hep-th/yymmnnn

Abstract. Twisted K-theory has received much attention recently in both mathematics and physics. We describe some models of twisted K-theory, both topological and geometric. Then we state a theorem which relates representations of loop groups to twisted equivariant K-theory. This is joint work with Michael Hopkins and Constantin Teleman. The loop group of a compact Lie group G is the space of smooth maps S 1 → G with multiplication defined pointwise. Loop groups have been around in topology for quite some time [Bo], and in the 1980s were extensively studied from the point of view of representation theory [Ka], [PS]. In part this was driven by the relationship to conformal field theory. The interesting representations of loop groups are projective, and with fixed projective cocycle τ there is a finite number of irreducible representations up to isomorphism. Considerations from conformal field theory [V] led to a ring structure on the abelian group R τ (G) they generate, at least for transgressed twistings. This is the Verlinde ring. For G simply connected R τ (G) is a quotient of the representation ring of G, but that is not true in general. At about this time Witten [W] introduced a three-dimensional topological quantum field theory in which the Verlinde ring plays an important role. Eventually it was understood that the fundamental object in that theory is a “modular tensor category ” whose Grothendieck group is the Verlinde ring. Typically it is a category of representations of a loop group or quantum group. For the special case of a finite group G the topological field theory is specified by a certain

Abstract. The central result here is an explicit computation of the Hochschild and cyclic homologies of a natural smooth subalgebra of stable continuous trace algebras having smooth manifolds X as their spectrum. More precisely, the Hochschild homology is identified with the space of differential forms on X, and the periodic cyclic homology with the twisted de Rham cohomology of X, thereby generalizing some fundamental results of Connes and Hochschild-Kostant-Rosenberg. The Connes-Chern character is also identified here with the twisted Chern character. 1.

Abstract. We prove the equivalence of two fundamental properties of algebraic stacks: being a quotient stack in a strong sense, and the resolution property, which says that every coherent sheaf is a quotient of some vector bundle. Moreover, we prove these properties in the important special case of orbifolds whose associated algebraic space is a scheme. (Mathematics Subject Classification: Primary 14A20, Secondary 14L30.) 1

We study a topological version of the T-duality relation between pairs consisting of a principal U(1)-bundle equipped with a degree-three integral cohomology class. We describe the homotopy type of a classifying space for such pairs and show that it admits a selfmap which implements a T-duality transformation. We give a simple derivation of a T-duality isomorphism for certain twisted cohomology theories. We conclude with some explicit computations of twisted K-theory groups and

We present an encompassing treatment of D–brane charges in supersymmetric SO(3) WZW models. There are two distinct supersymmetric CFTs at each even level: the standard bosonic SO(3) modular invariant tensored with free fermions, as well as a novel twisted model. We calculate the relevant twisted K–theories and find complete agreement with the CFT analysis of D–brane charges. The K–theoretical computation in particular elucidates some important aspects of N = 1 supersymmetric WZW models on non-simply connected Lie groups. March 2004Contents 1

We develop a group-theoretical approach to the formulation of generalized abelian gauge theories, such as those appearing in string theory and M-theory. We explore several applications of this approach. First, we show that there is an uncertainty relation which obstructs simultaneous measurement of electric and magnetic flux when torsion fluxes are included. Next we show how to define the Hilbert space of a self-dual field. The Hilbert space is Z2-graded and we show that, in general, self-dual theories (including the RR fields of string theory) have fermionic sectors. We indicate how rational conformal field theories associated to the two-dimensional Gaussian model generalize to (4k + 2)-dimensional conformal field theories. When our ideas are applied to the RR fields of string theory we learn that it is impossible to measure the K-theory class of a RR field. Only the reduction modulo torsion can be measured. May

The charges of D-branes in Kazama-Suzuki coset models are analyzed. We provide the calculation of the corresponding twisted equivariant K-theory, and in the case of Grassmannian cosets, su(n + 1)/u(n), compare this to the charge lattices that are derived from boundary conformal field theory.

The aim of this talk is to explain how symmetry breaking in a quantum field theory problem leads to a study of projective bundles, Dixmier-Douady classes, and associated gerbes. A gerbe manifests itself in different equivalent ways. Besides the cohomological description as a DD class, it can be defined in terms of a family of local line bundles or as a prolongation problem for an (infinite-dimensional) principal bundle, with the fiber consisting of (a subgroup of) projective unitaries in a Hilbert space. The prolongation aspect is directly related to the appearance of central extensions of (broken) symmetry groups. We also discuss the construction of twisted K-theory classes by families of supercharges for the supersymmetric Wess-Zumino-Witten model. This paper is based on a lecture at the meeting “La 70eme Rencontre entre

Abstract. We define the category of manifolds with extended tangent bundles, we study their symmetries and we consider the analogue of equivariant cohomology for actions of Lie groups in this category. We show that when the action preserves the splitting of the extended tangent bundle, our definition of extended equivariant cohomology agrees with the twisted equivariant de Rham model of Cartan, and for this case we show that there is localization at the fixed point set, à la Atiyah-Bott. 1.