Variational Principle for the Seiberg-Witten Equations

Abstract

Abstract. Originally, the Seiberg-Witten equations were described to be dual to the Yang-Mills equation. However, there is a Variational Principle from which the SW -equation and some of their analytical properties can be studied. IntroductionIn november of 1994, Edward Witten gave a lecture at MIT about N = 2 Supersymmetric Quantum Field Theory and the ideas concerning the S-duality developed in a joint work with Seiberg in [15]. In order to please the mathematicians in the audience, he applied the new ideas to the Yang-Mills Theory to show them a new pair of 1 st -order PDE, named SW -monopole equation, and conjectured that the SW -theory is dual to the Yang-Mills theory; the duality being at the quantum level, since the expectation values of dual theories are equals. In topology, this means that fixed a 4-manifold its Seiberg-Witten invariants are equal to the Donaldson invariants. After 10 years, it is believed the conjecture is true, some tour the force has been done in [11] to prove it, but in its generality it is still an open question. This new pair of equations has a simpler analytical nature than the Yang-Mills equations. Even though the open question, and the fact that the physical meaning of the Seiberg-Witten equations (SW α -eq.) is yet to be discovered, the mathematical usefulness of the equations is rather deep and efficient to understand one of the most basic phenomenon of differential topology in four dimension, namely, the existence of non-equivalent differential smooth structures on the same underlying topological manifold. It has not been efficient enough to solve either the smooth Poincaré conjecture in dimension 4 or the 11/8-conjecture, but they have been very useful to improve the understanding of the relation between 2 nd -homology classes and smooth structure, the sympletic structures on 4-manifolds and to give a construction of a large number of non-equivalent smooths structures on a compact smooth 4-manifold based on the isotopic classes of knots in S 3 [5]. Also, using the SW -theory, the Thom conjecture was proved in [9] and some results in [13] were obtained in a much easier way than the one using Y M -theory. Most of the simplicity coming from the SW -theory, as compared with the Y M -theory, came from the fact that the strutuctural group in SW -theory is the abelian U 1 , wherea in Y M -theory the group is the non-abelian SU 2 .During the 80's and earlies 90's, the Yang-Mills Theory was used to define a set of smooth invariants on 4-manifolds known as Donaldson invariants [2]. The Donaldson invariants sheded new light on the theory of smooth 4-manifolds, e.g.,