An index theorem for families of Dirac operators on odd-dimensional manifolds with boundary (Q1383635)

Let \(\Phi:M\to B\) be a fibration of Riemannian manifolds, \(B\) compact, the typical fibre a compact odd-dimensional manifold \(X\) with \(\partial X\neq\emptyset\), the fibres with smoothly varying spin structures and with smoothly varying Riemannian product structure near the boundary. Let \(D= \{D_z\}_z\) be the associated family of Dirac operators and \(D_0= \{D_{0,z}\}_z\) the boundary family. If \(D_{0,z}\) is invertible for each \(z\in B\) then, introducing Atiyah-Patodi-Singer boundary conditions, one gets a family of selfadjoint Fredholm operators and an element \(\text{Ind}(D)\in K^1(B)\). Bismut and Cheeger conjectured a formula for the Chern character \(\text{Ch(Ind} (D))\in H^{\text{odd}}(B)\). In this paper, the authors prove such a formula under very general conditions, for general Clifford bundles \(F\) and giving up the assumption of \(D_0\) invertible. Let \(\sigma\) be the parity operator of \(E_{\partial M}=E_0^+\oplus E_0^-\). The authors introduce the notion of a \(Cl(1)\) spectral section \(P\) satisfying \(\sigma P+P\sigma= \sigma\). Such a \(P\) exists since \(\text{Ind} (D_0)=0\) in \(K^1(B)\). The choice of \(P\) fixes a selfadjoint boundary condition, varying smoothly with the base point, \(Du=f\) in \(M\), \(P(u|_{\partial M})=0\). Moreover, \(P\) fixes \(\mathbb{Z}_2\)-graded finite rank smoothing operators \(A_p^0\) such that \(\{D_0+ A_p^0\}_Z\) is invertible for any \(z\in B\). Now it is possible to define an index class \(\text{Ind} (D,P)\in K^1(B)\). The authors establish the index formula \[ \text{Ch(Ind} (\partial P))= (2\pi i)^{-\frac{n+1}{2}} \int_{M/B} \widehat{A} (M/B)\cdot \text{Ch'}(E)- \tfrac 12 \eta_{\text{odd},P} \text{ in }H^{\text{odd}} (B). \] Here \[ \eta_{\text{odd},P}= \frac{1}{\sqrt{\pi}} \text{STr}_{\partial M} \Biggl( \frac{d\widetilde{\mathbb{B}}_u}{du} e^{-(\widetilde {\mathbb{B}}_u)^2} \Biggr) du, \] where \(\widetilde{\mathbb{B}}_u= u^{\frac 12} (\partial_0+ \chi(u) A_p^0)+ \mathbb{B}_{[1]}+ u^{-\frac 12} \mathbb{B}_{[2]}\) is the rescaled perturbed Bismut superconnection on the boundary fibration.