Introduction to Superstring Theory (Leuven Notes in Mathematical and Theoretical Physics): Elias Kiritsis

Description

Introduction to Superstring Theory Elias Kiritsis Lectures presented at the Catholic University of Leuven and at the University of Padova during the academic year 1996-97. To be published by Leuven University Press.

Contents
1 Introduction 5
2 Historical perspective 6
3 Classical string theory 9
3.1 The point particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Relativistic strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 Oscillator expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 Quantization of the bosonic string 23
4.1 Covariant canonical quantization . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Light-cone quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3 Spectrum of the bosonic string . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.4 Path integral quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.5 Topologically non-trivial world-sheets . . . . . . . . . . . . . . . . . . . . . 30
4.6 BRST primer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.7 BRST in string theory and the physical spectrum . . . . . . . . . . . . . . 33
5 Interactions and loop amplitudes 36
6 Conformal field theory 38
6.1 Conformal transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
6.2 Conformally invariant field theory . . . . . . . . . . . . . . . . . . . . . . . 41
6.3 Radial quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.4 Example: the free boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.5 The central charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.6 The free fermion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.7 Mode expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.8 The Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.9 Representations of the conformal algebra . . . . . . . . . . . . . . . . . . . 54
6.10 Affine algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.11 Free fermions and O(N) affine symmetry . . . . . . . . . . . . . . . . . . . 60
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6.12 N=1 superconformal symmetry . . . . . . . . . . . . . . . . . . . . . . . . 66
6.13 N=2 superconformal symmetry . . . . . . . . . . . . . . . . . . . . . . . . 68
6.14 N=4 superconformal symmetry . . . . . . . . . . . . . . . . . . . . . . . . 70
6.15 The CFT of ghosts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
7 CFT on the torus 75
7.1 Compact scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
7.2 Enhanced symmetry and the string Higgs effect . . . . . . . . . . . . . . . 84
7.3 T-duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.4 Free fermions on the torus . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.5 Bosonization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.6 Orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.7 CFT on higher-genus Riemann surfaces . . . . . . . . . . . . . . . . . . . . 97
8 Scattering amplitudes and vertex operators of bosonic strings 98
9 Strings in background fields and low-energy effective actions 102
10 Superstrings and supersymmetry 104
10.1 Closed (type-II) superstrings . . . . . . . . . . . . . . . . . . . . . . . . . . 106
10.2 Massless R-R states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
10.3 Type-I superstrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
10.4 Heterotic superstrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
10.5 Superstring vertex operators . . . . . . . . . . . . . . . . . . . . . . . . . . 117
10.6 Supersymmetric effective actions . . . . . . . . . . . . . . . . . . . . . . . . 119
11 Anomalies 122
12 Compactification and supersymmetry breaking 130
12.1 Toroidal compactifications . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
12.2 Compactification on non-trivial manifolds . . . . . . . . . . . . . . . . . . 135
12.3 World-sheet versus spacetime supersymmetry . . . . . . . . . . . . . . . . 140
12.4 Heterotic orbifold compactifications with N=2 supersymmetry . . . . . . . 145
12.5 Spontaneous supersymmetry breaking . . . . . . . . . . . . . . . . . . . . . 153
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12.6 Heterotic N=1 theories and chirality in four dimensions . . . . . . . . . . . 155
12.7 Orbifold compactifications of the type-II string . . . . . . . . . . . . . . . . 157
13 Loop corrections to effective couplings in string theory 159
13.1 Calculation of gauge thresholds . . . . . . . . . . . . . . . . . . . . . . . . 161
13.2 On-shell infrared regularization . . . . . . . . . . . . . . . . . . . . . . . . 166
13.3 Gravitational thresholds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
13.4 Anomalous U(1)'s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
13.5 N=1,2 examples of threshold corrections . . . . . . . . . . . . . . . . . . . 172
13.6 N=2 universality of thresholds . . . . . . . . . . . . . . . . . . . . . . . . . 175
13.7 Unification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
14 Non-perturbative string dualities: a foreword 179
14.1 Antisymmetric tensors and p-branes . . . . . . . . . . . . . . . . . . . . . . 183
14.2 BPS states and bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
14.3 Heterotic/type-I duality in ten dimensions. . . . . . . . . . . . . . . . . . . 186
14.4 Type-IIA versus M-theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
14.5 M-theory and the E8-E8 heterotic string . . . . . . . . . . . . . . . . . . . 196
14.6 Self-duality of the type-IIB string . . . . . . . . . . . . . . . . . . . . . . . 196
14.7 D-branes are the type-II R-R charged states . . . . . . . . . . . . . . . . . 199
14.8 D-brane actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
14.9 Heterotic/type-II duality in six and four dimensions . . . . . . . . . . . . . 205
15 Outlook 211
Acknowledgments 212
Appendix A: Theta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
Appendix B: Toroidal lattice sums . . . . . . . . . . . . . . . . . . . . . . . . . 216
Appendix C: Toroidal Kaluza-Klein reduction . . . . . . . . . . . . . . . . . . . 219
Appendix D: N=1,2,4, D=4 supergravity coupled to matter . . . . . . . . . . . 221
Appendix E: BPS multiplets and helicity supertrace formulae . . . . . . . . . . 224
Appendix F: Modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
Appendix G: Helicity string partition functions . . . . . . . . . . . . . . . . . . 234
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Appendix H: Electric-Magnetic duality in D=4 . . . . . . . . . .

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