An Introduction to Higher-Form Symmetries

Abstract

These notes are intended to be a pedagogical introduction to higher-form symmetries, which are symmetries whose charged objects are extended operators supported on lines, surfaces, and etc. This subject has been one of the most popular and effervescent topics of theoretical physics in recent years. Gauge theories are central in the study of higher-form symmetries, with Wilson and 't Hooft operators corresponding to the charged objects. Along these notes, we discuss in detail some basic aspects, including Abelian Maxwell and Chern-Simons theories, and $SU(N)$ non-Abelian gauge theories. We also discuss spontaneous breaking of higher-form symmetries.

arXiv comment73 pages, 13 figures, review notes, several typos corrected, references added, comments are welcome!

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• Lecture Notes on Generalized Symmetries and Applications
• Gapped Phases with Non-Invertible Symmetries: (1+1)d
• The Club Sandwich: Gapless Phases and Phase Transitions with Non-Invertible Symmetries
• Gauging modulated symmetries: Kramers-Wannier dualities and non-invertible reflections
• Hasse Diagrams for Gapless SPT and SSB Phases with Non-Invertible Symmetries
• (SPT-)LSM theorems from projective non-invertible symmetries
• Effective Field Theory for Spontaneously Broken Symmetry
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• Non-invertible duality and symmetry topological order of one-dimensional lattice models with spatially modulated symmetry
• Categorical-Symmetry Resolved Entanglement in CFT
• Fusion 3-Categories for Duality Defects
• Higher Gauge Theory
• Fractal Subsystem Symmetries, 't Hooft Anomalies, and UV/IR Mixing
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• Fractal Subsystem Symmetries, Anomalies, Boundaries, and Effective Field Theory
• Gravitational higher-form symmetries and the origin of hidden symmetries in Kaluza-Klein compactifications
• Dipole symmetries from the topology of the phase space and the constraints on the low-energy spectrum
• Higher-form symmetry and chiral transport in real-time Abelian lattice gauge theory
• (-1)-Form Symmetries and Anomaly Shifting from SymTFT
• Hydrodynamic fluctuations and topological susceptibility in chiral magnetohydrodynamics
• Graded Poisson and Graded Dirac structures
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• Antilinear superoperator, quantum geometric invariance, and antilinear symmetry for higher-dimensional quantum systems
• Generalized symmetry enriched criticality in (3+1)d
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• Defects in Wigner crystals: fracton-elasticity duality and vacancy proliferation
• Higher-form entanglement asymmetry and topological order
• Non-invertible symmetries of two-dimensional Non-Linear Sigma Models