The Metric Anomaly of Analytic Torsion on Manifolds with Conical Singularities

TL;DR

This paper investigates the behavior of analytic torsion on manifolds with conical singularities, revealing its invariance under certain metric deformations and identifying the spectral contributions from the cone's cross-section.

Contribution

It provides a detailed analysis of the metric anomaly of analytic torsion on conical manifolds, linking it to spectral data of the cross-section and extending previous computations.

Findings

Analytic torsion is invariant under higher order metric deformations near singularities.

The metric anomaly of analytic torsion is expressed in terms of spectral data of the cone's cross-section.

The spectral contribution exhibits a torsion-like invariant that scales with the cross-section's metric.

Abstract

In this paper we study the analytic torsion of an odd-dimensional manifold with isolated conical singularities. First we show that the analytic torsion is invariant under deformations of the metric which are of higher order near the singularities. Then we identify the metric anomaly of analytic torsion for a bounded generalized cone at its regular boundary in terms of spectral information of the cross-section. In view of previous computations of analytic torsion on cones, this leads to a detailed geometric identification of the topological and spectral contributions to analytic torsion, arising from the conical singularity. The contribution exhibits a torsion-like spectral invariant of the cross-section of the cone, which we study under scaling of the metric on the cross-section.