# If the Donaldson-Thomas invariants are deformation invariants, how are they B-model quantities?

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In algebraic geometry, the Donaldson-Thomas invariants are virtual counts of ideal sheaves (rank one, torsion-free, trivial determinant) on a Calabi-Yau threefold $X$.  We have

$\text{DT}_{\beta, n}(X) = \int_{[M_{\beta,n}(X)]^{\text{vir}}}1$

where $M_{\beta, n}(X)$ is the moduli space of ideal sheaves with Chern character $(1, 0, - \beta, -n)$.

In physics, I believe $\text{DT}_{\beta, n}(X)$ is the virtual count of BPS particles in 4d with charge $(1, 0, - \beta, -n)$ in Type IIA superstring theory compactified on $X$.  These are bound states of D2-D0 branes in a single D6-brane.  But I've also heard that DT invariants are quantities in the topological B-model on $X$.

My questions are the following:

1. Mathematically, the DT invariants are deformation invariants meaning they are unchanged under deformations of the complex structure on $X$ (though they can jump across walls!).  So how are they B-model quantities when the B-model depends on the complex moduli of $X$?

2. Secondly, and possibly related, I'm hoping someone can tell me if the following physical statements are correct:

We can curl up the time direction in $\mathbb{R}^{1, 3}$ to a circle of radius $R$ and perform T-duality.  This says that Type IIA in the $R \to \infty$ limit is equivalent to Type IIB in the $R \to 0$ limit.  This takes BPS particles coming from D6-D2-D0 bound states to BPS instantons coming from D5-D1-D(-1) branes, wrapping the same cycles in $X$.

Is this correct, and is this how DT invariants can be related to Type IIB theory?

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DT invariants are deformation invariant: they are unchanged under deformation of the complex structure of $X$. There is no jump across walls in this story.

I don't think that "DT invariants are quantities in the topological B-model on X" is correct. One might be tempted to say that because DT-invariants are counts of coherent sheaves, i.e. B-branes, boundary conditions of the B-model. But they are not physical quantities in the B-model in the sense that they are not numbers involved in the correlation functions of the B-model.

Your description in 2) is correct. Remark that one should not confuse the physical IIA and IIB string theories and the topological A and B models.

answered Jul 8 by (5,000 points)

Thanks a lot for your answer.  One reason I believed DT invariants were part of the B-model was this nice diagram on page 32 of the paper (https://arxiv.org/pdf/hep-th/0410018.pdf) of Klemm.  Do you feel he's saying the DT invariants are part of the B-model for the reason you're "tempted" to say so in your answer?

I think I do appreciate that they aren't physical quantities in the B-model.  Whereas the Gromov-Witten invariants definitely contribute to A-model correlation/partition functions.

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DT invariants are the A-model quantities, which are invariant under any deformations of the complex structure of the Calabi-Yau manifold $X$, on which the A-model is defined. However, they are related to the deformation of the complex structure of a certain CY manifold $Y$, if these two manifolds are related by mirror symmetry. It relates the A model on $X$ with the B-model on $Y$, and the latter is not invariant under the deformations of the complex structure.

It is a content of Kontsevich reformulation of mirror symmetry as an equivalence of category of coherent sheaves on $Y$ and the Fukaya category on $X$.

answered Jul 9 by (715 points)
edited Jul 9

Thanks for your response.  Can I maybe ask you to elaborate on how DT invariants are A-model quantities?  There's also a bit of an argument both ways on this thread (https://mathoverflow.net/questions/101314/are-donaldson-thomas-invariants-a-model-or-b-model), but the consensus seems to be that they are B-model quantities (ironically, with the exception of Thomas himself!).  As Thomas says, perhaps the confusion lies in a conflict with the math and physics notions.

@scpietromonaco It is discussed in section 3.3 of 1508.06642 with many references to original papers. The answer basically is that the DT and the GW invariants compute the same stuff.

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