Supersymmetry: Boundary conditions and edge states

Abstract

When spatial boundaries are inserted, supersymmetry (SUSY) can be broken. We have shown that in an N ¼ 2 supersymmetric theory, all local boundary conditions allowed by self-adjointness of the Hamiltonian break N ¼ 2 SUSY, while only a few of these boundary conditions preserve N ¼ 1 SUSY. We have also shown that for a subset of the boundary conditions compatible with N ¼ 1 SUSY, there exist fermionic ground states which are localized near the boundary. We also show that only very few nonlocal boundary conditions like periodic boundary conditions preserve full N ¼ 2 supersymmetry, but none of them exhibits edge states.

The ground state of the D = 11 supermembrane and matrix models on compact regions

“…In [8] the Hamiltonian and the wavefunction were given according to the symmetry group SO(9), so the above representation of the fields is explicit. For convenience the Majorana spinor is represented by means of linear combinations of elements of the subgroup SO(7)×U (1). In this way an explicit expansion λ α of the operator associated to the fermionic coordinates in terms of a unique complex spinor of eight components was obtained in [8].…”

“…Recently, supersymmetric boundary conditions have received renewed attention, due to their connection with quantum phase transition at the boundary of topological superconductors. We refer to [1] and [2], where a systematic analysis of a class of N = 2 supersymmetric theories subject to several boundary conditions was carried out.…”

BRST symmetry: Boundary conditions and edge states in QED

“…The conjugate momenta to the gauge fields A µ and the fermions ψ,ψ are given by 8) where the dot denotes derivation with respect to time. Notice that the field A 0 is not dynamical.…”

“…The presence of boundaries, in general, can reduce the symmetry of the system. As a reflection of this, all boundary conditions might not preserve the symmetry (as shown in [8]). Therefore, boundary conditions naturally assume significance in the discussion of gauge theories in manifolds with boundaries and their quantization using the BRST formalism.…”

Matrix model of QCD: Edge localized glueballs and phase transitions

“…For the Dirac operator, when the boundary conditions are of Atiyah-Patodi-Singer type, they lead to the η-invariant which has an impact on axial anomaly [11]. Such boundary conditions can also make or break supersymmetry or BRST invariance [12,13].…”