# Quantization of moduli spaces

### Aims

Introduction to topological quantum field theory, invariants of 3-manifolds, quantum moduli spaces, and conformal field theory.

### Contents

This course lasts one year. The course consists of three parts.

First part concerns the geometry of the moduli space of flat *G*-bundles
over surfaces (also known as representation varieties of the
fundamental group of a surface). There are several approaches to this
subject. Fock and Reshetikhin will explain the algebraic approach,
Andersen will explain how geometric quantization works in this
context.

In the second part, some known constructions of invariants of
3-manifolds will be explained. The functorial arrangement of these
invariants is known as a topological quantum field theory (TQFT). It
can be regarded as a functor from the category of cobordisms to the
category of vector spaces (linear systems).

The third part will start with the review of Segal's axioms of
conformal field theory, and then some of the constructions used in the
TQFT part will be extended to corresponding conformal field theories.

### Prerequisites

Topology

### Types of teaching

3-4 hours of lectures per week.

### Lecturers

Jørgen Ellegaard Andersen, Vladimir Fock and Nicolai Reshetikhin

### Teaching materials/Text-books

There is a vast literature on TQFT and conformal field theories. There
is also a substantial amount of literature on the quantization of moduli
spaces. As lectures will progress, the relevant references will be
given.

Some references for preliminary reading can be found at Andersen's web-site.