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

+ 2 like - 0 dislike

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?  

asked Jul 8, 2018 in Theoretical Physics by Benighted (310 points) [ no revision ]

2 Answers

+ 3 like - 0 dislike

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, 2018 by 40227 (5,140 points) [ no revision ]

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.  

+ 1 like - 0 dislike

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, 2018 by Andrey Feldman (904 points) [ revision history ]
edited Jul 9, 2018 by Andrey Feldman

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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