Classification of connected étale algebras in multiplicity-free modular fusion categories up to rank nine
Ken Kikuchi



Department of Physics, National Taiwan University, Taipei 10617, Taiwan

Classification of connected étale algebras in multiplicity-free modular fusion categories up to rank nine
    Ken KIKUCHI
    
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We classify connected étale algebras A's in multiplicity-free modular fusion categories ℬ's with rank(ℬ)≤9. We also identify categories ℬ_A's of right A-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-symmetry breaking.






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§ INTRODUCTION

Throughout the paper, C denotes a fusion category over the field C of complex numbers (see <cit.> for definitions). Its rank and simple objects are denoted ( C) and c_i's with i=1,2,…,( C), respectively. When a fusion category admits a braiding c, the braided fusion category (BFC) and its simple objects are denoted B and b_i's, respectively. A BFC with non-degenerate braiding c is called modular or modular fusion category (MFC). In a BFC, we can define an étale algebra A∈ B. (The definitions are collected in section <ref>.) The goal of this paper is to classify connected étale algebras in multiplicity-free MFCs up to rank nine. Our main results are summarized as the


Theorem. Connected étale algebras in multiplicity-free modular fusion categories up to rank nine are given by


Remark. For a reader's convenience, we also included known results <cit.> at rank six.

Remark. Some MFCs are realized by rational conformal field theories (RCFTs) such as Wess-Zumino-Witten (WZW) models or minimal models. In those cases, we collectively denote the MFCs sharing the same fusion ring by the realization, e.g., su(2)_6 or . Other MFCs are realized by subcategories of simple objects invariant under centers. We denote the MFCs by the realizations with p in their head, e.g., psu(2)_13. Note that this is not a `gauging' of their centers. For instance, the Z/2 Z center symmetry of su(2)_13 is anomalous, and cannot be gauged. (Mathematically, this means there is no Z/2 Z algebra.)

Remark. MFCs up to rank five have been classified in <cit.>. Classification of larger MFCs is not complete yet. Results we know are limited to multiplicity-free ones up to rank nine <cit.>, summarized in AnyonWiki <cit.>. We study these MFCs; there are four rank seven, 12 rank eight, and 10 rank nine multiplicity-free fusion rings.

Remark. The classification problem was initiated in <cit.>, but some results had been known before these works. Especially when MFCs admit realizations by WZW models, some classification results are known. (In this context, an MFC describing g_k WZW model is denoted C( g,k), and connected étale algebras in it are called quantum subgroups <cit.>. In another context, say, <cit.>, connected étale algebras are also called condensable or normal algebras.) For example, connected étale algebras were classified in <cit.> (for su(2)_k), in <cit.> (for su(3)_k), in <cit.> (for su(4)_k), and many more in <cit.>. Connected étale algebras in (pre-)MFCs up to rank five have been classified in <cit.>. When available, our results are consistent with them; C(A_6,1), C(A_1,6), C(A_1,7) are known to be completely anisotropic. Also, ⊠⊠ is known to be completely anisotropic when braiding structures of all factors are the same, while it can admit nontrivial connected étale algebras when two factors have the opposite braidings. Examples with nontrivial quantum subgroups are also consistent. The MFCs C(B_3,2), C(B_5,2), C(A_1,8) are known to have two quantum subgroups 1,1⊕ X where X is a Z/2 Z simple object in each MFC. The nontrivial connected étale algebras give conformal embeddings so(7)_2⊂su(7)_1,so(11)_2⊂su(11)_1,(g_2)_1×su(2)_8⊂(f_4)_1, respectively. The MFC C(B_4,2) has 1,1⊕ X and exotic[A quantum subgroup which is not a direct sum of invertible simple objects was called exotic in <cit.>.] quantum subgroups 1⊕ Z,1⊕ X⊕2Z,1⊕ Z⊕ V.[We thank Terry Gannon for teaching us this fact.] The nontrivial connected étale algebras give so(9)_2⊂su(9)_1,so(9)_2⊂so(16)_1,so(9)_2⊂(e_8)_1, and another so(9)_2⊂(e_8)_1, respectively.

Remark. Connected étale algebras have also been studied in physics. For example, in the context of anyon condensation <cit.>, physically natural conditions demand condensing object A be connected étale <cit.>.

Remark. Algebras in symmetry categories have another aspect; they give anomaly-free subsymmetries <cit.>. Therefore, these results also classify certain symmetries which can be gauged.



§ CLASSIFICATION

In this section, we classify connected étale algebras in multiplicity-free MFCs up to rank nine. Before we study the problem, we first collect relevant definitions in section <ref> and explain our classification method in section <ref>.


 §.§ Definitions

Let C be a fusion category over C, i.e., a finite C-linear semisimple abelian rigid monoidal category with bilinear monoidal product ⊗ and simple unit object 1. The monoidal product of C is specified by fusion matrices (N_i)_jk:=N_ij^k with N-coefficients

    c_i⊗ c_j≅⊕_k=1^( C)N_ij^kc_k.

We denote the fusion ring of C as K( C). On the other hand, we denote a fusion category with a fusion ring K as C(K). Since the entries of fusion matrices are non-negative, we can apply the Perron-Frobenius theorem to obtain the largest eigenvalue _ C(c_i) called the Frobenius-Perron dimension of a simple object c_i.[We add ambient categories in the subscript because the Frobenius-Perron dimension of a given object depend on the ambient categories.] The Frobenius-Perron dimension of the category C is defined as

    ( C):=∑_i=1^( C)(_ C(c_i))^2.

In a spherical fusion category (including MFCs), one can also define quantum dimension d_i of c_i by the quantum (or categorical) trace

    d_i:=(a_c_i),

where a:id_ C≅(-)^** is a pivotal structure. Its multiplication rules are the same as the fusion rules

    d_id_j=∑_k=1^( C)N_ij^kd_k.

When various fusion categories are involved, in order to avoid confusion, we denote a quantum dimension of c_i∈ C as d_ C(c_i). The squared sum of quantum dimensions define the categorical (or global) dimension

    D^2( C):=∑_i=1^( C)d_i^2.

Note that there are two D( C)'s, one positive and one negative, for each categorical dimension. A fusion category is called pseudo-unitary if D^2( C)=( C) and unitary if ∀ c_i∈ C, d_i=_ C(c_i).

An MFC B has additional structure, braiding c. It is a natural isomorphism between two bifunctors

    c_-,-:-⊗-∼⇒-⊗-

subject to hexagon axioms. More concretely, for b,b'∈ B (not necessarily simple), it is a family of natural isomorphisms

    c_b,b':b⊗ b'∼⇒ b'⊗ b.

A fusion category with a braiding is called braided fusion category (BFC). A BFC is a pair ( B,c) of a fusion category B and a braiding c, but we often write B. For a BFC B=( B,c), there exists another BFC B̃ with the reverse (or opposite) braiding c̃_b,b'=c_b',b^-1. The BFC B̃=( B,c̃) is called reverse BFC. The structure is specified by conformal dimensions h_i's of b_i's. For instance, the double braiding of two simple objects b_i,b_j∈ B is given by

    c_b_j,b_i· c_b_i,b_j≅∑_k=1^( B)N_ij^ke^2π ih_k/e^2π i(h_i+h_j)id_k,

where id_k is the identity morphism at b_k∈ B. (A BFC B is called symmetric if ∀ b,b'∈ B, c_b',b· c_b,b'≅ id_b⊗ b'.) Its quantum trace defines (unnormalized) S-matrix

    S_i,j:=(c_b_j,b_i· c_b_i,b_j)=∑_k=1^( B)N_ij^ke^2π ih_k/e^2π i(h_i+h_j)d_k.

A normalized S-matrix is defined by

    S_i,j:=S_i,j/D( B).

An MFC is defined as a spherical BFC (called pre-modular fusion category, pre-MFC) with non-degenerate S-matrix. It squares to charge conjugation matrix

    S^2=C.

The charge conjugation matrix is defined by

    C_i,j=δ_i,j  (b_i^*≅ b_j),

where b_i^*∈ B is the dual of b_i∈ B. It obeys
    S_i,j^*=(S_i,j)^*,

where the RHS is the complex conjugate of S_i,j. Another modular matrix T is also defined with conformal dimensions

    T_i,j:=e^2π ih_iδ_i,j.

The two modular matrices define an additive central charge c( B) mod 8 by

    (ST)^3=e^2π ic( B)/8S^2.


Given a BFC, we can define commutative algebras. An algebra in a fusion category C is a triple (A,μ,η) of an object A∈ C, multiplication morphism μ:A⊗ A→ A, and unit morphism η:1→ A subject to associativity and unit axioms. (We abuse the notation and an algebra is often denoted by A.) A category C_A of right A-modules consists of pairs (m,p) of an object m∈ C and morphism p:m⊗ A→ m subject to consistency conditions. A category _A C of left A-modules are defined analogously. (We also abuse the notation and an A-module is often denoted by m.) An algebra A∈ C is called separable if C_A is semisimple. An algebra A in a BFC with braiding c is called commutative if

    μ· c_A,A=μ.

A commutative separable algebra is called étale. Any étale algebra decomposes to a direct sum of connected ones <cit.>. Here, an algebra A in a fusion category C is called connected if _ C C(1,A)=1. A connected étale algebra A∈ B is called Lagrangian if (_ B(A))^2=( B). An example of a connected étale algebra is the unit object A≅1∈ B giving B_A≃ B. The connected étale algebra always exists, and is thus called trivial. A BFC is called completely anisotropic if it has no nontrivial connected étale algebra.

The category of right A-modules has an important subcategory B_A^0⊂ B_A. It consists of dyslectic (or local) modules <cit.> (m,p)∈ B_A obeying

    p· c_A,m· c_m,A=p.

(In the context of anyon condensation <cit.>, B_A, B_A^0 are called broken and deconfined phase, respectively.) The latter, the category B_A of right A-modules, is a left B-module category. Here, for a fusion category C, a left C-module category (or module category over C) <cit.> is a quadruple ( M,▹,m,l) of a category M, an action (or module product) bifunctor ▹: C× M→ M, and natural isomorphisms m_-,-,-:(-⊗-)▹-∼⇒-▹(-▹-) and l:1▹ M≃ M called module associativity constraint and unit constraint, respectively. They are subject to associativity and unit axioms. Let M_1,2 be C-module categories. The category M≃ M_1⊕ M_2 is called a direct sum of the module categories M_1,2. A C-module category is called indecomposable if it is not equivalent to a direct sum of nontrivial module categories. In our setup, the categories M≃ B_A (and also B_A^0) are fusion categories with monoidal products ⊗_A.[For an explanation, see the footnote 2 of <cit.>.] Then, the action of (left) B-module categories forms non-negative integer matrix representations (NIM-reps). Here is the reason. For any b∈ B,m∈ M, b▹ m is an object of M. Hence, it can be decomposed to a direct sum of simple objects in M with N-coefficients. The natural numbers assemble to r× r-matrices where r=( M). The NIM-rep is called r-dimensional.



 §.§ Method

In this section, we explain our classification method. It is based on <cit.>, but developed further.

Let B be an MFC and A∈ B a connected étale algebra. The category B_A^0 of dyslectic right A-modules is modular <cit.> obeying <cit.>

    ( B_A^0)   =( B)/(_ B(A))^2,
    
        e^2π ic( B_A^0)/8   =e^2π ic( B)/8.

Since we have <cit.>

    ∀ c∈ C,  _ C(c)≥1,

we obtain an inequality

    1≤(_ B(A))^2≤( B).

In addition, since A consists of simple objects of B, its general form is given by

    A≅⊕_i=1^( B)n_ib_i,

where n_i∈ N counts the number of b_i in A. As the direct sum is defined as (co)limit, the object is equipped with product projections p_i:A→ b_i and coproduct injections ι_i:b_i→ A. Its Frobenius-Perron dimension is given by the linear sum of those of simple objects:

    _ B(A)=∑_i=1^( B)n_i_ B(b_i).

In classifying connected algebras, we can set n_i=1 for b_i≅1 at the outset. Thus, for a given MFC B, we solve (<ref>) with an ansatz

    A≅1⊕⊕_b_i≇1n_ib_i  (n_i∈ N),

or

    _ B(A)=1+∑_b_i≇1n_i_ B(b_i).

As a result, we get a set of natural numbers n_i's. Each element gives a candidate for A. Then, for each candidate, we check whether it satisfies the axioms of connected étale algebra. Our strategy here is to check various necessary conditions to reduce the number of candidates. In particular, we check the commutativity (<ref>). In checking this axiom, it turns out useful to study a necessary condition

    μ· c_A,A· c_A,A=μ.

If A satisfies (<ref>), then by substituting the RHS in the LHS, we obtain the necessary condition. Since the condition is written in terms of double braiding, we can use the formula (<ref>). Here, the double braiding of the direct sum (<ref>) is computed by chasing commuting diagrams <cit.>:

    c_A,A· c_A,A≅∑_i,j=1^( B)n_in_j(ι_i⊗ι_j)· c_b_j,b_i· c_b_i,b_j·(p_i⊗ p_j).

(The morphisms ι,p are coproduct injection and product projection, respectively, introduced below (<ref>).) If there exists a term with nontrivial phase in (<ref>), it means the candidate is non-commutative, and it is ruled out. In particular, if a simple object b_i∈ B has 1∈ b_i⊗ b_i and e^-4π ih_i≠1, b_i cannot enter a commutative algebra. Thus, we can set n_i=0 for such simple objects at the outset. We will see the necessary condition (<ref>) is strong enough to rule out most of the simple objects. For the remaining candidates which pass the necessary condition, we check other necessary conditions such as an existence of MFC with the Frobenius-Perron dimension (<ref>) which matches the additive central charges. For a handful final candidates passing all necessary conditions, we check axioms of an algebra and directly compute the braiding <cit.>

    c_A,A≅∑_i,j=1^( B)n_in_j(ι_j⊗ι_i)· c_b_i,b_j·(p_i⊗ p_j)

to check (<ref>). Here, some facts save us from tedious computations.

Lemma 1. Let ( C,⊗,α,1,ι) be a monoidal category, ( C',⊗,α,1,ι) with C'⊂ C a monoidal subcategory, and (A,μ,η) an algebra in ( C',⊗,α,1,ι). The algebra in the monoidal subcategory C' is also an algebra in the larger monoidal category C.

Proof. By definition, the algebra obeys associativity

    μ·(id_A'⊗'μ)·α_A,A,A'=μ·(μ⊗'id_A'),

and unit

    id_A'· l_A'=μ·(η⊗'id_A'),   id_A'· r_A'=μ·(id_A'⊗'η)

axioms where quantities with primes belong to C'. Here, note that the monoidal subcategory has the same structure as the larger monoidal category by definition <cit.>. (We already used this fact in the statement.) Furthermore, by uniqueness of identity morphisms (up to isomorphism), we have

    id_A'≅ id_A.

Since left and right unit constraints l_A,r_A are defined with α_1,1,A,α_A,1,1,ι,id_A, they are also isomorphic on A:

    l_A'≅ l_A,   r_A'≅ r_A.

The axioms of the algebra (A,μ,η) now reduce to

    μ·(id_A⊗μ)·α_A,A,A=μ·(μ⊗ id_A),   id_A· l_A=μ·(η⊗ id_A),   id_A· r_A=μ·(id_A⊗η).

This is nothing but the definition of (A,μ,η) being an algebra in the larger monoidal category ( C,⊗,α,1,ι). □

Remark. Physically, this means anomaly-free symmetries in C' remain anomaly-free in a larger symmetry C⊃ C', as physicists know.

Furthermore, commutative algebras in braided fusion subcategory remain commutative in a larger BFC. This follows from the definition of braided fusion subcategory. Let ( B',c') be a braided fusion subcategory of a larger BFC ( B,c). Then, the braidings are equivalent on B'. Namely, ∀ b,b'∈ B', we have

    c'_b,b'≅ c_b,b'.

Therefore, a commutative algebra (A,μ,η) in a BFC ( B',c') obeying μ· c'_A,A=μ is a commutative algebra in a larger BFC ( B,c):

    μ· c_A,A=μ· c'_A,A=μ.

Sometimes, a direct check of commutativity axiom is difficult especially when braidings c_b_i,b_j are not known. In those cases, we indirectly find connected étale algebras manipulating the following fact:

Lemma 2. <cit.> The operations (not all) B→ B_A^0 are composable, associative, and invertible.

Remark. Note that not all operations are composable. For example, an operation B→ given by a Lagrangian algebra (if exists) cannot be composed with other operations. Therefore, the operations and their compositions fail to form a group.

How do we use this fact? Let A∈ B and A'∈ B_A^0 be connected étale algebras. The second algebra gives another MFC ( B_A^0)_A'^0. We get a sequence B→ B_A^0→( B_A^0)_A'^0. Since the two operations can be composed, we learn there should exist a connected étale algebra Ã∈ B giving B_Ã^0≃( B_A^0)_A'^0. Since B_A^0 has smaller Frobenius-Perron dimension (hence usually smaller rank) than B thanks to (<ref>), this can reduce the classification problem in larger MFC B to that in smaller MFC B_A^0. For an example of this lemma in action, see section <ref>.

Another fact we use is the

Lemma 3. <cit.> Let B be a modular fusion category. The modular fusion category[This special MFC is equivalent to the Drinfeld center (or quantum double) Z( B) <cit.>.] B⊠B̃ has a Lagrangian algebra A∈ B⊠B̃.

Even when we cannot check axioms directly, this fact helps us to ensure an existence of connected étale algebra. See section <ref> for our use of this lemma. We manipulate all these facts at our disposal.

In order to discuss (physical) applications of classification results, we also identify categories B_A^0, B_A of right A-modules. We do this in two steps. First, we identify the category B_A^0 of dyslectic right A-modules, and next we identify B_A which contains B_A^0 as a subcategory. In identifying B_A^0, the following fact turns out useful; the MFCs B and B_A^0 have the same topological twists <cit.>

    e^2π ih_b^ B=e^2π ih_b^ B_A^0,

where h_b^ B is the conformal dimension of b∈ B and h_b^ B_A^0 is that of b∈ B_A^0. Physically, this means anyon condensation preserves conformal dimensions (mod 1) of deconfined particles. We see this usually fixes B_A^0 uniquely.

When the category of dyslectic right A-modules is specified, the category B_A of right A-modules is highly constrained. The remaining contributions to the Frobenius-Perron dimensions come from confined particles. This sets an upper bound on ( B_A). Furthermore, we can get all candidate simple objects as follows. The free module functor

    F_A:=-⊗ A: B→ B_A

gives objects of B_A. The functor is surjective (or dominant) <cit.>. Namely, ∀ m∈ B_A, ∃ b∈ B such that m is a subobject of b⊗ A. Since B_A is a fusion category in our setup, ∀ b∈ B, b⊗ A∈ B_A can be decomposed as a direct sum of simple objects of B_A (definition of semisimplicity). All possible sets of simple objects are given by consistent collection of some of them. Some consistencies are correct Frobenius-Perron dimensions and closedness under actions b_j⊗- or monoidal products ⊗_A.

Given simple objects of B_A, we identify the category by computing monoidal products ⊗_A as follows. The free module functor satisfies two nice properties: 1) F_A is a tensor functor, and 2) it admits the forgetful functor U: B_A→ B as a right adjoint <cit.>[This second property is sometimes called the Frobenius reciprocity (see, say, <cit.>).]

    ∀ b∈ B, ∀ m∈ B_A,   B_A(F_A(b),m)≅ B(b,U(m)).

The first property in particular implies preservation of Frobenius-Perron dimensions

    ∀ b∈ B,  _ B(b)=_ B_A(F_A(b)),

and

    ∀ b,b'∈ B,   F_A(b)⊗_AF_A(b')≅ F_A(b⊗ b').

When b⊗ b' is a (finite) direct sum, we can distribute using the biexactness of monoidal product

    (b_1⊕ b_2⊕⋯⊕ b_n)⊗ A≅(b_1⊗ A)⊕(b_2⊗ A)⊕⋯⊕(b_n⊗ A).

With these facts, we can compute monoidal products ⊗_A to identify B_A. In addition, we can compute quantum dimensions. A right A-module m∈ B_A has <cit.>

    d_ B_A(m)=d_ B(m)/d_ B(A).



Before we start classifying connected étale algebras, let us see how our method works in a simple example. We take a symmetric pre-MFC Rep(S_3) as our ambient category B. (Note that this is not modular, but most of our classification method except (<ref>) still works here.) The pre-MFCs have three simple objects {1,X,Y} obeying monoidal products

They have

    _ B(1)=1=_ B(X),  _ B(Y)=2,

and

    ( B)=6.

It is known <cit.> that the pre-MFCs with (h_X,h_Y)=(0,0) mod 1 have a connected étale algebra A≅1⊕ Y giving B_A≃_ Z/2 Z^α. Let us see how our classification method reproduces this result.

Calculating b_j⊗ A, we get

    1⊗ A≅ A≅1⊕ Y,   X⊗ A≅ X⊕ Y,   Y⊗ A≅(1⊕ Y)⊕(X⊕ Y).

The result tells us B_A has two invertible simple objects[Logically, F_A(Y)=Y⊗ A can be simple. Let us rule out this possibility. Since F_A preserves Frobenius-Perron dimensions, we have _ B_A(F_A(Y))=2. If this is simple, it contributes 2^2=4 to ( B_A), which is a contradiction because it exceeds ( B_A)=2 we get below. Hence, F_A(Y)=Y⊗ A should be a direct sum of two simple objects with Frobenius-Perron dimensions one.]

    1⊕ Y,   X⊕ Y.

In B_A, they both have Frobenius-Perron dimensions one because _ B(1)=1=_ B(X) and the free module functor F_A:=-⊗ A preserves Frobenius-Perron dimensions. On the other hand, for a generic BFC B and a connected étale algebra A∈ B, we have the formula <cit.>

    _ B(A)=( B)/( B_A).

In our current example, this imposes

    ( B_A)=( B)/_ B(A)=2.

Therefore, no other simple objects can appear. We can identify B_A≃{1⊕ Y,X⊕ Y}. While we know rank two fusion category with Frobenius-Perron dimension two should be _ Z/2 Z^α, let us check this fact by computing monoidal products ⊗_A. First, we know <cit.> A is the monoidal unit of ⊗_A. Namely, ∀ m∈ B_A, m⊗_AA≅ m≅ A⊗_Am. Thus, we only have to compute (X⊕ Y)⊗_A(X⊕ Y). We have

    (X⊕ Y)⊗_A(X⊕ Y)   =F_A(X)⊗_AF_A(X)
       ≅ F_A(X⊗ X)
       ≅ F_A(1)≅ 1⊕ Y.

Therefore, the two right A-modules

    m_1≅1⊕ Y,   m_2≅ X⊕ Y

obey monoidal products

showing B_A≃_ Z/2 Z^α. Below, we apply this method to classify connected étale algebras in MFCs.



 §.§ Rank seven



  §.§.§ B≃ su(7)_1≃_ Z/7 Z^1

The MFCs have seven simple objects {1,X,Y,Z,U,V,W} obeying monoidal products

Thus, they have

    _ B(1)=_ B(X)=_ B(Y)=_ B(Z)=_ B(U)=_ B(V)=_ B(W)=1,

and

    ( B)=7.

Their quantum dimensions d_j's are solutions of the same multiplication rules d_id_j=∑_k=1^7N_ij^kd_k. There is only one solution

    (d_X,d_Y,d_Z,d_U,d_V,d_W)=(1,1,1,1,1,1).

Thus, all MFCs are unitary. The only categorical dimension is

    D^2( B)=7.

They have conformal dimensions[Naively, one also finds

    (h_X,h_Y,h_Z,h_U,h_V,h_W)=(1/7,1/7,2/7,2/7,4/7,4/7),(2/7,2/7,4/7,4/7,1/7,1/7),(5/7,5/7,3/7,3/7,6/7,6/7),(6/7,6/7,5/7,5/7,3/7,3/7)  (1).

However, they do not give new conformal dimensions because these are related to those in the main text by permutations (XUWYZV) of simple objects.]

    (h_X,h_Y,h_Z,h_U,h_V,h_W)=(3/7,3/7,6/7,6/7,5/7,5/7),(4/7,4/7,1/7,1/7,2/7,2/7)  (1).

The S-matrices are given by

    S=[         1         1         1         1         1         1         1;         1 e^∓2π i/7 e^±2π i/7 e^±6π i/7 e^∓6π i/7 e^±4π i/7 e^∓4π i/7;         1 e^±2π i/7 e^∓2π i/7 e^∓6π i/7 e^±6π i/7 e^∓4π i/7 e^±4π i/7;         1 e^±6π i/7 e^∓6π i/7 e^∓4π i/7 e^±4π i/7 e^±2π i/7 e^∓2π i/7;         1 e^∓6π i/7 e^±6π i/7 e^±4π i/7 e^∓4π i/7 e^∓2π i/7 e^±2π i/7;         1 e^±4π i/7 e^∓4π i/7 e^±2π i/7 e^∓2π i/7 e^±6π i/7 e^∓6π i/7;         1 e^∓4π i/7 e^±4π i/7 e^∓2π i/7 e^±2π i/7 e^∓6π i/7 e^±6π i/7 ].

(All signs are correlated. In other words, one S-matrix is given by choosing upper signs in all elements, and the other S-matrix is its complex conjugate.) They have additive central charges

    c( B)=-2   (1st h),
    +2   (2nd h).  (8)

There are

    1(quantum dimension)×2(conformal dimensions)×2(categorical dimensions)=4

MFCs, and all of them are unitary. We study connected étale algebras in all four MFCs simultaneously.

The most general form of a connected algebra is given by

    A≅1⊕ n_XX⊕ n_YY⊕ n_ZZ⊕ n_UU⊕ n_VV⊕ n_WW

with n_j∈ N. It has

    _ B(A)=1+n_X+n_Y+n_Z+n_U+n_V+n_W.

