + (2,2)3,3 internal conlormal field-theory. For these vacua methods developed by Kounnas, Ferrara, Oirardello and Porrati [12J are applicable a.nd a full determination of the Kaehler geometry that includes moduli, Es-charged fields a.nd Hl (End(T)) singlets is possible. Although the specific models in question are not phenomenologically
2032 interesting, being non-chiral (X "" 0), yet the result may be interesting since it sheds light on the general properties of the complete effective Lagrangian. The chapter is organized as follows; In section VI.10.2 we discuss the general structure of type II superstrings compactified on SU(2)3 groupfolds and, after fermioniza.tion we define the appropriate set of 20 + 20 free fermions. In section VI.l0.3 we explain our notations for the boundary vector group generators and we summarize the rules for the construction of modular invariants and GSO projectors in the case of type II theories. In section VI.10.4 we show how the generators of the (2,2)9,9 and of the (4, 4k6 + (2,2)3,3 superconformal algebras are constructed in terms of the free fermion fields describing the 8U(2)3 quantum dyna.mics. In section VI.lO.5 we describe the free fermion formulation of the h-map from type II to heterotic superstrings. In section VI.I0.6 we discuss the emission vertices o{ all the massless multiplets in a general (2,2h.9 compactification. In particular We discuss the abstract (1,1) and (2,1) harmonic forms, respectively identified with the crural-crural and cruralantichiral primary fields having conformal weights h = h :::: 1/2. Section VI.1O.6 fixes the conventions for the general compactification framework based on abstract supercon£ormal theories that includes our explicit 8U(2)3 constructions. In section VUO.7 we treat the emission vertices of massless states in an N=2 superstring based on a (4,4)~,6 + (2,2h.3 internal theory. In particular we discuss the (4,4 ke analogue of the chiral primary fields. In section Vl.lO.8 we show how to embed a (2,2h.& theory into the (4,4)6.6 + (2,2h,s system. This is important in order to understand those N=l vacua. that can be regarded as N=2 truncations. In section VI.10.9 we present the classification of the type II boundary vector systems corresponding to all possible SU(2)3 gtaupfold vacua.. Section VI.lO.I0 treats bosonization and illustra.tes the procedure by means of which the massless spectrum of the various vacua is analysed. The appendices Vl.lO.A,8,C are devoted to a somewhat detailed analysis of three explicit examples with scattered remarks on the corresponding effective Lagrangians.
VI.1O.2 Type II superstrings on SU(2)3 groupfolds In the case of type II superstrings the structure of conformal fields in the left sector is repeated identically in the right sector. Hence in the case of SU(2)3 groupfolds we end up with lL set of 20 + 20 free fermions. To make this statement clear we must recall that, at the quantum level, the group-manifold construction is reduced to a free fermion one through the fermionization of the currents (VI.1O.7). Explicitly we set:
J/1(z)
== 2v2eABCxf(z)xf(z)
(V [.10.12(1)
it( z) == 2v'2e ABCxf(z)xf (z)
(VI.lO.12b)
2033
where xf(z) and xf(z) are 9$9 free fermions in the adjoint representation of SU(2)3 ( the f eke fermions, in the lan~e of chapter VU) tha.t are to be added to the set of 9$9 free fermions ~t(z) and ~f(z) (the fTue fermions ill the same language) already present ill the q.modellagrangian and corresponding to the fermionic superpartners of the Kat-Moody currents (VI.lO.7). Upon use of this fermionization, we see that the internal degrees of freedom of the Type II superstring are described by a set of 18 left-moving free fermions:
xf(z) ,Af(z)
( A == 1,2,3 ; i
=1,2,3)
(V 1.10.13a)
i 6AB 6ij At(zPjB(w) = -- - 4z-w
+ reg.
(V I.IO.ISb)
i 6AB 6i j Xt(z)xjB(w) == -- - 4z-w
+ reg.
(V I.10.13c)
xt(zPf(w)
= regular
(V 1.IO.l3d)
plus an identical set of 18 right-moving free fermions:
xf(z) • Xf(z)
( A =1,2,3 ; i = 1,2,3 )
_ At(z) Xf(1ii) =
xt(i)xf(til) ==
i6AB 6;·
(V1.10.14a)
--4::---::? + reg. Z-w
(V l.lO.14b)
-4i 6z _Dijtil + reg.
(V1.10.14c)
AB
xf(z)Xf(tii) :: regular
(V1.l0.14d)
The explicit form of the stress-energy tensor and local supercurrent of this internal free fermion system is obtained via substitution of eq.(VI.lO.12) into the following formulae:
(V 1.10.15a)
T; .. I(Z) =:
t (~Jf ,=1
Jf -
2iAf8. At) :
(V [.lO.ISh)
(V I.10.15c)
(V I .1O.15d)
2034 that correspond to the Noether currents derived through variation of the classical (1,1) locally supersymmetric WZW action with respect to the two-dimensional gravitinos and vielbein [30b]. In this chapter we do not discuss the structure of this action, since the only information we need for the construction of type II superstring models is the form of the superconformal algebra. Clearly this latter is nothing else but the repetition, also in the right sector, of the algebra obtained in the left sector, whose derivation was given in chapters VI.3 and VI.5 for the case of the heterotic superstrings. The complete conformal superalgebra is obtained by summing to the generators of the internal one those corresponding to fiat MinkowslU space: GAl .....(z)
TM;"k(Z) ...
~PI'(z)PI'(z) GMink(Z)
TMin.(:)
= J2llit tP"(z)PI'(z) - itPl'(z)8.tPl'(z)
= .,I2lli~ ~"(z) Pl'(i)
= ~PI'(i)PI'(Z)
-
i~l'(i)8i~"(i)
(V 1.10.164) (V 1.IO.16b)
(V I.I0.16c) (V1.IO.I6J)
As stated in chapters VI.S and VI. 7, the available choices for the generators of the group B appearing in eq.(VI.I0.6) are completely determined from the requirements of space-time supersymmetry, multiloop modular invariance and world-sheet local 5upersymmetry, generated by the supercurrent GM;Dk(Z) + Gint(z). Furthermore the correspondence of these generators with suitable left boundary vector8 of free fermion boundary conditions can be given. These results can be utilized to derive the available choices for the generators of the group B appearing in eq.(VI.IO.6). This has been done by means of a computer program obtaining a set of boundary vector groups whose structure-we illustrate in the next sections. Here we recall the discussion of the B group given in chapters VI.S and VI. 7 (see in particular eq.s (VI.5.32-40). (VI.7.22) and (VI.7.33-36». Specifically the elements b, b E SU(2)3 out of which we c~struct_ the p.,airs (;b) E SU(~)3 @ SU(2)3 are elements of the discrete subgroup A = Al @ A2 0 A3 , where At C SU(2)i is a Z.. @Z•. In the adjoint representation of 8U(2) (which is two-to-one), the elements of are generated by rotations of 11' ,denoted eA, around the three coordinate axes (A =1,2,3) of SO(3) = SU(2)/Z2. As such these 3 x3 diagonal matrices eA generate a Z2 @ ~ subgroup of SO(3) and have the following explicit form:
a.
(eA)se
= 6BC (c5AS + 5AC -1)
(VI.IO.11)
Furthermore they satisfy the obvious relations:
eAts
= lc5AB + leABclec
In terms of these matrices the explicit form of eq.(VI.10.7) is:
(VI.IO.lS)
2035
Jt(ze21ri ) = Jf(z) eJ:A
it(ze-2'"I: '"
(VI.10.19a) (V I.10.19b)
iiB(Z)eJ:A
and. in order to specify an element (b,b) e B we just have to give two triplets of matrices cU; (i == 1,2,3) and eu; (i = 1,2,3). In the free fermion description of the type II superstring, the X's , the >"s and the 'I/1T's (T = 1,2) form the 20 left-moving (LM) world-sheet termions , while the 20 right-moving (RM) fermions are given by the same symbols with a tilde. A sector of the string is identified by the set of boundary conditions (BC) of the above 20+20 fermions • In full analogy with the fonna.1ism adopted for heterotic superstrings and following the conventions of chapter VI. 7 it is convenient to represent the Be of the world-sheet fermions in a given string sector by a. 4IJ..CODlponent boundary vector whose components are 0 or 1 (see the analogue eq.s (VI.7.72-73)): (VI.IO.2O) according to the rule: a... =
if 'P is & NS fermionj { 01 if 'P is a R fermion _
(VI.IO.21)
Relying on eq.(VI.I0.12) we see that every choice of a boundary vector corresponds to a choice oftwo triplets of matrices cU; (i ;: 1,2,3) , eu, (i '" 1,2,3) and hence, as explained above, to the choice of a generator (b,b) E B C SU(2)3 ® SU(2)3. Defining the scalar product of two boundary vectors as follows: 211 0.'
40
(VI.IO.22)
" a;{3i - " . b dd "" 'L.J L.J ' (l;Pi 1=1
i=21
modular invariance imposes the following constraints on the set E of boundary vectors defining a superstring model. First of all E should be an abelian group under the vector addition (each component is summed modulo two): this group is just the homomorphic image of B under the previously discussed correspondence. Secondly we should have (see the analogue eq.s(VI.7.83-88»): Va E E
0.2
Va,b E E
Vo.,b,ceE
(V 1.10.23b)
a·b == O(mod4)
a·b.c= !(Q..c+b.c-(o.+b).c) == 2
(V I.I0.23a.)
== a..Q. == 0 (modS) .
o(mod 2)
(VI.10.23c)
Thirdly the group E must contain the universal boundary vector 70 given below:
2036
(V 1.10.24(£)
[Ill} == 70 E S
where 1 denotes the vector ill which all the 20 fermions have Ramond boundary conditions, that is, all the entries are equal to one, This vector corresponds to no SU(2)3-twisting;
(V 1.10.24b) A computer anal)'$is reveals that, if we focus on the left boundary vectors (LBL), then there are just three sets of generators compatible among themselves, with worldsheet supersymmetry and with multiloop modular invariance. They are;
A == [1,b,b',8,s' ,s"] B == [I,b,b',8,,',c1 C = [1,b,b'",c,c'l
(VI.10.25a) (V 1.10.25b) (V 1.10.25c)
where the explicit form of thi: above vectors is completely specified by listing the set of Ramond fields from which listing we also read oft' the corresponding b generators:
(V 1.10.2&) b::: {x~
11 :::
Xr A1 >.~
{xh::f
(i=2,3)} ,." l®el®el A~ A: (i = 1,2)} ... el ® el ® 1
(V 1.10.26b) (V I.10.26e)
(VI.10.26d) (V I.IO.26e) rVI.10,26j)
(VI.I0.26g) As already stated, these results can be utilized to construct the genera.tors of the boundary group S in the case of type U superstrings. These generators have the form; 7A E E => '"fA :::
[a:ly]
(VI.IO.27)
where a:, (Y) are taken from the above sets of consistent left (respectively right) boundary vectors or Z2 linear combina.tions thereof. In the next section we show how this is explicitly done.
2037
VI.IO.3 • Construction or modular invariants and GSO projectors tor the Type II superstring The theories we will consider are based on a group =: g~erated by the following set of boundary vectors:
(i == 1, .... ,K) where K is some integer number. In all theories furthermore we set: ")'+
")'0
is given by eq.(VI.10.24) and
== [sIO] '" (i®i®i,ed.~el®el)
")'_ == [Oli]
N
(VI.I0.28)
(el®el®el,l®I®I)
(VI.IO.28a)
(V 1.10.28b)
The remaining generators vary instead from model to model. We note that 1+ and 1- are needed to get massless gravitinos in both the left- and right·moving sectors of the spectrum. We use the following convention for the boundary vector indices. The K + 3 generators are enumerated by capital Greek indices A, 1:, r = 0, +, -, i( == 1, ..., K). The generators from + to K are enumerated by capital Latin indices: "fA. ::
")'+, ")'- , ")'.
(VI.IO.29)
while the lower case indices i,j, k label tbe non-universal generators whose choice depends on the model. Once the generators have been selected the modular invariant still depends on a certain number of free signs which is given by:
. number of free SIgns
= 4 + K(K2+5)
(VI.IO.30)
The assignment of these signs can be understood as the assignment of a Zz cocycle on the boundary group E ,., (Zz)K+S. To be precise, we recall that the modular invariant partition function has the following structure: (VI.lO.31) where a, b is any pair of elements of S defining a spin strutture of the free fermion system and where the coefficients are Z:a elements:
(VI.I0.32) The relations imposed by modular invariance are (sce eq.s (VI.7.97) and (VI.7.109):
2038
C[ba] ; ; ;
6
'Yo
c[a+6+;0 a ] i",.2/
C[:1 ""
c[:Jc[;J
(V 1.10.33a)
c[!J
(V 1.10.33b)
c[b:C]
(V 1.1 O.S3c)
it,.·b == 5G
8
where 5.. is the statistics of the sector corresponding to the boundary vector a (54 is +1 jf a represents a boson, while it is -1 if a represents a fermion). The relations (VI.IO.33) are true for any type of fermiomc string, both heterotic and type II. Eqs. (VI.IO.33) allow the determination of all the coefficients c r:J for any pair of::: group elements if we assign the following data.. Let tA1; be a Z2·va]ued tensor (SAIl2 1) with the following symmetry:
=
(VI.IO.34) a.nd fulfilling the additional condition:
(VI.IO.35) Clearly the tensor tAl:) has a number of independent components equal to the one given by eq.(VI.IO.30). For instance as independent components we can choose the following ones:
(VI.IO.36) Given the tensor
tAIl
we set: (VI.lO.37)
and
(VI.1O.38) Inserting these relations into formulae (VLlO.33) we ca.n work out the value of for any pair of boundary vectors. In turn this implies that we the coefficients ca.n write the GSa projector for any sector of the theory, the projector being given by the following general formula:
c[:l
(VI.1O.39)
where, for any boundary vector a, we have a • F == E a,Fi I Fi being the fermion number of the i t 1t. world~sheet fermion !pi. More precisely, we can write (-It
INhs=i:t!p~tlP~
= {(_l)tr
fotO:i""O(NS); (-1 )N1 V21P~ ,Nh = E~=l IP~t IP~ for al == 1 (R) . N'
(V1.10.40)
where the zero mode IP& of the Ramond field is proportional to a suitable gamma. matrix. VI.IO.4 ~ SU(2)3 groupfolds and superconformal field theories
Given the universal boundary vectors i':.b the set of 18 left·moving (respectively right-moving) fermions is na.turally split into a. sextet plus a. 12'plet, the former coutaining the fermions that have Ramond boundary conditions in the 7:1: sector and the latter including those tha.t have Neveu-Schwarz boundary conditions in the same sector. Next we can make contact with six·dimensional orbifold picture by introducing the six bilinear combinations of the internal fermions that we list below:
PO(z}
= 2\"'2 (xMz) xti)(Z) + lli)(Z} lMz») =:
~
(Jli)(z)
+ 2\"'2e1BC lra1(z)Ar.){z}) (V I.I0.4Ie)
(i :;:: 1,2,3) pi' (z)
= 2\"'2 (Xfi)(Z) lfi)(Z) -
.
(i"" 1,2,3)
x:i)(z)A~i)(Z») (V 1.10.410)
similu definjtions for pi{z) and pi' (oi)
(V 1.10.41c)
As the reader can notice eqs. (VI.I0.41) involve only the 12 $12 fermions that have Neveu-Schwarz boundary conditions in the sectors 'Y:I:. The remaining 6 E9 6 fermions can be arranged into a six-vector of left·movers plus a six·vector of right-movers according to the following rules. Splitting the range of the SO(6}-vector index as:
x=
n"
:;:: 1,2,3 1*,2", 3"
=
(VI.10.42)
we set:
(i:::1,2,3) {i* =1",2*,3*} similar definitions for .,fox (oi)
(V 1.10.434) (V 1.10.430)
2040 Then it is easy to check that the 6 ED 6 currents (VI.lO.41) and the 6 ED 6 fermious (VI.1O.43) obey the OPEs corresponding to the direct product of a U(l)' left-moving times a U(l)B right-moving super Ka.c-Moody algebra:
(V I.I0.44a) (V I.I0.44b) (VI.I0.44c) with similar equations for the tilded quantities. Relying on eqs. (VI.I0.44) we can postulate the following bosonization rules:
1>i(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<2"}[~](T) + Z(2")[~](1')
(V [.lO.sac)
[!] (T) _ Z(2n)g](T)
(VI.IO.88d)
B~2")(1')
0::
z(2n>
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<2 ..)
