Lectures on Conformal Field Theory

Program

1. General coordinate, Weyl and Conformal invariance. Conformal group in D > 2 dimensions.
2. Conformal group in 2D. The Witt algebra. PSL(2,C). Introducing CFT: cylinder and plane. Radial quantization.
3. Example of CFT: free massless boson. OPE, holomorphic normal ordering.
4. Ward identities. Examples.
5. Energy-momentum tensor. Scaling dimensions. Primary fields. Examples. TT-OPE, central charge.
6. Virasoro algebra. "Commutator-residue" identities.
7. Mode expansions.
8. Highest weight states of Virasoro algebra,
9. Representations of Virasoro algebra, Kac determinant, unitarity, c=1, c=0 representations.
10. Correlation functions. Conformal Ward identity. PSL(2,C) invariance.
11. 2,3,4-point correlation functions. The use of PSL(2,C) invariance.
12. OPE and correlation functions. Conformal blocks. Introduction to Moore-Seiberg theory: Fusing-Braiding relations.
13. Another look on Kac determinant. Singular (Null) vectors. Constraints on OPE.
14. Kac table, Minimal models
15. Applications of CFT to Condenced Matter Physics.
16. CFT on a torus. Modular invariance.
17. Free massless fermion: Ramond and Neveu-Schwarz sectors. Modular invariant partition function. Identification with critical Ising model.
18. Verlinde formula for fusion rules.
19. Extensions of Virasoro algebra. WZNW theories, Superconformal algebras, W-algebras.
20. Detailed study of WZNW theory. Knizhnik-Zamolodchikov equation.
21. Coset theories.
22. Another look on Moore-Seiberg theory. Vafa relation. Rational CFT.
23. ...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:

• A.A. Belavin, A.M. Polyakov, A.B. Zamolodchikov "Infinite conformal symmetry in two-dimensional quantum field theory"
• P. Ginsparg "Applied Conformal Field Theory"
• A.N. Schellekens "Introduction to Conformal Field Theory"
• J. Bagger "Basic Conformal Field Theory"
• K. Gawedzki "Lectures on Conformal Field Theory"
• M. Gaberdiel "An Introduction to Conformal Field Theory"