MA5P8 - Conformal field theory

Lecturer: Dr. K. Wendland
Office: B1.34
Dr. K. Wendland


A 24 CAT module giving an introduction to conformal field theory for mathematicians, on beginning graduate student level.

Over the past decades, conformal field theory (CFT) has become one of the most active domains of interaction between mathematics and theoretical physics. Its roots date back at least as far as 1910, when Cunningham and Bateman observed that Maxwell's equations obey all conformal symmetries.
From a physicists' point of view, there are many reasons to study conformal field theory, like its applications in statistical mechanics, solid state physics and integrable systems. In string theory, the modern so far speculative attempt to unify the forces of Nature, CFTs describe possible string vacua. On the other hand, conformal field theory has developed to tempting mathematical beauty; its mathematical applications range from finite group theory over infinite-dimensional Lie algebras to parts of topology, modular forms, and algebraic geometry.
In 1998, R.E. Borcherds was awarded a fields medal for his work in automorphic forms and mathematical physics, where he in particular made use of conformal field theory. However, Borcherds is known to have said that either you know what CFTs are or you don't want to know. This statement must be viewed as a reaction to the current state of introductory literature, which can be particularly confusing from a mathematician's point of view.
This course is meant to give a digestible mathematical introduction to conformal field theory, assuming no background knowledge from quantum field theory. The aim is to motivate and explain the relevant terminology and in particular to discuss fundamental examples of CFTs in detail. This should enable the student to work with the existing literature by the end of the course.
More specifically, the Lie algebra associated to the infinitesimal conformal symmetries of the punctured Euclidean plane has a unique one-parameter family of nontrivial central extensions by C, the Virasoro algebra. The course will explain the role of the Virasoro algebra and its representation theory as fundamental building block for conformal field theories. We will construct the simplest examples of CFTs and study their properties, thereby motivating axiomatic approaches to CFT. By introducing specific moduli spaces of CFTs we will highlight some connections to string theory. If time allows, we will also introduce affine Kac-Moody algebras and non-trivial constructions of CFTs derived from them.
A detailed plan of the module is included in the module material below.

As prerequisites, basic undergraduate courses like linear algebra and group theory should suffice; some knowledge of complex analysis, is advisable; differential geometry and representation theory will be helpful but not required; no background knowledge in physics required.

Lecture notes will be provided as the module progresses.


Lectures are on
     Mondays and Tuesdays, 12:00-1:00, and on Wednesdays, 1:00-2:00 in MS05,
starting Friday, January 7, 2005.
Here is the collection of exercises which I set for this module in 2004: 1, 2, 3, 4, 5 (postscript); 1, 2, 3, 4, 5 (pdf).

Module Material (2004/2005)