= M T<2 ..+:4) M
(V1.10.91a)
S(2 ..)
= M ,s(2n+24) M
(V I.10.91b)
where the idempotent matrix M (M2 == 1) is:
Mij
0 1 0 ( 1 0 0 == 0 0 -1 o0 0
jJ
(VI.IO.92)
U in the expression for the modular invarisnt partition funetion of a type II superstring we make the following replacement: (VI.lO.93) it follows from eq.(VI.lO.91a) that we obtain the modular invariant partition function of a heterotic string model. In eq. (VI.IO.93) B~2)(1') are the characters oCthe SO(2) Kae-Moody algebra spanned by the transverse space-time fermioDs, while B~26)(1') are the characters of the SO(26) Kae-Moody algebra. Combining eq.(VI.10.93) with eq.(VI.10.88) we tan also write the above substitution in the equivalent way:
[~](T)
==?
Z(2') [:] (T)
(V I.10.94a)
Z(2)[~J(T)
==?
_Z(26}[~](1')
(V I.lO.94b)
z(2)
2054
Z(2)[~](T) Z(2)
[~J (T)
=;.
_Z(Z6)[~](T)
(VI.10.94c)
[~J (T)
(VI.1O.94d)
=;. - Z(26)
Eqs. (VI.10.94) can be formalized at the level of boundary vectors in the following way. Let us lirst subdivide the 44 right-moving fermions into the subsets:
if Xf (i;::: 1,2,3), 8P (p = 1, ...,10), e (I:::; 1, ... ,16)
xt
where are the fake fermions fermionizing the SU(2)3 bosons, while the 9 Xf plus the 10 81', plus the 16 complete the set of35 heterotic fermions. To the right-moving boundary vectors" ;' , ," , band b', defined in eqs. (VI.IO.26) and utilized in the construction of the type II generators lA, we can associate the following right-moving heterotic boundary vectors:
t!
e
==
{iLXf(i:::; 1,2,3), 81'(p = 1, ...,10), s.'
t!
e(I::: 1, ...,16)}
(V I.IO.9Sa)
=;.
{it,l~(i:::; 1,2), iP,j, 81 (p;::: 1, ...,lO),e(I = 1, ...,16)} (VI.IO.95b) s." =;.
t!
= {x~,lW:::;l,3),x.L1Lel'(p:::;1, ... ,lO),e(I:::;1, ... ,16)} (V1.10.95b)
(V I.I0.95d)
(V I.1O.95d) Given the genera.tors I~YI'.II of the E group in a. type II model, we immediately obtain the E genera.tors I~eter.,t;c in the corresponding heterotic model by performing the substitutions (VI.I0.95). Once defined on the genera.tors, the correspondence extends trivially to all the E-group-elements. Furthermore the heterotic c coefficients are related to the type II ones by the simple rule:
[:J
hct
C (a bhet }
=
C
[aJII exp [i1l'"4 (20.' "" + 2b· "" + (a. ",,)(b. "")) ]
(V 1.10.96)
where the vector"" is defined below:
(VI.1O.97)
2055
In this way, given the sectors and the GSO projectors of a type II superstring, we can immedia.tely write down the corresponding sectors and GSO projectors of the associa.ted heterotic theory based on the S.o(26) algebra. The extension of our rules to the SO(10)®Ee case is quite simple. The cb.aractera of this Kae-Moody algebra. can be written as follows:
Bi(SO(lO)@S.I(T)
~lO)( ) ...tE8 )( ) l1i l' IJ' T
_
-
(. -
~ -
0,v,s,s-)
(VI.I0.98)
where we have defined
and it can be immedia.tely checked that they transform under the modular group as the SO(26) characters, that is with the matrices 1'26) and S(26). This result follows from: B(Sa)(T
+ 1)
::: e- 2ht/ 3 JjB')(T)
B(Et) (_!) ::: B'l?:')(r) r In terms of fermion partition functions the repla.cem.ent B~2)(T) :::::;. Mij BjSO(lO)@Bs)(T)
(VI.10.100a) (V I.lO.lOOb)
(VI.lO.lOl)
corresponds to the substitution
Z(2)[~1(T):::::;' Z(lO)[~1(T}' B{Ea){T)
(V 1.10.1020.)
Z(2)[~](T)
:::::;.
-zt1G>m{T)' B(Ea)(T)
(V 1.10.1020)
Z(2)[~](r}
=l>
-zt1G>[!l(T)' B'Ee)(T)
(VI.I0.102c)
Z(2)[!)(T)
=l> _Z(lO)
[!](T) . B'Et)(T)
(VI.I0.102d)
Eq.s (VI.10.102) can be interpreted in terms of a two-step process. First we ma.p the type 11 model into the SO(26) model as described a.bove. Then we extend the ::: group through the addition of a boundary vector
A;:: [Oli]
(V 1.10.1034)
6 == {e{I==l, ...,l6)}
(V 1.10.103b)
2056
and we determine the additional
c[:l coefficients via the rule: (Vl.10.104)
Having established the general form of the h-map, in the next section we discuu the general $tructure of the massies. sector in a heterotic superstring compactified on a (2,2) superconforma1 field-theory.. VI.IO.6 Emission vertices of the massless multiplets in an N=l heterotic model based on a (2,2).,9 internal theory
Given the n=2 superconformal algebra in the normaIiaations of eq.s (VI.10.72), we denote the primary fields of a (2,2)e,E sUpe1'I:onformal theory by
t(h'~) (%,i) q,q where (h,h) are the conformal weights and (q,V are the U(l) charges:
T (z ) t
( h,h) _ (w,w_) f,q
(
t
(i)
tG:!)
(w,w)
(k'h) ( _)
= (z
h )2 t _ w,w + - w q,f
1 ) z - w
= (i
8\Ut(k,~) (w,w) + reg. q,q
! .C:;) tii)l
(V 1.10.1054) (w,w)
+
(_ 1 ) 8Iit(h,~) (w,w) + q,q
10
% -
reg.
(V1.10.105b)
i
(z)
t(h,~) (w,w) q,q
1'I (_) z t
= (
q
z -
) W
.(h,~) (w,w) f,q
+ reg.(VI.10.105e)
(h,ii) _ (-) w,w = r q _) t (h'h) _ (w,w-) + reg (VI.lO.105 d) f,q z - w q,q
We now introduce the "spectral How" of the 0=2 superconformal algebra. A one parameter twisting of the n=2 superconformal algebra (Vt10.72) is given by the boundary condition
2057
implying that T(z) and j(z) are periodic and therefore have integer modes. Setting '1 = 0 (mod 1) yields the Neveu-Schwarz sector while TJ "" (mod 1) corresponds to the Ramond sector. In fact there is a continuous I-parameter spectrum of sectors labelled by the parameter TJ corresponding to different representations of the n=2 superconformal algebra all of which, however, are isomorphic. Indeed by setting
t
=z%'1g:l:(z)
g;
(V1.IO.106a) 2
T,,(z)
==
T(z) +I'JJ(z) + ~2
JII(z) = J(z) + ;;
(V I.IO.10Gb) (V I.lO.106e)
{j;,
one easily verifies that the new operators T~, J v satisfy the same superconformal algebra (VI.lO.72). In terms of modes eq.s (VI.10.106) become:
=L.. +'1J" + ~1126...o
L'.,. == Uq L.,.U;l
(V I.lO.I07a)
J'.,. == U.,J..U;l = J.. + i116,..G
(V 1.1O.107h)
:1: - U O:l:U-1 0:1: 0'r=='Ir'l=r:l:'I
(V 1.1O.107c)
where Uv is the operator shifting the TJ = 0 representation of the n=2 supercon{ormal algebra to the 11 f:. 0 representation. Eq.s (VI.IU06) or (VI.IO.107) define the so called spectral flow of the n=2 superconformal algebra. In particular the O-mode eigenvalues hand q are related by c
h., = h + l1q + 6'12 q~
C
"" q+ 3'11
(V [.10.108a) (V I.I0.108b)
Notice that any unitary representation of the algebra must have h ?: 0 . Requiring h ?: 0 for all '1 we deduce from (VI.lO.108) the inequality
at
h> - 2e The same considerations can be applied to the right moving superconformal al· gebra, where the analogous quantities are tilded. In order to construct the U'I(Uv) operator we bosonize the U(l)-eurrent j(z)(](z»:
i (z)
fa iii
= i
; (z) =
8.!p(z)
8i
2058
where \I'(z} and y;(i) are free bosonic fields
!P(Z)\I'(W) :;: -In{z - w) + reg. y;(z) y;(w) ::: -In(z - 'iii) + reg.
v1
and the normalization is fixed by the OPE (VLIO.72£) • Given 8. U(l) neutral field X(s) it is easy to verify that the new field
ezPlil!q\l'(z)]x(Z) has charge qj indeed
J(Z) ezp1il!w(w)] x(w) ==
i~ 8tp(z) eZP[il!q\l'(W)] X(w)
:;: _9- up[i
z-w
~9!P(W)] x{w}
V~
Therefore the primary field t (!~) (%, f) satisfies the following 11 ==
t spectral
flow:
t(k,~) (z,z) q,g
""
'" exp[iQJ!\I'(Z)] i
[ii¥1y;(Z)]
(VI.IO.109)
where i(h- ~,,- aG>(z,z) has UtI) charges q :;: q :;: 0 and in primary field of the tI :;: c - 1 conformal field theory one obtains by deleting the free fields \1'( z) and y;{z). The conformal weights 11.0 and he of t are obtained from eq. (VI.I0.108) with .,,:; I and q :;: 0 (g == 0):
implying 3q2 11.0=11.-2c
and similarly for the right moving sector. Since the speetraliow shifts the charge by 7'Ji. every state of charge (q,g) flows to a new state of charge (q -
h,g - h) so tha.t the "\I' momentum"lfq. jicg
2059 becomes (jic(q - f1]), by
~(ij - h). It follows that the spectral operator U1/ is given ,: U'I ;;; fr:pl-i1].ft,(1fJ -
~)l
From eq. (VI.10.108) one concludes that to every primary field of a (2,2) superconformal field theory one CIl.I1 associate an entire new family of primary fields obtained by shifting the U(1) charges. Primary fields related to each other by spectral flow will be denoted by the same symbol, changing only the values of the conformal weights and of the central charges. We see that the spectral flow from 0 to ~ establishes a one-to-one correspondence between NS and R states. This is a strong hint of a connection between the n=2 superconformal symmetry and N=l space- time supersymmetry. It turns out that this is indeed the case jf the U(l) charges of the physical states are odd integers (see rei. [9]). The spectral flow within the internal (2,2h.9 theory corresponds, in space-time, to the completion of supersymmetric multiplets and, in group-space, to the completion of E. representations. We now review the concept of chiral-chiral and chital-antichiral primary fields that leads to the identification of the abstract (1,1) and (2,1) harmonic forlllll within the (2,2) theory. One ell.I1 show that the ehiral multiplets of II.I1 n=2 superconformal algebra are the short representations characterized by a lowest component that fulfills the condition: h
= iqi 2
Indeed let us consider the anticommutator C 2 1) _ {g>.+,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<" and their spin one-half superp&rtner, the dilatino Xa 3) The vector multiplets of the ga~e group (1, t) : G9"'''9.
= GB ® Ee ® E~
containing the corresponding gauge bosons and ga.uginos. 4) The matter Wess-ZunUno multiplets (0+,0-, that subdivide into 5 sets:
i)
40.) h1,l 27 - familie; of Ee 4b) 1.2,1 27 - anti/amilies of
Ee
4c) hl,l mo~li (Ee singlets, Gs charged) 4d) h2 ,! m~li (E6 singlets, Gs charged) 4e) N../ 1St singlets COTTesponding to non flat direction, Let us first see how the spectral flow is utilized to complete N= 1 supermultiplets in the left-moving sector and Ee representations in the right-moving sector. We begin with the graviton multiplet and with the vector multiplets corresponding to the gauge group Ee ® E~. All these states are associated with the spectral flow of the identity field lC::)(z.z) in the internal conformal field theory. Indeed utilizing 1(:;:)(2,z) and its spectral flow associates we can write the following massless vertices in the canonical picture:
Vi<~ (k, z, z)
= 2i.../4 eI9"(-) tjlI'(z) 1 (::~)(z,.i')P"(z)e(i" ·X(z,l)! (V 1.10.117)
V: (k, Zt z) == eh'-(z) S'" (z) 1 ct~)(Z'Z) 'pi«z)e[Il:.X(z,J)] (V I.10.uSa) val' (k, z, z) =
etfo"(z) S.. (z)
l( ;":) (z,%) Fi«z)e(ik
.X(.,tl]
(V I.IO.11Sb) vjU\ (k, z, z)
=:
2ei"/4ei."(~)"'i«z) l(~:~)(z,Z)jA(i)e!U'X(*.Z)1 (V [.10.119)
V: (h. z, z)
=e1."(z) S'" (z) lct~)(Z'Z) jA(z)e[i".X(z,zl! (V [.10.1200.)
V.; (k, z, i)
= e! .'·(z} sa (z)1 (i::)(Z'z) jA(z)e[ik.X(o,f»)
2064
(V1.10.120b) VI'- (k, z, z)
= 2 e;"/4 eW'(~) 1/I"(z) 1 (~:~)(z'Z)l (z) e(·k.X(z,.2)1 (VI .10.121)
V"A (k, z, z) = 2e·... /4 eW '(") 1/I"{z) 1(~: D(Z,
E) SA (E) e1ill,X(",!)1 (V I.10.122a.)
V"A(k,z,z) ""
=2e'''/4 eW'(z) 1/I"(z) 1 ( V; (k, z, z)
0, 13)(z,Z)SA (i)e{.k,X(Z,.i)J 0'-2 (V 1.10.122b)
= eH"(z) S"(Z)l(!i~o)(Z'i)] (z)elik,X(z,l)1 (V I.10.123a.)
Va. (k, z, z) = eH"(") S"(Z)l(!':)(Z,z)] (z)elu.X(.t,!)J .
2'
(V 1.10.123b) VcA (k, z, z) = e! 9"( ..) Sa(z) 1
(J' ~)(Z,i)SA (i) eli
A·X(.. ,i)l
2'2
(V I.IO.I24a)
vi (k, z, z)
=
ef~"(')Sa(Z)l( l'li)(z.!)SA(Z)eI.k.X( .....)1 2'2
(VI.IO.124b)
In the above formulae p.(!) denotes the Kac-Moody currents of the 50(10)0& group corresponding to the fermion bilinears 8P(i)OQ(f), e(.f)e(i) plus the spin field SM (E), 4>"( z) is the free bosonic field emerging from the Buperghost bosonization and PII(i) = -i01XII(Z,z). The vertex (VI.I0.117) creates the graviton, the dilaton and the aldon. The vertices (VI.l0.118) create the left-handed and right-handed parts, respectively, of both the gravitino and the dilatino. From the vertex (VI.lO.119) we obtain the gauge bosons of 50(10) 0 E; while, in eq.(VI.I0.1~) we give the emission vemces for the
2065
corresponding gauginos. In eq.s (VI.I0.121-122) we have the emission vertices for the gauge bosolls that promote SO(1O) to E.. To appreciate this point it suffices to recall the decompostion ofthe E. adjoint under 50(10) ® U(l). We have:
n
SO(~U(l) (ii, q = 0)
(16, q
=
e (1. q = 0)
De (l§, q
:0
(V 1.10.125)
-~)
Eq.s (VI.10.121-122) give the explicit form of the gauginos of E. in the coset directions &/50(10). We emphasize that the U(l) subgroup of Eo is generated by the U(l) current belonging to the right moving n=2 algebra. The vertices (VI.I0.117 VI.10.124) have the property that the total U(1) charge, defined by summing the U(I) charge of the internal theory to the U(I) charge of the space-time theory, or of the right-moving current algebra, is odd in the left sector and even in the right-sector. This fad corresponds to Gepner's formulation of the GSO projection. In the free fermion constructions this rule just comes out automatiea11y from the GSO projection derived by imposing modular invariance on the free fermion partition function. We also remind the reader that the vertices (VI.IO.117 - VI.10.l24) are massless because the total conformal weight is one in both the left and the right moving sectors. We can now discuss the massless vertices for the WZ multiplets belonging to the 27 families and 27·antifamilies. These are associated with the spectral flow of the (1,1) and (2,1) primary fields. To understand this point it suffices to recall the branching rule of the 27 and 27 under the subgroup 50(10) ® U(I): 27
SO(lO~U(l)
='*
(1!!, q == 1)
e 21 SO(l~U(l)
(1&, q ;; -0
e
(VI.IO.12&)
(l,q= -2)
(1!!, q:::: -1)
e
(16, q ::::
i)
(V 1.10.126&)
e(1,q=2) Correspondingly the emission vertices for the complex scalars 0+ are:
(!!l,q
= 1)
V,r (h,
%,
i) ==
eWf(z) e[ik.X( .. ,i)l
'f!
