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) 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.