Defects, Generalized Symmetry, and Categorical QFT

Defect fusion becomes a categorical structure only after junctions, associators, and their coherence are specified. This chapter connects that structure to generalized symmetry, gauging, boundary realizations, and the limits of classification.

Enter this chapter

Start with a defect network and label its strata and junction spaces. For gauging, also identify the action or algebra object that defines the operation. Helpful background. Non-Invertible Topological Defects and Fusion supplies the physical defect operators. Constructions from Gauging, Duality, and Condensation distinguishes the physical operations. Fusion, Junctions, and Endpoints supplies the geometry that the categorical maps encode.

Topics

1. Defects on Stratified Spacetimes and Higher-Categorical Composition — Turns stratified geometry into coherent higher composition.
2. Fusion Categories, Module Categories, and Bimodule Defects — Distinguishes fusion data from module actions and interfaces.
3. Invertible p-Form Symmetries and Topological Defects — Identifies group-like higher-form defects and their charges.
4. Noninvertible Symmetries, Fusion, and Junction Data — Keeps junction and associator data beyond a fusion ring.
5. Categorical Symmetries, Higher Representations, and Charges — Specifies how a categorical symmetry acts on a theory.
6. Gauging, Equivariantization, Orbifolds, and Condensation — States the extra anomaly and algebra conditions needed for each operation.
7. Symmetry TFTs, Anomaly Inflow, and Boundary Realizations — Packages symmetry and anomaly data as a relative bulk–boundary system.
8. Boundary Conditions and Interfaces in Functorial Field Theory — Relates admissible interfaces to functorial composition.
9. Generalized-Symmetry Sectors, Selection Rules, and Reconstruction — Separates sector selection from reconstruction of a theory.
10. Semisimple, Nonsemisimple, and Unitarity Boundaries — Identifies which finiteness and positivity assumptions fail in broader settings.
11. Categorical Classification: Equivalence and Completeness Limits — States the equivalence notion and realization domain of a classification.

How the ideas fit together

Fusion coefficients count junction channels; associators compare their compositions. A module category adds an action, while gauging or condensation adds still more algebraic data Ostrik 2003, §§ 2–3 (PDF); Carqueville and Runkel 2016, §§ 2–5 (PDF). Invertible higher-form symmetry is a special topological-defect setting Gaiotto et al. 2015, §§ 3–4 (PDF). The table separates abstract coherence from an actual action on a given QFT and from a positive continuum realization.

Hypotheses and licensed conclusions

Domains, hypotheses, licensed results, excluded upgrades, and adversarial checks for defects and categorical symmetry

Object and domain	Required hypotheses	Licensed result	Excluded converse or upgrade	Adversarial check
Defects on a stratified two-dimensional spacetime	Compatible phase labels on every link; typed line and point defects; gluing and isotopy data	A bicategory of phases, defects, and junctions with defined horizontal and vertical composition	A drawing of incident strata does not supply a junction or a field-theory amplitude	Mismatch the target phase of one wall with the source phase of the next
Fusion and module categories	Finite semisimplicity, rigidity, coherent associators and units; compatible module associator	Finite defect fusion, duals, boundary actions, bimodule composition, and a stated Morita relation	Fusion coefficients alone determine neither the associator nor a QFT realization	Keep the fusion table but violate a pentagon or snake identity
Invertible higher-form defect action	Topological codimension-$(p+1)$ support, group-like fusion, inverse, charge lattice, and global background data	A $p$-form symmetry action measured by oriented linking characters	A deformable noninvertible defect is not a group element	Fuse the candidate with every label and fail to obtain the transparent defect
Noninvertible categorical action	Junction spaces, all $F$-symbols, pentagon and triangle equations, duals, and a coherent module category	Allowed sector transitions and composition rules for topological defect endpoints	Eigenvalues of one fusion matrix are not complete generalized charges	Compare inequivalent module associators with identical action matrices
Gauging, equivariantization, or condensation	Anomaly-free crossed action; or a connected separable commutative algebra and local modules; retained global data	The corresponding equivariant, gauged, orbifold, or condensed category in the named finite setting	These constructions are not interchangeable and do not automatically preserve modularity	Condense a nonseparable or nonlocal algebra, or omit the crossed-sector obstruction
Symmetry TFT boundary or functorial interface	Specified bulk theory, polarization, tangential structure, anomaly matching, balanced actions, and adjunction data	Topological symmetry and inflow data, gauging interfaces, and coherent wall composition	The bulk topological theory does not determine boundary Hamiltonians or scaling dimensions	Hold the symmetry TFT fixed while changing boundary dynamics
Categorical classification or reconstruction	Named source class, target category, equivalence, realization functor, completeness assumptions, and semisimplicity or trace replacement	Exactly the stated fusion-ring, Morita, center, extended-TQFT, sector, or realization conclusion	Matching centers or Grothendieck rings does not imply physical QFT equivalence	Change associator, symmetry enrichment, local observables, or extension data while preserving the chosen invariant

Structured table data (JSON).

Review the chapter

Can two defect systems with identical fusion multiplicities still differ?

Solution

Yes. Their associators or braiding can differ while the dimensions of fusion channels agree. Those maps determine how junction compositions are identified and must satisfy coherence. Equality of the fusion ring therefore does not establish equivalence of the full categorical symmetry.

Where to go next

Continue to existence and classification frontiers to compare these categorical results with constructive and analytic existence problems.

References

• Carqueville, Nils, and Ingo Runkel. “Orbifold Completion of Defect Bicategories.” Quantum Topology 7 (2016): 203–279. DOI; Open PDF.
• Gaiotto, Davide, Anton Kapustin, Nathan Seiberg, and Brian Willett. “Generalized Global Symmetries.” Journal of High Energy Physics 2015, no. 2 (2015): 172. DOI; Open PDF.
• Ostrik, Victor. “Module Categories, Weak Hopf Algebras and Modular Invariants.” Transformation Groups 8 (2003): 177–206. DOI; Open PDF.