Solutions of the Einstein-Dirac and Seiberg-Witten Monopole Equations

TL;DR

This paper finds specific solutions to the Seiberg-Witten Monopole Equations on product manifolds, revealing configurations with constant curvature and spinors, and connecting to Einstein-Maxwell-Dirac solutions with cosmological implications.

Contribution

It introduces unique solutions to the Seiberg-Witten equations on product manifolds with covariantly constant curvature and spinors, linking these to Einstein-Maxwell-Dirac solutions.

Findings

Solutions with covariantly constant U(1) curvature and monopole spinors on product manifolds.

Self-dual electromagnetic fields when p_1 = p_2, leading to Einstein-Maxwell-Dirac solutions.

Metric derivable from a Kähler potential satisfying Monge-Ampère equations.

Abstract

We present unique solutions of the Seiberg-Witten Monopole Equations in which the U(1) curvature is covariantly constant, the monopole Weyl spinor consists of a single constant component, and the 4-manifold is a product of two Riemann surfaces of genuses p_1 and p_2. There are p_1 -1 magnetic vortices on one surface and p_2 - 1 electric ones on the other, with p_1 + p_2 \geq 2 p_1 = p_2= 1 being excluded). When p_1 = p_2, the electromagnetic fields are self-dual and one also has a solution of the coupled euclidean Einstein-Maxwell-Dirac equations, with the monopole condensate serving as cosmological constant. The metric is decomposable and the electromagnetic fields are covariantly constant as in the Bertotti-Robinson solution. The Einstein metric can also be derived from a K\"{a}hler potential satisfying the Monge-Amp\`{e}re equations.