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Non-invertible Symmetry Boundaries from Anyon Condensation: A Computational Classification

Adam Bazih

Published May 20, 2026• 0 citations
Publisher

Abstract

[NOTE, June 2026: Computations in this paper used mtctools <= 0.2.x, whose condensable-algebra finder has since been found to have false negatives on conformal-embedding (E-type) algebras (e.g. 1+(j=3) in SU(2)_10, within the level range covered here), and whose condensed-phase identification did not resolve the sixteen-fold way. The boundary counts below are therefore lower bounds where E-type algebras were missed; re-check against mtctools 0.3.0 and the corrected release at DOI 10.5281/zenodo.20618695. The text below is the original and is retained for the record.] We develop a systematic computational framework connecting anyon condensation in modular tensor categories to the classification of non-invertible symmetry boundaries in the Symmetry Topological Field Theory (SymTFT) paradigm. For each condensable algebra in a doubled MTC, we identify the corresponding symmetry boundary condition and classify the resulting non-invertible symmetry structure. We present explicit results for SU(2) Chern-Simons theories at levels k=1-8, SU(3) at levels k=1-3, and G₂ at level 1, covering 40+ modular tensor categories and identifying 100+ distinct symmetry boundary types.

DOI
10.5281/zenodo.20302896

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[NOTE, June 2026: Computations in this paper used mtctools <= 0.2.x, whose condensable-algebra finder has since been found to have false negatives on conformal-embedding (E-type) algebras, and whose condensed-phase identification did not resolve the sixteen-fold way (a D^2 = 4 condensed phase is determined only together with the central charge mod 8). Affected classifications should be re-checked against mtctools 0.3.0 and the corrected release at DOI 10.5281/zenodo.20618695. The text below is the original and is retained for the record.] We establish rigorous obstructions to fermionic anyon condensation in product modular tensor categories and analyze the lattice structure of condensable algebras. Our key results include: (1) a proof that fermionic condensation is impossible in SU(2)_k ⊗ SU(2)_k for all k ≤ 8, and provide the mechanism via conformal weight analysis; (2) computation of the full condensation lattice (partial order under inclusion) for SU(2) levels 1-8, revealing rich combinatorial structure; (3) identification of maximal condensable algebras as Lagrangian when they exist. These results contribute to the classification program for topological phases of matter and have implications for the construction of fermionic topological orders from bosonic ones.

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[CORRECTION NOTE, June 2026: The central uniqueness claim of this preprint is corrected. The object 1 + ((1,1);1) (d_A = 4) is NOT an algebra: its generator fails fusion closure. The correct classification has exactly two non-trivial condensable algebras, the Z3 simple-current algebra (d_A = 3) and the maximal algebra (d_A = 6), and the maximal condensation yields the nu=11 Ising-type category of the sixteen-fold way - the modular tensor category of SO(11)_1 (h_sigma = 11/16) - which shares its fusion rules with Ising and its fusion category with SU(2)_2 but is equivalent to neither as a braided category. See the corrected release at DOI 10.5281/zenodo.20618695 (concept 10.5281/zenodo.20128951). A corrected manuscript is under journal review. The text below is the original and is retained for the record.] We give a complete analysis of anyon condensation in the rank-30 modular tensor category C = SU(3)_3 ⊗ SU(2)_2. We prove that A = 1 ⊕ ((1,1);1) is the unique non-trivial condensable algebra by classifying all physical modular invariants. The local module category C^loc_A is computed explicitly: it is a rank-3 MTC with quantum dimensions (1,1,√2), total quantum dimension D² = 4, and Ising fusion rules. The F-symbol F^{σσσ}_σ = (1/√2)[[1,1],[1,-1]] is verified from SU(2)_2 quantum 6j-symbols, confirming C^loc_A ≅ Ising ≅ SU(2)_2 as modular tensor categories.

