Wavefunctions for Non-Abelian Vortices

TL;DR

This paper constructs exact wavefunctions for non-Abelian vortices in various geometries, revealing their equivalence to particles moving in covering spaces and connecting to the quantum mechanics of multiple particles on 2D surfaces.

Contribution

It provides explicit solutions for vortex wavefunctions on different geometries and links these to the fundamental group properties, advancing understanding of non-Abelian vortex quantum mechanics.

Findings

Wavefunctions for vortices on plane, cylinder, and torus constructed

Vortex physics shown to be equivalent to particles in covering spaces

Solutions relate to Abelian fundamental groups in simple cases

Abstract

We construct exact wavefunctions of two vortices on a plane, a single vortex on the cylinder and a vortex on the torus. In each case, the physics is shown to be equivalent to a particle moving in a covering space, something simple to solve in those examples. We describe how our solutions fit into the general theory of quantum mechanics of $N$ particles on a two-dimensional space and attribute our success to the fact that the fundamental groups are Abelian in those simple cases that we are considering.