Representation Varieties of Non-orientable Surfaces via Topological Quantum Field Theories

We study the $G$-representation varieties of non-orientable surfaces. By a geometric method using Topological Quantum Field Theories (TQFTs), we compute virtual classes of these $G$-representation varieties in the Grothendieck ring of varieties, for $G$ the groups of complex upper triangular matrices of rank 2 and 3. We discuss various ways to use conjugacy invariance to simplify the computations. Finally, we give a number of remarks on the resulting classes, and observe and explain the zero eigenvalues that appear in the TQFT.