Half-BPS impurity backgrounds and supersymmetry

Abstract We develop a rigid $$\mathcal {N}=(1,1)$$ N = ( 1 , 1 ) superspace framework for spatially inhomogeneous impurity deformations in $$D=1+1$$ 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.