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