2d Physics and Beyond
It is a part of
a program on Advanced Mathematical Physics for BSc/MSc
St. Petersburg Division of Steklov Mathematical Institute
Anton M. Zeitlin
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:
Infinite-dimensional algebras (Kac-Moody and Virasoro)
and their representations.
Vertex operator algebras.
(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.
The overview of the course:
- 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.
- 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.
- 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.
- Strings... Towards Strings.
- 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.
P. Di Francesco, P. Mathieu, D. Senechal "Conformal Field Theory" 1997, ch. 1-7
- J. Polchinski "String Theory" 1998, ch. 2,15