Let $X$ be a projective, smooth Calabi-Yau threefold and let $Z \subset X$ be a subscheme supported on curves and points. Its structure sheaf $\mathcal{O}_{Z}$ fits into the short exact sequence

$$0 \to \mathcal{I}_{Z} \to \mathcal{O}_{X} \to \mathcal{O}_{Z} \to 0.$$

One can show that the D-brane charge of $\mathcal{O}_{Z}$ is

$$\mathcal{Q}(\mathcal{O}_{Z}) = \text{PD}\bigg(\text{ch}(\mathcal{O}_{Z}) \sqrt{\text{td}X}\bigg) = (0,0, \beta, n),$$

where $\beta = [Z]_{\text{red}} \in H_{2}(X, \mathbb{Z})$ and $\chi(\mathcal{O}_{Z})=n$. Mathematically, the **Donaldson-Thomas invariants** are a (virtual) count of ideal sheaves corresponding to $\mathcal{O}_{Z}$ with fixed charge $\mathcal{Q}$.

Now for the physics...I believe one thinks of $\mathcal{O}_{Z}$ as a bound state of D2-D0 branes. The morphism $\mathcal{O}_{X} \to \mathcal{O}_{Z}$ is the coupling to a single D6-brane. One says (I think) that the Donaldson-Thomas invariants in B-model topological string theory computes BPS states associated to D6-D2-D0 B-brane bound states. My first question is:

1. *What kind of BPS states are these? Are they BPS particles in 4d?*

My second question is:

2. *What are Donaldson-Thomas invariants or the corresponding partition function in some physical string theory?* In the paper (https://arxiv.org/pdf/hep-th/0403167.pdf) even though they don't call them that, they're talking about DT invariants in Type IIB string theory. But they talk about them as D5-D1-D(-1) bound states. *How do these relate to the D6-D2-D0 bound states in the topological sector?* Obviously, there are subtle points translating between topological and physical branes. And in Type IIB, indeed the D$p$-branes must have $p$ odd.

My final question is of a different flavor than those above, but still interesting I think.

3. *Mathematically, the structure sheaves $\mathcal{O}_{Z}$ can have "thickened" or non-reduced structure. What role does this thickening play in physics? What is a "thickened" brane, if indeed one should think of it that way?*