Noncommutative geometry and conformal geometry. II: Connes-Chern character and the local equivariant index theorem
In this interesting paper, authors compute explicitly the Connes-Chern character of an equivariant Dirac spectral triple. 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 \textit{H. Donnelly} and \textit{V. K. Patodi} [Topology 16, 1--11 (1977; Zbl 0341.53023)], \textit{P. B. Gilkey} [Lect. Notes Pure Appl. Math. 48, 91--147 (1979; Zbl 0405.58044)] and \textit{T. Kawasaki} [An analytic Lefschetz fixed point formula and its application to \(V\)-manifolds: General defect formula and Hirzebruch signature theorem. Ann Arbor, MI: Johns Hopkins University (PhD thesis) (1976)]. 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, the authors compute the short-time limit of the JLO cocycle of an equivariant Dirac spectral triple. This approach may be generalized to prove the equivariant family index formula and the infinitesimal equivariant index formula.
scientific article; zbMATH DE number 6580413
scientific article