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

TL;DR

The paper establishes a canonical symplectic isomorphism between the moduli spaces $M_G^omega$ of flat $G$-bundles over a torus in non-trivial topological sectors and the moduli spaces $M_{G^omega}^1$ of flat bundles for a twisted simply connected group $G^omega$. By describing $M_G^omega$ via monodromies and folding to an orbit Lie algebra $\breve{\mathfrak g}$, the authors show $M_G^omega \cong M_{G^omega}^1$, and that the symplectic form satisfies $k\Omega_G = (k/N)\Omega_{\breve{\mathfrak g}}$, with $N$ the order of the automorphism $omega$. This result enables holomorphic quantization of the nontrivial sectors by translating them to the trivial sector of a different gauge group, linking geometric quantization to orbit Lie algebra structures and to applications in Chern-Simons theory, WZW models, and coset conformal field theories. A level quantization condition ($k$ multiple of $N$) emerges as the geometric counterpart of fixed-point phenomena in the associated quantum theories.

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.