Astérisque - Titles - 328 (2009) 255-296 < Titles < 2009 < 328

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Astérisque 328 (2009), xi+389 pages
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Presentation, Summary

The index of projective families of elliptic operators: the decomposable case
Varghese Mathai, Richard B. Melrose, Isadore M. Singer
Astérisque 328 (2009), 255-296

Résumé :
L'indice des familles projectives d'opérateurs elliptiques: le cas décomposable
Une théorie de l'indice pour des familles projectives d'opérateurs pseudodifférentiels elliptiques est développée sous les deux conditions suivantes: la classe de Dixmier-Douady est dans [Unparseable or potentially dangerous latex formula. Error 2 ], et la partie de degré deux est trivialisée sur l'espace total de la fibration. Le fibré d'Azumaya correspondant peut alors être raffiné en un fibré d'opérateurs régularisants. Les indices topologiques et analytiques d'une famille projective d'opérateurs elliptiques associée au fibré d'Azumaya lisse sont à valeurs dans la -théorie tordue de la base de la famille et le résultat principal est l'égalité de ces deux indices. Le caractère de Chern tordu de la famille est calculé par une variante de la théorie de Chern-Weil.

Mots-clefs : -théorie tordue, théorème de l'indice, invariant de Dixmier-Douady décomposable, fibré d'Azumaya lisse, caractère de Chern, cohomologie tordue

Abstract:
An index theory for projective families of elliptic pseudodifferential operators is developed under two conditions. First, that the twisting, i.e. Dixmier-Douady, class is in [Unparseable or potentially dangerous latex formula. Error 2 ] and secondly that the 2-class part is trivialized on the total space of the fibration. One of the features of this special case is that the corresponding Azumaya bundle can be refined to a bundle of smoothing operators. The topological and the analytic index of a projective family of elliptic operators associated with the smooth Azumaya bundle both take values in twisted -theory of the parameterizing space and the main result is the equality of these two notions of index. The twisted Chern character of the index class is then computed by a variant of Chern-Weil theory.

Keywords: Twisted K-theory, index theorem, decomposable Dixmier-Douady invariant, smooth Azumaya bundle, Chern Character, twisted cohomology

Class. math. : 19K56, 58G10, 58G12, 58J20, 58J22

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