D-branes, KK-theory and duality on noncommutative spaces
We present a new categorical classification framework for D-brane charges on noncommutative manifolds using methods of bivariant K-theory. We describe several applications including an explicit formula for D-brane charge in cyclic homology, a refinement of open string T-duality, and a general criterion for cancellation of global worldsheet anomalies.

title: D-Branes, RR-Fields and Duality on Noncommutative Manifolds
abstract: We develop some of the ingredients needed for string theory on noncommutative spacetimes, proposing an axiomatic formulation of T-duality as well as establishing a very general formula for D-brane charges. This formula is closely related to a noncommutative Grothendieck-Riemann-Roch theorem that is proved here. Our approach relies on a very general form of Poincaré duality, which is studied here in detail. Among the technical tools employed are calculations with iterated products in bivariant $$\mathsf K$$-theory and cyclic theory, which are simplified using a novel diagram calculus reminiscent of Feynman diagrams.

title: T-duality for circle bundles via noncommutative geometry
abstract: Recently Baraglia showed how topological T-duality can be ex- tended to apply not only to principal circle bundles, but also to non-principal circle bundles. We show that his results can also be recovered via two other methods: the homotopy-theoretic approach of Bunke and Schick, and the non- commutative geometry approach which we previously used for principal torus bundles. This work has several interesting byproducts, including a study of the K-theory of crossed products by e O(2) = Isom(R), the universal cover of O(2), and some interesting facts about equivariant K-theory for Z/2. In the final section of this paper, some of these results are extended to the case of bundles with singular fibers, or in other words, non-free O(2)-actions.

title: T-Duality for Orientifolds and Twisted KR-Theory
abstract: D-brane charges in orientifold string theories are classified by the KR-theory of Atiyah. However, this is assuming that all O-planes have the same sign. When there are O-planes of different signs, physics demands a “KR-theory with a sign choice” which up until now has not been studied by mathematicians (with the unique exception of Moutuou, who did not have a specific application in mind). We give a definition of this theory and compute it for orientifold theories compactified on S1 and T2. We also explain how and why additional “twisting” is implemented. We show that our results satisfy all possible T-duality relationships for orientifold string theories on elliptic curves, which will be studied further in subsequent work.

title: Amenability of crossed products ofC*-algebras
abstract: AbstractIt is shown that the class of amenable (resp. strongly amenable)C*-algebras is closed under the process of taking crossed products with discrete amenable groups. Under certain circumstances, amenability is also preserved under taking a “crossed product” with an amenable semigroup of linear endomorphisms. These facts are used to show that certain simpleC*-algebras
$$\mathcal{O}_n $$
 studied by J. Cuntz are amenable but not strongly amenable (thus answering a question of B. E. Johnson), yet are stably isomorphic to strongly amenable algebras.

title: Design of the Chicago Air Shower Array