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
DTβ,n(X)=∫[Mβ,n(X)]vir1
where Mβ,n(X) is the moduli space of ideal sheaves with Chern character (1,0,−β,−n).
In physics, I believe DTβ,n(X) is the virtual count of BPS particles in 4d with charge (1,0,−β,−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 R1,3 to a circle of radius R and perform T-duality. This says that Type IIA in the R→∞ limit is equivalent to Type IIB in the R→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?