+ 0- in a 27-family
(t D
(z,i) P(l) (V1.10.127a)
2066
VOlA (h, z, i) :::: e'·'·(z) e(ilo,xh,J)! ~~ (l~~ii) (z.i) SA (i) (V 1.10.127b)
(1,1=-2)
~ (h, z, z) = ei."(z) ef"·X(...Jl]~: C~~2) (z,z) (4 ;::; 1, ....... h1 •1)
(Vl.10.127c)
(V 1.10.127d)
while the emission vertices for the complex scalars 0+
+ 0- in a 27-antjfalllily are:
(lQ, q == -1)
Vi (h, z. z)
= e'."(6) efi.'X(~.i)1 w; (1:~1) (z,f) 81'(1) (Vl.10.12&)
(l,i
= 2)
V.. (k, z, z) == ei."(z) ef1i:, x h,l)1 w; (4 ...
0:;)
(z,E)
1, ......,h2•1 )
(VI.10.128c)
(V1.10.128d)
Utilizing the spectral flow in the left-moving sector, from the vertices (VI.IO.127) and (VI.10.128) one obtains the vertices of the corresponding spin one half superpartners. For instance the spin ~ partners of the (!J1, q = scalars are described by the vertices:
-1)
v:' (k, z, z)
::::
eH"(:r)
sa (z) q;! (!i:l) (z,z) 8P(z)e ",Xh,1) i
(V [.10.129b)
In a silllilar way one deals with all the other cases. In addition to th~ Et families and antifamilies one has the WZ-multiplets that are Et singlets. In full generality the emission vertices for the complex scalars belonging to these In11ltiplets have the form:
2067
(VI.I0.130) where 0.(1;;) (.;,z) is any pritoarYiiel~of (2,2),.& theory with the specified conformal weights and U(l) charges; the index i runs on the set of such primary fields. Obviously in the set ofthe prirnaryfields ni(f~) (.;,E) we retrieve aJso the moduli (z, z)
+;(j:;)
defined by equa.tion (VI.10.1l2d), yet not all the ~(t:;) (z,E) are moduli. Hence if we call N,·",let. the number of primary fields have: N'i",let. =
hl,l
Oi(N) (z.f), then we •
+ h2,1 + N'"
where N'" is the number of singlets that correspond to non·flat directions of the scalar potential. When the (2,2)'.9 theory admits .. geometrical interpretation as D'-model on a smooth Calabi-Yau a·fold, then we can write: N'"
= dimH1 (End{T» + dimGs
Indeed, since the moduli are GE-cilarged, by moving to a generic point in moduli space, we break GE via an ordinary Higgs mechanism. In this way we soak a number of singlets equal to dimas, needed to make the Gs·vector multiplets massive. The difference N'" - dim GE should be equal to the number of deformations of the tangent bundle, namely to the dimension ofthe first cohomology group 8 1 {End (T». The left-moving spectral flow on the vertices (VI.10.130) provides the vertices of the corresponding spin l partners in complete analogy to eq.s (VI.IO.l29). Finally we discuss the emission vertices of the enhuceznent group Gg. They are given by:
v,.l., (1:, z, E) == 2ei1tJ4 e',p·'(:o)w,.(z) XIs
C:!}
(!:!)
(i)
eIU'X(z,J)!
(VI.10.131)
where XIs (z) are U(l) neutral fields of conrormal weight h ::t 0, h = 1 belonging to the internal conformal field-theory (2,2) ••11 and spanning the Kat-Moody algebra of the Gs group. By left-moving spectral flow one easily obtains the vertices for the corresponding gauginos. • In sections VI.I0.9 -10 we discuss how the .:(l~±\)(ZI f), Oi(i;;) (z.z) and XIB C,~ (z) primary fields are explicitly constructed in the case of SU(2)3 groupfold compactifications. In the next sections we discuss the structure of the emission vertices in N=2 compa.ctifica.ti~ based on (4,4).,. + (2,2)s,s internal conformal field theories.
2068
VLlO.1 Emission vertices of the massless mUltiplets in N=2 heterotic models based on a {4,4)8,8 ED (2,2)8,8 internal theory We are specifically interested in N == 1 theories. In the free fermion a.pproach, however, there are N == 1 theories whose massless w:tor can be viewed as a trUDcation of the massless sector of an N "" 2 theory. This gives the advantage that, by following the procedure inttoduced by Ferrara, Kaunnas, Girardello and Porrati and based on quotients of quaternionic manifolds with respect to quatemionic isometrics, the effective Lagrangian of this class of N=l models can be completely determined.
For this reason in the present section we consider the general structure of the massless emission vertices in the case of N:o: 2 heterotic models based on (4,4}e,8 ED (2,2h,3 internal theories. Let us introduce some notations. For the normalizations of the n ... 4 supereonformal algebra we recall eq.s (VI.lO.T3) In the case c "" 6, we observe that the SU(2) Kac-Moody algebra contained in the n=4 superalgebra has level k/fP = 1 and, as such, is bosonizable in terms of a single free boson 1'(Z) obeying the standard OPE:
1'(Z} T(W) == -In{z -w} +rcg.
(V1.10.132)
jS(z) = ~8.1'(Z)
(V l.lO.133a)
j:l:(z) == jl(z) ±j2(z} =e:l:i v'2T {z)
(V I.I0.133b)
It suffices to set:
This bosoniza.tion inttoduees the analogue, for (4,4).,6 theories, ofthe spectral flow present in any (2,2) theory. Primary fields of a (4,4) theory fall into SU(2) multiplets and have the following structure: h
h1mm
• [J:i
(z,f)
where J and m are, respectively, the isospin and third isospin component of the left
sedor, while i and mare their right sector COUDterpartS:
(V I .10.l34a)
• mm
Z. [h,h] J,i
'3{-}
J
f
JRm
(_) m [h,lt] (w,w_) treg. w,w = (z-1i1). J;i
(V 1.10.134b)
In the pa.rticular case c=6 the bosonization (VI.IO.133) implies that we can write
(V 1.10.135) where t(h-m' ,l-",')(:,z) is an SU(2) singlet in both sectors. In complete analogy with (2,2) theories we conclude that to every primary field of a (4,4)6,6 theory one can asaociate an entire new family of primary fields obtained by shifting the eigenvalues oC j3 and j3. This is the n ::;;; 4 spectral flow. As we did before, primary fields related to each other by spectral flow will be denoted by the same symbol, and differ only by the values of the conformal weights and of the isospin labels. The free bosons bosonizing the U(l) currents J(:), j(z) of the (2,2)'.3 theory will be named O'(z), u(z) and according to eq.$ (2.2) we write:
(V I.10.136a)
j(i) =i 8zu(z)
(V 1.1 O.l36b)
The associated U(l) charges are denoted Q, (2,2)3,3 theory is written as follows:
Q. so that a primary field of the
(V [.10.137) The combined spectral flows of the n = 4 and the n :::: 2 theories correspond, in space-time, to the completion of N => 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, 2A{A "'" 1,2) and one graviphoton.
2) The mon vector multiplet (1,2(l),O+,O-) containing the axion hI'''' the dilaton tP, a dilatino, an axino and an axionphoton. 3) Tbe gauge vector multiplets dimG 0(1, 2a),O+ ,0-) containing two gauginos and one complex scalar for each gauge boson in the gauge group (V I.1O.139a)
(V I.IO.139b)
4) The hypermuItiplets (2( ~), 2(0+), 2(0-» that subdivide in the following sets 4a) ~hl.l 56-families of E7 4b) ~kl.1 moduli (singlets under E1. charged under GE) 4c)
iNfI./
E1 -singlets not corresponding to flat directions of the scalar potential.
The reason why we count the hypermultiplets as ik1.1 is that the complex scalars emerging from the compactifications combine pairwise to build a. single hypermultiplet.
In analogy with the (2,2)s.s-case one might be tempted to conclude that the number Nfl! - dimGs corresponds to the number of deformations of the tangent bundle to the smooth manifold over which the (4,4)6,& theory lives. This is not so since, in general, there are also hypermultiplets of E7 singlets whose vertices contain nontrivial fields of the (2,2h.3 theory. Here by nontrivial we mean fields that are not rela.ted to any deformation ofthe hyper Kahler manifold supporting the (4, 4)~.8 theory. Let us now discuss the emission vertices of the various multiplets: those of the graviton and vector multiplets are associated with the spectral flows of the identity fields in the (4,4}&.6 and the (2,2h.3 theories.
2072
Indeed we have:
1) Graviton ED DilatoD ED Axion vertex:
2) Gravitino ED Dilatino ED Axino vertex
V~ "'(k,z,z) =e1~"(z) Sa (%)I[i::r(Z)lct~)(Z) P"(z)e1U'X(s,J)! (VI,IO,140b)
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<&) (i ,.. 1, .. ,5) (V I.1O.183b) (i = 1, ..,8) (V I.10.183c)
Equations (VI.IO.183) make sense since all the O's and all the fs have the same boundary conditions in all the boundary vectors. Using the free bosons, any massless vertex of the theory takes, in the transverse gauge, the following general Corm:
(VI.IO.l84) where
(VI.IO.185) is &10+22 component vector. In particular the gravitino vertex (VI.1O.1l7) takes the form:
where
(VI.10.187) The choice of signs being decided by the GSO projection. When acting on the bosonized vertices (VI.10.184), the fermion number operator (_1)'Y4. F can be represented as: .
(V I.10.18B)
2087
Let us explain the symbols appearing in eq. (Vl.l0.18S). Since the fermions are arranged into permanent pairs it follows that every 20+44 component boundary vector be::: can be replaced by a. reduced 10+2~ component vector bthat has a one for each pair of Ra.mond fermions and a zero for each pair of N-S fermions. The meaning of the reduced vector b is the following. The vertex operators of the sector b are characterized by momentum vectors A[b] with the following property: ~. E Z
if bi ;:; 0
{ ~:= O,,,,lO}
~i E Z + ~ if bi :;:: 1
t=1 ...,22
(V 1.10.189)
This being clarified, the action of the fermion number (VI.IO.lS8) is well defined. Using (VI.lO.IS8) in (VI.1O.182) and choosing for b the gravitino sector 1'+ we find that the choices (VI.10.168) for the £+,\ signs imply the following result: the values of ).0. ~3, ).0,).9 that survi ve the projection are
1
).0
= 2-;
>'0
= -; 2
-1
).3 "" ).6
>'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
(V 1.10.192) Since the SO(6) fermions (6.Z) are bosoruzable in all BVS. it follows that equations (VI.IO.191) and (VI.10.192) can be utilized in all vacua of our classification. From these equations we obtain also the explicit values of the U(l) and SU(2) charges in terms of compactified momenta
q = >'3 +A. +>'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' dire<:tious are , the third components of two additional SU(2)' and SU(2) groups wbich appear ill the free fermion realizations of the (4,4)6,6 superconformal algebra. Their origin is very sim.ply understood: it suffices to recall that the proje<:tion from N=4 to N==2 supersymmctry breaks SO(6) to SO(4)®SO{2). One has:
50(4) = SU(2) ® SU(2)' e (4,4)6,& SO(2) = U(l) e (2,2)3,3
(V I .10.194a) (V1.10.194b)
We shall have more to say about the accidental SU'(2) symmetry wbile discu&sing the explicit examples. Let us now illustra.te how we have analysed the spectra of all the bosonizable (2,2) vacua ILlld obtained, by means of II. computer programme, the results displayed in Table VI.IO.VII. Each yacuurn is identified by a BVS equipped with .orne choice of the Z2·va!ued tensor fij (iJ=l,... ,K). To make sure that a given vacuum is (2,2) supersymmetric we just have to check that, while looking at it as a type II string vacuum, the following conditions are fulfilled: i) Both in the "1_ and "1+ sectors the GSO projector acting on the vertex of eq. (Vl.I0.l86) selects (VI.IO.I90) as the only surviving states. ii) There are no other surviving massless gravitino sectors besides 7- ILlld 7+. Both conditions are easily and systematically checked by a computer. Once we are sure that we are dealing with a (2,2) vacuum the information we want to extract from the BVS with signs fij is the following: A) The choice of the enhancement group Gs B) The number hl •l of 27-families and their Gs quantum numbers C) The number h2•1 or 27·families and their Gs quantum numbers D) The number N(·ingln.) of E6 singlets and their Gs quantum numbers. Relying on space-time supersymmetry we can restrict our attention to bosons, namely to gauge vectors and scalars. Furthermore, utilizing right-moving world-sheet supersymmetry we can count families and anti-families just by counting the scalars that are SO(10) decuplets, namely those emitted by the vertices (VI.I0.127a) and (V1.10.128a.). Hence the full massless spectrum is determined by looking at the vertices of type (VI.l0.13l) (enhancement group), (VI.IO.127a) (faJnilies), (VI.lO.128a) (antifamilies) and (VI.lO.130) (singlets).
2089
With a little thought one concludes that the sectors where such vertices can be located corespond, in type II notation to boundary vectors hE S of the form:
b=(O,O ........,.
'I'd ........,. 0,0 'Pal
Mid
(V [.10.195)
Mink
where the 18-component vector PL has one of the following two structures:
Ih = {~::l.
(V [.10.196)
while the 18 component Pa has one of the following three structures:
PIt;:: {~::lS
(V I.10.197) 02 11& Reca.illng that all the ,. generators are chosen to be RR vectors, the sectors of type (VI.lO.195) correspond to the following combinations:
ait ....... :::;: 'Ti,
ai, ...i' ..+1
""
+ ... +1ia"
7+ + 7- +7.,
+... +';'mH
hi, ...;,. :::; 10 +1+ +1- + ai, ...'.
(V I.IO.I98a) (VI.1O.198b) (VI.10.198c)
Hence for every BVS the computer has just to check which of the vectors (VI.l0.198) fulfills the conditions (VI.10.196) and (VI.10.l97). In these sectors we look for the families, antifamilies, singlets and enhancement gauge bosons that survive the GSO projection for each possible choice of the Zz-valued tensor (ij' Such a programme is easily (although with some expense of CPU time) carried through in the case of the bosonizable systems where we can use the form (VI.I0.188) of the fermion number operator. One should note that by inserting (VI.I0.188) into (VI.I0.182) and by using modular properties of the c[~l coefficients, we obtain:
Pb =
ft ~ (1
+ 01; c[1:] he( ei""A.A{bJ)
(V I.10.199)
so that, sector by sector, the surviving states are the solutions of a set of linear equa.tions on the Z2 field:
'YA.,\[bJ
+~
[!] =
0 mod 2
(VI.IO.200)
having defined:
(V 1.10.201)
2090
or
Carrying through a systematic computer scanning the bosouinble (2,2) vacua we have obtained the results displqed in Table VUO.VII In appendix A,B we discuss in detail two explicit examples showing the Gs quantum nUIllbers of a.Il the massless states. An example of non bosonizable yet LRP vacuum is worked out in appendix C. In the same appendices we make remarks on the eft'eetive lagrangian of the considered models.