For this to obey (<ref>), the natural numbers n_j's can only take seven values

    (n_X,n_Y,n_Z,n_U,n_V,n_W)=   (0,0,0,0,0,0),(1,0,0,0,0,0),(0,1,0,0,0,0),(0,0,1,0,0,0),
       (0,0,0,1,0,0),(0,0,0,0,1,0),(0,0,0,0,0,1).

The first solution is nothing but the trivial connected étale algebra A≅1 giving B_A^0≃ B_A≃ B. The other six solutions do not give connected étale algebras because they fail to satisfy the necessary condition (<ref>).

We conclude

That is, all the four MFCs B≃_ Z/7 Z^1's are completely anisotropic.



  §.§.§ B≃ su(2)_6

The MFCs have seven simple objects {1,X,Y,Z,U,V,W} obeying monoidal products

Thus, they have

    _ B(1)=1=_ B(X),  _ B(Y)=√(2+√(2))   =_ B(Z),
    _ B(U)=1+√(2)=_ B(V),  _ B(W)=   √(4+2√(2)),

and

    ( B)=8(2+√(2))≈27.3.

Their quantum dimensions d_j's are solutions of the same multiplication rules d_id_j=∑_k=1^7N_ij^kd_k. There are four (nonzero) solutions

    (d_X,d_Y,d_Z,d_U,d_V,d_W)=   (1,√(2-√(2)),√(2-√(2)),1-√(2),1-√(2),-√(4-2√(2))),
       (1,-√(2-√(2)),-√(2-√(2)),1-√(2),1-√(2),√(4-2√(2))),
       (1,-√(2+√(2)),-√(2+√(2)),1+√(2),1+√(2),-√(4+2√(2))),
       (1,√(2+√(2)),√(2+√(2)),1+√(2),1+√(2),√(4+2√(2)))

with categorical dimensions

    D^2( B)=8(2-√(2))(≈4.7),  8(2+√(2)),

respectively for each pair. Only the last quantum dimensions give unitary MFCs. Each pair has eight conformal dimensions:

    (h_X,h_Y,h_Z,h_U,h_V,h_W)=   (1/2,1/32,1/32,1/4,3/4,5/32),(1/2,7/32,7/32,3/4,1/4,3/32),
       (1/2,9/32,9/32,1/4,3/4,13/32),(1/2,15/32,15/32,3/4,1/4,11/32),
       (1/2,17/32,17/32,1/4,3/4,21/32),(1/2,23/32,23/32,3/4,1/4,19/32),
       (1/2,25/32,25/32,1/4,3/4,29/32),(1/2,31/32,31/32,3/4,1/4,27/32)  (1)

for the first and second quantum dimensions, and

    (h_X,h_Y,h_Z,h_U,h_V,h_W)=   (1/2,3/32,3/32,3/4,1/4,15/32),(1/2,5/32,5/32,1/4,3/4,25/32),
       (1/2,11/32,11/32,3/4,1/4,23/32),(1/2,13/32,13/32,1/4,3/4,1/32),
       (1/2,19/32,19/32,3/4,1/4,31/32),(1/2,21/32,21/32,1/4,3/4,9/32),
       (1/2,27/32,27/32,3/4,1/4,7/32),(1/2,29/32,29/32,1/4,3/4,17/32)  (1)

for the third and fourth quantum dimensions. The S-matrices are given by

    S=[     1   d_X   d_Y   d_Z   d_U   d_V   d_W;   d_X     1  -d_Y  -d_Z   d_U   d_V  -d_W;   d_Y  -d_Y   d_W  -d_W ± d_Y ∓ d_Y     0;   d_Z  -d_Z  -d_W   d_W ± d_Z ∓ d_Z     0;   d_U   d_U ± d_Y ± d_Z    -1    -1 ∓ d_W;   d_V   d_V ∓ d_Y ∓ d_Z    -1    -1 ± d_W;   d_W  -d_W     0     0 ∓ d_W ± d_W     0 ].

There are

    4(quantum dimensions)×8(conformal dimensions)×2(categorical dimensions)=64

MFCs, among which those 16 with the last quantum dimensions give unitary MFCs. We classify connected étale algebras in all 64 MFCs simultaneously.

An ansatz

    A≅1⊕ n_XX⊕ n_YY⊕ n_ZZ⊕ n_UU⊕ n_VV⊕ n_WW

with n_j∈ N has

    _ B(A)=1+n_X+√(2+√(2))(n_Y+n_Z)+(1+√(2))(n_U+n_V)+√(4+2√(2))n_W.

For this to obey (<ref>), the natural numbers can take only 20 values

    (n_X,n_Y,n_Z,n_U,n_V,n_W)=   (0,0,0,0,0,0),(1,0,0,0,0,0),(2,0,0,0,0,0),(3,0,0,0,0,0),
       (4,0,0,0,0,0),(2,1,0,0,0,0),(2,0,1,0,0,0),(1,1,0,0,0,0),
       (1,0,1,0,0,0),(1,0,0,1,0,0),(1,0,0,0,1,0),(1,0,0,0,0,1),
       (0,1,0,0,0,0),(0,2,0,0,0,0),(0,1,1,0,0,0),(0,0,1,0,0,0),
       (0,0,2,0,0,0),(0,0,0,1,0,0),(0,0,0,0,1,0),(0,0,0,0,0,1).

The first is nothing but the trivial connected étale algebra A≅1 giving B_A^0≃ B_A≃ B. Those with X do not give commutative algebra because X has (d_X,h_X)=(1,1/2) (mod 1 for h) and c_X,X≅-id_1 <cit.>. The others also fail to be commutative because they contain simple object(s) b_j≇1 with nontrivial conformal dimensions.

We conclude

That is, all the 64 MFCs B≃ su(2)_6's are completely anisotropic.



  §.§.§ B≃ so(7)_2

The MFCs have seven simple objects {1,X,Y,Z,U,V,W} obeying monoidal products

Thus, they have

    _ B(1)=1=_ B(X),     _ B(Y)=_ B(Z)=_ B(U)=2,
    _ B(V)=   √(7)=_ B(W),

and

    ( B)=28.

Their quantum dimensions d_j's are solutions of the same multiplication rules d_id_j=∑_k=1^7N_ij^kd_k. There are two (nonzero) solutions

    (d_X,d_Y,d_Z,d_U,d_V,d_W)=(1,2,2,2,-√(7),-√(7)),(1,2,2,2,√(7),√(7)).

Only the second quantum dimensions give unitary MFCs. They both have the categorical dimension

    D^2( B)=28.

They have four conformal dimensions[Naively, one finds 24 conformal dimensions, but the others are equivalent to the four in the main text by permutation (VW) or (YZU) of simple objects. For example, one also finds (h_X,h_Y,h_Z,h_U,h_V,h_W)=(0,1/7,2/7,4/7,5/8,1/8) (mod 1) is consistent, but it is the same as our first conformal dimension under a permutation (VW).]

    (h_X,h_Y,h_Z,h_U,h_V,h_W)=(0,1/7,2/7,4/7,1/8,5/8),(0,1/7,2/7,4/7,3/8,7/8),(0,3/7,6/7,5/7,1/8,5/8),(0,3/7,6/7,5/7,3/8,7/8).  (1)

The S-matrices are given by

    S=[     1     1     2     2     2   d_V   d_W;     1     1     2     2     2  -d_V  -d_W;     2     2     s    s'    s”     0     0;     2     2    s'    s”     s     0     0;     2     2    s”     s    s'     0     0;   d_V  -d_V     0     0     0 ± d_V ∓ d_W;   d_W  -d_W     0     0     0 ∓ d_W ± d_W ]

with

    s=-4sinπ/14,   s'=4sin3π/14,   s”=-4cosπ/7,

or their permutations (ss's”). They have additive central charges

    c( B)=2   (1st&2nd h),
    -2   (3rd&4th h).  (8)

There are

    2(quantum dimensions)×4(conformal dimensions)×2(categorical dimensions)=16

MFCs, among which those eight with the second quantum dimensions give unitary MFCs. We study connected étale algebras in all 16 MFCs simultaneously.

We work with an ansatz

    A≅1⊕ n_XX⊕ n_YY⊕ n_ZZ⊕ n_UU⊕ n_VV⊕ n_WW

with n_j∈ N. It has

    _ B(A)=1+n_X+2(n_Y+n_Z+n_U)+√(7)(n_V+n_W).

For this to obey (<ref>), the natural numbers can take only 24 values 

    (n_X,n_Y,n_Z,n_U,n_V,n_W)=   (0,0,0,0,0,0),(1,0,0,0,0,0),(2,0,0,0,0,0),(3,0,0,0,0,0),
       (4,0,0,0,0,0),(2,1,0,0,0,0),(2,0,1,0,0,0),(2,0,0,1,0,0),
       (1,1,0,0,0,0),(1,0,1,0,0,0),(1,0,0,1,0,0),(1,0,0,0,1,0),
       (1,0,0,0,0,1),(0,1,0,0,0,0),(0,2,0,0,0,0),(0,1,1,0,0,0),
       (0,1,0,1,0,0),(0,0,1,0,0,0),(0,0,2,0,0,0),(0,0,1,1,0,0),
       (0,0,0,1,0,0),(0,0,0,2,0,0),(0,0,0,0,1,0),(0,0,0,0,0,1).

The first is nothing but the trivial connected étale algebra A≅1 giving B_A^0≃ B_A≃ B. Next, the 19 candidates with Y,Z,U,V,W have nontrivial conformal dimensions, and they fail to be commutative. Thus, we are left with those four with only X's:

    n_X=1,2,3,4.

All but the first solution is ruled out by studying Frobenius-Perron dimension. The latter three have _ B(A)=3,4,5, and demand ( B_A^0)=28/9,7/4,28/25. However, there is no MFC with such Frobenius-Perron dimensions. Thus, the candidates are ruled out. We are left with just A≅1⊕ X.

The candidate is indeed a commutative algebra by the lemma 1 and c_X,X≅ id_1 because X has (d_X,h_X)=(1,0) <cit.>. Furthermore, it turns out to be separable, hence connected étale. Let us check this fact by identifying B_A.

It has _ B(A)=2, and demands

    ( B_A^0)=7,  ( B_A)=14.

We can identify the MFC as

    B_A^0≃_ Z/7 Z^1

from its Frobenius-Perron dimension. Taking the invariance of topological twists (<ref>) into account, we see central charges are also matched. What is a fusion category B_A containing the rank seven MFC _ Z/7 Z^1? It turns out that the category of right A-modules is

    B_A≃TY( Z/7 Z),

a Z/7 Z Tambara-Yamagami category <cit.>. One of the easiest ways to see this fact is to perform anyon condensation. It `identifies' 1 and X, and hence V and W. The other invariant simple objects Y,Z,U `split' into two each. As a result, we get seven invertible objects. The deconfined seven particles have the same conformal dimensions (mod 1) as simple objects in _ Z/7 Z^1 MFCs. Thus, they form B_A^0≃_ Z/7 Z^1. The category B_A of right A-modules have one additional non-invertible simple object with quantum dimension ±√(7). This is nothing but a Z/7 Z Tambara-Yamagami category.[More rigorously, we have to find NIM-reps. Indeed, we find an eight-dimensional NIM-rep

    n_1=1_8=n_X,   n_Y=[ 0 1 1 0 0 0 0 0; 1 0 0 0 0 1 0 0; 1 0 0 0 0 0 1 0; 0 0 0 0 1 1 0 0; 0 0 0 1 0 0 1 0; 0 1 0 1 0 0 0 0; 0 0 1 0 1 0 0 0; 0 0 0 0 0 0 0 2 ],   n_Z=[ 0 0 0 1 1 0 0 0; 0 0 0 0 1 0 1 0; 0 0 0 1 0 1 0 0; 1 0 1 0 0 0 0 0; 1 1 0 0 0 0 0 0; 0 0 1 0 0 0 1 0; 0 1 0 0 0 1 0 0; 0 0 0 0 0 0 0 2 ],
    
        n_U=[ 0 0 0 0 0 1 1 0; 0 0 1 1 0 0 0 0; 0 1 0 0 1 0 0 0; 0 1 0 0 0 0 1 0; 0 0 1 0 0 1 0 0; 1 0 0 0 1 0 0 0; 1 0 0 1 0 0 0 0; 0 0 0 0 0 0 0 2 ],   n_V=[ 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 1; 1 1 1 1 1 1 1 0 ]=n_W.

Denoting a basis of B_A by {m_1,m_2,m_3,m_4,m_5,m_6,m_7,m_8}, we obtain a multiplication table



[1 ][c]       
1.0 !
    b_j⊗\   m_1   m_2   m_3   m_4   m_5   m_6   m_7   m_8
1,X   m_1   m_2   m_3   m_4   m_5   m_6   m_7   m_8

    Y   m_2⊕ m_3   m_1⊕ m_6   m_1⊕ m_7   m_5⊕ m_6   m_4⊕ m_7   m_2⊕ m_4   m_3⊕ m_5   2m_8

    Z   m_4⊕ m_5   m_5⊕ m_7   m_4⊕ m_6   m_1⊕ m_3   m_1⊕ m_2   m_3⊕ m_7   m_2⊕ m_6   2m_8

    U   m_6⊕ m_7   m_3⊕ m_4   m_2⊕ m_5   m_2⊕ m_7   m_3⊕ m_6   m_1⊕ m_5   m_1⊕ m_4   2m_8

    V,W   m_8   m_8   m_8   m_8   m_8   m_8   m_8   ⊕_j=1^7m_j
.



From the multiplication rules, in this basis, we can identify

    m_1≅1⊕ X,   m_2≅ Y≅ m_3,   m_4≅ Z≅ m_5,   m_6≅ U≅ m_7,   m_8≅ V⊕ W.

In the category of right A-modules (or broken phase) B_A, they have quantum dimensions (<ref>)

    d_ B_A(m_1)=d_ B_A(m_2)=d_ B_A(m_3)=d_ B_A(m_4)=d_ B_A(m_5)=d_ B_A(m_6)=d_ B_A(m_7)=1,   d_ B_A(m_8)=±√(7).

Furthermore, employing the free module functor F_A (<ref>), one can compute monoidal products ⊗_A:

    m_j⊗_Am_8≅ m_8≅ m_8⊗_Am_j (j=1,2,…,7),   m_8⊗_Am_8≅⊕_j=1^7m_j.

This shows B_A≃TY( Z/7 Z).] Note that this example has

    ( B_A)>( B).

While B_A consists of objects of B and the former has Frobenius-Perron dimension no larger than the latter, in general, the former can have larger rank as in this example.

We conclude

All the 16 MFCs B≃ so(7)_2's fail to be completely anisotropic.



  §.§.§ B≃ psu(2)_13

The MFCs have seven simple objects {1,X,Y,Z,U,V,W} obeying monoidal products

Thus, they have

    _ B(1)=1,  _ B(X)=   sin2π/15/sinπ/15,  _ B(Y)=sin3π/15/sinπ/15,  _ B(Z)=sin4π/15/sinπ/15,
    _ B(U)=sin5π/15/sinπ/15,     _ B(V)=sin6π/15/sinπ/15,  _ B(W)=sin7π/15/sinπ/15,

and

    ( B)=15/4sin^2π/15≈86.8.

Their quantum dimensions are solutions of the same multiplication rules d_id_j=∑_k=1^7N_ij^kd_k. There are four (nonzero) solutions

    (d_X,d_Y,d_Z,d_U,d_V,d_W)=   (sinπ/15/cosπ/30,-sin6π/15/cosπ/30,-sin2π/15/cosπ/30,sin5π/15/cosπ/30,sin3π/15/cosπ/30,-sin4π/15/cosπ/30),
       (-sin7π/15/cos7π/30,sin3π/15/cos7π/30,sinπ/15/cos7π/30,-sin5π/15/cos7π/30,sin6π/15/cos7π/30,-sin2π/15/cos7π/30),
       (-sin4π/15/cos11π/30,sin6π/15/cos11π/30,-sin7π/15/cos11π/30,sin5π/15/cos11π/30,-sin3π/15/cos11π/30,sinπ/15/cos11π/30),
       (sin2π/15/sinπ/15,sin3π/15/sinπ/15,sin4π/15/sinπ/15,sin5π/15/sinπ/15,sin6π/15/sinπ/15,sin7π/15/sinπ/15)

with categorical dimensions

    D^2( B)=15/4cos^2π/30(≈3.8),  15/4cos^27π/30(≈6.8),  15/4cos^211π/30(≈22.7),  15/4sin^2π/15,

respectively. Each quantum dimension has two conformal dimensions

    (h_X,h_Y,h_Z,h_U,h_V,h_W)=(2/5,1/15,0,1/5,2/3,2/5),(3/5,14/15,0,4/5,1/3,3/5)   (1st quantum dimension),
    (1/5,8/15,0,3/5,1/3,1/5),(4/5,7/15,0,2/5,2/3,4/5)   (2nd quantum dimension),
    
    (2/5,11/15,0,1/5,1/3,2/5),(3/5,4/15,0,4/5,2/3,3/5)   (3rd quantum dimension),
    (1/5,13/15,0,3/5,2/3,1/5),(4/5,2/15,0,2/5,1/3,4/5)   (4th quantum dimension).  (1)

The S-matrices are given by

    S=[    1  d_X  d_Y  d_Z  d_U  d_V  d_W;  d_X -d_Z  d_V -d_W  d_U -d_Y    1;  d_Y  d_V  d_V  d_Y    0 -d_Y -d_V;  d_Z -d_W  d_Y    1 -d_U  d_V -d_X;  d_U  d_U    0 -d_U -d_U    0  d_U;  d_V -d_Y -d_Y  d_V    0 -d_V  d_Y;  d_W    1 -d_V -d_X  d_U  d_Y -d_Z ].

There are

    4(quantum dimensions)×2(conformal dimensions)×2(categorical dimensions)=16

MFCs, among which those four with the last quantum dimensions give unitary MFCs. We study connected étale algebras in all 16 MFCs simultaneously.

An ansatz

    A≅1⊕ n_XX⊕ n_YY⊕ n_ZZ⊕ n_UU⊕ n_VV⊕ n_WW

with n_j∈ N has

    _ B(A)=1+sin2π/15/sinπ/15n_X+sin3π/15/sinπ/15n_Y+sin4π/15/sinπ/15n_Z+sin5π/15/sinπ/15n_U+sin6π/15/sinπ/15n_V+sin7π/15/sinπ/15n_W.

For this to obey (<ref>), the natural number can take only 27 values

    (n_X,n_Y,n_Z,n_U,n_V,n_W)=   (0,0,0,0,0,0),(1,0,0,0,0,0),(2,0,0,0,0,0),
       (3,0,0,0,0,0),(4,0,0,0,0,0),(2,1,0,0,0,0),
       (2,0,1,0,0,0),(2,0,0,1,0,0),(1,1,0,0,0,0),
       (1,2,0,0,0,0),(1,0,1,0,0,0),(1,0,0,1,0,0),
       (1,0,0,0,1,0),(1,0,0,0,0,1),(0,1,0,0,0,0),
       (0,2,0,0,0,0),(0,1,1,0,0,0),(0,1,0,1,0,0),
       (0,1,0,0,1,0),(0,1,0,0,0,1),(0,0,1,0,0,0),
       (0,0,2,0,0,0),(0,0,1,1,0,0),(0,0,1,0,1,0),
       (0,0,0,1,0,0),(0,0,0,0,1,0),(0,0,0,0,0,1).

The first is nothing but the trivial connected étale algebra A≅1 giving B_A^0≃ B_A≃ B. Others with nontrivial simple objects b_j≇Z have nontrivial conformal dimensions, and do not give commutative algebras. Since Z has trivial conformal dimension h_Z=0 mod 1, it can give commutative algebra. However, those with Z's are also ruled out as follows. Apart from the trivial one, the natural number n_Z can take one or two. They have _ B=1+sin4π/15/sinπ/15,1+2sin4π/15/sinπ/15, and demand ( B_A^0)≈4.1,1.3, but there are no MFC with these Frobenius-Perron dimensions. Thus, the two candidates are ruled out.

We conclude

Namely, all the 16 MFCs B≃ psu(2)_13's are completely anisotropic.



 §.§ Rank eight



  §.§.§ B≃_ Z/2 Z× Z/2 Z× Z/2 Z^α≃_ Z/2 Z^-1⊠

The MFCs have eight simple objects {1,X,Y,Z,T,U,V,W} obeying monoidal products

Thus, they have

    _ B(1)=_ B(X)=_ B(Y)=_ B(Z)
       =_ B(T)=_ B(U)=_ B(V)=_ B(W)=1,

and

    ( B)=8.

Their quantum dimensions d_j's are solutions of the same multiplication rules d_id_j=∑_k=1^8N_ij^kd_k. There are eight solutions

    (d_X,d_Y,d_Z,d_T,d_U,d_V,d_W)=   (-1,-1,-1,-1,1,1,1),(-1,-1,1,1,-1,-1,1),
       (-1,1,-1,1,1,-1,-1),(-1,1,1,-1,-1,1,-1),
       (1,-1,-1,1,-1,1,-1),(1,-1,1,-1,1,-1,-1),
       (1,1,-1,-1,-1,-1,1),(1,1,1,1,1,1,1).

Only the last solution gives unitary MFCs. All quantum dimensions have the same categorical dimension

    D^2( B)=8.


In order to count the number of MFCs, let us list up conformal dimensions. Naively, the MFCs can have two structures _ Z/2 Z^-1⊠_ Z/2 Z^-1⊠_ Z/2 Z^-1 or _ Z/2 Z^-1⊠. However, by computing conformal dimensions, one finds they all have the structure _ Z/2 Z^-1⊠. (We have already specified this fact in the name.) In order to avoid double-counting, we choose _ Z/2 Z^-1={1,X} and ={1,U,V,W}. Then the other simple objects are given by

    Y≅ X⊗ W,   Z≅ X⊗ U,   T≅ X⊗ V.

We have h_X=1/4,3/4 (mod 1) and (h_U,h_V,h_W)=(0,0,1/2),(0,1/2,0),(1/2,0,0),(1/2,1/2,1/2) (mod 1). Depending on quantum dimensions, we have different symmetries.



  
(d_X,d_U,d_V,d_W)=(1,1,1,1). This quantum dimension gives unitary MFCs. Even after our choice, we still have permutations (UV),(UW),(VW) of simple objects. Different MFCs are given by

    (h_X,h_U,h_V,h_W)=(1/4,0,0,1/2),(1/4,1/2,1/2,1/2),(3/4,0,0,1/2),(3/4,1/2,1/2,1/2)  (1).

Including the two signs of categorical dimensions, we have eight unitary MFCs.



  
(d_X,d_U,d_V,d_W)=(1,1,-1,-1). We have smaller symmetry (VW). Thus, different MFCs are given by

    (h_X,h_U,h_V,h_W)   =(1/4,0,0,1/2),(1/4,1/2,0,0),(1/4,1/2,1/2,1/2),
           (3/4,0,0,1/2),(3/4,1/2,0,0),(3/4,1/2,1/2,1/2)  (1).

With two signs of categorical dimensions, we have 12 MFCs.



  
(d_X,d_U,d_V,d_W)=(-1,1,1,1). Different MFCs are given by

    (h_X,h_U,h_V,h_W)=(1/4,0,0,1/2),(1/4,1/2,1/2,1/2),(3/4,0,0,1/2),(3/4,1/2,1/2,1/2)  (1).

There are eight MFCs.



  
(d_X,d_U,d_V,d_W)=(-1,1,-1,-1). Different MFCs are given by

    (h_X,h_U,h_V,h_W)   =(1/4,0,0,1/2),(1/4,1/2,0,0),(1/4,1/2,1/2,1/2),
           (3/4,0,0,1/2),(3/4,1/2,0,0),(3/4,1/2,1/2,1/2)  (1).

There are 12 MFCs.

In total, there are

    8+12+8+12=40

MFCs, among which those eight in the first case are unitary. The S-matrices are given by

    S=[       1     d_X  d_Xd_W  d_Xd_U  d_Xd_V     d_U     d_V     d_W;     d_X      -1    -d_W    -d_U    -d_V  d_Xd_U  d_Xd_V  d_Xd_W;  d_Xd_W    -d_W      -1     d_W       1 -d_Xd_W    -d_X     d_X;  d_Xd_U    -d_U     d_W      -1     d_V     d_X -d_Xd_V -d_Xd_W;  d_Xd_V    -d_V       1     d_V      -1 -d_Xd_V     d_X    -d_X;     d_U  d_Xd_U -d_Xd_W     d_X -d_Xd_V       1    -d_V    -d_W;     d_V  d_Xd_V    -d_X -d_Xd_V     d_X    -d_V       1      -1;     d_W  d_Xd_W     d_X -d_Xd_W    -d_X    -d_W      -1       1 ].

They have additive central charges

    c( B)=c(_ Z/2 Z^-1)+c()  (8),

where

    c(_ Z/2 Z^-1)=1   (h_X=1/4),
    -1   (h_X=3/4),   c()=0   (one h=1/2&the other h=0),
    4   (all h=1/2).  (8)

We classify connected étale algebras in all 40 MFCs simultaneously.

An ansatz

    A≅1⊕ n_XX⊕ n_YY⊕ n_ZZ⊕ n_TT⊕ n_UU⊕ n_VV⊕ n_WW

with n_j∈ N has

    _ B(A)=1+n_X+n_Y+n_Z+n_T+n_U+n_V+n_W.

For this to obey (<ref>), the natural numbers can take only eight values

    (n_X,n_Y,n_Z,n_T,n_U,n_V,n_W)=   (0,0,0,0,0,0,0),(1,0,0,0,0,0,0),(0,1,0,0,0,0,0),(0,0,1,0,0,0,0),
       (0,0,0,1,0,0,0),(0,0,0,0,1,0,0),(0,0,0,0,0,1,0),(0,0,0,0,0,0,1).

