D-Branes, RR-Fields and Duality on Noncommutative Manifolds
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: A noncommutative sigma-model
abstract: We begin to study a sigma-model in which both the space-time manifold and the two-dimensional string world-sheet are made noncommutative. The most precise results apply to the case where both the space-time manifold and the two-dimensional string world-sheet are replaced by noncommutative tori. In this situation, we are able to determine when maps between such noncommutative tori exist, to derive the Euler-Lagrange equations, to classify many of the critical points of the Lagrangian, and to study the associated partition function.

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: On mysteriously missing T-duals, H-flux and the T-duality group
abstract: A general formula for the topology and H-flux of the T-duals of type II string theories with H-flux on toroidal compactifications is presented here. It is known that toroidal compactifications with H-flux do not necessarily have T-duals which are themselves toroidal compactifications. A big puzzle has been to explain these mysterious ``missing T-duals'', and our paper presents a solution to this problem using noncommutative topology. We also analyze the T-duality group and its action, and illustrate these concepts with examples.

title: K-theoretic matching of brane charges in S- and U-duality
abstract: We discuss K-theoretic matching of D-brane charges in the string duality between type I on T 4 and type IIA on K3. This case is more complex than the familiar case of IIA/IIB duality, which is already well understood, but it turns out that replacing K3 by its orbifold blow-down seems largely to resolve the apparent problems with the theory.

title: Dualities in Field Theories and the Role of K-Theory
abstract: It is now known (or in some cases just believed) that many quantum field theories exhibit dualities, equivalences with the same or a different theory in which things appear very different, but the overall physical implications are the same. We will discuss some of these dualities from the point of view of a mathematician, focusing on "charge conservation" and the role played by K-theory and noncommutative geometry. Some of the work described here is joint with Mathai Varghese and Stefan Mendez-Diez; the last section is new.

title: Design of the Chicago Air Shower Array