Relative torsion is the generalization of the ratio of the Reidemeister torsion $T_{\text{Reid}}$ and analytic torsion $T_{\text{an }}$ to the flat bundle ${\cal E}$ over a compact manifold $M$ associated to an arbitrary representation $\rho:\pi_1 (M)\to Gl_{\cal A}({\cal W})$.
In this paper, explicit form of the relative torsion as the integrals of differential forms, is given, assuming ${\cal W}$ is a finite type ${\cal A}$-Hilbert module where ${\cal A}$ is a finite von Neumann algebra, (Theorem 1.1). If ${\cal A}= \bbfC$, authors’ formula coincides to the main result of {\it J.-M. Busmut} and {\it W. Zhang}, An extension of a theorem by Cheeger and Müller, Astérisque 205, 235 p. (1992;

Zbl 0781.58039). In a previous paper, authors proved coincidence of $T_{\text{Reid}}$ and $T_{\text{an}}$ if $\rho$ is a unitary representation of determinant class [{\it D. Burghelea}, {\it L. Friedlander}, {\it T. Kappeler}, and {\it P. McDonald}, Geom. Funct. Anal. 6, No. 5, 751-859 (1996;

Zbl 0874.57025)]. This result follows from Theorem 1.1. In this paper $\rho$ needs not neither unitary nor determinant class. So Theorem 1.1 is a generalization of authors’ previous result and main result of Bismut-Zhang.
The relative torsion ${\cal R}$ is defined for the data $(M, {\cal T},g, \tau)$, $M$ a closed connected manifold, ${\cal T}=\{{\cal E}, \nabla, \mu\}$, where $\nabla$ and $\mu$ are the canonical flat connection and a Hermitian structure of ${\cal E}$, respectively, $g$ is a Riemannian metric of $M$ and $\tau= (h,g')$ is a generalized triangulation of $M$, where $h$ is a Morse function and $g'$ is a metric of $M$. By using $\mu$ and $g$, Hodge Laplacian $\Delta$ on the space of differential forms with coefficients in ${\cal E}$ is defined. On the other hand, by using $\tau$, combinatorial coboundary operator $\delta$ and its Laplacian $\Delta^{\text{comb}}$ are defined. Since authors work on general $\rho$, $\det\Delta$, etc. may not be defined. To overcome this difficulty, Sobolev metric on the space of differential forms with coefficients in ${\cal E}$ is introduced by using fractional powers of $\Delta+\text{Id}$. The operator $d(g_s)$ is defined by $$d(g_s)_k= \left(\matrix -\delta_{k-1} & g_{k;s}\\ 0 & d_k\endmatrix \right),$$ where $g_s=\text{Int}_s\circ (\Delta+ \text{Id})^{-s/2}$ is the modified integration map. The Laplacian $\Delta(g_s)$ of $g_s$ is similarly defined as Hodge Laplacian. It is shown, $\Delta(g_s)$ admits a nonvanishing regularized determinant. The relative torsion ${\cal R}$ is defined as $$\log{\cal R}=\frac 12\sum_K(-1)^{k+1} k\log\det\bigl( \Delta(g_s) \bigr)_k. $$ It is shown that this definition does not depend on $s$. Under the setting of this paper, $T_{\text{an}}$ and $T_{\text{Reid}}$ may not be defined. But if ${\cal E}$ is of determinant class, then they are defined and $\log{\cal R}=\log T_{\text{an}}- \log T_{\text{Reid}}$ holds [{\it A. L. Carey}, {\it V. Mathai} and {\it A. Mishchenko}, Nielsen theory and Reidemeister Torsion, Banach Cent. Publ. 49, 43-67 (1999;

