Ray-Singer Type Theorem for the Refined Analytic Torsion

TL;DR

This paper proves that the refined analytic torsion forms a holomorphic section over the representation space of a manifold's fundamental group, relates it to combinatorial torsion, and connects spectral flow with combinatorial topology.

Contribution

It establishes the holomorphic nature of refined analytic torsion, computes its ratio with Farber-Turaev torsion, and links eta-invariant with torsion phase, extending previous theorems.

Findings

Refined analytic torsion is a holomorphic section.

Ratio between refined analytic and Farber-Turaev torsion is calculated.

A formula relating eta-invariant and torsion phase is derived.

Abstract

We show that the refined analytic torsion is a holomorphic section of the determinant line bundle over the space of complex representations of the fundamental group of a closed oriented odd dimensional manifold. Further, we calculate the ratio of the refined analytic torsion and the Farber-Turaev combinatorial torsion. As an application, we establish a formula relating the eta-invariant and the phase of the Farber-Turaev torsion, which extends a theorem of Farber and earlier results of ours. This formula allows to study the spectral flow using methods of combinatorial topology.