Non-Invertible Higher-Form Symmetries

• Non-invertible higher-form symmetries are generalized symmetries defined by topological defects whose fusion is governed by fusion categories rather than groups.
• They are constructed via higher gauging and condensation defects, with fusion rules that sum over distinct defects and are encoded using higher-category theory.
• These symmetries impose strict constraints on QFT dynamics, often forbidding gapped symmetric phases and requiring gapless or symmetry-breaking behavior.

Non-invertible higher-form symmetries generalize the notion of global and generalized symmetries in quantum field theory by admitting topological symmetry defects whose fusion rules are governed by fusion categories or higher categories rather than groups. Unlike invertible symmetries, these operators do not possess inverses: their fusion may produce sums over distinct defects or be valued in lower-dimensional topological theories. Non-invertible higher-form symmetries critically constrain quantum field theory dynamics, vacuum structure, and phase transitions, and are a natural byproduct of gauging discrete higher-form or higher-group symmetries, mixing anomalies, and dualities. They can be systematically encoded using higher-category theory and symmetry topological field theories (SymTFTs), providing powerful algebraic and geometric frameworks for classification and calculation.

1. Fundamental Definition and Algebraic Structure

A $p$-form symmetry in $d$ dimensions is generated by topological defect operators $D_{d-p-1}(M_{d-p-1})$ supported on codimension-$(p+1)$ manifolds \cite{Gaiotto-Kapustin-Seiberg-Willett}. In the invertible (group-like) case, these operators form an abelian group: $$D_{d-p-1}^{(g)} \otimes D_{d-p-1}^{(h)} = D_{d-p-1}^{(gh)}$$ with inverses for each $g\in G^{(p)}$. For non-invertible symmetries, fusion closes only up to direct sums: $$D_q^{(a)} \otimes D_q^{(b)} = \bigoplus_c N_{ab}^c D_q^{(c)},\quad N_{ab}^c\in\mathbb{Z}_{\ge0}$$ with no requirement of an inverse. These structures are encoded in fusion categories for $p=0$ and fusion higher-categories for higher $p$ \cite(Schafer-Nameki, 2023){ICTP Lectures}, \cite(Bhardwaj et al., 2022){Bhardwaj et al}. The categorical approach captures not only fusion, but also higher junctions (associators, pentagonators) and the full network of topological defects.

2. Construction: Higher Gauging and Condensation Defects

Non-invertible higher-form symmetries arise fundamentally via "higher gauging": gauging a discrete $q$-form symmetry $G$ on a codimension-$p$ manifold in spacetime (with $p\le q+1$) produces a condensation defect $S(\Sigma)$ \cite(Roumpedakis et al., 2022){Roumpedakis-Seifnashri-Shao}. In path-integral language,

$$S(\Sigma) = \frac{1}{\sqrt{|H_q(\Sigma,G)|}} \sum_{[a]\in H_q(\Sigma,G)} U_a(\Sigma)$$

where $U_a$ are topological ($q$-form) operators. The fusion of such defects on a surface $\Sigma$ is determined by the intersection pairing and leads to fusion rules valued in TQFTs: $$S_N(\Sigma) \times S_N(\Sigma) = (\mathbb{Z}_{\gcd(k,N)})\,S_{\gcd(k,N)}(\Sigma)$$ with fusion "coefficients" being 1+1d TQFTs. This mechanism generalizes to condensation defects associated to higher-condensation of bulk lines or surfaces, and forms the backbone for realizing all non-invertible 0-form symmetries in 2+1d TQFT and their higher-categorical generalizations \cite(Bhardwaj et al., 2022){Universal Non-Invertible Symmetries}.

3. Classification and Intrinsic Non-Invertibility

The classification of non-invertible higher-form symmetries, particularly in class $\mathcal{S}$ theories, relies on the interplay between the symmetry group acting on charge lattices and the possible global forms of the theory \cite(Bashmakov et al., 2022){Symmetries of Class S}. For type $\mathfrak{a}_{p-1}$ class $\mathcal{S}$ theories, non-invertible symmetries originate from automorphisms $$F\in \text{Mod}(\Sigma_g)=\mathrm{Sp}(2g,\mathbb{Z})$$ acting on Riemann surfaces, and whether they can be related to invertible symmetries by discrete gauging. Intrinsically non-invertible symmetries occur when no choice of global form (maximal isotropic sublattice $L\subset H_1(\Sigma_g;\mathbb{Z}_p)$) solves $F(L)=L$. Algebraically, intrinsic non-invertibility is diagnosed by the absence of fixed isotropic $g$-planes under $F$; it reflects a fundamental obstruction to interpreting non-invertible physics in any invertible theory via anomalous symmetry \cite(Bashmakov et al., 2022){Symmetries of Class S}.

4. Symmetry TFTs and Anomaly Structure

The symmetry content and anomalies of non-invertible higher-form symmetries are encoded in symmetry topological field theories (SymTFTs) in one higher dimension. For invertible symmetries, SymTFTs are Dijkgraaf–Witten theories; for intrinsically non-invertible cases the SymTFT must be coupled to topological gravity, reflecting the necessity to sum over manifold surgeries or outer-automorphism twists \cite(Bashmakov et al., 2022){Symmetries of Class S}. Linking invariants in SymTFT provide explicit diagnostics for 't Hooft anomalies: if all linking phases between bulk and boundary defects cannot be removed by changing representatives, the non-invertible symmetry is anomalous and cannot be gauged. Anomalous non-invertible symmetries impose robust constraints on the IR dynamics, often forbidding symmetric gapped vacua and enforcing gapless or symmetry-breaking phases \cite(Kaidi et al., 2023){Symmetry TFTs and Anomalies}.

