Refined analytic torsion on manifolds with boundary

Abstract

We discuss the refined analytic torsion, introduced by M Braverman and T Kappeler as a canonical refinement of analytic torsion on closed manifolds. Unfortunately there seems to be no canonical way to extend their construction to compact manifolds with boundary. We propose a different refinement of analytic torsion, similar to Braverman and Kappeler, which does apply to compact manifolds with and without boundary. In a subsequent publication we prove a surgery formula for our construction. 58J52

Burghelea-Haller analytic torsion for manifolds with boundary

“…abs and the refined analytic torsion ρ an ( ∇) (we use the notation ρ an ( ∇) instead of ρ an (∇) in [23]) associated to ( D, ∇). Let h be the Hermitian metric induced by b as in Section 2.…”

“…Vertman [23] defined a different refinement of analytic torsion, similar to Braverman and Kappeler, which applied to compact manifolds with and without boundary. Inspired by this, in the present paper, we extend the Burghelea-Haller analytic torsion to compact connected Riemannian manifolds with boundary.…”

Cappell-Miller analytic torsion for manifolds with boundary

“…Inspired by the paper [27], in this section we generalize the construction of the Cappell-Miller analytic torsion to manifolds with boundary. …”

“…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.…”

“…In a subsequent paper [28] Vertman derived a gluing formula for the refined analytic torsion in this setting under the assumption that the Hermitian metric is flat. Inspired by the construction of [27], Su [23] extended the Burghelea-Haller analytic torsion to compact manifolds with boundary and compared it with the refined analytic torsion. In this paper we extend the construction of the Cappell-Miller analytic torsion to manifolds with boundary and compare the Cappell-Miller analytic torsion with the refined analytic torsion.…”