Seiberg-Witten Monopole Equations And Riemann Surfaces

TL;DR

This paper explores solutions to reduced Seiberg-Witten monopole equations on Riemann surfaces, revealing invariance properties under fractional SL(2,R) transformations and connections to algebraic functions and uniformization.

Contribution

It introduces new solutions to the reduced Seiberg-Witten equations involving arbitrary analytic functions and analyzes their invariance and geometric properties.

Findings

Solutions depend on parameters (b,c) and an analytic function f(z).

Invariance under fractional SL(2,R) transformations.

Connections to algebraic functions and Riemann surface uniformization.

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,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 \geq 2 , the solutions, now SL(2,R) invariant, are the same surfaces accompanied by a U(1) bundle of c_1=\pm (p-1) and a 1-component constant spinor.