Quantum Modularity for a Closed Hyperbolic 3-Manifold
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913639619887104 |
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| author | Wheeler, Campbell |
| author_facet | Wheeler, Campbell |
| contents | This paper proves quantum modularity of both functions from $\mathbb{Q}$ and $q$-series associated to the closed manifold obtained by $-\frac{1}{2}$ surgery on the figure-eight knot, $4_1(-1,2)$. In a sense, this is a companion to 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 $4_1(-1,2)$ is a counterexample to previous conjectures of Gukov-Manolescu relating the Witten-Reshetikhin-Turaev invariant and the $\widehat{Z}(q)$ series. This could be reformulated in terms of a "strange identity", which gives a volume conjecture for the $\widehat{Z}$ invariant. Using factorisation of state integrals, we give conjectural but precise $q$-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 $4_1(-1,2)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_03265 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quantum Modularity for a Closed Hyperbolic 3-Manifold Wheeler, Campbell Geometric Topology High Energy Physics - Theory This paper proves quantum modularity of both functions from $\mathbb{Q}$ and $q$-series associated to the closed manifold obtained by $-\frac{1}{2}$ surgery on the figure-eight knot, $4_1(-1,2)$. In a sense, this is a companion to 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 $4_1(-1,2)$ is a counterexample to previous conjectures of Gukov-Manolescu relating the Witten-Reshetikhin-Turaev invariant and the $\widehat{Z}(q)$ series. This could be reformulated in terms of a "strange identity", which gives a volume conjecture for the $\widehat{Z}$ invariant. Using factorisation of state integrals, we give conjectural but precise $q$-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 $4_1(-1,2)$ 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. |
| title | Quantum Modularity for a Closed Hyperbolic 3-Manifold |
| topic | Geometric Topology High Energy Physics - Theory |
| url | https://arxiv.org/abs/2308.03265 |