Seiberg-Witten monopole equations and Riemann surfaces

Abstract

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 1 2 are invariant under fractional SL(2, IR) 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 ≥ 2 , the solutions, now SL(2, IR) invariant, are the same surfaces accompanied by a U (1) bundle of c 1 = ±(p − 1) and a 1-component constant spinor.

Generalized Seiberg-Witten equations on Riemann surface

“…Seiberg-Witten gauge theory has been of interest mathematicians, for as a TQFT, it provides new topological invariants which may provide new directions leading towards the classification of smooth, four-dimensional manifolds. Dimensional reduction of Seiberg-Witten equations to two-dimensions has been studied by Martin & Restuccia [8], Saclioglua & Nergiza [12] and Dey [4]. Except for [4], the reduction does not involve any Higgs field.…”