What Symmetries are Preserved by a Fermion Boundary State?

Understand

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.