Cartan Connections and Integrable Vortex Equations

TL;DR

This paper shows that integrable abelian vortex equations on constant curvature surfaces can be understood through flat non-abelian Cartan connections, leading to higher-dimensional vortex analogues and new geometric insights.

Contribution

It introduces a novel interpretation of vortex equations via Cartan connections and extends the concept to higher dimensions using group manifolds.

Findings

Vortex equations can be reformulated as flat non-abelian Cartan connections.

Higher-dimensional vortex analogues are constructed on group manifolds.

The Cartan connection approach provides new geometric perspectives on vortex solutions.

Abstract

We demonstrate that integrable abelian vortex equations on constant curvature Riemann surfaces can be reinterpreted as flat non-abelian Cartan connections. By lifting to three dimensional group manifolds we find higher dimensional analogues of vortices. These vortex configurations are also encoded in a Cartan connection. We give examples of different types of vortex that can be interpreted this way, and compare and contrast this Cartan representation of a vortex with the symmetric instanton representation.