# Lectures on Conformal Field Theory

### Program

- General coordinate, Weyl and Conformal invariance.
Conformal group in D > 2 dimensions.
- Conformal group in 2D. The Witt algebra. PSL(2,C).
Introducing CFT: cylinder and plane. Radial quantization.
- Example of CFT: free massless boson.
OPE, holomorphic normal ordering.
- Ward identities. Examples.
- Energy-momentum tensor. Scaling dimensions.
Primary fields. Examples. TT-OPE, central charge.
- Virasoro algebra. "Commutator-residue" identities.
- Mode expansions.
- Highest weight states of Virasoro algebra,
- Representations of Virasoro algebra, Kac determinant,
unitarity, c=1, c=0 representations.
- Correlation functions. Conformal Ward identity.
PSL(2,C) invariance.
- 2,3,4-point correlation functions. The use of PSL(2,C) invariance.
- OPE and correlation functions. Conformal blocks.
Introduction to Moore-Seiberg theory:
Fusing-Braiding relations.
- Another look on Kac determinant. Singular (Null) vectors.
Constraints on OPE.
- Kac table, Minimal models
- Applications of CFT to Condenced Matter Physics.
- CFT on a torus. Modular invariance.
- Free massless fermion: Ramond and Neveu-Schwarz sectors.
Modular invariant partition function. Identification with
critical Ising model.
- Verlinde formula for fusion rules.
- Extensions of Virasoro algebra. WZNW theories,
Superconformal algebras, W-algebras.
- Detailed study of WZNW theory.
Knizhnik-Zamolodchikov equation.
- Coset theories.
- Another look on Moore-Seiberg theory.
Vafa relation. Rational CFT.
- ...Advanced topics.

### Recommended References:

#### Books:

- P. Di Francesco, P. Mathieu, D. Senechal "Conformal Field Theory" 1997.
- J. Polchinski "String Theory" 1998.

#### Links to Key Papers and Reviews: