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Introduction to Generalized Symmetries

Justin Kaidi

Abstract

These notes were prepared for a series of intensive lectures delivered at Hokkaido University, Nagoya University, Kyoto University, and Kyushu University. We begin with a brief review of higher-form symmetries, anomalies, and discrete gauge theories, before introducing non-invertible symmetries in $(1+1)$-dimensional systems. The basic structure of fusion categories is then discussed, including a discussion of categorical analogs of discrete gauging and representation theory. We subsequently turn to $(3+1)$-dimensional theories, where several physical applications of non-invertible symmetries are discussed. These notes are intended to be largely self-contained, and require no prior familiarity with subjects such as conformal field theory or lattice models.

Introduction to Generalized Symmetries

Abstract

These notes were prepared for a series of intensive lectures delivered at Hokkaido University, Nagoya University, Kyoto University, and Kyushu University. We begin with a brief review of higher-form symmetries, anomalies, and discrete gauge theories, before introducing non-invertible symmetries in -dimensional systems. The basic structure of fusion categories is then discussed, including a discussion of categorical analogs of discrete gauging and representation theory. We subsequently turn to -dimensional theories, where several physical applications of non-invertible symmetries are discussed. These notes are intended to be largely self-contained, and require no prior familiarity with subjects such as conformal field theory or lattice models.
Paper Structure (51 sections, 1 theorem, 319 equations, 10 figures, 2 tables)

This paper contains 51 sections, 1 theorem, 319 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Topological Defects and Higher-Form Symmetries
  3. Noether's theorem
  4. Ward-Takahashi identities
  5. Conserved charges and topological defects
  6. Example:
  7. Coupling a current to a gauge field
  8. One-form symmetries
  9. Example:
  10. Higher-form symmetries
  11. Example:
  12. Anomalies
  13. Example 1:
  14. Example 2:
  15. Discrete gauge theories
  16. ...and 36 more sections

Key Result

Theorem 1

For every continuous symmetry there is an associated (equivalence class of) conserved current(s).

Figures (10)

  • Figure 1: The pentagon identity for invertible defects. This condition translates to the closedness of $\omega$ in group cohomology.
  • Figure 2: Schematic picture of the SymTFT. By imposing Dirichlet (resp. Neumann) boundary conditions on the left and the non-topological boundary condition (\ref{['eq:dynamboundary']}) on the right, we may compactify to obtain ${\cal X}$ (resp. ${\cal X}/G$). In general, there exist other topological boundary conditions as well.
  • Figure 3: A theory with anomaly $\omega$ can be realized on the boundary of an invertible theory Inv$^\omega(A)$. This is possible if and only if the Symmetry TFT is a (generalized) DW theory.
  • Figure 4: The four sectors of a theory with ${\mathbb Z}_2$ global symmetry. The red planes represent the two boundaries of the SymTFT. The green dot represents a non-topological operator on the dynamical boundary.
  • Figure 5: Measuring the ${\mathbb Z}_2$ charge of the operators in the sector labelled by $L_{(1,0)}$. By making use of the bulk braidings (\ref{['eq:linkingZ2']}), we find that such operators are ${\mathbb Z}_2$ odd.
  • ...and 5 more figures

Theorems & Definitions (1)

  • Theorem 1: Noether's theorem