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There are things that SW theory sees more clearly than YM. As an analogy, try proving Kodaira's vanishing theorem using the sheaf theoretic description of cohomology but not the Hodge theoretic version. In this classical situation you have two theories describing the same object, but in concrete situations one theory may be more useful than the other. What is true is that the primary invariants of these theories are equal. There are further invariants notably, the homotopical SW invariants which are only defined in one of the theories; and each sees geometry which is not visible in the other.
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Kronheimer-Mrowka's proofs of "Property P for knots", or Andrei Teleman's results on curves in Class VII surfaces, say, use instantons to see thing that monopoles don't. I hope this post is not too confusing. There has been some work in a more mathematically oriented way, by Nekrasov and others arxiv. Nekrasov's paper is about giving mathematical meaning and proving of the Seiberg-Witten's calculation of the "prepotential F", which was physically defined as something like "beta-function But it seems it was unclear what this thing means mathematically before Nekrasov's work.
If I understand correctly, these papers do give a better understanding of the equivalence or "duality" between Donaldson and the Seiberg-Witten theories, which was originally motivated by considerations on the prepotential. There does seem to be a gap in the literature on this subject though, since I recall juggling a great deal of partially overlapping papers back when I was writing my master thesis on the subject.
However such connections are just the anti- self-dual solutions and correspond to global minima of the action.
Notes on Seiberg-Witten Theory
The action can have other critical points which are not global minima. Liviu Nicolaescu Liviu Nicolaescu Sign up or log in Sign up using Google. Sign up using Facebook.
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