Noncommutative geometry, conformal geometry, and the local equivariant index theorem

We prove a local index formula in conformal geometry by computing the Connes-Chern character for the conformal Dirac (twisted) spectral triple recently constructed by Connes-Moscovici. Following an observation of Moscovici, the computation reduces to the computation of the CM cocycle of an equivariant Dirac (ordinary) spectral triple. This computation is obtained as a straightforward consequence of a new proof of the local equivariant index theorem of Patodi, Donelly-Patodi and Gilkey. This proof is obtained by combining Getzler's rescaling with an equivariant version of Greiner's approach to the heat kernel asymptotic. It is believed that this approach should hold in various other geometric settings. On the way we give a geometric description of the index map of a twisted spectral in terms of (twisted) connections on finitely generated projective modules.