Fusion Algebra of Symmetries

• Fusion algebra of symmetries is an algebraic structure that defines how symmetry operations, including topological defects, combine via fusion categories and Frobenius algebras.
• It provides a unified framework to analyze gauge symmetries, defect fusion, and duality phenomena in quantum field theories, statistical models, and lattice systems.
• Recent advances utilize operator algebras, weak Hopf algebras, and higher category theory to connect symmetry actions with topological invariants and quantum dimensions.

Fusion algebra of symmetries refers to the algebraic structure governing the composition and interrelation of symmetry operations—often implemented as topological defect lines, objects, or higher morphisms—in quantum field theory, statistical mechanics, and category-theoretic settings. Both invertible and non-invertible symmetries are encoded by fusion categories, generalized to higher categories for intricate global and gauge symmetries. Fusion algebras explicitly quantify how simple symmetry objects combine, the associativity constraints among fusions, and the coherent action of 0-form (ordinary), 1-form (higher), and more generalized symmetries. Recent advances formalize these algebras in terms of Frobenius algebras, module categories, weak Hopf algebras, G-crossed braided categories, and higher tube algebras, providing unified frameworks for analyzing symmetry, gauging, defect fusion, and duality phenomena.

1. Algebraic Foundations: Semisimple Frobenius Structures and Fusion Categories

For semisimple 2d TQFTs, the fusion algebra is encoded in a commutative Frobenius algebra $A$, decomposed as $A = \bigoplus_{i} \mathbb{C} e_i$ where $e_i$ are primitive idempotents satisfying $(e_i, e_j) = \delta_{ij}$ and $e_i \times e_j = \delta_{ij} N_i e_i$, $N_i$ the quantum dimensions (fusion-eigenvalues). The algebra product and nondegenerate pairing structure (inner product) enable complete diagonalization of correlators—see $Z_g = \sum_i N_i^{2g-2}$ for genus-$g$ vacuum amplitudes (Gukov et al., 2021).

In general, fusion categories $\mathcal{C}$ are $\mathbb{C}$-linear rigid semisimple monoidal categories, with simple objects $\{X_i\}$ and fusion coefficients $N_{ij}^k \in \mathbb{N}$ such that $X_i \otimes X_j = \bigoplus_k N_{ij}^k X_k$. Associativity is governed by $F$-symbols (generalized $6j$-symbols) which satisfy the pentagon equations; in multiplicity-free cases, $N_{ij}^k = 0,1$ and associator spaces are $1$-dimensional (Perez-Lona et al., 2023). The fusion algebra $\mathrm{Fus}(\mathcal{C})$ is the Grothendieck ring of $\mathcal{C}$ with multiplication determined by fusion coefficients.

Braided fusion categories exhibit further topological structure via braidings $c_{X,Y}$, encoded in hexagon equations and related to modular $S$-matrices appearing in Verlinde-type formulas for structure constants. For higher symmetries, the algebraic formalism generalizes to fusion 2-categories (see section on $\Sigma\mathcal{B}$ below), offering compositional rules for objects, 1-morphisms, and 2-morphisms.

2. Symmetry Actions: 0-form, 1-form, and Higher Symmetries in the Fusion Algebra

Discrete symmetries in TQFT (0-form, 1-form, generalized global) act naturally on the fusion algebra of idempotents. In the fusion-eigenbasis, 0-form symmetries permute the $e_i$ among those with identical quantum dimensions: $g: e_i \mapsto e_{\sigma_g(i)}$, with $\sigma_g$ a permutation within the orbit of $N_i = N_j$ (Gukov et al., 2021). 1-form symmetries act as grading automorphisms via roots of unity: $\alpha: e_i \mapsto \chi_i(\alpha) e_i$, with $\chi_i$ a character of $G^{(1)}$.

