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.