A general families index theorem
The authors prove a families index theorem for a family of leafwise generalized Dirac operators associated with a foliation, having Hausdorff graph, of a compact Riemannian manifold. The proof is based on the definition of an appropriate Bismut superconnection and on the asymptotics associated with rescaling the manifold's metric by a factor of \(1/t\). A previous paper by \textit{J. L. Heitsch} [K-Theory 9, No. 6, 507-528 (1995; Zbl 0843.58117)] used the heat operator of the superconnection's curvature to define a Haefliger cohomology class that, in the \(t\rightarrow 0\) limit, gave the analogue of the classical cohomological families index formula. In the current paper, the authors identify regularity conditions (notably the condition that the Dirac operators' strong Novikov-Shubin invariants exceed three times the foliation's codimension) and show that these regularity conditions imply that the heat operator's \(t\rightarrow \infty\) limit gives the Chern character of the family's index bundle. A number of index theorems are special cases of the authors' general families index theorem. These include the Atiyah-Singer families index theorem and the Atiyah \(L^{2}\) covering index theorem. Indeed, the general families index theorem implies a common generalization of these two theorems. The order zero term of the general families index recovers A. Connes's index for measured foliations. (The regularity assumption is unnecessary here.) The relationship between the general families index theorem and the Connes-Skandalis longitudinal index theorem has not yet been determined.
scientific article; zbMATH DE number 1384367