Lecture Notes on Generalized Symmetries and Applications

Abstract

In this lecture note, we give a basic introduction to the rapidly developing concepts of generalized symmetries, from the perspectives of both high energy physics and condensed matter physics. In particular, we emphasize on the (invertible) higher-form and higher-group symmetries. For the physical applications, we discuss the geometric engineering of QFTs in string theory and the symmetry-protected topological (SPT) phases in condensed matter physics. The lecture note is based on a short course on generalized symmetries, jointly given by Yi-Nan Wang and Qing-Rui Wang in Feb. 2023, which took place at School of Physics, Peking University (https://indico.ihep.ac.cn/event/18796/).

83 pages

References in corpus (138)

• Topological Insulators
• Topological insulators and superconductors
• Quantum Spin Hall Effect in Graphene
• Fault-tolerant quantum computation by anyons
• Anyons in an exactly solved model and beyond
• Topological Insulators in Three Dimensions
• Classification of topological quantum matter with symmetries
• The galileon as a local modification of gravity
• Generalized Global Symmetries
• Entanglement spectrum of a topological phase in one dimension
• Symmetry protected topological orders and the group cohomology of their symmetry group
• Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
• Classification of Gapped Symmetric Phases in 1D Spin Systems
• Classifying quantum phases using Matrix Product States and PEPS
• Symmetry Protected Topological phases of Quantum Matter
• Braiding statistics approach to symmetry-protected topological phases
• Symmetry Fractionalization, Defects, and Gauging of Topological Phases
• Webs of (p,q) 5-branes, Five Dimensional Field Theories and Grid Diagrams
• Fermion Path Integrals And Topological Phases
• 2D symmetry protected topological orders and their protected gapless edge excitations
• Complete classification of 1D gapped quantum phases in interacting spin systems
• Five-Dimensional Supersymmetric Gauge Theories and Degenerations of Calabi-Yau Spaces
• Fermionic Symmetry Protected Topological Phases and Cobordisms
• Generalized Symmetries in Condensed Matter
• A symmetry principle for Topological Quantum Order
• A classification of symmetry enriched topological phases with exactly solvable models
• Symmetry-protected topological orders for interacting fermions -- Fermionic topological nonlinear $\sigma$ models and a special group supercohomology theory
• Webs of five-branes and N=2 superconformal field theories
• Higher Gauging and Non-invertible Condensation Defects
• 5-dim Superconformal Index with Enhanced En Global Symmetry
• Categorical symmetry and non-invertible anomaly in symmetry-breaking and topological phase transitions
• Non-Invertible Higher-Categorical Symmetries
• An Invitation to Higher Gauge Theory
• Reflection positivity and invertible topological phases
• Classifying symmetry-protected topological phases through the anomalous action of the symmetry on the edge
• Fermionic SPT phases in higher dimensions and bosonization
• On Geometric Classification of 5d SCFTs
• Symmetry and Topological Order
• Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology
• On the classification of topological orders
• On the Defect Group of a 6D SCFT
• Universal Non-Invertible Symmetries
• Anomalies of discrete symmetries in three dimensions and group cohomology
• Algebraic higher symmetry and categorical symmetry -- a holographic and entanglement view of symmetry
• 5-Brane Webs, Symmetry Enhancement, and Duality in 5d Supersymmetric Gauge Theory
• Non-Invertible Symmetries from Holography and Branes
• Symmetry enriched string-nets: Exactly solvable models for SET phases
• Generalized Elitzur's Theorem and Dimensional Reduction
• 5d Superconformal Field Theories and Graphs
• Generalized Global Symmetries of $T[M]$ Theories. I
• Coulomb and Higgs Branches from Canonical Singularities: Part 0
• Exactly Solvable Models for Symmetry-Enriched Topological Phases
• The Branes Behind Generalized Symmetry Operators
• Fibers add Flavor, Part I: Classification of 5d SCFTs, Flavor Symmetries and BPS States
• Symmetry Protected Topological phases and Generalized Cohomology
• The Global Form of Flavor Symmetries and 2-Group Symmetries in 5d SCFTs
• Classification and Properties of Symmetry Enriched Topological Phases: A Chern-Simons approach with applications to Z2 spin liquids
• Branes and Non-Invertible Symmetries
• A Hidden Symmetry of the Galileon
• 2-Group Global Symmetries and Anomalies in Six-Dimensional Quantum Field Theories
• 5d and 4d SCFTs: Canonical Singularities, Trinions and S-Dualities
• Topological invariants for gauge theories and symmetry-protected topological phases
• Unifying Constructions of Non-Invertible Symmetries
• Higher Form Symmetries and M-theory
• Three dimensional canonical singularity and five dimensional N=1 SCFT
• Classifying 5d SCFTs via 6d SCFTs: Arbitrary rank
• Quantized topological terms in weak-coupling gauge theories with symmetry and their connection to symmetry enriched topological phases
