On moduli spaces of flat connections with non-simply connected structure group

TL;DR

This paper studies the moduli space of flat G-bundles over a torus for non-simply connected Lie groups, revealing an isomorphism with trivial bundles via a change in structure group, with implications for physics.

Contribution

It demonstrates an isomorphism between moduli spaces of topologically non-trivial and trivial bundles for non-simply connected groups, linking topology and gauge group structure.

Findings

Connected components correspond to topologically non-trivial bundles.

Isomorphism as symplectic spaces with trivial bundle moduli spaces.

Potential applications in physical theories involving gauge groups.

Abstract

We consider the moduli space of flat G-bundles over the twodimensional torus, where G is a real, compact, simple Lie group which is not simply connected. We show that the connected components that describe topologically non-trivial bundles are isomorphic as symplectic spaces to moduli spaces of topologically trivial bundles with a different structure group. Some physical applications of this isomorphism which allows to trade topological non-triviality for a change of the gauge group are sketched.