The first is nothing but the trivial connected étale algebra A≅1 giving B_A^0≃ B_A≃ B. Among the other seven candidates, the second with X cannot be commutative due to our choice _ Z/2 Z^-1={1,X}. Accordingly, those three with Y,Z,T cannot be commutative either because they have h=1/4 (mod 1/2). Thus, we are left with three nontrivial candidates

    1⊕ U,  1⊕ V,  1⊕ W.

These are Z/2 Z algebras by the lemma 1, and they can also be commutative depending on quantum and conformal dimensions. A Z/2 Z algebra is commutative iff the nontrivial simple object has (d,h)=(1,0),(-1,1/2) <cit.>. Thus, the Z/2 Z algebras are commutative when

    c_U,U≅ id_1(d_X,d_U,d_V,d_W,h_X,h_U,h_V,h_W)   =(1,1,1,1,1/4,0,0,1/2),(1,1,1,1,3/4,0,0,1/2),
           (1,1,-1,-1,1/4,0,0,1/2),(1,1,-1,-1,3/4,0,0,1/2),
           (-1,1,1,1,1/4,0,0,1/2),(-1,1,1,1,3/4,0,0,1/2),
           (-1,1,-1,-1,1/4,0,0,1/2),(-1,1,-1,-1,3/4,0,0,1/2),
    c_V,V≅ id_1(d_X,d_U,d_V,d_W,h_X,h_U,h_V,h_W)   =(1,1,1,1,1/4,0,0,1/2),(1,1,1,1,3/4,0,0,1/2),
           (1,1,-1,-1,1/4,1/2,1/2,1/2),(1,1,-1,-1,3/4,1/2,1/2,1/2),
           (-1,1,1,1,1/4,0,0,1/2),(-1,1,1,1,3/4,0,0,1/2),
           (-1,1,-1,-1,1/4,1/2,1/2,1/2),(-1,1,-1,-1,3/4,1/2,1/2,1/2),
    c_W,W≅ id_1(d_X,d_U,d_V,d_W,h_X,h_U,h_V,h_W)   =(1,1,-1,-1,1/4,0,0,1/2),(1,1,-1,-1,1/4,1/2,1/2,1/2),
           (1,1,-1,-1,3/4,0,0,1/2),(1,1,-1,-1,3/4,1/2,1/2,1/2),
           (-1,1,-1,-1,1/4,0,0,1/2),(-1,1,-1,-1,1/4,1/2,1/2,1/2),
           (-1,1,-1,-1,3/4,0,0,1/2),(-1,1,-1,-1,3/4,1/2,1/2,1/2).
       (mod 1 for h)

Note that not all of these are separable. The obstruction is in quantum dimensions. For example, let us look at the third commutative algebra of 1⊕ V. If we naively use the formula (<ref>), a right A-module m≅1⊕ V has d_ B_A(m)=0, a contradiction.[In view of anyon condensation, this fact leads to the following observation. In this case, 1 and V which are naively `identified' under anyon condensation have different quantum dimensions. While anyon condensation can `identify' two (or more) simple objects with different conformal dimensions – in this case, the resulting object is confined – it may not be allowed to `identify' simple objects with quantum dimensions of different signs. Since most literature on anyon condensation studies only unitary MFCs, it seems this problem was unknown.]  This line of reasoning rules out the second and fourth lines of 1⊕ V, and all of 1⊕ W. Therefore, we find three connected étale algebras

    A≅1   (all MFCs),
    1⊕ U   (those in (<ref>)),
    1⊕ V   (d_U,d_V,d_W,h_U,h_V,h_W)=(1,1,1,0,0,1/2).  (1 for h)


Let us determine the categories of (dyslectic) right A-modules B_A^0, B_A. The nontrivial connected étale algebras have _ B(A)=2 and demand

    ( B_A^0)=2,  ( B_A)=4.

Since the only MFC with =2 is _ Z/2 Z^-1, we get

    B_A^0≃_ Z/2 Z^-1.

This identification also matches central charges because when the algebras are separable, we have c()=0 mod 8 and c( B_A^0) is determined by h_X as we will see. The category B_A of right A-modules contain this MFC as a subcategory and has =4. There are three candidates, , _ Z/2 Z× Z/2 Z^α, and _ Z/4 Z^α. Here, note that all simple objects of B has _ B=1 because the free module functor preserves Frobenius-Perron dimensions (<ref>). This observation rules out , and we are left with rank four candidates _ Z/2 Z× Z/2 Z^α,_ Z/4 Z^α. In order to find out the correct category, we search for four-dimensional NIM-reps. We start from A≅1⊕ U. We find a four-dimensional NIM-rep

    n_1=1_4=n_U,   n_X=[ 0 0 0 1; 0 0 1 0; 0 1 0 0; 1 0 0 0 ]=n_Z,   n_Y=[ 0 0 1 0; 0 0 0 1; 1 0 0 0; 0 1 0 0 ]=n_T,   n_V=[ 0 1 0 0; 1 0 0 0; 0 0 0 1; 0 0 1 0 ]=n_W.

Denoting a basis of B_A by {m_1,m_2,m_3,m_4}, we obtain a multiplication table

We can identify

    m_1≅1⊕ U,   m_2≅ V⊕ W,   m_3≅ Y⊕ T,   m_4≅ X⊕ Z.

They have

    d_ B_A(m_1)=1,   d_ B_A(m_2)=d_ B(V),   d_ B_A(m_3)=d_ B(X)d_ B(V),   d_ B_A(m_4)=d_ B(X).

One can also obtain these identifications via the free module functor F_A:

    F_A(1)≅1⊕ U≅ F_A(U),   F_A(X)≅ X⊕ Z≅ F_A(Z),   F_A(Y)≅ Y⊕ T≅ F_A(T),   F_A(V)≅ V⊕ W≅ F_A(W).

They obey the Z/2 Z× Z/2 Z monoidal products

This shows B_A≃_ Z/2 Z× Z/2 Z^α.

For the other nontrivial connected étale algebra, just the names of matrices change. For A≅1⊕ V, we have

    n_1=1_4=n_V,   n_X=[ 0 0 0 1; 0 0 1 0; 0 1 0 0; 1 0 0 0 ]=n_T,   n_Y=[ 0 0 1 0; 0 0 0 1; 1 0 0 0; 0 1 0 0 ]=n_Z,   n_U=[ 0 1 0 0; 1 0 0 0; 0 0 0 1; 0 0 1 0 ]=n_W.

It has identifications

    m_1≅1⊕ V,   m_2≅ X⊕ T,   m_3≅ Y⊕ Z,   m_4≅ U⊕ W.

They obey the same Z/2× Z/2 Z monoidal products

This shows B_A≃_ Z/2 Z× Z/2 Z^α.

We conclude

Namely, 16 MFCs B≃_ Z/2 Z× Z/2 Z× Z/2 Z^α≃_ Z/2 Z^-1⊠'s in (<ref>) fail to be completely anisotropic, while the other 24 with

    (d_X,d_U,d_V,d_W,h_X,h_U,h_V,h_W)   =(1,1,1,1,1/4,1/2,1/2,1/2),(1,1,1,1,3/4,1/2,1/2,1/2),
           (1,1,-1,-1,1/4,1/2,0,0),(1,1,-1,-1,1/4,1/2,1/2,1/2),
           (1,1,-1,-1,3/4,1/2,0,0),(1,1,-1,-1,3/4,1/2,1/2,1/2),
           (-1,1,1,1,1/4,1/2,1/2,1/2),(-1,1,1,1,3/4,1/2,1/2,1/2),
           (-1,1,-1,-1,1/4,1/2,0,0),(-1,1,-1,-1,1/4,1/2,1/2,1/2),
           (-1,1,-1,-1,3/4,1/2,0,0),(-1,1,-1,-1,3/4,1/2,1/2,1/2)  (1 for h)

are completely anisotropic.



  §.§.§ B≃_ Z/2 Z× Z/4 Z^α

The MFCs have eight simple objects {1,X,Y,Z,T,U,V,W} obeying monoidal products

(One can identify _ Z/2 Z^-1={1,X},_ Z/4 Z^α={1,Z,V,W}, and Y≅ X⊗ Z,T≅ X⊗ W,U≅ X⊗ V.) Thus, they have

    _ B(1)=_ B(X)=_ B(Y)=_ B(Z)
       =_ B(T)=_ B(U)=_ B(V)=_ B(W)=1,

and

    ( B)=8.

Their quantum dimensions d_j's are solutions of the same multiplication rules d_id_j=∑_k=1^8N_ij^kd_k. There are four solutions

    (d_X,d_Y,d_Z,d_T,d_U,d_V,d_W)=   (-1,-1,1,-1,-1,1,1),(-1,-1,1,1,1,-1,-1),
       (1,1,1,-1,-1,-1,-1),(1,1,1,1,1,1,1).

Only the last quantum dimensions give unitary MFCs. All the quantum dimensions have the same categorical dimension

    D^2( B)=8.

All quantum dimensions have the same four conformal dimensions[Naively, one finds 16 consistent conformal dimensions, but the others are equivalent to one of four in the main text under permutations (XY)(TU) or (XY)(VW) or (XY)(TVUW) of simple objects.]

    (h_X,h_Y,h_Z,h_T,h_U,h_V,h_W)=   (1/4,3/4,1/2,1/8,1/8,3/8,3/8),(1/4,3/4,1/2,1/8,1/8,7/8,7/8),
       (1/4,3/4,1/2,3/8,3/8,5/8,5/8),(1/4,3/4,1/2,5/8,5/8,7/8,7/8).  (1)

The S-matrices are given by

    S=[        1      d_X   d_Xd_Z      d_Z   d_Xd_W   d_Xd_V      d_V      d_W;      d_X       -1     -d_Z   d_Xd_Z     -d_W     -d_V   d_Xd_V   d_Xd_W;   d_Xd_Z     -d_Z       -1      d_X  -d_Zd_W      d_V  -d_Xd_V  -d_Xd_W;      d_Z   d_Xd_Z      d_X        1  -d_Xd_W  -d_Xd_V     -d_V     -d_W;   d_Xd_W     -d_W  -d_Zd_W  -d_Xd_W      ∓ i      ± i ∓ i· d_X ± i· d_X;   d_Xd_V     -d_V      d_V  -d_Xd_V      ± i      ∓ i ± i· d_X ∓ i· d_X;      d_V   d_Xd_V  -d_Xd_V     -d_V ∓ i· d_X ± i· d_X      ± i      ∓ i;      d_W   d_Xd_W  -d_Xd_W     -d_W ± i· d_X ∓ i· d_X      ∓ i      ± i ].

Regardless of quantum dimensions, they have additive central charges

    c( B)=2   (1st h),
    0   (2nd h),
    4   (3rd h),
    -2   (4th h).  (8)

There are

    4(quantum dimensions)×4(conformal dimensions)×2(categorical dimensions)=32

MFCs, among which those eight with the fourth quantum dimensions give unitary MFCs. We study connected étale algebras in all 32 MFCs simultaneously.

The most general form

    A≅1⊕ n_XX⊕ n_YY⊕ n_ZZ⊕ n_TT⊕ n_UU⊕ n_VV⊕ n_WW

with n_j∈ N has

    _ B(A)=1+n_X+n_Y+n_Z+n_T+n_U+n_V+n_W.

For this to obey (<ref>), the natural numbers can take only eight values

    (n_X,n_Y,n_Z,n_T,n_U,n_V,n_W)=   (0,0,0,0,0,0,0),(1,0,0,0,0,0,0),(0,1,0,0,0,0,0),(0,0,1,0,0,0,0),
       (0,0,0,1,0,0,0),(0,0,0,0,1,0,0),(0,0,0,0,0,1,0),(0,0,0,0,0,0,1).

The first is nothing but the trivial connected étale algebra A≅1 giving B_A^0≃ B_A≃ B. The other six candidates except the fourth with Z have nontrivial simple object, and they fail to be commutative. The fourth candidate 1⊕ Z does pass the necessary condition (<ref>), but since the simple object Z has (d_Z,h_Z)=(1,1/2) (mod 1 for h_Z), it has c_Z,Z≅-id_1 <cit.>. Thus, it does not give commutative algebra either.

We conclude

All the 32 MFCs B≃_ Z/2 Z× Z/4 Z^α's are completely anisotropic.



  §.§.§ B≃ su(8)_1≃_ Z/8 Z^α

The MFCs have eight simple objects {1,X,Y,Z,T,U,V,W} obeying monoidal products

Thus, they have

    _ B(1)=_ B(X)=_ B(Y)=_ B(Z)
       =_ B(T)=_ B(U)=_ B(V)=_ B(W)=1,

and

    ( B)=8.

Their quantum dimensions d_j's are solutions of the same multiplication rules d_id_j=∑_k=1^8N_ij^kd_k. There are two solutions

    (d_X,d_Y,d_Z,d_T,d_U,d_V,d_W)=(1,-1,-1,-1,-1,1,1),(1,1,1,1,1,1,1).

Only the second gives unitary MFCs. The two quantum dimensions both have the same four conformal dimensions[Naively, one finds eight consistent conformal dimensions, but the others are equivalent to one in the main text under permutations (YU)(ZT) of simple objects.]

    (h_X,h_Y,h_Z,h_T,h_U,h_V,h_W)=   (0,1/16,1/16,9/16,9/16,1/4,1/4),(0,3/16,3/16,11/16,11/16,3/4,3/4),
       (0,5/16,5/16,13/16,13/16,1/4,1/4),(0,7/16,7/16,15/16,15/16,3/4,3/4).  (1)

The S-matrices are given by

    S=[          1        d_X        d_Y        d_Z        d_T        d_U        d_V        d_W;        d_X          1       -d_Y       -d_Z       -d_T       -d_U        d_V        d_W;        d_Y       -d_Y ± e^±π i/4 ± e^∓π i/4 ∓ e^∓π i/4 ∓ e^±π i/4        ∓ i        ± i;        d_Z       -d_Z ± e^∓π i/4 ± e^±π i/4 ∓ e^±π i/4 ∓ e^∓π i/4        ± i        ∓ i;        d_T       -d_T ∓ e^∓π i/4 ∓ e^±π i/4 ± e^±π i/4 ± e^∓π i/4        ± i        ∓ i;        d_U       -d_U ∓ e^±π i/4 ∓ e^∓π i/4 ± e^∓π i/4 ± e^±π i/4        ∓ i        ± i;        d_V        d_V        ∓ i        ± i        ± i        ∓ i         -1         -1;        d_W        d_W        ± i        ∓ i        ∓ i        ± i         -1         -1 ].

They have additive central charges

    c( B)=
    1   (1st&3rd h),
    -1   (2nd&4th h).  (8)

There are

    2(quantum dimensions)×4(conformal dimensions)×2(categorical dimensions)=16

MFCs, among which those eight with the second quantum dimensions give unitary MFCs. We classify connected étale algebras in all 16 MFCs simultaneously.

We work with an ansatz

    A≅1⊕ n_XX⊕ n_YY⊕ n_ZZ⊕ n_TT⊕ n_UU⊕ n_VV⊕ n_WW

with n_j∈ N. It has

    _ B(A)=1+n_X+n_Y+n_Z+n_T+n_U+n_V+n_W.

For this to obey (<ref>), the natural numbers can take only eight values

    (n_X,n_Y,n_Z,n_T,n_U,n_V,n_W)=   (0,0,0,0,0,0,0),(1,0,0,0,0,0,0),(0,1,0,0,0,0,0),(0,0,1,0,0,0,0),
       (0,0,0,1,0,0,0),(0,0,0,0,1,0,0),(0,0,0,0,0,1,0),(0,0,0,0,0,0,1).

The first is nothing but the trivial connected étale algebra A≅1 giving B_A^0≃ B_A≃ B. Those six with Y,Z,T,U,V,W fail to satisfy the necessary condition (<ref>), and they are ruled out. We are left with the second n_X=1 or A≅1⊕ X. It is a Z/2 Z algebra by the lemma 1. It turns out that this is connected étale; it is commutative because X has (d_X,h_X)=(1,0) (mod 1 for h_X), and hence c_X,X≅ id_1 <cit.>. To check the separability, we identify B_A.

Since it has _ B(A)=2, it demands

    ( B_A^0)=2,  ( B_A)=4.

The category B_A^0 of dyslectic modules is identified as _ Z/2 Z^-1 because it is the only MFC with =2. This identification also matches central charges.[The Z/2 Z MFC has

    c(_ Z/2 Z^-1)=1   (h_ Z/2 Z=1/4),
    -1   (h_ Z/2 Z=3/4).

We will see B_A^0={1⊕ X,V⊕ W}. The nontrivial deconfined particle has h_V=1/4,3/4=h_W, and this determines the central charge c( B_A^0).] The category B_A of right A-modules should contain the MFC and have =4. It turns out that

    B_A≃_ Z/4 Z^α.

One of the easiest ways to see this fact is to perform anyon condensation. Under the procedure, we `identify' 1 and X, and hence the pairs (Y,U), (Z,T), and (V,W). Since they obey Z/4 Z monoidal products, we arrive the identification.[More rigorously, we should find NIM-reps. Indeed, we found a four-dimensional NIM-rep

    n_1=1_4=n_X,   n_Y=[ 0 0 0 1; 0 0 1 0; 1 0 0 0; 0 1 0 0 ]=n_U,   n_Z=[ 0 0 1 0; 0 0 0 1; 0 1 0 0; 1 0 0 0 ]=n_T,   n_V=[ 0 1 0 0; 1 0 0 0; 0 0 0 1; 0 0 1 0 ]=n_W.

Denoting a basis of B_A by {m_1,m_2,m_3,m_4}, we obtain a multiplication table




    b_j⊗\   m_1   m_2   m_3   m_4
1,X   m_1   m_2   m_3   m_4

    Y,U   m_4   m_3   m_1   m_2

    Z,T   m_3   m_4   m_2   m_1

    V,W   m_2   m_1   m_4   m_3
.



In this basis, we can identify

    m_1≅1⊕ X,   m_2≅ V⊕ W,   m_3≅ Z⊕ T,   m_4≅ Y⊕ U.

In the category B_A of right A-modules, they have quantum dimensions (<ref>):

    d_ B_A(m_1)=1=d_ B_A(m_2),   d_ B_A(m_3)=d_ B(Y)=d_ B_A(m_4).

Furthermore, they obey monoidal products




    ⊗_A   m_1   m_2   m_3   m_4
m_1   m_1   m_2   m_3   m_4
m_2      m_1   m_4   m_3
m_3         m_2   m_1
m_4            m_2
.




This shows B_A≃_ Z/4 Z^α.]

We conclude

That is, all the 16 MFCs B≃_ Z/8 Z^α's fail to be completely anisotropic.



  §.§.§ B≃⊠_ Z/2 Z× Z/2 Z^α

The MFCs have eight simple objects {1,X,Y,Z,T,U,V,W} obeying monoidal products

(One can identify ={1,W},_ Z/2 Z× Z/2 Z^α={1,X,Y,Z}, and T≅ Y⊗ W,U≅ X⊗ W,V≅ Z⊗ W.) Thus, they have

    _ B(1)=_ B(X)=_ B(Y)=_ B(Z)=1,
       _ B(T)=_ B(U)=_ B(V)=_ B(W)=ζ:=1+√(5)/2,

and

    ( B)=10+2√(5)≈14.5.

Their quantum dimensions d_j's are solutions of the same multiplication rules d_id_j=∑_k=1^8N_ij^kd_k. There are eight solutions

    (d_X,d_Y,d_Z,d_T,d_U,d_V,d_W)=   (-1,-1,1,-ζ,-ζ,ζ,ζ),(-1,-1,1,ζ^-1,ζ^-1,-ζ^-1,-ζ^-1),
       (-1,1,-1,-ζ^-1,ζ^-1,ζ^-1,-ζ^-1),(-1,1,-1,ζ,-ζ,-ζ,ζ),
       (1,-1,-1,-ζ,ζ,-ζ,ζ),(1,-1,-1,ζ^-1,-ζ^-1,ζ^-1,-ζ^-1),
       (1,1,1,-ζ^-1,-ζ^-1,-ζ^-1,-ζ^-1),(1,1,1,ζ,ζ,ζ,ζ).

Only the last quantum dimensions give unitary MFCs. They have categorical dimensions

    D^2( B)=10-2√(5)(≈5.5),  10+2√(5).

In order to list up conformal dimensions without double-counting, we perform case analysis. (This fixes redundancies in names of simple objects, and hence superficially makes descriptions asymmetric.)



  
B≃⊠_ Z/2 Z^-1⊠_ Z/2 Z^-1. In this case, the _ Z/2 Z^-1⊠_ Z/2 Z^-1 factor has three classes depending on quantum dimensions (d_X,d_Y) <cit.>. Together with d_W=ζ,-ζ^-1, we have six classes.

    
  * (d_X,d_Y,d_W)=(1,1,ζ). This gives unitary MFCs. Different MFCs are given by conformal dimensions
    
    (h_X,h_Y,h_W)=(1/4,1/4,2/5),(1/4,1/4,3/5),(1/4,3/4,2/5),(1/4,3/4,3/5),(3/4,3/4,2/5),(3/4,3/4,3/5)  (1).

    (The other conformal dimensions of b_k≅ b_i⊗ b_j is given by h_k=h_i+h_j mod 1.) Including two signs of categorical dimensions, we have 12 unitary MFCs.
    
  * (d_X,d_Y,d_W)=(1,1,-ζ^-1). Different MFCs are given by
    
    (h_X,h_Y,h_W)=(1/4,1/4,1/5),(1/4,1/4,4/5),(1/4,3/4,1/5),(1/4,3/4,4/5),(3/4,3/4,1/5),(3/4,3/4,4/5)  (1).

    With two signs of categorical dimensions, there are 12 MFCs.
    
  * (d_X,d_Y,d_W)=(1,-1,ζ). Different MFCs are given by
    
    (h_X,h_Y,h_W)=(1/4,1/4,2/5),(1/4,1/4,3/5),(1/4,3/4,2/5),(1/4,3/4,3/5),(3/4,1/4,2/5),(3/4,1/4,3/5),(3/4,3/4,2/5),(3/4,3/4,3/5)  (1).

    Taking two signs of categorical dimensions into account, we get 16 MFCs.
    
  * (d_X,d_Y,d_W)=(1,-1,-ζ^-1). Different MFCs are given by
    
    (h_X,h_Y,h_W)=(1/4,1/4,1/5),(1/4,1/4,4/5),(1/4,3/4,1/5),(1/4,3/4,4/5),(3/4,1/4,1/5),(3/4,1/4,4/5),(3/4,3/4,1/5),(3/4,3/4,4/5)  (1).

    With two categorical dimensions, we get 16 MFCs.
    
  * (d_X,d_Y,d_W)=(-1,-1,ζ). Different MFCs are given by
    
    (h_X,h_Y,h_W)=(1/4,1/4,2/5),(1/4,1/4,3/5),(1/4,3/4,2/5),(1/4,3/4,3/5),(3/4,3/4,2/5),(3/4,3/4,3/5)  (1).

    There are 12 MFCs.
    
  * (d_X,d_Y,d_W)=(-1,-1,-ζ^-1). Different MFCs are given by
    
    (h_X,h_Y,h_W)=(1/4,1/4,1/5),(1/4,1/4,4/5),(1/4,3/4,1/5),(1/4,3/4,4/5),(3/4,3/4,1/5),(3/4,3/4,4/5)  (1).

    There are 12 MFCs.

In total, there are

    12+12+16+16+12+12=80

MFCs B≃⊠_ Z/2 Z^-1⊠_ Z/2 Z^-1, among which those 12 with quantum dimensions (d_X,d_Y,d_W)=(1,1,ζ) are unitary. The S-matrices are given by

    S=[         1       d_X       d_Y    d_Xd_Y    d_Yd_W    d_Xd_W d_Xd_Yd_W       d_W;       d_X        -1         1      -d_X       d_W      -d_W   -d_Xd_W    d_Xd_W;       d_Y         1        -1      -d_Y      -d_W       d_W   -d_Yd_W    d_Yd_W;    d_Xd_Y      -d_X      -d_Y         1   -d_Yd_W   -d_Xd_W       d_W d_Xd_Yd_W;    d_Yd_W       d_W      -d_W   -d_Yd_W         1        -1       d_Y      -d_Y;    d_Xd_W      -d_W       d_W   -d_Xd_W        -1         1       d_X      -d_X; d_Xd_Yd_W   -d_Xd_W   -d_Yd_W       d_W       d_Y       d_X        -1   -d_Xd_Y;       d_W    d_Xd_W    d_Yd_W d_Xd_Yd_W      -d_Y      -d_X   -d_Xd_Y        -1 ].

They have additive central charges

    c( B)=c()+c(_ Z/2 Z^-1)+c(_ Z/2 Z^-1)  (8)

where

    c()=2/5   (h_W=1/5),
    -2/5   (h_W=4/5),
    14/5   (h_W=2/5),
    -14/5   (h_W=3/5).   c(_ Z/2 Z^-1)=1   (h_ Z/2 Z=1/4),
    -1   (h_ Z/2 Z=3/4).  (8)




  
B≃⊠. In this case, simple objects X,Y,Z can have conformal dimensions

    (h_X,h_Y,h_Z)=(0,0,1/2),(0,1/2,0),(1/2,0,0),(1/2,1/2,1/2)  (1).

Thus, we have four classes.

    
  * (d_X,d_Y,d_Z,d_W)=(1,1,1,ζ). This gives unitary MFCs. Different MFCs are given by
    
    (h_X,h_Y,h_Z,h_W)=(0,0,1/2,2/5),(0,0,1/2,3/5),(1/2,1/2,1/2,2/5),(1/2,1/2,1/2,3/5)  (1).

    With the two signs of categorical dimensions, there are eight unitary MFCs.
    
  * (d_X,d_Y,d_Z,d_W)=(1,1,1,-ζ^-1). Different MFCs are given by
    
    (h_X,h_Y,h_Z,h_W)=(0,0,1/2,1/5),(0,0,1/2,4/5),(1/2,1/2,1/2,1/5),(1/2,1/2,1/2,4/5)  (1).

    Including the two signs of categorical dimensions, we have eight MFCs.
    
