Geometry of representation spaces in SU(2)

TL;DR

This paper explores the structure of representation spaces of 3-manifold fundamental groups in SU(2), covering techniques like twisted cohomology, gauge theory, and geometric quantization, with applications to Chern-Simons theory and integrable systems.

Contribution

It provides detailed examples and develops methods for analyzing representation spaces, including twisted cohomology, gauge theory, and geometric quantization, with new insights into their geometric structure.

Findings

Description of the geometry of SU(2) representation spaces

Development of techniques for twisted cohomology and gauge theory

Application of geometric quantization to representation spaces

Abstract

These notes of a course given at IRMA in April 2009 cover some aspects of the representation theory of fundamental groups of manifolds of dimension at most 3 in compact Lie groups, mainly $\su$. We give detailed examples, develop the techniques of twisted cohomology and gauge theory. We review Chern-Simons theory and describe an integrable system for the representation space of a surface. Finally, we explain some basic ideas on geometric quantization. We apply them to the case of representation spaces by computing Bohr-Sommerfeld orbits with metaplectic correction.