A Representation-Theoretic Approach to -Characters
We raise the question of whether (a slightly generalized notion of) -characters can be constructed purely representation-theoretically. In the main example of the quantum toroidal ₁ algebra, geometric engineering of adjoint matter produces an explicit vertex operator which computes certain -charact...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2022 |
| Main Author: | |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2022
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211814 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | A Representation-Theoretic Approach to -Characters. Henry Liu. SIGMA 18 (2022), 090, 20 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862654471854620672 |
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| author | Liu, Henry |
| author_facet | Liu, Henry |
| citation_txt | A Representation-Theoretic Approach to -Characters. Henry Liu. SIGMA 18 (2022), 090, 20 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We raise the question of whether (a slightly generalized notion of) -characters can be constructed purely representation-theoretically. In the main example of the quantum toroidal ₁ algebra, geometric engineering of adjoint matter produces an explicit vertex operator which computes certain -characters, namely Hirzebruch χy-genera, completely analogously to how the -matrix computes -characters. We give a geometric proof of the independence of preferred direction for the refined vertex in this and more general non-toric settings.
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| first_indexed | 2026-03-15T18:30:30Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211814 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T18:30:30Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Liu, Henry 2026-01-12T10:16:34Z 2022 A Representation-Theoretic Approach to -Characters. Henry Liu. SIGMA 18 (2022), 090, 20 pages 1815-0659 2020 Mathematics Subject Classification: 17B37; 17B67; 14N35 arXiv:2203.07072 https://nasplib.isofts.kiev.ua/handle/123456789/211814 https://doi.org/10.3842/SIGMA.2022.090 We raise the question of whether (a slightly generalized notion of) -characters can be constructed purely representation-theoretically. In the main example of the quantum toroidal ₁ algebra, geometric engineering of adjoint matter produces an explicit vertex operator which computes certain -characters, namely Hirzebruch χy-genera, completely analogously to how the -matrix computes -characters. We give a geometric proof of the independence of preferred direction for the refined vertex in this and more general non-toric settings. This whole line of inquiry on -characters was inspired by similar questions of A. Okounkov, and benefited greatly from his and N. Nekrasov’s lectures at the 2019 Skoltech Summer School on Mathematical Physics. Discussions with N. Arbesfeld, C.-C.M. Liu, A. Okounkov, and R. Pandharipande were also important, particularly for the contents of Section 3.4. We also thank the anonymous referees for bringing [13] to our attention, and for numerous suggestions that improved the content and exposition in this paper. This research was supported by the Simons Collaboration on Special Holonomy in Geometry, Analysis, and Physics. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Representation-Theoretic Approach to -Characters Article published earlier |
| spellingShingle | A Representation-Theoretic Approach to -Characters Liu, Henry |
| title | A Representation-Theoretic Approach to -Characters |
| title_full | A Representation-Theoretic Approach to -Characters |
| title_fullStr | A Representation-Theoretic Approach to -Characters |
| title_full_unstemmed | A Representation-Theoretic Approach to -Characters |
| title_short | A Representation-Theoretic Approach to -Characters |
| title_sort | representation-theoretic approach to -characters |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211814 |
| work_keys_str_mv | AT liuhenry arepresentationtheoreticapproachtocharacters AT liuhenry representationtheoreticapproachtocharacters |