  * (d_X,d_Y,d_Z,d_W)=(1,-1,-1,ζ). Different MFCs are given by
    
    (h_X,h_Y,h_Z,h_W)=(0,0,1/2,2/5),(0,0,1/2,3/5),(1/2,0,0,2/5),(1/2,0,0,3/5),(1/2,1/2,1/2,2/5),(1/2,1/2,1/2,3/5)  (1).

    There are 12 MFCs.
    
  * (d_X,d_Y,d_Z,d_W)=(1,-1,-1,-ζ^-1). Different MFCs are given by
    
    (h_X,h_Y,h_Z,h_W)=(0,0,1/2,1/5),(0,0,1/2,4/5),(1/2,0,0,1/5),(1/2,0,0,4/5),(1/2,1/2,1/2,1/5),(1/2,1/2,1/2,4/5)  (1).

    There are 12 MFCs.

In total, there are

    8+8+12+12=40

MFCs B≃⊠, among which those eight with the quantum dimensions (d_X,d_Y,d_Z,d_W)=(1,1,1,ζ) are unitary. The S-matrices are given by

    S=[       1     d_X     d_Y     d_Z  d_Yd_W  d_Xd_W  d_Zd_W     d_W;     d_X       1    -d_Y    -d_Z -d_Yd_W     d_W -d_Zd_W  d_Xd_W;     d_Y    -d_Y       1      -1     d_W -d_Yd_W    -d_W  d_Yd_W;     d_Z    -d_Z      -1       1    -d_W -d_Zd_W     d_W  d_Zd_W;  d_Yd_W -d_Yd_W     d_W    -d_W      -1     d_Y       1    -d_Y;  d_Xd_W     d_W -d_Yd_W -d_Zd_W     d_Y      -1     d_Z    -d_X;  d_Zd_W -d_Zd_W    -d_W     d_W       1     d_Z      -1    -d_Z;     d_W  d_Xd_W  d_Yd_W  d_Zd_W    -d_Y    -d_X    -d_Z      -1 ].

They have additive central charges

    c( B)=c()+c()  (8)

where

    c()=2/5   (h_W=1/5),
    -2/5   (h_W=4/5),
    14/5   (h_W=2/5),
    -14/5   (h_W=3/5).   c()=0   (one h=1/2&the other h=0),
    4   (all h=1/2).  (8).


Including both B≃⊠_ Z/2 Z^-1⊠_ Z/2 Z^-1 and B≃⊠, we have

    80+40=120

MFCs with fusion ring K(⊠_ Z/2 Z× Z/2 Z^α). Among them, 12 from the first and eight from the second MFCs are unitary, 20 in total.

Having listed MFCs, we can classify connected étale algebras in them. We work with an ansatz

    A≅1⊕ n_XX⊕ n_YY⊕ n_ZZ⊕ n_TT⊕ n_UU⊕ n_VV⊕ n_WW

with n_j∈ N. It has

    _ B(A)=1+n_X+n_Y+n_Z+ζ(n_T+n_U+n_V+n_W).

For this to obey (<ref>), the natural numbers can take only 26 values

    (n_X,n_Y,n_Z,n_T,n_U,n_V,n_W)=   (0,0,0,0,0,0,0),(1,0,0,0,0,0,0),(2,0,0,0,0,0,0),
       (1,1,0,0,0,0,0),(1,0,1,0,0,0,0),(1,0,0,1,0,0,0),
       (1,0,0,0,1,0,0),(1,0,0,0,0,1,0),(1,0,0,0,0,0,1),
       (0,1,0,0,0,0,0),(0,2,0,0,0,0,0),(0,1,1,0,0,0,0),
       (0,1,0,1,0,0,0),(0,1,0,0,1,0,0),(0,1,0,0,0,1,0),
       (0,1,0,0,0,0,1),(0,0,1,0,0,0,0),(0,0,2,0,0,0,0),
       (0,0,1,1,0,0,0),(0,0,1,0,1,0,0),(0,0,1,0,0,1,0),
       (0,0,1,0,0,0,1),(0,0,0,1,0,0,0),(0,0,0,0,1,0,0),
       (0,0,0,0,0,1,0),(0,0,0,0,0,0,1).

The first is nothing but the trivial connected étale algebra A≅1 giving B≃ B_A^0≃ B_A. Whether the others can be commutative depends on quantum and conformal dimensions.

For B≃⊠_ Z/2 Z^-1⊠_ Z/2 Z^-1, X,Y,T,U,V,W have nontrivial conformal dimensions, and candidates with them fail to be commutative. Hence, we are left with candidates with just Z's. It has (d_Z,h_Z)=(d_Xd_Y,h_X+h_Y) (mod 1 for h_Z). Thus, those with (d_Z,h_Z)=(1,0),(-1,1/2) (mod 1 for h_Z) are commutative. More concretely, independent of quantum and conformal dimensions of W, we have

    c_Z,Z≅ id_1(d_X,d_Y,h_X,h_Y)=(1,1,1/4,3/4),(-1,-1,1/4,3/4),(1,-1,1/4,1/4),(1,-1,3/4,3/4)  (1 for h).

Therefore, we are left with two additional candidates for connected étale algebras

    A≅1⊕ Z,1⊕2Z.

The second candidate A≅1⊕2Z has _ B(A)=3, and in particular demands

    ( B_A^0)=5+√(5)/9≈1.61.

However, there is no MFC with this Frobenius-Perron dimension. Thus, the candidate is ruled out. We are left with the first candidate. It is a Z/2 Z algebra by the lemma 1. It also turns out separable. To show this, we identify B_A.

It has _ B(A)=2, and demands

    ( B_A^0)=5+√(5)/2,  ( B_A)=5+√(5).

Since the only MFC with =5+√(5)/2 is a Fibonacci MFC, we arrive

    B_A^0≃.

The matching of central charges makes the identification more precise. The Fibonacci category has

    c()=±2/5   (d_=-ζ^-1),
    ±14/5   (d_=ζ).  (4)

The _ Z/2 Z× Z/2 Z^α has

    c(_ Z/2 Z× Z/2 Z^α)=2   (_ Z/2 Z× Z/2 Z^α≃_ Z/2 Z^-1⊠_ Z/2 Z^-1 with h_X=h_Y),
    0   (_ Z/2 Z× Z/2 Z^α≃_ Z/2 Z^-1⊠_ Z/2 Z^-1 with h_X≠ h_Y).  (4)

We see the second _ Z/2 Z^-1⊠_ Z/2 Z^-1 MFCs with different conformal dimensions (mod 1) trivially match additive central charges, while the first _ Z/2 Z^-1⊠_ Z/2 Z^-1 with the same conformal dimensions cannot. Therefore, although (d_Z,h_Z)=(-1,1/2) in (<ref>) has trivial self braiding c_Z,Z≅ id_1, they do not give connected étale algebras. We conclude

    A≅1⊕ Z is connected étale(d_X,d_Y,h_X,h_Y)=(1,1,1/4,3/4),(-1,-1,1/4,3/4)  (1 for h).

The category B_A of right A-modules should contain this MFC as a subcategory. Together with the Frobenius-Perron dimension, it is identified as[A four-dimensional NIM-rep is given by

    n_1=1_4=n_Z,   n_X=[ 0 1 0 0; 1 0 0 0; 0 0 0 1; 0 0 1 0 ]=n_Y,   n_T=[ 0 0 1 0; 0 0 0 1; 1 0 0 1; 0 1 1 0 ]=n_U,   n_V=[ 0 0 0 1; 0 0 1 0; 0 1 1 0; 1 0 0 1 ]=n_W.

The solution gives identifications

    m_1≅1⊕ Z,   m_2≅ X⊕ Y,   m_3≅ T⊕ U,   m_4≅ V⊕ W.

The simple objects m_1,4 are deconfined, while m_2,3 are confined, i.e., B_A^0={m_1,m_4}. In B_A, they have

    d_ B_A(m_1)=1,   d_ B_A(m_2)=±1,   d_ B_A(m_3)=±ζ^±1,   d_ B_A(m_4)=±ζ^±1.

Signs in the last quantum dimension are correlated, but those in the third are not. Employing the free module functor F_A, one obtains monoidal products ⊗_A:




    ⊗_A   m_1   m_2   m_3   m_4
m_1   m_1   m_2   m_3   m_4
m_2      m_1   m_4   m_3
m_3         m_1⊕ m_4   m_2⊕ m_3
m_4            m_1⊕ m_4
.



This shows the identification.]

    B_A≃ su(2)_3.


For B≃⊠, T,U,V,W have nontrivial conformal dimensions, and candidates with them fail to be commutative. On the other hand, those with just X,Y,Z can have c_b_j,b_j≅ id_1 depending on their quantum and conformal dimensions. From the lists above, independent of h_W, we find

    c_X,X≅ id_1(d_X,d_Y,d_Z,d_W,h_X,h_Y,h_Z,h_W)   =(1,1,1,ζ,0,0,1/2,2/5),(1,1,1,ζ,0,0,1/2,3/5),
           (1,1,1,-ζ^-1,0,0,1/2,1/5),(1,1,1,-ζ^-1,0,0,1/2,4/5),
           (1,-1,-1,ζ,0,0,1/2,2/5),(1,-1,-1,ζ,0,0,1/2,3/5),
           (1,-1,-1,-ζ^-1,0,0,1/2,1/5),(1,-1,-1,-ζ^-1,0,0,1/2,4/5),
    c_Y,Y≅ id_1(d_X,d_Y,d_Z,d_W,h_X,h_Y,h_Z,h_W)   =(1,1,1,ζ,0,0,1/2,2/5),(1,1,1,ζ,0,0,1/2,3/5),
           (1,1,1,-ζ^-1,0,0,1/2,1/5),(1,1,1,-ζ^-1,0,0,1/2,4/5),
           (1,-1,-1,ζ,1/2,1/2,1/2,2/5),(1,-1,-1,ζ,1/2,1/2,1/2,3/5),
           (1,-1,-1,-ζ^-1,1/2,1/2,1/2,1/5),(1,-1,-1,-ζ^-1,1/2,1/2,1/2,4/5),
    c_Z,Z≅ id_1(d_X,d_Y,d_Z,d_W,h_X,h_Y,h_Z,h_W)   =(1,-1,-1,ζ,0,0,1/2,2/5),(1,-1,-1,ζ,0,0,1/2,3/5),
           (1,-1,-1,ζ,1/2,1/2,1/2,2/5),(1,-1,-1,ζ,1/2,1/2,1/2,3/5),
           (1,-1,-1,-ζ^-1,0,0,1/2,1/5),(1,-1,-1,-ζ^-1,0,0,1/2,4/5),
           (1,-1,-1,-ζ^-1,1/2,1/2,1/2,1/5),(1,-1,-1,-ζ^-1,1/2,1/2,1/2,4/5).
       (mod 1 for h)

How about the matching of central charges? We will see the Toric Code MFC has conformal dimensions (h_X,h_Y,h_Z)=(0,0,1/2) when it admits nontrivial connected étale algebras. Such MFCs have c()=0 mod 8, and the central charges are trivially matched because both c( B) and c() are determined by the same conformal dimension h_W.

Just as for B≃⊠_ Z/2 Z^-1⊠_ Z/2 Z^-1, the candidates with _ B=2 can admit MFCs B_A^0, but those with _ B=3 do not, and are ruled out. Note that not all candidates in (<ref>) are separable. The same observation as in section <ref> leads to

    A≅1   (all MFCs),
    1⊕ X   (those in (<ref>)),
    1⊕ Y   (d_X,d_Y,d_Z,h_X,h_Y,h_Z)=(1,1,1,0,0,1/2).  (1 for h)

In order to figure out B_A, we have to find NIM-reps. Here, since NIM-reps do not depend on conformal dimensions, we immediately learn[If one searches for NIM-reps, one realizes just names of matrices change. One finds

    n_1=1_4=n_X,   n_Y=[ 0 1 0 0; 1 0 0 0; 0 0 0 1; 0 0 1 0 ]=n_Z,   n_T=[ 0 0 1 0; 0 0 0 1; 1 0 0 1; 0 1 1 0 ]=n_V,   n_U=[ 0 0 0 1; 0 0 1 0; 0 1 1 0; 1 0 0 1 ]=n_W

for A≅ 1⊕ X, and

    n_1=1_4=n_Y,   n_X=[ 0 1 0 0; 1 0 0 0; 0 0 0 1; 0 0 1 0 ]=n_Z,   n_U=[ 0 0 1 0; 0 0 0 1; 1 0 0 1; 0 1 1 0 ]=n_V,   n_T=[ 0 0 0 1; 0 0 1 0; 0 1 1 0; 1 0 0 1 ]=n_W

for A≅ 1⊕ Y. They give identifications

    m_1≅1⊕ X,   m_2≅ Y⊕ Z,   m_3≅ T⊕ V,   m_4≅ U⊕ W,

and

    m_1≅1⊕ Y,   m_2≅ X⊕ Z,   m_3≅ U⊕ V,   m_4≅ T⊕ W,

respectively.]

    B_A^0≃,   B_A≃ su(2)_3.


We conclude

and

In particular, 64 MFCs B≃⊠_ Z/2 Z^-1⊠_ Z/2 Z^-1's not in (<ref>), and 24 MFCs B≃⊠'s with

    (d_X,d_Y,d_Z,d_W,h_X,h_Y,h_Z,h_W)   =(1,1,1,ζ,1/2,1/2,1/2,2/5),(1,1,1,ζ,1/2,1/2,1/2,3/5),
           (1,1,1,-ζ^-1,1/2,1/2,1/2,1/5),(1,1,1,-ζ^-1,1/2,1/2,1/2,4/5),
           (1,-1,-1,ζ,1/2,0,0,2/5),(1,1,1,ζ,1/2,0,0,3/5),
           (1,-1,-1,ζ,1/2,
        1/2,1/2,2/5),(1,1,1,ζ,1/2,1/2,1/2,3/5),
           (1,-1,-1,-ζ^-1,1/2,0,0,1/5),(1,-1,-1,-ζ^-1,1/2,0,0,4/5),
           (1,-1,-1,-ζ^-1,1/2,1/2,1/2,1/5),(1,-1,-1,-ζ^-1,1/2,1/2,1/2,4/5)  (1 for h)

are completely anisotropic.



  §.§.§ B≃⊠_ Z/4 Z^α

The MFCs have eight simple objects {1,X,Y,Z,T,U,V,W} obeying monoidal products

(One can identify ={1,U},_ Z/4 Z^α={1,X,Y,Z}, and T≅ X⊗ U,V≅ Z⊗ U,W≅ Y⊗ U.) Thus, they have

    _ B(1)=_ B(X)=_ B(Y)=_ B(Z)=1,
       _ B(T)=_ B(U)=_ B(V)=_ B(W)=ζ:=1+√(5)/2,

and

    ( B)=10+2√(5)≈14.5.

Their quantum dimensions d_j's are solutions of the same multiplication rules d_id_j=∑_k=1^8N_ij^kd_k. There are four solutions

    (d_X,d_Y,d_Z,d_T,d_U,d_V,d_W)=   (1,-1,-1,ζ,ζ,-ζ,-ζ),(1,-1,-1,-ζ^-1,-ζ^-1,ζ^-1,ζ^-1),
       (1,1,1,-ζ^-1,-ζ^-1,-ζ^-1,-ζ^-1),(1,1,1,ζ,ζ,ζ,ζ).

(Only the last quantum dimensions give unitary MFCs.) They have categorical dimensions

    D^2( B)=10-2√(5)(≈5.5),  10+2√(5).

They have eight conformal dimensions.

    
  * d_U=ζ. These have
    
    (h_X,h_Y,h_Z,h_T,h_U,h_V,h_W)   =(1/2,1/8,1/8,1/10,3/5,29/40,29/40),(1/2,1/8,1/8,9/10,2/5,21/40,21/40),
           (1/2,3/8,3/8,1/10,3/5,39/40,39/40),(1/2,3/8,3/8,9/10,2/5,31/40,31/40),
           (1/2,5/8,5/8,1/10,3/5,9/40,9/40),(1/2,5/8,5/8,9/10,2/5,1/40,1/40),
           (1/2,7/8,7/8,1/10,3/5,19/40,19/40),(1/2,7/8,7/8,9/10,2/5,11/40,11/40)  (1).

    
  * d_U=-ζ^-1. These have
    
    (h_X,h_Y,h_Z,h_T,h_U,h_V,h_W)   =(1/2,1/8,1/8,3/10,4/5,37/40,37/40),(1/2,1/8,1/8,7/10,1/5,13/40,13/40),
           (1/2,3/8,3/8,3/10,4/5,7/40,7/40),(1/2,3/8,3/8,7/10,1/5,23/40,23/40),
           (1/2,5/8,5/8,3/10,4/5,17/40,17/40),(1/2,5/8,5/8,7/10,1/5,33/40,33/40),
           (1/2,7/8,7/8,3/10,4/5,27/40,27/40),(1/2,7/8,7/8,7/10,1/5,3/40,3/40)  (1).


The S-matrices are given by

    S=[        1      d_X      d_Y      d_Z   d_Xd_U      d_U   d_Zd_U   d_Yd_U;      d_X        1     -d_Y     -d_Z   d_Xd_U      d_U  -d_Zd_U  -d_Yd_U;      d_Y     -d_Y      ± i      ∓ i  -d_Yd_U   d_Yd_U ∓ i· d_U ± i· d_U;      d_Z     -d_Z      ∓ i      ± i  -d_Zd_U   d_Zd_U ± i· d_U ∓ i· d_U;   d_Xd_U   d_Xd_U  -d_Yd_U  -d_Zd_U       -1     -d_X      d_Z      d_Y;      d_U      d_U   d_Yd_U   d_Zd_U     -d_X       -1     -d_Z     -d_Y;   d_Zd_U  -d_Zd_U ∓ i· d_U ± i· d_U      d_Z     -d_Z      ∓ i      ± i;   d_Yd_U  -d_Yd_U ± i· d_U ∓ i· d_U      d_Y     -d_Y      ± i      ∓ i ].

They have

    c( B)=c()+c(_ Z/4 Z^α)  (8)

where

    c()=2/5   (h_=1/5),
    -2/5   (h_=4/5),
    14/5   (h_=2/5),
    -14/5   (h_=3/5).   c(_ Z/4 Z^α)=1   (h_ Z/4 Z=1/8),
    3   (h_ Z/4 Z=3/8),
    -3   (h_ Z/4 Z=5/8),
    -1   (h_ Z/4 Z=7/8),  (8).

There are

    4(quantum dimensions)×8(conformal dimensions)×2(categorical dimensions)=64

MFCs, among which those 16 with the fourth quantum dimensions give unitary MFCs. We classify connected étale algebras in all 64 MFCs simultaneously.

We work with an ansatz

    A≅1⊕ n_XX⊕ n_YY⊕ n_ZZ⊕ n_TT⊕ n_UU⊕ n_VV⊕ n_WW

with n_j∈ N. It has

    _ B(A)=1+n_X+n_Y+n_Z+ζ(n_T+n_U+n_V+n_W).

For this to obey (<ref>), exactly the same as the previous example, the natural numbers can take only 26 values

    (n_X,n_Y,n_Z,n_T,n_U,n_V,n_W)=   (0,0,0,0,0,0,0),(1,0,0,0,0,0,0),(2,0,0,0,0,0,0),
       (1,1,0,0,0,0,0),(1,0,1,0,0,0,0),(1,0,0,1,0,0,0),
       (1,0,0,0,1,0,0),(1,0,0,0,0,1,0),(1,0,0,0,0,0,1),
       (0,1,0,0,0,0,0),(0,2,0,0,0,0,0),(0,1,1,0,0,0,0),
       (0,1,0,1,0,0,0),(0,1,0,0,1,0,0),(0,1,0,0,0,1,0),
       (0,1,0,0,0,0,1),(0,0,1,0,0,0,0),(0,0,2,0,0,0,0),
       (0,0,1,1,0,0,0),(0,0,1,0,1,0,0),(0,0,1,0,0,1,0),
       (0,0,1,0,0,0,1),(0,0,0,1,0,0,0),(0,0,0,0,1,0,0),
       (0,0,0,0,0,1,0),(0,0,0,0,0,0,1).

The first is nothing but the trivial connected étale algebra A≅1 giving B≃ B_A^0≃ B_A. Those with X's cannot give commutative algebras because X has (d_X,h_X)=(1,1/2) (mod 1 for h) and c_X,X≅-id_1 <cit.>. The others also fail to be commutative because nontrivial simple objects entering them have nontrivial conformal dimensions.

We conclude

That is, all the 64 MFCs B≃⊠_ Z/4 Z^α's are completely anisotropic.



  §.§.§ B≃_ Z/2 Z^-1⊠⊠

The MFCs have eight simple objects {1,X,Y,Z,T,U,V,W} obeying monoidal products

(One can identify _ Z/2 Z^-1={1,X},={1,T},{1,U}, and Y≅ X⊗ U,Z≅ X⊗ T,V≅ X⊗ T⊗ U,W≅ T⊗ U.) Thus, they have

    _ B(1)=1=_ B(X),     _ B(Y)=_ B(Z)=_ B(T)=_ B(U)=ζ,
       _ B(V)=3+√(5)/2=_ B(W),

and

    ( B)=15+5√(5)≈26.2.

Their quantum dimensions d_j's are solutions of the same multiplication rules d_id_j=∑_k=1^8N_ij^kd_k. There are eight solutions

    (d_X,d_Y,d_Z,d_T,d_U,d_V,d_W)
       =(-1,ζ^-1,ζ^-1,-ζ^-1,-ζ^-1,√(5)-3/2,3-√(5)/2),(1,-ζ^-1,-ζ^-1,-ζ^-1,-ζ^-1,3-√(5)/2,3-√(5)/2),
           (-1,-ζ,ζ^-1,-ζ^-1,ζ,1,-1),(1,ζ,-ζ^-1,-ζ^-1,ζ,-1,-1),
           (-1,ζ^-1,-ζ,ζ,-ζ^-1,1,-1),(1,-ζ^-1,ζ,ζ,-ζ^-1,-1,-1),
           (-1,-ζ,-ζ,ζ,ζ,-3+√(5)/2,3+√(5)/2),(1,ζ,ζ,ζ,ζ,3+√(5)/2,3+√(5)/2)

with categorical dimensions

    D^2( B)=15-5√(5)(≈3.8),  10,  15+5√(5).

Only the last quantum dimensions give unitary MFCs.

In order to list up their conformal dimensions without double-counting, we perform case analysis.

    
  * (d_X,d_T,d_U)=(1,ζ,ζ). This gives unitary MFCs. Different MFCs are given by
    
    (h_X,h_Y,h_Z,h_T,h_U,h_V,h_W)   =(1/4,13/20,13/20,2/5,2/5,1/20,4/5),(1/4,17/20,13/20,2/5,3/5,1/4,0),
           (1/4,17/20,17/20,3/5,3/5,9/20,1/5),(3/4,3/20,3/20,2/5,2/5,11/20,4/5),
           (3/4,7/20,3/20,2/5,3/5,3/4,0),(3/4,7/20,7/20,3/5,3/5,19/20,1/5)  (1).

    Including two signs of categorical dimensions, we have 12 unitary MFCs.
    
  * (d_X,d_T,d_U)=(-1,ζ,ζ). The sets of conformal dimensions are the same as the previous case. There are 12 MFCs.
    
  * (d_X,d_T,d_U)=(1,ζ,-ζ^-1). Different MFCs are given by
    
    (h_X,h_Y,h_Z,h_T,h_U,h_V,h_W)   =(1/4,9/20,13/20,2/5,1/5,17/20,3/5),(1/4,1/20,13/20,2/5,4/5,9/20,1/5),
           (1/4,9/20,17/20,3/5,1/5,1/20,4/5),(1/4,1/20,17/20,3/5,4/5,13/20,2/5),
           (3/4,19/20,3/20,2/5,1/5,7/20,3/5),(3/4,11/20,3/20,2/5,4/5,19/20,1/5),
           (3/4,19/20,7/20,3/5,1/5,11/20,4/5),(3/4,11/20,7/20,3/5,4/5,3/20,2/5)  (1).

    With two signs of categorical dimensions, there are 16 MFCs.
    
  * (d_X,d_T,d_U)=(-1,ζ,-ζ^-1). The sets of conformal dimensions are the same as the previous case. There are 16 MFCs.
    
  * (d_X,d_T,d_U)=(1,-ζ^-1,-ζ^-1). Different MFCs are given by
    
    (h_X,h_Y,h_Z,h_T,h_U,h_V,h_W)   =(1/4,9/20,9/20,1/5,1/5,13/20,2/5),(1/4,1/20,9/20,1/5,4/5,1/4,0),
           (1/4,1/20,1/20,4/5,4/5,17/20,3/5),(3/4,19/20,19/20,1/5,1/5,3/20,2/5),
           (3/4,11/20,19/20,1/5,4/5,3/4,0),(3/4,11/20,11/20,4/5,4/5,7/20,3/5)  (1).

    There are 12 MFCs.
    
  * (d_X,d_T,d_U)=(-1,-ζ^-1,-ζ^-1). The sets of conformal dimensions are the same as the previous case, and there are 12 MFCs.

The S-matrices are given by

    S=[         1       d_X    d_Xd_U    d_Xd_T       d_T       d_U d_Xd_Td_U    d_Td_U;       d_X        -1      -d_U      -d_T    d_Xd_T    d_Xd_U   -d_Td_U d_Xd_Td_U;    d_Xd_U      -d_U         1   -d_Td_U d_Xd_Td_U      -d_X       d_T   -d_Xd_T;    d_Xd_T      -d_T   -d_Td_U         1      -d_X d_Xd_Td_U       d_U   -d_Xd_U;       d_T    d_Xd_T d_Xd_Td_U      -d_X        -1    d_Td_U   -d_Xd_U      -d_U;       d_U    d_Xd_U      -d_X d_Xd_Td_U    d_Td_U        -1   -d_Xd_T      -d_T; d_Xd_Td_U   -d_Td_U       d_T       d_U   -d_Xd_U   -d_Xd_T        -1       d_X;    d_Td_U d_Xd_Td_U   -d_Xd_T   -d_Xd_U      -d_U      -d_T       d_X         1 ].

