The Chern-Connes character for the Dirac operator on manifolds with boundary

A cyclic cocycle is constructed for the Dirac operator on a compact spin manifold with boundary with the -invariant cochain introduced as the boundary correction term. This cocycle is seen to satisfy certain growth condition weaker than being entire and its pairing with the Chern characters of idempotents as well as the relevant index formulae are studied. The -cochain is a generalization of the Atiyah-Patodi-Singer -invariant and it carries information on the APS -invariants for Dirac operators twisted by bundles. It is also shown that one obtains the transgressed Chern character, defined by Connes and Moscovici, by applying the boundary operatorB in the cyclic bicomplex to the higher components of the -cochain.