Non-Invertible Symmetries and Boundaries for Two-Dimensional Fermions

TL;DR

This paper explores the relationship between boundary conditions and categorical symmetries in two-dimensional fermionic conformal field theories, identifying all anomaly-free invertible symmetries and their associated non-invertible defects.

Contribution

It classifies all anomaly-free invertible symmetries for two free Weyl fermions and analyzes their self-duality and boundary conditions in fermionic CFTs.

Findings

All anomaly-free invertible symmetries are of the form Z_k for Pythagorean triples.

The theory is self-dual under gauging these symmetries, leading to non-invertible topological defects.

Any conformal boundary condition preserving U(1)^2 symmetry can be constructed by dressing a trivial Dirichlet boundary.

Abstract

We study the relation between boundary conditions and categorical symmetries of two-dimensional fermionic conformal field theories. We determine all anomaly-free invertible global symmetries of two free complex Weyl fermions, which take the form $\mathbb{Z}_k$ for each primitive Pythagorean triple $a^2 + b^2 = k^2$. The theory is self-dual under gauging any of these symmetries, and so to each there is associated a non-invertible topological defect. We study the properties of these lines, and show that any conformal boundary condition of two Dirac fermions that preserves a $U(1)^2$ symmetry can be found by dressing a trivial Dirichlet boundary with one of them. We discuss two microscopic descriptions of these defects: fermions coupled to a quantum-mechanical rotor degree of freedom; and an abelian gauge theory that realises symmetric mass generation in a half-space.