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REVIEW 3 major objections 3 minor 66 references

Proliferation Transitions for Non-Abelian Anyons

T0 review · 3 major / 3 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Non-abelian anyon condensation becomes one boundary Higgsing step.

desk verdict The SymTFT framework for proliferation transitions is genuinely new and mostly solid for types (i) and (ii), but the abstract's 'arbitrary condensable algebra' outruns the delivery: the type (iii) terminal step is an explicit conjecture, and the paper says so in the text. read the letter →

arxiv 2608.12303 v1 pith:MWXYETLJ submitted 2026-08-12 cond-mat.str-el hep-thmath.QAquant-ph

Sakura Schafer-Nameki , Yunqin Zheng , Andrea Antinucci This is my paper · ORCID
classification cond-mat.str-elhep-thmath.QAquant-ph
keywords anyoncondensationproliferationtransitionSymTFTnon-abeliananyonsmodulartensorcategoryMügercenterboundaryHiggsinganomalous
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 any anyon condensation $T\to T/A$ in a 2+1d topological order can be realized as a genuine phase transition, including when the condensed anyons are non-abelian. The construction embeds $T$ in a 3+1d symmetry topological field theory (SymTFT) whose bulk is fixed by the transparent lines of the subcategory generated by the condensed anyons; when those transparent lines are bosonic they form $\mathrm{Rep}(G)$ for a finite group $G$. The transition is driven entirely on one boundary by scalar fields whose vacuum expectation values change the boundary condition, while the opposite boundary stays fixed as the input theory. If correct, this is the first systematic prescription for proliferation transitions for arbitrary condensable algebras: it reproduces the known abelian case, and produces explicit field theories for non-abelian examples such as $D(S_3)$ and $SU(2)_k$ theories.

What carries the argument

The load-bearing object is the Müger center $E_A=Z_2(\langle A\rangle)\cong\mathrm{Rep}(G)$ of the category generated by the condensable algebra; by Deligne's theorem it is $\mathrm{Rep}(G)$ whenever all transparent lines are bosons. This fixes the SymTFT bulk as 3+1d $G$ gauge theory, with the anyons of $A$ placed on a symmetry boundary stacked with $M_A=\langle A\rangle/\mathrm{Rep}(G)$, and the physical boundary stacked with $N_A=\langle A\rangle'/\mathrm{Rep}(G)$. The mechanism driving the transition is boundary Higgsing: scalar fields on the symmetry boundary terminate the Wilson lines of $A$, and their expectation values change the boundary condition from Neumann-type to Dirichlet-type (possibly with residual $M_A/\hat A$ and $H$ stacking). Algebraically, the transition is the replacement in eq. (3.35), combining a Landau-Ginzburg theory $S^{\mathrm{LG}}_{H\subset G}$ with a smaller transition $S_{M_A\to M_A/\hat A}$ that can be defined recursively until it terminates in type (i) or type (iii).

What would settle it

A concrete check: for $T=D(S_3)$ and $A=1\oplus 1_-$, the recipe predicts the $\mathbb{Z}_2$-gauged Ising ($\mathrm{Ising}^*$) transition; building a lattice model with the proposed boundary scalars and measuring critical exponents would confirm or refute that prediction, and the same comparison can be made for the $S_3$ Lagrangian algebras, where the transition is predicted to be first order.

Watch

Extended reading notes

Core claim

For a condensable algebra $A$ in a modular tensor category $T$, let $\langle A\rangle$ be the fusion subcategory that $A$ generates and $E_A=Z_2(\langle A\rangle)\cong\mathrm{Rep}(G)$ its Müger center. The central claim is that both $T$ and $T/A$ admit SymTFT sandwiches in 3+1d $G$ gauge theory with the same physical boundary, and that the proliferation transition $T\to T/A$ is the boundary phase transition in which the symmetry boundary is replaced by $(B_{\mathrm{Dir}}\boxtimes S_{M_A\to M_A/\hat A}\boxtimes S^{\mathrm{LG}}_{H\subset G})/G^{(0)}$, eq. (3.35), where $M_A=\langle A\rangle/\mathrm{Rep}(G)$, $N_A=\langle A\rangle'/\mathrm{Rep}(G)$, $\hat A$ is the image of $A$ in $M_A$, and $H$ is the stabilizer associated with the transparent part $A\cap E_A=\mathrm{Fun}(G/H)$. The endpoint is $T/A=(M_A/\hat A)\boxtimes N_A/H^{(0)}$. The claim covers abelian and non-abelian anyons, with algebras of Tannakian, mixed, and modular type; worked examples include every condensable algebra of $D(S_3)$, $SU(2)_k$ for $4|k$, the doubles $SU(2)_k\boxtimes SU(2)_{-k}$, and a canonical extension to anomalous anyons that cannot be gauged inside $T$ alone.

