Non-Invertible Symmetries

A topological defect can constrain a quantum theory without having a two-sided fusion inverse. This chapter explains that possibility through concrete fusion and action data, then separates constructions by gauging from anomaly and infrared constraints.

Enter this chapter

Begin with Non-Invertible Topological Defects and Fusion to test topologicality and invertibility independently.

Helpful background. Fusion, Junctions, and Endpoints and What Is an Anomaly? supply junction composition and the distinction between an anomaly and a symmetry property.

Topics

1. Non-Invertible Topological Defects and Fusion — Defines noninvertibility through a specified topological fusion law.
2. Actions, Generalized Charges, and Selection Rules — Replaces group conjugation by the action of defect networks.
3. Constructions from Gauging, Duality, and Condensation — Builds interfaces and defects through controlled gauging operations.
4. Anomalies, RG Constraints, and Framework Limits — Separates fusion, gaugeability, and constraints on candidate infrared phases.

How the ideas fit together

An inverse must fuse with a defect to the identity alone, in both orders. Orientation reversal or taking an adjoint does not guarantee that property: fusion may retain additional channels. The fusion comparison makes the extra channel explicit. Junction and associativity data are still needed to turn such a rule into a consistent defect system. See Schäfer-Nameki 2024, § 1.2.

Gauging on one side of a surface first makes an interface between theories. Obtaining a symmetry defect of one theory requires an appropriate identification of its two sides. Neither that identification nor anomaly freedom follows from a formal fusion polynomial.

Review the chapter

Suppose a simple defect $D$ in a unitary finite semisimple fusion setting obeys $D\otimes\bar D=1\oplus X$, with $X$ nonzero. Can $\bar D$ be its inverse?

Solution

No. Its product contains a second nonzero channel, so it is not isomorphic to the identity object. In that setting positive quantum dimensions also give $d_D^2=1+d_X>1$, whereas an invertible simple defect has dimension one.

Where to go next

Continue to Topological Gauge Theories and Symmetry TFT for controlled ways to organize defects, boundaries, and gauging.

References

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