REVIEW
On general background of quantum non-invertible symmetry in 2D
High Energy Physics - Theory · Submitted 9 Sep 2026 · 51 references
Pith's verdict: A clean representation-theoretic derivation of the Fourier kernel for non-invertible symmetry backgrounds, rigorous for finite groups and formally extended to Lie groups.
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Referee report
The paper
The paper claims
Explicit Fourier transform links flat gauge backgrounds to non-invertible symmetry networks in 2D.
The reading
What carries the argument
The central machinery is the construction of Fourier kernels W and C using the representation theory of G. For the forward kernel, the Wilson lines of irreducible representations ρ, σ are transported through the flat G background with commuting holonomies (g,h); the kernel W is the trace over E_ρ ⊗ E_σ of the composition: S_{σ,ρ} ∘ (σ(g) ⊗ ρ(e)) ∘ $I^{{∨ σ,ρ}}$_{λ;j} $I^{{λ;i}}$_{ρ,σ} ∘ (ρ($h^{{-1}}$) ⊗ σ(e)). For the inverse kernel, the regular Frobenius algebra A = ⊕_{σ} d_σ σ is decomposed using embeddings J_{ρ,r} and projections J^∨_{ρ,r} defined via the Peter-Weyl basis. Dressing the multiplication and comultiplication with right-regular actions R_y and R_{$x^{{-1}}$} produces modified maps that can be expanded in the same intertwiner bases, yielding coefficients M and M^∨ whose contraction gives the inverse kernel C. For Lie groups the sums become integrals, and the same construction reproduces the expected Cartan torus form of flat gauging on the regular stratum using Clebsch-Gordan coefficients and Peter-Weyl matrix elements.
Extended reading notes
Core claim
The paper establishes that the partition function of a 2D theory after gauging a non-abelian finite group G, in a general Rep(G) topological defect network on the torus, can be expressed as a sum over commuting pairs of holonomies (g,h) in G, weighted by a kernel derived by transporting the Wilson lines of the dual representations through the flat G background and contracting with intertwiners and co-intertwiners. Conversely, any flat G background partition function can be recovered from the set of Rep(G) defect network partition functions via an inverse kernel that emerges from decorating the regular Frobenius algebra A = Fun(G) with right-regular actions. For multiplicity-free fusion rules, the forward kernel reduces to a character projector multiplied by a proportionality factor; for higher-multiplicity cases (e.g., A4), the full junction space is resolved and the character projector alone is insufficient. The inverse kernel satisfies a simple relation to the forward kernel: C_{\rho,\$\sigma$}^{\$\lambda$;i,j}(x,y) = (d_\rho d_\$\sigma$ / d_\$\lambda$) W_{\$\sigma$,\rho}^{\$\lambda$;j,i}(y,x). The construction extends to compact Lie groups, where sums become Peter-Weyl integrals, and regauging Rep(SU(2)) formally yields the original flat SU(2) partition function, provided the measure is regularized by cutting off the sum over representations.
Load-bearing premise
The construction for compact Lie groups (e.g., Rep(SU(2))) assumes that the flat gauging procedure can be defined by cutting off the representation sum and that contributions from irregular strata (where the centralizer is larger than the Cartan torus) do not affect the final result; these contributions are not analytically controlled.
Editorial extensions
If this is right
- One can compute the partition function of an orbifold theory in any resolved Rep(G) defect network on the torus directly from the representation data of G, without solving modular bootstrap equations.
- The inverse transform provides a systematic way to recover any flat G background partition function from the set of all Rep(G) network partition functions, confirming that gauging and regauging are inverses on the physical image.
- The method extends straightforwardly to higher-genus Riemann surfaces by resolving the defect network into trivalent junctions and following the same transport rule.
- For finite groups with higher-multiplicity fusion rules (e.g., A4), the construction keeps the full multiplicity space explicit, avoiding the compression that occurs when only character projectors are used.
- The formal extension to Rep(SU(2)) shows that the Fourier transform approach can be applied to continuous groups, giving a candidate definition of flat gauging for non-abelian Lie groups in 2D.
Reading between the lines
- The Fourier kernels W and C provide a concrete dictionary between two descriptions of the same physical theory: one with ordinary group symmetry G and one with non-invertible symmetry Rep(G). This dictionary may allow the transfer of results from group-theoretic computations (e.g., anomaly matching) to non-invertible settings without directly handling the full fusion category data.
