Non-L2 solutions to the Seiberg–Witten equations

Abstract

Abstract:We show that a previous paper of Freund describing a solution to the SeibergWitten equations has a sign error rendering it a solution to a related but different set of equations. The non-L 2 nature of Freund's solution is discussed and clarified and we also construct a whole class of solutions to the Seiberg-Witten equations. An important vanishing theorem of [1], reminiscent of the Lichernowicz-Weitzenböck vanishing theorems, shows that there are no non-trivial solutions to the Seiberg-Witten equations on four manifolds with non-negative Riemannian scalar curvature. However one can have non-trivial solutions which are singular in some way-for example one could have a non-trivial solution which was not L 2 : in [3] Freund describes such a non-L 2 to the Seiberg-Witten equations on R 4 . Unfortunately a sign discrepancy in [3] means that the expressions given there obey equations which differ from the Seiberg-Witten equations in a certain sign. These other equations also admit L 2 solutions as well as non-L 2 ones and so Freund's equations are fundamentally different from the Seiberg-Witten equations.In § 2 we describe the Seiberg-Witten equations in a fairly explicit manner so as to expose notational conventions and matters of signs. In section § 3 we give the details concerning Freund's work and then in section § 4 we give an L 2 solution of Freund's equations and a class of solutions to the Seiberg-Witten equations. § 2. The Seiberg-Witten equations If M is an oriented Riemannian four manifold with metric g ij then the data we need for the Seiberg-Witten equations are a U (1) connection A i on M and a local spinor field M .If F ij is the curvature of A i , so that its self-dual part F + ij is given by F + ij = 1/2(F ij + ( √ g/2)ǫ ijkl F kl ), then the Seiberg-Witten equations are

Magnetic zero-modes, vortices and Cartan geometry

“…The equations (3.46) have been discussed in the literature as the dimensionally reduced Freund equations [2], while their charge-conjugates have appeared as the variational equations of a particular Dirac-Chern-Simons action [4].…”

“…AMN observed that their solutions can be expressed in terms of solutions of the Liouville equation on S 2 , and addressed the singularities in the resulting formulae. In [2], they also pointed out that the coupled Dirac and non-linear equation can be obtained as the dimensional reduction of a perturbed Seiberg-Witten equation on R 4 with a crucial sign flip (the resulting equation is often called the Freund equation).…”

Seiberg-Witten equations in R4: Lie symmetries, particular solutions, integrability

“…The four dimensional Seiberg-Witten equations are reduced to the three dimensional ones through some differential constraints, namely ‫ץ‬ i f i = 0 and ‫ץ‬ 0 f j =0. 22,23 We have already mentioned that the reduced three dimensional Seiberg-Witten equations are invariant under 10-parameter conformal group of three dimensional Euclidean space as well as under shifts along x 0 . So, the T 4 › ͑SO͑4͒ D͒ group goes over into the SO͑4,1͒ T 0 one.…”

“…By exploiting this result, some particular solutions to the Seiberg-Witten equations in R 3 were found. 21,22 Furthermore, the Freund's equations were proposed which differ from Seiberg-Witten ones in the sign of the quadratic term. 22 It was shown in Ref.…”

“…21,22 Furthermore, the Freund's equations were proposed which differ from Seiberg-Witten ones in the sign of the quadratic term. 22 It was shown in Ref. 23 that conformal group of threedimensional Euclidean space is a maximal group of Lie symmetries of the Seiberg-Witten equations in R 3 .…”