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A candidate reading of seven recent preprints from cond-mat.str-el and physics.atom-ph suggests that the classical toolkit for classifying gapped phases and their transitions—built on Landau symmetry breaking, local order parameters, and invertible group symmetries—requires systematic extension at the level of microscopic algebraic structure, not merely at the phenomenological level. The emerging picture, drawn from the corpus, is that *generalized symmetries* (non-invertible, categorical, and higher-form) impose precise algebraic constraints on what constitutes a valid order parameter, what boundary conditions are permissible, and which phase transitions can occur without any form of spontaneous symmetry breaking, even after arbitrary gauging. Concretely: (i) condensable algebras in 2+1d topological orders classify gapped boundaries, and twin algebras sharing identical anyon content but inequivalent algebra structure describe physically distinct phases with isomorphic ground-state degeneracy but inequivalent order parameters [corpus:arxiv:2605.31602]; (ii) these twin phases can undergo direct transitions that are provably beyond-Landau even after partial gauging [corpus:arxiv:2605.31601]; (iii) string order parameters in 1d systems organize into Lagrangian algebras in the Drinfel'd center of the symmetry category, making the fusion rule of the symmetry category—not just its representation theory—the operative diagnostic [corpus:arxiv:2605.27193]; (iv) non-invertible symmetry-breaking transitions can be mapped via generalized gauging to deconfined quantum critical points, and vice versa, establishing a precise duality [corpus:arxiv:2605.27672]; and (v) non-invertible symmetry enriched topological orders (NI-SETOs) can be constructed from string-net models, extending symmetry enrichment beyond the group-theoretic setting [corpus:arxiv:2605.28794]. Two additional corpus entries—a Rydberg spin-chain study [corpus:arxiv:2605.27166] and a topological lattice gauge theory paper [corpus:arxiv:2605.28688]—provide microscopic anchoring and structural context, respectively. The central falsification path is explicit: if a twin-phase transition can be shown to spontaneously break some hidden symmetry after arbitrary gauging, the framework collapses. Concrete tensor-network (DMRG/MPS) constructions of twin-phase order parameters in 1+1d would constitute a direct test. --- Authorship: Saluca Agentic AI Research Team (Saluca LLC). AI-drafted from arXiv preprint corpus on the date in the filename. Cited arXiv preprints: 2605.27166, 2605.27193, 2605.27672, 2605.28688, 2605.28794, 2605.29745, 2605.29875, 2605.30421, 2605.31185, 2605.31540, 2605.31601, 2605.31602

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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.

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Anyon Condensation in Modular Tensor Categories: Computational Classification and Algebraic Structure (v4)

Adam Bazih

Systematic computational survey of condensable algebras in product MTCs of the form SU(N1)_k1 x SU(N2)_k2. v2: corrected notation, fixed references, corrected quantum dimensions.

