Description

Conformal Field Theories are the special class of quantum field theories which have conformal symmetry. Quantum Field Theories, which are central to the modern description of physics at the smallest scales, are generally complicated systems with infinitely many degrees of freedom which interact with each other through nonlinear terms in the equations of motion. Whilst this rich structure is needed to describe the physics we see in nature, such theories are beyond exact mathematical analysis generically. Conformal symmetry in two dimensions is related to analytic maps which one meets in complex analysis. It is a huge symmetry, much larger than say rotational symmetry. Indeed, it is so big that it is able to tame the infinite degrees of freedom of a quantum field theory, and in some instances render it completely solvable.

Besides its intrinsic mathematical interest, two-dimensional conformal field theory has a number of important applications. In particular, it is at the heart of string theory, which as a consistent theory of quantum gravity, has been a dominant area of research in theoretical particle physics in recent years. Although such strings live in many dimensions (typically more than four!), as they move through space and time they sweep out a two-dimensional surface or worldsheet. It is through this worldsheet description of strings that two-dimensional conformal field theory comes into play.

The project will begin in the first term by covering the basic principles involved. This would include understanding conformal symmetry in two or more dimensions, infinitesimal conformal transformations and the infinite-dimensional Virasoro algebra, primary fields, operator product expansions and correlation functions. Minimal models will be the primary example of a conformal field theory that can be solved exactly.

More specialised topics might involve studying more advanced examples (WZW models and coset models), and corresponding extended conformal symmetries (Kac-Moody Algebras, W-algebras). Other areas to investigate would be the application of conformal field theories to statistical models at phase-transitions, or to string theory.

Prerequisites

Resources

• Conformal Field Theory (Di Francesco et al.)
• Introduction to Conformal Field Theory (Blumenhagen et al.)

• Applied Conformal Field Theory (Ginsparg)
• Introduction to ConformalField Theory (Schellekens)
• Introduction to Conformal Field Theory (Zuber)