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

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.