Virtual Classes of Representation Varieties of Upper Triangular Matrices via Topological Quantum Field Theories
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 Röt...
Saved in:
| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Date: | 2022 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2022
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211809 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| 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 |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862722934872735744 |
|---|---|
| author | Hablicsek, Márton Vogel, Jesse |
| author_facet | Hablicsek, Márton Vogel, Jesse |
| citation_txt | 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 |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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.
|
| first_indexed | 2026-03-21T05:20:19Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211809 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T05:20:19Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Hablicsek, Márton Vogel, Jesse 2026-01-12T10:14:33Z 2022 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 1815-0659 2020 Mathematics Subject Classification: 14D23; 14D21; 14C30; 14D20; 14D07; 57R56 arXiv:2008.06679 https://nasplib.isofts.kiev.ua/handle/123456789/211809 https://doi.org/10.3842/SIGMA.2022.095 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. The authors thank Bas Edixhoven and David Holmes for reading a previous version of this paper and giving valuable comments; Άngel González-Prieto, whose papers were the starting point of this research, and who was kind enough to answer any questions; and Sean Lawton for pointing out an error regarding χ. The authors also thank the reviewers for their detailed feedback and invaluable comments. The paper is part of the master’s thesis [33] of the second author. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Virtual Classes of Representation Varieties of Upper Triangular Matrices via Topological Quantum Field Theories Article published earlier |
| spellingShingle | Virtual Classes of Representation Varieties of Upper Triangular Matrices via Topological Quantum Field Theories Hablicsek, Márton Vogel, Jesse |
| title | Virtual Classes of Representation Varieties of Upper Triangular Matrices via Topological Quantum Field Theories |
| title_full | Virtual Classes of Representation Varieties of Upper Triangular Matrices via Topological Quantum Field Theories |
| title_fullStr | Virtual Classes of Representation Varieties of Upper Triangular Matrices via Topological Quantum Field Theories |
| title_full_unstemmed | Virtual Classes of Representation Varieties of Upper Triangular Matrices via Topological Quantum Field Theories |
| title_short | Virtual Classes of Representation Varieties of Upper Triangular Matrices via Topological Quantum Field Theories |
| title_sort | virtual classes of representation varieties of upper triangular matrices via topological quantum field theories |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211809 |
| work_keys_str_mv | AT hablicsekmarton virtualclassesofrepresentationvarietiesofuppertriangularmatricesviatopologicalquantumfieldtheories AT vogeljesse virtualclassesofrepresentationvarietiesofuppertriangularmatricesviatopologicalquantumfieldtheories |