They have additive central charges

    c( B)=c(_ Z/2 Z^-1)+c()+c()  (8)

where

    c(_ Z/2 Z^-1)=1   (h_X=1/4),
    -1   (h_X=3/4).,   c()=2/5   (h_T,U=1/5),
    -2/5   (h_T,U=4/5),
    14/5   (h_T,U=2/5),
    -14/5   (h_T,U=3/5).  (8)

There are

    2×12+2×16+2×12=80

MFCs, among which those 12 with the last quantum dimensions are unitary. We classify connected étale algebras in all 80 MFCs simultaneously.

An ansatz

    A≅1⊕ n_XX⊕ n_YY⊕ n_ZZ⊕ n_TT⊕ n_UU⊕ n_VV⊕ n_WW

with n_j∈ N has

    _ B(A)=1+n_X+ζ(n_Y+n_Z+n_T+n_U)+3+√(5)/2(n_V+n_W).

For this to obey (<ref>), the natural numbers can take only 31 values

    (n_X,n_Y,n_Z,n_T,n_U,n_V,n_W)=   (0,0,0,0,0,0,0),(1,0,0,0,0,0,0),(2,0,0,0,0,0,0),
       (3,0,0,0,0,0,0),(4,0,0,0,0,0,0),(2,1,0,0,0,0,0),
       (2,0,1,0,0,0,0),(2,0,0,1,0,0,0),(2,0,0,0,1,0,0),
       (1,1,0,0,0,0,0),(1,0,1,0,0,0,0),(1,0,0,1,0,0,0),
       (1,0,0,0,1,0,0),(1,0,0,0,0,1,0),(1,0,0,0,0,0,1),
       (0,1,0,0,0,0,0),(0,2,0,0,0,0,0),(0,1,1,0,0,0,0),
       (0,1,0,1,0,0,0),(0,1,0,0,1,0,0),(0,0,1,0,0,0,0),
       (0,0,2,0,0,0,0),(0,0,1,1,0,0,0),(0,0,1,0,1,0,0),
       (0,0,0,1,0,0,0),(0,0,0,2,0,0,0),(0,0,0,1,1,0,0),
       (0,0,0,0,1,0,0),(0,0,0,0,2,0,0),(0,0,0,0,0,1,0),
       (0,0,0,0,0,0,1).

The first is nothing but the trivial connected étale algebra A≅1 giving B_A^0≃ B_A≃ B. Next, those with X's cannot be commutative because the simple object has (d_X,h_X)=(±1,1/4) (mod 1/2 for h) and c_X,X≅± i· id_1 <cit.>. Thirdly, the simple objects Y,Z,T,U,V have nontrivial conformal dimensions, and candidates with them are ruled out. Thus, we are left with those with just W's. Concretely, the only nontrivial candidate is n_W=1 or A≅1⊕ W. When h_W=0 mod 1, it can be commutative. Indeed, it is known <cit.> to be a commutative algebra in ⊠. It further turns out to be separable. To show this point, we identify B_A.

It has _ B(A)=5+√(5)/2, and demands

    ( B_A^0)=2,  ( B_A)=5+√(5).

Calculating b_j⊗ A, we find simple objects

    1⊕ W,   X⊕ V,   Y⊕ Z⊕ V,   T⊕ U⊕ W.

This suggests B_A has rank four. Indeed, we find a four-dimensional NIM-rep

    n_1=1_4,   n_X=[ 0 1 0 0; 1 0 0 0; 0 0 0 1; 0 0 1 0 ],   n_Y=[ 0 0 1 0; 0 0 0 1; 1 0 0 1; 0 1 1 0 ]=n_Z,
    
        n_T=[ 0 0 0 1; 0 0 1 0; 0 1 1 0; 1 0 0 1 ]=n_U,   n_V=[ 0 1 1 0; 1 0 0 1; 1 0 0 2; 0 1 2 0 ],   n_W=[ 1 0 0 1; 0 1 1 0; 0 1 2 0; 1 0 0 2 ].

This gives identifications

    m_1≅1⊕ W,   m_2≅ X⊕ V,   m_3≅ Y⊕ Z⊕ V,   m_4≅ T⊕ U⊕ W.

The first two have _ B_A=1, and form B_A^0≃_ Z/2 Z^-1. This also matches central charges because c(⊠)=0 (mod 8) when h_W=0 (mod 1), and h_X=h_V gives the conformal dimension of m_2∈ B_A^0. One can identify B_A by working out the monoidal products:

We get

    B_A≃ su(2)_3.

Since this is semisimple, A is separable, hence étale:

    A≅1⊕ W  (d_T,d_U,h_T,h_U)=(ζ,ζ,2/5,3/5),(-ζ^-1,-ζ^-1,1/5,4/5).  (1 for h)


We conclude

Namely, those 16 in (<ref>) fail to be completely anisotropic, while the other 64 MFCs B≃_ Z/2 Z^-1⊠⊠'s are completely anisotropic.



  §.§.§ B≃ so(9)_2

The MFCs have eight simple objects {1,X,Y,Z,T,U,V,W} obeying monoidal products

Thus, they have

    _ B(1)=1=_ B(X),     _ B(Y)=_ B(Z)=_ B(T)=_ B(U)=2,
       _ B(V)=_ B(W)=3,

and

    ( B)=36.

Their quantum dimensions d_j's are solutions of the same multiplication rules d_id_j=∑_k=1^8N_ij^kd_k. There are two (nonzero) solutions

    (d_X,d_Y,d_Z,d_T,d_U,d_V,d_W)=(1,2,2,2,2,-3,-3),(1,2,2,2,2,3,3).

Only the second gives unitary MFCs. They both have the same categorical dimension

    D^2( B)=36,

and four conformal dimensions[Naively, one finds 24 conformal dimensions, but the other 20 are given by permutations (YTU) and (VW). Thus, different MFCs are labeled by the four conformal dimensions in the main text.]

    (h_X,h_Y,h_Z,h_T,h_U,h_V,h_W)   =(0,1/9,0,7/9,4/9,0,1/2),(0,1/9,0,7/9,4/9,1/4,3/4),
           (0,2/9,0,5/9,8/9,0,1/2),(0,2/9,0,5/9,8/9,1/4,3/4)  (1).

The S-matrices are given by

    S=[    1  d_X  d_Y  d_Z  d_T  d_U  d_V  d_W;  d_X  d_X  d_Y  d_Z  d_T  d_U -d_V -d_W;  d_Y  d_Y    s   -2   s'   s”    0    0;  d_Z  d_Z   -2    4   -2   -2    0    0;  d_T  d_T   s'   -2   s”    s    0    0;  d_U  d_U   s”   -2    s   s'    0    0;  d_V -d_V    0    0    0    0 -d_V  d_W;  d_W -d_W    0    0    0    0  d_W -d_W ]

with

    s=4sinπ/18,   s'=4cos2π/9,   s”=-4cosπ/9,

or their permutations (ss's”). They have additive central charges

    c( B)=0  (8).

There are

    2(quantum dimensions)×4(conformal dimensions)×2(categorical dimensions)=16

MFCs, among which those eight with the second quantum dimensions are unitary. We classify connected étale algebras in all 16 MFCs simultaneously.

An ansatz

    A≅1⊕ n_XX⊕ n_YY⊕ n_ZZ⊕ n_TT⊕ n_UU⊕ n_VV⊕ n_WW

with n_j∈ N has

    _ B(A)=1+n_X+2(n_Y+n_Z+n_T+n_U)+3(n_V+n_W).

For this to obey (<ref>), the natural numbers can take only 56 values. The sets contain one with all n_j's be zero. It is nothing but the trivial connected étale algebra A≅1 giving B_A^0≃ B_A≃ B. Next, let us study candidates with nontrivial simple object(s). We can rule out those with Y,T,U,W because they have nontrivial conformal dimensions and fail to meet the necessary condition (<ref>). Thus, we are left with candidates with just X,Z,V. Note that {1,X,Z} form symmetric pre-modular fusion subcategory Rep(S_3).

Solving (<ref>) by setting n_Y,n_T,n_U,n_W to zero, we get 16 sets

    (n_X,n_Z,n_V)   =(0,0,0),(1,0,0),(2,0,0),(3,0,0),
           (4,0,0),(5,0,0),(3,1,0),(2,1,0),
           (2,0,1),(1,1,0),(1,2,0),(1,0,1),
           (0,2,0),(0,1,1),(0,1,0),(0,0,1).

Some of these are ruled out by studying Frobenius-Perron dimensions. The three candidates

    (n_X,n_Z,n_V)=(3,0,0),(1,1,0),(0,0,1)

have _ B=4 and demand ( B_A^0)=36/16. However, there is no MFC with such Frobenius-Perron dimensions. Thus, the three candidates are ruled out. Similarly, four candidates

    (n_X,n_Z,n_V)=(4,0,0),(2,1,0),(0,2,0),(1,0,1)

are ruled out because there is no MFC with =36/25. Thus, we are left with nine candidates

    (n_X,n_Z,n_V)=(0,0,0),(1,0,0),(2,0,0),(5,0,0),(3,1,0),(2,0,1),(1,2,0),(0,1,1),(0,1,0).

The first gives A≅1, and we already found this above. Among the other eight, from the lemma 1, we know three with (n_X,n_Z,n_V)=(1,0,0),(1,2,0),(0,1,0) or A≅1⊕ X,1⊕ X⊕2Z,1⊕ Z are commutative algebras because they are so in A∈Rep(S_3)⊂ so(9)_2 for all conformal dimensions. It turns out all of these are also separable, hence étale. Let us check this point one after another by identifying B_A.



  
A≅1⊕ X. Since it has _ B(A)=2, it demands

    ( B_A^0)=9,  ( B_A)=18.

In view of anyon condensation, we `identify' 1,X, hence V,W. The other Y,Z,T,U `split' into two each. This suggests B_A have rank 10. Indeed, we find a 10-dimensional NIM-rep

    n_1=1_10=n_X,      n_Y=[ 0 1 1 0 0 0 0 0 0 0; 1 0 0 0 0 0 0 1 0 0; 1 0 0 0 0 0 0 0 1 0; 0 0 0 0 0 0 1 0 1 0; 0 0 0 0 0 1 0 1 0 0; 0 0 0 0 1 0 1 0 0 0; 0 0 0 1 0 1 0 0 0 0; 0 1 0 0 1 0 0 0 0 0; 0 0 1 1 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 2 ],   n_Z=[ 0 0 0 1 1 0 0 0 0 0; 0 0 0 0 0 1 0 0 1 0; 0 0 0 0 0 0 1 1 0 0; 1 0 0 0 1 0 0 0 0 0; 1 0 0 1 0 0 0 0 0 0; 0 1 0 0 0 0 0 0 1 0; 0 0 1 0 0 0 0 1 0 0; 0 0 1 0 0 0 1 0 0 0; 0 1 0 0 0 1 0 0 0 0; 0 0 0 0 0 0 0 0 0 2 ],
       n_T=[ 0 0 0 0 0 1 1 0 0 0; 0 0 0 1 0 0 1 0 0 0; 0 0 0 0 1 1 0 0 0 0; 0 1 0 0 0 0 0 1 0 0; 0 0 1 0 0 0 0 0 1 0; 1 0 1 0 0 0 0 0 0 0; 1 1 0 0 0 0 0 0 0 0; 0 0 0 1 0 0 0 0 1 0; 0 0 0 0 1 0 0 1 0 0; 0 0 0 0 0 0 0 0 0 2 ],   n_U=[ 0 0 0 0 0 0 0 1 1 0; 0 0 1 0 1 0 0 0 0 0; 0 1 0 1 0 0 0 0 0 0; 0 0 1 0 0 1 0 0 0 0; 0 1 0 0 0 0 1 0 0 0; 0 0 0 1 0 0 0 1 0 0; 0 0 0 0 1 0 0 0 1 0; 1 0 0 0 0 1 0 0 0 0; 1 0 0 0 0 0 1 0 0 0; 0 0 0 0 0 0 0 0 0 2 ],
       n_V=[ 0 0 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 0 0 1; 1 1 1 1 1 1 1 1 1 0 ]=n_W.

The solution gives identifications

    m_1≅1⊕ X,   m_2≅ Y≅ m_3,   m_4≅ Z≅ m_5,   m_6≅ T≅ m_7,   m_8≅ U≅ m_9,   m_10≅ V⊕ W.

They have quantum dimensions (<ref>)

    d_ B_A(m_j)=1 (j=1,2,…,9),   d_ B_A(m_10)=±3.

Furthermore, they obey monoidal products

    m_j⊗_Am_10≅ m_10≅ m_10⊗_Am_j (j=1,2,…,9),   m_10⊗_Am_10≅⊕_j=1^9m_j.

This shows

    B_A^0≃_ Z/9 Z^1,   B_A≃TY( Z/9 Z).

The identification also matches central charges. See section <ref>. Since TY( Z/9 Z) is semisimple, A is separable.



  
A≅1⊕ X⊕2Z. It has _ B(A)=6, and demands

    ( B_A^0)=1,  ( B_A)=6.

We can identify B_A^0≃. This matches central charges. Calculating b_j⊗ A, we find candidate simple objects of B_A

    1⊕ X⊕2Z,   Y⊕ T⊕ U,   V⊕ W

with Frobenius-Perron dimensions one. Logically, the latter two can have coefficients, but the possibilities are ruled out.[Here is a proof.

First, the third simple object should have coefficient one. Since it always appears in the form 3(V⊕ W) in b_j⊗ A, one can have 3(V⊕ W), or V⊕ W,2(V⊕ W) as simple object(s). The first possibility contributes 3^2 to ( B_A), and it exceeds six. The latter contributes 1^2+2^2, and together with other simple objects, the contributions exceed six. Thus, the coefficient of V⊕ W should be one.

Second, the second simple object should also have coefficient one. It can have coefficient two. Then, the three simple objects match Frobenius-Perron dimension 1^2+2^2+1^2=6. Furthermore, we find a three-dimensional NIM-rep

    n_1=1_3=n_X,   n_Y=n_T=n_U=[ 0 1 0; 2 1 0; 0 0 2 ],   n_Z=[ 2 0 0; 0 2 0; 0 0 2 ],   n_V=[ 0 0 3; 0 0 6; 1 1 0 ]=n_W

with the identifications. However, working out the monoidal products ⊗_A, we find the putative fusion ring is not isomorphic to any of rank three fusion categories. In particular, there is no fusion category with m_2⊗_Am_3≅2m_3. Thus, the second simple object should also have coefficient one.] In view of anyon condensation, the second and the third simple objects can `split' into two and three, respectively. In order to match the Frobenius-Perron dimension, they should `split.' Thus, we search for six-dimensional NIM-reps. Indeed, we find a solution

    n_1=1_6=n_X,   n_Y=n_T   =n_U=[ 0 1 1 0 0 0; 1 0 1 0 0 0; 1 1 0 0 0 0; 0 0 0 0 1 1; 0 0 0 1 0 1; 0 0 0 1 1 0 ],
    
        n_Z=[ 2 0 0 0 0 0; 0 2 0 0 0 0; 0 0 2 0 0 0; 0 0 0 2 0 0; 0 0 0 0 2 0; 0 0 0 0 0 2 ],     n_V=[ 0 0 0 1 1 1; 0 0 0 1 1 1; 0 0 0 1 1 1; 1 1 1 0 0 0; 1 1 1 0 0 0; 1 1 1 0 0 0 ]=n_W.

This gives identifications

    m_1≅1⊕ X⊕2Z,   m_2≅ Y⊕ T⊕ U≅ m_3,   m_4≅ m_5≅ m_6≅ V⊕ W.

In B_A, they have quantum dimensions

    d_1=d_2=d_3=1,   d_4=d_5=d_6=±1.

The result tells us B_A should have rank six. Let us try to figure out B_A.

As all simple objects have Frobenius-Perron dimensions one, it should be multiplicity-free. Thus, the fusion ring would be either FR^6,2_1 or FR^6,4_1.[We follow the notation of AnyonWiki <cit.>. A symbol FR^r,n_i denotes a fusion ring with rank r and n non-self-dual simple objects. The subscript i labels different fusion rings with the same (r,n).] We can also work out monoidal product ⊗_A, but the free module functor cannot fix it uniquely. What we can fix is

    m_1⊗_Am_j≅ m_j≅ m_j⊗_Am_1 (j=1,2,…,6),   m_2⊗_Am_2≅ m_3,   m_2⊗_Am_3≅ m_1≅ m_3⊗_Am_2,   m_3⊗_Am_3≅ m_2.

Namely, {m_1,m_2,m_3} form the Z/3 Z fusion ring. We also learn

    (m_2⊕ m_3)⊗_A(m_4⊕ m_5⊕ m_6)   ≅ F_A(Y⊗ V)
       ≅ F_A(V⊕ W)
       ≅2(m_4⊕ m_5⊕ m_6)≅(m_4⊕ m_5⊕ m_6)⊗_A(m_2⊕ m_3),
    
        (m_4⊕ m_5⊕ m_6)⊗_A(m_4⊕ m_5⊕ m_6)   ≅ F_A(V⊗ V)
       ≅ F_A(1⊕ Y⊕ Z⊕ T⊕ U)
       ≅3(m_1⊕ m_2⊕ m_3).

When m_4⊗_Am_4≅ m_5⊗_Am_5≅ m_6⊗_Am_6≅ m_1, using consistency and associativity, we get two fusion rings

or

depending on, say, m_4⊗_Am_5. Note that the non-commutativity appears automatically as a consequence of consistency and associativity. These are isomorphic to the fusion ring FR^6,2_1.[The identification with the notation in AnyonWiki <cit.> is given by

    m_1≅1,   m_2≅6,   m_3≅5,   m_4≅2,   m_5≅3,   m_6≅4,

or

    m_1≅1,   m_2≅5,   m_3≅6,   m_4≅2,   m_5≅3,   m_6≅4,

respectively.] On the other hand, when m_4⊗_Am_4≅ m_1,m_5⊗_Am_5≅ m_2,m_6⊗_Am_6≅ m_3, consistency and associativity give a unique fusion ring

Note that the commutativity also appears automatically as a consequence of consistency and associativity. This is isomorphic to the fusion ring FR^6,4_1.[The identification with the notation in AnyonWiki <cit.> is given by

    m_1≅1,   m_2≅5,   m_3≅6,   m_4≅2,   m_5≅3,   m_6≅4.
] (Since m_4,5,6 enter symmetrically, the other monoidal products are given by permutations of simple objects.) Therefore, we find

    B_A≃ C(FR^6,2_1) or  C(FR^6,4_1).

(It is unsatisfactory that we cannot fix B_A uniquely. However, since the two have the same ranks, the ambiguity does not affect our physical applications.) Since this is semisimple, A is separable, hence étale.



  
A≅1⊕ Z. It has _ B(A)=3, and demands

    ( B_A^0)=4,  ( B_A)=12.

Matching of additive central charges, Frobenius-Perron dimensions, and the invariance of topological twists only allow

    B_A^0≃_ Z/2 Z^-1⊠_ Z/2 Z^-1   (h_V,h_W)=(1/4,3/4),
       (h_V,h_W)=(0,1/2).  (1)

The category B_A of right A-modules has additional simple objects. A natural scenario has two more simple objects with Frobenius-Perron dimensions two so that 1+1+1+1+2^2+2^2=12. Indeed, we find a six-dimensional NIM-rep

    n_1=1_6,   n_X=[ 0 1 0 0 0 0; 1 0 0 0 0 0; 0 0 1 0 0 0; 0 0 0 1 0 0; 0 0 0 0 0 1; 0 0 0 0 1 0 ],   n_Y=n_T=n_U=[ 0 0 1 0 0 0; 0 0 1 0 0 0; 1 1 1 0 0 0; 0 0 0 1 1 1; 0 0 0 1 0 0; 0 0 0 1 0 0 ],
    
        n_Z=[ 1 1 0 0 0 0; 1 1 0 0 0 0; 0 0 2 0 0 0; 0 0 0 2 0 0; 0 0 0 0 1 1; 0 0 0 0 1 1 ],   n_V=[ 0 0 0 1 1 0; 0 0 0 1 0 1; 0 0 0 2 1 1; 1 1 2 0 0 0; 1 0 1 0 0 0; 0 1 1 0 0 0 ],   n_W=[ 0 0 0 1 0 1; 0 0 0 1 1 0; 0 0 0 2 1 1; 1 1 2 0 0 0; 0 1 1 0 0 0; 1 0 1 0 0 0 ].

The solution gives identifications

    m_1≅1⊕ Z,   m_2≅ X⊕ Z,   m_3≅ Y⊕ T⊕ U,   m_4≅ V⊕ W,   m_5≅ V,   m_6≅ W.

In view of anyon condensation, m_1,2,5,6 are deconfined, and form B_A^0. The other two simple objects m_3,4 are confined. In B_A, they have quantum dimensions

    d_ B_A(m_1)=1=d_ B_A(m_2),   d_ B_A(m_3)=2,   d_ B_A(m_4)=±2,   d_ B_A(m_5)=±1=d_ B_A(m_6).

Working out the monoidal products ⊗_A, we find

The fusion ring is isomorphic to FR^6,0_2.[The identification of simple objects with the notation in AnyonWiki is given by

    m_1≅1,   m_2≅4,   m_3≅6,   m_4≅5,   m_5≅2,   m_6≅3,

or its permutation (23).] This in particular implies 1⊕ Z is separable, and étale.

Now, we are left with (n_X,n_Z,n_V)=(2,0,0),(5,0,0),(3,1,0),(2,0,1),(0,1,1). We study these in turn. It turns out only the last, A≅1⊕ Z⊕ V, is connected étale.



  
A≅1⊕2X. It has _ B(A)=3, and demands

    ( B_A^0)=4,  ( B_A)=12.

Calculating b_j⊗ A, we find candidate simple objects of B_A:

    1⊕2X,  2⊕ X,  3Y,  3Z,  3T,  3U,   V,   W.

(The putative B_A may also have V⊕ W, but our discussion below does not depend on the presence/absence.) In a putative B_A, they have Frobenius-Perron dimensions

    1,  1,  2,  2,  2,  2,  1,  1,

respectively. In order to match additive central charges and Frobenius-Perron dimensions, B_A^0 should consist of {1⊕2X,2⊕ X,V,W}. The four additional objects contribute 4×2^2=16 to ( B_A), and exceeds 12.[One may wonder why it is not allowed to pick just two candidate simple objects from those four with Frobenius-Perron dimensions two. This can match Frobenius-Perron dimensions, but one finds the actions b_j⊗- cannot be closed.] Thus, the candidate is ruled out.



  
A≅1⊕5X. It has _ B(A)=6, and demands

    ( B_A^0)=1,  ( B_A)=6.

Computing b_j⊗ A, we find candidate simple objects of B_A  with the smallest Frobenius-Perron dimensions:

    1⊕5X,  5⊕ X,  3Y,  3Z,  3T,  3U,   V⊕ W,  … .

They have Frobenius-Perron dimensions

    1,  1,  1,  1,  1,  1,  1,  … ,

respectively. Again, their contributions to the Frobenius-Perron dimension exceed six, and the candidate is ruled out.




  
A≅1⊕3X⊕ Z. It has _ B(A)=6, and demands

    ( B_A^0)=1,  ( B_A)=6.

Calculating b_j⊗ A, we find candidate simple objects of B_A:

    1⊕3X⊕ Z,  3⊕ X⊕ Z,   Y⊕ T⊕ U,  3Y,  3T,  3U,  … .

Note that there should exist candidate simple objects made of V,W. They have Frobenius-Perron dimensions

    1,  1,  1,  1,  1,  1,  … ,

respectively. Again, their contributions to Frobenius-Perron dimension exceed six, and the candidate is ruled out.



  
A≅1⊕2X⊕ V. It has _ B(A)=6, and demands

    ( B_A^0)=1,  ( B_A)=6.

With the free module functor F_A(b_j)=b_j⊗ A, we find candidate simple objects of B_A:

    1⊕2X⊕ V,  2⊕ X⊕ W,  3Y,  3Z,  3T,  3U,   V⊕ W,  … .

There should exist more simple object, but they do not affect the following discussion. They have Frobenius-Perron dimensions

    1,  1,  1,  1,  1,  1,  1,  … ,

respectively. Their contributions to ( B_A) exceed six, and the candidate is ruled out.



  
A≅1⊕ Z⊕ V. It has _ B(A)=6, and demands

    ( B_A^0)=1,  ( B_A)=6.

Computing b_j⊗ A, we find candidate simple objects of B_A:

    1⊕ Z⊕ V,   X⊕ Z⊕ W,   Y⊕ T⊕ U⊕ V⊕ W.

They have Frobenius-Perron dimensions

    1,  1,  2,

respectively. Their contributions to ( B_A) matches, 1^2+1^2+2^2=6. This suggests B_A has rank three. Indeed, we find a three-dimensional NIM-rep

    n_1=1_3,   n_X=[ 0 1 0; 1 0 0; 0 0 1 ],   n_Y=n_T=n_U=[ 0 0 1; 0 0 1; 1 1 1 ],
       n_Z=[ 1 1 0; 1 1 0; 0 0 2 ],   n_V=[ 1 0 1; 0 1 1; 1 1 2 ],   n_W=[ 0 1 1; 1 0 1; 1 1 2 ].

The solution gives identifications

    m_1≅1⊕ Z⊕ V,   m_2≅ X⊕ Z⊕ W,   m_3≅ Y⊕ T⊕ U⊕ V⊕ W.

Computing their quantum dimensions (<ref>), we find the candidate can be separable only when B is unitary. Then, the right A-modules have quantum dimensions

    d_ B_A(m_1)=1=d_ B_A(m_2),   d_ B_A(m_3)=2.

