The Metric Anomaly of Analytic Torsion at the Boundary of an Even Dimensional Cone

TL;DR

This paper clarifies the third, complex term in the analytic torsion formula for even-dimensional cones, linking it to metric anomaly and isolating the singularity's contribution for a clearer understanding.

Contribution

It identifies the third term in the analytic torsion formula as a metric anomaly, refining the understanding of torsion in cones by separating singularity effects.

Findings

The third term is a metric anomaly related to boundary structure.

The analytic torsion can be expressed solely via Betti numbers and base torsion.

The singular contribution is isolated from the boundary metric anomaly.

Abstract

The formula for analytic torsion of a cone in even dimensions is comprised of three terms. The first two terms are well understood and given by an algebraic combination of the Betti numbers and the analytic torsion of the cone base. The third "singular" contribution is an intricate spectral invariant of the cone base. We identify the third term as the metric anomaly of the analytic torsion coming from the non-product structure of the cone at its regular boundary. Hereby we filter out the actual contribution of the conical singularity and identify the analytic torsion of an even-dimensional cone purely in terms of the Betti numbers and the analytic torsion of the cone base.