Existence theorems for non-Abelian Chern--Simons--Higgs vortices with flavor

TL;DR

This paper proves the existence of non-Abelian vortex solutions in a Chern--Simons--Higgs model with specific gauge and flavor symmetries, using variational methods and elliptic system analysis.

Contribution

It establishes the existence of genuine non-Abelian vortices with quantized fluxes in a SU(N)×U(1) Chern--Simons--Higgs model, including multiple solutions on a periodic domain.

Findings

Existence of vortex solutions over the full plane.

Multiple gauge-distinct solutions on a doubly periodic domain.

Solutions have quantized magnetic fluxes and electric charges.

Abstract

In this paper we establish the existence of vortex solutions for a Chern--Simons--Higgs model with gauge group $SU(N) \times U(1)$ and flavor SU(N), these symmetries ensuring the existence of genuine non-Abelian vortices through a color-flavor locking. Under a suitable ansatz we reduce the problem to a $2\times 2$ system of nonlinear elliptic equations with exponential terms. We study this system over the full plane and over a doubly periodic domain, respectively. For the planar case we use a variational argument to establish the existence result and derive the decay estimates of the solutions. Over the doubly periodic domain we show that the system admits at least two gauge-distinct solutions carrying the same physical energy by using a constrained minimization approach and the mountain-pass theorem. In both cases we get the quantized vortex magnetic fluxes and electric charges.