2091
APPENDIX VI.10.A
A bosonizable (2,2) vacuum or type A: Al(l,2,3,4)E13
From Table VI.lO.III we read off the boundary vectors whose transcription in bosonized form is given in Table VI.IO.VIII. There are 15 mazsless sectors of the type described by eqs. (VI.10.194.195). They are the following ones:
(VI.IO.AI) Before analysing their content it is convenient to discuss the enhancement group which is: GE
=U(lt ® SO(4}
(VI.IO.A2)
where the first two U(I) currents are common to all (2,2) vacua and are given by:
Pl)(Z) = iv'2 8.T'(z) ;:: i (8l cp3 - 8i cp6) ifa)(i} "" iv'6
(:a
8.1'(':) -
fs
=i (8.cp3 +8zcp' - 28 cp9) 1
(VI.I0.A3/l)
8i U(Z») (V I.IO.A3b)
while the remaining four U(l) currents are specinc to the model under consideration and can be labeled as follows:
,-
,i
(VI.I0.A4)
,j<M(z) ;:: i 8zcps(z). Finally the two Cartan subalgebra currents of the SO( 4) group are identified with: jC.l)(f) "" i 81 cp!(i)
(VI.10.ASa)
j(~)(.i) ;:: i
(VI.IO.A5b)
8j cpT(i)
The reason of these choices is easily understood by looking at Table VI.IO. VIII. The columns i,7 are the only pair of identical columns in the right sector. This means that the four fer1nions corresponding to cpl, cpT have identical boundary conditions
2092
in all boundary vectors and span an unbroken SOC4) cU1'l'el1t algebra which is the maximal non abelian factor in OB. Relying on these conventions every primary conformallield of the (2,2)9,9 theory can be labeled with the quantum numbers that specify its transformation properties under the Gs group:
91 =lS - X' 92 =X3 + X' - aX· til =X' ; ~ =X· ; Y3 =: X5 ; Y4 "" X8 WI
(VI.10.A6)
=Xl ; 102 ... X7
{~ti2'Yh"'U4} being the six U(l) charges and {1OltWa} the two component of SOme weight vector in the 50(4) weight lattice. Obviously the massless and massive states will be arranged into multiplets, each multiplet containing all the weights of an irreducible 0 B representation. We note that a current algebra G"',t similar to that generating OBis contributed by the left sector. olelt is not a. gauge symmetry of the effective lagrangian since the corresponding gauge bosons are massive, however it is an exact global symmetry. Hence it is very convenient to classify the states according to irreducible representations of
cr.i!:. 1 ® OB. In the model under consideration we ha.ve:
(VI.IO.A7) which is isomorphic to Gs. This is a frequent but not necessary occurrence. Once more two of the six U(l) currents are common to all (2,2) vacua and are left moving analogues of the curents (VI.I0.A3a-A.3b). The remaining four U(1) currents and the SO(4) currents are specific to the model. They are completely identiii.ed it we write the left moving analogues of eqs. (VUO.A6), namely if we identify the global U(l) charges and global SO(4) weights in terms of the left moving compact momenta. 'We have: ql =)..3 -
'A'
q, = 'A3 + .\6 +.\S
=A4 ; 1/3 "" 'AT; y, =),.2 TVI = 'AS ; 101 =).,8 111 = AJ
;
(VI.I0.AS)
1/2
Utilizing these notations we can now present the spectrum of Wess·Zumino multiplets appearing in the effective theory of this model. They are given in Table VI.10.VIII. As the reader can see we have explicitly displayed the Os quantum numbers of the multiplets only in a couple of seetors, just to show what they look liket listing £Or all the other sectors only the total number of families t anti-families and singlets.
2093 This was done only for reasons of space and in order to make the table readable.
If we denote by (J!,J2,X) an irreducible representation of SU(2)@SU(2)0E&, the full spectrum of WZ-multiplets is the following:
WZ ::;:5 (0,0,21) Ea 17 (0,0,27) Ea 90 (0,0,,61)Ea
20
G,~,l) Ea17
G,O,l) Ea11 (O,~,l)
(VI.IO.A9)
Therefore the complete Kabler manifold associated with the effective supergravity lagrangian of this model is:
SU(l,l)
MC"ml'le!. ::::
~0
Mma.tte ..
(VI.lO.AlO)
where Mrn"'tt ... is a Kahler manifold of complex dimension:
dimMmu.tt ... = 832
(VI.lO.All)
The manifold M",..tter contains the moduli, the charged fields and also the Higgs fields of the enhancement gauge group. Clearly our fermionic construction corresponds to a very specific point in the moduli space of a more general (2,2)9,9 theory characterized by hl,l 5 • h2 ,1 :::: 17. Yet if we are able to determine the Kabler metric of Mma.tteT by using the special features of the fermionic construction, then our result will provide the complete answer for the effective lagrangian in any point of the (2, 2h.9 moduli space, under the only condition that the point we consider is arcwise connected to the fermionizable point. Indeed it suffices to shift the moduli and, when necessary, to integrate on the singlets that are eaten by the GE gauge bosons. Now the relevant point is that, in the case of the fermionic constructions much more is known about the effective lagrangian than for other constructions. This is due to the possihility of regarding the N=l theory as a sequential GSO projection
=
N=4->N=2-N=1. The fields that are truncations of N=4 multipleh have a known metric. For the twisted multiplets appearing in the first projection a method to determine their metric was found in ref. [12j. It is only for the twisted states appearing in the last projection that a general method to derive the corresponding Kahler metric is not known at the present.
APPENDIX VI.lO.B
A bosonizable (2,2) vacuum of type B: B5(l,2,3,4)en
The reason why systems as the one we study here are interesting is that their effective N=l supergravity can be regarded as & pure truncation of an N=2 supergravity. This happens because the boundary vector that makes the final projection from N=2 to N==l space time supersymmetry does not introduce new massless sectors into the spectrum. This makes the method of reference 112] applicable, so that, for these (2,2) va.cua. the complete e1fective theory can, in principle, be determined. Let us then study, from this specific point of view, the vacuum B5(I,2,3,4)E23' A5 final N=1 projector we utilize the boundary vector "Yl , which has the aforementioned property of introducing only massive sectors. Hence, in order to understand the structure ofthe (2,2) va.cuum we consider its N=2 parent B5(2,3,4}f23 that leads to a (4,4)6.& ED (2,2)3.3 superconformal theory. The maximal system we consider is bosonizable, but, as we explained in the main text, the permanent pairs are not the same as in the type A vacua. Numbering the fermion! as in eq. (VI.IO.160a.), we can associa.te the 9+9 free bosons to the following pairs of fermions: 'PI +-+ (15,16) 'P2 ..... (2,8) 'P3 +-+ (5,11) 'P4 +-+ (1,7) 'Ps ..... (13,14) 'Pi +-+ (6,12) 'P1 +-+ (a, 9) 'P8 H (4,10) 'PI +-+ (17,18)
;1 H (is, i6) rF2 +-+ (ia, i'4) ;3 +-+ (5, il) ;4 +-+ (i,7) ;5 ..... (i,s) ;IS foot (6, f2)
;1 (3,9) foot
; ...... (4,10) ;, .... (i7,18)
(VI.10.BI)
After this, the boundary vectors can be transcribed in a bosonized form, exa.ctly as we did for the system Al in Table VI.IO.IX. The analogue table for B5 is not given, the reader can retrieve it with some work when necesa&ry. In both B5(1,2,3,4) and B5(2,3,4) the only candidate massless &ectors for families, anti-!a.milies and singlets are the following ODes: (VI.I0.B2)
This shows that the massless &ectors of the N=l model is indeed an N=2 truncation as we claimed.
2095
In order to appreciate in full the structure of the theory it is convenient to go back to the parent N=4 model obtained by removing not oruY71 but also ')'2. In this case the only massless sectors that remain.are: , (VI.lO.B3) As we remarked earlier the heterotic model baaed on the universal system U c:: {-ra.7+.7-} is an N=4 theory with gauge group: G(Ul N=4
=SO(12) @ Es @ &'
(VI.IO.B4)
where the Cman subalgebra. of the enhancement group GB ;;:: SO(12) is given by the currents fJijJi with i::::l,2,4,5,7,8. If we introduce the generators 73 and 14, the group 80(12) breaks into SO(S)@SO(4) since the set of 12 fermions with identital boundary conditions splits in two subsets of 8 and 4 fermions respectively. For a similar reason the SO(6) group needed to promote 50(10) to & also splits into SO(4) @ SO(2). Schematically we can write:
{V I.lO.BSa}
(VI.10.BSb) Then without the twisted states introduced by the b3 -sedor we would have the gauge group:
where Er is constructed in the usual way out of the following groups:
(VI.l0.B7) Note that under the symbol denoting each group we have written the free bosons that realize iu Carlan subalgebra. The actual gauge group, however, is not only (VI.lO.B6) since the ba- sedor contributes additional gauge bosons in the following representations:
( J 1 = !2' J111
""!2'
8.)
(VI.IO.B84) (VI.IO.B8h)
where 8. is the vector representation of SO{S) and 56 the fundamental of ET. The sequencial breaking of the gauge group paralleling the sequencial projections N =4 ..... N ... 2 ..... N = 1 is summarized in Table VI.I0.X.
2096
We can now cfiscuss the massless speclrum o! the N=2 vacuum distinguishing between twisted and untwisted states. Following ref. [121 we call untwisted the states that were already present in the N=4 theory, while we name twisted the new massless states that r.ppear after the projection. In the N=4 case the whole massless matter is given by the vector multiplets of the gauge group SO(12) ® Ea ® Ea'. The corresponding scalars span the manifold:
M N... =
SO{6,562~
SO(6) x SO(562)
(VI.lO.BIl)
where 562 ... 66
+248 +248
In the N=2 case we have both vector multiplet! and hypermultiplets. The vector multiplets scalars span a special Kahler manifold {211 while the hypermultiplet scalars span a quaternionic manifold. The full scalar manifold of the N=2 theory is given by: M N=2 = SU(1,1) ® SO(2,418) Q U(l) SO(2) x SO{418) ®
(VI.I0.B12)
where the first factor corresponds to the dilaton vector multiplet, the second to the gauge vector multiplets of SU(2)3 ® SO(8) ® ET ®Ea: 418
=3 + 3 + 3 + 28 + 133 + 248
(V I.10.B13)
and Qis the quaternionic manifold spanned by the hypermultiplets. Q must contr.in, the following submanifold:
SO(4,144) SO(4) x SO(I44) C Q
(V I.10.B14)
that corresponds to the untwisted hypermultiplets. Indeed from the decompositions (VI.10.B9) we conclude that in the projection N = 4 ..... N ... 2 the coset (VI.I0.Bll) disintegrates as follows:
SO(6,562) SO(2,418) SO(4,144) SO(6) X SO(562) -- SO(2) x SO(418) ® SO(4) x SO(144)
(V 1.1O.BIS)
the 144 untwisted hypermultiplets being in the (4,8) of 50(4) 0S0(8) and in the (2,56) of SU(2) ® E1• In addition we obviously have the twisted hypermultiplets whose number and charge assignements can not be predicted a priori. They are located in the ~3 and bu sectors. The full spectrum can be found by explicit calculation and is summarized in Table VI.10'xI. Let us stress that in calculating the spectrum it is essential to fix £33 -1 in order for the theory to be a true (4,4)6,6 ED (2,2)3,3" We conclude that the manifold Qhas quaternionic dimensions:
=
2097
(V I.I0.B16) and following 1121 we should be able to work out its metric. If we obtain such a result, the Kahler metric of the N=l model resulting Crom the 1'1 GSO projection is easily obtained by truncation. The spectrum of the WZ multiplets surviving the last N = 2 --+ N =1 projection is displayed in Table VI.10.XII. Note that we have not written the left moving charges hut only the charge assignements under the gauge group. As we see the N==l model is non chiral OX::: 0). This seems to be a general property of the B-type systems.
APPENDIX VI.10.C
An LRP (2,2) vacuum of type B: B26(l,2,3)
The vacuum studied in this appendix has the same property as the vacuum of the previous appendix, namely the massless sector is a pure N=2 truncation. However, from another viewpoint this vacuum provides an example of a left-right paired system that can not he completely bosonized. This fact implies, in particular, that the rank of the enhancement group is strictly smaller than eight; actually we find: 'fankGs =4
(VI.IO.Cl)
To treat vacua of this type at all mass levels, we cannot use just free bosons but we have to introduce also the conformal fields of the critical Ising model. However,
since at zero massless level the theory is a truncation of an N=2 theory that turns out to be bosonizable, it follows that the particular combinations of Ising fields that appear in the massless vertices can also be rewritten as Cree boson exponentials. The peculiar thing is that the a.belian currents 8!tpi(I) corresponding to these free bosons are projected out from the gauge sector of the theory. This is precisely the way how rankGs is reduced. If we delete the '11 genera.tor. the system B26(2,3) correponds to a (4. 4}&,. Gl (2.2)3,3 theory and it is fully bosonizable. In particular the free bosons can be associated to permanent pairs in the foUowing way: rp3,tp8.rp.,fh,fP"~9 are defined as in equation (VI.10.Bl). while all the others are defined as in equation (VI.10.177).
2098
The candidate massless sectors £or 1amilles, anti·1amilles and singlets are in both and B26(2,3), and are the following ones:
B26(1,2~)
{O},42, ba, 423
(VI.I0.C2)
If we remove also the 12 vector the resulting system B26(3) is an N",,4 theory, as in the previous example, however with a smaller gauge group than that corresponding to the universal system (see eq. (VI.lO.BI». Indeed the 1a-vec:tor makes a projeetion on the right moving supersynunetry, reducing the theory to 11.(6,4) one. Consequently the gauge group of the N=4 parent theory is in this ease given by:
(VI.lO.C3) When we introduce back "YlI we step down to N=2 and the group (VI.10.C3) is broken down to the following subgroup: GN =4 ::> GN =2 =SO(2)3 @SO(6) ® SU(2) ® & ® Ea'
(VI.IO.C4)
In the untwisted sector therefore we expect the following scalar manifold disintegra.tion: SO(6,418) SO(6) x SO(418)
-+
SO(2, (02) SO(4,16) S0(2) x SO(402} ® SO(4} x SO(16)
(VI.lO.C5)
the first factor in the l.h.s. of (VI.10.C5) being the scalar manifold associated with the vector mulliplets ~d the second factor being the quaternionic manifold of the untwi8ted hypermultiplets. These latter tall in the representations (6,2) of SO(6) ® SO(2) and (2,2) of SO(2) ® SO(2). In full analogy with eq. (VI.lO.BI2) the scalar manifold of the N=2 theory associated with the system under consideration is: M N =2 :: SU(l,l} ® SO(2,402) ®Q U(l) SO(2) X SO(402)
(VI.I0.C6)
where the quatemionic manifold Q has quatemionie dimension: dim('llUOtcmicmic:}Q:;;:
80
(VI.IO.C1)
and contains as a submanifold the tmtwisted hyperDlUltiplets coset SO(4,16} SO(4) x SO(16) the remaining 64 dimensions correspond to the twisted hypermultiplets. whoseeharge assignements are displa.yed in Table VI.I0.xm. Let us now explain what happens in the last projection fronl N=2 to N=l supersymmetry. Making reference to our previous rule for naming the free bosons and the
2099 ordering (VI.IO.160a) of the fermions, the permanent pairs tha.t survive the introduction of '11 into the BVS are those corresponding to the free bosons 1(>3, 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)+ ••
E;:(z,i)E::(z,i)t~(Z.i)E~(z,z)X~(i)
(V I.IO.CI5)
with
V(k, z.E} =eilt. X (r,l)eW ·{I)e(i/2)('I"(')+'I'e(rj±
(V I.IO.CI6)
2101
±l for 4 and 4 ofSO(6) respectively. Comparing (VI.10.C15) and (VI.lO.C16) we get an identification of the product of two Ising model twist fields with suitable free boson exponentials:
E3 '"
E~(.t,z)Ei(zl z)
"" e(i/2)[-"'3(.c):i:~s(£)1
l:~(ZIZ)l:~(Ztz) = e('/2)(Y>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<8)
16'li'J -
V'li"J -.
{is ®'8 ®i\,
(0300\ 0<3, ea ®el 0'3)
ea®ea®el)
(z.tz.t 0 (f.)
