Pre-quantization of the Moduli Space of Flat G-Bundles over a Surface

TL;DR

This paper investigates the conditions under which the moduli space of flat G-bundles over a surface can be pre-quantized, especially focusing on non-simply connected Lie groups and identifying the cohomological obstructions involved.

Contribution

It explicitly determines the obstruction class in H^3(G^2;Z) for non-simply connected groups and characterizes the levels that allow pre-quantization.

Findings

Identifies the cohomological obstruction to pre-quantization for non-simply connected G.

Provides explicit criteria for levels admitting pre-quantization for all such G.

Extends known results from simply connected to non-simply connected Lie groups.

Abstract

For a simply connected, compact, simple Lie group G, the moduli space of flat G-bundles over a closed surface is known to be pre-quantizable at integer levels. For non-simply connected G, however, integrality of the level is not sufficient for pre-quantization, and this paper determines the obstruction -- namely a certain cohomology class in H^3(G^2;Z) -- that places further restrictions on the underlying level. The levels that admit a pre-quantization of the moduli space are determined explicitly for all non-simply connected, compact, simple Lie groups G.