INTRODUCTION TO SUPERSTRING THEORY By Elias Kiritsis free download

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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 . . . . . . . . . . . . . . . . . . 240
References 243  

1 Introduction
String theory has been the leading candidate over the past years for a theory that consistently
unifies all fundamental forces of nature, including gravity. In a sense, the theory
predicts gravity and gauge symmetry around flat space. Moreover, the theory is UVfinite.
The elementary objects are one-dimensional strings whose vibration modes should
correspond to the usual elementary particles.
At distances large with respect to the size of the strings, the low-energy excitations can
be described by an effective field theory. Thus, contact can be established with quantum
field theory, which turned out to be successful in describing the dynamics of the real world
at low energy.
I will try to explain here the basic structure of string theory, its predictions and problems.
In chapter 2 the evolution of string theory is traced, from a theory initially built to
describe hadrons to a “theory of everything”. In chapter 3 a description of classical bosonic
string theory is given. The oscillation modes of the string are described, preparing the scene
for quantization. In chapter 4, the quantization of the bosonic string is described. All three
different quantization procedures are presented to varying depth, since in each one some
specific properties are more transparent than in others. I thus describe the old covariant
quantization, the light-cone quantization and the modern path-integral quantization. In
chapter 6 a concise introduction is given, to the central concepts of conformal field theory
since it is the basic tool in discussing first quantized string theory. In chapter 8 the
calculation of scattering amplitudes is described. In chapter 9 the low-energy effective
action for the massless modes is described.
In chapter 10 superstrings are introduced. They provide spacetime fermions and realize
supersymmetry in spacetime and on the world-sheet. I go through quantization again,
and describe the different supersymmetric string theories in ten dimensions. In chapter 11
gauge and gravitational anomalies are discussed. In particular it is shown that the superstring
theories are anomaly-free. In chapter 12 compactifications of the ten-dimensional
superstring theories are described. Supersymmetry breaking is also discussed in this context.
In chapter 13, I describe how to calculate loop corrections to effective coupling
constants. This is very important for comparing string theory predictions at low energy
with the real world. In chapter 14 a brief introduction to non-perturbative string connections
and non-perturbative effects is given. This is a fast-changing subject and I have
just included some basics as well as tools, so that the reader orients him(her)self in the
web of duality connections. Finally, in chapter 15 a brief outlook and future problems are
presented.
I have added a number of appendices to make several technical discussions self-contained.
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In Appendix A useful information on the elliptic ϑ-functions is included. In Appendix B,
I rederive the various lattice sums that appear in toroidal compactifications. In Appendix
C the Kaluza-Klein ansatz is described, used to obtain actions in lower dimensions after
toroidal compactification. In Appendix D some facts are presented about four-dimensional
locally supersymmetric theories with N=1,2,4 supersymmetry. In Appendix E, BPS states
are described along with their representation theory and helicity supertrace formulae that
can be used to trace their appearance in a supersymmetric theory. In Appendix F facts
about elliptic modular forms are presented, which are useful in many contexts, notably
in the one-loop computation of thresholds and counting of BPS multiplicities. In Appendix
G, I present the computation of helicity-generating string partition functions and
the associated calculation of BPS multiplicities. Finally, in Appendix H, I briefly review
electric–magnetic duality in four dimensions.
I have not tried to be complete in my referencing. The focus was to provide, in most
cases, appropriate reviews for further reading. Only in the last chapter, which covers
very recent topics, I do mostly refer to original papers because of the scarcity of relevant
reviews.