Refined analytic torsion as an element of the determinant line

Abstract

We construct a canonical element, called the refined analytic torsion, of the determinant line of the cohomology of a closed oriented odd-dimensional manifold M with coefficients in a flat complex vector bundle E . We compute the Ray-Singer norm of the refined analytic torsion. In particular, if there exists a flat Hermitian metric on E , we show that this norm is equal to 1. We prove a duality theorem, establishing a relationship between the refined analytic torsions corresponding to a flat connection and its dual.

Burghelea-Haller analytic torsion for manifolds with boundary

“…Braverman and Kappeler [4,5] defined the refined analytic torsion for the flat vector bundle over odd dimensional manifolds and showed that it equals the Turaev torsion (cf. [12,22]) up to a multiplication by a complex number of absolute value one.…”

Cappell-Miller analytic torsion for manifolds with boundary

“…Braverman and Kappeler [5,2,3,4,6] defined and studied the refined analytic torsion for (M, E), which can be viewed as a refinement of the Ray-Singer torsion [22] and an analytic analogue of the Farber-Turaev torsion [13,14,25,26]. It was shown that the refined analytic torsion is closely related to the Farber-Turaev torsion [5,2,6,16].…”

“…Braverman and Kappeler [5,2,3,4,6] defined and studied the refined analytic torsion for (M, E), which can be viewed as a refinement of the Ray-Singer torsion [22] and an analytic analogue of the Farber-Turaev torsion [13,14,25,26]. 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.…”

“…By combining the absolute and relative boundary conditions, Vertman [27] applied the original construction of Braverman-Kappeler [5,2] to a new setting. The proposed construction refines the square of the Ray-Singer torsion, but applies to compact manifolds with or without boundary.…”

Η-Invariant AND FLAT VECTOR BUNDLES II

“…14 In this article, we first show that our results in 14 lead to a direct derivation of Gilkey's variation formula Theorem 3.7. 13 The second purpose of this paper is to apply the results in 14 to examine the 7/-invariants appearing in the recent papers of Braverman-Kappeler [7][8][9] on refined analytic torsions. We show that the imaginary part of the 77-invariant appeared in these articles admits an explicit local expression which suggests an alternate formulation of the definition of the refined analytic torsion there.…”