Non-invertible Symmetries in Weyl Fermions, and Applications to Fermion-Boundary Scattering Problem

TL;DR

This paper constructs non-invertible topological defects in 2D Weyl fermion theories, explores their relation to symmetry gauging, and applies these concepts to fermion boundary scattering problems.

Contribution

It introduces a method to construct and analyze non-invertible defects in Weyl fermions using boundary conditions and symmetry gauging, including explicit algorithms and classifications.

Findings

Non-invertible defects are constructed from boundary conditions of Dirac fermions.

Duality defects relate to gauging finite Abelian groups, with explicit algorithms provided.

Certain non-Abelian symmetries cannot be realized as duality defects via finite Abelian gauging.

Abstract

We construct a family of non-invertible topological defects in two-dimensional theories of $n$ Weyl fermions. The construction relies on the existence of $G$-symmetric conformal boundary conditions for $n$ Dirac fermions. Upon unfolding, these boundary conditions become topological defects $\mathcal D$ of $n$ Weyl fermions that intertwine the two $G$-representations, and they are generically non-invertible. For $G=U(1)^n$, we show that $\mathcal D$ is a duality defect associated with gauging a finite Abelian group $\Gamma$, and we give an explicit algorithm for determining $\Gamma$ and its action on the fermions. We also show that the same finite-Abelian gauging description applies in certain restricted examples with non-Abelian $G$. By contrast, for certain non-Abelian symmetry structures, including the $G=SU(2)$ symmetry appearing in the $1$-$5$-$7$-$8$-$9$ problem, we prove that $\mathcal D$ cannot be realized as a duality defect for gauging any finite Abelian group. Finally, we explain how the duality-defect perspective gives a streamlined derivation of fermion scattering from a conformal boundary.