Quantum Modularity for a Closed Hyperbolic 3-Manifold

TL;DR

This work provides the first detailed quantum modularity analysis for a closed hyperbolic 3‑manifold, specifically $4_1(-1,2)$. By combining the WRT invariant and the $\widehat{Z}$ series within a unified modular/resurgent framework, it demonstrates quantum modular behavior, stationary‑phase mechanisms, and Borel resummation structures that tie volume growth to geometric data and connect to state integrals and the 3d index. The results yield explicit modular relations for both $\Sha$ and $\widehat{Z}$, reveal a refined matrix‑valued modularity, and offer conjectural but precise q‑hypergeometric formulas for Stokes constants, thereby extending modularity/ resurgence phenomena from knots to closed hyperbolic 3‑manifolds. These insights illuminate the relationship between volume conjectures, WRT asymptotics, and quantum invariants, providing a robust framework with computational tools and topological interpretations.

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.

Figures (7)

• Figure 1: Picture of the Borel plane labelled by the Stokes constants for the asymptotic series $\Phi^{(\rho_0)}$ associated to the trivial connection. The points are located at $\frac{\mathrm{V}_{\rho_1}}{2\pi {\rm i}}+(2\pi {\rm i})\mathbb Z$, $\frac{\mathrm{V}_{\rho_2}}{2\pi {\rm i}}+(2\pi {\rm i})\mathbb Z$, $\frac{\mathrm{V}_{\rho_3}}{2\pi {\rm i}}+(2\pi {\rm i})\mathbb Z$, $\frac{\mathrm{V}_{\rho_4}}{2\pi {\rm i}}+(2\pi {\rm i})\mathbb Z$, $\frac{\mathrm{V}_{\rho_5}}{2\pi {\rm i}}+(2\pi {\rm i})\mathbb Z$, $\frac{\mathrm{V}_{\rho_6}}{2\pi {\rm i}}+(2\pi {\rm i})\mathbb Z$, $\frac{\mathrm{V}_{\rho_7}}{2\pi {\rm i}}+(2\pi {\rm i})\mathbb Z$. See Section \ref{['stomat']} for more details.
• Figure 2: Plots of the WRT invariant of $4_1(-1,2)$ against the first order approximation in Witten's asymptotic expansion conjecture where $N\in\mathbb Z$ and $\mathbf e(x)=\exp(2\pi {\rm i}x)$.
• Figure 3: Plots of the logarithm of the coefficients of $\widehat{Z}$ on the left and the error in the asymptotics on the right.
• Figure 4: Plots of the logarithm of the WRT invariant and its modular quotient for the first ten thousand rational numbers in $(0,1)$ and ordered by denominators.
• Figure 5: The regions $R$ that arise from the stationary phase approximation.

Theorems & Definitions (32)

• Theorem 1.1
• Conjecture 1.2: Witten's asymptotic expansion conjecture Witten
• Conjecture 1.3: Chen--Yang's volume conjecture ChenYang
• Conjecture 1.4: $\widehat{Z}$ volume conjecture
• Conjecture 1.5: quantum modularity conjecture for the WRT invariant
• Conjecture 1.6: quantum modularity conjecture for $\widehat{Z}$
• Conjecture 1.7
• Proposition 2.1
• Proposition 2.2
• Theorem 3.1