Bismut-Cheeger eta form and higher spectral flow

TL;DR

This paper extends the mathematical framework of eta invariants and spectral flow to include equivariant Bismut-Cheeger eta forms under group actions, generalizing key formulas in geometric analysis.

Contribution

It introduces a generalization of the variation, embedding, and adiabatic limit formulas for eta invariants to the equivariant Bismut-Cheeger eta forms with fiberwise group actions.

Findings

Generalized formulas for eta invariants under group actions

Extended the spectral flow concept to equivariant settings

Provided new tools for analyzing fiberwise Dirac operators

Abstract

In this paper, using the equivariant version of the Dai-Zhang higher spectral flow, we generalize the variation formula, embedding formula and the adiabatic limit formula for the Atiyah-Patodi-Singer eta invariants to the equivariant Bismut-Cheeger eta forms for a fiberwise compact Lie group action when the kernels of corresponding fiberwise Dirac operators form equivariant vector bundles.