## Geometry & Topology

### 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 $3$–dimensional 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.

#### Article information

Source
Geom. Topol., Volume 17, Number 3 (2013), 1253-1315.

Dates
Revised: 25 October 2012
Accepted: 5 January 2013
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732608

Digital Object Identifier
doi:10.2140/gt.2013.17.1253

Mathematical Reviews number (MathSciNet)
MR3073925

Zentralblatt MATH identifier
1283.57017

#### Citation

Dimofte, Tudor; Garoufalidis, Stavros. The quantum content of the gluing equations. Geom. Topol. 17 (2013), no. 3, 1253--1315. doi:10.2140/gt.2013.17.1253. https://projecteuclid.org/euclid.gt/1513732608

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