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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

In 8 dimensional case we do not have direct analogous theory like 4 dimension case by strong weak duality. One can naively consturct the SW theory for CY4(line bundles with sections), but the theory turns out to be quite trivial since we need the virtual dimention of the moduli space is topological and we do not have much choice. In 4 dimensions, it is Kroheimer and Mrowka first showed that Donaldson polynomials have recurrence relations for simple type 4 mfds. Then Seiberg and Witten wanted to understand this from Physical perspective and finally got to SW theory. All Gauge theories in 4 dim are expected to be recovered by SW theory. But this is far from clear even for CY3(people seems only consider DT invs for curves(=GW by MNOP) and pts(computed by several groups) so far).

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