SS 2011 •
Prof. Dr. Christoph Bohle
Dr. Sebastian Heller
Ab der 2. Vorlesungswoche
Beginn der Übung: 3. Vorlesungswoche
One of the major problems of contemporary mathematics is to develop a
better understanding of quantum field theory, the theory on which
large parts of fundamental physics are based. Despite its success as a
physical theory, much of quantum field theory in its present form is
lacking a rigorous mathematical foundation. This situation is highly
dissatisfactory, considering that in the last decades quantum field
theory turned into a preeminent source of mathematical inspiration.
Conformal field theory is a part of quantum field theory which allows
a mathematically rigorous treatment. Although is encompasses only a
small portion of quantum field theory (that of 2D quantum field
theories with local conformal symmetry), it is a good starting point
for mathematicians to get acquainted with quantum field theory
ideas. This is not only due to its mathematical accessibility, but
also because of its connection to several important mathematical
We will essentially approach conformal field theory as a mathematical
theory. The main part of the lecture will be an introduction to vertex
algebras, the algebraic structure underlying conformal field theory.
In the last part of the lecture we plan to discuss some mathematical
applications related to integrable systems and the theory of vector
bundles on Riemann surfaces. We shamefully ignore all physical
applications (although they are plenty, most prominently in string
theory and statistical mechanics, where conformal field theory
describes critical phenomena).
Prerequisites: A basic knowledge of Lie algebras and Riemann surfaces
will be helpful. We do not assume any knowledge of physics, although a
deeper understanding surely needs some.