Load-bearing premise

The construction assumes that the transparent lines in the subcategory generated by the anyons being condensed are all bosons, so they form $\mathrm{Rep}(G)$ for a finite group $G$; if a transparent fermion appears, the bulk is no longer ordinary $G$ gauge theory and the stated boundary recipe does not directly apply.

Editorial extensions

If this is right

  • Every condensable algebra in a modular tensor category acquires a boundary-Higgsing transition whose two sides are exactly $T$ and $T/A$, so anyon condensation is realized as the long-distance limit of a local phase transition rather than only a topological operation.
  • For type (i) algebras the transition is a Landau-Ginzburg$^*$ model for $G\to H$; in $D(S_3)$, the $\mathrm{Rep}(\mathbb{Z}_2)$ transition is the $\mathbb{Z}_2$-gauged Ising transition and the Lagrangian algebras give the $S_3$-gauged Landau-Potts model.
  • For $SU(2)_k$ with $4|k$, the $A_D$ transition $SU(2)_k\to SO(3)_{k/2}$ falls in the Ising$^*$ universality class, and the diagonal Lagrangian algebra of $SU(2)_k\boxtimes SU(2)_{-k}$ is realized by condensing a bifundamental scalar, recovering the known self-dual Higgs transition.
  • Anomalous anyons that cannot be gauged inside $T$ can still be proliferated by stacking a canonically chosen minimal TQFT $M_X$ and gauging the diagonal; the endpoint is $N_X=\langle X\rangle'/E_X$, generalizing the abelian minimal-theory construction.
  • Type (ii) transitions decompose recursively into an LG$^*$ step followed by a strictly smaller proliferation, so the construction terminates after finitely many steps.

Reading between the lines

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

  • Extension: the predicted universality classes are directly testable in lattice realizations of the $D(S_3)$ and $SU(2)_8$ transitions; finite-size scaling of the order parameter would distinguish the Ising$^*$ and first-order predictions from generic first-order behavior.
  • Extension: if the recursive picture holds, any multi-step anyon condensation chain can be organized into sequential boundary Higgsings, which suggests a systematic Landau-Ginzburg potential for nested condensations rather than a case-by-case construction.
  • Extension: the transparent-fermion cases explicitly left out (for example $SU(2)_{10}$ with the $E_6$ algebra and the $k\equiv 2 \bmod 4$ doubles) are the natural next test; a spin-SymTFT generalization would likely need the super-Tannakian analogue of the Rep(G) identification as its input.
  • Extension: for anomalous $X$, the paper's factorization of any anomaly-cancelling theory into $M_X\boxtimes X'$ implies a form of uniqueness for the minimal auxiliary sector; classifying all such $X$ would show whether the endpoint depends only on $M_X$.
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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

3 major / 3 minor

Summary. The paper proposes a general SymTFT construction of proliferation phase transitions T → T/A for a condensable algebra A in a modular tensor category T. For each A the authors define the generated subcategory ⟨A⟩ and its Müger center EA ≅ Rep(G), take the bulk to be 3+1d G gauge theory, and build non-minimal boundaries from MA = ⟨A⟩/Rep(G) and NA = ⟨A⟩'/Rep(G). The transition is realized on the symmetry boundary by scalars that Higgs Rep(G) down to Rep(H) and condense the image â of A; the claimed endpoint is T/A = (MA/â) ⊠ NA/H^(0). The identities (3.23), (3.30) and (3.32) are supported by a proof in Appendix B and are checked in abelian examples, D(S3), SU(2)_k and its double, including a canonical treatment of anomalous anyons with a minimal auxiliary TQFT. Three algebra types are distinguished: type (i) is fully constructed as an LG* transition, type (ii) is reduced recursively, and type (iii) relies on an explicitly acknowledged, unproven scalar-per-generator proposal.

