The index of projective families of elliptic operators: The decomposable case
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Author(s)
Mathai, V.
Melrose, Richard B
Singer, Isadore Manuel
Date Issued
August 2009
Journal
Astérisque
Publisher
Société mathématique de France
Citation
Mathai, Varghese, Richard B. Melrose and Isadore M. Singer. "The index of projective families of elliptic operators: The decomposable case." From Probability to Geometry (II). Volume in honor of the 60th birthday of Jean-Michel Bismut. Astérisque 328 (2009): 255-296.
Version
Original manuscript
Abstract
An index theory for projective families of elliptic pseudodiffer-ential operators is developed under two conditions. First, that the twisting, i.e. Dixmier-Douady, class is in H² (Xℤ) ∪ H¹ (X; Z) ⊂ H³ (Zℤ) 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 K-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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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http://smf4.emath.fr/en/Publications/Asterisque/2009/328/html/smf_ast_328_255-296.php
