What is called analytic torsion or Ray-Singer torsion (Ray-Singer 73) is the invariant $T(X,g)$ of a Riemannian manifold $(X,g)$ given by a product of powers of the functional determinants $det_{reg} \Delta|_{\Omega^p}$ of the Laplace operators $\Delta|_{\Omega^p}$ of the manifold acting on the space of differential p-forms:
This analytic torsion is an analogue in analysis of the invariant of topological manifolds called Reidemeister torsion. The two agree for compact Riemannian manifolds (Cheeger 77).
According to (Morishita 09) the relation between Reidemeister torsion and analytic torsion is analogous to that between Iwasawa polynomials and zeta functions obtained by adelic integration. (…)
(Fried 86)
Analytic torsion appears as one factor in the perturbative path integral quantization of Chern-Simons field theory. See there at Quantization – Perturbative – Path integral quantization.
D. Ray, Isadore Singer, Analytic torsion for complex manifolds, Ann. Math. 98, 1 (1973), 154–177.
Jeff Cheeger, Analytic torsion and Reidemeister torsion, Proc. Natl. Acad. Sci. USA 74, No. 7, pp. 2651-2654 (1977), pdf
Wikipedia, Analytic torsion
A.A. Bytsenko, A.E. Goncalves, W. da Cruz, Analytic Torsion on Hyperbolic Manifolds and the Semiclassical Approximation for Chern-Simons Theory (arXiv:hep-th/9805187)
Review of the role played in the perturbative quantization of Chern-Simons theory includes
Discussion for hyperbolic manifolds in terms of the Selberg zeta function is due to
with further developments including