Magnetic Impurities, Integrable Vortices and the Toda Equation

TL;DR

This paper extends integrable vortex equations to include magnetic impurities, establishing conditions for integrability, and finds exact solutions linking these equations to Toda equations, including a novel solution with opposite sign.

Contribution

It introduces a generalized gauge theory framework incorporating magnetic impurities into integrable vortex equations and connects these to Toda equations with new exact solutions.

Findings

Generalized vortex equations remain integrable with impurities under certain conditions.

Derived a geometric condition to simplify vortex equations, similar to Taubes-Liouville reduction.

Discovered a new exact solution related to Toda equations with opposite sign.

Abstract

The five integrable vortex equations, recently studied by Manton, are generalized to include magnetic impurities of the Tong-Wong type. Under certain conditions these generalizations remain integrable. We further set up a gauge theory with a product gauge group, two complex scalar fields and a general charge matrix. The second species of vortices, when frozen, are interpreted as the magnetic impurity for all five vortex equations. We then give a geometric compatibility condition, which enables us to remove the constant term in all the equations. This is similar to the reduction from the Taubes equation to the Liouville equation. We further find a family of charge matrices that turn the five vortex equations into either the Toda equation or the Toda equation with the opposite sign. We find exact analytic solutions in all cases and the solution with the opposite sign appears to be new.