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

Márton Hablicsek ; Jesse Vogel

Symmetry, integrability and geometry: methods and applications, Tome 18 (2022)

Résumé

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.

Détail

Keywords: representation variety, character variety, topological quantum field theory, Grothendieck ring of varieties.

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