Complex valued Ray–Singer torsion II

Abstract

In this paper we extend Witten-Helffer-Sjöstrand theory from selfadjoint Laplacians based on fiber wise Hermitian metrics to nonselfadjoint Laplacians based on fiber wise non-degenerate symmetric bilinear forms. As an application we show that results about the asymptotics of the Ray-Singer torsion of self-adjoint Witten deformation, as well as the strategy proposed by Burghelea-Friedlander-Kappeler to derive the comparison of Ray-Singer and Reidemeister torsion, can be extended to nonself-adjoint Witten deformation. This is then used to conclude the equality of complex analytic and Milnor-Turaev torsion, at least for odd dimensional manifolds, up to sign.

Refined analytic torsion on manifolds with boundary

“…There an alternative definition of refined analytic torsion is considered and a conjecture on its relation to the Turaev torsion is formulated. This conjecture has been proved independently by Burghelea and Haller in [13] and Su and Zhang in [29].…”

Cappell-Miller analytic torsion for manifolds with boundary

“…Then the refined analytic torsion ρ an (∇), cf. [27, (4-6), (4-7), (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)], can be written as…”

“…It was shown that the refined analytic torsion is closely related to the Farber-Turaev torsion [5,2,6,16]. Burghelea and Haller [9,8,10] defined the complex-valued Ray-Singer torsion associated to a non-degenerate symmetric bilinear form on a flat vector bundle over a manifold of arbitrary dimension and made an explicit conjecture concerning the relationship between the Burghelea-Haller analytic torsion and the Farber-Turaev torsion. This conjecture was proved up to sign by BurgheleaHaller [10] and in full generality by Su-Zhang [24].…”

“…Burghelea and Haller [9,8,10] defined the complex-valued Ray-Singer torsion associated to a non-degenerate symmetric bilinear form on a flat vector bundle over a manifold of arbitrary dimension and made an explicit conjecture concerning the relationship between the Burghelea-Haller analytic torsion and the Farber-Turaev torsion. This conjecture was proved up to sign by BurgheleaHaller [10] and in full generality by Su-Zhang [24]. Cappell and Miller [11] used non-self-adjoint Laplace operators to define another complex-valued analytic torsion and used the method in [24] to prove an extension of the Cheeger-Müller theorem [1,12,19,20], which states that the Cappell-Miller analytic torsion is equal to the Reidemeister torsion of the bundle E ⊕ E * , where E * denotes the dual bundle to E.…”

A Cheeger-Müller theorem for symmetric bilinear torsions on manifolds with boundary

“…In [30], Su and Zhang used the approach developed by Bismut-Zhang [4,5], making use of the Witten deformation, proved the conjecture in full generality. In [9], Burghelea and Haller proved their conjecture, up to sign, in the case where X is of odd dimensional. Now consider X with the boundary Y = ∅.…”