#### Volume 17, issue 3 (2013)

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The quantum content of the gluing equations

### Tudor Dimofte and Stavros Garoufalidis

Geometry & Topology 17 (2013) 1253–1315
##### Abstract

The gluing equations of a cusped hyperbolic $3$–manifold $M$ are a system of polynomial equations in the shapes of an ideal triangulation $\mathsc{T}$ 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 $PSL\left(2,ℂ\right)$ 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 $PSL\left(2,ℂ\right)$ representation.

##### Keywords
volume, complex Chern–Simons theory, Kashaev invariant, gluing equations, Neumann–Zagier equations, Neumann–Zagier datum, hyperbolic geometry, ideal triangulations, $1$–loop, torsion, quantum dilogarithm, state integral, perturbation theory, Feynman diagram, formal Gaussian integration
##### Mathematical Subject Classification 2010
Primary: 57M25, 57N10