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arXiv:2609.12750

Superstrip Algebras and Nonperturbative Spectra in Fermionic Theories

Jin Chen, Zhihao Duan, Qiang Jia, Sungjay Lee

Published Sep 11, 2026• 0 citations
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Abstract

This is an extended version of arXiv:2511.22129. We develop a general construction of the superstrip algebra $\mathbf{sStr}_{\mathscr C}(\mathcal M)$, the fermionic counterpart of the strip algebra, for a gapped $(1+1)$-dimensional system with a fermionic superfusion category $\mathscr C$ and the vacuum structure described by the supermodule category $\mathcal M$. As illustrations, we work out the superstrip algebra for two simple examples: the fermionic variant of the Fibonacci fusion category and the q-type $\mathbb{Z}_2$ symmetry. We then apply the framework to the gapped IR phases of the $\mathcal N=2$ and $\mathcal N=1$ superconformal minimal models deformed by their least relevant operator and compute the Witten indices. We explain how the particle and soliton spectra, together with their fermion parity, are organized into representations of the corresponding superstrip algebras. A mixed 't Hooft anomaly between the spontaneously broken symmetry and the unbroken fermion parity $(-1)^F$ further forces each $\mathcal N =2$ soliton to carry a fractional fermion number. Finally, we set up the boundary SymTFT for fermionic systems and reproduce the results in several examples studied in this paper.

