BRST symmetry: Boundary conditions and edge states in QED

Abstract

In manifolds with spatial boundary, BRST formalism can be used to quantize gauge theories. We show that, in a U (1) gauge theory, only a subset of all the boundary conditions allowed by the self-adjointness of the Hamiltonian preserves BRST symmetry. Hence, the theory can be quantized using BRST formalism only when that subset of boundary conditions is considered. We also show that for such boundary conditions, there exist fermionic states which are localized near the boundary.

Edge states and anomalous SUSY in (2+1)-dimensional Maxwell Chern-Simons theory

“…The hypermultiplet on (2+1)-dimensional manifold M is comprised of complex scalar field φ and Dirac fermion ψ and SUSY is anomalously broken when Dirichlet φ . In this case, both the scalar modes φ k = c ± k e ±ikx1−|m|x2 and the fermion modes (30) with energy E = ±k satisfying the above boundary conditions are localized near the boundary and decays into the bulk for sufficiently large |m|. Consequently, there is a N = 1 SUSY among the edge states with a SUSY parameter sastisfying ǫ 1 = sgn(m)ǫ 2 .…”

“…Thus, when the fermionic edge states are present in the domain of H f are not paired with any bosonic modes. This can be elucidated by the fact that the fermionic edge states (30) do not satisfy (24) and hence cannot be compatible with the SUSY variation of (22). Hence, in the system with such fermionic edge states in the domains of H f , supersymmetry remains broken despite the invariance of S. In other words, the SUSY is anomalous for the fermionic boundary condition (32).…”

“…On the other hand, with bosonic boundary conditions (19)(20)(21)(22) and fermionic boundary conditions (32) N = 1 supersymmetry is anomalously broken due to the presence of the fermionic edge states. The edge states (30) are protected by the mass gap and consequently, there is no such anomalous SUSY in the massless theory.…”

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].…”