Fermionization of fusion category symmetries in 1+1 dimensions

Abstract

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 $\mathrm{Rep}(H)$ of a semisimple weak Hopf algebra $H$, the fermionized TQFT has a superfusion category symmetry $\mathrm{SRep}({\mathcal{H}}^u)$, which is the supercategory of super representations of a weak Hopf superalgebra ${\mathcal{H}}^u$. The weak Hopf superalgebra ${\mathcal{H}}^u$ depends not only on $H$ but also on a choice of a non-anomalous ${\mathbb{Z}}_2$ subgroup of $\mathrm{Rep}(H)$ that is used for the fermionization. We derive a general formula for ${\mathcal{H}}^u$ by explicitly constructing fermionic TQFTs with $\mathrm{SRep}({\mathcal{H}}^u)$ 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.

arXiv comment63 pages; v2: references added; v3: Appendices A and B added, many small changes throughout

References in corpus (71)

• Generalized Global Symmetries
• Topological Defect Lines and Renormalization Group Flows in Two Dimensions
• Interacting anyons in topological quantum liquids: The golden chain
• TFT construction of RCFT correlators I: Partition functions
• Duality and defects in rational conformal field theory
• Generalized Symmetries in Condensed Matter
• Generalised twisted partition functions
• On finite symmetries and their gauging in two dimensions
• Kramers-Wannier duality from conformal defects
• Kramers-Wannier-like duality defects in (3+1)d gauge theories
• Non-Invertible Duality Defects in 3+1 Dimensions
• Symmetries and Strings of Adjoint QCD${}_2$
• Higher Gauging and Non-invertible Condensation Defects
• Orbifold groupoids
• Topological Defects on the Lattice I: The Ising model
• Non-Invertible Higher-Categorical Symmetries
• Interacting topological phases and modular invariance
• Non-invertible Condensation, Duality, and Triality Defects in 3+1 Dimensions
• On the classification of topological orders
• Defect lines, dualities, and generalised orbifolds
• Spin TQFTs and fermionic phases of matter
• Bicategories for boundary conditions and for surface defects in 3-d TFT
• Algebraic higher symmetry and categorical symmetry -- a holographic and entanglement view of symmetry
• Topological defects for the free boson CFT
• A Web of 2d Dualities: ${\mathbf{Z}}_2$ Gauge Fields and Arf Invariants
• Non-Invertible Symmetries of $\mathcal{N}=4$ SYM and Twisted Compactification
• Construction of two-dimensional topological field theories with non-invertible symmetries
• Topological Transition on the Conformal Manifold
• Orbifold completion of defect bicategories
• Fermionic Matrix Product States and One-Dimensional Topological Phases
• Hamiltonian models for topological phases of matter in three spatial dimensions
• Fermionic minimal models
• On Continuous 2-Category Symmetries and Yang-Mills Theory
• Anyonic Chains, Topological Defects, and Conformal Field Theory
• Non-invertible self-duality defects of Cardy-Rabinovici model and mixed gravitational anomaly
• Lattice Topological Field Theory in Two Dimensions
• Symmetry as a shadow of topological order and a derivation of topological holographic principle
• Semi-Abelian gauge theories, non-invertible symmetries, and string tensions beyond $N$-ality
• Monoidal supercategories
• On lattice models of gapped phases with fusion category symmetries
• Global Anomalies on the Hilbert Space
• Topological dualities in the Ising model
• Topological conformal defects with tensor networks
• Classification of topological phases with finite internal symmetries in all dimensions
• A classification of 2D fermionic and bosonic topological orders
• Non-Invertible Symmetries from Discrete Gauging and Completeness of the Spectrum
• Numerical evidence for a Haagerup conformal field theory
• Fermion condensation and super pivotal categories
• A critical lattice model for a Haagerup conformal field theory
• Orbifolds of n-dimensional defect TQFTs
• Topological field theories and symmetry protected topological phases with fusion category symmetries
• Discrete torsion defects
• Duality Defects in $E_8$
• 3-dimensional defect TQFTs and their tricategories
• Lorentzian Dynamics and Factorization Beyond Rationality
• Line and surface defects in Reshetikhin-Turaev TQFT
• Matrix product states and equivariant topological field theories for bosonic symmetry-protected topological phases in (1+1) dimensions
• Classifying phases protected by matrix product operator symmetries using matrix product states
• Fermionic CFTs and classifying algebras
• Topological Models on the Lattice and a Remark on String Theory Cloning
• Topological Field Theory with Haagerup Symmetry
• Gapped boundary theories in three dimensions
• Defects in the Tri-critical Ising model
• Spin Topological Field Theory and Fermionic Matrix Product States
• Orbifolds and topological defects
• Two More Fermionic Minimal Models
• Fermionization and boundary states in 1+1 dimensions
• Fermionization of conformal boundary states
• A (Dummy's) Guide to Working with Gapped Boundaries via (Fermion) Condensation
• Topological Field Theory and Matrix Product States
• Operator fusion from wavefunction overlaps: Universal finite-size corrections and application to Haagerup model

Cited by in corpus (16)

• Mixed Anomalies, Two-groups, Non-Invertible Symmetries, and 3d Superconformal Indices
• Gapped Phases with Non-Invertible Symmetries: (1+1)d
• Anomalies of Non-Invertible Symmetries in (3+1)d
• Topological Defect Lines in Two Dimensional Fermionic CFTs
• Fusion Surface Models: 2+1d Lattice Models from Fusion 2-Categories
• Quantum Phases and Transitions in Spin Chains with Non-Invertible Symmetries
• Non-invertible symmetry-protected topological order in a group-based cluster state
• The holography of non-invertible self-duality symmetries
• Generalized cluster states from Hopf algebras: non-invertible symmetry and Hopf tensor network representation
• Fermionic Non-Invertible Symmetries in (1+1)d: Gapped and Gapless Phases, Transitions, and Symmetry TFTs
• Fusion 3-Categories for Duality Defects
• Weak Hopf non-invertible symmetry-protected topological spin liquid and lattice realization of (1+1)D symmetry topological field theory
• Non-Invertible Gauss Law and Axions
• Bosonization and Anomaly Indicators of (2+1)-D Fermionic Topological Orders
• B-brane transport in anomalous (2,2) models and localization
• Subsystem Symmetry-Protected Topological Phases from Subsystem SymTFT of 2-Foliated Exotic Tensor Gauge Theory