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1 A first look at strings 1

1.1 Why strings? 1

1.2 Action principles 9

1.3 The open string spectrum 16

1.4 Closed and unoriented strings 25

Exercises 30

2 Conformal field theory 32

2.1 Massless scalars in two dimensions 32

2.2 The operator product expansion 36

2.3 Ward identities and Noether's theorem 41

2.4 Conformal invariance 43

2.5 Free CFTs 49

2.6 The Virasoro algebra 52

2.7 Mode expansions 58

2.8 Vertex operators 63

2.9 More on states and operators 68

Exercises 74

3 The Polyakov path integral 77

3.1 Sums over world-sheets 77

3.2 The Polyakov path integral 82

3.3 Gauge fixing 84

3.4 The Weyl anomaly 90

3.5 Scattering amplitudes 97

3.6 Vertex operators 101

3.7 Strings in curved spacetime 108

Exercises 118

4 The string spectrum 121

4.1 Old covariant quantization 121

4.2 BRST quantization 126

4.3 BRST quantization of the string 131

4.4 The no-ghost theorem 137

Exercises 143

5 The string S-matrix 145

5.1 The circle and the torus 145

5.2 Moduli and Riemann surfaces 150

5.3 The measure for moduli 154

5.4 More about the measure 159

Exercises 164

6 Tree-level amplitudes 166

6.1 Riemann surfaces 166

6.2 Scalar expectation values 169

6.3 The bc CFT 176

6.4 The Veneziano amplitude 178

6.5 Chan-Paton factors and gauge interactions 184

6.6 Closed string tree amplitudes 192

6.7 General results 198

Exercises 204

7 One-loop amplitudes 206

7.1 Riemann surfaces 206

7.2 CFT on the torus 208

7.3 The torus amplitude 216

7.4 Open and unoriented one-loop graphs 222

Exercises 229

8 Toroidal compactification and T-duality 231

8.1 Toroidal compactification in field theory 231

8.2 Toroidal compactification in CFT 235

8.3 Closed strings and T-duality 241

8.4 Compactification of several dimensions 249

8.5 Orbifolds 256

8.6 Open strings 263

8.7 D-branes 268

8.8 T-duality of unoriented theories 277

Exercises 280

9 Higher order amplitudes 283

9.1 General tree-level amplitudes 283

9.2 Higher genus Riemann surfaces 290

9.3 Sewing and cutting world-sheets 294

9.4 Sewing and cutting CFTs 300

9.5 General amplitudes 305

9.6 String field theory 310

9.7 Large order behavior 315

9.8 High energy and high temperature 317

9.9 Low dimensions and noncritical strings 323

Exercises 327

Appendix A:A short course on path integrals 329

A.1 Bosonic fields 329

A.2 Fermionic fields 341

Exercises 345

References 347

Glossary 359

Index 389