Refined analytic torsion as analytic function on the representation variety and applications

TL;DR

This paper demonstrates that refined analytic torsion varies analytically over the representation variety of a manifold with boundary, establishes a gluing formula for certain components, and offers a new proof of a related torsion gluing formula.

Contribution

It proves the analyticity of refined analytic torsion as a section over the representation variety and introduces a novel proof of the Bruening-Ma gluing formula using temporal gauge transformation.

Findings

Refined analytic torsion is an analytic section over the representation variety.

A gluing formula for refined analytic torsion on specific components is established.

A new proof of the Bruening-Ma gluing formula using temporal gauge transformation is provided.

Abstract

We prove that refined analytic torsion on a manifold with boundary is an analytic section of the determinant line bundle over the representation variety. As a fundamental application we establish a gluing formula for refined analytic torsion on connected components of the complex representation space which contain a unitary point. Finally we provide a new proof of Bruening-Ma gluing formula for the Ray-Singer torsion associated to a non-Hermitian connection. Our proof is quite different from the one given by Bruening and Ma and uses a temporal gauge transformation.