Equivariant eta forms and equivariant differential $K$-theory

TL;DR

This paper develops an advanced analytic framework for equivariant differential K-theory, extending spectral theory tools to compact Lie group actions and orbifolds, and addresses key conjectures in the field.

Contribution

It introduces a new analytic model of equivariant differential K-theory for manifolds with finite stabilizers, extending spectral section theory and spectral flow to the equivariant setting.

Findings

Proved anomaly formula and functoriality of equivariant eta forms.

Extended Melrose-Piazza spectral section to equivariant case.

Constructed an analytic model of equivariant differential K-theory for orbifolds.

Abstract

In this paper, for a compact Lie group action,we prove the anomaly formula and the functoriality of the equivariant Bismut-Cheeger eta forms with perturbation operators when the equivariant family index vanishes. In order to prove them, we extend the Melrose-Piazza spectral section and its main properties to the equivariant case and introduce the equivariant version of the Dai-Zhang higher spectral flow for arbitrary dimensional fibers.Using these results, we construct a new analytic model of the equivariant differential K-theory for compact manifolds when the group action has finite stabilizers only,which modifies the Bunck-Schick model of the differential K-theory. This model could also be regarded as an analytic model of the differential K-theory for compact orbifolds. Especially, we answer a question proposed by Bunke and Schick about the well-definedness of the push-forward map.