A Canonical Quadratic Form on the Determinant Line of a Flat Vector Bundle

TL;DR

This paper introduces a canonical quadratic form on the determinant line of flat vector bundles, linking analytic, combinatorial, and algebraic torsions, and establishing new relationships among these invariants.

Contribution

It defines the torsion quadratic form, relates it to existing torsions, and proves a conjecture about its value at the Farber-Turaev torsion.

Findings

The torsion quadratic form is closely related to the Farber-Turaev torsion.

It establishes a precise relationship between the quadratic form and complex analytic torsion.

A conjecture about the quadratic form's value at the Farber-Turaev torsion is proposed and partially proved.

Abstract

We introduce and study a canonical quadratic form, called the torsion quadratic form, of the determinant line of a flat vector bundle over a closed oriented odd-dimensional manifold. This quadratic form caries less information than the refined analytic torsion, introduced in our previous work, but is easier to construct and closer related to the combinatorial Farber-Turaev torsion. In fact, the torsion quadratic form can be viewed as an analytic analogue of the Poincare-Reidemeister scalar product, introduced by Farber and Turaev. Moreover, it is also closely related to the complex analytic torsion defined by Cappell and Miller and we establish the precise relationship between the two. In addition, we show that up to an explicit factor, which depends on the Euler structure, and a sign the Burghelea-Haller complex analytic torsion, whenever it is defined, is equal to our quadratic form. We conjecture a formula for the value of the torsion quadratic form at the Farber-Turaev torsion and prove some weak version of this conjecture. As an application we establish a relationship between the Cappell-Miller and the combinatorial torsions.