[Paper Review] Quantum Modularity for a Closed Hyperbolic 3-Manifold

Campbell Wheeler|arXiv (Cornell University)|Aug 7, 2023

Geometric and Algebraic TopologyMathematics3 citations

TL;DR

This paper establishes quantum modularity for the Witten–Reshetikhin–Turaev (WRT) invariant and the $\widehat{Z}$ series of the closed hyperbolic 3-manifold obtained by $-1/2$ surgery on the figure-eight knot, $4_1(-1,2)$. It proves these invariants form a matrix-valued quantum modular form, unifies Chen–Yang's volume conjecture with Witten's asymptotic expansion conjecture, and provides precise $q$-hypergeometric formulae for Stokes constants via state integral factorisation, linking them to the 3d index.

ABSTRACT

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.

Motivation & Objective

• To establish quantum modularity for quantum invariants of a closed hyperbolic 3-manifold, a case where such structures were previously unexplored.
• To unify Chen–Yang’s volume conjecture and Witten’s asymptotic expansion conjecture through the lens of quantum modularity.
• To provide precise $q$-hypergeometric formulae for Stokes constants of $4_1(-1,2)$ using factorisation of state integrals.
• To extend the 3d index conjecture from knots to closed hyperbolic 3-manifolds via the $\widehat{Z}$ invariant.
• To demonstrate that $4_1(-1,2)$ is a counterexample to prior conjectures relating $\widehat{Z}$ and WRT invariants, revealing a 'strange identity'.

Proposed method

• Proving quantum modularity of the WRT invariant and $\widehat{Z}$ series using stationary phase analysis and Borel resummation of asymptotic expansions.
• Employing the circle method and $q$-difference equations to analyze radial asymptotics of $\widehat{Z}$ at roots of unity.
• Using factorisation of state integrals to isolate contributions from non-abelian and trivial flat connections in the $\mathrm{SL}_2(\mathbb{C})$-character variety.
• Deriving $q$-hypergeometric expressions for generating series of Stokes constants via $q$-series expansions and modular properties.
• Computing Stokes matrices in different regions of the complex plane using matrix factorisations and numerical verification up to $O(q^4)^{100}$.
• Relating the generating series of Stokes constants to the 3d index of $4_1(-1,2)$, extending a conjecture previously known only for knots.

Experimental results

Research questions

• RQ1 Does the WRT invariant and $\widehat{Z}$ series of a closed hyperbolic 3-manifold exhibit quantum modularity?
• RQ2 Can the quantum modularity of these invariants unify Chen–Yang’s volume conjecture and Witten’s asymptotic expansion conjecture?
• RQ3 Do the Stokes constants of $4_1(-1,2)$ admit a precise $q$-hypergeometric description via state integral factorisation?
• RQ4 Is the generating series of Stokes constants for $4_1(-1,2)$ related to its 3d index, as conjectured for knots?
• RQ5 Does $4_1(-1,2)$ serve as a counterexample to prior conjectures linking $\widehat{Z}$ and WRT invariants, implying a 'strange identity'?

Key findings

• The WRT invariant and $\widehat{Z}$ series of $4_1(-1,2)$ form a matrix-valued quantum modular form, confirming quantum modularity for a closed hyperbolic 3-manifold.
• The manifold $4_1(-1,2)$ provides a counterexample to prior conjectures relating $\widehat{Z}$ and WRT invariants, revealing a 'strange identity' that implies a volume conjecture for $\widehat{Z}$.
• Precise $q$-hypergeometric formulae are conjectured for the generating series of Stokes constants, derived via factorisation of state integrals.
• The generating series of Stokes constants for $4_1(-1,2)$ is shown to be related to the 3d index, extending a conjecture previously valid only for knots.
• Numerical verification confirms the Stokes matrices in different regions of the complex plane, with corrections appearing at order $O(q^4)^{100}$.
• The radial asymptotics of $\widehat{Z}$ at roots of unity are computed, and the constant terms are shown to be modular-like, supporting quantum modularity.