  The gluing equations of a cusped hyperbolic 3-manifold $M$ are a system of
polynomial equations in the shapes of an ideal triangulation $\calT$ 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 $\PSL(2,\BC)$ character variety of $M$. We provide a
computer implementation of the first three terms of the series using the
standard {\tt 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 3d 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
$\PSL(2,\BC)$ representation.
