Fermionic Non-invertible Symmetry Behind Supersymmetric ADE Solitons

TL;DR

This paper introduces the superstrip algebra as a new framework to understand fermionic non-invertible symmetries in 1+1 dimensional gapped systems, revealing their role in soliton spectra and quantum numbers.

Contribution

It proposes the superstrip algebra to systematically classify categorical symmetry data in massive fermionic models, applicable beyond supersymmetric or integrable systems.

Findings

Superstrip algebra encodes vacuum and soliton structure.

Application to $ ext{N}=2$ minimal models shows preservation of non-invertible superfusion category.

Explains the origin of $ADE$-type soliton spectrum and fractional fermion numbers.

Abstract

The non-perturbative constraints imposed by intrinsic fermionic non-invertible symmetries in 1+1 dimensional gapped systems remain largely unexplored. In this letter, we propose the superstrip algebra as a unified framework to catalog the categorical symmetry data in a massive fermionic model. The algebra and its representations explicitly encode the vacuum structure, soliton degeneracies, and their quantum numbers. As a demonstration, we apply this framework to the $\mathcal N=2$ minimal models with their least relevant deformation. We show that this specific deformation alone preserves a non-invertible superfusion category, a fermionic variant of $\text{SU}(2)_k$ known to underlie the $ADE$ classification of critical theories. Its superstrip algebra then accounts for the origin of the resulting $ADE$-type soliton spectrum and their fractional fermion number. Although our primary examples are supersymmetric and integrable, our framework itself relies on neither property, providing a new powerful tool for studying a broad class of strongly-coupled fermionic systems.