An index for gauge-invariant operators and the Dixmier-Douady invariant
نویسنده
چکیده
Let G → 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 G(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 G → B. As an application, we define a gauge-equivariant index for a family of elliptic operators (Pb)b∈B invariant with respect to the action of G → B, which, in this approach, is an element of K G(B). We then give another definition of the gauge-equivariant index as an element of K0(C (G)), the K-theory group of the Banach algebra C(G). We prove that K0(C (G)) ≃ K G(G) and that the two definitions of the gauge-equivariant index are equivalent. The algebra C(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 proved themselves useful in the study of Ramond-Ramond fields.
منابع مشابه
Bundles of C*-categories, II: C*-dynamical systems and Dixmier-Douady invariants
We introduce a cohomological invariant arising from a class in nonabelian cohomology. This invariant generalizes the Dixmier-Douady class and encodes the obstruction to a C*-algebra bundle being the fixed-point algebra of a gauge action. As an application, the duality breaking for group bundles vs. tensor C*-categories with non-simple unit is discussed in the setting of Nistor-Troitsky gauge-eq...
متن کاملFractional Analytic Index
For a finite rank projective bundle over a compact manifold, so associated to a torsion, Dixmier-Douady, 3-class, w, on the manifold, we define the ring of differential operators ‘acting on sections of the bundle’ in a formal sense. In particular any oriented even-dimensional manifold carries a projective spin Dirac operator in this sense. More generally the corresponding space of pseudodiffere...
متن کاملShift Invariant Spaces and Shift Preserving Operators on Locally Compact Abelian Groups
We investigate shift invariant subspaces of $L^2(G)$, where $G$ is a locally compact abelian group. We show that every shift invariant space can be decomposed as an orthogonal sum of spaces each of which is generated by a single function whose shifts form a Parseval frame. For a second countable locally compact abelian group $G$ we prove a useful Hilbert space isomorphism, introduce range funct...
متن کاملSome relations between twisted K-theory and E8 gauge theory
Recently, Diaconescu, Moore and Witten provided a nontrivial link between K-theory and M-theory, by deriving the partition function of the Ramond-Ramond fields of Type IIA string theory from an E8 gauge theory in eleven dimensions. We give some relations between twisted K-theory and M-theory by adapting the method of [1], [2]. In particular, we construct the twisted K-theory torus which defines...
متن کاملTopology of the C*{algebra bundles
Using the methods of the noncommutative geometry and the K–theory, we prove that the well–known Dixmier–Douady invariant of continuous–trace C –algebras and the Godbillon– Vey invariant of the codimension–1 foliations on compact manifolds coincide in a class of the so–called ”foliation derived” C–algebra bundles. Moreover, with the help of such bundles both of the above invariants admit an eleg...
متن کامل