Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 004, 74 pages      arXiv:2308.03265      https://doi.org/10.3842/SIGMA.2025.004

Quantum Modularity for a Closed Hyperbolic 3-Manifold

Campbell Wheeler
Institut des Hautes Études Scientifiques, Le Bois-Marie, Bures-sur-Yvette, France

Received January 11, 2024, in final form December 23, 2024; Published online January 08, 2025

Abstract
This paper proves quantum modularity of both functions from $\mathbb{Q}$ and $q$-series associated to the closed manifold obtained by $-\smash{\frac{1}{2}}$ 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 $\smash{\widehat{Z}(q)}$ series. This could be reformulated in terms of a ''strange identity'', which gives a volume conjecture for the $\smash{\widehat{Z}}$ 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.

Key words: 3d index; asymptotic expansions; Borel resummation; character varieties; Chern-Simons invariants; circle method; closed three-manifolds; cocycles; dilogarithm; duality; Faddeev quantum dilogarithm; factorisation; flat connections; hyperbolic manifolds; modularity; perturbative invariants; $q$-difference equations; $q$-hypergeometric functions; quadratic relations; quantum invariants; quantum modular forms; resurgence; surgery; state integrals; stationary phase; Stokes constants; Stokes phenomenon; strange identity; three-manifolds; volume conjecture; Witten-Reshetikhin-Turaev invariants; $\smash{\widehat{Z}}$ invariants.

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