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In this paper, we study the $G$-representation and character varieties ofnon-orientable closed surfaces. By means of a geometric method based on aTopological Quantum Field Theory (TQFT), we compute the virtual classes ofthese 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 inthe case of orientable closed surfaces, and we extend it to the case ofnon-orientable surfaces. Furthermore, we provide a practical approach forexplicitly computing the TQFT, allowing for more simplified and structuredcomputations. Finally, for $G = \textrm{SL}_2$ we describe and explain therelationship between the representation varieties of the orientable andnon-orientable closed surfaces.

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