T'
Z';iZ;
(2,2)
= N ",2
2107 TABLE
v1.10.ll1
101... 1"01 ,y.t .... of boundory veotors Al
A3 AS 117 A9 All AI3 AIS A17 AIS A21 A23 81 83 85 87
sa B1I
813 SIS 817 819 821 B23 !l25 827 629 831 833 835 837 839 641 843 845 847 849 BSI aS3 855 857
aSg 861 863 865 867 869 871 973 875
em CI C3
e5 C7
C9
683030 663663 663635 663366 663300 663636 663366 663663 663030 663333 663030 663300 116416 116426 116115 116115 118125 116126 116145 116126 116115 116145 116126 t 162 I 6 116226 116216 t16115 116115 116145 116115 116145 116416 116115 116115 116126 116115 116216 116126 116425 116445 116445 116425 116445 11 6446 116426 116445 I t 6425 116226 11 62 16 116215 116225 116116 116116 116116 116116 116116
636030 636333 636355 636536 836030 636030 636368 636333 636030 636553 636030 636030 553300 553300 5!;3663 553663 553553 553635 553663 553635 553653 553663 553635 553300 553300 553300 553553 553553 553563 553553 553563 553300 553663 553663 553635 553653 553300 553$35 553300 553300 553300 553300 553300 553300 553300 553333 553333 553300 553:).00 553300 553300 454542 454M2 454641 454$41 454$41
000606 000506 033065 033560 033660 033660 000605 000660 000033 000330 000303 033660 000065 033650 000550 000660 033033 000560 000330 000605 000650 033033 033650 000055 033550 033550 000550 000660 033033 033033 033660 033660 000550 033650 0330:>5 033660 033560 033055 000033 000033 000033 000033 000303 000330 000330 000330 000303 000330 000055 000303 000303 000330 000330 000033 000330 000330
0336!;O 303303 303033 303506 303606 303605 033660 303066 033056 330033 055330 303530 033605 303560 330000 330000 303303 330000 330000 033055 330000 303303 303330 033505 303660 215476 330660 330000 303103 303303 303550 303550 000330 303560 303505 303550 303650 303605 000303 000303 330303 330303 330033 330000 330000 :),30033 330033 330000 330000 330000 330033 033000 330000 303000 033000 330000
'
226416 330000 226115 226126 226115 22&216 226126 330000 330033
112 A4 A6 A8 Al0 A12 A14 A16 AlB 1120 1122
663030 663663 663635 663366 663300 663636 663366 663663 663030 653333 663030
636030 63633J 6363S5 636536 636030 638030 636366 636333 636030 636553 636030
000606 000506 033065 033560 033660 033660 00060~
000660 000033 000330 000303
033650 303303 303033 303506 303608 303805 033660 303066 033056 330033 055330
112 11 64 16 553300 000065 033605 84 116426 5~3300 033650 303560 t 16115 553663 000550 330000 86 116115 553663 000660 330000 88 810 116125 5536$3 033033 303303 812 116126 553635 000560 330000 814 116145 553663 000330 330000 516 116126 55363~ 000605 OU055 B18 116115 553653 000650 330000 820 116145 553663 033033 303303 822 116126 553635 033650 303330 824 116216 551300 000055 033505 B2& 116226 553300 033550 303660 B28 116216 553300033550 215476 B30 116115 553553 000550 330660 832 116115 553553 000680 330000 834 116145 553583 033033 303303 835 116115 553553 033033 303303 8lB 116145 553583 033660 l03550 840 116416 553300 033660 303550 842 116115 553663 000550 000330 844 11611 5 ~53663 033650 303560 846 1\$126 553635 033055 303505 648 116115 553653 033660 303550 850 116216 553300 033560 303650 852 116126 553535 033055 303605 654 116425 553300 000033 000303 aS6 116445 553300 000033 000303 B58 1\ 6445 553300 000033 330303 860 116425 553300 000033 330303 862 116445 553300 000303 330033 664 116445 $53300 000330 330000 S66 116426 553300 000330 330000 868 116445 553333 000330 330033 870 116425 553333 000303 330033 812 116226 553300 000330 330000 874 116216 553300 000055 330000 676 116215 553300 000303 330000 878 116225 553300 000303 330033 116116 454542 000330 033000 (;2 116116 454542 000330 330000 C4 116116 454641 000033 303000 C6 116116 454641 000330 033000 (;8 CIO 116116 454641 000330 330000
226418 330000 22611 11 226128 226115 226218 226126 330000
2108 CII 116116 454641 000303 303000 CIl 116116 454641 033303 115125 CIS CI7 C19 C21 C23 C2S C27 029 e31 C33 C35 CJ7 C39 C41 C43 045 C41
116116 454641 033033 303330 118118 454641 033330 303033 116116 454542 033303 2152211 116116 4$4542 033033 303303 116116 454542 033330 303303 116118 454764 000033 303000 11111111 454764 0~3303 303033 116116 454784 033033 303303 116116 454784 033330 303303 116146 454464 000330 033000 116148 454464 000330 330000 116116 .54454 000605 303000 116115 454461 000303 033000 116115 454461 000303 J30000 116118 454464 000330 303000 116116 454464 000055 033000 116118 464464 000055 330000 049 116118 454464 000605 303000 CSI 116116 464464 033055 215248 C33 USU6 454484 033055 215746 CSS 116116 454484 033605 303055 C57 116116 454464 033550 303055 CS9 116125 464451 033303 303033 C61 116125 454451 033330 303033 cn 116116 454461 000650 033000 065 118116 454461 000650 .330000 C67 116125 454452 033033 215726 C69 116126 454461 033605 303660 C71 116116 454461 033550 303805 C7J 116116 454464 000055 303000 C75 11614$ 454461 033033 215246 en 118145 454461 033033 215746 C19 118115 454451 033033 215725 C81 116116 454454 033605 303065 C8J 116146 454464 033605 303650 CBS 116116454454 033660 303065 C87 116146 454464 033650 303055 cot 1I6t45 454461 033303 lOJ033 C91 116145 454461 033330 303033 C9J 116116 454454 000660 033000 C9S 116116 454454 000660 330000 C97 116146 454464 000650 303000 C 99 11Gt 15 454452 000303 033000 454452 000303 330000 Cl0l 1161 CI03 116146 454461 000330 303000 elos 116116 454451 000330 033000 CI07 116116 454461 000330 330000 CIOS 116116 45445 I 000660 J03OO0 ClIl 116145454462 033033 215246 C113 116145 454462 033033 215746 C115 116115 464452 033033 215225 ClI7 116115 454452 033033 215725 CI19 116116 454451 033605 303065 C121 116118454451 033605215126 0123 11611S 454451 033860 J03065 CI25 116145 4$4461 033650 3030SS CI27 116145 454462 033303 303033 et29 116115 454452 033303 215225
U'
CI2 CI4 CI$ CUI
C20 C2a 024
C26
C28 C30 C32
C34 C36 C38
C40 C42 C44 046 048
CSO CS2 CS4 CS8 C5I C60 062 C64 C66 C68 010 012 C74
C16 C78 C60 C62 CII4
cat C88
C90 C92 C94 C9S C90 Cloo CI02 C104 CI06 CIOB C110 CI12 CI14 C1l6 CII8 CI20 CI22 C124 C126 e128 C130
116116 454641 118116 454641 11611$ 454641 116118 454641 11611& 454542 116116 454542 116116 454542 1181111 454784 118118 454164 116116 454784 t 16116 464164 116146 454464 116148 464464 11&116 454454 1 UIII5 464461 116115 454461 116126 454454 tl611' 464464 11' 116 464484 116116 454464 116118 464464 116116 454464 118116 454464 118118 454464 116125 454451 116125 464451 116116 454461 11611S 454461 116125 454452 U6126 454451 116116 454461 116116 454454 118145 454461 116145 454461 116115 454451 118116 464454 116148 464464 116116 464454 116146 454464 116145 454461 116145 454461 116116 454454 118116 454454 116146 454464 USIIS 454452 116115 454452 116148 454461 U6116 454451 116116 454451 11&116 454461 116145 454462 118145 454462 116115454452 116115 454452 116118 454451 118116 45445\ H611e 454461 116146 454461 116145 454462 116115 4544S2
000303 303000 033303 215125 033033 303330 033330 303033 033303 21522& 033033 303303 033330 303303 000033 303000 033303 303033 033033 303303 033330 303303 000330 033000 000330 330000 Q00605 .103000 000303 033000 000303 330000 000330 303000 Coo05S 033000 00005$ 330000 000605 .103000 03305S 215246 033055 215746 033605 303055 03J650 303055 033303 303033 033330 303033 000650 033000 000650 330000 033033 21 S726 033605 303660 033650 303605 000065 303000 03303.3 21$246 033033 215746 033033 21$725 033605 303085 033605 303650 033660 303065 033650 303055 033303 303033 033330 303033 000660 033000 000660 ~3OOO0 000650 303000 000303 033000 000303 J30000 000330 303000 000330 033000 000330 330000 000660 303000 033033 215246 033033 215746 033033 215225 033033 215725 033605 303065 033605 215126 033660 303065 033650 30305S 033303 303033 033303 215225
2109 C131 C133 C135 Cll7 C139 C141 CI43 CI4S
cal C149 C1S1 CI53 Cll15 C1S7 CI59 em C163 CIS5 CIS7 C,I69 cm C173 C175 CI77 C179 CISI CI83 CI85 CIS? CI89 C191 CI93 CI95 CI97 C199 C201 C203 C205 C207 C209 C21l C213 C215 e217 C219 C221 C223 C2lS C227 e229 C231 C233 C%l5 C237 C239 C241 C243 C245 C247 C249
11&145 454462 033330 303033 118115 454452 033330 303033 116115 454672 000303 033000 116115 454672 000303 330000 116116 454574 000605 033000 116116 454574 00060l! 330000 118125 4a4571 033330303303 116116 454641 000033 033033 033330 116116 454641 000330 303330 330303 116116 ~S4S42 000033 303033 330303 116116 454542 000033 033033 303303 116116 454542 000033 330033 330330 116118 454542 033303 303033 226116 116116 454641 000330 125226 215226 116118 454641 000033 125125 21!j125 11S11& 454764 000033 033033 303303 1161\ 6 454764 000033 303033 330303 116116 454757 033303 303033 303330 116116 454751 033303 303330 215116 116116 454757 033033 303330 2261\6 116125 454451 000303 303303 330033 116116 454464 000055 0330$5 033605 116125 454451 000033303033 330303 116116 454464 000650 303650 330055 116116 454461 000650 303650 330055 116145 454461 000303 303303 330033 116116 454454 000065 033065 033605 116145 454461 000033 303033 330303 116146 454464 000605 30360$ 3300~5 116145 454462 000303 303303 330033 116145 454462 000330 303330 330033 116116 454451 000660 30:.1660 330065 116115 454452 000303 033303 303033 116145 454462 000303 303303 303330 116145 454462 000303 330303 330330 116145 454462 000330 033330 303033 116115 454452 000330 033330 303033 116116 454451 000065 033065 303605 116116 454451 000065125116215116 116116 454451 000065 330065 330505 116115 454452 000033 033033 303303 116145 454461 000055 303055 303605 116116 454451 000605 033605 303065 116116 454451 000605 125116 215116 116115 454451 000660 125116 215116 116146 454461 000650 125146 215146 116116 454451 033065 303660 226116 116145 ~54462 033033 303330 226145 115145 454461 033055 303650 226146 116115 454452033033 303330 226115 116146 454461 03360S 303650 225146 116115 454452 033303 303330 226115 116145 454451 000330033330 303033 116146 454454 000065 033065 303605 116126 454464 000650 125126 215126 116146 454454 000660 125146 215148 116145 454451 033033 303303 226H5 116146 454454 033065 303S05 226146 116116 454264 000550 033550 303055 116116 454264 000605 033605 303055
CI32 C134 CIlS CllS CI40 C142 C144 C146 CI48 elS0 CIS2 Cl54 C156 C158 C1S0 C162 Cli4 C166 CI6S CI70 CI72 C174 C176 el7S C1S0 C1S2 CIS4 CIS6 et88 CI9a C192 CI94 CI96 C198 C200 C202 C204 C205 C20S ClIO C212 C214 C216 C218 C220 C222 Cl24 C2l6 C22S C230 C2l2 C234 e236 C2la C240 C242 C244 C246 C24S e250
116145 4S4462 033330 303033 116115 454452 033330 303033 116115 454612 (0031)3 033000 118115 451612 000303 330000 116116 454574 00060~ 033000 116116 454574 00060S 330000 116125 454511 0333~0 303303 116116 454641 000033 033033 033330 116116 454641 000330 303330 330303 116116 454542 000033 303033 330303 116116 454542 0000~3 033033 303303 116116 454542 000033 330033 330330 118116 454542 033303 303033 226116 116116 454641 0003JO 125226 215226 116116 454641 000033 125125 215125 116116 454764000033033033303303 116116 454764 000033 303033 330303 118116 454757 033303 303033 303330 116116 454757 033303 303330 215116 116116 454757 033033 303330 226116 116125 454451 000303 303303 330033 116116 454464 000055 033055 033605 116125 454451 000033 303033 330303 116116 454464 ooono 303650 330055 116116 45446 I 000650 303650 330055 116145 454451 000303 303303 330033 116116 454454 OOOO~5 033065 033605 116145 454461 000033 303033 330303 116146 454464 000605 303605 330055 116145 454462 Oa03()3 303303 330033 116145 454462 000330 303330 330033 116116454451000660303660330065 116115 454452 000303 033303 303033 It 6145 454462 000303 303303 303330 116145 454462 000303 330303 330330 116145 454462 000330 033330 303033 1 HIllS 454452 000330 033330 303033 116116 454451 000065 033065 303605 116116 454451 000065 125116215116 116116 454451 000065 330055 33060S 116115 454452 000033 033033 303303 116146 454451 0000$5 303055 303605 116116 454451 00061)5 033605 303055 116116 454451 00060S 125116215116 116116 454451 000660 125116215116 116146 454461 0006S0 125146 215146 116116 454451 ()3306S 303660 226116 116145 454462 033033 303330 226145 116146 454461 0330$5 303650 226146 116115 454452 033033 303330 226115 116146 4S4461 033605 303650 226146 116115 454452 033303 303330 226115 116145 454451 000330 033330 303033 116146 454454 000065 033065 303605 116126 454464 OOOS5O 125126 215126 116146 454454 000660 125146 215146 116145 454451 033033 303303 225145 116146 454454 033065 303605 226146 116116 454264 0005S0 033550 303055 116116 454264 000605 033605 303055
2110 d51 C253 C25S C257 C25t C261 C263 C265 C267 C26& C271 C27l C275 C277 C279 C2I1 C263 C265
116116 454264 ooo50S omos J030SS 116116 454264 0:53055 J03&50 226118 116116 454264 0J3055 303550 226118 116115 454872 000303 125216 215216
\\6\16 454871 ~'S055 ~03550 116116 454611 033505 J030S5 1111116 454671 ~ 303550 116116 454671 ~ 303505 116116 454574 0330S5 303&05 "il16 454574 0330" 3Ol6$O 116125 454571 033033 303303 116125 454571 033033 303310 1111116 454514 033&05 J03055 116116 454574 033&50 303055 H&I25 454571 033303 303330 11611& 454571 033055303605 11&116 454571 033055 J03&5O 116116454571 033&05303&50
21511' 215116 22611& 215118 215118 215116
lOJ330 215125 215116
303605 226125 303650 tl5H6 226116
d52 116116 454264 OOOSOS 033S0S 303055
d54 lIS11' 454264 C256 11&118 454264 C258 118115 454672 C2$O 116118 45461t C262 11811& 454671 C264 116116 454671 C266 116116 454871 C266 118116 454574
O3J055
~5O U~l1l\\
03J055 303550 226116 000303 033055 03350S 033505 033550
125216 215216
303550 215116 303055 215116 ~ 2261111 303505 215118 0330SS J03&OS 2151111 C270 116118 454574 033055 303&50 215116 C272 116125 454571 03J033 303303 303330
C274 C216 C278 C260
H4I2S 454571 116116 454S74 11611& 454574 118125 454571 d82 11&116454571 C264 H61\6 4S457\ C286 116116454571
033033 3033JO 215125
033605 J03OS5 lU511S 033&50 303055 303805 033303 303330 226125 0330S5 3Ol605 303650 \}331)5$ 303650 21511& 03360S 303650 226116
2111 Table VI.1o.IV
SU(2) 111m paliuJIS ia Ramoltd .sector code 0
1 2 3
X' 1
A~
0 1 0
1
4
1
S 6
0 1 0
'1
1 0 0 0
0 1 1
)0'
Xl
Jo1
1
1
1
0 1 0 0 1 0 1
1 1 1 1 0 0 0
1 1 0 0 1 1 .0
0
0
'" 0
0 1 1 0 0
Table VLIO.V
MaJdmaI s,rstems of bOllndarr veelon T;ype
N_berof
(2,0)
(2,2)
(2,4)
(2,6) 1
maxI~ 8)'Stems
A
23
15
5
B
18
62
26
C
286
286
2
Table VUO.VI
(2,2) s,rsems Dr bolUld8l1 YfClors
Type
#-
Bosonizable systems m + "N:2 Jnc.~&i....