• Five-dimensional SCFTs and gauge theory phases: an M-theory/type IIA perspective
• Fibers add Flavor, Part II: 5d SCFTs, Gauge Theories, and Dualities
• Anomalies of discrete symmetries in various dimensions and group cohomology
• Non-Invertible Symmetry Webs
• On the 6d origin of discrete additional data of 4d gauge theories
• Higher Anomalies, Higher Symmetries, and Cobordisms I: Classification of Higher-Symmetry-Protected Topological States and Their Boundary Fermionic/Bosonic Anomalies via a Generalized Cobordism Theory
• Tropical Geometry and Five Dimensional Higgs Branches at Infinite Coupling
• Symmetry as a shadow of topological order and a derivation of topological holographic principle
• 3d Coulomb branch and 5d Higgs branch at infinite coupling
• On the cobordism classification of symmetry protected topological phases
• Trifectas for $T_N$ in 5d
• Symmetry-protected topological invariants of symmetry-protected topological phases of interacting bosons and fermions
• Classifying 5d SCFTs via 6d SCFTs: Rank one
• Non-Invertible Defects in 5d, Boundaries and Holography
• Higher-Form Symmetries and Their Anomalies in M-/F-Theory Duality
• Dualities and 5-brane webs for 5d rank 2 SCFTs
• Twisted Circle Compactifications of 6d SCFTs
• Phases of 5d SCFTs from M-/F-theory on Non-Flat Fibrations
• Symmetry TFTs for 3d QFTs from M-theory
• Coulomb and Higgs Branches from Canonical Singularities, Part 1: Hypersurfaces with Smooth Calabi-Yau Resolutions
• Discrete and higher-form symmetries in SCFTs from wrapped M5-branes
• 2-Group Symmetries and M-Theory
• 6d SCFTs, 5d Dualities and Tao Web Diagrams
• Duality and enhancement of symmetry in 5d gauge theories
• String Universality and Non-Simply-Connected Gauge Groups in 8d
• Bosonic Anomalies, Induced Fractional Quantum Numbers and Degenerate Zero Modes: the anomalous edge physics of Symmetry-Protected Topological States
• Holographic theory for continuous phase transitions -- the emergence and symmetry protection of gaplessness
• Generalized Charges, Part I: Invertible Symmetries and Higher Representations
• Short-range entanglement and invertible field theories
• Generalized symmetries and holography in ABJM-type theories
• An Introduction to Higher-Form Symmetries
• Higher categorical symmetries and gauging in two-dimensional spin systems
• A mathematical theory of gapless edges of 2d topological orders. Part I
• 0-Form, 1-Form and 2-Group Symmetries via Cutting and Gluing of Orbifolds
• A classification of 2D fermionic and bosonic topological orders
• Holography, 1-Form Symmetries, and Confinement
• Bulk-boundary correspondence for three-dimensional symmetry-protected topological phases
• Higher Form Symmetries of Argyres-Douglas Theories
• Classification of 5d N=1 gauge theories
• Higher Form Symmetries TFT in 6d
• Lattice models that realize ${\mathbb{Z}}_n$-1-symmetry protected topological states for even $n$
• Instantons, Symmetries and Anomalies in Five Dimensions
• Minimalist approach to the classification of symmetry protected topological phases
• Lattice Model for Fermionic Toric Code
• Maruyoshi-Song Flows and Defect Groups of $D_p^b(G)$ Theories
• Topological non-linear $\sigma$-model, higher gauge theory, and a realization of all 3+1D topological orders for boson systems
• Introduction to Generalized Global Symmetries in QFT and Particle Physics
• Relative Anomalies in (2+1)D Symmetry Enriched Topological States
• 5D and 6D SCFTs from ${\mathbb{C}}^3$ orbifolds
• Gauging Noninvertible Defects: A 2-Categorical Perspective
• A mathematical theory of gapless edges of 2d topological orders. Part II
• 1-Form Symmetry, Isolated N=2 SCFTs, and Calabi-Yau Threefolds
• Chiral symmetry on the edge of 2D symmetry protected topological phases
• Flavor Symmetry of 5d SCFTs, Part 2: Applications
• Generalized Symmetries in F-theory and the Topology of Elliptic Fibrations
• SymTFTs and Duality Defects from 6d SCFTs on 4-manifolds
• Higher representations for extended operators
• Flavor Symmetry of 5d SCFTs, Part 1: General Setup
• Higher Anomalies, Higher Symmetries, and Cobordisms II: Lorentz Symmetry Extension and Enriched Bosonic/Fermionic Quantum Gauge Theory
• Reflections on the Matter of 3d $\mathcal{N}=1$ Vacua and Local $Spin(7)$ Compactifications
• On Enriching the Levin-Wen model with Symmetry
• A unified view on symmetry, anomalous symmetry and non-invertible gravitational anomaly
• Global anomaly detection in two-dimensional symmetry-protected topological phases
• Anomalous symmetry protected topological states in interacting fermion systems
• Domain Wall Decorations, Anomalies and Spectral Sequences in Bosonic Topological Phases
• Exactly solvable models for U(1) symmetry-enriched topological phases
• Representation theory for categorical symmetries
• Homotopy Theoretic Classification of Symmetry Protected Phases
• Flavor Symmetries and Automatic Enhancement in the 6d Supergravity Swampland
• 6D Heterotic Little String Theories and F-theory Geometry: An Introduction
• The Pontrjagin Dual of 4-Dimensional Spin Bordism

