[Paper Review] Lecture Notes on Generalized Symmetries and Applications

Ran Luo, Qing-Rui Wang+1 more|arXiv (Cornell University)|Jul 18, 2023

Topological Materials and Phenomena180 references12 citations

TL;DR

Introductory lecture notes on generalized symmetries, focusing on invertible higher-form and higher-group symmetries, with applications to string theory and symmetry-protected topological phases.

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/).

Motivation & Objective

• Motivate generalized symmetry beyond 0-form and establish its relevance in high energy and condensed matter physics.
• Define higher-form symmetries via topological operators and extended objects.
• Connect generalized symmetries to ’t Hooft anomalies and symmetry-protected topological (SPT) phases.
• Explain gauging of higher-form symmetries and discuss anomaly inflow and related structures.
• Present applications in geometric engineering of QFTs in string theory and in condensed matter realizations.

Proposed method

• Introduce topological operators U_g supported on codimension-(p+1) manifolds and their action on p-dimensional objects.
• Provide concrete Maxwell theory examples to illustrate U(1)_E and U(1)_M 1-form symmetries and their Wilson/t Hooft objects.
• Discuss gauging of higher-form symmetries and the resulting dual symmetries, including anomaly obstructions (’t Hooft anomalies).
• Describe the descent formalism and anomaly polynomials (I_d, I_{d+1}) and the anomaly inflow mechanism.
• Outline three equivalent perspectives of higher-form symmetry: topological defect networks, flat connections, and classifying spaces.
• Extend the discussion toward higher groups, crossed modules, and non-invertible/categorical symmetries.
• Present applications to string theory (AdS/d+1 QFT, M-theory backgrounds) and condensed matter (toric code, Z_2 SPTs).

Experimental results

Research questions

• RQ1 How do higher-form and higher-group symmetries generalize ordinary (0-form) symmetries?
• RQ2 How are ’t Hooft anomalies for p-form symmetries classified and related to SPT phases?
• RQ3 What are the procedures and obstructions for gauging higher-form symmetries?
• RQ4 What are explicit physical applications of generalized symmetries in string theory and condensed matter systems?

Key findings

• Maxwell theory in four dimensions exhibits a U(1)_E x U(1)_M 1-form symmetry with electric and magnetic topological operators acting on Wilson and ’t Hooft lines.
• Gauging a p-form symmetry yields a dual d−p−2 symmetry; gauging can be obstructed by ’t Hooft anomalies, described via anomaly polynomials and inflow.
• Anomalies can be captured by descent relations involving I_d and I_{d+1}, linking boundary theories to bulk TQFTs and SPT phases.
• Concrete examples illustrate anomaly inflow and background-field gauging in Maxwell theory with background fields, and in 3d Chern-Simons theory with Z_k 1-form symmetry.
• The discussion connects 0+1D and 1+1D anomaly structures to 1+1D and 2+1D SPT phases, highlighting the role of cohomology and categorified symmetries.
• Three equivalent viewpoints of higher-form symmetry—topological defect networks, flat connections, and classifying spaces—provide a unified framework for analysis.