A
23
23+0
B
106
1+31
C
661
m +"N==2:
LRP + 'PN:4 &1'11'"
LRU
C
1+33+0
40
0+91+118
342
2112 TAe~E
Vl.l0.VlI
~IST
OF BOSOIHZABLE (2.2) COMPACTlFlCATIONS OF THE HETEROTIC SIJPERSTRINO ON SU{2)"3 GROIJPI'OLDS
IIG
NF NA END
0 0 0 0 0 0 0 0 0 0
0
0 0 0 0 0 0 0 0 0 0 0 4
1 3 3 3 3
,
68 3 68 3 76 3100 3196 5 5 611 S 68 5 5 76 5 5100 5 5 108 7 7 68 7 7 76 7 7 92 7 7 100 7 7 108 9 9 196
11 It 68 II II 100 II 11 196 11 II 204 1313196 IS 15 196 15 IS 220 19 19 195 I 5 68 I
5
4
I
4
:5 3 3
5100 7 16 7100 7 loa 9 100 9 108 ~ 76 9 68 9 68 9100 9 108
4 4
4 8
8 8 8 8 8
S S I
I
76
8 8
1 3 t1 68 :5 11 76 :5 11 100 3 II 100
a
3 11 108
a
8
a 8
12 12 12 12 16 16
5 13 100 7 15 76 7 15 100 715108 I 13 100 1 13 108 :1 IS 196 5 112041 17 100 I 17 108
GAUGE GROUP G(E)
r.IODEL
SICNS
$0(2)"8 $0(2)"8 50(2)"6 SO(2)"S SO(l)"S SO(2)"a 50(2)"$ SO(2»'6 SO(2)"a SO(2)"& 50(2)"8 SO(2)"& 50(2)"5 $0(2)"8 SO(2)"6 SO{Z)"S 50(2)"8 50(2)"8 50(2)"8 SO(2)"6 $0(2)"'8 50(2)"8 50(2)"5 SO(2}"8 5O(2}"a 50(2)"6 50(2)"8 50(2)"6 50(2)"8 50(2)"6 50(2)"8 SO(2}"S 50(2)"6 50(Z)"(; 50(2)"8 SO(2) "8 SO(2) "6 50(2)"8 SO(2)"6 SO(2)"6 SO(2)"8 50(2)"6 SO{2)"S 50(2)"6 SO(2)"8 $0(2)"6 50(2)"8 50(2)"6 SO{2)"S SO{2)"& 50(2)"8 SO(2)"6
51\(1234) 4A(l234) 111(1234) 411(1234) 811(1234) 511(1234) lA{I234) 311(1234) 4A(1234) IA{U34) 8A(I234) 3A(1234) 2A(IZ3} SA(1234) 3A(1234) 4A(1234) SA(1234} 811(1234) 811(1234) 3A(1234) 5A(1234) 8A(1234) 211(123) 8A(1234) 4A(1234) IA(12J4) 4A(1234) tA(I2J4) 4A(1234) lA(1234} 5A(12J4) lA(l234) lA(1234) lA(1234) 5A(1234) 4A(1234) IA(1234) 8A(1234) 3A(1234) IA(12l4) 8A{1234} lA(1234) 511(1234) 3A(1234) 8A(1234} 3A(1234) 5A(1234) IA(12l4} 4A(1234) lA(12l4) 5A(1234} IA(I234)
11 22 13 22 14 22 13 23 33 24 22 23 23 24 23
.50(4)
x 50(4) x SOC 4) x 50(4) • 50(4) • $0(6) • 50(4)
• 50(4)
• 50(6)
x SO(4} • 50(4)
x SO(4) • 50(4) x SO(4} • 50(4)
• SOC 4)
x 50(4) • 50(4) x 50(4)
• 50(4) • SO(4) x SOC 4)
• 50(4) • 50(4)
24 22 3J 23 24 22 33 44 33 24 24 33 33 44
23 24 33 44 24 23 44
34 33 34
22 24 24 33 14 24 23 14 24 34 23 24 22 -23(24) 23 33 '34 22 'Z4(23) 33 44 13 22 (34) '33(44) 23 '34 44 '23(14) 44 11 22 23 '34 44 22 24 33 (34) U 22 23 24 '34 -14 22 'Z3 '23{24) 44 13 23 24 33 '44 '14 22 '23 34 '14 23 24 l3 34 13 23 24 "J4 44 14 '2l{24) '34 44 22 33 '34 '14(23} 33 II "14 23 24 34 24 '33 44 '14 23 24 3444 '14 '23 44 13 22 44 (34) '23(24} 23 '34 "4 '23 33 II 22 24 33 (34)
2113 16
16 16 16 24 24
24 24 24 48
1 17 132 3 19 68 J 19 100 3 19 loe 1 25196 3 27 100 3 27 196 :$ 27 228 7 31 204 351 195
50(2)"6 SO(2)"8 SO(2}"S SO(2)"6 50(2) "s SO(2)"8 So(a}"s SO(2}"6 SO(2) "6 50(2)"8
x 50(4)
• SO{4)
" 50(4) x SOC 4)
lA(1234) 8A(1234) 8A(1234) 3A(1234) SA( 1234) 8A( 1234) BA(IZ34} IA(1234) 3A(1234) SA,(12l4}
13 22 23 '34 44 '13 '14 22 '23 '14 '23 33 34 '14 23 24 33 44 '14 '23 '13 '14 '23 44 '14 '23 34 13 23 24 'J4 '14 23 24 34 '1,3 '14 '23
N • 2 TRUNCAT IONS NG
0 .0 0
NF NA (ND
.. 1 1
()
0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0
0
NO N, NA END
1
3 3 3
3 3 3 3 3 3 3 5 5 7 7 7 11
4 12 20 36 44 1 52 3 0 3 3 18 3 20 3 32 3 36 3 44 3 48 3 52 3 84 5 52 5 68 5 16 7 52 7 100 7 108 11 S4
GAUGE GROIIP G(E) 50(2)"4 50(2)"8 SO(2)"S SO(2}"S 50(2) "8 50(2)"8 SO(2)"8 SO(2)"8 $0(2)"e SO(2)"e SO{2)"S 50(2)"8 50(2)"4 SO(2)"8 SO(2)··8 SO(2)·'8 SO(2)*'8 SO(2)'·S SO(2}'o, 50(2)"8 SO(2)"S 50(2)0'4 SO(2)"8 SO{t} ..s
x SO(4)"2
x SO(4}"%
x SO(4)"2
x 50(4)"2
""'DEl
SIGNS
58( 1234) 41B( 12345) 41B(12345) 6B{1234} 418(12345) 418(12345) 68(1234) 418(12345) 415(12345) 418(12345) 68(1234) 418(12345) 56(1234) 41B(1234) 418(12345) 68(1234) 58(1234) 6B(1234) S6(1234) 29B(1234)P 6S(1234) 58(1234) 41a(1234) 6S(1234)
24 33 44 34 44 33 2433 24 33 22 33 24 33 44 23 24 44 23 34 33 44 33 34 34 44 34 44 23 24 34 44 33 23 22 23 33 3344 22 23 33 3334 44 34 34 23 2434
• # of gonorotlo••• 1/2 (Eulor charaeterl.tlc) • NI'-NA • I 0' lami lies 27 • abstract h(l,l) • I of onti/omll i •• 27 • Qbstr~1 h(I,I) .. (f of E(5) s!ngleto)-NF-NA-{# of gonerators of G(E» • .. , of tongent bundl. dofo ...... tlons E.d(T)
G(E). gouge group .f enhe.c.",.nt, The total gouge group i. C(E) • £(6) • £(8), SIGNS; we list the pol r. IJ of tho Sign. E(l,J)--1. Thero 10 c comp I .. t. sy...st ry Nil -> -NG, To obtaIn the "ro•• r.ed" models. with Nf <-> NA, cancel tho p"i r. ·IJ and I.troduee t~. po! ts (lJ)
2114 Table VI.10.VIU Generalors or th. system Al{I,:J,3,4) ell pen.ralOI ~~~l 1 I 70 1 0 'Y+ 1-
1'1 1'1 'T)
1'.
~!~
~.~s ~6~' ~8~e ~O~I ~2~a ~.~. ~6~7 ~8 ~9
ill 1 1 1 1 () 0 1 0 o I (I 0 00 o 0 I 0 I 0 00 o I 1 j o0 o0 1 I o0 1 1 1 I I 1 1 111 I j 1 I o0 I 0 o1
~o
11 1 1 1 11 1 1 1
o 0 00 o 0 o 0 o 0 I 0 () I o 0 1 0 o 1 1 1 I 1 o0 I 1 1 1 1 1 1 I 00 I 1 1 1 I 1 a I) 1 1 1 1 tl I) 1 1 1) o 1 I I 1
o
o
Table VUO.IX
Spectrum althe massless WZ multipets in tb. (2,2) l'aCuum Al(1,2,3,4) U(I)!
f;eCIOI
qql ¥1¥2 YaY. J I
to} <>1 a2
ba <112
414
to~o.l
~2iYl YzYa iY4 J1 i2
o0 o0
0 0
oI o ·1
· ... · ... · ... · ...
1 0 ~q 00 10 00 00 ! 0 I 0 00 00 10 1 0 ~ \l \ 1111 00 10 1 0 1 f) 1 o o 00 I 0 1 () ~ I) ~ 00 1 0 1 I) ! o 00
o0 10 }12 o 0 o 1 o 1 o I)
~
10 10 10 1 () I 0 1 (l
io} lq
lH to!
"
.. · .
t
· . · .. · ... · . b128 · ... blat .... bz3t · ... · .. bl • s• ... · . bls
50(4) :eO!>I,1 hz,t
·. . · . .· . · . ·. · . · . · ...
·
,
ha
]2 q~l
1 1 1 0 00 0 00 00 1 1 I 0 000 Oil 00
!O! t t
41>\
U(I}I
50(4)
lO I !~
~
t I H
I 0 0 0 4 0 0 4 I 0 0 o 16
11 1 0 ~ 2
0 0 1 I
2
I 00 27 11
!
1, 12
!
~ 00 ;
! iq
27
(I
I
I)
11 I o 13 (I 0 1 3 , () 1 o 00 1 o 1 0 13 r1: 111 IiI 01) 1 00 o 0 ! 00 I 00 3 12 o 0 ~ 00 I o 1o 1 00 J I) 10 1 ;1:10 00 I I)
q
q
•
• o:qt •
· ... · ... · ... · ... · ... · ... · ...
h N'
0
0 \)
0
0 I)
0
I
o 16 o 18 I)
18
1 ·1 0 1 ·1 0 () 0 4 0 0 4 0 0 2 !) tl 2 0 0 1 0 0 1 0 0 :I 0 0 2
0 2 ·2 1S 0 4 ·4 20 0 0
o 16 o 18
1
1
:I
0 2 18
0 :I ·2 18
"
·4 36 0 & 11 1 23
<"
2115 Table VI.lO.x Sequential breai:illg od tile galtge 8toUP in tile syslem B$(l,2,3,4) N 4
2
I UM I
SU(2)1
1
qauge gtOUp SO(12) 50(8) 5U(2)111 I SU(2)" U(I)/I U(l)11l I soW I SO(4)11
I
~I
E.
I
I
~I ~I
E1 U(I)IV
I
Be
Teble VI.10.xI Massless spec:IzU", oIille DIOde! BJi(2,3,4) tJ,couesp<mtlins to a ISCNd,B, theall
sectoJ
~gen
S0(6) 0(2 SU(2) ~U(2)
{O}
2 2 2 2 2 2
3 3
,
_lor
28
multiplets
3
~3l a4E
6;
2
h,
:I
&u &u
2 2 2 :I
~ ~
SO(I2) N=2 Be ~ U(2) ~U(2)ll1 50(8 SU(2)1l Er~ Ill" multiplet.
2 :I
:I
2
2 2 :I 2
56
untwisted 4 ypetmult
56
4
56
4
8
2 2
8
2
8
2
2
2 2
twisted ypenuult.
Table VI.10.XII Massl_ WZ muhiptls ill tile (2,2) _aulD 85(1,2,3,4) £28 ~to U(I)I U(l¥lJ ~0(4) ~0(4)Il U(I)l1 ~(l)ll'
Be h"l 0 0
:1:1
:1:1
from
4 ~
:1:1
-2 2
multo from 1IIltwiP.
:1:1 :1:1
:i:l :1:1
:1:1 :1:1 :1:1
:1:1
±l
4
±1
4
:U
:1:1
JPeEmu
~.
4 ±1
N'
0
2 2 16 2
0 0 0
0 0
21 2 2'1 0
0
27 2 2'1 0 0 0
0
0 2 2
:1:3
0
1 ·1
2T :I
0
::1:1
4
:1:1 :1:1
h
0 0 0 0 0 1
0 0 0
:1:1
:1:1 from
:i:3 1 .1 1 -1
:1:1 :1:1
27 0
4
&,
from twisted 62$ liypetDlU
27 1
4
~ypermul
iwist~
0 0
4
{OJ
h2,1
:1:1
:1:3
27
0 0 0
1 .J
0
0 0 0
16
2 ·2
0
16 4 0
0
2 ·2
0
0
16 16
0
0
0
4
2 -2
II 0 16 16
0
2
0 0
0
0
0
0
0
0 0
4
2116 Table Vl.lo.xm
Massless specuum of tM model B26(23} , corU8JH»1ding 10 a ISCIl =3,S. tlteorJ' ~ctOI
SO(6)
E.
SO{I2)
lYO(2 ~U(2) ~U(2)'!s0(2 ~O(2 ~O(2 ~O(6 ~U(2) Er
{OJ
.Bi N=2 .Bi lUIII!iple!s
3
2 2 2 2 2 2 2
15
veclor mUltiplet.
2
2 3
:%
3
2
"3
2 2 2 2
<13
2
43
2
"n
2
"'S
2 2
{OJ {OJ
2
4Z
claa
p41 untwisted ypermult
6
2 ~
2
2
2 2
4 4
:%
2
2 2
:a
..
:%
4
2
2
2 2
2 2
twisted ~ypermult.
2 2
11
56
Table VLlO. XIV
Massless WZ muJ/ipelS in Ilte (2,2) VlICuum B26(1,3,3) !recto !u(l) !u(l)'50(3) 150(3)'Be ~ 1 :1:1 1 vector {O} :1:1 I multo 3 3 from
t
L
.~
.!
2
from {O} untwist. ypermul
2
twisted
Gu
2
Cln
;!;l 2
423
2T
;
2'i
tl
¥
8 2 2 4
2
2
1 1 1 2 2
4
tl
4n ClU
2
I
1 .~
42.
2'i
3
"2S
~permul 423
!
3
4, I'IQm
zr
3
¥ .u 2
3 3
3
1
1 1 1 1
t.1 hJ,1
0 0 0
0 0 0
ix
N'
0
2
0
2 9 1
0
0
0 0 0 0
(I
0 0
0 0 0 0 0 0
0
0
0
32
2
0 2
2 ·2
0
0 0 0 0
0 0 0
0
0 I)
0 0 0 0
0 0 0 0 0 0
(l
0
6 6
0 8 8 6 8 6 6
2117
Historical Remarks and References for Part VI
It is proper to start off by saying that a few reference books on superstrings do already exist and contain excellent bibliographies. In particular the reader is referred to the book by Green t Schwarz and Witten [11 and to the reprint volume, t~e First Fifteen Years of Superstring Theory" edited by J.H. Schwarz (21. Here we just want to put the history of superstring theory into the modern perspective adopted by our presentation and, £01' the reader's benefit, fist the key references corresponding to the various stages of the theory development. The very beginning of the whole subject dates back to 1968 to the seminal paper by Gabriele Veneziano [3] where a four-particle scattering amplitude was guessed that had the property of duality. namely, displayed the same resonant poles in ihe s and in the t channels. Veneziano's formula had its origin within a cultural stream completely different from that of the unified theories of all interactions. In the sixties, indeed, there were two philosophical schools in elementary particle physiCS with quite opposite credos. On the one hand the "field theory" school continued the development of quantum field theories sticking to the belief that the framework successfully describing electrodynamics and based on the geometrical principle of gauge invatiance could eventually be applied also to the theories of the weak and strong interactions. On the other hand, the liS-matrix" school advocated the intrinsic difference of strong interactions and, relying on the enormous richness of the hadronic spectrum, considered the local field theory inadequate for their description, proposing various alternative approaches to the direct construction of the scattering matrix for hadrons. Vene~iano's four-point function was a step in that direction and, in the 1968 perspective, represented the accomplishment of a programme begun eight years before with the introduction of Regge poles [4] in high energy physics.
