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This thesis studies the pre-quantization of quasi-Hamiltonian group actionsfrom a cohomological viewpoint. The compatibility of pre-quantization withsymplectic reduction and the fusion product are established, and are used tounderstand the sufficient conditions for the pre-quantization of $M_G(\Sigma)$,the moduli space of flat $G$-bundles over a closed surface $\Sigma$. For asimply connected, compact, simple Lie group $G$, $M_G(\Sigma)$ is known to bepre-quantizable at integer levels. For non-simply connected $G$, however,integrality of the level is not sufficient for pre-quantization, and thisthesis 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 determinedexplicitly for all non-simply connected, compact, simple Lie groups $G$.Partial results are obtained for the case of a surface $\Sigma$ with markedpoints. Also, it is shown that via the bijective correspondence betweenquasi-Hamiltonian group actions and Hamiltonian loop group actions, thecorresponding notions of pre-quantization coincide.

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