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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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2025 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2025
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/214173 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Ẑ and Splice Diagrams. Sergei Gukov, Ludmil Katzarkov and Josef Svoboda. SIGMA 21 (2025), 073, 30 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| Summary: | 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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| ISSN: | 1815-0659 |