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 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.