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An index for gauge-invariant operators and the Dixmier-Douady invariant

Authors: Victor Nistor and Evgenij Troitsky
Journal: Trans. Amer. Math. Soc. 356 (2004), 185-218
MSC (2000): Primary 46L80
Published electronically: August 25, 2003
MathSciNet review: 2020029
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Abstract: Let $\mathcal{G}\to B$ be a bundle of compact Lie groups acting on a fiber bundle $Y \to B$. In this paper we introduce and study gauge-equivariant $K$-theory groups $K_\mathcal{G}^i(Y)$. These groups satisfy the usual properties of the equivariant $K$-theory groups, but also some new phenomena arise due to the topological non-triviality of the bundle $\mathcal{G}\to B$. As an application, we define a gauge-equivariant index for a family of elliptic operators $(P_b)_{b \in B}$ invariant with respect to the action of $\mathcal{G}\to B$, which, in this approach, is an element of $K_\mathcal{G}^0(B)$. We then give another definition of the gauge-equivariant index as an element of $K_0(C^*(\mathcal{G}))$, the $K$-theory group of the Banach algebra $C^*(\mathcal{G})$. We prove that $K_0(C^*(\mathcal{G})) \simeq K^0_\mathcal{G}(\mathcal{G})$ and that the two definitions of the gauge-equivariant index are equivalent. The algebra $C^*(\mathcal{G})$ is the algebra of continuous sections of a certain field of $C^*$-algebras with non-trivial Dixmier-Douady invariant. The gauge-equivariant $K$-theory groups are thus examples of twisted $K$-theory groups, which have recently turned out to be useful in the study of Ramond-Ramond fields.

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Additional Information

Victor Nistor
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802

Evgenij Troitsky
Affiliation: Department of Mechanics and Mathematics, Moscow State University, 119992 Moscow, Russia

Received by editor(s): April 22, 2002
Published electronically: August 25, 2003
Additional Notes: The first author was partially supported by NSF Young Investigator Award DMS-9457859 and NSF Grants DMS 991981 and 0200808
The second author was partially supported by RFFI Grant 99-01-01202 and Presidential Grant 00-15-99263.
Article copyright: © Copyright 2003 American Mathematical Society