Bismut superconnections and the Chern character for Dirac operators on foliated manifolds

Abstract

In this paper, we show how to define a Bismut superconnection for generalized Dirac operators defined along the leaves of a compact foliated manifold M. Using the heat operator of the curvature of the superconnection, we define a (nonnormalized) Chem character for the Dirac operator, which lies in the Haefliger cohomology of the foliation. Rescaling the metric on M by 1/a and letting a -+ 0, we obtain the analog of the classical cohomologicaI formula for the index of a family of Dirac operators. In certain special cases, we can also compute the limit as a ~ oo and show that it is the Chern character of the 'index bundle' given by the kernel of the Dirac operator. Finally, we discuss the relation of our results with the Chem character in cyclic cohomology. Key wards: Bismut superconnection, generalized Dirac operators, foliations. 1. Bismut Superconnection for Foliations In this section, we follow pages 454-458 of (BV) making the necessary changes to adapt it to the foliation case. Let M be a C °° n dimensional manifold and F a C a p dimensional foliation on M. Let go be a metric on M. This induces a metric gr on each leaf L of F. We assume that the tangent bundle TF of F is oriented and that it is spin with a fixed spin structure. Assume that p is even and denote by So = S + ® S o the bundle of spinors along the leaves of F. Given a leaf L of F, denote by V ° the Levi-Civita connection on L and on SOIL. We denote also by V ° the connection on TF over M given by the orthogonal projection of the Levi-Civita connection for go on TM. Note that V ° on TF over M restricted to a leaf L is just the V ° above. Let £ be a complex vector bundle over M with Hermitian metric and connection. Denote by V ° the tensor product connection on So ® £JL. These data determine a smooth f amily D @ £ = (DL ® g) of Dirac operators, where DL @ g acts on sections of So @ £1L. They also determine another family of Dirac operators, D 6 g = (D (~ gx)~:eM where D @ £~ acts on sections of ~r*(S0 @ gIL~) where 7r : Lx -+ Lz is the holomony cover of the leaf through x. Thus D@£ is a smooth family of Dirac operators on g, the graph of F,