Non-Invertible Symmetries in QFT

• Non-invertible symmetries are generalized symmetry structures defined by topological defect operators whose fusion yields a sum of operators rather than a single inverse.
• They are realized in gauge theories with disconnected groups and discrete gauging, where folded Gukov–Witten operators exhibit quantum dimensions greater than one and non-group fusion rules.
• These symmetries impose novel selection rules and constraints in QFT and string theory, shedding light on the absence of exact global symmetries in quantum gravity.

Non-invertible symmetries are generalized symmetry structures in quantum field theory (QFT), statistical mechanics, and string theory, characterized by the existence of topological operators whose fusion does not obey the group law—meaning that these operators do not possess inverses in the usual algebraic sense. Unlike conventional symmetries implemented by groups, non-invertible symmetries are encoded in extended topological defects (operators supported on codimension one or higher submanifolds) whose fusion relations are categorical, exhibit quantum dimensions larger than unity, and generate selection rules and constraints on the dynamics that are not accessible using group symmetry notions. Their realization is especially prominent in gauge theories involving disconnected gauge groups and in constructions involving discrete gauging, outer automorphisms, and higher-form generalizations.

1. Algebraic Formulation and Fusion Structure

A non-invertible symmetry is defined by the property that its associated (topological) defect operators do not generate an ordinary group under fusion. Instead, the fusion of two such symmetry defects typically yields a direct sum of defects, often with multiplicities:

$D \times D' = \sum_{i} n_{i} D_{i}$

The quantum dimension $\mathrm{qdim}(D)$ of a defect operator is defined via the linking coefficients—e.g., counting the number of group elements or conjugacy classes relevant to the operator. In the case of the "folded" Gukov–Witten (GW) operators obtained in disconnected gauge groups such as $\widetilde{SU}(N) = SU(N) \rtimes \mathbb{Z}_2$, the non-trivial GW operators have quantum dimension two. This non-triviality in quantum dimension implies the absence of an inverse: for a defect $T$ with $\mathrm{qdim}(T) > 1$, there is no defect $T^{-1}$ such that $T \times T^{-1} = 1$.

One canonical example of non-invertible fusion is realized in folded GW operators:

$$T_k^{\widetilde{S}} \cdot T_{k'}^{\widetilde{S}} = T_{k+k'}^{\widetilde{S}} + T_{|k-k'|}^{\widetilde{S}}$$

This "categorical" (i.e., higher-categorical or fusion-category) structure defines the algebraic backbone of non-invertible symmetries in QFTs, where the invertible case would correspond to $T_k\cdot T_{k'}=T_{k+k'}$.

2. Disconnected Gauge Groups and Discrete Gauging

A sharp realization arises in pure gauge theories with disconnected gauge groups $G$ such as $\widetilde{SU}(N)$ or $\widetilde{U}(N)$. Here, $G$ is constructed as a semidirect product of a connected Lie group (such as $SU(N)$) and a discrete automorphism group ($\mathbb{Z}_2$ corresponding to charge conjugation):

$G = G^0 \rtimes \mathbb{Z}_2$

Gauging the discrete outer automorphism (e.g., charge conjugation) leads to identification of certain symmetry operators and the emergence of non-invertible 1-form symmetries. The "folded" GW operator is constructed by summing over the orbit of the automorphism:

$$T_k^{\widetilde{S}}(M^{d-2}) = T_k^{SU}(M^{d-2}) + T_{-k}^{SU}(M^{d-2})$$

The fusion rules—for instance, in $O(2)$, $$T_\theta^{O(2)} T_\phi^{O(2)} = T_{\theta+\phi}^{O(2)} + T_{\theta-\phi}^{O(2)}$$—are inherently non-invertible as the quantum dimension becomes greater than one, and the fusion closes only up to sums of operators with no inverse.

Non-invertibility is a direct consequence of both (a) the parent charge-conjugation acting on the 1-form symmetry generators, and (b) the gauging process, which "folds" the GW operators into sums, thus enhancing their quantum dimension.

3. Dual (d–2)-Form Symmetries and Twist Vortices

In these theories, the disconnectedness of the gauge group, $\pi_0(G) \cong \mathbb{Z}_2$, automatically endows the system with a global (d–2)-form symmetry. This dual symmetry is described by topological Wilson lines associated with the representations of $\pi_0(G)$:

• Trivial representation: $W_1^G$
• Sign representation: $W_{\text{sign}}^G$

Their fusion forms a copy of the group algebra of $\mathbb{Z}_2$, i.e., $W_1^G \cdot W_1^G = W_1^G$, $W_1^G \cdot W_{\text{sign}}^G = W_{\text{sign}}^G$, $W_{\text{sign}}^G \cdot W_{\text{sign}}^G = W_1^G$. This part of the symmetry is invertible in contrast to the non-invertible 1-form symmetry discussed above.

