What Symmetries are Preserved by a Fermion Boundary State?
Philip Boyle Smith, David Tong
TL;DR
The paper classifies boundary states for 2N Majorana fermions in 1+1 dimensions that preserve a chiral U(1)^N symmetry and investigates when chiral fermion parity can be preserved at the boundary. It introduces the charge lattice Lambda[R] defined by R = Qbar^{-1} Q and analyzes enhanced non-abelian symmetry through the root system Delta[R], establishing that a Z2 x Z2 chiral parity is possible precisely when Lambda[R] is an even lattice, which requires N to be a multiple of 4 and recovers the Z8 classification of 2+1D SPTs from a boundary-state perspective. The Maldacena-Ludwig SO(8) boundary state emerges as a maximal-symmetry example within this framework, and a family of dyon states provides explicit realizations with varying central charges. The discrete Z2 x Z2 parity cannot augment the continuous symmetry group, a result shown via anomaly considerations and exact sequences, with detailed constructions and central-charge spectra illustrated for N=4 and generalized to higher N.
Abstract
Usually, a left-moving fermion in d=1+1 dimensions reflects off a boundary to become a right-moving fermion. This means that, while overall fermion parity $(-1)^F$ is conserved, chiral fermion parity for left- and right-movers individually is not. Remarkably, there are boundary conditions that do preserve chiral fermion parity, but only when the number of Majorana fermions is a multiple of 8. In this paper we classify all such boundary states for $2N$ Majorana fermions when a $U(1)^N$ symmetry is also preserved. The fact that chiral-parity-preserving boundary conditions only exist when $2N$ is divisible by 8 translates to an interesting property of charge lattices. We also classify the enhanced continuous symmetry preserved by such boundary states. The state with the maximum such symmetry is the $SO(8)$ boundary state, first constructed by Maldacena and Ludwig to describe the scattering of fermions off a monopole