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Gauging Non-Invertible Symmetries in (2+1)d Topological Orders

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Pith's one-line read Gauging a non-invertible symmetry turns toric code into the non-Abelian theory D(D6).

desk verdict A serious, mostly convincing framework for gauging non-invertible symmetries in (2+1)d TQFTs; the constructive core looks sound, but the main no-go constraint rests on an unproven near-group assumption and needs repair before the paper can be accepted. read the letter →

arxiv 2507.01142 v1 pith:UK7JHE6E submitted 2025-07-01 hep-th cond-mat.str-elmath-phmath.MPquant-ph

Mahesh K. N. Balasubramanian , Matthew Buican , Clement Delcamp , Rajath Radhakrishnan This is my paper
classification hep-thcond-mat.str-elmath-phmath.MPquant-ph
keywords non-invertiblesymmetriesgaugingtopologicalordercondensationdefectsfusion2-categoriesMoritadualitytoriccodequantumdoublemodels
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that gauging a non-invertible 0-form symmetry in a (2+1)d topological order is exactly the dual operation to gauging a 1-form symmetry given by a condensable algebra of lines, so both can be computed with the same surface-algebra machinery. The central object is an algebra of surfaces $A_S$, a separable algebra in the fusion 2-category of condensation defects, recovered from a topological interface $I$ by $A_S \simeq I^\dagger \otimes I$. Using this, the authors compute that gauging the non-invertible symmetry $S_1 \boxplus S_e$ of the toric code $D(\mathbb{Z}_2)$ produces the non-Abelian quantum double $D(D_6)$, and they give a recipe that turns large classes of Abelian topological orders into non-Abelian ones. They also generalize symmetry fractionalization and discrete torsion to the non-invertible setting, prove a fixed-point theorem for twisted sector lines, and derive constraints that rule out many would-be gaugings. If right, the paper makes non-invertible gauging a practical tool rather than a formal curiosity.

What carries the argument

The machine is the algebra of surfaces $A_S$: a separable algebra in the fusion 2-category $\mathrm{Mod}(\mathcal{B})$ of module 1-categories over the anyon category $\mathcal{B}$. Its underlying object is a condensation surface—a codimension-one defect built by higher-gauging a 1-form symmetry, i.e., inserting a mesh of lines for a condensable algebra $A_L$. Two relations carry the argument: $A_S \simeq I^\dagger \otimes I$ and $\widehat{A}_S \simeq I \otimes I^\dagger$, where $I$ is the topological interface that half-gauges $A_L$; associativity of interface fusion then gives $A_S \otimes I \simeq I \otimes \widehat{A}_S$. The Morita theory of fusion 2-categories supplies the dualization: gauging $A_S$ in $\mathrm{Mod}(\mathcal{B})$ yields $\mathrm{Mod}(\mathcal{Z}_{\mathcal{B}}(A_S))$, and the constraint $\mathcal{Z}(A_S) \simeq \mathcal{B}_1 \boxtimes \mathcal{B}_2$ encodes gaugeability. A generalized fixed-point theorem equates the rank of the twisted-sector fusion 1-category with the number of fixed points of the surface action on lines, and this equality is what forces dual surface algebras to have matching numbers of twisted lines.

What would settle it

Compute the full fusion 1-category of twisted sector lines for $\widehat{A}_S = S_1 \boxplus S_{D(H),\psi}$ in the untwisted Abelian Dijkgraaf–Witten theory with $H = \mathbb{Z}_2 \times \mathbb{Z}_3$ and verify whether the non-trivial object obeys the near-group rule with $n$ in the forbidden range $0 < n < |H|^2 - 1$; finding any consistent fusion category or braided tensor functor $D(H) \to \mathcal{Z}(\widehat{A}_S)$ for a non-invertible $S_{D(H),\psi}$ would falsify Constraint 4.4.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes that the distinction between gauging a 0-form and a 1-form symmetry is not fundamental: both are realized by summing over a network of condensation surfaces, with the choice of a fusion 1-category on the bounding lines specifying generalized symmetry fractionalization and generalized discrete torsion. The duality is carried by a topological interface $I$ between two topological orders: fusing $I$ with its orientation reversal gives the surface algebra $A_S \simeq I^\dagger \otimes I$ in the first theory, while $I \otimes I^\dagger$ gives the dual algebra $\widehat{A}_S$ in the second, and gauging either reproduces the other. The concrete flagship computation is that gauging $\widehat{A}_S = S_1 \boxplus S_e$ in $D(\mathbb{Z}_2)$, the toric code, yields $D(D_6)$, the non-Abelian quantum double of the dihedral group of order six; the paper also works out $D(\mathbb{Z}_2) \to Z(\mathrm{Ising})$, $D(\mathbb{Z}_2) \to D(D_4)$, $D(\mathbb{Z}_2) \to D(\mathbb{Z}_4)$, $D(\mathbb{Z}_2) \to D(D_8)$, and the twisted $D_6$ family. A generalized fixed point theorem states that the number of twisted sector lines bounding a condensation surface equals the number of lines it fixes, and a set of constraints delimits which surface algebras can be gaugeable.

