# BPS soliton-impurity models and supersymmetry

## TL;DR

This paper develops supersymmetric extensions of soliton-impurity models in (1+1) and (2+1) dimensions, preserving partial supersymmetry and providing new insights into BPS states and their topological properties.

## Contribution

It introduces novel supersymmetric BPS soliton-impurity models in various dimensions, including the $CP^1$ and Abelian Higgs models, with different fractions of preserved supersymmetry.

## Key findings

- Half-BPS soliton-impurity models in (1+1) dimensions preserve half of $
abla$ supersymmetry.
- In (2+1) dimensions, the $CP^1$ model's impurity generalization preserves one-quarter of $
abla$ SUSY.
- The Abelian Higgs model admits impurity generalizations preserving one-quarter and one-half of $
abla$ SUSY.

## Abstract

We find supersymmetric extensions of the half-BPS soliton-impurity models in (1+1) dimensions which preserve half of the $\mathcal{N}=1$ supersymmetry. This is related to the fact that in the bosonic sector (i.e., the half-BPS soliton-impurity model), only one soliton (for example, the kink) is a BPS configuration which solves the pertinent Bogomolnyi equation and saturates the topological energy bound. On the other hand, the topological charge conjugate state (the antikink) is not a BPS solution. This means that it obeys the full Euler-Lagrange equation and does not saturate the topological energy bound. The supersymmetric approach also allows us to construct half-BPS soliton-impurity models in (2+1) dimensions. Concretely, in the case of the $CP^1$ model, its BPS impurity generalisation preserves one-quarter of the $\mathcal{N}=2$ SUSY, while for the Abelian Higgs model at critical coupling both impurity generalisations preserving one-quarter (the case of a new, so-called Higgs impurity) as well as one-half of the $\mathcal{N}=2$ SUSY (the case of the previously known magnetic impurity) are possible. We also discuss a possible relation between the BPS $CP^1$-impurity model and the Dzyaloshinskii-Moriya interaction energy.