Quantum Modularity for a Closed Hyperbolic 3-Manifold
This paper proves quantum modularity of both functions from ℚ and -series associated to the closed manifold obtained by −1/2 surgery on the figure-eight knot, 4₁(−1, 2). In a sense, this is a companion to the work of Garoufalidis-Zagier, where similar statements were studied in detail for some simpl...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2025 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2025
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/212887 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Quantum Modularity for a Closed Hyperbolic 3-Manifold. Campbell Wheeler. SIGMA 21 (2025), 004, 74 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862713337689669632 |
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| author | Wheeler, Campbell |
| author_facet | Wheeler, Campbell |
| citation_txt | Quantum Modularity for a Closed Hyperbolic 3-Manifold. Campbell Wheeler. SIGMA 21 (2025), 004, 74 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | This paper proves quantum modularity of both functions from ℚ and -series associated to the closed manifold obtained by −1/2 surgery on the figure-eight knot, 4₁(−1, 2). In a sense, this is a companion to the work of Garoufalidis-Zagier, where similar statements were studied in detail for some simple knots. It is shown that quantum modularity for closed manifolds provides a unification of Chen-Yang's volume conjecture with Witten's asymptotic expansion conjecture. Additionally, we show that 4₁(−1, 2) is a counterexample to previous conjectures of Gukov-Manolescu relating the Witten-Reshetikhin-Turaev invariant and the Ẑ() series. This could be reformulated in terms of a ''strange identity'', which gives a volume conjecture for the Ẑ invariant. Using factorisation of state integrals, we give conjectural but precise -hypergeometric formulae for generating series of Stokes constants of this manifold. We find that the generating series of Stokes constants is related to the 3d index of 4₁(−1, 2) proposed by Gang-Yonekura. This extends the equivalent conjecture of Garoufalidis-Gu-Mariño for knots to closed manifolds. This work appeared in a similar form in the author's Ph.D. Thesis.
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| first_indexed | 2026-03-19T21:53:37Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-212887 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-19T21:53:37Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Wheeler, Campbell 2026-02-13T13:51:02Z 2025 Quantum Modularity for a Closed Hyperbolic 3-Manifold. Campbell Wheeler. SIGMA 21 (2025), 004, 74 pages 1815-0659 2020 Mathematics Subject Classification: 57N10; 57K16; 57K14; 57K10 arXiv:2308.03265 https://nasplib.isofts.kiev.ua/handle/123456789/212887 https://doi.org/10.3842/SIGMA.2025.004 This paper proves quantum modularity of both functions from ℚ and -series associated to the closed manifold obtained by −1/2 surgery on the figure-eight knot, 4₁(−1, 2). In a sense, this is a companion to the work of Garoufalidis-Zagier, where similar statements were studied in detail for some simple knots. It is shown that quantum modularity for closed manifolds provides a unification of Chen-Yang's volume conjecture with Witten's asymptotic expansion conjecture. Additionally, we show that 4₁(−1, 2) is a counterexample to previous conjectures of Gukov-Manolescu relating the Witten-Reshetikhin-Turaev invariant and the Ẑ() series. This could be reformulated in terms of a ''strange identity'', which gives a volume conjecture for the Ẑ invariant. Using factorisation of state integrals, we give conjectural but precise -hypergeometric formulae for generating series of Stokes constants of this manifold. We find that the generating series of Stokes constants is related to the 3d index of 4₁(−1, 2) proposed by Gang-Yonekura. This extends the equivalent conjecture of Garoufalidis-Gu-Mariño for knots to closed manifolds. This work appeared in a similar form in the author's Ph.D. Thesis. This work was carried out throughout my doctoral studies and therefore owes agreat debt to my supervisors Stavros Garoufalidis and Don Zagier. Their constant sharing of ideas and perspectives has been invaluable to the completion of this work. I would also like to thank Dongmin Gang, Mauricio Romo, and Masahito Yamazaki for sharing some of their code early on in this project. Again, I would like to thank Dongmin Gang for sharing a computation of the 3d index of 4₁(−1, 2) proposed in [26]and the question of whether this could be found in the Stokes constants. I thank Jie Gu, Marcos Mariño, and Matthias Storzer for various conversations throughout this project and Sergei Gukov for suggestions to improve the manuscript. Finally, I would like to thank the referees for many helpful suggestions. The work of the author has been supported by the Max-Planck-Gesellschaft with the Max Planck Institute for Mathematics in Bonnand the Southern University of Science and Technology's International Center for Mathematics in Shenzhen. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Quantum Modularity for a Closed Hyperbolic 3-Manifold Article published earlier |
| spellingShingle | Quantum Modularity for a Closed Hyperbolic 3-Manifold Wheeler, Campbell |
| title | Quantum Modularity for a Closed Hyperbolic 3-Manifold |
| title_full | Quantum Modularity for a Closed Hyperbolic 3-Manifold |
| title_fullStr | Quantum Modularity for a Closed Hyperbolic 3-Manifold |
| title_full_unstemmed | Quantum Modularity for a Closed Hyperbolic 3-Manifold |
| title_short | Quantum Modularity for a Closed Hyperbolic 3-Manifold |
| title_sort | quantum modularity for a closed hyperbolic 3-manifold |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212887 |
| work_keys_str_mv | AT wheelercampbell quantummodularityforaclosedhyperbolic3manifold |