Half-BPS Impurity Backgrounds and Supersymmetry

TL;DR

This paper develops a superspace framework for inhomogeneous impurity deformations in 1+1 dimensions, identifying conditions for supersymmetry preservation and deriving BPS equations with energy bounds.

Contribution

It introduces a systematic supersymmetric approach to inhomogeneous impurity backgrounds, including half-BPS conditions and BPS equations in 1+1D.

Findings

Derived the half-BPS condition and supersymmetry projector for impurity backgrounds.

Established a Bogomol'nyi bound for static configurations.

Commented on how coordinate dependence affects the BPS structure.

Abstract

We develop a rigid $\mathscr{N} =(1,1)$ superspace framework for spatially inhomogeneous impurity deformations in $D=1+1$ dimensions by embedding the impurity profile into a real background superfield (spurion). This spurionic completion provides a manifestly supersymmetric description at the level of the action and offers a systematic route to identify which inhomogeneous backgrounds preserve a nontrivial subset of supercharges. Focusing on static interface-type configurations, we determine the half-BPS condition on the spurion background and the corresponding supersymmetry projector. In the resulting half-BPS sector we derive the associated first-order BPS equation for static bosonic matter configurations and establish an exact Bogomol'nyi completion of the static energy, yielding a sharp bound saturated by BPS solutions. We further comment on how explicit coordinate dependence and derivative-dependent impurity couplings can obstruct the Bogomol'nyi structure, thereby motivating spurionic extensions that retain supersymmetric control over inhomogeneous deformations.