The String Coffee Table

The main result presented is a generalization of the formula for D-brane charge known from

Ruben Minasian, Gregory Moore
K-theory and Ramond-Ramond charge
hep-th/9710230,

which is roughly the Chern class of the Chan-Paton bundle times the square root of the Todd class.

The authors of the present paper manage to find an analogue of the Todd class for general $\mathrm{spin}^c$ $C^*$-algebras, such that in terms of this the D-brane charge in the corresponding noncommutative geometry is still given by a formula of the familiar form.

Before starting, I would like to record some of the definitions.

Let $X$ be a manifold - spacetime - assumed to be spin, of dimension 10 and equipped with a Riemannian metric.

First, an elementary definition, for the sake of completeness. The goal is to very precisely state what we are talking about.

Definition 1) An (non-twisted) D-brane in $X$ is an embedded $\mathrm{spin}^c$-manifold

together with a K-theory class

specifying the Chan-Paton bundle on the D-brane according to

Edward Witten
Overview Of K-Theory Applied To Strings
hep-th/0007175.

The point of this being “non-twisted” is that, more generally, there is a Kalb-Ramond field on $X$, manifested in terms of an abeluian gerbe with connection and curving. In the presence of non-trivial such gerbes

Daniel S. Freed, Edward Witten
Anomalies in String Theory with D-Branes
hep-th/9907189

tells us that we have a twisted Chan-Paton bundle, with the twist given by the class of the gerbe pulled back to the D-brane and equal to the third Stieffel-Whitney-class of $W$ ($\to$).

So we really need a more refined definition of D-brane. (And I should maybe remark that we are talking here about what one could call “geometric D-branes”, those coming from submanifolds, as opposed to the more general ones arising as arbitrary CFT boundary conditions. No CFT is considered explicitly in the present case.)

Definition 2) A (possibly twisted) D-brane in $X$ in the presence of a Kalb-Ramond background field, i.e. in the presence of an abelian gerbe with connection and curving $\mathbf{G} \to X$ ($\to$) on $X$, with characteristic class

is an embedded oriented submanifold

together with a Chan-Paton K-theory class

such that the Freed-Witten anomaly cancellation condition

holds, where $W_3$ denotes the third Stieffel/Whitney class.

There are various ways how to encode the gerbe $\mathbf{G} \to X$ in terms of ordinary differential geometric structures. As before in the context of topological T-duality, one here finds it convenient to think of the gerbe in terms of its (possibly infinite-rank) gerbe modules ($\to$), hence $PU(H)\simeq BU(1) \simeq \mathbb{C}P^\infty$-bundles, for $H$ some seperable Hilbert space, or rather the associated vector bundles of compact operators on $H$. I believe we can think of these as coming from the endomorphism bundle of our gerbe module.

This latter point of view will be the one best suited for the $C^*$-algebraic setup, since we may associate to such algebra bundles naturally the $C^*$-algebra of sections of the bundle.

(I am still wondering if from this algebra one can get a globally well defined version of the Seiberg-Witten Moyal star, which is locally given by something like $\theta \propto (g \pm B)^{-1}$, for $B$ the local gerbe connection 2-form. But nobody seems to know.)

In order to say something about the charges of these D-branes, we need a couple of pairings involving K-cohomology and -homology ($\to$).

A D-brane as defined above, can be taken to define a class in the K-homology of $X$

or, more precisely, in the presence of a Kalb-Ramond-gerbe, a class in the twisted K-homology

Next, we want to define the RR-charge of a given D-brane. In words, this is computed by

$\bullet$ pushing the Chan-Paton bundle from the worldvolume forward (the “wrong way” ($\to$)) intto spacetime $X$

$\bullet$ computing its Chern class there

$\bullet$ cup-multiplying this Chern class with half of the Atiyah-Hirzebruch class of spacetime

$\bullet$ pulling the result back to the world-volume $W$

$\bullet$ and evaluating it on $W$.

If we denote the ordinary Chern class of a bundle $N$ with $\mathrm{ch}(N)$, and the modified Chern class by

where $\hat A$ is the Atiyah-Hirzebruch class, then the above reads in formulas

Definition 3) The RR-charge of a D-brane with world volume $\phi : W \to X$ and Chan-Paton bundle $E$ is

Understanding this formula in more detail crucially involves Poincaé duality. The goal is to generalize this notion of Poincaré duality to K-theory over noncommutative spaces. The authors of the above paper argue that the best language for that is Kasparov’s KK-Theory It turns out that the notion of Poincaré duality we wish to find amounts KK bimodule composition with invertible $A \otimes B$-$\C$-bimodules.

The authors emphasize the usefulness of a certain diagrammatic notation for dealing with the bimodule-like product on Kasparov bimodules. To my mind, it seems that this notation is nothing but the obvious string diagram notation for dealing with categoriy of bimodules internal to braided categories. Essentially, all this lives in fact in a 3-category of the sort discussed here.

[ more later….]