An index for gauge-invariant operators and the Dixmier-Douady invariant
| dc.creator | Nistor, Victor | |
| dc.creator | Troitsky, Evgenij | |
| dc.date | 2002-01-22 | |
| dc.date | 2002-02-18 | |
| dc.date.accessioned | 2026-07-07T04:46:02Z | |
| dc.date.available | 2026-07-07T04:46:02Z | |
| dc.description | 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. | |
| dc.description | 28 pages, LaTeX | |
| dc.identifier | https://arxiv.org/abs/math/0201207 | |
| dc.identifier | http://arxiv.org/abs/math/0201207 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/63174 | |
| dc.subject | K-Theory and Homology | |
| dc.subject | Operator Algebras | |
| dc.title | An index for gauge-invariant operators and the Dixmier-Douady invariant | |
| dc.type | text |