# Is there a Seiberg-Witten version of Donaldson-Thomas theory?

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.

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There is a conjecture (by Iqbal, Nekrasov, Okounkov and Vafa) that Donaldson-Thomas theory is the same as Gromov-Witten theory under a certain change of variables ($-q = e^{iu}$). The papers "Gromov–Witten theory and Donaldson–Thomas theory" I,II by Maulik, Nekrasov Okounkov and Pandharipande, seem like a good place to start reading. I'm not sure about the current state of this conjecture. If my memory serves me right, the conjecture involves some steps using Seiberg-Witten theory. How this relates to your exact question is unclear to me, this is very far outside my field of expertise. –  Jan Jitse Venselaar Jul 3 '12 at 10:40