Ph. D. Thesis: Pre-quantization of the moduli space of flat G-bundles

TL;DR

This thesis investigates the conditions for pre-quantizing the moduli space of flat G-bundles, focusing on cohomological obstructions and explicit level criteria for various Lie groups, including non-simply connected cases.

Contribution

It identifies the cohomological obstructions to pre-quantization for non-simply connected groups and explicitly determines admissible levels for all such compact simple Lie groups.

Findings

Pre-quantization is compatible with symplectic reduction and fusion.

Obstructions are characterized by a specific cohomology class in H^3(G×G;Z).

Explicit levels for pre-quantization are provided for all non-simply connected simple Lie groups.

Abstract

This thesis studies the pre-quantization of quasi-Hamiltonian group actions from a cohomological viewpoint. The compatibility of pre-quantization with symplectic reduction and the fusion product are established, and are used to understand the sufficient conditions for the pre-quantization of $M_G(\Sigma)$, the moduli space of flat $G$-bundles over a closed surface $\Sigma$. For a simply connected, compact, simple Lie group $G$, $M_G(\Sigma)$ 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 thesis determines the obstruction---namely a certain cohomology class in $H^3(G\times G;\Z)$---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 $G$. Partial results are obtained for the case of a surface $\Sigma$ with marked points. Also, it is shown that via the bijective correspondence between quasi-Hamiltonian group actions and Hamiltonian loop group actions, the corresponding notions of pre-quantization coincide.