Seiberg-Witten monopole equations and Riemann surfaces

The twice dimensionally reduced Seiberg-Witten monopole equations admit solutions depending on two real parameters (b,c) and an arbitrary analytic function f(z) determining a solution of Liouville's equation. The U(1) and manifold curvature 2-forms F and R-2(1) are invariant under fractional SL(2,R) transformations of f(z). When b = 1/2 and c = 0 and f(z) is the Fuchsian function uniformizing an algebraic function whose Riemann surface has genus p greater than or equal to 2, the solutions, now completely SL(2,R) invariant, are the same surfaces accompanied by a U(1) bundle of c(1) = +/-(p-1) and a 1-component constant spinor. (C) 1997 Elsevier Science B.V.