What's Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries

Abstract

We survey recent developments in a novel kind of generalized global symmetry, the non-invertible symmetry, in diverse spacetime dimensions. We start with several different but related constructions of the non-invertible Kramers-Wannier duality symmetry in the Ising model, and conclude with a new interpretation for the neutral pion decay and other applications. These notes are based on lectures given at the TASI 2023 summer school "Aspects of Symmetry."

95 pages. Many figures, tables, Lagrangians, and Hamiltonians, with a hint of category theory. v2: minor changes

Cited by in corpus (40)

• Majorana chain and Ising model -- (non-invertible) translations, anomalies, and emanant symmetries
• Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space
• Gapped Phases with Non-Invertible Symmetries: (1+1)d
• Quantum Phases and Transitions in Spin Chains with Non-Invertible Symmetries
• Non-invertible and higher-form symmetries in 2+1d lattice gauge theories
• The Club Sandwich: Gapless Phases and Phase Transitions with Non-Invertible Symmetries
• Gauging modulated symmetries: Kramers-Wannier dualities and non-invertible reflections
• Emergent generalized symmetries in ordered phases and applications to quantum disordering
• Lattice T-duality from non-invertible symmetries in quantum spin chains
• Realizing triality and $p$-ality by lattice twisted gauging in (1+1)d quantum spin systems
• Hasse Diagrams for Gapless SPT and SSB Phases with Non-Invertible Symmetries
• Fractionalization of Coset Non-Invertible Symmetry and Exotic Hall Conductance
• Gapped Phases in (2+1)d with Non-Invertible Symmetries: Part I
• (SPT-)LSM theorems from projective non-invertible symmetries
• Gauging non-invertible symmetries on the lattice
• Fermionic Non-Invertible Symmetries in (1+1)d: Gapped and Gapless Phases, Transitions, and Symmetry TFTs
• SymTFT for (3+1)d Gapless SPTs and Obstructions to Confinement
• Reading between the rational sections: Global structures of 4d $\mathcal{N}=2$ KK theories
• Fractionally Charged Particles at the Energy Frontier: The SM Gauge Group and One-Form Global Symmetry
• Boundary SymTFT
• Hamiltonian Truncation Crafted for UV-divergent QFTs
• Asymptotic safety, quantum gravity, and the swampland: a conceptual assessment
• Ferromagnets, a New Anomaly, Instantons, and (noninvertible) Continuous Translations
• Anomaly inflow for CSS and fractonic lattice models and dualities via cluster state measurement
• Monopole Breaking of Chern-Weil Symmetries
• Noninvertible symmetry and topological holography for modulated SPT in one dimension
• The 3d $A$-model and generalised symmetries, Part I: bosonic Chern-Simons theories
• Remarks on effects of projective phase on eigenstate thermalization hypothesis
• Higher-form symmetry and chiral transport in real-time Abelian lattice gauge theory
• Hydrodynamic fluctuations and topological susceptibility in chiral magnetohydrodynamics
• Massless chiral fields in six dimensions
• (2+1)d Lattice Models and Tensor Networks for Gapped Phases with Categorical Symmetry
• Spacetime symmetry-enriched SymTFT: from LSM anomalies to modulated symmetries and beyond
• Fusion of Low-Entanglement Excitations in 2D Toric Code
• Les Houches Lecture Notes on Tensor Networks
• Intertwined order of generalized global symmetries
• From gauging to duality in one-dimensional quantum lattice models
• Non-invertible symmetries of two-dimensional Non-Linear Sigma Models
• Higher-form anomalies and state-operator correspondence beyond conformal invariance
• Generalized symmetry enriched criticality in (3+1)d