Decomposing instantons in two dimensions

Abstract

We study the Bogomol’nyi–Prasad–Sommerfeld (BPS) vortices in the (1+1)-dimensional supersymmetric U(1) gauged nonlinear sigma model. We use the moduli matrix approach to analytically construct the moduli space of solutions and solve numerically the BPS equations. We identify two topologically inequivalent types of magnetic vortices, which we call S and N vortices. Moreover, we discuss their relation to instantons (lumps) present in the ungauged case. In particular, we describe how a lump is split into a couple of component S–N vortices after gauging. We extend this analysis to the case of the extended Abelian Higgs model with two flavors, which is known to admit semi-local vortices. After gauging the relative phase between fields, semi-local vortices are also split into component vortices. We discuss interesting applications of this simple set-up. Firstly, the gauging of nonlinear sigma models reveals a semiclassical ‘partonic’ nature of instantons in 1+1 dimensions. Secondly, weak gauging provides for a new interesting regularization of the metric of semi-local vortices.