1 Introduction ................................................. 1
2 Basics in Conformal Field Theory ............................. 5
2.1 The Conformal Group ..................................... 5
2.1.1 Conformal Invariance ............................. 5
2.1.2 Conformal Group in d ≥ 3 ......................... 8
2.1.3 Conformal Group in d = 2 ........................ 12
2.2 Primary Fields ......................................... 17
2.3 The Energy-Momentum Tensor ............................. 19
2.4 Radial Quantisation .................................... 20
2.5 The Operator Product Expansion ......................... 23
2.6 Operator Algebra of Chiral Quasi-Primary Fields ........ 29
2.6.1 Conformal Ward Identity ......................... 29
2.6.2 Two- and Three-Point Functions .................. 30
2.6.3 General Form of the OPE for Chiral Quasi-
Primary Fields .................................. 32
2.7 Normal Ordered Products ................................ 37
2.8 The CFT Hilbert Space .................................. 41
2.9 Simple Examples of CFTs ................................ 44
2.9.1 The Free Boson .................................. 44
2.9.2 The Free Fermion ................................ 56
2.9.3 The (b,c) Ghost Systems ......................... 67
2.10 Highest Weight Representations of the Virasoro
Algebra ................................................ 70
2.11 Correlation Functions and Fusion Rules ................. 76
2.12 Non-Holomorphic OPE and Crossing Symmetry .............. 81
2.13 Fusing and Braiding Matrices ........................... 84
Further Reading ............................................. 86
3 Symmetries of Conformal Field Theories ...................... 87
3.1 Kac-Moody Algebras ..................................... 87
3.2 The Sugawara Construction .............................. 88
3.3 Highest Weight Representations of  (2)k ............... 92
3.4 The (N)1 Current Algebra ............................. 97
3.5 The Knizhnik-Zamolodchikov Equation .................... 99
3.6 Coset Construction .................................... 102
3.7  Algebras ........................................... 106
Further Reading ............................................ 1ll
4 Conformal Field Theory on the Torus ........................ 113
4.1 The Modular Group of the Torus and the Partition
Function .............................................. 114
4.2 Examples for Partition Functions ...................... 120
4.2.1 The Free Boson ................................. 120
4.2.2 The Free Boson on a Circle ..................... 122
4.2.3 The Free Boson on a Circle of Radius
R = √2k ........................................ 126
4.2.4 The Free Fermion ............................... 130
4.2.5 The Free Boson Orbifold ........................ 138
4.3 The Verlinde Formula .................................. 142
4.4 The (2)k Partition Functions ........................ 146
4.5 Modular Invariants of Virc<1 .......................... 149
4.6 The Parafermions ...................................... 152
4.7 Simple Currents ....................................... 156
4.8 Additional Topics ..................................... 164
4.8.1 Asymptotic Growth of States in RCFTs ........... 164
4.8.2 Dilogarithm Identities ......................... 166
Further Reading ............................................ 167
5 Supersymmetric Conformal Field Theory ...................... 169
5.1  = 1 Superconformal Models .......................... 169
5.2 = 2 Superconformal Models .......................... 175
5.3 Chiral Ring ........................................... 181
5.4 Spectral Flow ......................................... 184
5.5 Coset Construction for the = 2 Unitary Series ...... 187
5.6 Gepner Models ......................................... 190
5.7 Massless Modes of Gepner Models ....................... 201
Further Reading ............................................ 203
6 Boundary Conformal Field Theory ............................ 205
6.1 The Free Boson with Boundaries ........................ 206
6.1.1 Boundary Conditions ............................ 206
6.1.2 Partition Function ............................. 211
6.2 Boundary States for the Free Boson .................... 213
6.2.1 Boundary Conditions ............................ 214
6.2.2 Tree-Level Amplitudes .......................... 220
6.3 Boundary States for RCFTs ............................. 225
6.4 CFTs on Non-Orientable Surfaces ....................... 229
6.5 Crosscap States for the Free Boson .................... 239
6.6 Crosscap States for RCFTs ............................. 245
6.7 The Orientifold of the Bosonic String ................. 248
Further Reading ............................................ 256
Concluding Remarks ......................................... 257
General Books on CFT and String Theory ..................... 259
Index ......................................................... 261
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