Cited by in corpus (39)

• Gapped Phases with Non-Invertible Symmetries: (1+1)d
• On the Holographic Dual of a Topological Symmetry Operator
• 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
• Non-invertible symmetries act locally by quantum operations
• Emergent generalized symmetries in ordered phases and applications to quantum disordering
• Hasse Diagrams for Gapless SPT and SSB Phases with Non-Invertible Symmetries
• Generalized cluster states from Hopf algebras: non-invertible symmetry and Hopf tensor network representation
• (SPT-)LSM theorems from projective non-invertible symmetries
• 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
• Non-invertible duality and symmetry topological order of one-dimensional lattice models with spatially modulated symmetry
• Generalized symmetries in singularity-free nonlinear $\sigma$ models and their disordered phases
• Fusion 3-Categories for Duality Defects
• Notes on gauging noninvertible symmetries, part 2: higher multiplicity cases
• Topological aspects of brane fields: solitons and higher-form symmetries
• Kramers-Wannier self-duality and non-invertible translation symmetry in quantum chains: a wave-function perspective
• Quantum Cluster State Model with Haagerup Fusion Category Symmetry
• Monopole Breaking of Chern-Weil Symmetries
• Higher-Matter and Landau-Ginzburg Theory of Higher-Group Symmetries
• Fractal Subsystem Symmetries, Anomalies, Boundaries, and Effective Field Theory
• Weak Hopf non-invertible symmetry-protected topological spin liquid and lattice realization of (1+1)D symmetry topological field theory
• Higher-form symmetry and chiral transport in real-time Abelian lattice gauge theory
• The 3d $A$-model and generalised symmetries, Part I: bosonic Chern-Simons theories
• Hydrodynamic fluctuations and topological susceptibility in chiral magnetohydrodynamics
• Strings as Hyper-Fractons
• Generalized symmetry constraints on deformed 4d (S)CFTs
• Subsystem Symmetry-Protected Topological Phases from Subsystem SymTFT of 2-Foliated Exotic Tensor Gauge Theory
• Spacetime symmetry-enriched SymTFT: from LSM anomalies to modulated symmetries and beyond
• Antilinear superoperator, quantum geometric invariance, and antilinear symmetry for higher-dimensional quantum systems
• Exponential $U(1)$ Symmetry-Breaking Phase as a Disorder-Free Quantum Glass
• The Minimal Supersymmetric Standard Model with Non-Invertible Selection Rules
• Brane Symmetries Revisited: Symmetries of Tensile and Tensionless Branes in Possibly Degenerate Metrics and their Manifestations
• Generalized symmetry enriched criticality in (3+1)d
• Non-invertible symmetries of two-dimensional Non-Linear Sigma Models
• Superselection Rules, Bosonization Duality in 1+1 Dimensions and Momentum-Space Entanglement
• More on 5d Wilson Loops in Higher-Rank Theories and Blowup Equations