Real embedding and equivariant eta forms

Bismut and Zhang (Math Ann 295(4):661–684, 1993) establish a modZ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm {mod}}\, \mathbb {Z}$$\end{document} embedding formula of Atiyah–Patodi–Singer reduced eta invariants. In this paper, we explain the hidden modZ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm {mod}}\, \mathbb {Z}$$\end{document} 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.