A lax monoidal topological quantum field theory for representation varieties

Abstract

We construct a lax monoidal Topological Quantum Field Theory that computes Deligne-Hodge polynomials of representation varieties of the fundamental group of any closed manifold into any complex algebraic group G. We also extend the result to the parabolic case with any number of punctures and arbitrary monodromies. As a byproduct, we obtain formulas for these polynomials in terms of homomorphisms between the space of mixed Hodge modules on G. The construction is developed in a categorical-theoretic framework that can be applied to other situations.