Noncommutative geometry and conformal geometry. II: Connes-Chern character and the local equivariant index theorem
type: Item
instance of: scholarly article
title: Noncommutative geometry and conformal geometry. II: Connes-Chern character and the local equivariant index theorem
zbMATH Open document ID: 1343.58004
DOI: 10.4171/JNCG/235
author: Raphaël Ponge
author: Hang Wang
published in: Journal of Noncommutative Geometry
publication date: 2016-05-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1411.3703
review text: 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.
reviewed by: Yong Wang
Mathematics Subject Classification ID: 58B34
Mathematics Subject Classification ID: 58J20
Mathematics Subject Classification ID: 58J35
zbMATH DE Number: 6580413
zbMATH Keywords: noncommutative geometry
zbMATH Keywords: equivariant index theory
zbMATH Keywords: heat kernel techniques
zbMATH Keywords: Connes-Chern character
MaRDI profile type: Publication
arXiv ID: 1411.3703
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