  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 $G$. As 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 allowing its application to other situations.
