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

Adam Bazih

Published May 12, 2026• 0 citations
Publisher

Abstract

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.
DOI
10.5281/zenodo.20128951

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Condensable Algebras in Product Modular Tensor Categories: A Comprehensive Survey of SU(N₁)_{k₁} ⊗ SU(N₂)_{k₂}

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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.

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Fermionic Condensation Obstruction and Condensation Lattices in Product Modular Tensor Categories

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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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The local module category of SU(3)_3 ⊗ SU(2)_2: an Ising condensed phase

Adam Bazih

[CORRECTION NOTE, June 2026: This companion paper inherits the corrected classification: the object 1 + ((1,1);1) (d_A = 4) is NOT an algebra (its generator fails fusion closure); the correct algebras are the Z3 simple-current algebra (d_A = 3) and the maximal algebra (d_A = 6). The condensed phase is the nu=11 Ising-type category of the sixteen-fold way - the modular tensor category of SO(11)_1 (h_sigma = 11/16), fixed by the preserved central charge c = 11/2 mod 8 - which shares its fusion rules with Ising and its fusion category with SU(2)_2 but is equivalent to neither as a braided category (the claim 'matching the Ising MTC (equivalently SU(2)_2)' below is corrected accordingly). See DOI 10.5281/zenodo.20618695 (concept 10.5281/zenodo.20128951). The text below is the original and is retained for the record.] We compute the local module category ClocA for the condensable algebra A = 1 ⊕ ((1,1);1) in the product modular tensor category SU(3)3 ⊗ SU(2)2. Starting from the rank-30 parent theory with total quantum dimension D2 = 144, we identify the unique modular invariant Z commuting with the modular S- and T-matrices, extract a 30×3 branching matrix, and compute the condensed S-matrix via Sloc = (1/dA)bTSb. The result is a rank-3 modular tensor category with quantum dimensions (1, 1, √2), total quantum dimension D2 = 4, and modular data matching the Ising MTC (equivalently SU(2)2) to machine precision. The D2 discrepancy between intrinsic (= 4) and extrinsic (= 9) values is resolved by the embedding dimension factor (d0embed)2 = 9/4. This is a companion paper to 'Anyon condensation in SU(3)3 ⊗ SU(2)2: condensable algebras, monodromy spectrum, and the Z6 fusion ring' (Bazih, 2026), resolving its Remark 3.4 on the structure of ClocA.

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

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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 (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.

•May 20, 2026•0 citations
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.
[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: This companion paper inherits the corrected classification: the object 1 + ((1,1);1) (d_A = 4) is NOT an algebra (its generator fails fusion closure); the correct algebras are the Z3 simple-current algebra (d_A = 3) and the maximal algebra (d_A = 6). The condensed phase is the nu=11 Ising-type category of the sixteen-fold way - the modular tensor category of SO(11)_1 (h_sigma = 11/16), fixed by the preserved central charge c = 11/2 mod 8 - which shares its fusion rules with Ising and its fusion category with SU(2)_2 but is equivalent to neither as a braided category (the claim 'matching the Ising MTC (equivalently SU(2)_2)' below is corrected accordingly). See DOI 10.5281/zenodo.20618695 (concept 10.5281/zenodo.20128951). The text below is the original and is retained for the record.] We compute the local module category ClocA for the condensable algebra A = 1 ⊕ ((1,1);1) in the product modular tensor category SU(3)3 ⊗ SU(2)2. Starting from the rank-30 parent theory with total quantum dimension D2 = 144, we identify the unique modular invariant Z commuting with the modular S- and T-matrices, extract a 30×3 branching matrix, and compute the condensed S-matrix via Sloc = (1/dA)bTSb. The result is a rank-3 modular tensor category with quantum dimensions (1, 1, √2), total quantum dimension D2 = 4, and modular data matching the Ising MTC (equivalently SU(2)2) to machine precision. The D2 discrepancy between intrinsic (= 4) and extrinsic (= 9) values is resolved by the embedding dimension factor (d0embed)2 = 9/4. This is a companion paper to 'Anyon condensation in SU(3)3 ⊗ SU(2)2: condensable algebras, monodromy spectrum, and the Z6 fusion ring' (Bazih, 2026), resolving its Remark 3.4 on the structure of ClocA.
[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.