Non-Invertible Global Symmetries

• Non-invertible global symmetries are generalized symmetry structures in QFT where topological operators lack group-like inverses.
• They diagnose incomplete spectra by revealing unbreakable charge-carrying defects, such as Wilson lines and cosmic strings, with practical implications.
• Their interplay with deformations like Higgsing and Chern–Simons couplings provides a unifying framework for understanding constraints in quantum gravity and the Swampland.

Non-invertible global symmetries are generalized symmetry structures in quantum field theory (QFT) generated by topological operators whose fusion algebra does not obey group-like invertibility. Such operators, rather than possessing an inverse with respect to composition, may fuse into direct sums of other operators with non-negative integer coefficients—defining a richer, categorical symmetry algebra. The existence of non-invertible global symmetries provides highly sensitive diagnostics for incompleteness in the spectrum of allowed charged objects (particles, lines, vortices, or defects), with consequences for both field-theoretic consistency and quantum gravity. The interplay between the completeness of the spectrum and the absence of non-invertible topological symmetries has emerged as a central organizing principle in modern gauge theory and the Swampland program.

1. Definition and Characterization via Topological Operators

Non-invertible global symmetries are implemented by topological extended operators—typically of codimension-2 or higher—whose fusion rules are not group-like. For invertible symmetry operators $U(g)$, the fusion (composition) law is $U(g_1) U(g_2) = U(g_1 g_2)$, and every $g$ has an inverse. In contrast, for non-invertible operators $T_a$, the fusion takes the form

$T_a \times T_b = \sum_c N_{a,b}^{(c)} T_c$

with $N_{a,b}^{(c)} \in \mathbb{Z}_{\geq 0}$, so there is generally no inverse. These topological operators (such as Gukov–Witten surface operators in $d\geq 4$) measure and constrain the spectrum of charged operators via topological linking (e.g., producing group-theoretic phases or more general "linking coefficients"). A crucial property is that these defects can be deformed freely as long as they do not cross nontrivial charged objects. The spectrum is incomplete when not every charge-carrying defect (such as a Wilson line) is "endable"—precisely when not all such lines can terminate on a local operator.

The canonical example is found in pure finite gauge group theories, such as $\mathbb{Z}_n$ gauge theory, where line and surface operators labeled by different representations or conjugacy classes exist. Their fusion need not correspond to group multiplication when the spectrum is incomplete; indeed, the presence of robust unbreakable lines or surfaces signals a non-invertible global symmetry.

2. Equivalence Between Completeness and Absence of Non-Invertible Symmetry Operators

The central theorem establishes that, in gauge theories with compact (possibly disconnected) gauge groups, the completeness of the spectrum of gauge charges is equivalent to the complete breaking of all (potentially non-invertible) topological symmetry operators. For compact connected gauge group $G$, completeness of the spectrum (i.e., every allowed representation is realized by dynamical states) implies the breaking of the electric 1-form symmetry generated by the center $Z(G)$. However, for finite or disconnected gauge groups, it is not sufficient to check only the invertible symmetries: the full set of topological defects, which includes non-invertible ones, must be considered.

In this setting, fusion rules of symmetry operators (e.g., Gukov–Witten surfaces) can take the form

$$T_{[g]} \times T_{[g']} = \sum_{[g'']} N_{[g],[g']}^{([g''])} T_{[g'']}$$

where $[g]$ denotes a conjugacy class. If there is any non-endable Wilson line or defect—hence an incomplete spectrum—then some topological (generally non-invertible) symmetry persists. On the magnetic side, analogous statements hold for 't Hooft operators ('t Hooft lines or their higher-codimension generalizations).

The completeness hypothesis of quantum gravity, which posits that every possible gauge charge should be realized, maps directly onto the absence of all non-invertible topological symmetry operators.

3. Illustrative Examples: Finite and Disconnected Gauge Groups, Twist Vortices

Finite Gauge Groups

In pure $\mathbb{Z}_n$ gauge theory, all Wilson lines and surface operators are robust and topological in the absence of dynamical charged matter. Only when enough charged dynamical fields are introduced and every Wilson line becomes "endable" do all the topological symmetry operators disappear, destroying the non-invertible structure.

Disconnected Gauge Groups

For example, in $O(2)$ gauge theory (realizable as $U(1)$ with charge conjugation gauged), Gukov–Witten operators are labeled by an angle $\theta \in [0, \pi]$: $T_\theta$ with quantum dimension $2$ for generic $\theta$, showing non-invertibility. Introduction of matter in all appropriate representations breaks all such non-invertible operators and leaves no generalized global symmetry.

Twist Vortices

In four dimensions, codimension-2 twist vortices (such as cosmic strings) classified by discrete holonomies serve as another test: completeness in their spectrum precisely corresponds to the absence of topological operator obstructions, and therefore, to the absence of non-invertible symmetries. The existence of unbreakable cosmic strings implies an incomplete spectrum and thus a non-invertible symmetry.

4. Deformations: Higgsing and Chern–Simons Couplings

The framework remains robust under symmetry-breaking deformations or modifications by topological terms.

Higgsing

Upon Higgsing (breaking $G \to H$), UV (ultraviolet) completeness and absence of UV non-invertible symmetry operators descend to the IR (infrared) via branching rules. If a non-invertible symmetry exists in the UV, it will induce non-invertible structures in the IR, possibly after dressing or splitting into multiple operators.

Chern–Simons and BF Couplings

In the presence of Chern–Simons terms or their generalizations (such as $BF$ couplings or promoting $\theta$ angles to dynamical axions), higher-form symmetry structures become nontrivial or higher-group-like. Regardless, the criterion persists: if all extended operators can be terminated by dynamical objects, then all candidate topological (even non-invertible) symmetry operators are non-topological. For example, in axion electrodynamics, appearance of new topological operators associated with axion winding or magnetic symmetry is matched precisely by the completeness of the spectrum including worldvolume degrees of freedom.

5. Consequences for the Swampland Program and Phenomenology

The correspondence between completeness and the absence of non-invertible global symmetries unifies the constraints often postulated in the Swampland program: that quantum gravity admits no global symmetries (including non-invertible ones) and always realizes a complete spectrum of gauge charges. If any non-invertible topological global symmetry remains, it signals a forbidden incompleteness in the spectrum. These constraints extend beyond the purely field-theoretic domain to implications for cosmic string phenomenology and other extended defects, with the potential for both direct detection and theoretical exclusion of candidate quantum gravitational theories.

6. Summary Table: Fusion Rules and Diagnostic Criteria

Scenario	Existence of Non-Invertible Symmetry	Completeness of Spectrum
Pure finite gauge theory	Yes	Incomplete
All Wilson/’t Hooft lines endable	No	Complete
O(2) (disconnected group) w/o matter	Yes	Incomplete
Twisted vortices (cosmic strings) absent	No	Complete

The table summarizes the correspondence: non-invertible symmetries exist precisely when, and only when, the spectrum is incomplete.

7. Outlook and Unified Framework

The identification of non-invertible global symmetries with failures of spectrum completeness highlights a powerful diagnostic for quantum field theory and gravity. The requirement that all extended topological symmetry operators—potentially of non-invertible, categorical type—must be breakable (by endable extended objects) provides a unifying principle connecting symmetry, spectrum, and dualities. This framework applies across deformations, including Higgsing and topological couplings, and admits generalizations to other settings, such as brane constructions and string-theoretic embeddings. The methodology strengthens constraints on model building and deepens the understanding of the algebraic and categorical structure underlying quantum field theory and quantum gravity.