A lax monoidal Topological Quantum Field Theory for representation varieties

TL;DR

This paper develops a lax monoidal Topological Quantum Field Theory that calculates Deligne-Hodge polynomials of representation varieties, providing a categorical framework applicable to various contexts.

Contribution

It introduces a novel categorical construction of a TQFT for representation varieties, linking algebraic geometry and topological quantum field theory.

Findings

Formulas for Deligne-Hodge polynomials in terms of mixed Hodge modules

A categorical framework for TQFT construction

Application to fundamental groups of closed manifolds

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 $G$. 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.