Load-bearing premise

The proof of Constraint 4.4 assumes that the twisted sector lines of $S_1 \boxplus S_{D(H),\psi}$ form a near-group fusion category with a single non-trivial object $\rho$ satisfying $\rho\otimes\rho = \oplus_{(m,e)\in H\oplus\hat{H}}(m,e)\oplus n\cdot\rho$; if the same object admits a different fusion category, the contradiction that rules out such gaugeable symmetries does not follow.

Editorial extensions

If this is right

  • Non-invertible 0-form gauging can be performed by the same recipe as 1-form gauging: pick $A_L$, form $A_S$, then gauge it; the flagship example maps $D(\mathbb{Z}_2)$ to $D(D_6)$ and can be checked directly by decomposing the resulting line spectrum.
  • Generalized symmetry fractionalization is a choice of fusion ring for the twisted sector lines of $A_S$, and generalized discrete torsion is a choice of associator; both are constrained by the requirement that a braided tensor functor $\mathcal{B} \to \mathcal{Z}(A_S)$ exist.
  • The fixed-point theorem means the dual surface algebra $\widehat{A}_S$ must have exactly as many twisted sector lines as $A_S$; this consistency condition rules out otherwise plausible gaugings.
  • Iterated gauging of invertible symmetries reproduces a single non-invertible gauging exactly when the surface algebra admits an increasing sequence of sub-algebras, with examples $D(\mathbb{Z}_2) \to D(D_8)$ decomposing as $D(\mathbb{Z}_2) \to D(D_4) \to D(D_8)$, while $D(\mathbb{Z}_2) \to D(D_6)$ does not decompose.
  • Constraints in Section 4 delimit gaugeable symmetries in Abelian discrete gauge theories: for instance, a surface algebra $S_1 \boxplus S_{D(H),\psi}$ with non-invertible $S_{D(H),\psi}$ is not gaugeable when $|H|$ is a product of distinct primes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the paper reduces non-invertible 0-form gauging to 1-form gauging data, its recipes could plausibly be translated into explicit local operations on stabilizer codes, making the $D(\mathbb{Z}_2) \to D(D_6)$ step a candidate circuit primitive rather than just a formal map.
  • The framework suggests there should be a cohomological classification of generalized symmetry fractionalization and discrete torsion for surface algebras; a concrete next step would be to match the paper's three fusion categories on the $D_6$ example with an appropriate categorified cohomology theory.
  • Constraint 4.4's reliance on near-group fusion categories implies that finding even one gaugeable non-invertible $S_1 \boxplus S_{D(H),\psi}$ in a theory with more prime factors would open a new family of dualities, giving a testable extension of the paper's no-go result.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper develops a framework for gauging non-invertible symmetries in (2+1)d topological orders, representing such symmetries as algebras of surfaces in the fusion 2-category Mod(B). The central mechanism is the relation AS ≃ I†⊗I and bAS ≃ I⊗I† for an interface I, so that gauging a non-invertible 0-form symmetry is dual to 1-form gauging of a condensable algebra AL. The authors generalize symmetry fractionalization and discrete torsion, prove a fixed-point theorem for condensation surfaces, derive constraints on gaugeable surface algebras (Constraints 4.1–4.5), analyze when non-invertible gaugings resolve into invertible steps, and illustrate the method on D(Z2), including the maps D(Z2)→D(D6), D(Z2)→Z(Ising), and the twisted D6 theories.