Working out the monoidal products ⊗_A, we find

We can identify B_A≃Rep(S_3). Since Rep(S_3) is semisimple, A is separable.

Here, a careful reader would notice that we have not shown commutativity of the candidate. For V to give commutative algebras, we need to limit our MFCs to those with the first and third conformal dimensions. Then, since these are conformal dimensions of C(B_4,2) or its conjugate, the commutativity can in principle be proven using the F- and R-symbols obtained in <cit.>, and computing multiplication morphism μ. However, without these computations, we can conclude this should be connected étale from the lemma 2. We have already seen 1⊕ Z gives so(9)_2→ (for the first and third conformal dimensions). The latter MFC is known <cit.> to have two (for unitary ) or one (for non-unitary ) nontrivial connected étale algebra(s). This implies, for unitary , there are two inequivalent operations

    →.

Therefore, for B≃ so(9)_2 which admits unitary B_A^0≃, there should exist two inequivalent operations

    so(9)_2→.

We have already found one nontrivial connected étale algebra, A≅1⊕ X⊕2Z, giving one such operation.[Note that

    (1⊕ X)⊗(1⊕ Z)≅1⊕ X⊕2Z.
] From our considerations so far, the only candidate which could give another operation is A≅1⊕ Z⊕ V. Thus, we learn the candidate is connected étale. Indeed, this matches known results in C(B_4,2).[We thank Terry Gannon for teaching this fact to us.] On the other hand, when A≅1⊕ Z gives non-unitary B_A^0≃, the Toric Code MFC admits <cit.> only one nontrivial connected étale algebra. Thus, A≅1⊕ Z⊕ V cannot be connected étale. This conclusion is consistent with our result above that the candidate is not separable in non-unitary so(9)_2. We found a connected étale algebra

    A≅1⊕ Z⊕ V  (d_V,d_W,h_V,h_W)=(3,3,0,1/2).  (1 for h)


We conclude

All the 16 MFCs B≃ so(9)_2's fail to be completely anisotropic.



  §.§.§ B≃Rep(D(D_3))

The MFCs have eight simple objects {1,X,Y,Z,T,U,V,W} obeying monoidal products

Thus, they have

    _ B(1)=1=_ B(X),     _ B(Y)=_ B(Z)=_ B(T)=_ B(U)=2,
       _ B(V)=3=_ B(W),

and

    ( B)=36.

Their quantum dimensions d_j's are solutions of the same multiplication rules d_id_j=∑_k=1^8N_ij^kd_k. There are two (nonzero) solutions

    (d_X,d_Y,d_Z,d_T,d_U,d_V,d_W)=(1,2,2,2,2,-3,-3),(1,2,2,2,2,3,3)

with the same categorical dimension

    D^2( B)=36.

They both have the same four conformal dimensions[Naively, one finds 72 consistent conformal dimensions, but the others give the same MFCs with the one in the main text under permutations of simple objects (YZ),(YT),(YU),(ZT),(ZU),(TU),(VW).]

    (h_X,h_Y,h_Z,h_T,h_U,h_V,h_W)   =(0,0,0,1/3,2/3,0,1/2),(0,0,0,1/3,2/3,1/4,3/4),
           (0,1/3,1/3,2/3,2/3,0,1/2),(0,1/3,1/3,2/3,2/3,1/4,3/4)  (1).

They have S-matrices

    S=[    1    1    2    2    2    2  d_V  d_W;    1    1    2    2    2    2 -d_V -d_W;    2    2    s   s'   -2   -2    0    0;    2    2   s'    s   -2   -2    0    0;    2    2   -2   -2   -2    4    0    0;    2    2   -2   -2    4   -2    0    0;  d_V -d_V    0    0    0    0   ±3   ∓3;  d_W -d_W    0    0    0    0   ∓3   ±3 ],

with

    (s,s')=(4,-2)   (1st and 2nd h),
    (-2,4)   (3rd and 4th h).

They have additive central charges

    c( B)=0   (1st&2nd h),
    4   (3rd&4th h).  (8).

There are

    2(quantum dimensions)×4(conformal dimensions)×2(categorical dimensions)=16

MFCs, among which those eight with the second quantum dimensions are unitary. We classify connected étale algebras in all 16 MFCs simultaneously.

An ansatz

    A≅1⊕ n_XX⊕ n_YY⊕ n_ZZ⊕ n_TT⊕ n_UU⊕ n_VV⊕ n_WW

with n_j∈ N has

    _ B(A)=1+n_X+2(n_Y+n_Z+n_T+n_U)+3(n_V+n_W).

For this to obey (<ref>), the natural numbers can take only 56 values just as in the previous example. The sets contain the one with all n_j's be zero. It is the trivial connected étale algebra A≅1 giving B_A^0≃ B_A≃ B. Other candidates with T,U,W have nontrivial conformal dimensions, and they fail to be commutative. Thus, we are left with candidates with just X,Y,Z,V. Note that {1,X,Y},{1,X,Z} form Rep(S_3)⊂Rep(D(D_3)). Apart from the trivial one, there are 24 candidates

    (n_X,n_Y,n_Z,n_V)   =(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1),
           (2,0,0,0),(1,1,0,0),(1,0,1,0),(1,0,0,1),
           (0,2,0,0),(0,1,1,0),(0,1,0,1),(0,0,2,0),
           (0,0,1,1),(3,0,0,0),(2,1,0,0),(2,0,1,0),
           (2,0,0,1),(1,2,0,0),(1,1,1,0),(1,0,2,0),
           (4,0,0,0),(3,1,0,0),(3,0,1,0),(5,0,0,0).

Some of these are ruled out by studying Frobenius-Perron dimensions. The four candidates

    (n_X,n_Y,n_Z,n_V)=(0,0,0,1),(1,1,0,0),(1,0,1,0),(3,0,0,0)

have _ B=4 and demand ( B_A^0)=9/4. However, there is no MFC with such Frobenius-Perron dimension. Thus, the candidates are ruled out. Similarly, seven candidates

    (n_X,n_Y,n_Z,n_V)=(1,0,0,1),(0,1,1,0),(0,2,0,0),(0,0,2,0),(2,1,0,0),(2,0,1,0),(4,0,0,0)

are ruled out because there is no MFC B_A^0 with Frobenius-Perron dimension ( B_A^0)=36/25. Thus, we are left with 13 nontrivial candidates

    (n_X,n_Y,n_Z,n_V)   =(1,0,0,0),(0,1,0,0),(0,0,1,0),(2,0,0,0),(0,1,0,1),(0,0,1,1),(2,0,0,1),
           (1,2,0,0),(1,1,1,0),(1,0,2,0),(3,1,0,0),(3,0,1,0),(5,0,0,0).

Thanks to the lemma 1, we know five of them (n_X,n_Y,n_Z,n_V)=(1,0,0,0),(0,1,0,0), (0,0,1,0),(1,2,0,0),(1,0,2,0) or A≅1⊕ X,1⊕ Y,1⊕ Z,1⊕ X⊕2Y,1⊕ X⊕2Z are commutative algebras (for certain conformal dimensions) because they are so in (symmetric) braided fusion subcategory Rep(S_3)⊂ B <cit.>. Let us see whether they are separable or not by identifying B_A. As a result, we find all of them are separable, hence étale (for certain conformal dimensions).



  
A≅1⊕ X. The simple object X has (d_X,h_X)=(1,0) (mod 1 for h_X), and it has trivial braiding c_X,X≅ id_1 <cit.>. Thus it is a commutative algebra for all four conformal dimensions. Furthermore, it turns out separable, hence étale. Let us check this point by identifying B_A.

Since it has _ B(A)=2, it demands

    ( B_A^0)=9,  ( B_A)=18.

Employing the matching of central charges (<ref>) and the invariance of topological twists (<ref>), we identify

    B_A^0≃_ Z/3 Z^1⊠_ Z/3 Z^1.

(See section <ref>.) What is the category B_A of right A-modules? We find

    B_A≃TY( Z/3 Z× Z/3 Z).

One of the easiest ways to find this fact is to perform anyon condensation. It `identifies' 1,X, and hence V,W. The resulting 1⊕ X and V⊕ W have Frobenius-Perron dimensions one, three in B_A. The other simple objects Y,Z,T,U split into two each. They all have Frobenius-Perron dimensions one in B_A. In total, we have nine invertible simple objects forming Z/3× Z/3 MFC (<ref>), and one additional non-invertible simple object with _ B_A=3. The fusion category is identified with a Z/3 Z× Z/3 Z Tambara-Yamagami category (<ref>).[More rigorously, we should find NIM-reps. Indeed, we find four 10-dimensional solutions. One of them is given by

    n_1=1_10=n_X,   n_Y=   [ 0 1 1 0 0 0 0 0 0 0; 1 0 1 0 0 0 0 0 0 0; 1 1 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 1 0 1 0; 0 0 0 0 0 1 0 1 0 0; 0 0 0 0 1 0 0 1 0 0; 0 0 0 1 0 0 0 0 1 0; 0 0 0 0 1 1 0 0 0 0; 0 0 0 1 0 0 1 0 0 0; 0 0 0 0 0 0 0 0 0 2 ],   n_Z=[ 0 0 0 1 1 0 0 0 0 0; 0 0 0 0 0 0 1 1 0 0; 0 0 0 0 0 1 0 0 1 0; 1 0 0 0 1 0 0 0 0 0; 1 0 0 1 0 0 0 0 0 0; 0 0 1 0 0 0 0 0 1 0; 0 1 0 0 0 0 0 1 0 0; 0 1 0 0 0 0 1 0 0 0; 0 0 1 0 0 1 0 0 0 0; 0 0 0 0 0 0 0 0 0 2 ],
    
    n_T=   [ 0 0 0 0 0 1 1 0 0 0; 0 0 0 0 1 0 0 0 1 0; 0 0 0 1 0 0 0 1 0 0; 0 0 1 0 0 0 0 1 0 0; 0 1 0 0 0 0 0 0 1 0; 1 0 0 0 0 0 1 0 0 0; 1 0 0 0 0 1 0 0 0 0; 0 0 1 1 0 0 0 0 0 0; 0 1 0 0 1 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 2 ],   n_U=[ 0 0 0 0 0 0 0 1 1 0; 0 0 0 1 0 1 0 0 0 0; 0 0 0 0 1 0 1 0 0 0; 0 1 0 0 0 1 0 0 0 0; 0 0 1 0 0 0 1 0 0 0; 0 1 0 1 0 0 0 0 0 0; 0 0 1 0 1 0 0 0 0 0; 1 0 0 0 0 0 0 0 1 0; 1 0 0 0 0 0 0 1 0 0; 0 0 0 0 0 0 0 0 0 2 ],
    n_V=   [ 0 0 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 0 0 1; 1 1 1 1 1 1 1 1 1 0 ]=n_W.

The NIM-rep gives identifications

    m_1≅1⊕ X,   m_2≅ Y≅ m_3,   m_4≅ Z≅ m_5,   m_6≅ T≅ m_7,   m_8≅ U≅ m_9,   m_10≅ V⊕ W.

(The other three solutions give the same identifications.) They have

    d_ B_A(m_j)=1 (j=1,2,…,9),   d_ B_A(m_10)=±3.

Working out the monoidal products ⊗_A, we find

    m_j⊗_Am_10≅ m_10≅ m_10⊗_Am_j (j=1,2,…,9),   m_10⊗_Am_10≅⊕_j=1^9m_j,

showing (<ref>).] Since TY( Z/3 Z× Z/3 Z) is a fusion category, it is semisimple. Therefore, A is separable, and étale.



  
A≅1⊕ Y. For the first and second conformal dimensions, 1⊕ Y gives commutative algebra. What is left is to check its separability. It has _ B(A)=3, and demands

    ( B_A^0)=4,  ( B_A)=12.

Computing b_j⊗ A, we find candidate simple objects of B_A:

    1⊕ Y,   X⊕ Y,   Z⊕ T⊕ U,   V⊕ W,   V,   W

with some multiplicities. In B_A, they have Frobenius-Perron dimensions

    1,  1,  2,  2,  1,  1,

respectively. The invariance of topological twists, matching of central charges, and Frobenius-Perron dimensions demand

    B_A^0={1⊕ Y,X⊕ Y,V,W}≃   (h_V,h_W)=(0,1/2),
    _ Z/2 Z^-1⊠_ Z/2 Z^-1   (h_V,h_W)=(1/4,3/4).  (1)

The two additional simple objects give correct Frobenius-Perron dimensions for them to match ( B_A)=12. Indeed, we find four six-dimensional NIM-reps. One of them is given by

    n_1=1_6,   n_X=[ 0 1 0 0 0 0; 1 0 0 0 0 0; 0 0 1 0 0 0; 0 0 0 1 0 0; 0 0 0 0 0 1; 0 0 0 0 1 0 ]   ,   n_Y=[ 1 1 0 0 0 0; 1 1 0 0 0 0; 0 0 2 0 0 0; 0 0 0 2 0 0; 0 0 0 0 1 1; 0 0 0 0 1 1 ],
    
        n_Z=n_T=n_U=[ 0 0 1 0 0 0; 0 0 1 0 0 0; 1 1 1 0 0 0; 0 0 0 1 1 1; 0 0 0 1 0 0; 0 0 0 1 0 0 ],   n_V=   [ 0 0 0 1 1 0; 0 0 0 1 0 1; 0 0 0 2 1 1; 1 1 2 0 0 0; 1 0 1 0 0 0; 0 1 1 0 0 0 ],   n_W=[ 0 0 0 1 0 1; 0 0 0 1 1 0; 0 0 0 2 1 1; 1 1 2 0 0 0; 0 1 1 0 0 0; 1 0 1 0 0 0 ].

The solution gives the identifications

    m_1≅1⊕ Y,   m_2≅ X⊕ Y,   m_3≅ Z⊕ T⊕ U,   m_4≅ V⊕ W,   m_5≅ V,   m_6≅ W.

(The other three solutions give the same identifications.) In B_A, they have quantum dimensions

    d_ B_A(m_1)=d_ B_A(m_2)=d_ B_A(m_3)=1,   d_ B_A(m_4)=±2,   d_ B_A(m_5)=±1=d_ B_A(m_6),

where signs are correlated. Working out the monoidal product ⊗_A, we find

Just as in the previous MFC, the category B_A of right A-modules is isomorphic to a fusion category C(FR^6,0_2). Thus, 1⊕ Y is separable, hence étale.



  
A≅1⊕ Z. Since the ambient category B≃Rep(D(D_3)) is invariant under the change of names Y↔ Z, our previous analysis implies A≅1⊕ Z is a connected étale algebra for the first and second conformal dimensions. It gives B_A≃ C(FR^6,0_2).



  
A≅1⊕ X⊕2Y. For the first and second conformal dimensions, the algebra is commutative. Furthermore, it turns out separable, hence étale. So as to check this point, we identify B_A.

Calculating b_j⊗ A, we find candidate simple objects:

    1⊕ X⊕2Y,   Z⊕ T⊕ U,   V⊕ W,

with Frobenius-Perron dimensions

    1,  1,  1,

respectively. Logically, the latter two can have coefficients, but the possibilities are ruled out.[The proof is the same as in the footnote <ref>.] For them to saturate ( B_A)=6, the second and third should `split' into two and three, respectively. Therefore, we search for six-dimensional NIM-reps. Indeed, we find four solutions. One of them is given by

    n_1=1_6=n_X,   n_Y=[ 2 0 0 0 0 0; 0 2 0 0 0 0; 0 0 2 0 0 0; 0 0 0 0 1 1; 0 0 0 1 0 1; 0 0 0 1 1 0 ],   n_Z=[ 0 1 1 0 0 0; 1 0 1 0 0 0; 1 1 0 0 0 0; 0 0 0 0 1 1; 0 0 0 1 0 1; 0 0 0 1 1 0 ]=n_T,
    
        n_U=[ 0 1 1 0 0 0; 1 0 1 0 0 0; 1 1 0 0 0 0; 0 0 0 2 0 0; 0 0 0 0 2 0; 0 0 0 0 0 2 ],   n_V=[ 0 0 0 1 1 1; 0 0 0 1 1 1; 0 0 0 1 1 1; 1 1 1 0 0 0; 1 1 1 0 0 0; 1 1 1 0 0 0 ]=n_W.

The NIM-rep gives identifications

    m_1≅1⊕ X⊕2Y,   m_2≅ Z⊕ T⊕ U≅ m_3,   m_4≅ m_5≅ m_6≅ V⊕ W.

(The other three solutions give the same identifications.) They have

    d_ B_A(m_1)=d_ B_A(m_2)=d_ B_A(m_3)=1,   d_ B_A(m_4)=d_ B_A(m_5)=d_ B_A(m_6)=±1.

The result tells us B_A should have rank six. Since F_A(b)⊗_AF_A(b') are the same as in 1⊕ X⊕2Z∈ so(9)_2, we immediately learn B_A≃ C(FR^6,2_1) or  C(FR^6,4_1). Since these are semisimple, A is separable, hence étale.



  
A≅1⊕ X⊕2Z. The symmetry Y↔ Z and the previous analysis imply A≅1⊕ X⊕2Z is a connected étale algebra for the first and second conformal dimensions. It gives B_A≃ C(FR^6,2_1) or  C(FR^6,4_1).

Now, we are left with eight candidates

    (n_X,n_Y,n_Z,n_V)=(2,0,0,0),(0,1,0,1),(0,0,1,1),(2,0,0,1),(1,1,1,0),(3,1,0,0),(3,0,1,0),(5,0,0,0)

or A≅1⊕2X,1⊕ Y⊕ V,1⊕ Z⊕ V,1⊕2X⊕ V,1⊕ X⊕ Y⊕ Z,1⊕3X⊕ Y,1⊕3X⊕ Z,1⊕5X. We study these in turn. It turns out that all but two fail to be étale.



  
A≅1⊕2X. It has _ B(A)=3, and demands

    ( B_A^0)=4,  ( B_A)=12.

Calculating b_j⊗ A, we find candidate simple objects of B_A:

    1⊕2X,  2⊕ X,   3Y,  3Z,  3T,  3U,   V,   W,  … .

They have Frobenius-Perron dimensions

    1,  1,  2,  2,  2,  2,  1,  1,  … ,

respectively. In order to match Frobenius-Perron dimensions, B_A^0 should consist of {1⊕2X,2⊕ X,V,W}. The four additional candidate simple objects contribute 4×2^2=16 to Frobenius-Perron dimension, and exceeds 12. Thus, the candidate 1⊕2X is ruled out.



  
A≅1⊕ Y⊕ V. For the first and third conformal dimensions, V has h_V=0 (mod 1), and the candidate can be commutative. Taking Y into account, we learn this candidate can be commutative only for the first conformal dimension. It has _ B(A)=6, and demands

    ( B_A^0)=1,  ( B_A)=6.

Computing b_j⊗ A, we find candidate simple objects:

    1⊕ Y⊕ V,   X⊕ Y⊕ W,   Z⊕ T⊕ U⊕ V⊕ W.

They have Frobenius-Perron dimensions

    1,  1,  2,

respectively. Their contributions to ( B_A) match, 1^2+1^2+2^2=6. The only possibility for the category of dyslectic (right) A-modules is

    B_A^0≃.

This identification also matches central charges. For B_A, the analysis above suggests it has rank three. Indeed, we find a three-dimensional NIM-rep

    n_1=1_3,   n_X=   [ 0 1 0; 1 0 0; 0 0 1 ],   n_Y=[ 1 1 0; 1 1 0; 0 0 2 ],
    
        n_Z=n_T=n_U=[ 0 0 1; 0 0 1; 1 1 1 ],      n_V=[ 1 0 1; 0 1 1; 1 1 2 ],   n_W=[ 0 1 1; 1 0 1; 1 1 2 ].

The solution gives identifications

    m_1≅1⊕ Y⊕ V,   m_2≅ X⊕ Y⊕ W,   m_3≅ Z⊕ T⊕ U⊕ V⊕ W.

By computing quantum dimensions, we find the candidate can be separable only when d_V=3=d_W. Then, the right A-modules have

    d_ B_A(m_1)=1=d_ B_A(m_2),   d_ B_A(m_3)=2.

Working out the monoidal products, we find

We can identify

    B_A≃Rep(S_3).

Since Rep(S_3) is semisimple, A is separable.

We still need to show commutativity. However, since F- and R-symbols of the MFCs are unknown (to the best of our knowledge), we take the indirect path through the lemma 2. We have already found A≅1⊕ Y,1⊕ Z are connected étale, giving B_A^0≃_ Z/2 Z× Z/2 Z^α. The MFCs further admit connected étale algebras A'∈ B_A^0 giving ( B_A^0)_A'^0≃. More precisely, for the first conformal dimensions, we get

    B_A^0≃,

and it admits one (when  is non-unitary) or two (when  is unitary) connected étale algebra(s). (Taking both A≅1⊕ Y,1⊕ Z into account, we have two or four operations Rep(D(D_3))→, respectively.) One such operation Rep(D(D_3))→ is given by A≅1⊕ X⊕2Y or A≅1⊕ X⊕2Z we found above. These exist regardless of quantum dimensions, and we can identify them as connected étale algebras giving the composition

    Rep(D(D_3))→ non-unitary →.

When A≅1⊕ Y,1⊕ Z give unitary , i.e., d_V=3=d_W, we need two more connected étale algebras. The only possibility is this and its symmetric partner. Therefore, we conclude the candidate A≅1⊕ Z⊕ V is connected étale for the first conformal dimension. It gives

    Rep(D(D_3))→.

We found

    A≅1⊕ Y⊕ V  (d_V=3=d_W&1st h).




  
A≅1⊕ Z⊕ V. The symmetry Y↔ Z and the previous analysis imply this candidate is connected étale

    A≅1⊕ Z⊕ V  (d_V=3=d_W&1st h),

giving

    B_A^0≃,   B_A≃Rep(S_3).




  
A≅1⊕2X⊕ V. For the first and third conformal dimensions, this can be commutative. It has _ B(A)=6, and demands

    ( B_A^0)=1,  ( B_A)=6.

With the free module functor F_A(b_j)=b_j⊗ A, we find candidate simple objects

    1⊕2X⊕ V,  2⊕ X⊕ W,  3Y,  3Z,  3T,  3U,   V⊕ W,  … ,

with additional simple objects which do not affect our discussion below. In B_A, they have Frobenius-Perron dimensions

    1,  1,  1,  1,  1,  1,  1,  …,

respectively. Their contributions to ( B_A) exceed six, and the candidate is ruled out.



  
A≅1⊕ X⊕ Y⊕ Z.
It has _ B(A)=6, and demands

    ( B_A^0)=1,  ( B_A)=6.

Computing b_j⊗ A, we find candidate simple objects

    1⊕ X⊕ Y⊕ Z,  1⊕ X⊕2Y,  1⊕ X⊕2Z,   Y⊕ T⊕ U,   Z⊕ T⊕ U,   V⊕ W,

with Frobenius-Perron dimensions

    1,  1,  1,  1,  1,  1,

respectively. Their contributions to Frobenius-Perron dimension match. We proceed to search for a rank six fusion category. Employing the free module functor, one can work out monoidal product ⊗_A. As a result, we find

    m_j⊗_Am_6≅ m_6≅ m_6⊗_Am_j (j=1,2,…,5).

We conclude 1⊕ X⊕ Y⊕ Z cannot be étale. The reason is as follows. If it were étale, B_A should be a fusion category. As all simple objects m_j have _ B_A(m_j)=1, the fusion ring should be multiplicity-free and listed in <cit.> or AnyonWiki. However, there is no fusion category with such monoidal product, a contradiction. Thus, the candidate is ruled out.



  
A≅1⊕3X⊕ Y. It has _ B(A)=6, and demands

    ( B_A^0)=1,  ( B_A)=6

as in the previous candidate. Calculating b_j⊗ A, we find candidate simple objects (assuming the smallest possible Frobenius-Perron dimensions)

    1⊕3X⊕ Y,  3⊕ X⊕ Y,  1⊕ X⊕2Y,  3Y,   Z⊕ T⊕ U,  3Z,  3T,  3U,  2V,  2W,  …

with Frobenius-Perron dimensions

    1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  … ,

respectively. Their contributions to Frobenius-Perron dimension exceed six, and the candidate is ruled out.



  
A≅1⊕3X⊕ Z. Symmetry (YZ) of our ambient category B≃Rep(D(D_3)) and the previous analysis rule out the candidate.



  
A≅1⊕5X. It has _ B(A)=6, and demands

    ( B_A^0)=1,  ( B_A)=6.

Computing b_j⊗ A, we find candidate simple objects (assuming the smallest possible Frobenius-Perron dimensions)

    1⊕5X,  5⊕ X,  3Y,  3Z,  3T,  3U,  2V,   V⊕ W,  2W,

with Frobenius-Perron dimensions

    1,  1,  1,  1,  1,  1,  1,  1,  1,

respectively. Their contributions to Frobenius-Perron dimension exceed six, and the candidate is ruled out.

We conclude

All the 16 MFCs B≃Rep(D(D_3))'s fail to be completely anisotropic. It may be useful to summarize these results in a “cascade” of conformal embeddings:

Let us see the consistency of our results. 



  
3rd and 4th conformal dimensions. For these conformal dimensions, the only connected étale algebra is A≅1⊕ X. Correspondingly, the only path is Rep(D(D_3))→_ Z/3 Z^1⊠_ Z/3 Z^1. Since the resulting MFC has central charge (“anomaly”) different from , it cannot descend to  and the path terminates there.



  
2nd conformal dimension. For the second conformal dimensions, there are two additional paths given by A≅1⊕ Y,1⊕ Z. The compositions of the two operations

    Rep(D(D_3))→_ Z/3 Z^1⊠_ Z/3 Z^1,  Rep(D(D_3))→_ Z/2 Z^-1⊠_ Z/2 Z^-1

give two inequivalent paths

    Rep(D(D_3))→_ Z/3 Z^1⊠_ Z/3 Z^1→.

