Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2206.13159 · Journal of High Energy Physics

Fermionization of fusion category symmetries in 1+1 dimensions

Kansei Inamura

Published Oct 17, 2023• 33 citations
arXivHTMLPDF

Abstract

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( $$ \mathcal{H} $$ H u ), which is the supercategory of super representations of a weak Hopf superalgebra $$ \mathcal{H} $$ H u . The weak Hopf superalgebra $$ \mathcal{H} $$ 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 $$ \mathcal{H} $$ H u by explicitly constructing fermionic TQFTs with SRep( $$ \mathcal{H} $$ 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.

DOI
10.1007/jhep10(2023)101

Related papers

On lattice models of gapped phases with fusion category symmetries

Kansei Inamura

Abstract We construct topological quantum field theories (TQFTs) and commuting projector Hamiltonians for any 1+1d gapped phases with non-anomalous fusion category symmetries, i.e. finite symmetries that admit SPT phases. The construction is based on two-dimensional state sum TQFT whose input datum is an H-simple left H-comodule algebra, where H is a finite dimensional semisimple Hopf algebra. We show that the actions of fusion category symmetries $$ \mathcal{C} $$ C on the boundary conditions of these state sum TQFTs are represented by module categories over $$ \mathcal{C} $$ C . This agrees with the classification of gapped phases with symmetry $$ \mathcal{C} $$ C . We also find that the commuting projector Hamiltonians for these state sum TQFTs have fusion category symmetries at the level of the lattice models and hence provide lattice realizations of gapped phases with fusion category symmetries. As an application, we discuss the edge modes of SPT phases based on these commuting projector Hamiltonians. Finally, we mention that we can extend the construction of topological field theories to the case of anomalous fusion category symmetries by replacing a semisimple Hopf algebra with a semisimple pseudo-unitary connected weak Hopf algebra.

arXiv:2110.12882•Journal of High Energy Physics•Mar 1, 2022•77 citations

(2+1)d lattice models and tensor networks for gapped phases with categorical symmetry

Kansei Inamura, Sheng-Jie Huang, Apoorv Tiwari +1 more

Gapped phases in 2+1 dimensional quantum field theories with fusion 2-categorical symmetries were recently classified and characterized using the Symmetry Topological Field Theory (SymTFT) approach [L. Bhardwaj et al., SciPost Phys. 19, 056 (2025); L. Bhardwaj et al., arXiv: 2502.20440]. In this paper, we provide a systematic lattice model construction for all such gapped phases. Specifically, we consider “all-boson type” fusion 2-category symmetries, all of which are obtainable from 0-form symmetry groups G G (possibly with an ’t Hooft anomaly) via generalized gauging—that is, by stacking with an H H -symmetric TFT and gauging a subgroup H H . The continuum classification directly informs the lattice data, such as the generalized gauging that determines the symmetry category, and the data that specifies the gapped phase. We construct commuting projector Hamiltonians and ground states applicable to any non-chiral gapped phase with such symmetries. We also describe the ground states in terms of tensor networks. In light of the length of the paper, we include a self-contained summary section presenting the main results and examples.

arXiv:2506.09177•SciPost Physics•Feb 13, 2026•3 citations

Quantum Cluster State Spin Chain with Ising Fusion Category Symmetry: A Perspective from Weak Hopf SymTFT

