[论文解读] Lectures on Generalized Symmetries

Lakshya Bhardwaj, Lea E. Bottini+5 more|arXiv (Cornell University)|Jul 14, 2023

Cosmology and Gravitation Theories被引用 9

一句话总结

These lecture notes introduce generalized global symmetries in quantum field theory, focusing on invertible higher-form and higher-group symmetries, their anomalies, gauging, symmetry topological field theories (SymTFTs), and connections to holography and string theory, with gauge theory examples.

ABSTRACT

These are a set of lecture notes on generalized global symmetries in quantum field theory. The focus is on invertible symmetries with a few comments regarding non-invertible symmetries. The main topics covered are the basics of higher-form symmetries and their properties including 't Hooft anomalies, gauging and spontaneous symmetry breaking. We also introduce the useful notion of symmetry topological field theories (SymTFTs). Furthermore, an introduction to higher-group symmetries describing mixings of higher-form symmetries is provided. Some advanced topics covered include the encoding of higher-form symmetries in holography and geometric engineering constructions in string theory. Throughout the text, all concepts are consistently illustrated using gauge theories as examples.

研究动机与目标

• Motivate generalized global symmetries as tools for probing strongly coupled QFT and phase structure beyond perturbation theory.
• Define higher-form and higher-group symmetries and establish their basic topological and invertible properties.
• Explain ’t Hooft anomalies, gauging, and spontaneous symmetry breaking for generalized symmetries.
• Introduce symmetry topological field theories (SymTFTs) and their role in organizing symmetry data.
• Provide connections to string theory constructions and holography through practical gauge theory examples.

提出的方法

• Formulate p-form symmetries as topological, invertible codimension-(p+1) operators U(g) acting on extended operators.
• Describe the action of p-form symmetries on q-dimensional operators and derive the concept of q-charges via representations and Pontryagin duals.
• Illustrate with Maxwell theory, discrete higher-form gauge theories, and non-Abelian gauge theories to reveal general properties and fusion rules.
• Discuss the abelian nature of higher-form symmetry groups for p ≥ 1 and derive charges via linking and operator insertions.
• Introduce symmetry topological field theories (SymTFTs) and link higher-form symmetries to anomalies and gauging.
• Outline higher-group symmetries as mixings of higher-form symmetries and hint at their roles in gauge theories and string-theoretic constructions.

实验结果

研究问题

• RQ1 What are the defining properties and mathematical structures of higher-form and higher-group symmetries in QFT?
• RQ2 How do ’t Hooft anomalies, SPT phases, and gauging work for generalized symmetries?
• RQ3 How do generalized symmetries act on extended operators and what are their representations (including Pontryagin duals)?”
• RQ4 What is the role of SymTFTs in encoding and manipulating symmetry data?
• RQ5 How do higher-form symmetries arise in holography and geometric engineering in string theory?

主要发现

• Higher-form symmetries generalize ordinary global symmetries via topological, invertible codimension-(p+1) operators.
• Higher-form symmetry groups G^(p) are abelian for p ≥ 1, and their action on p-dimensional operators is described by one-dimensional representations and charges belonging to the Pontryagin dual G^(p) 。
• p-form symmetries act on extended operators of dimension q ≥ p, with q-charges captured by (q+1)-representations of the (p+1)-group associated to G^(p).
• The notes introduce SymTFTs as a framework to encode symmetry data, anomalies, and gauging procedures for generalized symmetries.
• The text provides concrete gauge-theory examples (Maxwell, Abelian and non-Abelian gauge theories) to illustrate higher-form symmetries, anomalies, gauging, and spontaneous symmetry breaking.
• Higher-group symmetries are presented as mixings of higher-form symmetries, with discussions of continuous and discrete variants and their potential connections to holography and string theory.