Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 095, 38 pages      arXiv:2008.06679      https://doi.org/10.3842/SIGMA.2022.095

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

Márton Hablicsek and Jesse Vogel
Mathematical Institute, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands

Received February 28, 2022, in final form November 28, 2022; Published online December 06, 2022

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.

Key words: representation variety; character variety; topological quantum field theory; Grothendieck ring of varieties.

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