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?


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.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.