Significance. If the type (iii) gap is closed, the paper would provide the first systematic construction of proliferation transitions for non-abelian anyons, with no free parameters and with endpoint formulas derived from Müger’s and Deligne’s theorems rather than from any fitting. The clean separation of bulk and boundary data, the explicit treatment of non-minimal boundary conditions, and the worked D(S3) and SU(2)_k examples are valuable and make the proposal falsifiable. However, as stated, the main theorem is not fully established for type (iii), and the abstract’s unqualified “arbitrary condensable algebra” overstates the Tannakian-boson restriction. These are load-bearing caveats in an otherwise well-constructed framework.

major comments (3)
  1. [Section 3.3, Eqs. (3.34)–(3.35)] The factor S_{M_A → M_A/â} in the general transition theory is not constructed. The paper itself states, “We are not aware of a universal construction for S_{M_A → M_A/â},” and the subsequent scalar-per-generator proposal is a conjecture, not a theorem. This is load-bearing because for type (iii) algebras, where EA = Vec and G = 1, the LG* factor is absent and (3.35) reduces to exactly this unknown factor. No direct type (iii) example is treated by the proposed algorithm: the two D(S3) type (iii) algebras are instead obtained by the two-step condensation (5.20). Thus the central claim that (3.35) realizes T → T/A for every condensable algebra is established only modulo an open terminal step.
  2. [Section 6.1, k=28] The required transition SO(3)_14 → (G2)_1 is deferred with “We leave this type (iii) transition for future exploration.” This is a load-bearing omission because the E8 algebra at k=28 lies inside the paper’s main Tannakian scope (its Müger center is Rep(Z2)), and the transition theory (3.35) cannot be completed for this example without the missing S factor. The example therefore does not currently demonstrate the advertised construction, and it leaves the type (iii) protocol without a single direct bosonic test case.
  3. [Abstract and Section 2, footnote 18] The paper claims to treat arbitrary condensable algebras, but the general construction assumes a Tannakian Müger center EA = Rep(G), excluding super-Tannakian centers Rep(G,z) with transparent fermions. This restriction is stated honestly, but it contradicts the unqualified “arbitrary” claim and it can affect intermediate steps of the recursive type (ii) protocol if a substep lands on a super-Tannakian center. The claims should be re-scoped to condensable algebras with Tannakian Müger center, or the super-Tannakian cases should be treated.
minor comments (3)
  1. [Section 5, Eq. (5.4)] Equation (5.4) writes the transition as P_{D(S3)→D(Z2)}, but the preceding line identifies the endpoint as D(Z3); please correct this inconsistency.
  2. [Section 4] In the sentence “This is preciselt in the general discussion,” the word “preciselt” should be “precisely.”
  3. [Table I] The transition-theory row for type (iii) lists “One scalar per generating anyon” without indicating that this is a proposal rather than a derived result; adding an explicit marker would prevent readers from over-counting the degree of proof.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the T/A endpoint is derived from Müger/condensation-transitivity and checked on known examples; the type (iii) terminal transition is an admitted gap, not a circular reduction.

full rationale

The paper's derivation chain is self-contained rather than circular. The endpoint formula T/A = (M_A/hat A) ⊖ N_A / H^(0), displayed as (3.30) and proved in Appendix B, is obtained from condensation transitivity (2.3), Müger's dimension/centralizer theorem (Theorem 2.4), and the properties of the tensor functor F; it is not assumed as an input. The factorization T = (M_A ⊖ N_A)/G^(0) in (3.18) is likewise derived from Müger's theorem, not from the target transition. The transition theory (3.35) is a compositional/recursive construction: the LG* factor S^LG_{H⊆G} is an independently known Landau-Ginzburg model, and the factor S_{M_A→M_A/hat A} is explicitly identified as the same problem for a smaller pair, with the paper admitting no universal construction and proposing a scalar-per-generating-anyon ansatz in the terminal type (iii) case. That is a completeness gap, explicitly flagged in Section 3.3 and in the deferred SU(2)_28 example in Section 6.1, but it is not a hidden self-reference or fitted-input-called-prediction. The construction is checked against external/known benchmarks (abelian U(1)_8 → U(1)_2 Ising*, D(S_3) gauged endpoints, SU(2)_k quotients), and the self-cited SymTFT framework is supported by external references [54,55]; no load-bearing step reduces equation to equation by definition. Score 0 is therefore appropriate on the circularity scale.