Significance. If correct, the paper gives a practical and conceptually unifying recipe for non-invertible gauging and for producing non-Abelian from Abelian topological orders. Its strengths are the explicit examples, which include concrete line decompositions, spin matching, and Morita equivalences, and the systematic use of separable algebras in fusion 2-categories. The advertised D(Z2)→D(D6) example is not merely an existence claim: the authors exhibit the action of bAS, the splitting of lines, and identify the resulting spectrum. The negative constraints, especially Constraint 4.4, are less securely established and require repair.

major comments (2)
  1. [Sec. 4, Eq. (4.14)] Constraint 4.4 is load-bearing for the paper's classification claims, as it is used in Sec. 6.8 to rule out surface algebras of the form S1 ⊞ Sme and S1 ⊞ S̃me. The proof assumes without derivation that the twisted-sector fusion category of bAS ≃ S1 ⊞ S_{D(H),ψ} is a near-group category of the form H ⊕ H + n. Given only that the D(H) lines form a pointed subcategory and that they act trivially on ρ under fusion, the most general self-fusion is ρ⊗ρ ≃ ⊕_{(m,e)} N_{(m,e)} (m,e) ⊕ n·ρ with integer multiplicities N_{(m,e)}. Nothing in the text establishes N_{(m,e)} = 1. The classification theorems [63–65] invoked to obtain the contradiction (4.20) apply to the N=1 near-group family, so if multiplicities differ, the inequality 0 < n < |H|^2 − 1 need not follow and Constraint 4.4 collapses. Please derive (4.14) from the module-category data defining S_{D(H),ψ}, or replace the no-go statement by a conditional one with a direct classification of the possible fusion categories, for instance for H = Z_p.
  2. [Sec. 5, Claims 5.5–5.6] The sequential-decomposition results are advertised as a main contribution, but the proofs are presented as sketches. Claim 5.5 is established by an induction whose base case appeals to [59, Remark 4.12], and Claim 5.6 is stated after a short paragraph invoking Theorem 4.10 of [59] and Sec. 3.1 of [69] without spelling out how the G-grading on bAS yields invertible gaugings at each step. Since these claims are used in Sec. 6.6 to conclude that D(Z2)→D(D8) can be decomposed into a sequence of invertible gaugings, please either state the relevant imported theorems precisely and complete the induction, or explicitly label these results as conjectural.
minor comments (3)
  1. [Sec. 4, before Eq. (4.14)] The sentence 'we loose no generality' should read 'we lose no generality.'
  2. [Secs. 6.5 and 6.6] The notation 'D(AS)' appears without definition; since AS is used both for a surface algebra and for the associated fusion category, the intended object (presumably the Drinfeld center Z(AS)) should be defined explicitly.
  3. [Sec. 6.3] The verification that the gauged theory is D(D6) lists quantum dimensions and the spins of only the first five of the eight lines; presenting the fusion ring or the full T-matrix of the equivalence classes would make the 'agrees with untwisted Dijkgraaf–Witten theory' check complete.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: Constraint 4.4's chain uses an unproven near-group ansatz, an independent published prior result by the authors ([21]), and external classifications; the D(Z2) to D(D6) example is computed and verified rather than imposed, and no output equals an input by construction or fit.

