Algebraic deformations of moduli spaces of flat connections

Andras Szenes (MIT)

The moduli space of flat connections on a Riemann surface with a simple structure group is an algeraic variety. There is a canonical non-commutative deformation of this variety based on quantum groups and the Reshsetikhin-Turaev invariants. In joint work with P. Roche we construct a trace on this algebra which contains information about the Hilbert polynomial of the moduli spaces in a surprising manner, via asymptotic expansions of certain defective modular forms. This allows us to formulate a conjecture interpreting the deformation in the formal non-commutative Riemann-Roch theory.