Hyperbolic vortices and Dirac fields in 2+1 dimensions

TL;DR

This paper explores the geometric and algebraic structures of hyperbolic vortices in 2+1 dimensions, revealing their connection to Dirac fields and gauge theories through advanced mathematical frameworks.

Contribution

It introduces a novel geometric interpretation of hyperbolic vortices using Cartan connections and links these to solutions of Dirac and Seiberg-Witten equations in 2+1 dimensions.

Findings

Vortex configurations correspond to solutions of the Dirac equation with magnetic coupling.

Lifting hyperbolic vortices to AdS space yields solutions to Lorentzian Seiberg-Witten equations.

The approach unifies geometric vortex descriptions with gauge and spinor field theories.

Abstract

Starting from the geometrical interpretation of integrable vortices on two-dimensional hyperbolic space as conical singularities, we explain how this picture can be expressed in the language of Cartan connections, and how it can be lifted to the double cover of three-dimensional Anti-de Sitter space viewed as a trivial circle bundle over hyperbolic space. We show that vortex configurations on the double cover of AdS space give rise to solutions of the Dirac equation minimally coupled to the magnetic field of the vortex. After stereographic projection to (2+1)-dimensional Minkowski space we obtain, from each lifted hyperbolic vortex, a Dirac field and an abelian gauge field which solve a Lorentzian, (2+1)-dimensional version of the Seiberg-Witten equations.