Analytic surgery and gluing of the Bismut-Lott torsion form and eta form

TL;DR

This paper investigates the behavior of Bismut-Lott torsion and eta forms under analytic surgery, deriving a gluing formula that decomposes these invariants into additive and error components.

Contribution

It introduces a new gluing formula for Bismut-Lott torsion and eta forms, extending the understanding of their behavior under analytic surgery.

Findings

Rescaled heat kernel remains non-singular under surgery

Torsion and eta forms decompose into logarithmic, b-forms, and error terms

Igusa additivity property satisfied by the logarithmic term

Abstract

Given a fiber bundle with closed connected fibers, and a family of separating hypersurfaces, we study the behavior of the Bismut-Lott analytic torsion form, and the eta form for a duality bundle, under analytic surgery in the sense of Hassell, Mazzeo and Melrose. We find that under the surgery limit, the rescaled heat kernel is non-singular, while both the Bismut-Lott analytic torsion form and eta form can be written as the sum of a logarithmic term, which satisfies the Igusa additivity property, the b- Bismut-Lott analytic torsion form (respectively the b- eta form), and an error term coming from the reduced normal operator. Hence we obtain a gluing formula for these invariants.