It is a part of
a program on Advanced Mathematical Physics for BSc/MSc
students

at
St. Petersburg Division of Steklov Mathematical Institute

http: //www.ipme.ru/zam.html

The aim of this course is to provide the necessary background in Conformal Field Theory (CFT) for the further study of different topics in the following modern Math/Phys and Phys/Math directions, such as:

**Math**:

Infinite-dimensional algebras (Kac-Moody and Virasoro)
and their representations.

Vertex operator algebras.

Boson-fermion correspondence.

**Phys**:

(Super) String theory and a lot of related things.

(Solvable) Lattice models.

The lectures are destined for the BSc/MSc students (3-5 years of study and more) from the Mathematical Physics and High Energy Physics Departments of St.Petersburg State University.

- Basics from the classical field theory:

The Euler-Lagrange equations, Noether currents. Energy momentum tensor. Field theory in curved spaces. Differential-geometrical background (Lie derivatives, Killing vectors, Symmetries (Diff-invariance as an important example)). - Weyl, scale and conformal invariance in D dimensions. Infinite-dimensional conformal symmetry in 2d. Global and Local transformations. Classical conformal invariance: conserved currents. Examples of invariant Lagrangians.
- Towards quantum conformal invariance: Polyakov's bootstrap conjecture. The Operator Product Expansion (OPE).
- Massless scalars in 2d: Holomorphic ordering and examples of the OPE.
- Conservaton of currents on the quantum level: Ward identities. Examples.
- The cylinder and the plane: time and radial ordering. Correlation functions and quantum conformal invariance. Holomorphic currents and Contour-commutator trick. The mode expansion of energy-momentum tensor and Virasoro algebra. Central extensions.
- Primary fields and states. Massless scalars in 2d and the construction of primary fields. Descendant states.
- Conformal vacuum. The adjoint operator. 2,3-point correlation functions. Zamolodchikov's inner product.
- Oscillator mode expansion for the free massless scalars in 2d. Holomorphic and Fock ordering. The Fock space of states. The Virasoro generators in terms of oscillators.
- Some facts from the representation theory of the Virasoro algebra. Unitarity. Singular vectors.
- Singular vectors and the constraints on the correlation functions and the OPE. Minimal models and Rational CFT (RCFT) - a review. Lattice models.
- Strings... Towards Strings.

**Exercises:**

- Massless scalars on a circle. Space of states. The dependence on the radius of the circle.
- Massless scalars with a dilaton. Energy-momentum tensor. Central charge. Primary fields.
- b,c ghost system. Energy-momentum tensor. Central charge. Mode expansion. The construction of the conformal vacuum.
- Free massless Majorana fermion. Energy-momentum tensor, OPE, central charge. R and NS sectors: mode expansion.

**Recommended References:**

- P. Di Francesco, P. Mathieu, D. Senechal "Conformal Field Theory" 1997, ch. 1-7
- J. Polchinski "String Theory" 1998, ch. 2,15