Moduli spaces of representations of surface groups (aka character varieties) are very interesting spaces due to their tight relation with moduli spaces of Higgs bundles and flat connections. Nowadays, several approaches are available in the literature to understand the geometry of these character varieties constructed via geometric invariant theory quotients. Despite these advances, the geometry of character stacks, where roughly speaking the group action is not quotiented but still tracked, remains a mystery.

To address this problem, in this talk we shall construct a lax monoidal topological quantum field theory that computes the virtual classes of G-representation stacks in the Grothendieck ring of BG-stacks. This tool gives rise to an effective computational method for these virtual classes based on topological recursion on the genus of the surface. Time permitting, we will also discuss how this construction provides evidence that lax monoidal TQFTs represent a new hope in the quantization of algebraic invariants.

Joint work with M. Hablicsek and J. Vogel.