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

Symmetry Integrability and Geometry-Methods and Applications

article

Márton HablicsekJesse Vogel¹ 

[1] Mathematical Institute

关键词: representation variety;    character variety;    topological quantum field theory;    Grothendieck ring of varieties.;   

DOI  :  10.3842/SIGMA.2022.095

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown