Mathematics – K-Theory and Homology
Scientific paper
2002-01-22
Mathematics
K-Theory and Homology
28 pages, LaTeX
Scientific paper
Let $\GR \to B$ 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 $K_\GR^i(Y)$. 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 $\GR \to B$. 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 $\GR \to B$, which, in this approach, is an element of $K_\GR^0(B)$. We then give another definition of the gauge-equivariant index as an element of $K_0(C^*(\GR))$, the $K$-theory group of the Banach algebra $C^*(\GR)$. We prove that $K_0(C^*(\GR)) \simeq K^0_\GR(\GR)$ and that the two definitions of the gauge-equivariant index are equivalent. The algebra $C^*(\GR)$ 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 proved themselves useful in the study of Ramond-Ramond fields.
Nistor Victor
Troitsky Evgenij
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