Conformal Field Theory A.N. Schellekens [Word cloud by www.worldle.net] Last modified 16 Januari 2017 1 Contents 1 Classical Conformal Invariance7 1.1 Symmetries...................................7 1.1.1 General coordinate invariance.....................7 1.1.2 Weyl invariance.............................7 1.1.3 Conformal invariance..........................8 1.2 Conformal transformations in d dimensions.................9 1.3 The conformal group.............................. 11 1.4 Relation to stereographic projections..................... 12 1.5 The conserved current............................. 13 1.6 An example: the free boson.......................... 13 1.7 The conformal algebra in two dimensions.................. 14 1.8 Complexification and Wick rotation..................... 16 1.9 The global subgroup.............................. 16 1.10 Tensors in complex coordinates........................ 18 1.11 Conformal fields in two dimensions...................... 19 1.12 Relation to string theory........................... 20 1.13 Free bosons in complex coordinates...................... 20 2 Quantum Conformal Invariance 21 2.1 Radial quantization.............................. 21 2.2 Radial ordering................................. 24 2.3 The generator of conformal transformations................. 24 2.4 Quantization of the free boson........................ 26 2.5 The free boson propagator........................... 28 2.6 The normally ordered energy momentum tensor............... 30 2.7 Operator products for free bosons...................... 30 2.8 The Virasoro algebra.............................. 31 3 Virasoro Representation Theory 32 3.1 Unitarity.................................... 33 3.2 Highest weight representations........................ 33 3.3 The vacuum.................................. 34 3.4 Positivity of c and h .............................. 34 3.5 States and conformal fields.......................... 35 3.6 Descendant fields................................ 36 3.7 The Kac determinant............................. 36 3.8 The Kac determinant at level 2........................ 38 3.9 The Kac determinant at level 3 and 4.................... 40 3.10 The discrete series of minimal unitary Virasoro models........... 41 2 4 Correlation Functions 42 4.1 Correlation functions on the Riemann sphere................ 43 4.2 Two-point functions.............................. 44 4.3 Three-point functions............................. 45 4.4 Four-point functions.............................. 46 4.5 Conformal Ward identities........................... 47 4.6 Correlators of descendants........................... 47 4.7 Null state decoupling............................. 49 4.8 Operator products............................... 50 4.9 Duality..................................... 51 5 Conformal Field Theory on a Torus 52 5.1 Parametrization of the torus......................... 52 5.2 The partition function............................. 53 5.3 The cylinder versus the Riemann sphere................... 55 5.4 Virasoro characters............................... 57 5.5 Modular invariance............................... 57 5.6 Modular transformations of the characters.................. 59 5.7 Conditions for modular invariance...................... 59 5.8 The diagonal invariant............................. 60 5.9 Integration over moduli............................ 60 5.10 Operator products and locality........................ 60 5.11 Fusion rules................................... 61 5.12 The Verlinde formula............................. 62 5.13 Higher genus partition functions....................... 63 6 Extensions of the Virasoro Algebra 64 6.1 Rational conformal field theories....................... 64 6.2 Currents.................................... 65 6.3 Fermionic currents............................... 65 6.4 Mode expansions................................ 66 6.5 Integer and half-integer modes........................ 66 6.6 Types of chiral algebra extensions...................... 67 6.7 Properties of extended Virasoro representations............... 67 6.8 Charge conjugation.............................. 69 6.9 Characters and modular transformations................... 69 6.10 Virasoro tensor products........................... 71 6.11 Extensions and off-diagonal partition functions............... 71 6.12 The new S and T matrices.......................... 72 6.13 Extensions and automorphisms........................ 73 6.14 Simple currents................................. 73 3 7 Free Fermions 74 7.1 The propagator................................. 74 7.2 Energy-Momentum tensor and central charge................ 74 7.3 Mode expansion................................ 75 7.4 The spin field.................................. 75 7.5 Free fermion characters............................ 76 7.5.1 Neveu-Schwarz states.......................... 76 7.5.2 Neveu-Schwarz characters....................... 78 7.5.3 Ramond states............................. 78 7.5.4 The Ramond ground state....................... 80 7.5.5 Ramond characters........................... 80 7.6 The partition function............................. 80 7.7 Theta-functions................................ 81 7.8 Modular transformations........................... 82 7.9 The modular invariant partition function.................. 83 7.10 Ising characters................................. 83 7.11 The matrix S and the fusion rules...................... 84 7.12 Multi-fermion systems............................. 84 8 Free Boson Partition Functions 84 8.1 The spectrum.................................. 85 8.2 The characters and the diagonal invariant.................. 85 8.3 Chiral bosons.................................. 86 8.4 Further extensions of the chiral algebra................... 87 8.5 Representations of the extended algebra................... 88 8.6 The matrix S .................................. 89 8.7 Relation with circle compactification..................... 91 8.8 R 2=R duality................................ 92 8.9 Rationality! ................................... 92 8.10 Theories with more than one free boson................... 92 8.11 Orbifolds.................................... 93 8.12 Orbifolds as singular manifolds........................ 93 8.13 Orbifolds in conformal field theory...................... 93 8.14 Orbifolds of the circle............................. 94 8.14.1 The twist fields............................. 94 8.15 Projecting on the invariant states....................... 95 8.16 The partition function............................. 97 8.17 The geometric description........................... 97 8.18 The c = 1 conformal field theories...................... 98 8.19 Moduli and marginal deformations...................... 99 4 9 Kac-Moody Algebras 100 9.1 Spin one operator products.......................... 100 9.2 Intermezzo: some Lie algebra facts...................... 100 9.3 The central term................................ 102 9.4 Modes...................................... 103 9.5 Twisted and untwisted affine algebras.................... 103 9.6 Primary fields................................. 104 9.7 The Sugawara tensor.............................. 104 9.8 Highest weight representations........................ 105 9.9 U(1) theories.................................. 106 9.10 The SU(2) Kac-Moody algebra........................ 106 9.11 SU(2) at level 1................................. 108 9.12 Generalization to other Kac-Moody algebras................. 108 9.13 The Frenkel-Kac construction......................... 111 9.14 The WZW-model............................... 111 9.15 Modular transformation properties...................... 112 9.16 Modular invariant partition functions for SU(2)............... 113 9.17 Fusion rules and simple currents....................... 113 9.18 Modular invariant partition functions for other Kac-Moody algebras... 114 9.19 Coset conformal field theories......................... 114 9.20 The minimal discrete series as a coset theory................ 116 9.21 Field Identification............................... 117 9.22 Other coset models............................... 117 10 Superconformal Algebras 118 10.1 The N =1 algebra............................... 118 10.2 The N =2 algebra............................... 120 5 Preface Conformal field theory has been an important tool in theoretical physics during the last decades. Its origins can be traced back on the one hand to statistical mechanics, and on the other hand to string theory. Historically the most important impetus came from statistical mechanics, where it described and classified critical phenomena. Mainly after 1984 the subject went through a period of rapid development because of its importance for string theory. In addition there has been important input from mathematics, in particular through the work of Kac and collaborators. One can distinguish yet another separate origin of some ideas, namely from work on rigorous approaches to quantum field theory. At present the subject still continues to develop, and it is still important in all the fields mentioned, plus a few additional branches of mathematics. These lectures are mainly on two-dimensional CFT. Recently conformal field theory appeared in yet another context, namely the \AdS/CFT-correspondence", where also higher dimensional (super)conformal invariance is relevant. I tried to include references to most relevant papers, but the emphasis was on papers I consider to be worth reading even today, and not on papers that are mainly of historical interest.