The quantum content of the gluing equations

TL;DR

The paper constructs a full quantum content for the gluing equations of cusped hyperbolic 3-manifolds by formulating a formal power series $\mathcal{Z}_M(\hbar)$ from Neumann-Zagier data. It proves topological invariance of the 1-loop torsion term $\tau_M$, extends the construction to the PSL(2, C) character variety, and provides explicit Feynman diagram based recipes to compute higher-loop invariants $S_{M,n}$ with results lying in the invariant trace field. The authors implement the first three terms using SnapPy and validate the torsion against nonabelian Reidemeister torsion across thousands of hyperbolic knots, illustrating a deep link between quantum Chern-Simons theory and hyperbolic geometry. They also connect the combinatorial construction to the state-integral model and discuss natural generalizations to multi-cusp and nonhyperbolic cases, highlighting both theoretical structure and practical computation.

Abstract

The gluing equations of a cusped hyperbolic 3-manifold $M$ are a system of polynomial equations in the shapes of an ideal triangulation $\calT$ of $M$ that describe the complete hyperbolic structure of $M$ and its deformations. Given a Neumann-Zagier datum (comprising the shapes together with the gluing equations in a particular canonical form) we define a formal power series with coefficients in the invariant trace field of $M$ that should (a) agree with the asymptotic expansion of the Kashaev invariant to all orders, and (b) contain the nonabelian Reidemeister-Ray-Singer torsion of $M$ as its first subleading "1-loop" term. As a case study, we prove topological invariance of the 1-loop part of the constructed series and extend it into a formal power series of rational functions on the $\PSL(2,\BC)$ character variety of $M$. We provide a computer implementation of the first three terms of the series using the standard {\tt SnapPy} toolbox and check numerically the agreement of our torsion with the Reidemeister-Ray-Singer for all 59924 hyperbolic knots with at most 14 crossings. Finally, we explain how the definition of our series follows from the quantization of 3d hyperbolic geometry, using principles of Topological Quantum Field Theory. Our results have a straightforward extension to any 3-manifold $M$ with torus boundary components (not necessarily hyperbolic) that admits a regular ideal triangulation with respect to some $\PSL(2,\BC)$ representation.

Figures (9)

• Figure 1: Diagrams contributing to $S_{M,2}$ with symmetry factors. The top row of diagrams have exactly two loops, while the bottom row have fewer loops and additional $1$-vertices and $2$-vertices.
• Figure 2: Diagrams with three loops contributing to $S_3$.
• Figure 3: Diagrams with $1$-vertices and $2$-vertices contributing to $S_3$.
• Figure 4: An ideal tetrahedron
• Figure 5: The boundary of the cusp neighborhood for the figure-eight knot.

Theorems & Definitions (28)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 1.7
• Conjecture 1.8
• Definition 1.9
• Remark 1.10
• Remark 1.11
• Definition 2.1