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

Let script G sign → B be a bundle of compact Lie groups acting on a fiber bundle Y → B. In this paper we introduce and study gauge-equivariant K-theory groups K _{script G sign} 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 script G sign → B. As an application, we define a gauge-equivariant index for a family of elliptic operators (P _b) _b∈B invariant with respect to the action of script G sign → B, which, in this approach, is an element of K _{script G sign} ⁰(B). We then give another definition of the gauge-equivariant index as an element of K ₀(C* (script G sign)), the K-theory group of the Banach algebra C*(script G sign). We prove that K ₀(C*(script G sign)) ≃ K _{script G sign} ⁰(scrit G sign) and that the two definitions of the gauge-equivariant index are equivalent. The algebra C*(script G sign) 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.