Adiabatic Limit and Analytic Torsion of Vector Bundles

TL;DR

This paper explores the relationship between the index and analytic torsion of vector bundles over manifolds with adiabatic metrics, establishing new connections between these invariants and the Quillen metric.

Contribution

It introduces an analog of analytic torsion for vector bundles using the Witten Laplacian and proves the equivalence of Quillen metrics for the bundle and the base manifold.

Findings

Index of the vector bundle equals the index of the manifold.

Analytic torsion for the bundle is defined via the Witten Laplacian.

Quillen metrics for the bundle and the base manifold coincide.

Abstract

For a vector bundle $E^{n+k}$ over a closed manifold $M^n$ with $k$ even and $n$ odd, we equip the metric with an adiabatic parameter, and prove that the index of $E$ is the same as the index of $M$. We also introduce an analog of analytic torsion on $E$ using the Witten Laplacian. Moreover, we prove that the Quillen metric associated with this analytic torsion coincides with that of $M$.