•May 12, 2026•0 citations
[NOTE, June 2026: Computations in this paper used mtctools <= 0.2.x, whose condensable-algebra finder has since been found to have false negatives on conformal-embedding (E-type) algebras (e.g. 1+(j=3) in SU(2)_10, within the level range covered here), and whose condensed-phase identification did not resolve the sixteen-fold way. The boundary counts below are therefore lower bounds where E-type algebras were missed; re-check against mtctools 0.3.0 and the corrected release at DOI 10.5281/zenodo.20618695. The text below is the original and is retained for the record.] We develop a systematic computational framework connecting anyon condensation in modular tensor categories to the classification of non-invertible symmetry boundaries in the Symmetry Topological Field Theory (SymTFT) paradigm. For each condensable algebra in a doubled MTC, we identify the corresponding symmetry boundary condition and classify the resulting non-invertible symmetry structure. We present explicit results for SU(2) Chern-Simons theories at levels k=1-8, SU(3) at levels k=1-3, and G₂ at level 1, covering 40+ modular tensor categories and identifying 100+ distinct symmetry boundary types.
[NOTE, June 2026: Computations in this paper used mtctools <= 0.2.x, whose condensable-algebra finder has since been found to have false negatives on conformal-embedding (E-type) algebras, and whose condensed-phase identification did not resolve the sixteen-fold way (a D^2 = 4 condensed phase is determined only together with the central charge mod 8). Affected classifications should be re-checked against mtctools 0.3.0 and the corrected release at DOI 10.5281/zenodo.20618695. The text below is the original and is retained for the record.] We establish rigorous obstructions to fermionic anyon condensation in product modular tensor categories and analyze the lattice structure of condensable algebras. Our key results include: (1) a proof that fermionic condensation is impossible in SU(2)_k ⊗ SU(2)_k for all k ≤ 8, and provide the mechanism via conformal weight analysis; (2) computation of the full condensation lattice (partial order under inclusion) for SU(2) levels 1-8, revealing rich combinatorial structure; (3) identification of maximal condensable algebras as Lagrangian when they exist. These results contribute to the classification program for topological phases of matter and have implications for the construction of fermionic topological orders from bosonic ones.
[CORRECTION NOTE, June 2026: The central uniqueness claim of this preprint is corrected. The object 1 + ((1,1);1) (d_A = 4) is NOT an algebra: its generator fails fusion closure. The correct classification has exactly two non-trivial condensable algebras, the Z3 simple-current algebra (d_A = 3) and the maximal algebra (d_A = 6), and the maximal condensation yields the nu=11 Ising-type category of the sixteen-fold way - the modular tensor category of SO(11)_1 (h_sigma = 11/16) - which shares its fusion rules with Ising and its fusion category with SU(2)_2 but is equivalent to neither as a braided category. See the corrected release at DOI 10.5281/zenodo.20618695 (concept 10.5281/zenodo.20128951). A corrected manuscript is under journal review. The text below is the original and is retained for the record.] We give a complete analysis of anyon condensation in the rank-30 modular tensor category C = SU(3)_3 ⊗ SU(2)_2. We prove that A = 1 ⊕ ((1,1);1) is the unique non-trivial condensable algebra by classifying all physical modular invariants. The local module category C^loc_A is computed explicitly: it is a rank-3 MTC with quantum dimensions (1,1,√2), total quantum dimension D² = 4, and Ising fusion rules. The F-symbol F^{σσσ}_σ = (1/√2)[[1,1],[1,-1]] is verified from SU(2)_2 quantum 6j-symbols, confirming C^loc_A ≅ Ising ≅ SU(2)_2 as modular tensor categories.
We present a systematic computational survey of condensable algebras (connected étale algebras) in product modular tensor categories of the form SU(N₁)_{k₁} ⊗ SU(N₂)_{k₂} at small rank. Using the Verlinde formula and modular invariant classification, we identify all non-trivial condensable algebras across 10 product MTCs with rank ≤ 36. We verify the Shimizu-Yadav conditions ensuring that the local module categories are modular, and compute the condensation completion structure including domain walls and defects using the Décoppet-Yu framework. Our survey reveals a rich landscape: simple current algebras of types Z₂, Z₃, and Z₂×Z₂ dominate, with condensed phases including the Ising MTC. We also correct an error in a previous classification of SU(3)₃ ⊗ SU(2)₂, where the maximal condensable algebra has d_A = 6 (not 4), yielding the Ising MTC as the condensed phase. All computational code is included.
A candidate reading of seven recent preprints from cond-mat.str-el and physics.atom-ph suggests that the classical toolkit for classifying gapped phases and their transitions—built on Landau symmetry breaking, local order parameters, and invertible group symmetries—requires systematic extension at the level of microscopic algebraic structure, not merely at the phenomenological level. The emerging picture, drawn from the corpus, is that *generalized symmetries* (non-invertible, categorical, and higher-form) impose precise algebraic constraints on what constitutes a valid order parameter, what boundary conditions are permissible, and which phase transitions can occur without any form of spontaneous symmetry breaking, even after arbitrary gauging. Concretely: (i) condensable algebras in 2+1d topological orders classify gapped boundaries, and twin algebras sharing identical anyon content but inequivalent algebra structure describe physically distinct phases with isomorphic ground-state degeneracy but inequivalent order parameters [corpus:arxiv:2605.31602]; (ii) these twin phases can undergo direct transitions that are provably beyond-Landau even after partial gauging [corpus:arxiv:2605.31601]; (iii) string order parameters in 1d systems organize into Lagrangian algebras in the Drinfel'd center of the symmetry category, making the fusion rule of the symmetry category—not just its representation theory—the operative diagnostic [corpus:arxiv:2605.27193]; (iv) non-invertible symmetry-breaking transitions can be mapped via generalized gauging to deconfined quantum critical points, and vice versa, establishing a precise duality [corpus:arxiv:2605.27672]; and (v) non-invertible symmetry enriched topological orders (NI-SETOs) can be constructed from string-net models, extending symmetry enrichment beyond the group-theoretic setting [corpus:arxiv:2605.28794]. Two additional corpus entries—a Rydberg spin-chain study [corpus:arxiv:2605.27166] and a topological lattice gauge theory paper [corpus:arxiv:2605.28688]—provide microscopic anchoring and structural context, respectively. The central falsification path is explicit: if a twin-phase transition can be shown to spontaneously break some hidden symmetry after arbitrary gauging, the framework collapses. Concrete tensor-network (DMRG/MPS) constructions of twin-phase order parameters in 1+1d would constitute a direct test. --- Authorship: Saluca Agentic AI Research Team (Saluca LLC). AI-drafted from arXiv preprint corpus on the date in the filename. Cited arXiv preprints: 2605.27166, 2605.27193, 2605.27672, 2605.28688, 2605.28794, 2605.29745, 2605.29875, 2605.30421, 2605.31185, 2605.31540, 2605.31601, 2605.31602
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(S3)D(S_3)D(S3​) and SU(2)kSU(2)_kSU(2)k​ Chern-Simons theories, and extend it to anomalous anyons.