Abstract

A General Families Index Theorem

We consider the heat operator of a Bismut superconnection for a family of generalized Dirac operators defined along the leaves of a foliation with Hausdorff graph. We assume that the strong Novikov-Shubin invariants of the Dirac operators are greater than three times the codimension of the foliation. We compute the t → ∞ asymptotics associated to a rescaling of the metric by 1/t and show that the heat operator converges to the Chern character of the index bundle of the operator. Combined with previous results, this gives a general families index theorem for such operators.