  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.
