Multiple Vortices in the Aharony–Bergman–Jafferis–Maldacena Model

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 BPS vortex equations, arising in the dual-layered Chern–Simons field theory developed by Aharony, Bergman, Jafferis, and Maldacena, over \({\mathbb{R}^2}\) and on a doubly periodic domain. In the full-plane setting, we show that the solution realizing a prescribed distribution of vortices exists and is unique. In the compact setting, we show that a solution realizing n prescribed vortices exists over a doubly periodic domain \({\Omega}\) if and only if the condition $$n < \frac{\lambda |\Omega|}{2 \pi}$$holds, where \({\lambda >0 }\) is the Higgs coupling constant. In this case, if a solution exists, it must be unique. Our methods are based on calculus of variations.