Boundary States for Chiral Symmetries in Two Dimensions

TL;DR

The paper develops a boundary CFT framework to classify and construct boundary states for $N$ Dirac fermions in $1+1$ dimensions while preserving a $U(1)^N$ chiral symmetry free of mixed anomalies. It introduces a lattice-theoretic data set, notably the preserved symmetry matrix ${ m R}$ and the associated charge lattice $ ext{Λ[R]}$, to compute the boundary central charge $g_{ m R}=igl[ ext{Vol}( ext{Λ[R]})igr]^{1/2}$ and the ground-state degeneracy $G[ ext{R}, ext{R}']$ on an interval. A central result is that all boundary states fall into two SPT-related classes, $ ext{V}$ and $ ext{A}$, with integer degeneracy when end conditions lie in the same class and a Majorana zero mode (yielding degeneracy in $ ext{√2} imes ext{Z}$) when they lie in different classes. The work also provides concrete, modularly consistent constructions of boundary states via Ishibashi states and establishes a robust connection between boundary gapping, SPT phases, and Majorana physics, with detailed treatments for both general and special (e.g., two-fermion) cases.

Abstract

We study boundary states for Dirac fermions in d=1+1 dimensions that preserve Abelian chiral symmetries, meaning that the left- and right-moving fermions carry different charges. We derive simple expressions, in terms of the fermion charge assignments, for the boundary central charge and for the ground state degeneracy of the system when two different boundary conditions are imposed at either end of an interval. We show that all such boundary states fall into one of two classes, related to SPT phases supported by (-1)^F, which are characterised by the existence of an unpaired Majorana zero mode.