Geometric construction of Hopf cyclic characteristic classes

In earlier joint work with A. Connes on transverse index theory on foliations, cyclic cohomology adapted to Hopf algebras has emerged as a decisive tool in deciphering the total index class of the hypoelliptic signature operator. We have found a Hopf algebra H(n), playing the role of a `quantum structure group' for the `space of leaves' of a codimension n foliation, whose Hopf cyclic cohomology is canonically isomorphic to the Gelfand-Fuks cohomology of the Lie algebra of formal vector fields. However, with a few low-dimensional exceptions, no explicit construction was known for its Hopf cyclic classes. This paper provides an effective method for constructing the Hopf cyclic cohomology classes of H(n) and of H(n) relative to O(n), in the spirit of the Chern-Weil theory, which completely elucidates their relationship with the characteristic classes of foliations.