The String Coffee Table

Since Witten’s original claim that D-branes are described by K-theory

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

several refinements of the precise relationship have been discussed. Usually, the decategorification and Grothedieck group completion performed in forming K-theory from the category of vector bundles is identified with the physical process of partial mutual annihilation of space-filling D9 brane and anti-brane pairs, thereby realizing all lower-dimensional branes as decay products of D9-brane configurations.

T. Asakawa, S. Sugimoto and S. Terashima in their paper promote an alternative point of view, which, as they aim to demonstrate, makes more direct contact with the conception of K-theory in terms of $C^*$-algebras. Namely, they use the fact that one can go the other way around, and realize all higher-dimensional branes as composites of non-BPS D(-1) branes, using a certain flavor of what is called Matrix Theory.

From this point of view the world is described by $N\to \infty$ objects called non-BPS D-instantons, whose geometric configuration is encoded by ten operators

called the scalar fields, as well as an operator

called the tachyon field, all represented on some seperable Hilbert space

The mathematically inclined reader not familiar with this might (or might not) get an impression of what is going on here by looking at

Alain Connes, Michael R. Douglas, Albert Schwarz
Noncommutative Geometry and Matrix Theory
hep-th/9711162.

The dynamics of these funny objects is encoded by a functional which contains terms like

This is something one derives from boundary string field theory.

One expects to be able to make full sense of this only if at least

is trace class, hence $T$ is an unbounded operator, and

are in

the space of bounded operators on $\mathcal{H}$.

Anyone who has run across a spectral triple before should now have a déjà vu. If we like Fredholm modules better than spectral triples, we may instead consider the normalized tachyon operator

This is now bounded and $e^{-T^2}$ being trace class implies that $T_b^2 - 1$ is compact. Similarly the $[T,\Phi^\mu]$ are now required to be compact operators - and we have obtained a Fredholm module from our spectral triple describing D-instanton dynamics.

This story is thought to extend to an entire dictionary, which should look roughly like this.

Asakawa, Sugimoto and Terashima are mainly interested in identifying the very last entry on the right. They argue in

T. Asakawa, S. Sugimoto, S. Terashima
D-branes and KK-theory in Type I String Theory
hep-th/0202165

that the KK-theory $KK(A,B)$ describes D-brane configurations on product spaces $A\otimes B$ (with $A$ and $B$ the corresponding $C^*$-algebras.) I am not sure that I precisely follow this, but it is certainly compelling to associate KK-theory to pairs of D-brane configurations. Given that D-branes “are modules” ($\to$), KK-theory seems to describe the associated bimodules.

But before I get into this KK issue I should try to explain some of the entries of the above dictionary.

Tachyon fields as Dirac operators

One of the crucial keys for unlocking the above dictionary is the fact that the tachyon field operators that we are talking about indeed are Dirac operators in situations where they describe “geometric” D-branes.

Seiji Terashima
A Construction of Commutative D-branes from Lower Dimensional Non-BPS D-branes
hep-th/0101087.

D-branes as spectral triples or Fredholm modules

Recall the target space definition of a D-brane ($\to$), as a $\mathrm{spin}^\mathbb{C}$ manifold $W$ with a Chan-Paton K-class $E$ over it embedded into spacetime by means of $\phi : W \to X$.

This data is equivalently encoded in a spectral triple $(\mathcal{H},\rho : C(X) \to B(\mathcal{H}),D)$ or, alternatively, a Fredholm module (hence a class in K-homology) by setting

where $S_W$ is the spinor bundle on $W$, by letting the $*$-homomorphism

be given by precomposition with $\phi$ and taking $D$ to be the Dirac operator on $W$.

Notice that in particular the embedding of the D-brane is indeed encoded in an algebra homomorphism, as stated in the above table (which is of course nothing but a result of applying the contravariant functor from topological spaces to $C^*$-algebras).

We don’t want to distinguish D-branes up to gauge transformations, up continuous deformations of the tachyon potential $\simeq D$ and up to addition of $D$-brane configurations which disappear after tachyon condensation. Dividing out the space of Fredholm modules by these three equivalence relations, following the above dictionary, yields the K-homology group, and our D-brane defines a class of that. Which K-homology group precisely (which degree, real or complex, depends on whether we are in ty IIA, IIB or I, and on details which I have not mentioned yet.)

“Bi-branes” as bimodules

Let’s follow Asakawa et al. again and think of the Hilbert space $\mathcal{H}$ appearing in our spectral triples as the space $\mathcal{H} \simeq l^2(N,\C)$ of square summable series with amplitudes valued in $\mathbb{C}$, one for each of the $N$ Chan-Paton labels. Since $\mathbb{C}$ is the $C^*$-algebra of a point, we might imagine that it makes sense to consider Chan-Paton amplitudes indexed by a more general algebra $B$, encoding not just a point but some other space. This would lead to a $B$-Hilbert space $l^2(N,B)$, where linearity with respect to the complex numbers is replaced by linearity with respect to some $C^*$-algebra $B$.

As I said, Askawa et al. argue that this does describe D-branes on product manifolds. While I am wondering if it should not rather describe configurations with open strings stretched between a space encoded by $A$ and one encoded by $B$, I am willing to accept that it seems not incredibly unnatural to extend the above table and generalize Fredholm modules to bimodules:

Kasparov’s KK-Theory

So we start with two $C^*$-algebra $A$ and $B$. We want to construct something like an $A$-$B$-bimodule with a Fredholm operator represented on it. The fact that left and right bimodule action commute can be rephrased as saying that $A$ acts $B$-linearly, if we like. In particular, if $B=\C$, this would just say that $A$ acts by ordinary linear maps on some vector space.

So generalizing this, one says that a Hilbert $B$-module is a $B$-module with a $B$-valued (instead of $C$-valued) inner product. An (odd) Kasparaov $A$-$B$ bimodule is then a Hilbert $B$-module on which $A$ acts by $B$-linear self-adjoint operators, together with a self-adjoint operator $F$, playing the role of a Fredholm operator.

The definition (inlcuding the details which I skipped) is such that for $B = \C$ this reduces to an ordinary Fredholm module.

The duality between K-theory and K-homology in this language amounts to switching between Kasparav $A$-$\C$ and $\C$-$A$ bimodules.

The important point is that on Kasparov bimodules we do have a product induced from the ordinary tensor product of modules:

While I am not sure yet about the physical interpretation of $KK_\bullet(A,B)$ for $A$ and $B$ both different from the ground field, KK-theory certainly provides us with a very natural way to obtain K-cohomology as the dual to the Fredholm-module K-cohomology description of D-branes described above.

We simply observe that $KK(\mathbb{C},\mathbb{C})$ is the group of equivalence classes of $\mathbb{C}$-modules equipped with a $\mathbb{C}$-action. But this are nothing but vector spaces, hence the K-theory of a point, hence isomorphic to the natural numbers $\mathbb{Z}$.

Therefore the index pairing of K-theory with K-homology ($\to$) now essentially just becomes the product of a $\mathbb{C}$-$A$ bimodule (a K-theory class) with a $A$-$\mathbb{C}$-bimodule (a K-homology class).

The real power of the KK-formuation, though, as emphasized by Brodzki, Mathai, Rosenberg and Szabo, is that, since we are in a braided context, we may perform the KK-product only over certain factors of the left and right algebras. This allows to encode Poincaré duality. Maybe I’ll talk about that another time.