Higher-Form and Higher-Group Symmetries

A symmetry can act on lines or surfaces rather than only on local operators. Higher-form symmetry organizes that action through topological linking operators; higher-group structure describes additional mixing between symmetry layers. This chapter develops both without identifying their distinct background data.

Enter this chapter

Begin with Higher-Form Symmetry from Operators and Linking before using current or background formulas.

Helpful background. Support, Codimension, and Operator Data and Differential Forms, Integration, Orientation, and Stokes Theorem supply support dimensions and differential-form degrees.

Topics

1. Higher-Form Symmetry from Operators and Linking — Defines the action of topological operators on extended charges.
2. Higher-Form Currents, Charges, Backgrounds, and Ward Identities — Relates current, charge, and background degrees with their Ward identities.
3. Electric and Magnetic One-Form Symmetries — Uses compact Maxwell theory to distinguish electric and magnetic examples.
4. Breaking Higher-Form Symmetry and Diagnosing Phases — Tests long-distance behavior of genuine charged operators.
5. Gauging a Higher-Form Symmetry — Specifies higher-gauge fields, sums, and anomaly conditions.
6. Higher-Group Symmetry and Coupled Backgrounds — Identifies nonindependent transformations of different background fields.
7. Higher-Group Operators, Gauging, and Anomalies — Traces the resulting restrictions on defects, gauging, and anomalies.

How the ideas fit together

For a $p$-form symmetry in $d$ dimensions, charged operators have $p$-dimensional support and symmetry operators have codimension $p+1$. Continuous examples couple to a $(p+1)$-form background. The degree dictionary keeps these dimensions distinct and checks the ordinary $p=0$ case.

Coexisting zero-form and one-form symmetries need not form a nontrivial higher group. Mixing requires coupled transformation or extension data that cannot be removed by an allowed redefinition. Screening must also be checked: a line that can end on dynamical charged matter cannot carry an unscreened exact one-form charge. See Schäfer-Nameki 2024, §§ 2.3–2.4 and 2.8.

Review the chapter

In four dimensions, what supports act as generators and charged objects for a one-form symmetry?

Solution

The charged objects are lines. The topological symmetry operators have codimension two, hence are surfaces, and a continuous symmetry uses a two-form background. Their linking supplies the action; the two kinds of support must not be interchanged.

Where to go next

Continue to Non-Invertible Symmetries when defect fusion has no group inverse.

References

• Sakura Schäfer-Nameki, “ICTP Lectures on (Non-)Invertible Generalized Symmetries,” Physics Reports 1063 (2024), 1–55. DOI. Open PDF.