Adiabatic limit, Bismut-Freed connection, and the real analytic torsion form

TL;DR

This paper investigates the adiabatic limit of the Bismut-Freed connection for deformed sub-signature operators on fibered manifolds, revealing the natural emergence of the Bismut-Lott analytic torsion form.

Contribution

It provides a detailed computation of the adiabatic limit of the Bismut-Freed connection and links it to the Bismut-Lott analytic torsion form in the context of complex flat vector bundles.

Findings

The adiabatic limit of the Bismut-Freed connection is explicitly computed.

The Bismut-Lott analytic torsion form appears naturally in this limit.

The results deepen understanding of the interplay between geometric analysis and topological invariants.

Abstract

For a complex flat vector bundle over a fibered manifold, we consider the 1-parameter family of certain deformed sub-signature operators introduced by Ma-Zhang. We compute the adiabatic limit of the Bismut-Freed connection associated to this family and show that the Bismut-Lott analytic torsion form shows up naturally under this procedure.