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

Abstract

LetG→B\mathcal {G}\to Bbe a bundle of compact Lie groups acting on a fiber bundleY→BY \to B. In this paper we introduce and study gauge-equivariantKK-theory groupsKGi(Y)K_\mathcal {G}^i(Y). These groups satisfy the usual properties of the equivariantKK-theory groups, but also some new phenomena arise due to the topological non-triviality of the bundleG→B\mathcal {G}\to B. As an application, we define a gauge-equivariant index for a family of elliptic operators(Pb)b∈B(P_b)_{b \in B}invariant with respect to the action ofG→B\mathcal {G}\to B, which, in this approach, is an element ofKG0(B)K_\mathcal {G}^0(B). We then give another definition of the gauge-equivariant index as an element ofK0(C∗(G))K_0(C^*(\mathcal {G})), theKK-theory group of the Banach algebraC∗(G)C^*(\mathcal {G}). We prove thatK0(C∗(G))≃KG0(G)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 algebraC∗(G)C^*(\mathcal {G})is the algebra of continuous sections of a certain field ofC∗C^*-algebras with non-trivial Dixmier-Douady invariant. The gauge-equivariantKK-theory groups are thus examples of twistedKK-theory groups, which have recently turned out to be useful in the study of Ramond–Ramond fields.