Shape modes of $\mathbb{C}P^1$ vortices

TL;DR

This paper studies internal vibrational modes of vortices in the gauged al{C}P^1 sigma model, revealing at least one shape mode per vortex and suggesting weakly bound modes are common.

Contribution

It introduces a geometric formalism based on the Bogomol'nyi decomposition to analyze vortex shape modes, extending to models with similar energy decompositions.

Findings

Proved existence of at least one shape mode for al{C}P^1$ vortices.

Numerically identified shape modes and frequencies for radially symmetric vortices.

Shape mode eigenvalues are close to the scattering threshold, indicating weakly bound modes.

Abstract

In this paper we investigate the existence of internal modes of vortices in the gauged $\mathbb{C}P^1$ sigma model. We develop a clean geometric formalism that highlights the symmetries of the Jacobi operator, obtained from the second variation of the energy functional. The formalism and subsequent results fundamentally rely on the Bogomol'nyi decomposition of the energy functional, and can therefore be extended to other models with such a decomposition. We prove the existence of at least one shape mode for a general $\mathbb{C}P^1$ vortex solution on $\mathbb{R}^2$, and find numerically the shape modes and corresponding frequencies of a radially symmetric vortex. A surprising result is that the shape mode eigenvalues are very close to the scattering threshold, suggesting weakly bound shape modes could be characteristic of the $\mathbb{C}P^1$ model.