A new proof of a Bismut-Zhang formula for some class of representations

TL;DR

This paper provides a simplified proof of the Bismut-Zhang formula for certain representations by leveraging the holomorphic nature of refined analytic torsion and the Cheeger-Mueller theorem.

Contribution

It demonstrates that the Bismut-Zhang formula can be derived from the Cheeger-Mueller theorem for representations near unitary ones, using holomorphicity of refined torsion.

Findings

The Bismut-Zhang formula holds for representations in connected components containing unitary representations.

Refined analytic torsion is a holomorphic function on the space of representations.

The proof simplifies previous approaches by connecting to the Cheeger-Mueller theorem.

Abstract

Bismut and Zhang computed the ratio of the Ray-Singer and the combinatorial torsions corresponding to non-unitary representations of the fundamental group. In this note we show that for representations which belong to a connected component containing a unitary representation the Bismut-Zhang formula follows rather easily from the Cheeger-Mueller theorem, i.e. from the equality of the two torsions on the set of unitary representations. The proof uses the fact that the refined analytic torsion is a holomorphic function on the space of representations.