APS boundary conditions, eta invariants and adiabatic limits (Q2750946)

Let \(\pi:X\rightarrow B\) be a twisted product where the fiber \(Y\) is a closed manifold but the base \(B\) is a compact manifold with smooth (possibly) non-empty boundary. Assume \(X\) and \(B\) are spin manifolds, that the metrics on \(B\) and \(X\) are product near the boundary, and that \(\pi\) is a Riemannian submersion. Express \(g_X=\pi^*g_B+g_Y\) and let \(g_X(\varepsilon):=\varepsilon^{-2}\pi^*(g_B)+g_Y\) for \(\varepsilon>0\) be the canonical variation. Assume the family of Dirac operators on the fibers is invertible and impose spectral boundary conditions to define the eta invariant \(\eta_X(\varepsilon)\) of \(X\). Let \(\widetilde\eta\) be the eta form of \textit{J. M. Bismut} and \textit{J. Cheeger} [J. Am. Math. Soc. 2, No. 1, 33-70 (1989; Zbl 0671.58037)]. NEWLINENEWLINENEWLINEThe author generalizes the adiabatic limit formula of Bismut-Cheeger to this context by showing that NEWLINE\[NEWLINE\lim_{\varepsilon\rightarrow 0}\eta_X(\varepsilon)=\int_B\widehat A(g_B)\wedge\widetilde\eta;NEWLINE\]NEWLINE the primary new difficulty is dealing with the associated boundary conditions.