Monopole Condensates in Seiberg-Witten Theory

TL;DR

This paper explores solutions to Seiberg-Witten monopole equations using Riemann surfaces, revealing how monopole condensates relate to electromagnetic fields, vortices, and cosmological constants.

Contribution

It introduces explicit solutions involving Riemann surfaces with equal genus, linking monopole condensates to cosmological constants and vortex configurations.

Findings

Solutions exist for equal genus Riemann surfaces

Monopole condensates act as a cosmological constant

Solutions are unique for given surface genera

Abstract

A product of two Riemann surfaces of genuses p_1 and p_2 solves the Seiberg-Witten monopole equations for a constant Weyl spinor that represents a monopole condensate. Self-dual electromagnetic fields require p_1=p_2=p and provide a solution of the euclidean Einstein-Maxwell-Dirac equations with p-1 magnetic vortices in one surface and the same number of electric vortices in the other. The monopole condensate plays the role of cosmological constant. The virtual dimension of the moduli space is zero, showing that for given p_1 and p_2, the solutions are unique.