Relative torsion for representations in finite type Hilbert modules

Let M be a closed manifold, ρ a representation of π1(M) on an A-Hilbert model W of finite type (A a finite von Neumann algebra) and μ a Hermitian structure on the flat bundle E → M associated to ρ. The relative torsion, first introduced by Carey, Mathai and Mishchenko, [CMM], associates to any pair (g, τ), consisting of a Riemannian metric g on M and a generalized triangulation τ = (h, g), a numerical invariant, R(M,ρ, μ, g, τ). Unlike the analytic torsion Tan, associated to (M,ρ, μ, g), or the Reidemeister torsion TReid, associated to F = (M,ρ, μ, g, τ), which are defined only when the pair (M,ρ) is of determinant class, R is always defined and when (M,ρ) is of determinant class is equal to the quotient Tan/TReid. The purpose of this paper is to prove the following Theorem: Theorem (i) There exists a density αF on M \Cr(h), which is a local quantity so that if μ is parallel in a smooth neighborhood of Cr(h) then αF vanishes on the neighborhood of the critical points and logR = ∫ M\Cr(h) αF . (ii) If μ is parallel then R = 1. An exact formula for R is also provided. This theorem can be viewed as an extension of our result [BFKM] which says that in the case where (M,ρ) is of determinant class and μ is parallel then R = 1.