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

The construction introduces no new fundamental entities: the boundary scalar fields are standard dynamical fields, and the auxiliary TQFT in the anomalous-anyon section is a derived object built from the Müger center data. No numbers are fitted to data. The main assumptions are standard mathematical theorems and the stated restriction to bosonic transparent lines.

assumptions (5)
  • domain assumption The topological order T is described by a unitary modular tensor category of anyons.
    Throughout the paper, T is taken to be a unitary MTC; this is the standard mathematical model for 2+1d topological order. Assumed in the introduction and section 2.
  • standard math Deligne's theorem: a Tannakian braided fusion category with trivial twists is equivalent to Rep(G) for a finite group G.
    Used to identify the Müger center E_A as Rep(G), equation (2.8), and to determine the SymTFT bulk as G gauge theory. Cited to [17].
  • standard math Müger's dimension and centralizer theorems: dim(B)dim(B') = dim(T), B'' = B, and B ⊠ B' = T when Z2(B) = Vec.
    Theorem 2.4 is the backbone of the factorization (3.18) and the derivation of the SymTFT sandwich. Cited to [27].
  • domain assumption Condensation transitivity: for condensable algebras A ⊆ B, B/A is condensable in T/A and T/A/(B/A) = T/B.
    Used in equation (2.3) and in the multi-step transitions in sections 5 and 6. Cited to [30].
  • ad hoc to paper The Müger center E_A is Tannakian, i.e. all transparent lines are bosons, not fermions.
    The paper explicitly restricts to transparent bosons 'for clarity of exposition' (footnote 18 and section 2), excluding super-Tannakian Müger centers. This assumption is load-bearing because the SymTFT bulk is then the ordinary G gauge theory.

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Cite this review

Pith. "Pith review of Proliferation Transitions for Non-Abelian Anyons." pith.science (2026). https://pith.science/paper/MWXYETLJ

@misc{pith2026260812303,
  author       = {Pith},
  title        = {Pith review of: Proliferation Transitions for Non-Abelian Anyons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MWXYETLJ}},
  note         = {Machine review of arXiv:2608.12303}
}
abstract

We construct phase transitions that proliferate condensable anyons in general 2+1d topological orders, including non-abelian ones. The central tool that provides a systematic approach to this question is the Symmetry Topological Field Theory (SymTFT). For a given topological order, we identify the relevant symmetry from the transparent lines generated by the condensable anyons, and thereby realize the topological order in terms of a 3+1d SymTFT sandwich. The proliferation phase transition is realized by coupling scalar fields to the anyons purely on the symmetry boundary of the SymTFT. We illustrate the construction for abelian theories, as well as non-abelian ones, $D(S_3)$ and $SU(2)_k$ Chern-Simons theories, and extend it to anomalous anyons.

Figures

Figures reproduced from arXiv: 2608.12303 by the authors.

Figure 1
Figure 1. The condensable algebras of D(S3), partially or￾dered by the subalgebra relation. The cyan ones form subcat￾egories, orange ones have a generated sub-category Rep(G), the uncolored are mixed algebras of type (iii). A type ⟨A⟩ ϕ H T /A 1 ⊕ 1− (i) A = Rep(Z2) ϕ− 1 D(Z3) 1 ⊕ E (i) Rep(S3) e ϕE Z2 D(Z2) 1 ⊕ [a] (i) Rep(S3) m ϕ[a] Z ′ 2 D(Z2) 1 ⊕ 1− ⊕ 2E (i) A = Rep(S3) e (ϕE, ϕ−) 1 Vec 1 ⊕ 1− ⊕ 2[a] (i) A = Rep(S3) m (ϕ… view at source ↗
Figure 2
Figure 2. The Hasse diagrams of condensable algebras of [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

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Works this paper leans on

66 extracted references · 33 canonical work pages

  1. [1]

    We refer to both the topological line and its endpoint as an anyon, when it does not lead to confusion

    work page

  2. [2]

    ForD(S 3) the pure charges are irreps: the trivial 1, the sign 1 −, and the 2d irrepE

    However, a gen- eral analysis for any finiteGwhere all possible condens- able algebras are known and the associated generated subcategories can be explicitly determined will appear in [38]. ForD(S 3) the pure charges are irreps: the trivial 1, the sign 1 −, and the 2d irrepE. The fluxes are the conjugacy classes [a] and [b]. The condensable algebrasA are ...

    work page

  3. [3]

    The first step is to realizeTas a SymTFT for a suitable symmetry that then allows proliferation of the anyons inA

    PROLIFERA TION TRANSITIONS FROM THE SYMTFT Our starting point is an MTCTand a condensable al- gebraAinT. The first step is to realizeTas a SymTFT for a suitable symmetry that then allows proliferation of the anyons inA. We give a quick crashcourse on the SymTFT focusing on 2+1d theories, in appendix A. In section 3.1 we give a more physical description of...

    work page

  4. [4]