full rationale

The derivation chain is not circular in any of the flagged senses. The core identifications AS = I^+ (x) I and bAS = I (x) I^+ (Eqs. 3.3-3.4) are imported from Decoppet's internal-hom construction [23]; the Morita-duality formulas (3.35)-(3.37) and the sandwich decompositions (Figs. 13, 20) come from [32] and [20, 26] - external works with no author overlap. The principal no-go, Constraint 4.4, proceeds from three premises: the near-group ansatz (4.14) ('we lose no generality in assuming...'), the bound d_AL < |H|^2 + 1 citing the authors' own [21], and the external near-group classification [63-65]. The skeptic's objection - that rho (x) rho is restricted to the form with unit multiplicities without derivation, so [63-65] need not apply - identifies a genuine proof gap, but a gap in justification is not a circular reduction: (4.14) is an asserted ansatz, not an input defined in terms of the no-go claim it supports, and no equation in the chain equals another by construction or is tuned to force the contradiction. The self-citation to [21] is repeated and genuinely load-bearing in the constraint proofs, but under the evidentiary rule it is real evidence: a peer-reviewed, published (Comm. Math. Phys.) derivation of condensation-defect properties whose stated assumptions do not include Constraint 4.4, the D(D6) example, or any quantity fitted in this paper, so it does not raise the circularity score. The constructive flagship result - gauging S1 boxplus Se in D(Z2) to obtain D(D6) - is computed rather than imposed: from the fixed algebra AL = ([1],1) oplus ([1],pi) the paper derives the module category D(D6)_AL, the dual surface bAS, the action (6.26), and the line splittings (6.29)-(6.32), then matches the resulting dimensions and spins against the independently known fusion and spin data (6.18)-(6.19); the same verified round-trip pattern holds for Z(Ising), D(D4), D(D8), and the twisted-D6 family. No fitted parameter is renamed as a prediction, and the 'uniqueness' invoked in footnote 17 is Ostrik's external rank-2 classification, not an author-imported forcing theorem. The generalized fixed-point theorem (Sec. 3.2) extends [19]'s invertible result with a new proof rather than rebranding it; App. C's Abelian re-proof leans on [21], again an independent published source. The paper is self-contained against external benchmarks, so the honest finding is no circularity; the near-group assumption gap should be tracked as a correctness risk, not as circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central constructions rest on known results in fusion 2-category theory and on the sandwich decomposition of interfaces. No free parameters or invented entities appear. The near-group fusion rule in Constraint 4.4 is an ad hoc assumption not fully derived.

assumptions (5)
  • domain assumption Every gapped topological interface between MTCs B1 and B2 can be decomposed into a sandwich of 1-form gauging interfaces and an invertible surface (Fig. 20).
    Used in Sec. 3.1 and Sec. 4 to identify the condensable algebras AL and BL and to derive constraints on gaugeable symmetries; imported from [20,26].
  • standard math The internal hom construction gives AS ≃ I†⊗I and bAS ≃ I⊗I† for surface algebras dual to an interface I.
    Used extensively in Sec. 3.1; from [23, Prop 4.1.1].
  • standard math Gauging a separable algebra AS in Mod(B) yields Mod(B)^*_{Mod(AS)} ≃ Mod(Z_B(AS)), and Z(CI) ≃ Z(AS) ≃ B1 ⊠ B2.
    Used in Sec. 3.3 and 3.6; from [31,32].
  • standard math In a 2d TQFT, the dimension of the space of local operators equals the number of simple boundary conditions.
    Used in the proof of the fixed point theorem in Sec. 3.2; from [27,28].
  • ad hoc to paper The fusion category of twisted sector lines for bAS ≃ S1 ⊞ S_{D(H),ψ} is a near-group fusion category of the form H ⊕ H + n with ρ⊗ρ = ⊕_{(m,e)} (m,e) ⊕ n·ρ (Eq. 4.14).
    Assumed without full derivation in the proof of Constraint 4.4; if other fusion structures exist, the contradiction does not follow.

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Pith. "Pith review of Gauging Non-Invertible Symmetries in (2+1)d Topological Orders." pith.science (2026). https://pith.science/paper/UK7JHE6E

@misc{pith2026250701142,
  author       = {Pith},
  title        = {Pith review of: Gauging Non-Invertible Symmetries in (2+1)d Topological Orders},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UK7JHE6E}},
  note         = {Machine review of arXiv:2507.01142}
}
read the original abstract

We present practical and formal methods for gauging non-invertible symmetries in (2+1)d topological quantum field theories. Along the way, we generalize various aspects of invertible 0-form gauging, including symmetry fractionalization, discrete torsion, and the fixed point theorem for symmetry action on lines. Our approach involves two complementary strands: the fusion of topological interfaces and Morita theory of fusion 2-categories. We use these methods to derive constraints on gaugeable symmetries and their duals while unifying the prescription for gauging non-invertible 0-form and 1-form symmetries and various higher structures. With a view toward recent advances in creating non-Abelian topological orders from Abelian ones, we give a simple recipe for non-invertible 0-form gauging that takes large classes of the latter to the former. We also describe conditions under which iterated gauging of invertible 0-form symmetries is equivalent to a single-step gauging of a non-invertible symmetry. We conclude with a set of concrete examples illustrating these various phenomena involving gauging symmetries of the infrared limit of the toric code.

Figures

Figures reproduced from arXiv: 2507.01142 by the authors.

