Donaldson and DT invariants

Question

Let $X$ be a smooth projective surface. Then, using the compactified moduli space of anti self-dual connections or torsion free sheaves we can construct Donaldson invariants of $X$. Similarly, one can take a CY3-fold and by slanting elements of the universal sheaf with elements of the Chow group of the 3fold, construct the DT (Donaldson-Thomas invariants) - at least morally I think this is the idea.

If CY3 is the canonical bundle of $X$ do the DT invariants of $K_X$ and the Donaldson invariants of $X$ relate to each other in any sense?

More generally, is there any short of relation between Donaldson and Donaldson-Thomas invariants?

Answer 1

In some sense this is the topic of Vafa-Witten theory for complex surfaces; see the many recent papers of Göttsche-Kool on the subject.

In DT theory the virtual dimension is 0, so you don't usually use insertions (or the slant product) -- you just get one number. It is (a virtual version of) the Euler characteristic of the Donaldson moduli space, and Göttsche-Kool have shown that can be expressed in terms of classical Donaldson invariants.

Then there are refined invariants, which recover (a virtual version of) the Hirzebruch $\chi^{\ }_y$-genus of the Donaldson moduli space. I do not know if that can be expressed in terms of Donaldson invariants, but it seems reasonable to expect it can.