Zhian Jia

In this work, we present a construction of a cluster state lattice Hamiltonian that exhibits the symmetry of the Ising fusion algebra. This construction is formulated within the framework of weak Hopf symmetry topological field theory (SymTFT), where we assign smooth and rough boundaries to the weak Hopf quantum double model, thereby extending the conventional cluster state model. Central to our construction is the weak Hopf Ising boundary tube algebra $\mathcal{T}_{\mathsf{Ising}}$, whose representation category is equivalent to the Ising fusion category $\mathsf{Ising}$. We take this algebra as the input data for the weak Hopf quantum double model. The resulting model exhibits Ising fusion symmetry on both open and closed $1\text{d}$ manifolds. On open manifolds, the symmetry is governed by $\mathcal{T}_{\mathsf{Ising}} \otimes \mathcal{T}_{\mathsf{Ising}}^{\vee}$; on closed manifolds, it reduces to $\operatorname{Cocom}(\mathcal{T}_{\mathsf{Ising}}) \otimes \operatorname{Cocom}(\mathcal{T}_{\mathsf{Ising}}^{\vee})$. Since the Ising fusion algebra embeds into $\operatorname{Cocom}(\mathcal{T}_{\mathsf{Ising}}^{\vee})$, the model faithfully realizes the symmetry of the Ising fusion category.

arXiv:2508.02424•Aug 4, 2025•0 citations

Fusion category symmetry. Part II. Categoriosities at c = 1 and beyond

Ryan Thorngren, Yifan Wang

A bstract We study generalized symmetries of quantum field theories in 1+1D generated by topological defect lines with no inverse. This paper follows our companion paper on gapped phases and anomalies associated with these symmetries. In the present work we focus on identifying fusion category symmetries, using both specialized 1+1D methods such as the modular bootstrap and (rational) conformal field theory (CFT), as well as general methods based on gauging finite symmetries, that extend to all dimensions. We apply these methods to c = 1 CFTs and uncover a rich structure. We find that even those c = 1 CFTs with only finite group-like symmetries can have continuous fusion category symmetries, and prove a Noether theorem that relates such symmetries in general to non-local conserved currents. We also use these symmetries to derive new constraints on RG flows between 1+1D CFTs.

Journal of High Energy Physics•Jul 5, 2024•99 citations

Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases

Ryan Thorngren, Yifan Wang

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.

arXiv:1912.02817•Dec 5, 2019•107 citations

Topological field theories and symmetry protected topological phases with fusion category symmetries

Kansei Inamura

A bstract Fusion category symmetries are finite symmetries in 1+1 dimensions described by unitary fusion categories. We classify 1+1d time-reversal invariant bosonic symmetry protected topological (SPT) phases with fusion category symmetry by using topological field theories. We first formulate two-dimensional unoriented topological field theories whose symmetry splits into time-reversal symmetry and fusion category symmetry. We then solve them to show that SPT phases are classified by equivalence classes of quintuples ( Z , M , i , s , ϕ ) where ( Z , M , i ) is a fiber functor, s is a sign, and ϕ is the action of orientation- reversing symmetry that is compatible with the fiber functor ( Z , M , i ). We apply this classification to SPT phases with Kramers-Wannier-like self-duality.

