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

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.

💡 Research Summary

The paper addresses the problem of pre‑quantization of the moduli space of flat G‑bundles over a closed oriented surface Σ when the structure group G is a compact, simple Lie group that is not simply connected. For simply connected groups the symplectic form on the moduli space, given by the Atiyah‑Bott construction, represents an integral cohomology class precisely when the level k is an integer, and therefore a pre‑quantum line bundle exists for every integer k. The situation changes dramatically when π₁(G)≠0. In that case the moduli space ℳₖ(Σ,G) is still a symplectic orbifold, but the obstruction to lifting the symplectic form to an integral class lives in the third cohomology group H³(G²;ℤ).

The authors first recall that a pre‑quantum line bundle exists if and only if the cohomology class