The quantum content of the gluing equations

Abstract

The gluing equations of a cusped hyperbolic 3-manifold M are a system of polynomial equations in the shapes of an ideal triangulation T 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, C) character variety of M . We provide a computer implementation of the first three terms of the series using the standard 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, C) representation.

The 3D index of an ideal triangulation and angle structures

“…• Chern-Simons perturbation theory (that fits well with the earlier work on quantum Riemann surfaces of [4] and the later work on the perturbative invariants of [7]), • categorification and Khovanov Homology, that fits with the earlier work [28]. Although the gauge theory depends on the ideal triangulation T , and the 3D index in general may not converge, physics predicts that the gauge theory ought to be a topological invariant of the underlying 3-manifold M .…”

“…Consider two ideal triangulations T and T with N and N + 1 tetrahedra, respectively, related by a 2 − 3 move shown in Figure 8. The above figure matches the conventions of [7,Sec.3.6]. For a variable, matrix or vector f associated to T , we will denote by f the corresponding variable, matrix or vector associated to T .…”

Expanding 3d $$ \mathcal{N} $$ = 2 theories around the round sphere

“…The perturbative expansion at the value b = 1 gives infinitely many topological invariants of the 3-manifold. Note that our expansion is different from the expansion around b = 0, which has been studied in the context of the generalized volume conjectures [19][20][21].…”

3d N = 2 $$ \mathcal{N}=2 $$ minimal SCFTs from wrapped M5-branes

“…We use the one developed by Dimofte and incorporate Dehn filling into the state-integral model to cover more general class of 3-manifolds such as closed hyperbolic 3-manifolds. One systematic way of specifying the gluing rule of an ideal triangulation is using (generalized) Neunmann-Zagier (NZ) datum (A, B, C, D; f, f , ν, ν p ), refer to [35] for the definition, where A, B, C, D are T × T matrices forming Sp(2T, Q) 8) and (f, f , ν, ν p ) are vectors of length T . From these datum, the state-integral (SI) for the link complement is given by [35] Z SI (S 3 \K; X 1 , .…”