# Real embedding and equivariant eta forms

## TL;DR

This paper extends Bismut and Zhang's 1993 embedding formula for eta invariants to the equivariant family case, revealing a spectral flow interpretation and introducing an equivariant higher spectral flow concept.

## Contribution

It introduces an equivariant extension of the eta form embedding formula and interprets the mod Z term as a spectral flow, generalizing to the equivariant setting.

## Key findings

- Revealed the mod Z term as spectral flow
- Extended the embedding formula to equivariant families
- Introduced equivariant Dai-Zhang higher spectral flow

## Abstract

In 1993, Bismut and Zhang establish a mod Z embedding formula of Atiyah-Patodi-Singer reduced eta invariants. In this paper, we explain the hidden mod Z term as a spectral flow and extend this embedding formula to the equivariant family case. In this case, the spectral flow is generalized to the equivariant chern character of some equivariant Dai-Zhang higher spectral flow.