On character varieties of singular manifolds

TL;DR

This paper develops a Topological Quantum Field Theory to compute algebraic invariants of representation varieties over manifolds with singularities, extending to complex curves and specific groups like SL2.

Contribution

It introduces a lax monoidal TQFT framework for calculating virtual classes of G-representation varieties on singular manifolds, including parabolic cases and specific groups.

Findings

Constructed a TQFT for virtual class computation

Computed virtual classes for SL2 character varieties on nodal surfaces

Extended methods to manifolds with conic singularities

Abstract

In this paper, we construct a lax monoidal Topological Quantum Field Theory that computes virtual classes, in the Grothendieck ring of algebraic varieties, of $G$-representation varieties over manifolds with conic singularities, which we will call nodefolds. This construction is valid for any algebraic group $G$, in any dimension and also in the parabolic setting. In particular, this TQFT allow us to compute the virtual classes of representation varieties over complex singular planar curves. In addition, in the case $G = \mathrm{SL}_{2}(k)$, the virtual class of the associated character variety over a nodal closed orientable surface is computed both in the non-parabolic and in the parabolic scenarios.