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arXiv:2312.13353

Classification of connected étale algebras in modular fusion categories up to rank five

Ken Kikuchi

Published Dec 20, 2023• 2 citations
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Abstract

We classify connected étale algebras in (possibly non-unitary) modular fusion categories $\mathcal B$'s with $\text{rank}(\mathcal B)\le5$. We also comment on Lagrangian algebra, anyon condensation, and physical applications. Concretely, we prove certain spontaneous $\mathcal B$-symmetry breaking and predict ground state degeneracies in massive renormalization group flows from non-unitary minimal models.

DOI
10.48550/arxiv.2312.13353

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We classify connected étale algebras in (possibly non-unitary) modular fusion categories B\mathcal BB's with rank(B)≤5\text{rank}(\mathcal B)\le5rank(B)≤5. We also comment on Lagrangian algebra, anyon condensation, and physical applications. Concretely, we prove certain spontaneous B\mathcal BB-symmetry breaking and predict ground state degeneracies in massive renormalization group flows from non-unitary minimal models.
We classify connected étale algebras AAA 's in multiplicity-free modular fusion categories (MFCs) B\mathcal{B}B's at rank six, namely rank⁡(B)=6\operatorname{rank}(\mathcal{B})=6rank(B)=6. There are eight MFCs in total and the result indicates that only so(5)2s o(5)_2so(5)2​ has nontrivial connected étale algebra. We briefly mention anyon condensation as it is used to determine the category of right AAA-modules in so(5)2s o(5)_2so(5)2​. Finally, we discuss physical applications, specifically proving spontaneous B\mathcal{B}B-symmetry breaking (SSB) of these MFCs. The discussion also includes predicting ground state degeneracies and SSB in massive renormalization group flows from two non-unitary minimal models.
We classify connected étale algebras AAA's in multiplicity-free modular fusion categories B\mathcal BB's with rank(B)≤9\text{rank}(\mathcal B)\le9rank(B)≤9. We also identify categories BA\mathcal B_ABA​'s of right AAA-modules. The results have physical applications in constraining renormalization group flows. As demonstration, we study massive renormalization group flows from non-unitary minimal models to predict ground state degeneracies and prove spontaneous B\mathcal BB-symmetry breaking.
We develop a systematic method to classify connected étale algebras AAA's in (possibly degenerate) pre-modular category B\mathcal BB. In particular, we find the category of AAA-modules, BA\mathcal B_ABA​, have ranks bounded from above by ⌊FPdim(B)⌋\lfloor\text{FPdim}(\mathcal B)\rfloor⌊FPdim(B)⌋. For demonstration, we classify connected étale algebras in some B\mathcal BB's, which appear in physics. Physically, the results constrain (or fix) ground state degeneracies of (certain) B\mathcal BB-symmetric gapped phases. We study massive deformations of rational conformal field theories such as minimal models and Wess-Zumino-Witten models. In most of our examples, the classification suggests the symmetries B\mathcal BB's are spontaneously broken.
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.
[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.
Anyons in 2+12+12+1 dimensions are not only characterized by exotic braiding statistics but also by intricate fusion properties. Two anyons may fuse into multiple topological charge sectors, and associativity of fusing three anyons to produce a fixed charge sector is governed by FFF-symbols. While braiding invariants, such as the modular data, are typically complex valued, a complete description of general anyon models requires understanding the arithmetic properties of its fusion associativity data as well. The FFF-symbols for many of the most common 2+12+12+1d topological orders, including all Abelian anyon models as well as Fibonacci and Ising anyons, can be made real valued. We show this phenomenon is not universal by exhibiting braided fusion categories whose FFF-symbols cannot be made real. We call such FFF-symbols "inherently complex." The examples we study lack a charge-conjugation symmetry and our results are therefore consistent with the converse of a statement proved in a companion work linking real FFF-symbols in braided fusion categories with the existence of a suitable charge-conjugation symmetry. We analyse the smallest-rank braided fusion categories we know of with inherently complex FFF-symbols: Rep(Z7⋊Z3){\rm Rep}(\mathbb{Z}_7\rtimes\mathbb{Z}_3)Rep(Z7​⋊Z3​) and Rep(Z5⋊Z4){\rm Rep}(\mathbb{Z}_5\rtimes\mathbb{Z}_4)Rep(Z5​⋊Z4​). Consequently, the corresponding Z(Rep(Z7⋊Z3))\mathcal Z({\rm Rep}(\mathbb{Z}_7\rtimes\mathbb{Z}_3))Z(Rep(Z7​⋊Z3​)) and Z(Rep(Z5⋊Z4))\mathcal Z({\rm Rep}(\mathbb{Z}_5\rtimes\mathbb{Z}_4))Z(Rep(Z5​⋊Z4​)) anyon models also have inherently complex FFF-symbols. Our presentation connects these examples with recent results on classifying anyons beyond modular data.