    This case has been studied also in [2], and so serves as a point of comparison

    PROLIFERA TING ABELIAN ANYONS We first apply our general proposal to abelian anyons, with no anomaly – the anomalous case will be discussed later. This case has been studied also in [2], and so serves as a point of comparison. For abelian TOs, the condensable algebrasAare finite abelian groups, and so A∼= Rep(A), withA ∼= Zn1×···×Z ns decomposed into ssim...

    work page

  5. [5]

    We will here focus onG=S 3 =Z a 3 ⋊Zb

    PROLIFERA TING NON-ABELIAN ANYONS: T=D(S 3) Group quantum doublesD(G) for non-abelian finite groupsGare the simplest examples of non-abelian TOs. We will here focus onG=S 3 =Z a 3 ⋊Zb

    work page

  6. [6]

    PROLIFERA TING NON-ABELIAN ANYONS: T=SU(2) k AND ITS DOUBLE The Chern-Simons theoriesSU(2) k and their doubles Tk =SU(2) k ⊠SU(2) −k realize all three types of con- densable algebras (i)-(iii), and is thus a very suitable class of examples. Furthermore, which type of algebra occurs depends onkin a very simple way: The anyons ofSU(2) k are the isospins (j)...

    work page

  7. [7]

    (6.49) Again the phase transitionS MA→Vec is realized by cou- pling a scalar in the bi-vector, i.e

    ⊠SMA→Vec ⊠S LG Z1⊂Z2 2 (Z2×Z 2)(0) Z(Z2 2) BNeu(Z2 2) ! . (6.49) Again the phase transitionS MA→Vec is realized by cou- pling a scalar in the bi-vector, i.e. (1,1), representation ofSO(3)×SO(3), and condense it. 6.2.5. Discussion Transparent fermions.Fork≡2 mod 4, consider Aad. The anyonsaandbare fermions, which becomes part of the M¨ uger centerEA =Rep(Z...

    work page

  8. [8]

    An anomalous 1-form symmetry ofTcannot be gauged withinTby itself

    PROLIFERA TING ANOMALOUS ANYONS So far we focused on condensable algebrasA, which by definition are non-anomalous. An anomalous 1-form symmetry ofTcannot be gauged withinTby itself. It can nevertheless be combined or stacked with an auxil- iary TQFTXcarrying the opposite anomaly, after which an anomaly-free diagonal can be gauged. The choice of Xis arbitr...

    work page

Show all 66 references
  1. [9]

    transparency

    CONCLUSIONS AND OUTLOOK We made a proposal for proliferation transitionsTto T/Afor any condensable algebraAin an MTCT. The key new insight is to utilize the SymTFT to address this question systematically and in generality such that it is applicable to any MTCT, including non-a...

  2. [10]

    Proliferation transitions from a topological phase in 2 + 1 dimensions,

    M. Cheng and N. Seiberg, “Proliferation transitions from a topological phase in 2 + 1 dimensions,” arXiv:2603.00245 [cond-mat.str-el]

    arXiv

  3. [11]

    Taming the Conformal Zoo,

    G. W. Moore and N. Seiberg, “Taming the Conformal Zoo,”Phys. Lett. B220(1989) 422–430

    1989

  4. [12]

    Comments on 23 One-Form Global Symmetries and Their Gauging in 3d and 4d,

    P.-S. Hsin, H. T. Lam, and N. Seiberg, “Comments on 23 One-Form Global Symmetries and Their Gauging in 3d and 4d,”SciPost Phys.6no. 3, (2019) 039, arXiv:1812.04716 [hep-th]

    2019 arXiv

  5. [13]

    See [2] for details

    To be precise, one also need to deform the theory by monopole operators. See [2] for details

  6. [14]

    Condensability ofAis the statement that this symmetry is non-anomalous, i.e

    Anyons are topological line operators, and the lines in Agenerate a symmetry acting on the other anyons by braiding: a 1-form symmetry, non-invertible as soon as a constituent has quantum dimension larger than one. Condensability ofAis the statement that this symmetry is non-a...

