Existence of Non-Abelian Vortices in the Aharony--Bergman--Jafferis--Maldacena Theory

TL;DR

This paper proves existence and uniqueness theorems for non-Abelian vortex solutions in the Aharony--Bergman--Jafferis--Maldacena theory, using variational and fixed-point methods, with explicit conditions based on physical parameters.

Contribution

It provides the first rigorous mathematical results on existence and uniqueness of non-Abelian vortex solutions in this theoretical framework.

Findings

Existence of vortex solutions under specific parameter conditions

Uniqueness of solutions in doubly periodic domains

Explicit criteria for solution existence based on physical parameters

Abstract

Vortices in non-Abelian gauge field theory play important roles in confinement mechanism and are governed by systems of nonlinear elliptic equations of complicated structures. In this paper, we present a series of existence and uniqueness theorems for multiple vortex solutions of the non-Abelian BPS vortex equations over R^2 and on a doubly periodic domain. Our methods are based on calculus of variations and a fixed-point argument. The necessary and sufficient conditions for the existence of a unique solution in the doubly periodic situation are explicitly expressed in terms of several physical parameters involved.