arXiv:2103.15588•Journal of High Energy Physics•May 1, 2021•59 citations
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.
Abstract We construct topological quantum field theories (TQFTs) and commuting projector Hamiltonians for any 1+1d gapped phases with non-anomalous fusion category symmetries, i.e. finite symmetries that admit SPT phases. The construction is based on two-dimensional state sum TQFT whose input datum is an H-simple left H-comodule algebra, where H is a finite dimensional semisimple Hopf algebra. We show that the actions of fusion category symmetries C \mathcal{C} C C on the boundary conditions of these state sum TQFTs are represented by module categories over C \mathcal{C} C C . This agrees with the classification of gapped phases with symmetry C \mathcal{C} C C . We also find that the commuting projector Hamiltonians for these state sum TQFTs have fusion category symmetries at the level of the lattice models and hence provide lattice realizations of gapped phases with fusion category symmetries. As an application, we discuss the edge modes of SPT phases based on these commuting projector Hamiltonians. Finally, we mention that we can extend the construction of topological field theories to the case of anomalous fusion category symmetries by replacing a semisimple Hopf algebra with a semisimple pseudo-unitary connected weak Hopf algebra.
Gapped phases in 2+1 dimensional quantum field theories with fusion 2-categorical symmetries were recently classified and characterized using the Symmetry Topological Field Theory (SymTFT) approach [L. Bhardwaj et al., SciPost Phys. 19, 056 (2025); L. Bhardwaj et al., arXiv: 2502.20440]. In this paper, we provide a systematic lattice model construction for all such gapped phases. Specifically, we consider “all-boson type” fusion 2-category symmetries, all of which are obtainable from 0-form symmetry groups G G (possibly with an ’t Hooft anomaly) via generalized gauging—that is, by stacking with an H H -symmetric TFT and gauging a subgroup H H . The continuum classification directly informs the lattice data, such as the generalized gauging that determines the symmetry category, and the data that specifies the gapped phase. We construct commuting projector Hamiltonians and ground states applicable to any non-chiral gapped phase with such symmetries. We also describe the ground states in terms of tensor networks. In light of the length of the paper, we include a self-contained summary section presenting the main results and examples.
In this work, we present a construction of a cluster state lattice Hamiltonian that exhibits the symmetry of the Ising fusion algebra. This construction is formulated within the framework of weak Hopf symmetry topological field theory (SymTFT), where we assign smooth and rough boundaries to the weak Hopf quantum double model, thereby extending the conventional cluster state model. Central to our construction is the weak Hopf Ising boundary tube algebra TIsing\mathcal{T}_{\mathsf{Ising}}TIsing​, whose representation category is equivalent to the Ising fusion category Ising\mathsf{Ising}Ising. We take this algebra as the input data for the weak Hopf quantum double model. The resulting model exhibits Ising fusion symmetry on both open and closed 1d1\text{d}1d manifolds. On open manifolds, the symmetry is governed by TIsing⊗TIsing∨\mathcal{T}_{\mathsf{Ising}} \otimes \mathcal{T}_{\mathsf{Ising}}^{\vee}TIsing​⊗TIsing∨​; on closed manifolds, it reduces to Cocom⁡(TIsing)⊗Cocom⁡(TIsing∨)\operatorname{Cocom}(\mathcal{T}_{\mathsf{Ising}}) \otimes \operatorname{Cocom}(\mathcal{T}_{\mathsf{Ising}}^{\vee})Cocom(TIsing​)⊗Cocom(TIsing∨​). Since the Ising fusion algebra embeds into Cocom⁡(TIsing∨)\operatorname{Cocom}(\mathcal{T}_{\mathsf{Ising}}^{\vee})Cocom(TIsing∨​), the model faithfully realizes the symmetry of the Ising fusion category.
A bstract We study generalized symmetries of quantum field theories in 1+1D generated by topological defect lines with no inverse. This paper follows our companion paper on gapped phases and anomalies associated with these symmetries. In the present work we focus on identifying fusion category symmetries, using both specialized 1+1D methods such as the modular bootstrap and (rational) conformal field theory (CFT), as well as general methods based on gauging finite symmetries, that extend to all dimensions. We apply these methods to c = 1 CFTs and uncover a rich structure. We find that even those c = 1 CFTs with only finite group-like symmetries can have continuous fusion category symmetries, and prove a Noether theorem that relates such symmetries in general to non-local conserved currents. We also use these symmetries to derive new constraints on RG flows between 1+1D CFTs.
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.
A bstract Fusion category symmetries are finite symmetries in 1+1 dimensions described by unitary fusion categories. We classify 1+1d time-reversal invariant bosonic symmetry protected topological (SPT) phases with fusion category symmetry by using topological field theories. We first formulate two-dimensional unoriented topological field theories whose symmetry splits into time-reversal symmetry and fusion category symmetry. We then solve them to show that SPT phases are classified by equivalence classes of quintuples ( Z , M , i , s , ϕ ) where ( Z , M , i ) is a fiber functor, s is a sign, and ϕ is the action of orientation- reversing symmetry that is compatible with the fiber functor ( Z , M , i ). We apply this classification to SPT phases with Kramers-Wannier-like self-duality.