2118
Then in 1969 came the generalization, by Koba and Nielsen, of the Veneziano amplitude to the case of N-external particles [51 (previous formulations of the N-point amplitudes had also appeared [6] but did not exhibit the crucial projective invariance we discuss below). The KobaNielsen formula, involving an integration on N-3 variables, defined what was then called a "dual model", It led to very significant developments in a very short time, essentially in the same year. Indeed, in order to study the factorization properties of the dual amplitude in all its resonant channels. Fubini. Gordon and Veneziano introduced an operator formalism !7} based on an infinite number of creation and absorption operators, which almost immediately suggested to Nambu the mechanical interpretation of the underlying system in terms of a one-dimensional object propagating in space-time (the string) [S]. At the same time, Gliozzi put into evidence the deep connection between the cyclic symmetry of the dual amplitude and the projective invariance of the Koba-Nielsen integrand under MObius 50(2,1) transformations [9]. This observation was the germ of all future developments leading to the identifications of string vacua with 2D conformal field theories. In relation with this observation we must stress that, within the Fubini-Veneziano operator formalism, one was immediately led to introduce the concept of vertex operator, nowadays identified with the concept of conformal priJllary field, and that the Koba~Nielsen integrand was seen as an expectation value of a string of vertex operators ("conformal corre~ lator" in modem language, multiperipheral configuration in 1969 jargon). The concept of vertex operator was extended by Sciuto [lO} and, independently, also by Caneschi, Schwimmer and Veneziano [11J who constructed a new three-Reggeon vertex that in modern language simply describes the interaction of three arbitrary string states (to be precise, the splitting of one string into two strings). This vertex has been converted into a BRST invariant object only recently [12] and in this form it has become the essential ingredient in a fruitful research programme leading to an operatorial construction of multi loop string amplitudes. The next step on the road which eventually linked Veneziano'S original formula to the phYSics of an one-dimensional extended
2119
object (the string) and to what we nowadays call 2D conformal field theories was taken in 1970 by Miguel Virasoro. In a paper [13], that, curiously, was written without knOwing about G1iozzi ' s result, he extended the finite Gliozzi algebra, encompassing the three operators LO' L1 and L_ 1, to the infinite one containing all the Ln which bears his name. Since its introduction, the Virasoro algebra was seen as a gauge algebra of constraints, similar to those of electrodynamics, that had to be imposed on physical states: L Iphys. state> .. 0 n
(for n > 0) •
(VI. II. 1)
The Fock space generated by the infinite harmonic oscillators of the operator formalism was immediately seen to contain an infinite number of negative norm states. Virasoro conjectured that they could be removed by the condition (VI.A.I). Brower, Goddard and Thorn proved in 1972 that this ~~s true if and only if the dimension of the spacetime where the string propagated was D.. 26 [141. Actually the critical number of space-time dimensions, D" 26, had already made its appearance in the literature sometime before. in a paper by Lovelace [15] where it was sho~n that. in that very peculiar number of dimenSions, the branch cut singularities appearing in the calculation of the dual model loops became resonant poles and did not violate unitarity. A major effort in the dual theory was therefore dedicated to the search for new dual models which was characterized by lower critical dimensions and which possibly included fermions in their spectrum. In January 1971 Pierre Ramond introduced, using his own words, wave equations for free fermions ••• based on the structure of the dual theory for bosons [16J. In the same paper appeared, as an algebra of constraints, the superalgebra of Eqs. (VI.5.47). Henceforth Ref. [16] is not only at the origin of fermionic string theories, but, in a sense. also at the origin of the whole family of supersymmetric theories. Indeed, as was already stressed in the historical remarks for Part TWo, when Wess and Zummo, unaware of the 1971 results of Gal' fand and
2120
Likhtman, introduced in 1974 the supersymmetric field theoretical model that bears their names, they were explicitly aiming at a four~dimen~ siona1 generalization of the gauge superalgebra of dual models. A couple of months after the Ramond paper had been submitted for publication, Andre Neveu and John Schwarz introduced another bosonic dual model, called at that time the "dual pion model" [17), whose gauge algebra of constraints was the superalgebra given in Eqs. (VI.9.46). In 1972, Schwarz [18] and independently Goddard and Thorn [19J showed that the critical dimensions for the dual pion model was D=10.
While the development of the various dual models was going on in the way we have indicated (we should also mention the Shapiro~Virasoro model introduced in 1969-1970 in Refs. [20] and [21] and later to be identified with closed strings), the string picture advocated by Nambu \~as for some time left aside. Its essential role in the interpretation of the dual theory was finally clarified by the seminal paper of Goddard, Goldstone, Rebbi and Thorn [22], who analyzed all the conse~ quences of the string action introduced by Nambu-Goto and traced the origin of the Virasoro algebra to the reparametrization invariance of the string Lagrangian. They were also the first to utilize Dirac's theory of constrained canonical systems in this context and, in so doing, paved the way to the future BRST reformulation of the quantization problem. Up to 1974 the string theory or, equivalently dual models, were regarded a~ models of hadrons. Correspondingly, the Regge slope, identified with the string tension Za', . was given the value:
I:2a' = 10- 13 cm .
(VI.ll.2)
Yet, already in 1971 and 1972 there had been a glimmering that, perhaps, this was the wrong way to look at the theory. First Joel Scherk (23J and then Scherk and Neveu [24] considered the expansion of string ampli~ tudes in powers of the slope parameter a' around a limiting low-energy point-particle theory. In so doing ~hey showed that the massless vector particles, which are present in both the 26-dimensional model and the
2121
lO-dimensional model, behaved like quanta of a Yang-Mills field in this limit. It was also known that in the closed string sector one had a massless spin-2 particle. rather than vector bosons as in the open string case. FurtheX'DlOre. in a 1974 paper by Ademollo. D' Acida. Df Auria, Napolitano, Sciuto. 01 Vecch1a. Gliozzi. Musto and Nicodemi, it was shown, among other things, that this particle coupled to the stressenergy tensor like what a graviton would do [25]. In the same year it was independently shown by Yonera [26J and by Scherk and Schwarz [27J that. in the zeroth slope limit. these massless states do indeed couple as true gravitons. identifying. the Einstein Lagrangian of the gravitational field with the effective low-energy Lagrangian of the strings. In the same paper [27]. Scherk and Schwarz argued that henceforth string theories should be regarded as theories of quarks. leptons. gauge bosons. and gravitons rather than theories of hadronic resonances, thus shifting the value of the string tension to the Planck length: (VL 11. 3)
What they proposed is what has essentially happened in the literature since that moment.
By that same time, in fact. dual models had been completely superseded by Quantum Chromodynamics as theories of hadrons, and in 1974 the paper by Wess-Zumino started the development of supersymmetric field theories, ",ilich culminated in 1976 with the discovery of Supergravity by FerrAra. Freedman and van Nieuwenhuizen. The history of supergravity has been included at the end of Part III. Six months after supergravity was introduced. another very eventful and seminal paper appeared. which, from a conceptual point of view. is at the origin of the complete fusion, happened eight years later, of the theories of strings with those of supergravity. We refer to the 1976 paper by Gliozzi, Olive and Scherk that introduced the projector bearing their names and defined. for the first time, a string model with unbroken supersymmetry in the target space-time {28]. Reference (28] can be properLY regarded as the beginning of the superstring theory.
2122
In retrospective space-time supersymmetry has been shown to be related to extended superconformal algebras on the world-sheet. These algebras were discovered a year before by Ademollo et a!. in [29]. In the years from 1976 to 1984 superstring theory failed to be the focus of interest that shifted instead to supergravity. In particular. great hopes were raised that some supergravity IlIOdel might. eventually. be finite and yield a consistent quantum unification of gravity with all the other interactions. If such a field-theory model existed and could predict, maybe after dimensional reduction, the spectrum of quarks, leptons and gauge fields, one could explain the laws of Nature in terms of point-like objects excluding strings and superstrings. The progrSillme. however, failed for the reasons we have explained in Part V and also for the fact that although less divergent than gravity, all supergravities are nonetheless not finite. This means that. at some pOint, one was bound to come back to superstrings whose finiteness has not yet been established in a definite way, but which has, anyhow, been motivated by numerous arguments and a long series of partial results (see the book by Green, Schwarz and Witten and all the papers by Mandelstam quoted therein). The failure of linear supergravity became completely evident in the period going from the fall of 1983 to the spring of 1984 when the possibility of finding a realistic SU(3) x SU(2) x U(l) fermion spectrum within the compactification schemes of D=11 supergravity was ruled out by D'Auria, Fre [30]. More or less at the same time, Witten [3lJ showed that Kaluza·Klein compactifications of higher dimensional theories that do not contain elementary gauge fields could not yield chiral fermions upon compactification to D=4. This development took place only a few months before the rekindling of interest in superstrings exploded with the Green-Schwarz discovery of the anomaly cancellation (see below). In the years 1976-1984, although studied only by very few people, superstring theory continued to be developed. It is on the results obtained during that time that most of what was done after the fall of 1984 is based. First of all we had, already in 1976, the derivation of the locally supersymmetric and reparametrization invariant action for
2123
the spinning string by Brink, Di Vecchia and Howe [32] and also by Deser and Zumino [33]. Secondly,. and most prominently, we have the 1981 seminal papers by Polyakov [34J, who exploited the auxiliary 2dimensional metric field in order to introduce a functional integral formulation of quantum strings. Polyakov's approach provided an enhanced understanding of the critical dimensions and led to the identification of the Virasoro central charge with the conformal anomaly, a step taken by Friedan in his 1982 Les Houches lectures [35]. It should be noted that the occurrence of Schwinger terms in the operator product expansion of the stress energy tensor of two-dimensional quantum field theories (yielding the central extension of the Virasoro algebra) had been pointed out in the early seventies by Ferrara, Gatto, Grillo l36) and also by Fubini, Hanson and Jackiw [37]. Reference [34J inspired an extensive revisitation of string theory within a functional formulation and led to a proper geometrical understanding of the string loops in terms of Riemann surface moduli. In relation with this a very clarifying role was played by the 1983 article of Orlando Alvarez [38].
on the other hand, in 1981, Green and Schwarz introduced their explicitly target space supersymmetrical formalism (see Sect. VI.9.7) that played an important role in all the subsequent developments. Other two important developments took place in 1983 and at the beginning of 1984. On the other hand, Kato and Ogawa [39] and independently Hwang [40] applied the BRST quantization procedure to the string model and introduced the ghost-antighost fields, that is, the items that were still missing in order to get a modern field-theoretical approach to the whole subject. On the other hand, Belavin. Polyakov and Zamolodchikov introduced the general concept and formalism of conformal 2Dquantum field theories [411 whose importance in the modern approach to superstrings can never be sufficiently stressed. Finally we must mention the development by Goddard and Olive [42J of the vertex operator construction of Kac-Moody algebras that, later on, played a key role in the discovery of the heterotic superstrings. In the years just before 1984, while the D=11 model was the favorite choice of supergravity theorists, the 0=10 model also received
2124
particular attention. In 1982, while formulating the Lagrangian of the 0=10 super Yang-Mills theory in superspace using the rheonomy framework, D'Auria, Pre and da Silva [43] were the first to discover the coupling of the gauge Chern-Simons form. In the same year Bergshoeff, de Roo, de Wit and van Nieuwenhuizen wrote the action for N=l, 0=10 matter coupled supergravity in the case of an abelian gauge multipl~t [44]. The year after (1983) Chapline and Manton generalized that action to the non-abelian case emphasizing the role of the Chern-Simons form ~~. In connection with Witten's results on the need for elementary gauge fields to generate chiral fermions, the Chapline-Manton theory received attention, within the Kaluza-Klein approach, as a viable alternative to the D=11 model. The fundamental work of Alvarez Gaume and Witten ~n gravitational anomalies [461, however, showed that such a theory.~as in general plagued by mixed gauge and Lorentz anomalies that made it meaningless at the quantum level. Since it was obvious from Gliozzi-Scherk-Olive's paper of 1976 that the Chapline-Manton theory was the effective lagrangian of superstrings, this observation seemed to put a spell also on superstrings. This challenge motivated the 1984 analysis by Green and Schwarz of the superstring anomaly, culminating in the discovery of its cancellation for the group 50(32) based on the Lorentz Chern-Simons mechanism ~~. The ne~t step was the 1985 construction by D.J. Gross, J.A. Harvey, E. Martinec and R. Rohm of the heterotic superstring model, incorporating the choice of the anomaly free gauge group in a built in way [48]. The same year Candelas, Horowitz, Strominger and Witten [49] showed that the effective theory of the heterotic superstring could be compactified on a Calabi-Yau space leading to a chiral, almost realistic model in D=4. The major developments of 1986, at least as we perceive them, are the following two. On the one hand, we have the covariant construction of the fermion emission vertex, the bosonization of the superghosts, and the final casting of string vertices and amplitudes in the language of
2125
20 conformal field theory, results which were all obtained by Friedan, Martinec and Shenker and reported in [SO J. On the other hand we have the clarification by Selberg and Witten [51] of the role of spin structures on the world-sheet and of their relations with the modular group, the global anomalies and the GSO projector. This point was further elaborated and clarified by Alvarez Gaume, Moore and Vafa [52J. The proper treatment of spin structures and the i!llplementation of modular invariance has led. in the fall of 1986 and in early 1987. to what historically and conceptually appears to us as the major and most exciting event of the long story we have been reporting; the direct construction of heterotic superstrings in critical dimensions 0=4. This development, which was for many years the dream of whoever worked on string theory, is the simultaneous achievement of several scientists working in different collaborations: Narain, Sarmadi, Kawai, Lewellen, Tye, Lerche, Lust, Schellekens, Antoniadis, Sachas, KOUlmas [53] . With this event we close our historical report feeling that we have already approached times too recent to get an appropriate historical perspective.
2\26
KEY REFERENCES
[lJ I2}
[3] [4J
[5] [6]
[7] [8J [9] [10] [11] [12] [13] [14] [15] [16] [17J [18] [19] [201 [21J
M.B. Green, J.H. Schwarz, and E. Witten, Superstring Theory (in two volumes), (Cambridge University Press, 1987). The First Fifteen Years of Superstring Theory, reprint volume edited by J.H. Schwarz, (World Scientific (1?86)). G. Veneziano, Nuovo Cimento 57A (1968) 190. For Regge Pole Physics, see for instance the book by G.F. Chew: The Analytic S-Matrix: A Basis for Nuclear Democracy, (Benjamin, 1964). Z. Koba, and H.B. Nielsen, Nucl. Phys. BI0 (1969) 633. H.M. Chan, Phys. Lett. 28B (1969) 425; H.M. Chan, and Tsou S. Tsun, Phys. Lett. 28B (1969) 485; C. Goebel, and B. Sakita, Phys. Lett. 22 (1969) 257. S. Fubini, D. Gordon, and G. Veneziano, Phys. Lett. 29B (1969) 679. Y. Nambu, Prec. International Conference on Symmetries and Quark Models, Detroit, 1969 (Gordon and Breach, 1970), 269. F. G1iozzi, Nuovo Cimento Lett. 2 (1969) 846. S. Sciuto, Lettere a1 Nuovo Cimento 2 (1969) 411. L. Caneschi, A. Schwimmer, and G. Veneziano, Phys. Lett. 30B (1969) 351. P. Di Vecchia, R. Nakayama, J.L. Petersen, and S. Sciuto, Nucl. Phys. B282 (1987) 103. M.A. Virasoro, Phys. Rev. D1 (1970) 2933. R.C. Brower, Phys. Rev. 06 (1972) 1655; P. Goddard, and C.B. Thorn, Phys. Lett. 40B (1972) 235. C. Lovelace, Phys. Lett. 348 (1971) 500. P. Ramond, phys. Rev. 03 (197l) 53. A. Neveu, and J.H. Schwarz, Nucl. Phys. B31 (1971) 86. J.H. Schwarz, Nucl. Phys. B46 (1972) 61. P. Goddard, and C.B. Thorn, Phys. Lett. 40B (1972) 235. M.A. Virasoro, Phys. Rev. 177 (1969) 2309. J.A. Shapiro, Phys. Lett. 33B (1970) 361.