This is consistent with the result that _ Z/3 Z^1⊠_ Z/3 Z^1 with such conformal dimensions admits two connected étale algebras. (See section <ref>.) The other paths via _ Z/2 Z^-1⊠_ Z/2 Z^-1 also lead to  because the MFC admits a Lagrangian algebra. The two inequivalent compositions

    Rep(D(D_3))→_ Z/2 Z^-1⊠_ Z/2 Z^-1→

are given by two additional connected étale algebras A≅1⊕ X⊕2Y,1⊕ X⊕2Z.[Note that

    (1⊕ Y)⊗(1⊕ X)≅1⊕ X⊕2Y,  (1⊕ Z)⊗(1⊕ X)≅1⊕ X⊕2Z.
]



  
1st conformal dimension. For the first conformal dimensions, we still have A≅1⊕ X,1⊕ Y,1⊕ Z,1⊕ X⊕2Y,1⊕ X⊕2Z, and the story about the path

    Rep(D(D_3))→_ Z/3 Z^1⊠_ Z/3 Z^1→

is the same as the last case. However, a difference appears in the other paths. For non-unitary Rep(D(D_3)) with the first conformal dimensions, A≅1⊕ Y,1⊕ Z lead to . It admits one Lagrangian algebra giving →. The two inequivalent compositions

    Rep(D(D_3))→→

are given by A≅1⊕ X⊕2Y,1⊕ X⊕2Z. On the other hand, for a unitary Rep(D(D_3)), A≅1⊕ Y,1⊕ Z lead to  with two Lagrangian algebras. The other inequivalent paths are given by additional connected étale algebras A≅1⊕ Y⊕ V,1⊕ Y⊕ V. In total, there are four inequivalent paths

    Rep(D(D_3))→→

given by four Lagrangian algebras A≅1⊕ X⊕2Y,1⊕ X⊕2Z,1⊕ Y⊕ V,1⊕ Z⊕ V.



  §.§.§ B≃ su(2)_7

The MFCs have eight simple objects {1,X,Y,Z,T,U,V,W} obeying monoidal products

(One can identify _ Z/2 Z^-1={1,X},psu(2)_7={1,Y,T,V}, and Z≅ X⊗ Y,U≅ X⊗ T,W≅ X⊗ V.) Thus, they have

    _ B(1)=1=_ B(X),  _ B(Y)=sin2π/9/sinπ/9=_ B(Z),
    _ B(T)=sin3π/9/sinπ/9=_ B(U),  _ B(V)=sin4π/9/sinπ/9=_ B(W),

and

    ( B)=9/2sin^2π/9≈38.5.

Their quantum dimensions d_j's are solutions of the same multiplication rules d_id_j=∑_k=1^8N_ij^kd_k. There are six (nonzero) solutions

    (d_X,d_Y,d_Z,d_T,d_U,d_V,d_W)   =(-1,-sinπ/9/cosπ/18,sinπ/9/cosπ/18,-sin3π/9/cosπ/18,sin3π/9/cosπ/18,sin2π/9/cosπ/18,-sin2π/9/cosπ/18),
           (1,-sinπ/9/cosπ/18,-sinπ/9/cosπ/18,-sin3π/9/cosπ/18,-sin3π/9/cosπ/18,sin2π/9/cosπ/18,sin2π/9/cosπ/18),
           (-1,-sin4π/9/cos5π/18,sin4π/9/cos5π/18,sin3π/9/cos5π/18,-sin3π/9/cos5π/18,-sinπ/9/cos5π/18,sinπ/9/cos5π/18),
           (1,-sin4π/9/cos5π/18,-sin4π/9/cos5π/18,sin3π/9/cos5π/18,sin3π/9/cos5π/18,-sinπ/9/cos5π/18,-sinπ/9/cos5π/18),
           (-1,sin2π/9/sinπ/9,-sin2π/9/sinπ/9,sin3π/9/sinπ/9,-sin3π/9/sinπ/9,sin4π/9/sinπ/9,-sin4π/9/sinπ/9),
           (1,sin2π/9/sinπ/9,sin2π/9/sinπ/9,sin3π/9/sinπ/9,sin3π/9/sinπ/9,sin4π/9/sinπ/9,sin4π/9/sinπ/9),

with categorical dimensions

    D^2( B)=9/2cos^2π/18(≈4.6),  9/2cos^25π/18(≈10.9),  9/2sin^2π/9,

respectively. Only the last quantum dimensions give unitary MFCs. They each have four conformal dimensions

    (h_X,h_Y,h_Z,h_T,h_U,h_V,h_W)
       =(1/4,1/3,7/12,8/9,5/36,2/3,11/12),(1/4,2/3,11/12,1/9,13/36,1/3,7/12),(3/4,1/3,1/12,8/9,23/36,2/3,5/12),(3/4,2/3,5/12,1/9,31/36,1/3,1/12)   (1st&2nd),
    (1/4,1/3,7/12,5/9,29/36,2/3,11/12),(1/4,2/3,11/12,4/9,25/36,1/3,7/12),(3/4,1/3,1/12,5/9,11/36,2/3,5/12),(3/4,2/3,5/12,4/9,7/36,1/3,1/12)   (3rd&4th),
    (1/4,1/3,7/12,2/9,17/36,2/3,11/12),(1/4,2/3,11/12,7/9,1/36,1/3,7/12),(3/4,1/3,1/12,2/9,35/36,2/3,5/12),(3/4,2/3,5/12,7/9,19/36,1/3,1/12)   (5th&6th).  (1)

The S-matrices are given by

    S=[       1     d_X     d_Y  d_Xd_Y     d_T  d_Xd_T     d_V  d_Xd_V;     d_X      -1  d_Xd_Y    -d_Y  d_Xd_T    -d_T  d_Xd_V    -d_V;     d_Y  d_Xd_Y    -d_V -d_Xd_V     d_T  d_Xd_T      -1    -d_X;  d_Xd_Y    -d_Y -d_Xd_V     d_V  d_Xd_T    -d_T    -d_X       1;     d_T  d_Xd_T     d_T  d_Xd_T       0       0    -d_T -d_Xd_T;  d_Xd_T    -d_T  d_Xd_T    -d_T       0       0 -d_Xd_T     d_T;     d_V  d_Xd_V      -1    -d_X    -d_T -d_Xd_T     d_Y  d_Xd_Y;  d_Xd_V    -d_V    -d_X       1 -d_Xd_T     d_T  d_Xd_Y    -d_Y ].

There are

    6(quantum dimensions)×4(conformal dimensions)×2(categorical dimensions)=48

MFCs, among which those eight with the last quantum dimensions give unitary MFCs. We classify connected étale algebras in all 48 MFCs simultaneously.

We work with an ansatz

    A≅1⊕ n_XX⊕ n_YY⊕ n_ZZ⊕ n_TT⊕ n_UU⊕ n_VV⊕ n_WW

with n_j∈ N. It has

    _ B(A)=1+n_X+sin2π/9/sinπ/9(n_Y+n_Z)+sin3π/9/sinπ/9(n_T+n_U)+sin4π/9/sinπ/9(n_V+n_W).

For this to obey (<ref>), the natural numbers can take only 43 values. The sets contain the one with all n_j's be zero. This is the trivial connected étale algebra A≅1 giving B_A^0≃ B_A≃ B. The other 42 candidates contain nontrivial simple object(s) b_j≇1. They all fail to be commutative because the nontrivial simple objects have nontrivial conformal dimensions.

We conclude

All the 48 MFCs B≃ su(2)_7's are completely anisotropic.



  §.§.§ B≃⊠⊠

The MFCs have eight simple objects {1,X,Y,Z,T,U,V,W} obeying monoidal products

(One can identify ={1,X},{1,Y},{1,Z}, and T≅ X⊗ Z,U≅ Y⊗ Z,V≅ X⊗ Y,W≅ X⊗ Y⊗ Z.) Thus, they have

    _ B(1)=1,  _ B(X)=_ B(Y)=_ B(Z)=ζ,
    _ B(T)=_ B(U)=_ B(V)=3+√(5)/2,  _ B(W)=2+√(5),

and

    ( B)=25+10√(5)≈47.4.

Their quantum dimensions d_j's are solutions of the same multiplication rules d_id_j=∑_k=1^8N_ij^kd_k. There are eight solutions

    (d_X,d_Y,d_Z,d_T,d_U,d_V,d_W)
       =(-ζ^-1,-ζ^-1,-ζ^-1,3-√(5)/2,3-√(5)/2,3-√(5)/2,2-√(5)),(-ζ^-1,-ζ^-1,ζ,-1,-1,3-√(5)/2,ζ^-1),
           (-ζ^-1,ζ,-ζ^-1,3-√(5)/2,-1,-1,ζ^-1),(ζ,-ζ^-1,-ζ^-1,-1,3-√(5)/2,-1,ζ^-1),
           (-ζ^-1,ζ,ζ,-1,3+√(5)/2,-1,-ζ),(ζ,-ζ^-1,ζ,3+√(5)/2,-1,-1,-ζ),
           (ζ,ζ,-ζ^-1,-1,-1,3+√(5)/2,-ζ),(ζ,ζ,ζ,3+√(5)/2,3+√(5)/2,3+√(5)/2,2+√(5)).

Only the last quantum dimensions give unitary MFCs. The first, second to fourth, fifth to seventh, and the eighth have categorical dimensions

    D^2( B)=25-10√(5)(≈2.6),  25-5√(5)/2(≈6.9),  25+5√(5)/2(≈18.1),  25+10√(5),

respectively.

In order to list up their conformal dimensions without double-counting, we perform case analysis. Depending on quantum dimensions ζ,-ζ^-1 of Fibonacci objects, we have four classes.

    
  * (d_X,d_Y,d_Z)=(ζ,ζ,ζ). This class gives unitary MFCs. Different MFCs are given by
    
    (h_X,h_Y,h_Z,h_T,h_U,h_V,h_W)   =(2/5,2/5,2/5,4/5,4/5,4/5,1/5),(2/5,2/5,3/5,0,0,4/5,2/5),
           (2/5,3/5,3/5,0,1/5,0,3/5),(3/5,3/5,3/5,1/5,1/5,1/5,4/5).  (1)

    Including the two signs of categorical dimensions, we have eight unitary MFCs.
    
  * (d_X,d_Y,d_Z)=(ζ,ζ,-ζ^-1). Different MFCs are given by
    
    (h_X,h_Y,h_Z,h_T,h_U,h_V,h_W)   =(2/5,2/5,1/5,3/5,3/5,4/5,0),(2/5,2/5,4/5,1/5,1/5,4/5,3/5),
           (2/5,3/5,1/5,3/5,4/5,0,1/5),(2/5,3/5,4/5,1/5,2/5,0,4/5),
           (3/5,3/5,1/5,4/5,4/5,1/5,2/5),(3/5,3/5,4/5,2/5,2/5,1/5,0).  (1)

    With two categorical dimensions, there are 12 MFCs.
    
  * (d_X,d_Y,d_Z)=(ζ,-ζ^-1,-ζ^-1). Different MFCs are given by
    
    (h_X,h_Y,h_Z,h_T,h_U,h_V,h_W)   =(2/5,1/5,1/5,3/5,2/5,3/5,4/5),(2/5,1/5,4/5,1/5,0,3/5,2/5),
           (2/5,4/5,4/5,1/5,3/5,1/5,0),(3/5,1/5,1/5,4/5,2/5,4/5,0),
           (3/5,1/5,4/5,2/5,0,4/5,3/5),(3/5,4/5,4/5,2/5,3/5,2/5,1/5).  (1)

    There are 12 MFCs.
    
  * (d_X,d_Y,d_Z)=(-ζ^-1,-ζ^-1,-ζ^-1). Different MFCs are given by
    
    (h_X,h_Y,h_Z,h_T,h_U,h_V,h_W)   =(1/5,1/5,1/5,2/5,2/5,2/5,3/5),(1/5,1/5,4/5,0,0,2/5,1/5),
           (1/5,4/5,4/5,0,3/5,0,4/5),(4/5,4/5,4/5,3/5,3/5,3/5,2/5).  (1)

    There are eight MFCs.

The S-matrices are given by

    S=[         1       d_X       d_Y       d_Z    d_Xd_Z    d_Yd_Z    d_Xd_Y d_Xd_Yd_Z;       d_X        -1    d_Xd_Y    d_Xd_Z      -d_Z d_Xd_Yd_Z      -d_Y   -d_Yd_Z;       d_Y    d_Xd_Y        -1    d_Yd_Z d_Xd_Yd_Z      -d_Z      -d_X   -d_Xd_Z;       d_Z    d_Xd_Z    d_Yd_Z        -1      -d_Z      -d_Y d_Xd_Yd_Z   -d_Xd_Y;    d_Xd_Z      -d_Z d_Xd_Yd_Z      -d_Z         1   -d_Xd_Y   -d_Yd_Z       d_Y;    d_Yd_Z d_Xd_Yd_Z      -d_Z      -d_Y   -d_Xd_Y         1   -d_Xd_Z       d_X;    d_Xd_Y      -d_Y      -d_X d_Xd_Yd_Z   -d_Yd_Z   -d_Xd_Z         1       d_Z; d_Xd_Yd_Z   -d_Yd_Z   -d_Xd_Z   -d_Xd_Y       d_Y       d_X       d_Z        -1 ].

They have additive central charge

    c( B)=c()+c()+c()  (8)

where

    c()=2/5   (h_=1/5),
    -2/5   (h_=4/5),
    14/5   (h_=2/5),
    -14/5   (h_=3/5).  (8)

There are

    8+12+12+8=40

MFCs, among which those eight in the first class are unitary. We classify connected étale algebras in all 40 MFCs simultaneously.

An ansatz

    A≅1⊕ n_XX⊕ n_YY⊕ n_ZZ⊕ n_TT⊕ n_UU⊕ n_VV⊕ n_WW

with n_j∈ N has

    _ B(A)=1+ζ(n_X+n_Y+n_Z)+3+√(5)/2(n_T+n_U+n_V)+(2+√(5))n_W.

For this to obey (<ref>), the natural numbers can take only 60 values. It contains the one with all n_j's be zero. It is the trivial connected étale algebra A≅1 giving B_A^0≃ B_A≃ B. Those with X,Y,Z cannot be commutative because they have nontrivial conformal dimensions. Setting n_X=n_Y=n_Z=0, apart from the trivial one A≅1, we get 10 sets

    (n_T,n_U,n_V,n_W)   =(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1),(2,0,0,0),
           (1,1,0,0),(1,0,1,0),(0,2,0,0),(0,1,1,0),(0,0,2,0).

Some of them are ruled out due to their Frobenius-Perron dimensions. The six candidates (n_T,n_U,n_V,n_W)=(2,0,0,0),(1,1,0,0),(1,0,1,0),(0,2,0,0),(0,1,1,0),(0,0,2,0) have _ B=4+√(5), and demands ( B_A^0)≈1.2. However, there is no MFC with such Frobenius-Perron dimension. Thus, the six candidates are ruled out. Also, a candidate (n_T,n_U,n_V,n_W)=(0,0,0,1) or A≅1⊕ W is ruled out because there is no MFC with ( B_A^0)≈1.7. Thus, we are left with three candidates

    (n_T,n_U,n_V,n_W)=(1,0,0,0),(0,1,0,0),(0,0,1,0).

They all have _ B(A)=5+√(5)/2, and demands

    ( B_A^0)=5+√(5)/2,  ( B_A)=15+5√(5)/2.

As we recalled in section <ref>, these are commutative algebras when h=0 mod 1 <cit.>. What is left is to check separability of them. In order to judge this point, we identify B_A. We start from A≅1⊕ T.

The Frobenius-Perron dimensions only allow

    B_A^0≃,   B_A≃⊠.

This in particular suggests B_A has rank four. Indeed, we find a four-dimensional NIM-rep

    n_1=1_4,   n_X   =[ 0 1 0 0; 1 1 0 0; 0 0 0 1; 0 0 1 1 ]=n_Z,   n_Y=[ 0 0 1 0; 0 0 0 1; 1 0 1 0; 0 1 0 1 ],
    
        n_T=[ 1 1 0 0; 1 2 0 0; 0 0 1 1; 0 0 1 2 ],      n_U=[ 0 0 0 1; 0 0 1 1; 0 1 0 1; 1 1 1 1 ]=n_V,   n_W=[ 0 0 1 1; 0 0 1 2; 1 1 1 1; 1 2 1 2 ].

The solution gives identifications

    m_1≅1⊕ T,   m_2≅ X⊕ Z⊕ T,   m_3≅ Y⊕ W,   m_4≅ U⊕ V⊕ W.

For the candidate to be commutative, we need h_T=0 mod 1. This happens when d_X=d_Z. Then, the right A-modules have

    d_ B_A(m_1)=1,   d_ B_A(m_2)=d_X,   d_ B_A(m_3)=d_Y,   d_ B_A(m_4)=d_Xd_Y.

For MFCs with h_T=0 mod 1, m_1,3 are deconfined. They obey the monoidal products ⊗_A:


We see

    B_A^0={m_1,m_3}≃,   B_A={m_1,m_2,m_3,m_4}≃⊠.

The identification also matches central charges because two  generated by X,Z have opposite central charges, and both c( B) and c( B_A^0) are determined by h_Y. The identification of B_A shows A is separable (hence étale) because ⊠ is semisimple:

    A≅1⊕ T  (d_X,d_Z,h_X,h_Z)=(ζ,ζ,2/5,3/5),(-ζ^-1,-ζ^-1.1/5,4/5)  (1 for h)


How about the other two candidates A≅ 1⊕ U,1⊕ V? Since T,U,V enter symmetrically, we immediately learn they give connected étale algebras:

    A≅1⊕ U     (d_Y,d_Z,h_Y,h_Z)=(ζ,ζ,2/5,3/5),(-ζ^-1,-ζ^-1,1/5,4/5),
    
        A≅1⊕ V     (d_X,d_Y,h_X,h_Y)=(ζ,ζ,2/5,3/5),(-ζ^-1,-ζ^-1,1/5,4/5).
      (1 for h)


We conclude

Namely, those 16 MFCs B≃⊠⊠'s in (<ref>,<ref>) fail to be completely anisotropic, while the other 24 are completely anisotropic.



  §.§.§ B≃⊠ psu(2)_7

The MFCs have eight simple objects {1,X,Y,Z,T,U,V,W} obeying monoidal products

(One can identify ={1,X},psu(2)_7={1,Y,Z,T}, and U≅ X⊗ Y,V≅ X⊗ Z,W≅ X⊗ T.) Thus, they have

    _ B(1)=1,  _ B(X)=ζ,     _ B(Y)=sin2π/9/sinπ/9,  _ B(Z)=sin3π/9/sinπ/9,
    _ B(T)=sin4π/9/sinπ/9,  _ B(U)=ζsin2π/9/sinπ/9,     _ B(V)=ζsin3π/9/sinπ/9,  _ B(W)=ζsin4π/9/sinπ/9,

and

    ( B)=45+9√(5)/8sin^2π/9≈69.6.

Their quantum dimensions d_j's are solutions of the same multiplication rules d_id_j=∑_k=1^8N_ij^kd_k. There are six (nonzero) solutions

    (d_X,d_Y,d_Z,d_T,d_U,d_V,d_W)   =(-ζ^-1,-sinπ/9/cosπ/18,-sinπ/3/cosπ/18,1-sinπ/9/cosπ/18,ζ^-1sinπ/9/cosπ/18,ζ^-1sinπ/3/cosπ/18-ζ^-1+ζ^-1sinπ/9/cosπ/18),
           (-ζ^-1,-cosπ/18/sin2π/9,sinπ/3/sin2π/9,1-cosπ/18/sin2π/9,ζ^-1cosπ/18/sin2π/9,-ζ^-1sinπ/3/sin2π/9,-ζ^-1+ζ^-1cosπ/18/sin2π/9),
           (ζ,-sinπ/9/cosπ/18,-sinπ/3/cosπ/18,1-sinπ/9/cosπ/18,-ζsinπ/9/cosπ/18,-ζsinπ/3/cosπ/18,ζ-ζsinπ/9/cosπ/18),
           (ζ,-cosπ/18/sin2π/9,sinπ/3/sin2π/9,1-cosπ/18/sin2π/9,-ζcosπ/18/sin2π/9,ζsinπ/3/sin2π/9,ζ-ζcosπ/18/sin2π/9),
           (-ζ^-1,sin2π/9/sinπ/9,sin3π/9/sinπ/9,sin4π/9/sinπ/9,-ζ^-1sin2π/9/sinπ/9,-ζ^-1sin3π/9/sinπ/9,-ζ^-1sin4π/9/sinπ/9),
           (ζ,sin2π/9/sinπ/9,sin3π/9/sinπ/9,sin4π/9/sinπ/9,ζsin2π/9/sinπ/9,ζsin3π/9/sinπ/9,ζsin4π/9/sinπ/9),

with categorical dimensions

    D^2( B)=45-9√(5)/8cos^2π/18(≈3.2)   (1st),
    45-9√(5)/8sin^22π/9(≈7.5)   (2nd),
    45+9√(5)/8cos^2π/18(≈8.4)   (3rd),
    45+9√(5)/8sin^22π/9(≈19.7)   (4th),
    45-9√(5)/8sin^2π/9(≈26.6)   (5th),
    45+9√(5)/8sin^2π/9   (6th),

respectively. Each of them has four conformal dimensions:

    (h_X,h_Y,h_Z,h_T,h_U,h_V,h_W)
       =(1/5,1/3,8/9,2/3,8/15,4/45,13/15),(1/5,2/3,1/9,1/3,13/15,14/45,8/15),(4/5,1/3,8/9,2/3,2/15,31/45,7/15),(4/5,2/3,1/9,1/3,7/15,41/45,2/15)   (1st),
    (1/5,1/3,5/9,2/3,8/15,34/45,13/15),(1/5,2/3,4/9,1/3,13/15,29/45,8/15),(4/5,1/3,5/9,2/3,2/15,16/45,7/15),(4/5,2/3,4/9,1/3,7/15,11/45,2/15)   (2nd),
    (2/5,1/3,8/9,2/3,11/15,13/45,1/15),(2/5,2/3,1/9,1/3,1/15,23/45,11/15),(3/5,1/3,8/9,2/3,14/15,22/45,4/15),(3/5,2/3,1/9,1/3,4/15,32/45,14/15)   (3rd),
    (2/5,1/3,5/9,2/3,11/15,43/45,1/15),(2/5,2/3,4/9,1/3,1/15,38/45,11/15),(3/5,1/3,5/9,2/3,14/15,7/45,4/15),(3/5,2/3,4/9,1/3,4/15,2/45,14/15)   (4th),
    (1/5,2/3,7/9,1/3,13/15,44/45,8/15),(1/5,1/3,2/9,2/3,8/15,19/45,13/15),(4/5,2/3,7/9,1/3,7/15,26/45,2/15),(4/5,1/3,2/9,2/3,2/15,1/45,7/15)   (5th),
    (2/5,2/3,7/9,1/3,1/15,8/45,11/15),(2/5,1/3,2/9,2/3,11/15,28/45,1/15),(3/5,2/3,7/9,1/3,4/15,17/45,14/15),(3/5,1/3,2/9,2/3,14/15,37/45,4/15)   (6th).  (1)

The S-matrices are given by

    S=[       1     d_X     d_Y     d_Z     d_T  d_Xd_Y  d_Xd_Z  d_Xd_T;     d_X      -1  d_Xd_Y  d_Xd_Y  d_Xd_T    -d_Y    -d_Z    -d_T;     d_Y  d_Xd_Y    -d_T     d_Z      -1 -d_Xd_T  d_Xd_Z    -d_X;     d_Z  d_Xd_Z     d_Z       0    -d_Z  d_Xd_Z       0 -d_Xd_Z;     d_T  d_Xd_T      -1    -d_Z     d_Y    -d_X -d_Xd_Z  d_Xd_Y;  d_Xd_Y    -d_Y -d_Xd_T  d_Xd_Z    -d_X     d_T    -d_Z       1;  d_Xd_Z    -d_Z  d_Xd_Z       0 -d_Xd_Z    -d_Z       0     d_Z;  d_Xd_T    -d_T    -d_X -d_Xd_Z  d_Xd_Y       1     d_Z    -d_Y ].

There are

    6(quantum dimensions)×4(conformal dimensions)×2(categorical dimensions)=48

MFCs, among which those eight with the sixth quantum dimensions are unitary. We classify connected étale algebras in all 48 MFCs simultaneously.

An ansatz

    A≅1⊕ n_XX⊕ n_YY⊕ n_ZZ⊕ n_TT⊕ n_UU⊕ n_VV⊕ n_WW

with n_j∈ N has

    _ B(A)=1+ζ n_X+sin2π/9/sinπ/9n_Y+sin3π/9/sinπ/9n_Z+sin4π/9/sinπ/9n_T+ζsin2π/9/sinπ/9n_U+ζsin3π/9/sinπ/9n_V+ζsin4π/9/sinπ/9n_W.

For this to obey (<ref>), the natural numbers can take only 54 values. The sets contain the one with all n_j's zero. This corresponds to the trivial connected étale algebra A≅1 giving B_A^0≃ B_A≃ B. The other 53 candidates fail to be commutative because they contain nontrivial simple objects with nontrivial conformal dimensions.

We conclude

All the 48 MFCs B≃⊠ psu(2)_7's are completely anisotropic.



  §.§.§ B≃ psu(2)_15

The MFCs have eight simple objects {1,X,Y,Z,T,U,V,W} obeying monoidal products

Thus, they have

    _ B(1)=1,  _ B(X)=sin2π/17/sinπ/17,     _ B(Y)=sin3π/17/sinπ/17,  _ B(Z)=sin4π/17/sinπ/17,
    _ B(T)=sin5π/17/sinπ/17,  _ B(U)=sin6π/17/sinπ/17,     _ B(V)=sin7π/17/sinπ/17,  _ B(W)=sin8π/17/sinπ/17,

and

    ( B)=17/4sin^2π/17≈125.9.