Figure 1
Figure 1. Gauging Sg identifies the lines a and ρg(a). discussion shows that the simple lines related by the action of the surfaces being gauged are identified by gauging the surfaces. Consider the junction of two a lines on the surface S := ⊞g∈GSg. Let V a S(a) be the vector space of local operators at this junction. Note that dim V a S(a) > 0, because S contains the trivial surface. Upon gauging the symmetry G, the surface … view at source ↗
Figure 2
Figure 2. Action of I on line operators. In particular, S acts trivially on the trivial line 1. Therefore, we have 1 Gauging G −−−−−−→ M π∈Irr(Rep(G)) 1π . (2.4) This logic shows that the TQFT, TG, contains a subset of lines that form the category Rep(G) under fusion. This category is the dual 1-form symmetry whose gauging takes us back to the TQFT, T . The corresponding condensable algebra is the algebra C G equipped with a … view at source ↗
Figure 3
Figure 3. Fusion of a line x in B1 from the left on the interface produces a line operator, IL(x), on it. One can similarly consider the action of a line in B2 from the right. More precisely, let CI be the category of lines on the interface. Then, for x ∈ CI, there is an action of B1 and B2 on CI by fusion on the interface from the left and right, respectively. Of course, the fusion of lines does not change the interface itse… view at source ↗
Figures from the paper (25 more)
Figure 4
Figure 4. Figure 4: Fusion of a surface, S ∈ Mod(B1), with the interface I from the left produces a new interface, I ′ . Mod(B1) and Mod(B2). As shown in [23, Theorem 5.3.4], any interface I ∈ I can be obtained by gauging an algebra AS in Mod(B1) in order to obtain Mod(B2). Physically, th…
Figure 5
Figure 5. Figure 5: Unlike the case of fusion of general surfaces in Mod(B1) with I, the fusion of AS with the interface I leaves it unchanged. Note that AS typically does not correspond to an indecomposable surface. In general AS will decompose into a number of indecomposable surfaces de…
Figure 6
Figure 6. Figure 6: Pinching I to the left shows that there is a non-trivial interface between AS and I † ⊗ I. AS = I I † Mod(B1) Mod(B1) Mod(B1) Mod(B2) Mod(B1) x Ox [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: Local operators on AS correspond to lines that can end on both I and I † . We can repeat the above argument by starting with Mod(B2) and constructing Mod(B1) by inserting a network built from the surface AbS in Mod(B2). This surface is then determined by the fusion of …
Figure 8
Figure 8. Figure 8: Fusing a twisted sector line operator x of AS, on I produces a line on it. twisted sector lines of AbS are precisely those that become genuine lines in B1 upon gauging AS. Moreover, passing a fusion vertex consisting of three lines in B2 through I produces twisted sect…
Figure 9
Figure 9. Figure 9: Pinching the interface and fusing in the presence of a line operator, y ∈ CI, produces a twisted sector line operator x bounding a surface in AS. It will be useful for us to keep track of how the simple genuine lines in B1 map to those in B2 through the interface I. To…
Figure 10
Figure 10. Figure 10: Passing a line a ∈ B1 through the interface I produces a line x bounding a surface in AbS. If x is supported on the trivial surface in AbS, then x is a genuine line in B2. I = I a b c x y z AbS AbS AbS Mod(B1) Mod(B2) Mod(B1) Mod(B2) [PITH_FULL_IMAGE:figures/full_fig…
Figure 11
Figure 11. Figure 11: Passing a point operator at the junction of three lines through the interface I produces a point operator at the junction of three generically non-genuine lines bounding surfaces in AbS. We have suppressed the label of the point operator at the point junction for clar…
Figure 12
Figure 12. Figure 12: The top figure shows that two lines a, b ∈ B1 can form a junction on the surface AS if and only if there exists some intermediate line x ∈ B2 such that a and b are connected through the interfaces I and I † . The bottom figure shows that, to find point junctions of tw…
Figure 13
Figure 13. Figure 13: Fig.13 [PITH_FULL_IMAGE:figures/full_fig_p020_13.png]
Figure 14