  7. [15]

    Hierarchy construction for non-Abelian fractional quantum Hall states via anyon condensation,

    C. Zhang, A. Vishwanath, and X.-G. Wen, “Hierarchy construction for non-Abelian fractional quantum Hall states via anyon condensation,”Phys. Rev. B112 no. 12, (2025) 125116,arXiv:2406.12068 [cond-mat.str-el]

    2025 arXiv

  8. [16]

    Self-dual Higgs transitions: Toric code and beyond,

    W. Ji, R. A. Lanzetta, Z. Zhou, and C. Wang, “Self-dual Higgs transitions: Toric code and beyond,” arXiv:2601.20945 [cond-mat.str-el]

  9. [17]

    Higgsing transitions from topological field theory and noninvertible symmetry in Chern-Simons matter theories,

    C. Cordova, D. Garc´ ıa-Sep´ ulveda, and K. Ohmori, “Higgsing transitions from topological field theory and noninvertible symmetry in Chern-Simons matter theories,”Phys. Rev. D114no. 2, (2026) 025020, arXiv:2504.03614 [hep-th]

    2026 arXiv

  10. [18]

    Generalized Level-Rank Duality, Holomorphic Conformal Field Theory, and Non-Invertible Anyon Condensation,

    C. Cordova, D. Garc´ ıa-Sep´ ulveda, and J. A. Harvey, “Generalized Level-Rank Duality, Holomorphic Conformal Field Theory, and Non-Invertible Anyon Condensation,”arXiv:2512.24419 [hep-th]

  11. [19]

    Self-dualS 3 gauge theory in 2+1d: lattice model and topological phase transitions,

    D.-C. Lu, C. Wang, and A. Vishwanath, “Self-dualS 3 gauge theory in 2+1d: lattice model and topological phase transitions,”arXiv:2608.05294 [cond-mat.str-el]

    arXiv

  12. [20]

    Generalized charges, part II: Non-invertible symmetries and the symmetry TFT,

    L. Bhardwaj and S. Schafer-Nameki, “Generalized charges, part II: Non-invertible symmetries and the symmetry TFT,”SciPost Phys.19no. 4, (2025) 098, arXiv:2305.17159 [hep-th]

    2025

  13. [21]

    Gapped phases in (2+1)d with non-invertible symmetries: Part I,

    L. Bhardwaj, D. Pajer, S. Schafer-Nameki, A. Tiwari, A. Warman, and J. Wu, “Gapped phases in (2+1)d with non-invertible symmetries: Part I,”SciPost Phys. 19no. 2, (2025) 056,arXiv:2408.05266 [hep-th]

    2025 arXiv

  14. [22]

    String condensation and topological holography for 2+1D gapless SPT,

    R. Wen, “String condensation and topological holography for 2+1D gapless SPT,”arXiv:2408.05801 [cond-mat.str-el]

    arXiv

  15. [23]

    Gapped Phases in (2+1)d with Non-Invertible Symmetries: Part II,

    L. Bhardwaj, S. Schafer-Nameki, A. Tiwari, and A. Warman, “Gapped Phases in (2+1)d with Non-Invertible Symmetries: Part II,” arXiv:2502.20440 [hep-th]

    arXiv

  16. [24]

    We will assume in the general discussion, that the transparent lines are bosons, though the case of transparent fermions can be studied in a similar fashion

  17. [25]

    Cat´ egories tensorielles,

    P. Deligne, “Cat´ egories tensorielles,”Mosc. Math. J.2 (2002) 227–248

    2002

  18. [26]

    We will in this paper not consider this option in detail

    In general the theorem by Deligne states that the category is either Tannakian or super-Tannakian EA =Rep(G,z), wherezis a transparent fermion, namely it commutes with all the other lines ofE A but has spinθ=−1. We will in this paper not consider this option in detail

  19. [27]

    ICTP lectures on (non-)invertible generalized symmetries,

    S. Schafer-Nameki, “ICTP lectures on (non-)invertible generalized symmetries,”Phys. Rept.1063(2024) 1–55, arXiv:2305.18296 [hep-th]

    2024 arXiv

  20. [28]

    What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries,

    S.-H. Shao, “What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries,” inTheoretical Advanced Study Institute in Elementary Particle Physics 2023: Aspects of Symmetry. 8, 2023. arXiv:2308.00747 [hep-th]

    2023 arXiv

  21. [29]

    Gapless Phases in (2+1)d with Non-Invertible Symmetries,

    L. Bhardwaj, Y. Gai, S.-J. Huang, K. Inamura, S. Schafer-Nameki, A. Tiwari, and A. Warman, “Gapless Phases in (2+1)d with Non-Invertible Symmetries,”SciPost Phys.21(2026) 019, arXiv:2503.12699 [cond-mat.str-el]

    2026 arXiv

  22. [30]

    Condensate induced transitions between topologically ordered phases,

    F. A. Bais and J. K. Slingerland, “Condensate induced transitions between topologically ordered phases,”Phys. Rev. B79(2009) 045316,arXiv:0808.0627 [cond-mat.mes-hall]

