Volume 17, issue 3 (2013)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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 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, ) 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, ) 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
References
Publication
Received: 6 March 2012
Revised: 25 October 2012
Accepted: 5 January 2013
Published: 30 May 2013
Proposed: Walter Neumann
Seconded: Danny Calegari, Jean-Pierre Otal
Authors
Tudor Dimofte
School of Natural Sciences
Institute for Advanced Study
Einstein Drive
Princeton, NJ 08540
USA
Stavros Garoufalidis
School of Mathematics
Georgia Institute of Technology
686 Cherry Street
Atlanta, GA 30332-0160
USA
http://www.math.gatech.edu/~stavros