# What are Donaldson-Thomas Invariants in Physical String Theory?

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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?

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1) Compactifying IIA string theory on a Calabi-Yau 3-fold $X$, we get in the four non-compact dimensions a theory with $\mathcal{N}=2$ supersymmetry. D6-D2-D0 branes wrapped on $X$, with one non-compact direction (time direction), define BPS particles in 4d.

2) The spectrum of BPS particles in a $\mathcal{N}=2$ 4d theory depends on the vector multiplet background (in the vector multiplet moduli spaces, there are real codimension one walls along which the BPS spectrum changes). For IIA on X, the vector multiplet moduli space is the stringy Kähler moduli space, which near the large volume limit, is parametrized by the Kähler class on $X$ and the B-field. Donaldson-Thomas invariants (in the geometric sense of counts of ideal sheaves) are counts (more precisely supersymmetric indices) of the BPS states, of charge $(1,0,\beta,n)$, in the large volume limit (large Kähler class) and large B-field.

If one considers IIB on $X$, one gets another $\mathcal{N}=2$ 4d theory, and  D5-D1-D(-1) branes completely wrapped on $X$ define instantons (in the broad sense of spacetime localized objects) in 4d. These instantons give quantum corrections to the hypermultiplet moduli space, which is a torus fibration over the stringy Kähler moduli space of $X$. These instanton corrections depend on the precise locus in the stringy Kähler moduli spaces and Donaldson-Thomas invariants enter these instanton corrections in the large volume and large B-field story.

These two stories, IIA or IIB on $X$, with Donaldson-Thomas invariants counting either particles or instantons in 4d, are compatible, as is clear by a further compactification on a circle. Compactifying IIA on $X \times S^1$, the vector multiplet moduli space becomes a torus bundle over the stringy Kähler moduli space, with 3d instantons corrections coming from BPS particles in 4d wrapping the circle direction. Applying T-duality along the circle, one exchanges IIA and IIB and we get exactly the IIB hypermultiplet moduli space, with corrections now coming from instantons in 4d.

3) The thickening has to do with multiple branes coming together and with possibly non-trivial Higgs vev turned on (see eg https://arxiv.org/abs/hep-th/0309270 )

answered Dec 25, 2017 by (5,120 points)

I'm not completely sure I understand your second to last paragraph about reconciling the IIA and IIB picture.  Perhaps part of my confusion, do D$p$-branes still have a $(p+1)$-dimensional worldvolume even if they give rise to instantons in 4d?  I understand the distinction between IIA and IIB, as well as the difference between BPS particles and BPS instantons, but I'm afraid I'm still a bit confused how the dimension-counting works in reconciling things.  But thanks a lot for your great response.

Upon reflection, I think I understand your second to last paragraph after all.  Unless I'm mistaken, if you have a D$p$-brane in IIA or IIB wrapping a $p$-cycle $\Sigma$ in $X$ giving rise to a BPS particle in 4d, you can compactify on the time circle and apply T-duality to get a BPS instanton in 3d in the dual theory.  The BPS instanton in 3d arises from a "Euclidean brane" ED($p$-1) with a $p$-dimensional worldvolume $\Sigma$.  This explains how D6-D2-D0 bound states in IIA turn into D5-D1-D(-1) bound states in IIB, yet with the same support in the Calabi-Yau.  Does that sound right?

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