  When the index bundle of a longitudinal Dirac type operator is transversely
smooth, we define its Chern character in Haefliger cohomology and relate it to
the Chern character of the $K-$theory index. This result gives a concrete
connection between the topology of the foliation and the longitudinal index
formula. Moreover, the usual spectral assumption on the Novikov-Shubin
invariants of the operator is improved.
