Gluing formula of real analytic torsion forms and adiabatic limit

TL;DR

This paper proves a gluing formula for real analytic torsion forms in the context of smooth fibrations using the adiabatic method, assuming vanishing fiberwise twisted cohomology and product metrics near the hypersurface.

Contribution

It introduces a new proof of the gluing formula for real analytic torsion forms using the adiabatic method under specific geometric assumptions.

Findings

Established the gluing formula for real analytic torsion forms.

Applied the adiabatic method to the problem.

Assumed vanishing fiberwise twisted cohomology groups.

Abstract

In this article we use the adiabatic method to prove the gluing formula of real analytic torsion forms for a flat vector bundle on a smooth fibration under the assumption that the fiberwise twisted cohomology groups associated to the fibration of the cutting hypersurface are vanished. In this paper we assume that the metrics have product structures near the cutting hypersurface.