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arXiv:2402.00403 · Advances in Theoretical and Mathematical Physics

Classification of connected étale algebras in multiplicity-free modular fusion categories at rank six

Ken Kikuchi, Kah-Sen Kam, Fu-Hsiang Huang

Published Jan 1, 2026• 2 citations
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Abstract

We classify connected étale algebras $A$ 's in multiplicity-free modular fusion categories (MFCs) $\mathcal{B}$'s at rank six, namely $\operatorname{rank}(\mathcal{B})=6$. There are eight MFCs in total and the result indicates that only $s o(5)_2$ has nontrivial connected étale algebra. We briefly mention anyon condensation as it is used to determine the category of right $A$-modules in $s o(5)_2$. Finally, we discuss physical applications, specifically proving spontaneous $\mathcal{B}$-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.

DOI
10.4310/atmp.260604113634

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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 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 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.
[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.
We discuss anyon condensation in mixed-state topological order. The phases were recently conjectured to be classified by pre-modular fusion categories. Just like anyon condensation in pure-state topological order, a bootstrap analysis shows condensable anyons are given by connected étale algebras. We explain how to perform generic anyon condensation including non-invertible anyons and successive condensations. Interestingly, some condensations lead to pure-state topological orders. We clarify when this happens. We also compute topological invariants of equivalence classes.