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

TL;DR

This paper extends the use of Topological Quantum Field Theory to compute the virtual classes of character varieties of non-orientable surfaces, providing new methods and insights into their structure and relationships with orientable cases.

Contribution

It introduces a TQFT-based geometric method for non-orientable surfaces, expanding previous orientable surface techniques, and offers explicit computational approaches.

Findings

Computed virtual classes in the Grothendieck ring for specific groups

Extended TQFT methods to non-orientable surfaces

Explained relationships between orientable and non-orientable representation varieties

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 $\textrm{AGL}_1$ 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.