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The
quantum content of the gluing equations
Tudor Dimofte and Stavros Garoufalidis |
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Geometry & Topology 17 (2013) 1253–1315
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Abstract |
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The gluing equations of a cusped hyperbolic –manifold are a system of polynomial equations in the shapes of an ideal triangulation of that describe the complete hyperbolic structure of 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 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 as its first subleading “–loop” term. As a case study, we prove topological invariance of the –loop part of the constructed series and extend it into a formal power series of rational functions on the character variety of . 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 hyperbolic knots with at most 14 crossings. Finally, we explain how the definition of our series follows from the quantization of –dimensional hyperbolic geometry, using principles of topological quantum field theory. Our results have a straightforward extension to any –manifold with torus boundary components (not necessarily hyperbolic) that admits a regular ideal triangulation with respect to some 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
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Mathematical Subject Classification 2010
Primary: 57M25, 57N10
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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
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Authors |
| Tudor Dimofte | |
| School of Natural Sciences Institute for Advanced Study Einstein Drive Princeton, NJ 08540 USA |
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| Stavros Garoufalidis | |
| School of Mathematics Georgia Institute of Technology 686 Cherry Street Atlanta, GA 30332-0160 USA |
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| http://www.math.gatech.edu/~stavros | |
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