DOI
10.48550/arxiv.2609.12750

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This is an extended version of arXiv:2511.22129. We develop a general construction of the superstrip algebra sStrC(M)\mathbf{sStr}_{\mathscr C}(\mathcal M)sStrC​(M), the fermionic counterpart of the strip algebra, for a gapped (1+1)(1+1)(1+1)-dimensional system with a fermionic superfusion category C\mathscr CC and the vacuum structure described by the supermodule category M\mathcal MM. As illustrations, we work out the superstrip algebra for two simple examples: the fermionic variant of the Fibonacci fusion category and the q-type Z2\mathbb{Z}_2Z2​ symmetry. We then apply the framework to the gapped IR phases of the N=2\mathcal N=2N=2 and N=1\mathcal N=1N=1 superconformal minimal models deformed by their least relevant operator and compute the Witten indices. We explain how the particle and soliton spectra, together with their fermion parity, are organized into representations of the corresponding superstrip algebras. A mixed 't Hooft anomaly between the spontaneously broken symmetry and the unbroken fermion parity (−1)F(-1)^F(−1)F further forces each N=2\mathcal N =2N=2 soliton to carry a fractional fermion number. Finally, we set up the boundary SymTFT for fermionic systems and reproduce the results in several examples studied in this paper.
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 N=2\mathcal N=2N=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 SU(2)k\text{SU}(2)_kSU(2)k​ known to underlie the ADEADEADE classification of critical theories. Its superstrip algebra then accounts for the origin of the resulting ADEADEADE-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.
A bstract We discuss the fermionization of fusion category symmetries in two-dimensional topological quantum field theories (TQFTs). When the symmetry of a bosonic TQFT is described by the representation category Rep( H ) of a semisimple weak Hopf algebra H , the fermionized TQFT has a superfusion category symmetry SRep( H \mathcal{H} H H u ), which is the supercategory of super representations of a weak Hopf superalgebra H \mathcal{H} H H u . The weak Hopf superalgebra H \mathcal{H} H H u depends not only on H but also on a choice of a non-anomalous ℤ 2 subgroup of Rep( H ) that is used for the fermionization. We derive a general formula for H \mathcal{H} H H u by explicitly constructing fermionic TQFTs with SRep( H \mathcal{H} H H u ) symmetry. We also construct lattice Hamiltonians of fermionic gapped phases when the symmetry is non-anomalous. As concrete examples, we compute the fermionization of finite group symmetries, the symmetries of finite gauge theories, and duality symmetries. We find that the fermionization of duality symmetries depends crucially on F -symbols of the original fusion categories. The computation of the above concrete examples suggests that our fermionization formula of fusion category symmetries can also be applied to non-topological QFTs.
Classification of 2D Fermionic Systems with a Z2\mathbb Z_2Z2​ Flavor Symmetry
We classify superfusion categories describing two-dimensional fermionic systems equipped with the universal fermion-parity symmetry, implemented by a topological defect line (TDL) ZZZ, and an additional Z2\mathbb{Z}_2Z2​ flavor symmetry generated by a WWW TDL. Depending on whether WWW is m-type or q-type, its fusion rules lead to three distinct classes, and solving the super-pentagon equations yields 16 consistent superfusion categories. These are labeled by invariants (νW,νZ,νWZ)(ν_W,ν_Z,ν_{WZ})(νW​,νZ​,νWZ​), which determine the Z8\mathbb{Z}_8Z8​ anomaly classes of the symmetries generated by WWW, ZZZ, and WZWZWZ. We also provide explicit realizations using multiple Majorana fermions and comment on implications for fermionic CFTs and gapped phases.
We develop a systematic theory of symmetry fractionalization for fermionic topological phases of matter in (2+1)D(2+1)\mathrm{D}(2+1)D with a general fermionic symmetry group Gf{G}_{f}Gf​. In general, Gf{G}_{f}Gf​ is a central extension of the bosonic symmetry group Gb{G}_{b}Gb​ by fermion parity, ${(\ensuremath{-}1)}^{F}$, characterized by a nontrivial cohomology class $[{\ensuremath{\omega}}_{2}]\ensuremath{\in}{\mathcal{H}}^{2}({G}_{b},{\mathbb{Z}}_{2})$. We show how the presence of local fermions places a number of constraints on the algebraic data that defines the action of the symmetry on the supermodular tensor category that characterizes the anyon content. We find two separate obstructions to defining symmetry fractionalization, which we refer to as the bosonic and fermionic symmetry localization obstructions. The former is valued in H3(Gb,K(C)){\mathcal{H}}^{3}({G}_{b},K(\mathcal{C}))H3(Gb​,K(C)), while the latter is valued in either ${\mathcal{H}}^{3}({G}_{b},\mathcal{A}/{1,\ensuremath{\psi}})$ or Z2(Gb,Z2){Z}^{2}({G}_{b},{\mathbb{Z}}_{2})Z2(Gb​,Z2​) depending on additional details of the theory. K(C)K(\mathcal{C})K(C) is the Abelian group of functions from anyons to U(1)\mathrm{U}(1)U(1) phases obeying the fusion rules, A\mathcal{A}A is the Abelian group defined by fusion of Abelian anyons, and $\ensuremath{\psi}$ is the fermion. When these obstructions vanish, we show that distinct symmetry fractionalization patterns form a torsor over ${\mathcal{H}}^{2}({G}_{b},\mathcal{A}/{1,\ensuremath{\psi}})$. We study a number of examples in detail; in particular, we provide a characterization of fermionic Kramers degeneracy arising in symmetry class DIII within this general framework, and we discuss fractional quantum Hall and Z2{\mathbb{Z}}_{2}Z2​ quantum spin liquid states of electrons.
We study generalized discrete symmetries of quantum field theories in 1+1D generated by topological defect lines with no inverse. In particular, we describe 't Hooft anomalies and classify gapped phases stabilized by these symmetries, including new 1+1D topological phases. The algebra of these operators is not a group but rather is described by their fusion ring and crossing relations, captured algebraically as a fusion category. Such data defines a Turaev-Viro/Levin-Wen model in 2+1D, while a 1+1D system with this fusion category acting as a global symmetry defines a boundary condition. This is akin to gauging a discrete global symmetry at the boundary of Dijkgraaf-Witten theory. We describe how to "ungauge" the fusion category symmetry in these boundary conditions and separate the symmetry-preserving phases from the symmetry-breaking ones. For Tambara-Yamagami categories and their generalizations, which are associated with Kramers-Wannier-like self-dualities under orbifolding, we develop gauge theoretic techniques which simplify the analysis. We include some examples of CFTs with fusion category symmetry derived from Kramers-Wannier-like dualities as an appetizer for the Part II companion paper.
We show how 1+11+11+1-dimensional fermionic symmetry-protected topological states (SPTs, i.e., nontrivial short-range entangled gapped phases of quantum matter whose boundary exhibits 't Hooft anomaly and whose bulk cannot be deformed into a trivial tensor product state under finite-depth local unitary transformations only in the presence of global symmetries), indeed can be unwound to a trivial state by enlarging the Hilbert space via adding extra degrees of freedom and suitably extending the global symmetries. The extended projective global symmetry on the boundary can become supersymmetric in a specific sense, i.e., it contains group elements that do not commute with the fermion number parity ${(\ensuremath{-}1)}^{F}$, while the antiunitary time-reversal symmetry becomes fractionalized. This also means we can uplift and remove certain exotic fermionic anomalies (e.g., ``parity'' anomaly in time-reversal or reflection symmetry) via appropriate supersymmetry extensions in terms of group extensions. We work out explicit examples for multilayers of 1+1d1+1\mathrm{d}1+1d Majorana fermion chains, then comment on models with Sachdev-Ye-Kitaev (SYK) interactions, intrinsic fermionic gapless SPTs protected by supersymmetry, and generalizations to higher space-time dimensions via a cobordism theory.