Twist vortices, or Alice strings in four dimensions, are codimension-2 defects around which the gauge field undergoes monodromy in $G/G^0$ (e.g., $\mathbb{Z}_2$). These defects play a dual role:

• They serve as endpoints for the non-invertible GW operators, demonstrating the "higher-categorical" nature of the symmetry.
• Their presence explicitly breaks the (d–2)-form global symmetry by rendering the associated Wilson lines non-topological.

When such twist vortices are included, the only topological GW operators that remain are those in the simply connected case or in the "unfolded" version of the gauge group.

4. Implications for Quantum Gravity and String Theory

A central conjecture in quantum gravity is the non-existence of exact global symmetries, known as the "No Global Symmetry" hypothesis (or part of the Swampland program). The studied gauge theories, when embedded into string theory (for instance, on the worldvolumes of D-branes with orientifolds), provide a mechanism wherein all putative global symmetries are either gauged or explicitly broken. In these constructions:

• The embedding of $\widetilde{U}(N)$ or $\widetilde{SU}(N)$ gauge theories is achieved using brane systems with orientifolds (e.g., $N$ D3-branes and an O3-plane).
• The unavoidable appearance of twist vortices (due to the string-theoretic defects) ensures the automatic breaking of the global (d–2)-form symmetry.
• The "folded" GW operators persist as non-invertible generators in the pure gauge theory but are rendered non-topological in the full embedding, precluding the existence of any exact global symmetry.

This analysis supports the idea that any exact global symmetry—even a higher-form or non-invertible one—must be absent in a theory with gravity.

5. Key Formulas, Fusion Rules, and Operator Structure

The following table summarizes key operator constructions and fusion rules:

Operator Type	Construction / Labeling	Fusion Rule
GW operator	$T_k^{\widetilde{S}} = T_k^{SU} + T_{-k}^{SU}$	$$T_k^{\widetilde{S}} T_{k'}^{\widetilde{S}} = T_{k+k'}^{\widetilde{S}} + T_{|k - k'|}^{\widetilde{S}}$$
Wilson operator	$W_1^G$, $W_{\text{sign}}^G$ (from $\pi_0(G) \cong \mathbb{Z}_2$)	$W_1 W_1 = W_1$, $W_1 W_{\text{sign}} = W_{\text{sign}}$, $W_{\text{sign}} W_{\text{sign}} = W_1$
Twist vortex / Alice string	Codimension-2 defect with monodromy in $G/G^0$	Breaks (d–2)-form $\mathbb{Z}_2$ symmetry

The non-invertibility is manifest in the quantum dimension (e.g., 2 for non-trivial GW operators in the “folded” theory), and the associated fusion algebra is not a group.

6. Broader Context and Implications

Non-invertible symmetries forged via disconnected gauge groups, discrete gauging, and their associated defect operators form a fundamental component of the emerging landscape of categorical symmetry in QFT. The mathematical formalism aligns with fusion categories and higher-categorical structures. These symmetries impose robust selection rules, ensure IR constraints on the spectrum (e.g., via anomaly matching or consistency requirements), and have deep implications for dualities, RG flows, and the interplay between global topology and local operator content.

In string-theoretic and gravitational contexts, these structures serve as explicit models exhibiting the mechanism by which even subtle forms of global symmetry are removed in quantum gravity, with the interplay between D-brane configuration, orientifold type, and the associated higher-form symmetry breaking forming a concrete realization of swampland constraints.

7. Concluding Summary

Gauge theories with disconnected groups such as $\widetilde{SU}(N)$, $\widetilde{U}(N)$ realize non–invertible electric 1–form symmetries wherein the resulting “folded” GW operators possess quantum dimensions greater than one and satisfy categorical fusion rules without inverses. The dual (d–2)–form symmetry, which would be invertible ($\mathbb{Z}_2$), is necessarily broken by the presence of twist vortices (Alice strings), both in the pure gauge context and mandatorily upon string theory embedding. This structure is entirely consistent with the absence of exact global symmetries in quantum gravity and provides a precise demonstration of how discrete gauging leads not only to enhanced categorical symmetry structure but also to its ultimate dissolution in settings of gravitational consistency (Arias-Tamargo et al., 2022).