  Theories with 3D $\mathcal{N}=2$ bulk supersymmetry may preserve a 2D
$\mathcal{N}=(0,2)$ subalgebra when a boundary is introduced, possibly with
localized degrees of freedom. We propose generalized supercurrent multiplets
with bulk and boundary parts adapted to such setups. Using their structure, we
comment on implications for the $\overline{Q}_+$-cohomology. As an example, we
apply the developed framework to Landau-Ginzburg models. In these models, we
study the role of boundary degrees of freedom and matrix factorizations. We
verify our results using quantization.
