Refined Analytic Torsion on Manifolds with Boundary

TL;DR

This paper introduces a new refinement of analytic torsion applicable to compact manifolds with boundary, overcoming previous limitations and setting the stage for a future gluing formula.

Contribution

It proposes a novel refinement of analytic torsion that extends to manifolds with boundary, unlike previous approaches.

Findings

New refinement applicable to manifolds with boundary

Establishment of a gluing formula in subsequent work

Extension of Braverman and Kappeler's construction

Abstract

The refined analytic torsion, defined by M. Braverman and T. Kappeler on closed manifolds, can be viewed as a refinement of the Ray-Singer torsion, since it is a canonical choice of an element with Ray-Singer norm one, in case of unitary representations. The complex phase of the refinement is given by the rho-invariant of the odd-signature operator. Unfortunately there seems to be no canonical way to extend the construction of Braverman and Kappeler to compact manifolds with boundary. In particular a gluing formula seems to be out of reach. 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 establish a gluing formula for our construction, which in fact can also be viewed as a gluing law for the original definition of refined analytic torsion by Braverman and Kappeler.