Abstract
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+p22 (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 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 Kähler potential satisfying the Monge-Ampère equations.
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