Pre-quantization of the moduli space of flat G-bundles over a surface

For a simply connected, compact, simple Lie group GG, the moduli space of flat GG-bundles over a closed surface ΣΣ is known to be pre-quantizable at integer levels. For non-simply connected GG, however, integrality of the level is not sufficient for pre-quantization, and this paper determines the obstruction–namely a certain cohomology class in H3(G2;Z)H3(G2;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 GG.