    2009 arXiv

  23. [31]

    Hasse diagrams for gapless SPT and SSB phases with non-invertible symmetries,

    L. Bhardwaj, D. Pajer, S. Schafer-Nameki, and A. Warman, “Hasse diagrams for gapless SPT and SSB phases with non-invertible symmetries,”SciPost Phys. 19no. 4, (2025) 113,arXiv:2403.00905 [cond-mat.str-el]

    2025

  24. [32]

    Module categories over the Drinfeld double of a finite group,

    V. Ostrik, “Module categories over the Drinfeld double of a finite group,”arXiv Mathematics e-prints(Feb.,

  25. [33]

    The witt group of non-degenerate braided fusion categories,

    A. Davydov, M. M¨ uger, D. Nikshych, and V. Ostrik, “The witt group of non-degenerate braided fusion categories,”arXiv preprint arXiv:1009.2117(2010)

    2010 arXiv

  26. [34]

    On fusion categories,

    P. Etingof, D. Nikshych, and V. Ostrik, “On fusion categories,”Annals of Mathematics(2005) 581–642

    2005

  27. [35]

    Etingof, S

    P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor categories, vol. 205 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2015. https://doi.org/10.1090/surv/205

    2015 doi

  28. [36]

    On the Structure of Modular Categories,

    M. M¨ uger, “On the Structure of Modular Categories,” Proc. Lond. Math. Soc.87no. 2, (2003) 291–308, arXiv:math/0201017

    2003 arXiv

  29. [37]

    Anyon condensation and tensor categories,

    L. Kong, “Anyon condensation and tensor categories,” Nucl. Phys. B886(2014) 436–482,arXiv:1307.8244 [cond-mat.str-el]

    2014 arXiv

  30. [38]

    Twin Phases: Intrinsic Deconfined Quantum Criticality,

    A. Warman, Y. Gai, and S. Schafer-Nameki, “Twin Phases: Intrinsic Deconfined Quantum Criticality,” arXiv:2605.31601 [cond-mat.str-el]

    arXiv

  31. [39]

    On the structure of the Witt group of braided fusion categories,

    A. Davydov, D. Nikshych, and V. Ostrik, “On the structure of the Witt group of braided fusion categories,”Selecta Math. (N.S.)19(2013) 237–269, arXiv:1109.5558 [math.QA]

    2013 arXiv

  32. [40]

    This assumes we only have transparent bosont, whereas there is also the possibility of a transparent fermion, in which this isRep(G,z)

  33. [41]

    Modular extensions of unitary braided fusion categories and 2+ 1d topological/spt orders with symmetries,

    T. Lan, L. Kong, and X.-G. Wen, “Modular extensions of unitary braided fusion categories and 2+ 1d topological/spt orders with symmetries,” Communications in Mathematical Physics351no. 2, (2017) 709–739

    2017

  34. [42]

    Renormalization of the Potts Model,

    D. J. Amit, “Renormalization of the Potts Model,”J. Phys. A9(1976) 1441

    1976

  35. [43]

    Note that this fixes the subgroupHonly up to conjugation

  36. [44]

    For instance, Ising∗ universality class is related to Ising university class by gaugingZ 2

    Following the standard convention, the star meansGis 24 gauged. For instance, Ising∗ universality class is related to Ising university class by gaugingZ 2

  37. [45]

    There is also the possibility to have transparent fermions that requires a separate discussion, which we do not pursue systematically

  38. [46]

    Galois theory for braided tensor categories and the modular closure,

    M. M¨ uger, “Galois theory for braided tensor categories and the modular closure,”Adv. Math.150(1999) 151–201,arXiv:math/9812040

    1999 arXiv

  39. [47]

    Gai and S

    Y. Gai and S. Schafer-Nameki. Proliferation Transitions for Group Quantum Doubles. To appear

  40. [48]

    Multicritical Landau-Potts field theory,

    A. Codello, M. Safari, G. P. Vacca, and O. Zanusso, “Multicritical Landau-Potts field theory,”Phys. Rev. D 102no. 12, (2020) 125024,arXiv:2010.09757 [cond-mat.stat-mech]

    2020 arXiv

  41. [49]

    Holographic theory for continuous phase transitions: Emergence and symmetry protection of gaplessness,

    A. Chatterjee and X.-G. Wen, “Holographic theory for continuous phase transitions: Emergence and symmetry protection of gaplessness,”Phys. Rev. B108no. 7, (2023) 075105,arXiv:2205.06244 [cond-mat.str-el]

