Analytical index and eta forms for Dirac operators with one-dimensional kernel over a hypersurface

TL;DR

This paper extends the transgression formula for eta forms to certain Dirac operators on fiber bundles with odd-dimensional fibers, where the operators have at most one eigenvalue crossing zero transversally.

Contribution

It generalizes existing formulas for eta forms to a broader class of Dirac operators with specific spectral properties on fiber bundles.

Findings

Extended transgression formula for eta forms applicable to new class of Dirac operators.

Provided explicit analytical index and eta form computations in the generalized setting.

Enhanced understanding of spectral flow and index theory for Dirac operators with simple eigenvalue crossings.

Abstract

We generalize the transgression formula for the eta form of Bismut, Cheeger and Berline, Getzler, Vergne for vertical Dirac operators on a fibre bundle with odd dimensional fibres where the Dirac operators have locally at most one eigenvalue of multiplicity one crossing zero transversally.