A Topological Quantum Field Theory for Character Varieties of Non-orientable Surfaces

In this paper, we study the $G$-representation and character varieties of non-orientable closed surfaces. By means of a geometric method based on a Topological Quantum Field Theory (TQFT), we compute the virtual classes of these varieties in the Grothendieck ring of varieties for $G$ equal to $\textrm{AGL}_1$ and $\textrm{SL}_2$. This method was already known and used in the case of orientable closed surfaces, and we extend it to the case of non-orientable surfaces. Furthermore, we provide a practical approach for explicitly computing the TQFT, allowing for more simplified and structured computations. Finally, for $G = \textrm{SL}_2$ we describe and explain the relationship between the representation varieties of the orientable and non-orientable closed surfaces.