Solutions of the 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 p(1) and p(2). There are p(1) - 1 magnetic vorticeson one surface and p(2) - 1 electric ones on the other, with p(1) + p(2) greater than or equal to 2 (p(1) = p(2) = 1 being excluded). When p(1) = p(2), 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 a 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 Kahler potential satisfying the Monge-Ampere equations.