A TREATISE ON PHYSICS
CONTENTS 1 INTRODUCTION 2 BASICS IN CONFORMAL FIELD THEORY THE CONFORMAL GROUP CONFORMAL INVARIANCE CONFORMAL GROUP IN D ?3 CONFORMAL GROUP IN D = 2 PRIMARY FIELDS THE ENERGY–MOMENTUM TENSOR RADIAL QUANTISATION THE OPERATOR PRODUCT EXPANSION OPERATOR ALGEBRA OF CHIRAL QUASI-PRIMARY FIELDS CONFORMAL WARD IDENTITY TWO- AND THREE-POINT FUNCTIONS GENERAL FORM OF THE OPE FOR CHIRAL QUASI-PRIMARY FIELDS NORMAL ORDERED PRODUCTS THE CFT HILBERT SPACE SIMPLE EXAMPLES OF CFTS THE FREE BOSON THE FREE FERMION THE (B,C) GHOST SYSTEMS 2 10 HIGHEST WEIGHT REPRESENTATIONS OF THE VIRASORO ALGEBRA 2 11 CORRELATION FUNCTIONS AND FUSION RULES 2 12 NON-HOLOMORPHIC OPE AND CROSSING SYMMETRY 2 13 FUSING AND BRAIDING MATRICES FURTHER READING 3 SYMMETRIES OF CONFORMAL FIELD THEORIES KA?C–MOODY ALGEBRAS THE SUGAWARA CONSTRUCTION HIGHEST WEIGHT REPRESENTATIONS OF SU(2)K THE SO(N)1 CURRENT ALGEBRA THE KNIZHNIK–ZAMOLODCHIKOV EQUATION COSET CONSTRUCTION W ALGEBRAS 4 CONFORMAL FIELD THEORY ON THE TORUS THE MODULAR GROUP OF THE TORUS AND THE PARTITION FUNCTION EXAMPLES FOR PARTITION FUNCTIONS THE FREE BOSON ON A CIRCLE THE FREE BOSON ON A CIRCLE OF RADIUS R = ? 2K THE FREE BOSON ORBIFOLD THE VERLINDE FORMULA THE SU(2)K PARTITION FUNCTIONS MODULAR INVARIANTS OF VIRC<1 THE PARAFERMIONS SIMPLE CURRENTS ADDITIONAL TOPICS ASYMPTOTIC GROWTH OF STATES IN RCFTS DILOGARITHM IDENTITIES 5 SUPERSYMMETRIC CONFORMAL FIELD THEORY N = 1 SUPERCONFORMAL MODELS N = 2 SUPERCONFORMAL MODELS CHIRAL RING SPECTRAL FLOW COSET CONSTRUCTION FOR THE N = 2 UNITARY SERIES GEPNER MODELS MASSLESS MODES OF GEPNER MODELS 6 BOUNDARY CONFORMAL FIELD THEORY THE FREE BOSON WITH BOUNDARIES BOUNDARY CONDITIONS PARTITION FUNCTION BOUNDARY STATES FOR THE FREE BOSON TREE-LEVEL AMPLITUDES BOUNDARY STATES FOR RCFTS CFTS ON NON-ORIENTABLE SURFACES CROSSCAP STATES FOR THE FREE BOSON CROSSCAP STATES FOR RCFTS THE ORIENTIFOLD OF THE BOSONIC STRING CONCLUDING REMARKS