A Topological Quantum Field Theory for Representation Varieties of Torus Links

We construct a Topological Quantum Field Theory (TQFT) that computes the virtual classes of $G$-representation varieties associated with the fundamental group of a torus link complement. Following the framework established by González-Prieto, Logares and Muñoz for $G$-representation varieties of surface groups, we introduce a suitable bordism whose iteration enables a recursive topological evaluation of these virtual classes. Via this TQFT, we explicitly compute the virtual class of these varieties for the groups $G={\mathrm{A}\mathrm{G}\mathrm{L}}_1(\mathbb{C})$ and $G={\mathrm{S}\mathrm{L}}_2(\mathbb{C})$.