Introduction to Generalized Symmetries

arXiv:2603.08798 · hep-th, cond-mat.str-el · Submitted 2026-03-09 · Read on arXiv

hep-th, cond-mat.str-el

Submitted: 2026-03-09

Updated: 2026-09-18

Comments: 105 pages; v2: references added; v3: minor corrections

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Terminology

• Hilbert Space — 18× in this paper · explained in 2 episodes
• Topological Defects — 17× in this paper · explained in 2 episodes
• Quantization — 10× in this paper · explained in 12 episodes
• Redundancy — 9× in this paper · explained in 2 episodes
• Generalization — 6× in this paper · explained in 13 episodes
• Consistency — 3× in this paper · explained in 3 episodes
• Holonomy — 3× in this paper · explained in 2 episodes
• Symmetry Breaking — 3× in this paper · explained in 3 episodes
• Anyons — 2× in this paper · explained in 2 episodes
• Domain Walls — 1× in this paper · explained in 2 episodes
• Phase Transitions — 1× in this paper · explained in 2 episodes

Sources

• Generalized Symmetries in Condensed Matter
• Introduction to topological symmetry in QFT
• Lectures on Generalized Symmetries
• An Introduction to Higher-Form Symmetries
• ICTP Lectures on (Non-)Invertible Generalized Symmetries
• Lecture Notes on Generalized Symmetries and Applications
• Field Theories With a Vector Global Symmetry
• Anomalies and Invertible Field Theories
• Reflection positivity and invertible topological phases
• A Modern Point of View on Anomalies
• Relative quantum field theory
• What's Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries
• Introduction to Generalized Global Symmetries in QFT and Particle Physics
• Coupling a QFT to a TQFT and Duality
• An Invitation to Higher Gauge Theory
• Exploring 2-Group Global Symmetries
• On 2-Group Global Symmetries and Their Anomalies
• Generalized Global Symmetries
• Theta, Time Reversal, and Temperature
• The Bond-Algebraic Approach to Dualities

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