Supersymmetry: Boundary Conditions and Edge States

TL;DR

This paper investigates how boundary conditions affect supersymmetry in $ ext{N}=2$ theories, showing most break SUSY while some preserve $ ext{N}=1$, with localized fermionic boundary states.

Contribution

It demonstrates that boundary conditions generally break $ ext{N}=2$ SUSY but can preserve $ ext{N}=1$, and identifies localized fermionic states near boundaries.

Findings

Most boundary conditions break $ ext{N}=2$ SUSY.

Some boundary conditions preserve $ ext{N}=1$ SUSY.

Localized fermionic states exist near boundaries.

Abstract

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.