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

Abstract

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 ${\text{AGL}}_1$ and ${\text{SL}}_2$. 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 $G={\text{SL}}_2$ we describe and explain the relationship between the representation varieties of the orientable and non-orientable closed surfaces.

arXiv comment27 pages