# Heterotically Deformed Sigma Models on the World Sheet of Semilocal Strings in SQED

## TL;DR

This paper introduces a new 2D supersymmetric sigma model describing semi-local vortex strings in SQED, revealing heterotic deformations and supersymmetry breaking effects similar to non-Abelian vortices.

## Contribution

It develops an explicit sigma model for semi-local strings, analyzes heterotic deformations, and compares their moduli structures with non-Abelian vortices under supersymmetry breaking.

## Key findings

- A new sigma model describes semi-local vortex dynamics.
- Heterotic deformation couples zero modes, breaking supersymmetry.
- Analytic solutions for BPS equations are obtained.

## Abstract

A new two dimensional $\mathcal{N}=(0,2)$ Supersymmetric Non-Linear Sigma Model describes the dynamics of internal moduli of the BPS semi-local vortex string supported in four dimensional $\mathcal{N}=2$ SQED. While the core of these strings is very similar to Abrikosov-Nielsen-Olesen vortices, they are defined with a characteristic size modulus, much like the instanton lump size. This entails that the constituting fields of the vortex do not decay exponentially, as one goes far away from the core of the string, but as a rational function. The appearance of an extra scale in the problem also allows for an explicit, analytic, approximate solution to be written for the BPS equation, surprisingly. Despite the conceptually large differences between semi-local and non-Abelian vortices, it appears that the moduli structures have one main common feature, both undergo the same kind of heterotic deformation when a supersymmetry breaking potential term is added to the spacetime theory, moving from $\mathcal{N}=2$ to $\mathcal{N}=1$. By adding a mass term for the gauge scalar multiplet, a heterotic deformation develops on the worldsheet, which breaks supersymmetry down to $(0,2)$ by coupling supertranslational fermionic zero modes to supersize ones. Such an interaction between zero modes of two different sectors was already hypothesized and subsequently found for non-Abelian strings, providing a neat way of circumventing accidental supersymmetry enhancement via Zumino's theorem. We find that, for small values of the spacetime mass term, an entirely analogous term develops on the worldsheet of semi-local strings.