The comparison of two constructions of the refined analytic torsion on compact manifolds with boundary

TL;DR

This paper compares two different constructions of refined analytic torsion on compact manifolds with boundary, analyzing their equivalence using advanced mathematical techniques under specific conditions.

Contribution

It provides a detailed comparison of two distinct approaches to refined analytic torsion, employing the BFK-gluing formula, adiabatic limits, and deformation methods.

Findings

The two constructions are shown to be equivalent under certain conditions.

The comparison utilizes the BFK-gluing formula for zeta-determinants.

Results apply when the odd signature operator arises from a Hermitian flat connection with vanishing de Rham cohomologies.

Abstract

The refined analytic torsion on compact Riemannian manifolds with boundary has been discussed by B. Vertman and the authors, but these two constructions are completely different. Vertman used a double of de Rham complex consisting of the minimal and maximal closed extensions of a flat connection and the authors used well-posed boundary conditions ${\mathcal P}_{-, {\mathcal L}_{0}}$, ${\mathcal P}_{+, {\mathcal L}_{1}}$ for the odd signature operator. In this paper we compare these two constructions by using the BFK-gluing formula for zeta-determinants, the adiabatic method for stretching cylinder part near boundary and the deformation method used in [6], when the odd signature operator comes from a Hermitian flat connection and all de Rham cohomologies vanish.