Cartan connections for an infinite family of integrable vortices

TL;DR

This paper explores a family of integrable vortex equations through Cartan geometry, revealing their interpretation as flat non-Abelian connections and linking solutions to magnetic zero-modes of a Dirac operator.

Contribution

It introduces a new geometric framework for understanding integrable vortex equations parametrized by positive real numbers, extending previous models.

Findings

Vortex equations are interpreted as flatness of non-Abelian connections.

Solutions produce magnetic zero-modes for a Dirac operator.

The family of equations is parametrized by positive real numbers, generalizing standard cases.

Abstract

An infinite family of integrable vortex equations is studied and related to the Cartan geometry of the underlying Riemann surfaces. This Cartan picture gives an interpretation of the vortex equations as the flatness of a non-Abelian connection. Solutions of the vortex equations also give rise to magnetic zero-modes for a certain Dirac operator on the lifted geometry. The family of integrable vortex equations is parametrised by a positive number $n$, that is equal to unity in the standard case and an integer in the case of polynomial vortex equations; finally, it may be extended to any positive real number.