Virtual Classes of Representation Varieties of Upper Triangular Matrices via Topological Quantum Field Theories

Abstract

In this paper, we use a geometric technique developed by González-Prieto, Logares, Muñoz, and Newstead to study the $G$-representation variety of surface groups ${\mathfrak{X}}_G({\Sigma }_g)$ of arbitrary genus for $G$ being the group of upper triangular matrices of fixed rank. Explicitly, we compute the virtual classes in the Grothendieck ring of varieties of the $G$-representation variety and the moduli space of $G$-representations of surface groups for $G$ being the group of complex upper triangular matrices of rank $2$, $3$, and $4$ via constructing a topological quantum field theory. Furthermore, we show that in the case of upper triangular matrices the character map from the moduli space of $G$-representations to the $G$-character variety is not an isomorphism.

References in corpus (4)

• On the E-polynomials of a family of Character Varieties
• Virtual classes of parabolic ${\mathrm{SL}}_2(\mathbb{C})$-character varieties
• Topological Quantum Field Theories for character varieties
• Representation variety for the rank one affine group

Cited by in corpus (1)

• Quantization of algebraic invariants through Topological Quantum Field Theories