Ẑ and Splice Diagrams
We study quantum q-series invariants of 3-manifolds Ẑσ of Gukov-Pei-Putrov-Vafa, using techniques from the theory of normal surface singularities such as splice diagrams. We show that the (suitably normalized) sum of all Ẑσ depends only on the splice diagram, and in particular, it agrees for manifol...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2025 |
| Автори: | , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2025
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/214173 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Ẑ and Splice Diagrams. Sergei Gukov, Ludmil Katzarkov and Josef Svoboda. SIGMA 21 (2025), 073, 30 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862623155832487936 |
|---|---|
| author | Gukov, Sergei Katzarkov, Ludmil Svoboda, Josef |
| author_facet | Gukov, Sergei Katzarkov, Ludmil Svoboda, Josef |
| citation_txt | Ẑ and Splice Diagrams. Sergei Gukov, Ludmil Katzarkov and Josef Svoboda. SIGMA 21 (2025), 073, 30 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We study quantum q-series invariants of 3-manifolds Ẑσ of Gukov-Pei-Putrov-Vafa, using techniques from the theory of normal surface singularities such as splice diagrams. We show that the (suitably normalized) sum of all Ẑσ depends only on the splice diagram, and in particular, it agrees for manifolds with the same universal abelian cover. We use these ideas to find simple formulas for Ẑσ invariants of Seifert manifolds. Applications include a better understanding of the vanishing of the q-series Ẑσ. Additionally, we study moduli spaces of flat SL₂(ℂ) connections on Seifert manifolds and their relation to spectra of surface singularities, extending a result of Boden and Curtis for Brieskorn spheres to Seifert rational homology spheres with 3 singular fibers and to Seifert homology spheres with any number of fibers.
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| first_indexed | 2026-03-21T12:21:40Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-214173 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T12:21:40Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Gukov, Sergei Katzarkov, Ludmil Svoboda, Josef 2026-02-19T16:04:40Z 2025 Ẑ and Splice Diagrams. Sergei Gukov, Ludmil Katzarkov and Josef Svoboda. SIGMA 21 (2025), 073, 30 pages 1815-0659 2020 Mathematics Subject Classification: 57K31; 32S50; 32S25; 32S55 arXiv:2304.00699 https://nasplib.isofts.kiev.ua/handle/123456789/214173 https://doi.org/10.3842/SIGMA.2025.073 We study quantum q-series invariants of 3-manifolds Ẑσ of Gukov-Pei-Putrov-Vafa, using techniques from the theory of normal surface singularities such as splice diagrams. We show that the (suitably normalized) sum of all Ẑσ depends only on the splice diagram, and in particular, it agrees for manifolds with the same universal abelian cover. We use these ideas to find simple formulas for Ẑσ invariants of Seifert manifolds. Applications include a better understanding of the vanishing of the q-series Ẑσ. Additionally, we study moduli spaces of flat SL₂(ℂ) connections on Seifert manifolds and their relation to spectra of surface singularities, extending a result of Boden and Curtis for Brieskorn spheres to Seifert rational homology spheres with 3 singular fibers and to Seifert homology spheres with any number of fibers. We would like to thank Denis Auroux, Martin Čech, Shimal Harichurn, Mrunmay Jagadale, Maxim Kontsevich, Slava Krushkal, András Némethi, Sunghyuk Park, Pavel Putrov, and Nikolai Saveliev for helpful conversations and Zuzana Urbanova for the help with computer experiments. We thank the anonymous referees for their valuable suggestions and comments. S. Gukov was partially supported by NSF grant DMS-1664227, DOE grant DE-SC0011632. S. Gukov and J. Svoboda were partially supported by the Simons Grant, New structures in low-dimensional topology. L. Katzarkov was supported by NSF Grant Theory of Atoms, Simons investigators grant HMF Simons Foundation, grant SFI-MPS-T-Institutes-00007697, and the Ministry of Education and Science of the Republic of Bulgaria, grant DO1-239/10.12.2024. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Ẑ and Splice Diagrams Article published earlier |
| spellingShingle | Ẑ and Splice Diagrams Gukov, Sergei Katzarkov, Ludmil Svoboda, Josef |
| title | Ẑ and Splice Diagrams |
| title_full | Ẑ and Splice Diagrams |
| title_fullStr | Ẑ and Splice Diagrams |
| title_full_unstemmed | Ẑ and Splice Diagrams |
| title_short | Ẑ and Splice Diagrams |
| title_sort | ẑ and splice diagrams |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/214173 |
| work_keys_str_mv | AT gukovsergei zandsplicediagrams AT katzarkovludmil zandsplicediagrams AT svobodajosef zandsplicediagrams |