D-branes, KK-theory and duality on noncommutative spaces

This article centres in part around the following physical question: What is a D-brane? More precisely, given a closed string background X, what are the possible states of D-branes in X? At the level of the worldsheet field theory of open strings in X, this is a problem of finding the consistent boundary conditions in the underlying boundary conformal field theory. As we will discuss, when this worldsheet perspective is combined with the target space classification of D-branes in terms of Fredholm modules over a suitable C * -algebra, a powerful categorical description of D-branes and their charges emerges. This is particularly useful for those boundary states which have no geometric description. In certain instances these "nongeometric" backgrounds can be interpreted as noncommutative manifolds, i.e., as separable noncommutative C * -algebras. The formalism that we review in the following was developed in detail in refs. [1,2], and it allows for the construction of general charge vectors for D-branes on these noncommutative spaces.

This point of view becomes particularly fruitful for considerations involving compactifications with H-flux. Consider, for example, a principal torus bundle X → M with constant H-flux.

Applying a T-duality transformation along the fibre gives a space which does not always admit a global riemannian description. Instead, one can double the dualized directions and use elements of the T-duality group as transition functions between local patches. This is called a “T-fold” [3].

In some examples, one can show [4] that the open string metric on a T-fold is precisely the metric on an associated continuous field of stabilized noncommutative tori fibred over M which corresponds to a certain crossed product C * -algebra [5,6]. Thus the open string version of a T-fold can be generally regarded as a globally defined, noncommutative C * -algebra. Performing additional T-duality transformations along the base leads in some instances to nonassociative tori in the fibre directions [7]- [9]. This example, wherein the action of T-duality is realized by taking a certain crossed product algebra, motivates an axiomatic definition of topological open string T-duality. This generalizes and refines the more common examples of T-duality between noncommutative spaces in terms of Morita equivalence [10] to a special type of “KKequivalence”, which defines a T-duality action that is of order two up to Morita equivalence.

From a purely mathematical perspective of noncommutative geometry, the framework needed to achieve the physical constructions above develops more tools for dealing with noncommutative spaces in general. These include the appropriate noncommutative versions of Poincaré duality and orientation, topological invariants of noncommutative spaces such as the Todd genus, and a noncommutative version of the Grothendieck-Riemann-Roch theorem which is directly linked to the general formula for D-brane charge. All of this is defined and developed in the purely algebraic framework of separable C * -algebras.

It is well-known that D-brane charges and Ramond-Ramond fields in Type II superstring theory without H-flux are classified topologically by the complex K-theory of spacetime X [11]- [17]. We will begin by briefly reviewing some salient features of this classification that we will generalize later on to more generic noncommutative settings.

Let X be a compact spin c manifold. Poincaré duality in cohomology states that the natural bilinear pairing (x, y)

between cohomology classes x, y of X in complementary degree is non-degenerate. If α, β are de Rham representatives of x, y, then this pairing is just (x, y) H = X α ∧ β. On the other hand, in K-theory the natural bilinear pairing between complex vector bundles E, F → X is given by the index of the twisted Dirac operator

associated to the spin c structure on X. The Chern character gives a natural, Z 2 -graded ring isomorphism

but it doesn’t preserve these bilinear forms. However, by the Atiyah-Singer index theorem one has index

so we get an isometry by replacing the isomorphism (2.3) with the “twisted” Chern character ch -→ Todd(X) ⌣ ch .

(2.5)

Here Todd(X) ∈ H(X, Q) is the invertible Todd characteristic class of the tangent bundle of X, which can be expressed in terms of the Pontrjagin classes of X along with a degree two characteristic class c 1 ∈ H 2 (X, Z) whose reduction modulo 2 is the second Stiefel-Whitney class w 2 (X). This almost trivial observation plays a crucial role in what follows.

A natural geometric description of a D-brane in X is provided by a topological K-cycle (W, E, f ) in X [18]- [22], where f : W ֒→ X is a closed, embedded spin c submanifold of X (the brane worldvolume), and E → W is the Chan-Paton gauge bundle equipped with a hermitean connection and regarded as an element of the topological K-theory group K 0 (W ). The collection of K-cycles forms an additive category under disjoint union. The quotient of this c

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