2127
I22J P. Goddard, J. Goldstone, C. Rebbi, and C.B. Thorn, Nucl. Phys. 856 (1973) 109. [23J J. Scherk, Nucl. Phys. B31 (1971) 222. [24J A. Neveu, and J. Scherk, Nuc!. Phys. B36 (1972) 155. [25] M. Ademollo. A. D'Adda, R. O'Auria, E. Napolitano, S. Sciuto, P. Oi Vetchia, F. Gliozzi, R. Musto, and F. Nicodemi, Nuovo Cimento 21A (1974) 77. [26J T. Yoneya, Prog. Theor. Phys. 51 (1974) 176. [27J J. Scherk, and J.H. Schwarz, Nuc1. Phys. B91 (1974) 118. [28] F. Gliozzi, J. Scherk, and D. Olive, Phys. Lett. 65B (1976) 282; Nutl. Phys. B122 (1977) 253. [29] M. Ademollo, L. Brink, A. D'Adda, R. D'Auria, E. Napolitano, S. Sciuto, E. Del Giudice, P. Oi Vecchia, S. Ferrara, F. Gliozzi, R. Musto, and R. Pettorino, Phys. Lett. 62B (1976) 105. 130] R. D'Auria, and P. Fre, Ann. Phys. 157 (1984) 1. [SI] E. Witten, "Fermion quantum numbers in Kaluza-Klein theory," in Proe. of the II Shelter Island Meeting, eds. R. Jackiw, N. Knuri, S. Weinberg, and c. Witten (MIT Press, 1985). [32] L. Brink, P. Di Vecchia, and P. Howe, Phys. Lett. 65B (1976) 471. [33] S. Oeser, and B. Zumino, Phys. Lett. 658 (1976) 369. [34J A.M. Polyakov, Phys. Lett. 103S (1981) 287; Phys. Lett. 103S (1981) 291. [S5] D.H. Friedan, "Introduction to Polyakov's String Model", in Les Houehes Summer School 1982, edited by J.B. Zuber and R. Stora (North-Holland, 1984). [S6] S. Ferrara, R. Gatto, and A.F. Grillo, Nuovo Cimento 12A (1972) 959. [37] S. Fubini. A.J. Hanson, and R. Jackiw, Phys. Rev. D7 (1973) 1732. [38] O. Alvarez, Nucl. Phys. B216 (1983) 125. 139J M. Kato, and K. Ogawa, Nucl. Phys. B212 (1983) 443. [40J S. Hwang, Phys. Rev. 028 (1983) 2614. [41] A.A. Selavin, A.M. Polyakov, and A.B. Zamolodchikov, Nucl. Phys. B241 (1984) 333.
2128
[42]
[43J [44J [45J [46] {47] [48] 149]
[sol [51] [52] [53]
P. Goddard, and D. Olive, "Algebra, Lattices and Strings," talk given at the Workshop on Vertex Operators in Mathematics and Physics, Berkeley, California, 1983. R. D'Auria, P. Fre, and A.J. da Silva, Nucl. Phys. B196 (1982) 205. E. Bergshoeff, M. de Roo, B. de Wit, and P. van Nieuwenhuzen, Nucl. Phys. B195 (1982) 97. G.F. Chapline, and N. Manton, Phys. Lett. 120B (1983) lOS. L. Alvarez Gaume, and E. Witten, Nucl. Phys. B234 (1983) 269. M.B. Green, and J.H. Scl\l~arz, Phys. Lett. 149B (1984) 117. D.J. Gross, J.A. Harvey, E. Martinec, and R. Rohm, Nucl. Phys. B256 (1985) 253. P. Candelas, G. Horowitz, A. Strominger, and E. Witten, Nucl. Phys. B258 (1985) 46. D.oFriedan, E. Martinec, and S. Shenker, Nuc!. Phys. B271 (1986) 93. N. Seiberg, and E. Witten, Nucl. Phys. B276 (1986) 272. L. Alvarez Gaume, G. Moore, and C. Va£a, Commun. Math. Phys. 106 (1986) . K. Narain, Phys. Lett. 169B (1986) 41; K.S. Narain. M.H. Sarmadi, and E. Witten, Nucl. Phys. B279 (1986) 96; H. Kawai, O. Lewellen, and S. Tye, Phys. Rev. Lett. 57 (1986) 1832; Nucl. Phys. B288 (1987); Phys. Lett. 1915 (1987) 63; W. Lerche, D. Lust, and A.N. Schellekens, Nucl. Phys. B287 (1~87) 477; W. Lerche, B.E.W. Nilsson, and A.N. Schellekens, Nucl. Phys. B294 (1987) 136; I. Antoniadis, C. Bachas, and K. Kounnas, Nucl. Phys. B289 (1987) 87.
2129
INDEX abelian differentials 1432, 1464 of the first kind 1433 of the second kind 1433 of the third kind 1433 abstract Lie algebra 647 action describing the coupling of gravity to spin 3/2 matter 617 action geometrical 663 action of 'Va on the basic harmonics 1270 action of D = 11 supergravity 881 action of the heterotic IT-model 1535 action principle 508 active challge of the metric tensor under a. general coordinate transformation 68 adjoint multiplet 1797 adjoint representation of G 108, 1178 AdSxM 7 1194 AdS4 xS1 1254 background 1195 affine connection 92 algebraic topology 254 almost complex structure 930 or' 1855 amended transformation of the spin connection 613 An, Dn , Ee, E7, Es 1823 analytic automorphism group 1401 analytic function f"''' (z) 1376 angular momentum operators 284 anholonomic basis (Da, Dor) 497 anholonomic cotangent frame 379 anholonomic or tangent space indices 54 anholonomic supervielbein basis 655 anholonomized derivative 171 anomalies 1365, 1559 anomalies, cancellations of 1366 anomaly free supergravity 1838, 1854 anti de Sitter background 308 anti de Sitter group 327 anti de Sitter N-extended superfields 370 anti de Sitter radius R(AdS) 1151 anti de Sitter space, particles 426 anti de Sitter space, radius of 356 anti de Sitter supermultiplet 448 anti de Sitter supersymmetnc vacuum 959 anti de Sitter vacuum 178,712
2130
antighost 1782 asymptotic state 1783 Atiyah-Hirzebruch theorem 1365 atlas 31 ant(C} 1410 aut(IH) 1410 aut(D) 14.08 automorphism group 1033, 1408, 1786 ofthe metric 1420 auxiliary field fotmulation 1108 auxiliary fields 476, 586, 657, 952, 965, 1034, 1109, 1982 existence of 507 auxiliary O-form 728, 737 axion 1839, 1854, 1982 field strength 1838, 1855, 1859 ba.c:kground chaxge 1806, 1807 background fields 1777 base point 1633 basis of I·forms 43 beha.viour of k-forms under mappings 49 beta functions 1532, 1854 Betti number 260, 1231 Betti vector multiplet 1240 BF lowest states 1296, 1298, 1299 Bianchiidentities 84,85,306,507,688,1863 contracted 277 generalized 807 Bose-subalgebra 689 bosonic coordinates 339 bosOllic spin structure 1721 bosonic string, classical action 1646 bosonic vacuum 1081 bosonization of the superghost system 1804 boundaries 257 boundary COlIditiOllS, periodic 1502 boundary values of the superspace superfields 650 boundary vector 1131, 1832 bracket operation 62 Brans-Dicke theory 273 breaking of N 2 supersymmetry 1079 Breiten1ohner-Freedman 1242 bound 462 BRST charge 1378, 1515, 1577, 1583, 1596 BRST covariant quantization 1558 BRST current 1579
=
2131
BasT ill.vwnce 1381, 1722, 1770 ohhe vertices 1777 BRST-invariant hamiltonian 1565 BRST opera.tor Q 1562 Calabi-Ya.u 3-fold 1382 canonical bundle K 1483 canonical picture 1840 Caltan basis 1292, 1600 Cartau integrable system 796 Cartan subalgebra 1601, 1719 Cartan-Weyl basis 1798, 1800 Casimir operator 1181, 1281 Cauchy-Riemann equations 660, 932 Cd cohomology 1477 center of the Lie algebra 1824 central charge 322.357,1381, 1397, 1510,1548, 1580, 1607 central charges Z(+ )AB 328 central extension 1165, 1510 change of coordinates 31, 50 in second-older formalism 1955 ChapJine-Manton formulation 1858 Chapline-Manton Lagrangian 1848, 1859,1942 chatacteristic class 2012 characteristics of the Riemann-theta function 1475 chat&eter of the irreducible module h(,,) 1718 chatacte18 of a Ka.c·Moody algebra 1707, 1717 charge conjugation matrix 318, 523, 1265, 1512, 1804 chart 30 Chern class 1382 Chern-Simons folm 596, 1862 Chevalley cochains 781 Chevalley cohomology classes 799 ChevaI1.ey-Eilenberg theorem 1 804 Chevalley-Eilenberg theorem 2 805 chira! cl1nents, conserved 1529 chira! determinant of the operator 1690 chira! Dirac operator 1475 chila! fermions 358,1377 chirality 1156, 1363 chira! multiplet 1983 chira! potential 1000 chira! projectiOllS 502 chila! superfields 503 chiJ:a! target-space fermions 1511 classical confonnal field theory 1507
2132 classical superconformal theories 1386, 1582 classical Virasoro algebra 1396, 1509, 1554 classification of coset manifolds 195 Clifford algebra 316, 519 Clifford vacuum 396, 1725 closed form in snperspace 681 closed string 1501 closure of the exterior differential operator 504 cn-compatible charts 31 coadjoint covadant derivative 808 coadjoint representation 109 coadjoint set of O-forms 807 coboundaries b of the matter fields 1780 coboundaries of the Virasoro algebra 1598 coboundary 799, 1600 cocycle 799, 1792, 1862 cocycle matrix Mij 1802 cocycle of the SO{l,9}-Lie algebra 1861 coeycle of the super Poincare algebra 1963 cocycle operator 1792 eoeydes and coboundaries 1596 coexact 1430 eohomological equation 2014 cohomological operators 1875, 2013 cohomology classes 1564 cohomology group Hd(Md ,lR.) 1193 cohomology groups, relative 804 colinearity conditions 1844 color group 1305 commutation relations of the Poincare group in D-dimensions 113 commutator of two snpersymmetries 484 compact coset space 199 compactilied manifold Md 1377 compensator 381,489 complete functional basis 11 75 completeness of the field equations 704 complex conjugation 336 complex projective space q;P n 209 complex structure 1446 of the torus 1446 conformal anomalies 1380, 1387, 1554, 1576, 1583, 1589, 1703, 1708 cancellation 1520 conformal automorphism group 1672 conformal boosts 1513 conformal classes of the world-sheet 1393 conformal dass or complex structure 1397
2133
conformal equivalence 1398 conformal field 1710 theories 1381 conformal gauge 1508, 1569 conformal group 181 conformal invariall.ce 1228 conformal K'illing group 1673 conformal Killing spmors 1809 conformal Killing subgroup of ~he difeomorphism group 1662 conformal Killing vectors 1215, 1420, 1472, 1662, 1697, 1809 conformaDyequivalence 185 conformal supergravity 330, 1391 conformal theories 1505 conformal transformation 181, 1396 conformal weights (A,A) 1509,1770 conical points 1412 conjugacycl~
1824,1825,1830
con$iltent truncations 1156,1357 constrained systems 1560 constraints 1558 on superspace geometry 1522 contractible algebn 797 contractible generators 869 contraction 25 function 1723 of forms with' tangent vectors 46 contravariant tensor 12 coordinate basis 655 coordinate invariance 158, 690 coordinates on G/R 197 correction, transformation law in second order formalism 628 correlation function of the ghost 1813 correlation functions 1846 correspondence principle 1547 coset coordinate 1178 coset functions 1179 coset genera.tors 194, 198 coset manifold E7(_7)/SU(8) 1115 coset manifold SU(p,q)/SU(p)xSU(q)x U(l) 1071 coset manifolds 190, 191 coset manifolds, non-compact li90 coset representative 192, 198, 1177 coset space 191 coset spaces Mpqr 246, 1302 coset spaces Npqr 1251 coset spaces, parametrization 195
2134
cosmological constant 727, 959, 1151, 1154, 1377, 1635, 1653, 1713 cosmological term 169 cotangent basis 111 cotangent frame on superspa.ce 535 cotangentspa.ce 44 to superspa.ce 537 cotangent vector space T"'(MP/q) 343 covariant deriva.tive 80 of an adjoint set of p+l-lorms 807 of a spinorial p-lotm 89 of a vector field 94 opera.tor 123 covariant diferential 1523 covaria.nt exterior derivative 88 covaria.nt lattice approach 1384 covariant lattice construction 1819 covariant lattice of the ES0Es superstring 1832 covariant lattice of the SO(32) superstring 1832 covariant lattices 1769 covariant tensor 12, 1404 covariant vector: 51 covariant weight lattice D1~) 1829 covector field 44 covering space, universal 1401, 1532 covering spaces 1252 CPT invariance 415, 420 c;Pn 207 creation-absorption operators 1650 creation-annihilation operators 396 cross-sections of bundles 1473 current moding 1593 curvature 230, 688, 806 curvature 2-lorm 79, 82, 122, 227 curvature of the Kihler connection 948 curvature scalar 91, 1405, 1425 curvatures, of the gange-multiplet 988 curved indices 55 cycles 257 cyclic identity 90
=
D 4 low energy 8upergra.vity 1355 D := 4 massless spectrum of heterotic superstrings 1983 D = 5, pure gra.vity 1157 . D = 5, N = 2 Maxwell-Einstein systems 1131 D = 5, N = 4 Maxwell-Einstein systems 1134 D = 5, N = 2 second order Lagrangian 788
2135
D ::: 5, N "" 2 supergravity 755 D::: 5, N "" 2 sllper$pac~ 555 D = 6, N "" 2 sllpergravity 1135 D = 6, 2ojJ·Pierzing formula 836 D = 7, N "" 4 gauged supergravity 1137 D = 8, gauged N ::: 2 supergravity 1137 D = 9, N = 1 supergravity coupled to n vector multiplets 1137 D = 10, N = 1 action of the Green-Schwarz heterotic sigma-model 1964 D = 10, N = 1 para.metrization of the curvatures 1888 D = 10, N ::: 1 super Poincare Lie algebra 1860 D = 10 supergravity coupled to D = 10 super Yang-Mills 1952 D 11 superspace 563 D = 11 supersymmetry transformations 1195 d'Alembertian 174 decomposition of the exterior derivative in D =: 10 1877 decontraction of minimal algebra 870 dedekind eta function 1651 definite parity 165 d~.finition of massless particles 428 degree of a divisor 1460 degree of the canonical class 1466 Dehn twist 1449 de Rahm cohomology 258 classes 1214 derivative currents 1789 derivative interactions 1855 descendents 1779 de Sitter cosmological constant 1189 de Sitter manifold 959 de Sitter metric 295 determinant of the Dinc operator 1677, 1680 de Wit-Nicolai theory 1120 di-creation operators 1285, 1286 diff (E5) 1399 diffeomorphism 31,38 group 1379. 1398, 1658 diifeomorphisms, beha.viour of functions, vector fields, differential forms and tensor fields 59 diffeomorphism, infinitesimal 126, 678, 1418 diffeomorphisms connected to the identity 1412 diffeomorphisms of a manifold M 59 diifeomorphisms, one parameter group of 61 differentiable manifold of class C 31 differentia.II-forms on }vI1>/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
