Decomposing Instantons in Two Dimensions

TL;DR

This paper analyzes BPS string solutions in gauged sigma models, revealing a decomposition of instantons into component vortices, and explores implications for understanding instanton structure and regularization in gauge theories.

Contribution

It introduces a moduli matrix approach to decompose instantons into vortices in gauged sigma models, extending to semi-local vortices and revealing a partonic structure of instantons.

Findings

Identification of two topologically distinct vortex types, S and N.

Demonstration of instanton splitting into vortices upon gauging.

Regularization of semi-local vortex metrics preserving supersymmetry.

Abstract

We study BPS string-like solutions in the 3+1 dimensional gauged CP(1) non-linear sigma model. The same analysis can be applied to study instantons in 2 euclidean dimensions. We use the moduli matrix approach to construct analytically the moduli space and and solve numerically the BPS equations. We identify two topologically inequivalent type of magnetic vortices, which we call S and N vortices. Moreover we discuss their relation to "lump-string" solutions present in the un-gauged 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. When we gauge the relative phase between fields, semi-local vortices are also split into component vortices. We discuss interesting applications of this simple set-up. First, gauging of non-linear sigma models reveals a "partonic" nature of instantons in 1+1 dimensions, an idea long studied also in connection with four dimensional instantons. Second, weak gauging provides for an interesting regularization of the metric of semi-local vortices which preserves supersymmetry and does not lift the moduli space of the string.