- The fact that the inverse kernel is proportional to the forward kernel with swapped indices suggests a kind of duality relation that may hold more generally for any pair of fusion categories related by a Morita equivalence, for example between Rep(G) and the category of G-graded vector spaces.
- The regularisation required for the Lie group case (cutoff in representation sum to handle the Dirac delta on the commuting pair moduli space) indicates that the flat gauging of a continuous group is not a purely topological operation but involves an infinite-dimensional Hilbert space structure; the final result depends on how the cutoff is taken, which may be related to the choice of the Frobeniu
Original abstract
We study the dual quantum symmetry $\mathrm{Rep}(G)$ in the two-dimensional theory $\widetilde{\mathfrak{T}}_{\mathrm{Rep}(G)}$ obtained by gauging a non-abelian symmetry $G$ of $\mathfrak{T}_G$, where for Lie groups $G$ the gauging is understood as flat gauging. We develop a general framework for computing partition functions of $\widetilde{\mathfrak{T}}_{\mathrm{Rep}(G)}$ in arbitrary non-invertible symmetry backgrounds, represented by topological defect networks of $\mathrm{Rep}(G)$, in terms of the partition functions of the original theory $\mathfrak{T}_G$. We also derive the inverse transformation, expressing partition functions of $\mathfrak{T}_G$ in $G$ backgrounds in terms of those of $\widetilde{\mathfrak{T}}_{\mathrm{Rep}(G)}$. We test the framework in several examples, including finite groups with multiplicity-free and higher-multiplicity fusion rules, and discuss a formal extension to compact Lie groups, focusing on $\mathrm{Rep}(SU(2))$.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We study the dual quantum symmetry $\mathrm{Rep}(G)$ in the two-dimensional theory $\widetilde{\mathfrak{T}}_{\mathrm{Rep}(G)}$ obtained by gauging a non-abelian symmetry $G$ of $\mathfrak{T}_G$, where for Lie groups $G$ the gauging is understood as flat gauging. We develop a general framework for computing partition functions of $\widetilde{\mathfrak{T}}_{\mathrm{Rep}(G)}$ in arbitrary non-invertible symmetry backgrounds, represented by topological defect networks of $\mathrm{Rep}(G)$, in terms of the partition functions of the original theory $\mathfrak{T}_G$. We also derive the inverse transformation, expressing partition functions of $\mathfrak{T}_G$ in $G$ backgrounds in terms of those of $\widetilde{\mathfrak{T}}_{\mathrm{Rep}(G)}$. We test the framework in several examples, including finite groups with multiplicity-free and higher-multiplicity fusion rules, and discuss a formal extension to compact Lie groups, focusing on $\mathrm{Rep}(SU(2))$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study the dual quantum symmetry $\mathrm{Rep}(G)$ in the two-dimensional theory $\widetilde{\mathfrak{T}}_{\mathrm{Rep}(G)}$ obtained by gauging a non-abelian symmetry $G$ of $\mathfrak{T}_G$, where for Lie groups $G$ the gauging is understood as flat gauging. | contribution 0.78 | 0.65 | 0.07 | asserted 0.95 |
| s1 | We develop a general framework for computing partition functions of $\widetilde{\mathfrak{T}}_{\mathrm{Rep}(G)}$ in arbitrary non-invertible symmetry backgrounds, represented by topological defect networks of $\mathrm{Rep}(G)$, in terms of the partition functions of the original theory $\mathfrak{T}_G$. | contribution 0.83 | 0.96 | 0.37 | asserted 1.00 |
| s2 | We also derive the inverse transformation, expressing partition functions of $\mathfrak{T}_G$ in $G$ backgrounds in terms of those of $\widetilde{\mathfrak{T}}_{\mathrm{Rep}(G)}$. | contribution 0.76 | 0.95 | 0.15 | asserted 1.00 |
| s3 | We test the framework in several examples, including finite groups with multiplicity-free and higher-multiplicity fusion rules, and discuss a formal extension to compact Lie groups, focusing on $\mathrm{Rep}(SU(2))$. | contribution 0.85 | 0.95 | 0.11 | asserted 0.95 |
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Discussion
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