Gauging a finite abelian 0-form symmetry $G$ reconfigures the basis: surviving idempotents are organized by $G$-orbits and irreducible representations of stabilizer subgroups, yielding new fusion-eigenvalues $N_i / |G|$ for orbit elements (Gukov et al., 2021). Gauging 1-form symmetry $H$ projects out idempotents on which the character is nontrivial and rescales quantum dimensions by $N_i \to N_i / |H|$.

Fusion of topological symmetry defects is organized as a $G$-crossed braided fusion category: defect lines labeled by $g\in G$ obey fusion rules $$D_g \circ D_h \simeq \bigoplus_{k \in G} N_{g,h}^k D_k$$, with associativity and (twisted) commutativity up to $H^3(G,U(1))$ cohomology (Gukov et al., 2021).

Discrete gauge theories, such as string-net models with $Z_N$ fusion algebra, realize symmetry, gauging, and SPT phases through the branching constraint $N_{ijk} = \delta_{i+j+k,0}$ and $F$-symbol cocycles. Classification by $H^3(Z_N,U(1))$ group cohomology covers all gauge and SPT variants (Hung et al., 2012).

3. Higher Categorical Generalizations: Fusion 2-Category Symmetries

Fusion 2-category symmetry arises from condensation and completion of a braided fusion category $\mathcal{B}$, yielding $\Sigma\mathcal{B}$ (Xi et al., 2023). The structure:

• Objects: Separable algebras $A \subset \mathcal{B}$, assembled into Morita classes.
• 1-Morphisms: $B$-$A$-bimodules in $\mathcal{B}$.
• 2-Morphisms: Bimodule maps commuting with left/right actions.

Monoidal structure is induced via the braiding and tensor product in $\mathcal{B}$, with intertwining isomorphisms, associators, and pentagonators determined by canonical algebra isomorphisms and bimodule composition. The fusion rules and structure constants are extracted by decomposing tensor products of objects and mapping via bases of section/retraction morphisms (Clebsch-Gordan coefficients). Associativity is tracked by generalized $10j$-symbols arising from comparison of different parenthesizations in tensoring.

The explicit example $\Sigma\,\mathrm{sVec}$ features objects $1$, $A=1\oplus f$, with fusion group $\mathbb{Z}_2$, nontrivial $10j$-symbol matrices capturing the associator data, and quantum dimensions $d(f)=\sqrt{2}$, $d(\mu)=d(1)=d(A)=1$ (Xi et al., 2023).

4. Frobenius Algebra Objects and the Fusion Categorical Diagonal

In fusion categories of the form $$\mathrm{Rep}(G)\boxtimes \mathrm{Rep}(G) \cong \mathrm{Rep}(G\times G)$$, the canonical diagonal Frobenius algebra $A = \bigoplus_{\rho} \rho \boxtimes \rho^*$ generalizes the group diagonal embedding (Robbins et al., 2024). The algebra is specified by unit, multiplication, counit, and comultiplication maps that implement the permutation action on cosets $\left(G\times G\right) / G_{\mathrm{diag}}$, yielding fusion structure constants: $$N^{\tau \boxtimes \tau^*}_{\rho \boxtimes \rho^*,\,\sigma \boxtimes \sigma^*} = \delta_{\sigma, \rho^*} \delta_{\tau, \rho}$$ These rules extend field-theoretical constructions depending on diagonal subgroups to contexts with non-invertible symmetries, explicitly enabling gauging and topological defect insertion in non-invertible settings. Morita equivalence for Frobenius algebra objects clarifies when two gaugings yield equivalent symmetry-breaking patterns (Robbins et al., 2024).

5. Fusion Algebra, Gauging, and SPT Phases

The fusion algebra is central to both TQFT gauge procedures and quantum lattice models. Gauging a symmetry corresponds to choosing a Frobenius algebra object ($A$) and forming the module category $\mathcal{C}_A$. In multiplicity-free categories (e.g., $\mathrm{Rep}(H)$ for certain Hopf algebras), modular invariant partition functions are constructed from crossing kernels and associator $F$-symbols, ensuring invariance under torus modular transformations; details are explicit for $$\mathrm{Rep}(S_3), \mathrm{Rep}(D_4), \mathrm{Rep}(Q_8)$$ (Perez-Lona et al., 2023).

