What is called analytic torsion or Ray-Singer torsion is an invariant of Riemannian manifolds which is an analytic analogue of the invariant of topological manifolds called Reidemeister torsion. It has been proved by Cheeger that the two agree for the compact Riemannian manifolds.
D. Ray, I. 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