$\widehat{Z}$ and Splice Diagrams
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914004503363584 |
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| author | Gukov, Sergei Katzarkov, Ludmil Svoboda, Josef |
| author_facet | Gukov, Sergei Katzarkov, Ludmil Svoboda, Josef |
| contents | We study quantum $q$-series invariants of 3-manifolds $\widehat{Z}_σ$ 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 $\widehat{Z}_σ$ 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 $\widehat{Z}_σ$ invariants of Seifert manifolds. Applications include a better understanding of the vanishing of the $q$-series $\widehat{Z}_σ$. Additionally, we study moduli spaces of flat $\operatorname{SL}_2(\mathbb{C})$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_00699 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $\widehat{Z}$ and Splice Diagrams Gukov, Sergei Katzarkov, Ludmil Svoboda, Josef Geometric Topology Algebraic Geometry Quantum Algebra We study quantum $q$-series invariants of 3-manifolds $\widehat{Z}_σ$ 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 $\widehat{Z}_σ$ 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 $\widehat{Z}_σ$ invariants of Seifert manifolds. Applications include a better understanding of the vanishing of the $q$-series $\widehat{Z}_σ$. Additionally, we study moduli spaces of flat $\operatorname{SL}_2(\mathbb{C})$ 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. |
| title | $\widehat{Z}$ and Splice Diagrams |
| topic | Geometric Topology Algebraic Geometry Quantum Algebra |
| url | https://arxiv.org/abs/2304.00699 |