Einstein-maxwell-dirac and Seiberg-witten Monopole Equations

We present unique solutions of the Seiberg-Witten Monopole Equations in which the U(1) curvature is covariantly constant, the monopole Weyl spinor consists of a single constant component, and the 4-manifold is a product of two Riemann surfaces of genuses p1 and p2. There are p1 − 1 magnetic vortices on one surface and p2−1 electric ones on the other, with p1 + p2 ≥ 2 (p1 = p2 = 1 being excluded). When p1 = p2, the electromagnetic fields are self-dual and one also has a solution of the coupled euclidean Einstein-Maxwell-Dirac equations, with the monopole condensate serving as cosmological constant. The metric is decomposable and the electromagnetic fields are covariantly constant as in the Bertotti-Robinson solution. The Einstein metric can also be derived from a Kähler potential satisfying the Monge-Ampère equations.