    2023 arXiv

  42. [50]

    Critical Behavior of the Continuous N Component Potts Model,

    R. K. P. Zia and D. J. Wallace, “Critical Behavior of the Continuous N Component Potts Model,”J. Phys. A 8(1975) 1495–1507

    1975

  43. [51]

    Categorical symmetry and noninvertible anomaly in symmetry-breaking and topological phase transitions,

    W. Ji and X.-G. Wen, “Categorical symmetry and noninvertible anomaly in symmetry-breaking and topological phase transitions,”Phys. Rev. Res.2no. 3, (2020) 033417,arXiv:1912.13492 [cond-mat.str-el]

    2020 arXiv

  44. [52]

    The phase transition in the continuous potts model,

    D. J. Amit and A. Shcherbakov, “The phase transition in the continuous potts model,”Journal of Physics C: Solid State Physics7no. 5, (Mar, 1974) L96

    1974

  45. [53]

    On q-analog of McKay correspondence and ADE classification of sl(2) conformal field theories,

    J. Kirillov, Alexander and V. Ostrik, “On q-analog of McKay correspondence and ADE classification of sl(2) conformal field theories,”Adv. Math.171(2002) 183–227,arXiv:math/0101219

    2002 arXiv

  46. [54]

    Modular invariant partition functions in two dimensions,

    A. Cappelli, C. Itzykson, and J. B. Zuber, “Modular invariant partition functions in two dimensions,”Nucl. Phys. B280(1987) 445–465

    1987

  47. [55]

    Twin Algebras: Condensable Algebras beyond Anyons,

    Y. Gai, S. Schafer-Nameki, and A. Warman, “Twin Algebras: Condensable Algebras beyond Anyons,” arXiv:2605.31602 [cond-mat.str-el]

    arXiv

  48. [56]

    Ordering the topological order in the fractional quantum Hall effect,

    M. Cheng, S. Musser, A. Raz, N. Seiberg, and T. Senthil, “Ordering the topological order in the fractional quantum Hall effect,”Phys. Rev. B113 no. 11, (2026) 115103,arXiv:2505.14767 [cond-mat.str-el]

    2026

  49. [57]

    The holography of non-invertible self-duality symmetries,

    A. Antinucci, F. Benini, C. Copetti, G. Galati, and G. Rizi, “The holography of non-invertible self-duality symmetries,”JHEP03(2025) 052,arXiv:2210.09146 [hep-th]

    2025

  50. [58]

    Cheng and N

    M. Cheng and N. Seiberg. Peeling off Abelian anyons. To appear

  51. [59]

    M. Cheng. Work in Progress

  52. [61]

    Symmetry TFTs from String Theory,

    F. Apruzzi, F. Bonetti, I. Garc´ ıa Etxebarria, S. S. Hosseini, and S. Schafer-Nameki, “Symmetry TFTs from String Theory,”Commun. Math. Phys.402no. 1, (2023) 895–949,arXiv:2112.02092 [hep-th]

    2023 arXiv

  53. [62]

    Topological symmetry in quantum field theory,

    D. S. Freed, G. W. Moore, and C. Teleman, “Topological symmetry in quantum field theory,” arXiv:2209.07471 [hep-th]

    arXiv

  54. [63]

    Towards All Categorical Symmetries in 2+1 Dimensions,

    M. Bullimore and J. J. Pearson, “Towards All Categorical Symmetries in 2+1 Dimensions,” arXiv:2408.13931 [hep-th]

    arXiv

  55. [64]

    Local modules in braided monoidal 2-categories,

    T. D. D´ ecoppet and H. Xu, “Local modules in braided monoidal 2-categories,”J. Math. Phys.65no. 6, (2024) 061702,arXiv:2307.02843 [math.CT]

    2024 arXiv

  56. [65]

    (3+1)-TQFTs and topological insulators,

    K. Walker and Z. Wang, “(3+1)-TQFTs and topological insulators,”Front. Phys. (Beijing)7no. 2, (2012) 150–159

    2012

  57. [66]

    Minimal nondegenerate extensions,

    T. Johnson-Freyd and D. Reutter, “Minimal nondegenerate extensions,”J. Am. Math. Soc.37no. 1, (2023) 81–150,arXiv:2105.15167 [math.QA]

    2023

  58. [2002]

    math/0202130,arXiv:math/0202130 [math.QA]

    arXiv

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Reviewed August 16, 2026 · model on record in the stance chip above.