Supersymmetry: Boundary Conditions and Edge States

When spatial boundaries are inserted, SUSY can be broken. We show that in an $\mathcal{N}=2$ supersymmetric theory, all the boundary conditions allowed by self-adjointness of the Hamiltonian break $\mathcal{N}=2$ SUSY while only a few of these boundary conditions preserve $\mathcal{N}=1$ SUSY. We also show that for a subset of the boundary conditions compatible with $\mathcal{N}=1$ SUSY, there exist fermionic ground states which are localized near the boundary.