Vortices on Higher Genus Surfaces

TL;DR

This paper studies the topological behavior and quantum properties of vortices on surfaces with handles, revealing how surface topology influences vortex states and their dynamics.

Contribution

It introduces a framework for understanding vortex interactions on higher genus surfaces, including flux on handles and the impact on quantum states.

Findings

Handles can carry magnetic flux and Cheshire charge.

Vortex motion is described by the surface's braid group.

Topological constraints influence vortex configurations.

Abstract

We consider the topological interactions of vortices on general surfaces. If the genus of the surface is greater than zero, the handles can carry magnetic flux. The classical state of the vortices and the handles can be described by a mapping from the fundamental group to the unbroken gauge group. The allowed configurations must satisfy a relation induced by the fundamental group. Upon quantization, the handles can carry ``Cheshire charge.'' The motion of the vortices can be described by the braid group of the surface. How the motion of the vortices affects the state is analyzed in detail.