Quantum Modularity for a Closed Hyperbolic 3-Manifold

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2025
• Main Author: Wheeler, Campbell
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2025
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/212887
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Quantum Modularity for a Closed Hyperbolic 3-Manifold. Campbell Wheeler. SIGMA 21 (2025), 004, 74 pages

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.