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

Adam Bazih

Published May 10, 2026• 0 citations
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Abstract

[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.

DOI
10.5281/zenodo.20103636

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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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[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.
[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.
[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.
[NOTE, June 2026: This paper is part of the AME(4,6)/TOE series whose headline claims were partially retracted. In particular the "zero free parameters" framing and several numerical predictions are withdrawn; see the Status Note at DOI 10.5281/zenodo.20618848 and the maintained mathematics at DOI 10.5281/zenodo.20128951. The text below is unchanged and remains as the record of the programme's history.] 2-page rigorous technical brief explaining the derivation of the SU(3)_3 x SU(2)_2 modular tensor category from the AME(4,6) quantum error-correcting code via Z_6 = Z_3 x Z_2 grading and quantum group structure at roots of unity. Companion to AME(4,6) Theory of Everything series.
[NOTE, June 2026: This paper is part of the AME(4,6)/TOE series whose headline claims were partially retracted. In particular the "zero free parameters" framing and several numerical predictions are withdrawn; see the Status Note at DOI 10.5281/zenodo.20618848 and the maintained mathematics at DOI 10.5281/zenodo.20128951. The text below is unchanged and remains as the record of the programme's history.] We present the complete R-matrix braiding catalog for SU(3)_3 modular tensor category, comprising all 144 eigenvalues across 55 anyon pairs. Each eigenvalue takes the form exp(in*pi/18) where n is an integer, establishing that pi/18 serves as the universal phase quantum. The catalog is obtained through direct Sage computation (127 eigenvalues), charge conjugation symmetry (15 eigenvalues), and a novel Hopf link derivation for the multiplicity-2 channel (1,1) x (1,1) -> (1,1) (2 eigenvalues). We verify consistency through the Verlinde formula (144/144 channels confirmed) and charge conjugation symmetry (71/71 pairs, zero failures). Universal patterns discovered include: (i) the adjoint anyon (1,1) produces a universal elastic phase of -pi/2 for all non-simple-current anyons, (ii) adjoint self-braiding yields exclusively fourth roots of unity, and (iii) the simple current (3,0) generates phases that are exact multiples of pi/3. This catalog provides the foundational braiding data for the AME(4,6) theory of everything based on the Walker-Wang model.