Transgressed Chern forms for Dirac operators
type: Item
instance of: scholarly article
title: Transgressed Chern forms for Dirac operators
zbMATH Open document ID: 0646.46067
DOI: 10.1016/0022-1236(88)90075-4
author: Jean-Michel Bismut
published in: Journal of Functional Analysis
publication date: 1988-01-01T00:00:00Z
review text: \textit{D. Quillen} has introduced in his work [Topology 24, 89--95 (1985; Zbl 0569.58030)] a Chern-Weil theory with superconnections which, in the finite dimensional case, gives us the differential form representatives for the Chern character of a difference bundle. Using this construction in infinite dimensional situations by probabilistic methods, \textit{J.-M. Bismut} [Invent. Math. 83, 91--151 (1986; Zbl 0592.58047)] generalized his heat kernel proof of the Index theorem to give a local heat kernel proof of the Index theorem for a family of Dirac operators. For general p-summable Fredholm modules, \textit{A. Connes} and \textit{H. Moscovici} [C. R. Acad. Sci., Paris, Sér. I 303, 913--918 (1986; Zbl 0617.46075)] proved that the transgressed version of the Chern character forms associated with Quillen's superconnections exactly represents the Chern character cocycle in cyclic cohomology. These results of A. Connes, H. Moscovici and the author are exploited to calculate the Chern character cyclic cocycle in the case of Fredholm modules associated with Dirac operators.
Mathematics Subject Classification ID: 46L80
zbMATH DE Number: 4054352
zbMATH Keywords: Chern-Weil theory with superconnections
zbMATH Keywords: differential form representatives for the Chern character of a difference bundle
zbMATH Keywords: local heat kernel proof of the Index theorem for a family of Dirac operators
zbMATH Keywords: p-summable Fredholm modules
zbMATH Keywords: transgressed version of the Chern character forms
zbMATH Keywords: Chern character cocycle in cyclic cohomology
reviewed by: Do Ngoc Diep
MaRDI profile type: Publication
full work available at URL: 0022-1236(88)90075-4
OpenAlex ID: W1984172675
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description: scientific article; zbMATH DE number 4054352