Conformal Field Theory in Two Dimensions

Annotation

This is the follow-up course on conformal field theory in two dimensions (CFT2). We explicitly describe representation-theoretical structures underlying CFT2 operator content like (ir)reducible Verma modules of Virasoro algebra, operator product algebras, fusion, etc. As an application we describe a class of exactly solvable CFT2s called minimal models. For instance, this includes the Ising model in the critical point known from condensed matter physics. We also discuss an additional constraint on the operator content imposed by the modular invariance which allows for a consistent definition of CFT2 on any Riemann surface. As an example, we study CFT2 defined on a torus and study various implications of the modular invariance.

Course plan

1. Degenerate primary operators and singular vectors in Verma modules. 
2. Belavin-Polyakov-Zamolodchikov equation. Fusion for 3-point correlation function.
3. Fusion rules for general operators. Operator algebra truncation.
4. Reducible Verma modules. Verma characters and their properties. Characters for minimal models. Fusion algebra.
5. Minimal models, Kac table, operator algebra. Unitary minimal models. 
6. Examples: Lee-Yang model, Ising model, tri-critical Ising model (supersymmetry).
7. Introduction to torus CFT. Metric on two-dimensional torus, lattice realization. Modular transformations. 
8. Triple Jacobi identity and fermion-antifermion partition function.  
9. Poisson resummation formula. Jacobi theta-functions: periodicity properties, modular properties, product representation. Dedekind eta function. 
10. Torus partition functions. Modular invariance and operator spectra in minimal models.
11. Verlinde formula. Cardy formula. 
12. Free boson and fermion on a torus and their modular properties.

Literature

1. P. Di Francesco, P. Mathieu, D. Senechal. Conformal Field Theory, Springer 1997
2. R. Blumenhagen,  E. Plauschinn. Introduction to conformal field theory (with applications to string theory), Springer 2009 
3. P. Ginsparg. Applied conformal field theory. Les Houches Summer School 1988:1-168, e-Print:hep-th/9108028 [hep-th]
4. M. Schottenloher. A Mathematical Introduction to Conformal Field Theory (Lecture Notes in Physics, 759)