Transgressed Chern forms for Dirac operators (Q1103839)

\textit{D. Quillen} has introduced in his work [Topology 24, 89--95 (1985; Zbl 0569.58030)] a Chern-Weil theory with superconnections which, in the finite dimensional case, gives us the differential form representatives for the Chern character of a difference bundle. Using this construction in infinite dimensional situations by probabilistic methods, \textit{J.-M. Bismut} [Invent. Math. 83, 91--151 (1986; Zbl 0592.58047)] generalized his heat kernel proof of the Index theorem to give a local heat kernel proof of the Index theorem for a family of Dirac operators. For general p-summable Fredholm modules, \textit{A. Connes} and \textit{H. Moscovici} [C. R. Acad. Sci., Paris, Sér. I 303, 913--918 (1986; Zbl 0617.46075)] proved that the transgressed version of the Chern character forms associated with Quillen's superconnections exactly represents the Chern character cocycle in cyclic cohomology. These results of A. Connes, H. Moscovici and the author are exploited to calculate the Chern character cyclic cocycle in the case of Fredholm modules associated with Dirac operators.