Almost commuting elements in compact Lie groups

TL;DR

This paper characterizes the structure of the moduli space of commuting elements in compact Lie groups, computes associated Chern-Simons invariants, and confirms a conjecture relating these invariants to the moduli space components.

Contribution

It provides a detailed description of the moduli space components using Dynkin diagrams and verifies a conjecture connecting Chern-Simons invariants with moduli space dimensions.

Findings

Components of the moduli space are described via extended Dynkin diagrams.

Chern-Simons invariants are computed for flat bundles over the three-torus.

A conjecture relating invariants and moduli space dimensions is verified.

Abstract

We describe the components of the moduli space of conjugacy classes of commuting pairs and triples of elements in a compact Lie group. This description is in terms of the extended Dynkin diagram of the simply connected cover, together with the coroot integers and the action of the fundamental group. In the case of three commuting elements, we compute Chern-Simons invariants associated to the corresponding flat bundles over the three-torus, and verify a conjecture of Witten which reveals a surprising symmetry involving the Chern-Simons invariants and the dimensions of the components of the moduli space.