Edge modes and boundary conditions are classified by module categories, with fusion actions encoded as NIM-representations in subfactor principal graphs (Yu et al., 7 Apr 2025). Generalized orbifold groupoids and self-dualities arise from admissible gaugings, with principal and dual principal graphs classifying vacuum degeneracies and particle-soliton spectra.

In SPT phases, string-net models and their dual spin models demonstrate fusion algebra via exactly solvable Hamiltonians. SPT distinctions correspond to inequivalent $F$-symbol cocycles in $H^3(G,U(1))$ (Hung et al., 2012), and lattice realization connects fusion rules to commuting projector Hamiltonians (Inamura, 2021).

6. Operator Algebraic and Conditional Expectation Formalism

Fusion algebra symmetry is deeply connected to operator algebraic techniques. Given a boundary subalgebra $B \subset A$ of quasi-local observables, the fusion category $\mathcal{C}$ emerges as the DHR category of bimodules, with the fusion ring acting by unital completely positive quantum channels $\phi_X$ satisfying normalized fusion laws: $$\phi_X \circ \phi_Y = \sum_{Z} \frac{d_X d_Y}{d_Z} N_{XY}^Z \phi_Z$$ The fixed-point subalgebra recovers $B$ (Evans et al., 7 Jul 2025).

A conditional expectation $E:A \to B$ can be constructed using Haar integrals for weak Hopf algebras arising from fusion category symmetry (Ahmad et al., 22 Sep 2025). Its index $\operatorname{Ind}(E)$ provides a quantum-information-theoretic order parameter for symmetry breaking, saturating bounds dictated by the fusion category quantum dimension.

These frameworks rigorously connect categorical symmetry data to quantum lattice realizability, gapped/gapless phase constraints, and duality anomalies—e.g., Lieb-Schultz-Mattis-type theorems dictate that non-anomalous fusion category symmetries permit only gapped pure symmetric states if a fiber functor exists, while Kramers-Wannier type anomalous dualities force gapless states (Evans et al., 7 Jul 2025).

7. Higher Algebraic Structures: Tube Algebras, 2-Groups, and Generalized Actions

Beyond ordinary fusion categories, higher tube algebras and 2-categorical generalizations describe non-invertible symmetries in D$\geq$3 dimensions. The tube algebra $\mathrm{Tube}(\mathcal{C})$ assembles topological junctions of symmetry defects, and its $*$-representations classify twisted sector operators. The irreducible representations correspond bijectively to Drinfeld center objects (bulk lines), diagonalized by higher $S$-matrices (Bartsch, 6 Feb 2025).

Fusion actions of module categories on morphism spaces yield a categorical generalization of Schur-Weyl duality, with generalized Frobenius-Schur indicators as character data (Dong et al., 18 Aug 2025). A categorical Galois correspondence relates intermediate fusion subcategories to condensable subalgebras, with direct applications to orbifold and vertex operator algebra extensions.

Conclusion

Fusion algebra of symmetries unifies the algebraic and categorical description of symmetry, duality, and topological phases in quantum systems. It provides the underpinning for symmetry defect fusion, gauging, SPT classification, operator algebraic constraints, and higher categorical generalizations. These structures are now systematically computable and applicable to a wide array of problems in mathematical physics, condensed matter, and quantum information (Gukov et al., 2021, Xi et al., 2023, Robbins et al., 2024, Hung et al., 2012, Evans et al., 7 Jul 2025, Bartsch, 6 Feb 2025, Dong et al., 18 Aug 2025, Ahmad et al., 22 Sep 2025).