Donaldson invariants are a count of instantons (the solutions to a particular elliptic PDE) on 4-manifolds. One thing which makes the theory difficult is a lack of compactness for the moduli spaces of instantons: sequences of instantons can end up singular in the limit, near points where they form ''bubbles''. Many of the applications of Donaldson theory to 4-manifold topology can be recovered in the analytically simpler Seiberg-Witten framework, where a Weitzenböck identity allows you to prove that the moduli spaces of solutions to the Seiberg-Witten equations are actually a priori compact.

(One aspect of) Donaldson-Thomas theory is a complexification of Donaldson theory, counting certain generalised instantons on Calabi-Yau 4-folds (see Donaldson & Thomas, *Gauge theory in higher dimensions*, Equation 9: available **here**). A major problem which people seem to be working hard on solving (for example **this recent preprint of Walpuski**) is a lack of compactness for these equations. My question is:

Does there exist a Seiberg-Witten analogue of Donaldson-Thomas theory which circumvents these compactness issues?

If so the trick can't be as simple as the Weitzenböck identity, which allows you to bound the $L^{\infty}$-norm of the (spinor part of the) solution in terms of the scalar curvature, because the scalar curvature vanishes on a Calabi-Yau 4-fold.