D-branes as Coherent Sheaves

Question

I know that ultimately, we have the definition/conjecture that the D-brane category in B-model topological string theory is actually the (bounded) derived category of coherent sheaves on a Calabi-Yau threefold $X$.  For this particular question, I don't want to worry about the derived category or K-theory, I would just like to understand how a single D-brane (or stack of branes) is modeled as a coherent sheaf on $X$.  

I know very roughly how the story goes.  The D-branes are non-perturbative objects who support open string endpoints.  Remarkably, the endpoints appear to an observer in the brane as a particle in the QFT sense.  This leads one to model a brane as a holomorphic subvariety $Z$ of $X$ with a line bundle on $Z$.  One then sees for a variety of reasons that a stack of $N$ D-branes should correspond to a rank $N$ bundle, instead of a line bundle.  

Clearly, a vector bundle on a subvariety pushes forward to $X$ as a coherent sheaf.  But there are tons of coherent sheaves which do not arise as pushforwards of bundles.  Some degenerate behavior of a coherent sheaf $\mathcal{F}$ is:

1. $\mathcal{F}$ need not be pure dimensional: I think I am okay with this.  It simply corresponds to branes of different dimensions.  For example, if $\mathcal{F}$ is supported on curves and points, it should be thought of as a bound state of D0-D2 branes.  

2. A locally-free sheaf is thought of as a space-filling brane.  But there are also torsion-free sheaves $\mathcal{F}$ supported on all of $X$ which are not locally-free.  For example, ideal sheaves.  Do these have an interpretation as D-branes?  I know that passing to the derived category, ideal sheaves are maybe thought of as roughly "anti-branes" perhaps.  But I'm wondering if they play a role when sticking to merely the category of coherent sheaves.  

3. Finally, there are torsion sheaves supported on subvarieties of dimension larger than one, which need not arise as the pushforward of a vector bundle.  Can these be interpreted as D-branes wrapping the subvariety?  

Is it possible that these degenerations are thrown in on top of pushforwards of vector bundles to get a nice D-brane moduli space?  I would be happy with this, but I'm especially worried about something like an ideal sheaf.  This seems to me to have no nice D-brane interpretation outside of the derived category, or K-theory or something.  

Comments

Nice question, but could you provide any references regarding your comments? Maybe it will help to give an answer since I understand vector bundles as sheaves (in a specific sense) but I am not so familiar with the D-brane picture.

---

@conformal_gk

The canonical referene for this is the old article

Paul S. Aspinwall, "D-Branes on Calabi-Yau Manifolds" arXiv:hep-th/0403166

I believe there is still no real update on this story. But would enjoy being shown to be wrong on this.

Answer 1

As far as I am aware, the state of the art of the "derivation" of the derived category of coherent sheaves as the home of the B-branes from the usual Lagrangian input of topologically twisted superconformal sigma-models is still

Paul S. Aspinwall, "D-Branes on Calabi-Yau Manifolds", arXiv:hep-th/0403166

This involves some serious handwaving.

The magic happens on the first page of section 5.4: It is stated that we happen to need an abelian category to define a derived category (in the usual way), it is left implicit that we already believe that the B-brane category has to be a derived category, and since locally free sheaves don't form one, they must be completed to coherent sheaves. The paragraph ends with "We have thus proven that", but it's really a bit circular at this point.

A better argument is hidden in the section 5.6 "Anti-branes and K-theory": By the famous old arguments (here) about physical D-branes being cycles in K-homology we believe that if a D-brane carrying some locally free Chan-Paton sheaf \(E\)and containing an anti-D-brane carrying an anti-Chan-Paton sheaf maping into the original anti-Chan-Paton sheaf \(A \overset{\phi}{\longrightarrow} E\), then there is a cancellation by D-brane annihiliation. The result is something like the cokernel \(coker(\phi)\) of the original CP-sheaf by the anti-CP-sheaf. That cokernel in general does not exist as a locally free sheaf, but it will exist as a coherent sheaf.

That's roughly the idea: D-branes do not really carry vector bundles (aka locally free sheaves), what they really carry is the result under brane/anti-brane annihilation of locally free sheaves, and these are kernels/cokernels of locally free sheaves, which are coherent sheaves.

This is still not a rigorous derivation, but this gets pretty close to what should be going on.

A derivation from first principle of D-brane charge in something like K-theory and/or derived categories is still missing. I just made some comments on this at StringMath2017 (here).

Comments

Thanks a lot Urs!  Perhaps I've been mistaken all this time.  My very naive understanding of why one thinks of D-branes as subvarieties with vector bundles was that to an "observer" in the brane, the endpoint of an open string would look like a particle.  Upon quantizing the theory, the particle should become a particle in the sense of gauge theory, so should be thought really as a section of some vector bundle.  Is this at all equivalent to what you call a 'Chan-Paton sheaf'?  I know my idea is far from refined, but is it okay as far as a rough understanding?  

---

Yes, that's the same thing. Chan and Paton had originally given labels to the endpoints of open strings indicating which of a "stack" of coincident D-branes they sit on. Therefore the vector bundles that these endpoints, which are particles as seen in the D-branes, yes, couple to, are called "Chan-Paton bundles". I said "Chan-Paton sheaves", because in the above discussion it is the sheaves of sections of vector bundles that matters.