Shing-Tung Yau

Contents

Preface page xi

1 Introduction 1

1.1 Historical origins 2

1.2 General features 3

1.3 Basic string theory 6

1.4 Modern developments in superstring theory 9

2 The bosonic string 17

2.1 p-brane actions 17

2.2 The string action 24

2.3 String sigma-model action: the classical theory 30

2.4 Canonical quantization 36

2.5 Light-cone gauge quantization 48

3 Conformal ¬eld theory and string interactions 58

3.1 Conformal ¬eld theory 58

3.2 BRST quantization 75

3.3 Background ¬elds 81

3.4 Vertex operators 85

3.5 The structure of string perturbation theory 89

3.6 The linear-dilaton vacuum and noncritical strings 98

3.7 Witten™s open-string ¬eld theory 100

4 Strings with world-sheet supersymmetry 109

4.1 Ramond“Neveu“Schwarz strings 110

4.2 Global world-sheet supersymmetry 112

4.3 Constraint equations and conformal invariance 118

4.4 Boundary conditions and mode expansions 122

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viii Contents

4.5 Canonical quantization of the RNS string 124

4.6 Light-cone gauge quantization of the RNS string 130

4.7 SCFT and BRST 140

5 Strings with space-time supersymmetry 148

5.1 The D0-brane action 149

5.2 The supersymmetric string action 155

5.3 Quantization of the GS action 160

5.4 Gauge anomalies and their cancellation 169

6 T-duality and D-branes 187

6.1 The bosonic string and Dp-branes 188

6.2 D-branes in type II superstring theories 203

6.3 Type I superstring theory 220

6.4 T-duality in the presence of background ¬elds 227

6.5 World-volume actions for D-branes 229

7 The heterotic string 249

7.1 Nonabelian gauge symmetry in string theory 250

7.2 Fermionic construction of the heterotic string 252

7.3 Toroidal compacti¬cation 265

7.4 Bosonic construction of the heterotic string 286

8 M-theory and string duality 296

8.1 Low-energy e¬ective actions 300

8.2 S-duality 323

8.3 M-theory 329

8.4 M-theory dualities 338

9 String geometry 354

9.1 Orbifolds 358

9.2 Calabi“Yau manifolds: mathematical properties 363

9.3 Examples of Calabi“Yau manifolds 366

9.4 Calabi“Yau compacti¬cations of the heterotic string 374

9.5 Deformations of Calabi“Yau manifolds 385

9.6 Special geometry 391

9.7 Type IIA and type IIB on Calabi“Yau three-folds 399

9.8 Nonperturbative e¬ects in Calabi“Yau compacti¬cations 403

9.9 Mirror symmetry 411

9.10 Heterotic string theory on Calabi“Yau three-folds 415

9.11 K3 compacti¬cations and more string dualities 418

9.12 Manifolds with G2 and Spin(7) holonomy 433

10 Flux compacti¬cations 456

10.1 Flux compacti¬cations and Calabi“Yau four-folds 460

10.2 Flux compacti¬cations of the type IIB theory 480

Contents ix

10.3 Moduli stabilization 499

10.4 Fluxes, torsion and heterotic strings 508

10.5 The strongly coupled heterotic string 518

10.6 The landscape 522

10.7 Fluxes and cosmology 526

11 Black holes in string theory 549

11.1 Black holes in general relativity 552

11.2 Black-hole thermodynamics 562

11.3 Black holes in string theory 566

11.4 Statistical derivation of the entropy 582

11.5 The attractor mechanism 587

11.6 Small BPS black holes in four dimensions 599

12 Gauge theory/string theory dualities 610

12.1 Black-brane solutions in string theory and M-theory 613

12.2 Matrix theory 625

12.3 The AdS/CFT correspondence 638

12.4 Gauge/string duality for the conifold and generalizations 669

12.5 Plane-wave space-times and their duals 677

12.6 Geometric transitions 684

Bibliographic discussion 690

Bibliography 700

Index 726

Preface

String theory is one of the most exciting and challenging areas of modern

theoretical physics. It was developed in the late 1960s for the purpose of de-

scribing the strong nuclear force. Problems were encountered that prevented

this program from attaining complete success. In particular, it was realized

that the spectrum of a fundamental string contains an undesired massless

spin-two particle. Quantum chromodynamics eventually proved to be the

correct theory for describing the strong force and the properties of hadrons.

New doors opened for string theory when in 1974 it was proposed to identify

the massless spin-two particle in the string™s spectrum with the graviton, the

quantum of gravitation. String theory became then the most promising can-

didate for a quantum theory of gravity uni¬ed with the other forces and has

developed into one of the most fascinating theories of high-energy physics.

The understanding of string theory has evolved enormously over the years

thanks to the e¬orts of many very clever people. In some periods progress

was much more rapid than in others. In particular, the theory has experi-

enced two major revolutions. The one in the mid-1980s led to the subject

achieving widespread acceptance. In the mid-1990s a second superstring

revolution took place that featured the discovery of nonperturbative duali-

ties that provided convincing evidence of the uniqueness of the underlying

theory. It also led to the recognition of an eleven-dimensional manifesta-

tion, called M-theory. Subsequent developments have made the connection

between string theory, particle physics phenomenology, cosmology, and pure

mathematics closer than ever before. As a result, string theory is becoming

a mainstream research ¬eld at many universities in the US and elsewhere.

Due to the mathematically challenging nature of the subject and the

above-mentioned rapid development of the ¬eld, it is often di¬cult for some-

one new to the subject to cope with the large amount of material that needs

to be learned before doing actual string-theory research. One could spend

several years studying the requisite background mathematics and physics,

but by the end of that time, much more would have already been developed,

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xii Preface

and one still wouldn™t be up to date. An alternative approach is to shorten

the learning process so that the student can jump into research more quickly.

In this spirit, the aim of this book is to guide the student through the fasci-

nating subject of string theory in one academic year. This book starts with

the basics of string theory in the ¬rst few chapters and then introduces the

reader to some of the main topics of modern research. Since the subject is

enormous, it is only possible to introduce selected topics. Nevertheless, we

hope that it will provide a stimulating introduction to this beautiful subject

and that the dedicated student will want to explore further.