Twisted Analytic Torsion

TL;DR

This paper reviews classical torsion theories and introduces a new twisted analytic torsion for principal circle bundles, exploring its invariance properties and behavior under T-duality.

Contribution

It defines a novel twisted analytic torsion for invariant forms on circle bundles and analyzes its invariance and duality properties.

Findings

Twisted torsion is invariant under certain deformations in even dimensions.

Under T-duality, twisted torsions of invariant forms are inverses.

The paper extends classical torsion concepts to twisted settings with flux forms.

Abstract

We review the Reidemeister torsion, Ray-Singer's analytic torsion and the Cheeger-M"uller theorem. We describe the analytic torsion of the de Rham complex twisted by a flux form introduced by the current authors and recall its properties. We define a new twisted analytic torsion for the complex of invariant differential forms on the total space of a principal circle bundle twisted by an invariant flux form. We show that when the dimension is even, such a torsion is invariant under certain deformation of the metric and the flux form. Under T-duality which exchanges the topology of the bundle and the flux form and the radius of the circular fiber with its inverse, the twisted torsions of invariant forms are inverse to each other for any dimension.