Figure 14. Figure 14: Folding the fixed point action of S on a. a ⊠ a and the boundary is precisely Na S(a) . Note that the trivial surface yields the canonical boundary MB, of the folded theory. Clearly, all lines of the form a ⊠ a can end on this gapped boundary, and Na S1(a) = 1 for all…
Figure 15
Figure 15. Figure 15: If a is a fixed point of S, then the line a ⊠ a can end on both MB and MS. consider the configuration in [PITH_FULL_IMAGE:figures/full_fig_p022_15.png]
Figure 16
Figure 16. Figure 16: On folding, a boundary condition of the surface S becomes an interface between the gapped boundaries MS and MB. interface x and compactify to get a (1+1)d TQFT with each line x ∈ AS specifying a simple gapped boundary condition of TS (see [PITH_FULL_IMAGE:figures/ful…
Figure 17
Figure 17. Figure 17: The interface x between the gapped boundaries MS and MB gives a 1d gapped boundary of the (1+1)d TQFT TS upon folding. Applying this theorem to the algebra of surfaces AS and AbS and using (3.10), we find that the fusion 1-categories of lines bounding surfaces in AS a…
Figure 18
Figure 18. Figure 18: Gauging surfaces introduces new junctions between simple lines. form a junction on the surface AS. Suppose Na AS(a) is the dimension of the vector space of local operators at the junction of AS with the two a lines. After gauging AS, the line a hosts this vector space…
Figure 19
Figure 19. Figure 19: When B2 ≃ (B1) loc AL is obtained from gauging a line AL ∈ B1, then shrinking a slab of B2 theory within B1 gives a surface created from a network of AL lines. This construction shows that AS ≃ SAL is the condensation surface created by higher-gauging AL. In fact, we …
Figure 20
Figure 20. Figure 20: We can decompose a gapped domain wall, I, between B1 and B2 in terms of surfaces IeAL and Ie† BL that implement 1-form symmetry gauging corresponding to condensable algebras AL in B1 and BL in B2, respectively. These surfaces sandwich a third surface, J, with an inver…
Figure 21
Figure 21. Figure 21: Line operators which admit a junction with the trivial line on the interface I form a condensable algebra that is gauged in B2. algebra AL (i.e., AL is associated with a 1-form symmetry that can be gauged in the full spacetime). Combining this constraint with our disc…
Figure 22
Figure 22. Figure 22: We resolve the I † and I surfaces appearing in AbS ≃ I† ⊗ I in terms of invertible surfaces sandwiched between surfaces implementing 0-form and 1-form symmetry gauging. Now, let us consider the configuration shown in [PITH_FULL_IMAGE:figures/full_fig_p041_22.png]
Figure 19
Figure 19. Figure 19: Then for any SBL , SCL ∈ AbS, BL and CL must braid trivially with themselves and with each other. In Sec. 6, we apply the results of Sec. 3 and Sec. 4 to the gauging of various symmetries of the toric code (or, more precisely, the low energy limit of the toric code) a…
Figure 23
Figure 23. Figure 23: Decomposing a 1-form symmetry gauging into n steps. Now, let us consider a simple condition for a non-invertible B loc AL → B 0-form gauging, with AL a condensable algebra in B, to admit a decomposition into n invertible steps. From the discussion above, we know that …
Figure 24
Figure 24. Figure 24: Decomposing 0-form gauging into n steps From these algebras we have the sequential gauging B loc AL AeS2 −−−→ Bloc AL1 AeS1 −−−→ Bloc AL0 ≃ B . (5.9) Conversely, if we have an increasing sequence of surface algebras as in (5.8), then Claim 5.2 again applies, and the s…
Figure 25
Figure 25. Figure 25: The fusion rules of topological local operators on AS is given by the vertical fusion of lines x1, x2 ∈ AL that can end on both I and I † . The trivalent junctions are evaluated using the multiplication of the algebra AL. be the multiplication of this algebra. It is c…
Figure 26
Figure 26. Figure 26: Condition for the existence of a junction of the line a with a condensation surface. We resolve the LHS of [PITH_FULL_IMAGE:figures/full_fig_p084_26.png]
Figure 27
Figure 27. Figure 27: Working out the hom-space condition. References [1] N. Carqueville, I. Runkel & G. Schaumann, “Orbifolds of Reshetikhin-Turaev TQFTs”, Theor. Appl. Categor. 35, 513 (2020), arXiv:1809.01483 [math.QA] [2] S. X. Cui, M. S. Zini & Z. Wang, “On generalized symmetries and …

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.