Zbl 0941.19005)]. Definition and properties of determinant class are exposed in Appendix B together with examples of nondeterminant class morphisms.
Let $Cr(h)$ be the set of critical points of $h$, $X=-\text{grad}_{g'}h$. Then it is shown there exist a closed 1-form $\theta=\theta (\rho,\mu)$, an orientation bundle valued $(n-1)$-form $\Psi= \Psi(TM,g)$ on $TM\smallsetminus M$ and a smooth function $V=V(\rho, \mu_1, \mu_2)$ such that $$\multline \log{\cal R}=(-1)^{n+1} \int_{M \smallsetminus Cr(h)} \theta(\rho, \mu_0)\wedge X^*\bigl(\Psi (TM,g)\bigr)+\\ +\int_M V(\rho, \mu, \mu_0)e(M,g)- \sum_{x\in Cr(h)} (-1)^{\text{ind} (x)}V(\rho, \mu,\mu_0)(x), \endmultline$$ where $e$ is the Euler form and $\mu_0$ is parallel in the neighborhood of $Cr(h)$. If $M$ is of odd dimension, then this formula simplifies $$\log{\cal R}=\int_{M\smallsetminus Cr(h)}\theta (\rho,\mu_0)\wedge X^*\bigl(\Psi(TM,g)\bigr)-\sum_{x\in Cr(h)}(-1)^{\text{ind} (x)}V(\rho,\mu, \mu_0)(x)$$ (Theorem 1.1). To show Theorem 1.1, Witten deformation $d(t)=e^{-th} de^{th}$ is used. The deformation of relative torsion by this deformation is denoted by ${\cal R}(t)$. Then it is shown $$\log{\cal R}(t)=\log{\cal R}+t \dim {\cal W}\int_M he(M,g)$$(Theorem 3.1). This formula and computation of asymptotic form of ${\cal R}(t)$, $t\to\infty$, applying Witten-Helfer-Sjöstrand theory [{\it M. Helffer} and {\it J. Sjöstrand}, Commun. Partial Differ. Eq. 10, 245-340 (1985;

Zbl 0597.35024)] show the existence of a local density $\alpha$ on $M\smallsetminus Cr(h)$ when $\mu$ is parallel with respect to $\nabla$ in the neighborhood of $Cr(h)$ such that $\log{\cal R}= \int_{M \smallsetminus Cr(h)} \alpha$ (Proposition 1.1. It is also shown ${\cal R}=1$ if $\mu$ is parallel). Proof of Proposition 1.1 under the assumption $g= g'$ is given in Section 4. Proof of Proposition 1.1 without the assumption $g= g'$ needs to compute variations of ${\cal R}(M,\rho,\mu,g,\tau)$ with respect to $\mu$ and $g$ (Hermitian and metric anomalies). They are computed by using a smooth function $V$ and Euler form (Hermitian anomaly) and Chern-Simons form (metric anomaly) (Section 5). Theorem 1.1 is proved in Section 6 by these results and computation of anomaly with respect to the triangulation, given in Section 5. In [{\it D. Burghelea}, Lett. Math. Phys. 47, 149-158 (1999;

Zbl 0946.58026)] it was shown, $\log{\cal R}(M,\rho,\mu,\tau,g)$ gave a function $F(M,E)(\rho)$ on $\text{Rep}(\pi_1(M),Gl_{\cal A}({\cal W}))$, where $E$ is an Euler structure in the sense of Turaev [{\it V. Turaev}, Math. USSR, Izv. 34, No. 3, 627-662 (1990); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 3, 607-643 (1989;

Zbl 0707.57003)]. If $N$ is an even dimensional simply connected manifold, for a suitable Euler structure $E_0$ of $S^1\times N$, $$F (S^1 \times N,E_0) (\rho)=-{\chi(N) \over 2}\log\det \bigl(\rho(1)^* \rho(1) \bigr)^{1/2},$$ is shown as an application of Theorem 1.1 (Proposition 6.1). This shows nontriviality of $F$ and authors say it might be a useful source of topological and geometric invariants of $M$.
Other parts of the paper are as follows; Section 1 gives summary of the results and definitions of analytic torsion, Reidemeister torsion and relative torsion, and related terminologies. To define torsions, determinants of Laplacian etc. are needed. Since these determinants can not be defined directly, in general, regularization procedures are necessary. Introducing the notion $sF$ (strong Fredholm) type operator, these are done in Section 2. Proof of additive property of relative torsion (Lemma 2.6, a slightly stronger version of a Lemma due to Carey-Mathai-Mishchenko) is given in Appendix A (here Lemma 2.6 is misquoted as Lemma 2.14). By using $sF$ type operator, $\zeta$-regular operator and complex are defined. Then showing (de Rham and simplicial) complexes used in this paper become $\zeta$-regular complexes under the regularization by $\text{(Id}+ \Delta)^{-s/2}$ for $s$ is sufficiently large, relative torsion is defined in Section 3. Section 3 also study Witten deformation of the relative torsion which is crucial to the proof of Proposition 1.1.