Symmetry spans enforce gaplessness when a symmetry E embedded into two larger symmetries C and D has no compatible gapped phase that restricts from both.
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Fr¨ ohlich, J
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abstract
We demonstrate that the fusion algebra of conformal defects of a two-dimensional conformal field theory contains information about the internal symmetries of the theory and allows one to read off generalisations of Kramers-Wannier duality. We illustrate the general mechanism in the examples of the Ising model and the three-states Potts model.
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New dualities in 3d TQFTs are derived via non-invertible anyon condensation, generalizing level-rank dualities and providing new presentations for parafermion theories, c=1 orbifolds, and SU(2)_N.
Constructs non-invertible duality defects for one-form symmetries in 3+1D by partial gauging, derives fusion rules, proves incompatibility with trivial gapped phases, and realizes explicitly in Maxwell theory and lattice models.
Z_k symmetries from Pythagorean triples in two free Weyl fermions yield non-invertible defects that generate all U(1)^2-preserving boundaries for two Dirac fermions.
Generalized family anomalies for broken higher-group and non-invertible symmetries constrain RG flows and IR phases of QFT families, with explicit application to deformed 4d QCD.
Defect charges under generalized symmetries correspond one-to-one with gapped boundary conditions of the Symmetry TFT Z(C) on Y = Σ_{d-p+1} × S^{p-1} via dimensional reduction.
Higher gauging of 1-form symmetries on surfaces in 2+1d QFT yields condensation defects whose fusion rules involve 1+1d TQFTs and realizes every 0-form symmetry in TQFTs.
The Ising fusion category lattice model features a symmetric critical phase equivalent to the Ising model, a categorical ferromagnetic phase with threefold degeneracy, and a critical categorical antiferromagnetic phase with fourfold degeneracy described by an Ising CFT.
Non-invertible symmetries define quantum gates with generalized complexity distances, and simple objects in symmetry categories turn out to be computationally complex in concrete 4D and 2D QFT examples.
Classification of 2D fermionic systems with Z2 flavor symmetry yields 16 consistent superfusion categories labeled by anomaly invariants (ν_W, ν_Z, ν_WZ).
Imposing a duality-inspired fusion rule that forbids the [ε] sector from appearing in the [ε] × [ε] OPE yields numerical bounds on (Δ_σ, Δ_ε) that include the 2d Ising model but exclude the 3d Ising model.
Non-invertible symmetry defects from 11D supergravity descend to Type IIA, splitting the Bianchi sector into invertible H[3] and twisted non-invertible F[4] parts with a BF-type auxiliary sector.
Topological defects constrain the allowed RG flows of N=1 superconformal minimal models, first via a bosonic coset description and then for the full superconformal case.
Constructs Z_N extended fusion rings and modular partition functions for nonanomalous subgroups, extending to multicomponent systems and orbifoldings in CFTs.
An algebraic RG formalism for topological orders uses ideals in fusion rings to encode noninvertible symmetries and condensation rules between anyons.
Authors introduce a TFT-based framework for finite topological symmetries in QFT, including gauging, condensation defects, and duality defects, with an appendix on finite homotopy theories.
A survey of non-invertible symmetries with constructions in the Ising model and applications to neutral pion decay and other systems.
Lecture notes explain non-invertible generalized symmetries in QFTs as topological defects arising from stacking with TQFTs and gauging diagonal symmetries, plus their action on charges and the SymTFT framework.
This review summarizes transformative examples of generalized symmetries in QFT and their applications to anomalies and dynamics.
Lecture notes that systematically introduce higher-form symmetries, SymTFTs, higher-group symmetries, and related concepts in QFT using gauge theory examples.
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New dualities in 3d TQFTs are derived via non-invertible anyon condensation, generalizing level-rank dualities and providing new presentations for parafermion theories, c=1 orbifolds, and SU(2)_N.
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Z_k symmetries from Pythagorean triples in two free Weyl fermions yield non-invertible defects that generate all U(1)^2-preserving boundaries for two Dirac fermions.
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Generalized Families of QFTs
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Defect Charges, Gapped Boundary Conditions, and the Symmetry TFT
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Higher Gauging and Non-invertible Condensation Defects
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Symmetry breaking phases and transitions in an Ising fusion category lattice model
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Generalized Complexity Distances and Non-Invertible Symmetries
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Classification of 2D fermionic systems with Z2 flavor symmetry yields 16 consistent superfusion categories labeled by anomaly invariants (ν_W, ν_Z, ν_WZ).
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Conformal Bootstrap with Duality-Inspired Fusion Rule
Imposing a duality-inspired fusion rule that forbids the [ε] sector from appearing in the [ε] × [ε] OPE yields numerical bounds on (Δ_σ, Δ_ε) that include the 2d Ising model but exclude the 3d Ising model.
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Non-Invertible Symmetries in Compactified Supergravities
Non-invertible symmetry defects from 11D supergravity descend to Type IIA, splitting the Bianchi sector into invertible H[3] and twisted non-invertible F[4] parts with a BF-type auxiliary sector.
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Defects in N=1 minimal models and RG flows
Topological defects constrain the allowed RG flows of N=1 superconformal minimal models, first via a bosonic coset description and then for the full superconformal case.
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Extending fusion rules with finite subgroups: A general construction of $Z_{N}$ extended conformal field theories and their orbifoldings
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Homomorphism, substructure, and ideal: Elementary but rigorous aspects of renormalization group or hierarchical structure of topological orders
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Topological symmetry in quantum field theory
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What's Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries
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ICTP Lectures on (Non-)Invertible Generalized Symmetries
Lecture notes explain non-invertible generalized symmetries in QFTs as topological defects arising from stacking with TQFTs and gauging diagonal symmetries, plus their action on charges and the SymTFT framework.
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Snowmass White Paper: Generalized Symmetries in Quantum Field Theory and Beyond
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