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

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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