MA5P8  Conformal field theory



Overview
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 infinitedimensional 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 oneparameter 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 KacMoody
algebras and nontrivial 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
Lectures are on
Mondays and Tuesdays, 12:001:00,
and on Wednesdays, 1:002:00
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)