Seiberg-Witten monopole equations and Riemann surfaces Original Research Article

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 Liouvilleʹs equation. The U(1) and manifold curvature 2-forms Fand R21 are invariant under fractional SL(2,R) transformations of f (z). When b = 12 and c= 0 and f(z) is the Fuchsian function uniformizing an algebraic function whose Riemann surface has genus p ≥ 2, the solutions, now completely SL (2, R) invariant, are the same surfaces accompanied by a U(1) bundle of c1 = ±(p − 1) and a 1-component constant spinor.