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

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2022
• Main Authors: Hablicsek, Márton, Vogel, Jesse
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2022
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/211809
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Virtual Classes of Representation Varieties of Upper Triangular Matrices via Topological Quantum Field Theories. Márton Hablicsek and Jesse Vogel. SIGMA 18 (2022), 095, 38 pages

In this paper, we use a geometric technique developed by González-Prieto, Logares, Muñoz, and Newstead to study the -representation variety of surface groups G(Σ) of arbitrary genus for being the group of upper triangular matrices of fixed rank. Explicitly, we compute the virtual classes in the Röthendieck ring of varieties of the -representation variety and the moduli space of -representations of surface groups for 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 -representations to the -character variety is not an isomorphism.