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

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• 5-dim Superconformal Index with Enhanced En Global Symmetry
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• Branes and Non-Invertible Symmetries
• Topological invariants for gauge theories and symmetry-protected topological phases
• Unifying Constructions of Non-Invertible Symmetries
• Quantized topological terms in weak-coupling gauge theories with symmetry and their connection to symmetry enriched topological phases
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• Non-Invertible Symmetry Webs
• Symmetry as a shadow of topological order and a derivation of topological holographic principle
• Non-Invertible Defects in 5d, Boundaries and Holography
• Symmetry TFTs for 3d QFTs from M-theory
• 2-Group Symmetries and M-Theory
• Duality and enhancement of symmetry in 5d gauge theories
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Cited by in corpus (38)

• 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
• Topological aspects of brane fields: solitons and higher-form symmetries
• Notes on gauging noninvertible symmetries, part 2: higher multiplicity cases
• 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
• Weak Hopf non-invertible symmetry-protected topological spin liquid and lattice realization of (1+1)D symmetry topological field theory
• Higher-Matter and Landau-Ginzburg Theory of Higher-Group Symmetries
• Fractal Subsystem Symmetries, Anomalies, Boundaries, and Effective 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
• Strings as Hyper-Fractons
• Hydrodynamic fluctuations and topological susceptibility in chiral magnetohydrodynamics
• Generalized symmetry constraints on deformed 4d (S)CFTs
• Spacetime symmetry-enriched SymTFT: from LSM anomalies to modulated symmetries and beyond
• Subsystem Symmetry-Protected Topological Phases from Subsystem SymTFT of 2-Foliated Exotic Tensor Gauge Theory
• More on 5d Wilson Loops in Higher-Rank Theories and Blowup Equations
• Superselection Rules, Bosonization Duality in 1+1 Dimensions and Momentum-Space Entanglement
• Non-invertible symmetries of two-dimensional Non-Linear Sigma Models
• Generalized symmetry enriched criticality in (3+1)d
• Brane Symmetries Revisited: Symmetries of Tensile and Tensionless Branes in Possibly Degenerate Metrics and their Manifestations
• The Minimal Supersymmetric Standard Model with Non-Invertible Selection Rules
• Exponential $U(1)$ Symmetry-Breaking Phase as a Disorder-Free Quantum Glass