The action in the intrinsically non-invertible case takes the form: $$S_{5d} = \frac{2\pi}{2N} \int b^T \cup_\eta K\,\delta_\eta\,b + \frac{2\pi}{|F|} \int x\,\delta\eta$$ where $\eta$ is the automorphism gauge field, $K$ the higher-form BF matrix, and $x$ a Lagrange multiplier.

5. Fusion Category, Higher Representation Theory, and Higher Categories

Non-invertible higher-form symmetries are mathematically characterized by fusion (higher-)categories of projective higher-representations of the symmetry group or higher-group. Upon gauging higher-form or higher-group symmetries, the symmetry sector becomes a $(d-1)$-category whose objects are lower-dimensional TQFTs, 1-morphisms are topological interfaces, and 2-morphisms are junctions, with fusion rules induced from the categorical structure \cite(Bartsch et al., 2022){Higher Representation Theory II}. For example, in 3d, gauging a finite group $G$ gives rise to the fusion 2-category $2\mathrm{Rep}(G)$; for higher-groups or groups with mixed anomalies, one obtains higher group-theoretic fusion categories. In 4d theories with mixed 0-form/1-form symmetry and anomalies, the simple objects after gauging are labeled by projective representations and cocycles determined by spectral sequence data.

Associators and higher fusion data (F-symbols, pentagonators) measure the failure of strict associativity in the fusion algebra and are themselves lower-dimensional TQFTs or cocycle data in group cohomology \cite(Kim et al., 24 Sep 2025){Higher structure from Lagrangian descriptions}. This higher structure is essential for classifying and computing fusion networks in non-invertible symmetry sectors.

6. Physical Consequences: Obstructions, Dynamics, and Applications

Non-invertible higher-form symmetries impose rigorous constraints on QFT dynamics—most notably, they can forbid symmetric gapped phases and force gapless or symmetry-breaking behavior. For self-dual theories with finite one-form symmetry $\mathbb{Z}_N^{(1)}$, gapped symmetry-preserving vacua exist only if $N=k^2\ell$ with $-1$ a quadratic residue mod $\ell$; otherwise, the theory is gapless or spontaneously breaks the self-duality symmetry. Similar constraints apply to triality and beyond; the fusion rules and corresponding Gauss-sum identities govern the existence of topological phases \cite(Apte et al., 2022){Obstructions to Gapped Phases}.

In class $\mathcal{S}$, $N$-ality condensation defects provide explicit constructions of codimension-one interfaces realizing non-invertible higher-form symmetry actions, and their fusion rules encode the underlying algebraic structure. Intrinsically non-invertible defects require summing over topological gravity sectors, reflecting their fundamentally twisted nature.

Non-invertible higher-form symmetries are realized in lattice gauge models, topological orders, generalized SPT phases, and string/M-theory compactifications, often dictating the structure of ground state degeneracy, quantum memory, and classical information retention in quantum double codes \cite(Song et al., 29 Sep 2025){Strong-to-weak spontaneous symmetry breaking}. The categorical framework unifies these manifestations and enables explicit calculations in both continuum and lattice settings.

7. Generalizations, Higher Groups and Future Directions

Non-invertible higher-form symmetries extend naturally to higher-categorical symmetries, including 2-groups and relative TQFTs. The symmetry category after gauging a finite $k$-form or higher-group symmetry is the full $(d-1)$-category of lower-dimensional TQFTs with that symmetry. The framework admits further generalization to continuous non-invertible symmetries via dressing symmetry operators with lower-dimensional TQFTs, the emergence of fusion category symmetries at special points (e.g., infinite families labeled by $\mathbb{Q}/\mathbb{Z}$ in 3d Goldstone-Maxwell models), and the interplay of non-invertible defects with higher-form anomalies and topological gravity sectors \cite(Damia et al., 2022){Continuous Generalized Symmetries}.

One outstanding problem is the mathematical classification of fusion higher-categories, their centers, and the full spectrum of non-invertible higher-form symmetries in generic QFTs, including strongly coupled and gravitational settings. The universality of the SymTFT and higher-category formalism suggests a systematic roadmap for these developments.

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References

• (Bashmakov et al., 2022) Non-invertible Symmetries of Class $\mathcal{S}$ Theories
• (Apte et al., 2022) Obstructions to Gapped Phases from Non-Invertible Symmetries
• (Roumpedakis et al., 2022) Higher Gauging and Non-invertible Condensation Defects
• (Bartsch et al., 2022) Non-invertible Symmetries and Higher Representation Theory II
• (Kaidi et al., 2023) Symmetry TFTs and Anomalies of Non-Invertible Symmetries
• (Bhardwaj et al., 2022) Non-Invertible Higher-Categorical Symmetries
• (Schafer-Nameki, 2023) ICTP Lectures on (Non-)Invertible Generalized Symmetries
• (Bhardwaj et al., 2022) Universal Non-Invertible Symmetries
• (Kim et al., 24 Sep 2025) Higher structure of non-invertible symmetries from Lagrangian descriptions
• (Sela, 2024) Emergent non-invertible symmetries in $\mathcal{N}=4$ Super-Yang-Mills theory
• (Song et al., 29 Sep 2025) Strong-to-weak spontaneous symmetry breaking of higher-form non-invertible symmetries in Kitaev's quantum double model