Fermionic Non-invertible Symmetry Behind Supersymmetric ADE Solitons

TL;DR

This work introduces the superstrip algebra as a unifying framework to capture intrinsically fermionic non-invertible categorical symmetries in 1+1D gapped systems. By applying it to the least relevant deformation of 2D ${ m N}=2$ minimal models, the authors identify an $ ext{SU}(2)_k^{\mathcal{N}=2}$-like superfusion category whose representations organize the vacua and soliton spectra, enforcing an ADE-type soliton structure and half-integer fermion numbers via mixed ’t Hooft anomalies. Notably, the mechanism relies on categorical symmetry data rather than SUSY or integrability, suggesting broad applicability to strongly coupled fermionic systems and providing a categorical bridge to 4D via vortex-string correspondences. The results offer a principled explanation for soliton degeneracies and fractional quantum numbers and propose a road map for extending these ideas to other fermionic RCFTs and beyond.

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.

Figures (8)

• Figure 1: The duality between TIM and $\mathcal{N}=1$$A_2$ minimal model (MM). The two columns are related by fermionization and bosonization.
• Figure 2: The quiver diagram of the representation $\mathcal{R}_{\eta}$.
• Figure 3: Commutative diagram to compute the preserved superfusion category $\tilde{\mathscr C}_{A_{k+1}}$ in $A_{k+1}^{\rm def}$.
• Figure 4: Degeneracies between fermionic/bosonic states in soliton/soliton sectors $\mathcal{H}_{r,r+ 1}\leftrightarrows\mathcal{H}_{r+1,r+2}$, and soliton/anti-soliton ones $\mathcal{H}_{r,r+1}\leftrightarrows\mathcal{H}_{r+1,r}$ for odd $r$.
• Figure 5: The $\mathcal{R}_{\widetilde{\mathcal{L}}_1}$ superstrip algebra representation for the $D^{\rm def}_{k+2}$ and $E^{\rm def}_7$ models.