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arXiv:1908.07128

Classification of Rank 6 Modular Categories with Galois Group $\langle (012)(345)\rangle$

Green, David

Published Aug 20, 2019• 4 citations
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Abstract

Modular Tensor Categories (MTC's) arise in the study of certain condensed matter systems. There is an ongoing program to classify MTC's of low rank, up to modular data. We present an overview of the methods to classify modular tensor categories of low rank, applied to the specific case of a rank 6 category with Galois group $\langle(012)(345)\rangle$, and show that certain symmetries in this case imply nonunitarizable (hence, nonphysical) MTC's. We show that all the rank 6 MTC's with this Galois group have modular data conjugate to either the product of the semion category with $(A_1, 5)_{\frac{1}{2}}$ or a certain modular subcategory of $\mathcal{C}(\mathfrak{so}_5, 9, e^{jπi/9})$ with gcd$(18, j) = 1$.

DOI
10.48550/arxiv.1908.07128

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Modular Tensor Categories as Algebraic Framework for Topological Quantum Computation — E8 Intelligence Research

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FINDING: Modular tensor categories (MTCs) provide the algebraic backbone for topological quantum computation, with S and T matrices encoding anyon braiding and fusion rules. | MATH: MTC data: fusion rules \(N^k_{ij}\), \(S\)-matrix (modular transformation \(\tau \to -1/\tau\)), \(T\)-matrix (diagonal, \(\theta_a = e^{2\pi i h_a}\) topological spins), satisfying \(S^2 = C\) (charge conjugation), \((ST)^3 = e^{2\pi i c/8} S^2\) (where \(c\) is the chiral central charge). The entries of \(S\) and \(T\) lie in a cyclotomic field \(\mathbb{Q}(\zeta_N)\), \(\zeta_N = e^{2\pi i/N}\). For the Fibonacci anyon MTC: \(S = \frac{1}{\sqrt{\phi+2}}\begin{pmatrix} 1 & \phi \\ \phi & -1 \end{pmatrix}\), \(T = \text{diag}(1, e^{4\pi i/5})\), where \(\phi = (1+\sqrt{5})/2\). | CONNECTION: **Direct hit**: The Fibonacci MTC's \(S\)-matrix contains \(\phi = 1.618\) and \(\phi+2 = \phi^2+1 = 3.618\), whose inverse square root yields \(1/\sqrt{3.618} \approx 0.5257\). The golden ratio appears intrinsically. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Condensable algebras and the Ising condensed phase in SU(3)_3 ⊗ SU(2)_2

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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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Classification of Rank 6 Modular Categories with Galois Group ⟨(012)(345)⟩\langle (012)(345)\rangle⟨(012)(345)⟩
Modular Tensor Categories (MTC's) arise in the study of certain condensed matter systems. There is an ongoing program to classify MTC's of low rank, up to modular data. We present an overview of the methods to classify modular tensor categories of low rank, applied to the specific case of a rank 6 category with Galois group ⟨(012)(345)⟩\langle(012)(345)\rangle⟨(012)(345)⟩, and show that certain symmetries in this case imply nonunitarizable (hence, nonphysical) MTC's. We show that all the rank 6 MTC's with this Galois group have modular data conjugate to either the product of the semion category with (A1,5)12(A_1, 5)_{\frac{1}{2}}(A1​,5)21​​ or a certain modular subcategory of C(so5,9,ejπi/9)\mathcal{C}(\mathfrak{so}_5, 9, e^{jπi/9})C(so5​,9,ejπi/9) with gcd(18,j)=1(18, j) = 1(18,j)=1.
We use the computer algebra system GAP to classify modular data up to rank 12. This extends the previously obtained classification of modular data up to rank 6. Our classification includes all the modular data from modular tensor categories up to rank 12, with a few possible exceptions at rank 12 and levels 5,75,75,7 and 141414. Those exceptions are eliminated up to a certain bound by an extensive finite search in place of required infinite search. Our list contains a few potential unitary modular data which are not known to correspond to any unitary modular tensor categories (such as those from Kac-Moody algebra, twisted quantum doubles of finite group, as well as their Abelian anyon condensations). It remains to be shown if those potential modular data can be realized by modular tensor categories or not. We provide some evidence that all may be constructed from centers of near-group categories or gauging group symmetries of known modular tensor categories, with the exception of a total of five cases at rank 11 (with D2=1964.590D^2 =1964.590D2=1964.590) and 12 (with D2=3926.660D^2 =3926.660D2=3926.660). The classification of modular data corresponds to a classification of modular tensor categories (up to modular isotopes which are not expected to be present at low ranks). The classification of modular tensor categories leads to a classification of gapped quantum phases of matter in 2-dimensional space for bosonic lattice systems with no symmetry, as well as a classification of generalized symmetries in 1-dimensional space.
We classify all unitary modular tensor categories (UMTCs) of rank ≤ 4. There are a total of 35 UMTCs of rank ≤ 4 up to ribbon tensor equivalence. Since the distinction between the modular S -matrix S and − S has both topological and physical significance, so in our convention there are a total of 70 UMTCs of rank ≤ 4. In particular, there are two trivial UMTCs with S = (±1). Each such UMTC can be obtained from 10 non-trivial prime UMTCs by direct product, and some symmetry operations. Explicit data of the 10 non-trivial prime UMTCs are given in Sect. 5. Relevance of UMTCs to topological quantum computation and various conjectures are given in Sect. 6.
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.
[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.
FINDING: Modular tensor categories (MTCs) provide the algebraic backbone for topological quantum computation, with S and T matrices encoding anyon braiding and fusion rules. | MATH: MTC data: fusion rules NijkN^k_{ij}Nijk​, SSS-matrix (modular transformation τ→−1/τ\tau \to -1/\tauτ→−1/τ), TTT-matrix (diagonal, θa=e2πiha\theta_a = e^{2\pi i h_a}θa​=e2πiha​ topological spins), satisfying S2=CS^2 = CS2=C (charge conjugation), (ST)3=e2πic/8S2(ST)^3 = e^{2\pi i c/8} S^2(ST)3=e2πic/8S2 (where ccc is the chiral central charge). The entries of SSS and TTT lie in a cyclotomic field Q(ζN)\mathbb{Q}(\zeta_N)Q(ζN​), ζN=e2πi/N\zeta_N = e^{2\pi i/N}ζN​=e2πi/N. For the Fibonacci anyon MTC: S=1ϕ+2(1ϕϕ−1)S = \frac{1}{\sqrt{\phi+2}}\begin{pmatrix} 1 & \phi \\ \phi & -1 \end{pmatrix}S=ϕ+2​1​(1ϕ​ϕ−1​), T=diag(1,e4πi/5)T = \text{diag}(1, e^{4\pi i/5})T=diag(1,e4πi/5), where ϕ=(1+5)/2\phi = (1+\sqrt{5})/2ϕ=(1+5​)/2. | CONNECTION: **Direct hit**: The Fibonacci MTC's SSS-matrix contains ϕ=1.618\phi = 1.618ϕ=1.618 and ϕ+2=ϕ2+1=3.618\phi+2 = \phi^2+1 = 3.618ϕ+2=ϕ2+1=3.618, whose inverse square root yields 1/3.618≈0.52571/\sqrt{3.618} \approx 0.52571/3.618​≈0.5257. The golden ratio appears intrinsically. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
[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.