Noncommutative geometry and conformal geometry, II. Connes-Chern character and the local equivariant index theorem

This paper is the second part of a series of papers on non-commutative geometry and conformal geometry. In this paper, we compute explicitly the Connes-Chern character of an equivariant Dirac spectral triple. The formula that we obtain is used in the first paper of the series. The computation has two main steps. The first step is the justification that the CM cocycle represents the Connes-Chern character. The second step is the computation of the CM cocycle as a byproduct of a new proof of the local equivariant index theorem of Donnelly-Patodi, Gilkey and Kawasaki. The proof combines the rescaling method of Getzler with an equivariant version of the Greiner-Hadamard approach to the heat kernel asymptotics. Finally, as a further application of this approach, we compute the short-time limit of the JLO cocycle of an equivariant Dirac spectral triple.