Their quantum dimensions d_j's are solutions of the same multiplication rules d_id_j=∑_k=1^8N_ij^kd_k. There are eight solutions

    (d_X,d_Y,d_Z,d_T,d_U,d_V,d_W)   =(-sinπ/17/cosπ/34,-sin7π/17/cosπ/34,sin2π/17/cosπ/34,sin6π/17/cosπ/34,-sin3π/17/cosπ/34,-sin5π/17/cosπ/34,sin4π/17/cosπ/34),
           (sin3π/17/cos3π/34,-sin4π/17/cos3π/34,-sin6π/17/cos3π/34,sinπ/17/cos3π/34,sin8π/17/cos3π/34,sin2π/17/cos3π/34,-sin5π/17/cos3π/34),
           (-sin5π/17/cos5π/34,-sinπ/17/cos5π/34,sin7π/17/cos5π/34,-sin4π/17/cos5π/34,-sin2π/17/cos5π/34,sin8π/17/cos5π/34,-sin3π/17/cos5π/34),
           (sin7π/17/cos7π/34,sin2π/17/cos7π/34,-sin3π/17/cos7π/34,-sin8π/17/cos7π/34,-sin4π/17/cos7π/34,sinπ/17/cos7π/34,sin6π/17/cos7π/34),
           (-sin8π/17/cos9π/34,sin5π/17/cos9π/34,-sinπ/17/cos9π/34,-sin3π/17/cos9π/34,sin7π/17/cos9π/34,-sin6π/17/cos9π/34,sin2π/17/cos9π/34),
           (sin6π/17/cos11π/34,sin8π/17/cos11π/34,sin5π/17/cos11π/34,sin2π/17/cos11π/34,-sinπ/17/cos11π/34,-sin4π/17/cos11π/34,-sin7π/17/cos11π/34),
           (-sin4π/17/cos13π/34,sin6π/17/cos13π/34,-sin8π/17/cos13π/34,sin7π/17/cos13π/34,-sin5π/17/cos13π/34,sin3π/17/cos13π/34,-sinπ/17/cos13π/34),
           (sin2π/17/sinπ/17,sin3π/17/sinπ/17,sin4π/17/sinπ/17,sin5π/17/sinπ/17,sin6π/17/sinπ/17,sin7π/17/sinπ/17,sin8π/17/sinπ/17)

with categorical dimensions

    D^2( B)   =17/4cos^2π/34(≈4.3),  17/4cos^23π/34(≈4.6),  17/4cos^25π/34(≈5.3),  17/4cos^27π/34(≈6.7),
           17/4cos^29π/34(≈9.4),  17/4cos^211π/34(≈15.3),  17/4cos^213π/34(≈32.6),  17/4sin^2π/17,

respectively. They have two conformal dimensions each:

    (h_X,h_Y,h_Z,h_T,h_U,h_V,h_W)=(6/17,16/17,13/17,14/17,2/17,11/17,7/17),(11/17,1/17,4/17,3/17,15/17,6/17,10/17)   (1st),
    (1/17,14/17,5/17,8/17,6/17,16/17,4/17),(16/17,3/17,12/17,9/17,11/17,1/17,13/17)   (2nd),
    (4/17,5/17,3/17,15/17,7/17,13/17,16/17),(13/17,12/17,14/17,2/17,10/17,4/17,1/17)   (3rd),
    (8/17,10/17,6/17,13/17,14/17,9/17,15/17),(9/17,7/17,11/17,4/17,3/17,8/17,2/17)   (4th),
    (3/17,8/17,15/17,7/17,1/17,14/17,12/17),(14/17,9/17,2/17,10/17,16/17,3/17,5/17)   (5th),
    (2/17,11/17,10/17,16/17,12/17,15/17,8/17),(15/17,6/17,7/17,1/17,5/17,2/17,9/17)   (6th),
    (7/17,13/17,1/17,5/17,8/17,10/17,11/17),(10/17,4/17,16/17,12/17,9/17,7/17,6/17)   (7th),
    (5/17,2/17,8/17,6/17,13/17,12/17,3/17),(12/17,15/17,9/17,11/17,4/17,5/17,14/17)   (8th).  (1)

The S-matrices are given by

    S=[    1  d_X  d_Y  d_Z  d_T  d_U  d_V  d_W;  d_X -d_Z  d_U -d_W  d_V -d_T  d_Y   -1;  d_Y  d_U  d_W  d_T  d_X   -1 -d_T -d_V;  d_Z -d_W  d_T   -1 -d_Y  d_V -d_U  d_X;  d_T  d_V  d_X -d_Y -d_W -d_Z    1  d_U;  d_U -d_T   -1  d_V -d_Z -d_X  d_W -d_Y;  d_V  d_Y -d_T -d_U    1  d_W  d_X -d_T;  d_W   -1 -d_V  d_X  d_U -d_Y -d_T  d_Z ].

There are

    8(quantum dimensions)×2(conformal dimensions)×2(categorical dimensions)=32

MFCs, among which those four with the eighth quantum dimensions are unitary. We classify connected étale algebras in all 32 MFCs simultaneously.

An ansatz

    A≅1⊕ n_XX⊕ n_YY⊕ n_ZZ⊕ n_TT⊕ n_UU⊕ n_VV⊕ n_WW

with n_j∈ N has

    _ B(A)=1+sin2π/17/sinπ/17n_X+sin3π/17/sinπ/17n_Y+sin4π/17/sinπ/17n_Z+sin5π/17/sinπ/17n_T+sin6π/17/sinπ/17n_U+sin7π/17/sinπ/17n_V+sin8π/17/sinπ/17n_W.

For this to obey (<ref>), the natural numbers can take only 55 values. The sets contain the one with all n_j's zero. This corresponds to the trivial connected étale algebra A≅1 giving B_A^0≃ B_A≃ B. The other 54 candidates fail to be commutative because they contain nontrivial simple objects with nontrivial conformal dimensions.

We conclude

All the 32 MFCs B≃ psu(2)_15's are completely anisotropic.



 §.§ Rank nine



  §.§.§ B≃ su(9)_1≃_ Z/9 Z^1

The MFCs have nine simple objects {1,X,Y,Z,S,T,U,V,W} obeying monoidal products

(One can identify _ Z/3 Z^1={1,X,Y}.) Thus, they have

    _ B(1)=_ B(X)=_ B(Y)=_ B(Z)=_ B(S)
       =_ B(T)=_ B(U)=_ B(V)=_ B(W)=1,

and

    ( B)=9.

Their quantum dimensions d_j's are solutions of the same multiplication rules d_id_j=∑_k=1^9N_ij^kd_k. The only solution is

    (d_X,d_Y,d_Z,d_S,d_T,d_U,d_V,d_W)=(1,1,1,1,1,1,1,1)

with categorical dimension

    D^2( B)=9.

Thus, all MFCs are unitary. They have two[Naively, one finds six consistent conformal dimensions. However, the other four are related to one of the two in the main text by permutations (XY)(ZUWSTV) of simple objects.] conformal dimensions

    (h_X,h_Y,h_Z,h_S,h_T,h_U,h_V,h_W)=(0,0,1/9,1/9,7/9,7/9,4/9,4/9),(0,0,8/9,8/9,2/9,2/9,5/9,5/9)  (1).

The S-matrices are given by

    S=[         1       d_X       d_Y       d_Z       d_S       d_T       d_U       d_V       d_W;       d_X         1         1 e^∓2π i/3 e^±2π i/3 e^∓2π i/3 e^±2π i/3 e^±2π i/3 e^∓2π i/3;       d_Y         1         1 e^±2π i/3 e^∓2π i/3 e^±2π i/3 e^∓2π i/3 e^∓2π i/3 e^±2π i/3;       d_Z e^∓2π i/3 e^±2π i/3 e^±4π i/9 e^∓4π i/9 e^∓2π i/9 e^±2π i/9 -e^∓π i/9 -e^±π i/9;       d_S e^±2π i/3 e^∓2π i/3 e^∓4π i/9 e^±4π i/9 e^±2π i/9 e^∓2π i/9 -e^±π i/9 -e^∓π i/9;       d_T e^∓2π i/3 e^±2π i/3 e^∓2π i/9 e^±2π i/9 -e^±π i/9 -e^∓π i/9 e^∓4π i/9 e^±4π i/9;       d_U e^±2π i/3 e^∓2π i/3 e^±2π i/9 e^∓2π i/9 -e^∓π i/9 -e^±π i/9 e^±4π i/9 e^∓4π i/9;       d_V e^±2π i/3 e^∓2π i/3 -e^∓π i/9 -e^±π i/9 e^∓4π i/9 e^±4π i/9 e^∓2π i/9 e^±2π i/9;       d_W e^∓2π i/3 e^±2π i/3 -e^±π i/9 -e^∓π i/9 e^±4π i/9 e^∓4π i/9 e^±2π i/9 e^∓2π i/9 ].

(All signs are correlated. In other words, one S-matrix is given by choosing upper signs, and the other is its complex conjugate.) They have additive central charges

    c( B)=0  (8).

There are

    1(quantum dimension)×2(conformal dimensions)×2(categorical dimensions)=4

MFCs, which are all unitary. We classify connected étale algebras in them.

We work with an ansatz

    A≅1⊕ n_XX⊕ n_YY⊕ n_ZZ⊕ n_SS⊕ n_TT⊕ n_UU⊕ n_VV⊕ n_WW

with n_j∈ N. It has

    _ B(A)=1+n_X+n_Y+n_Z+n_S+n_T+n_U+n_V+n_W.

For this to obey (<ref>), the natural numbers can take only 45 values. The sets contain the one with all n_j's be zero. It corresponds to the trivial connected étale algebra A≅1 giving B_A^0≃ B_A≃ B.

The candidates with simple object(s) Z,S,T,U,V,W fail to be commutative because they have nontrivial conformal dimensions. Thus, we are left with nontrivial candidates with simple objects X,Y. Since any étale algebra A∈ B is self-dual A^*≅ A <cit.>, the only candidate is

    A≅1⊕ X⊕ Y.

In fact, this is a commutative algebra; the braidings were computed in <cit.>, and it was found they are trivial. Thanks to the lemma 1, A∈_ Z/3 Z^1⊂_ Z/9 Z^1 is a commutative algebra. It also turns out separable, hence étale. Let us check this point. The algebra has _ B=3, and demands

    ( B_A^0)=1,  ( B_A)=3.

One identifies B_A^0≃. This identification also matches central charges. The category B_A of right A-modules is identified as

    B_A≃_ Z/3 Z^1.

To find this, we search for NIM-reps. Since there is no fusion category with Frobenius-Perron dimension three up to rank two, we start from three-dimensional NIM-reps. Indeed, we find a three-dimensional NIM-rep

    n_1=n_X=n_Y=1_3,   n_Z=n_T=n_W=[ 0 0 1; 1 0 0; 0 1 0 ],   n_S=n_U=n_V=[ 0 1 0; 0 0 1; 1 0 0 ].

Denoting a basis of B_A by {m_1,m_2,m_3}, we get a multiplication table

In this basis, we can identify

    m_1≅1⊕ X⊕ Y,   m_2≅ S⊕ U⊕ V,   m_3≅ Z⊕ T⊕ W.

In B_A, they have quantum dimensions (<ref>)

    d_ B_A(m_1)=d_ B_A(m_2)=d_ B_A(m_3)=1,

and obey the Z/3 Z monoidal products

This shows B_A is a fusion category. Thus, A is separable, showing the fact.

We conclude

All the four MFCs B≃_ Z/9 Z^1's fail to be completely anisotropic.



  §.§.§ B≃_ Z/3 Z× Z/3 Z^α

The MFCs have nine simple objects {1,X,Y,Z,S,T,U,V,W} obeying monoidal products

(We have X^*≅ Y,Z^*≅ S,T^*≅ U,V^*≅ W. One can identify _ Z/3 Z^1={1,T,U},{1,V,W}, and X≅ U⊗ W,Y≅ T⊗ V,Z≅ U⊗ V,S≅ T⊗ W.) Thus, they have

    _ B(1)=_ B(X)=_ B(Y)=_ B(Z)=_ B(S)
       =_ B(T)=_ B(U)=_ B(V)=_ B(W)=1,

and

    ( B)=9.

Their quantum dimensions d_j's are solutions of the same multiplication rules d_id_j=∑_k=1^9N_ij^kd_k. There is only one solution

    (d_X,d_Y,d_Z,d_S,d_T,d_U,d_V,d_W)=(1,1,1,1,1,1,1,1)

with categorical dimension

    D^2( B)=9.

Thus, all MFCs are unitary. They have two conformal dimensions[Naively, one finds 18 consistent conformal dimensions, but the other 16 are equivalent to one of the two in the main text under permutations of simple objects.]

    (h_X,h_Y,h_Z,h_S,h_T,h_U,h_V,h_W)=(0,0,0,0,1/3,1/3,2/3,2/3),(1/3,1/3,1/3,1/3,2/3,2/3,2/3,2/3)  (1).

The S-matrices are given by

    S=[                1              d_X              d_Y           d_Xd_W           d_Yd_V           d_Yd_W           d_Xd_V              d_V              d_W;              d_X        e^∓2π i/3        e^±2π i/3     e^∓2π i/3d_W     e^±2π i/3d_V     e^±2π i/3d_W     e^∓2π i/3d_V           d_Xd_V           d_Xd_W;              d_Y        e^±2π i/3        e^∓2π i/3     e^±2π i/3d_W     e^∓2π i/3d_V     e^∓2π i/3d_W     e^±2π i/3d_V           d_Yd_V           d_Yd_W;           d_Xd_W     e^∓2π i/3d_W     e^±2π i/3d_W e^∓2π i/3∓2π i/3 e^±2π i/3±2π i/3 e^±2π i/3∓2π i/3 e^∓2π i/3±2π i/3     e^±2π i/3d_X     e^∓2π i/3d_X;           d_Yd_V     e^±2π i/3d_V     e^∓2π i/3d_V e^±2π i/3±2π i/3 e^∓2π i/3∓2π i/3 e^∓2π i/3±2π i/3 e^±2π i/3∓2π i/3     e^∓2π i/3d_Y     e^±2π i/3d_Y;           d_Yd_W     e^±2π i/3d_W     e^∓2π i/3d_W e^±2π i/3∓2π i/3 e^∓2π i/3±2π i/3 e^∓2π i/3∓2π i/3 e^±2π i/3±2π i/3     e^±2π i/3d_Y     e^∓2π i/3d_Y;           d_Xd_V     e^∓2π i/3d_V     e^±2π i/3d_V e^∓2π i/3±2π i/3 e^±2π i/3∓2π i/3 e^±2π i/3±2π i/3 e^∓2π i/3∓2π i/3     e^∓2π i/3d_X     e^±2π i/3d_X;              d_V           d_Xd_V           d_Yd_V     e^±2π i/3d_X     e^∓2π i/3d_Y     e^±2π i/3d_Y     e^∓2π i/3d_X        e^∓2π i/3        e^±2π i/3;              d_W           d_Xd_W           d_Yd_W     e^∓2π i/3d_X     e^±2π i/3d_Y     e^∓2π i/3d_Y     e^±2π i/3d_X        e^±2π i/3        e^∓2π i/3 ].

They have additive central charges

    c( B)=c(_ Z/3 Z^1)+c(_ Z/3 Z^1)  (8)

where

    c(_ Z/3 Z^1)=2   (h_ Z/3 Z=1/3),
    -2   (h_ Z/3 Z=2/3).  (8)

There are

    1(quantum dimension)×2(conformal dimensions)×2(categorical dimensions)=4

MFCs, and all of them are unitary. We classify connected étale algebras in them.

An ansatz

    A≅1⊕ n_XX⊕ n_YY⊕ n_ZZ⊕ n_SS⊕ n_TT⊕ n_UU⊕ n_VV⊕ n_WW

with n_j∈ N has

    _ B(A)=1+n_X+n_Y+n_Z+n_S+n_T+n_U+n_V+n_W.

For this to obey (<ref>), the natural numbers can take only 45 values just as in the previous example. The sets contain the one with all n_j's be zero. It is the trivial connected étale algebra A≅1 giving B_A^0≃ B_A≃ B. The others contain nontrivial simple object(s). Those 30 with T,U,V,W fail to be commutative because they have nontrivial conformal dimensions. Together with the self-duality <cit.> of étale algebras A^*≅ A, we are left with two nontrivial candidates

    A≅1⊕ X⊕ Y,1⊕ Z⊕ S.

Since {1,X,Y} or {1,Z,S} form Z/3 Z pre-modular fusion subcategories, they were studied in <cit.>. As a result, the candidates fail to be commutative for the second conformal dimension, while they are commutative for the first conformal dimension.

Let us check their separability by identifying B_A. Since the commutative algebras have _ B(A)=3, they demand

    ( B_A^0)=1,  ( B_A)=3.

The MFC B_A^0 is identified as

    B_A^0≃.

This identification also matches central charges. The category B_A of right A-modules is identified as

    B_A≃_ Z/3 Z^1.

To show this, we search for NIM-reps. We start from A≅1⊕ X⊕ Y. We find a three-dimensional NIM-rep

    n_1=n_X=n_Y=1_3,   n_Z=n_T=n_W=[ 0 0 1; 1 0 0; 0 1 0 ],   n_S=n_U=n_V=[ 0 1 0; 0 0 1; 1 0 0 ].

Denoting a basis of B_A by {m_1,m_2,m_3}, we obtain a multiplication table

In this basis, we can identify

    m_1≅1⊕ X⊕ Y,   m_2≅ S⊕ U⊕ V,   m_3≅ Z⊕ T⊕ W.

They have

    d_ B_A(m_1)=d_ B_A(m_2)=d_ B_A(m_3)=1.

Similarly, for the other commutative algebra A≅ 1⊕ Z⊕ S, just names of matrices change. We find a three-dimensional NIM-rep

    n_1=n_Z=n_S=1_3,   n_X=n_T=n_V=[ 0 0 1; 1 0 0; 0 1 0 ],   n_Y=n_U=n_W=[ 0 1 0; 0 0 1; 1 0 0 ].

We obtain identifications

    m_1≅1⊕ Z⊕ S,   m_2≅ X⊕ T⊕ V,   m_3≅ Y⊕ U⊕ W.

They both obey the Z/3 Z monoidal products

This shows B_A≃_ Z/3 Z^1 for the two commutative algebras. Since B_A is semisimple, this means the A's are separable, hence étale.

We conclude

That is, two MFCs B≃_ Z/3 Z× Z/3 Z^α's with the second conformal dimensions are completely anisotropic, while the other two are not.



  §.§.§ B≃_ Z/3 Z^1⊠

The MFCs have nine simple objects {1,X,Y,Z,S,T,U,V,W} obeying monoidal products

(One can identify _ Z/3 Z^1={1,Y,Z},={1,X,U}, and S≅ X⊗ Z,T≅ X⊗ Y,V≅ Y⊗ U,W≅ Z⊗ U.) Thus, they have

    _ B(1)=_ B(X)   =_ B(Y)=_ B(Z)=_ B(S)=_ B(T)=1,
       _ B(U)=_ B(V)=_ B(W)=√(2),

and

    ( B)=12.

Their quantum dimensions d_j's are solutions of the same multiplication rules d_id_j=∑_k=1^9N_ij^kd_k. There are two (nonzero) solutions

    (d_X,d_Y,d_Z,d_S,d_T,d_U,d_V,d_W)=(1,1,1,1,1,-√(2),-√(2),-√(2)),(1,1,1,1,1,√(2),√(2),√(2))

with categorical dimension

    D^2( B)=12.

Only the second quantum dimension gives unitary MFCs. They both have the same 16 conformal dimensions

    (h_X,h_Y,h_Z,h_S,h_T,h_U,h_V,h_W)   =(1/2,1/3,1/3,5/6,5/6,1/16,19/48,19/48),(1/2,1/3,1/3,5/6,5/6,3/16,25/48,25/48),
           (1/2,1/3,1/3,5/6,5/6,5/16,31/48,31/48),(1/2,1/3,1/3,5/6,5/6,7/16,37/48,37/48),
           (1/2,1/3,1/3,5/6,5/6,9/16,43/48,43/48),(1/2,1/3,1/3,5/6,5/6,11/16,1/48,1/48),
           (1/2,1/3,1/3,5/6,5/6,13/16,7/48,7/48),(1/2,1/3,1/3,5/6,5/6,15/16,13/48,13/48),
           (1/2,2/3,2/3,1/6,1/6,1/16,35/48,35/48),(1/2,2/3,2/3,1/6,1/6,3/16,41/48,41/48),
           (1/2,2/3,2/3,1/6,1/6,5/16,47/48,47/48),(1/2,2/3,2/3,1/6,1/6,7/16,5/48,5/48),
           (1/2,2/3,2/3,1/6,1/6,9/16,11/48,11/48),(1/2,2/3,2/3,1/6,1/6,11/16,17/48,17/48),
           (1/2,2/3,2/3,1/6,1/6,13/16,23/48,23/48),(1/2,2/3,2/3,1/6,1/6,15/16,29/48,29/48)  (1).

The S-matrices are given by

    S=[             1           d_X           d_Y           d_Z        d_Xd_Z        d_Xd_Y           d_U        d_Yd_U        d_Zd_U;           d_X             1        d_Xd_Y        d_Xd_Z           d_Z           d_Y          -d_U          -d_Y          -d_Z;           d_Y        d_Xd_Y     e^∓2π i/3     e^±2π i/3 e^±2π i/3 d_X  e^∓2π i/3d_X        d_Yd_U  e^∓2π i/3d_U  e^±2π i/3d_U;           d_Z        d_Xd_Z     e^±2π i/3     e^∓2π i/3  e^∓2π i/3d_X  e^±2π i/3d_X        d_Zd_U  e^±2π i/3d_U  e^∓2π i/3d_U;        d_Xd_Z           d_Z  e^±2π i/3d_X  e^∓2π i/3d_X     e^∓2π i/3     e^±2π i/3       -d_Zd_U -e^±2π i/3d_U -e^∓2π i/3d_U;        d_Xd_Y           d_Y  e^∓2π i/3d_X  e^±2π i/3d_X     e^±2π i/3     e^∓2π i/3       -d_Yd_U -e^∓2π i/3d_U -e^±2π i/3d_U;           d_U          -d_U        d_Yd_U        d_Zd_U       -d_Zd_U       -d_Yd_U             0             0             0;        d_Yd_U          -d_Y  e^∓2π i/3d_U  e^±2π i/3d_U -e^±2π i/3d_U -e^∓2π i/3d_U             0             0             0;        d_Zd_U          -d_Z  e^±2π i/3d_U  e^∓2π i/3d_U -e^∓2π i/3d_U -e^±2π i/3d_U             0             0             0 ].

(All signs are correlated. In other words, one S-matrix is given by choosing upper signs, and the other is its complex conjugate.) There are

    2(quantum dimensions)×16(conformal dimensions)×2(categorical dimensions)=64

MFCs, among which those 32 with the second quantum dimensions are unitary. We classify connected étale algebras in all 64 MFCs simultaneously.

An ansatz

    A≅1⊕ n_XX⊕ n_YY⊕ n_ZZ⊕ n_SS⊕ n_TT⊕ n_UU⊕ n_VV⊕ n_WW

with n_j∈ N has

    _ B(A)=1+n_X+n_Y+n_Z+n_S+n_T+√(2)(n_U+n_V+n_W).

For this to obey (<ref>), the natural numbers can take only 39 values. The sets contain the one with all n_j's be zero. It is the trivial connected étale algebra A≅1 giving B_A^0≃ B_A≃ B. The other 38 candidates contain nontrivial simple object(s) with nontrivial conformal dimensions, and they all fail to be commutative. (Those with just X's do satisfy the necessary condition (<ref>), but it has c_X,X≅-id_1 <cit.> and fail to be commutative.)

We conclude

All the 64 MFCs B≃_ Z/3 Z^1⊠'s are completely anisotropic.



  §.§.§ B≃⊠

The MFCs have nine simple objects {1,X,Y,Z,S,T,U,V,W} obeying monoidal products

(One can identify ={1,X,T},{1,Z,S}, and Y≅ X⊗ Z, U≅ Z⊗ T,V≅ X⊗ S,W≅ S⊗ T.) Thus, they have

    _ B(1)=_ B(X)=_ B(Y)=_ B(Z)=1,
    _ B(S)=   _ B(T)=_ B(U)=_ B(V)=√(2),  _ B(W)=2,

and

    ( B)=16.

Their quantum dimensions d_j's are solutions of the same multiplication rules d_id_j=∑_k=1^9N_ij^kd_k. There are four (nonzero) solutions

    (d_X,d_Y,d_Z,d_S,d_T,d_U,d_V,d_W)   =(1,1,1,-√(2),-√(2),-√(2),-√(2),2),(1,1,1,-√(2),√(2),√(2),-√(2),-2),
           (1,1,1,√(2),-√(2),-√(2),√(2),-2),(1,1,1,√(2),√(2),√(2),√(2),2)

with categorical dimension

    D^2( B)=16.

Only the last quantum dimension gives unitary MFCs. They have many conformal dimensions. In order to avoid double-counting, we perform case analysis.



  
(d_S,d_T)=(√(2),√(2)). These give unitary MFCs. Different MFCs are labeled by 20 conformal dimensions[Naively, one has 64 conformal dimensions, but two conformal dimensions related by permutations (SV),(TU),(XZ)(STVU),(XZ)(SUVT) give the same MFC. By studying some examples, one finds conformal dimensions are determined by two pairs (h_S,h_V),(h_T,h_U). In the 64 sets, one conformal dimension (h_S,h_T,h_U,h_V) appears in eight different orders. The number eight is given by

    8=2(1st pair in/outside)×2(order within 1st pair)×2(order within 2nd pair).

If two combinations give the same h_W, they give the same MFC. Then, the 20 conformal dimensions are given by 10 pairs

    (1/16,9/16)(1/16,9/16),(1/16,9/16)(3/16,11/16),(1/16,9/16)(5/16,13/16),(1/16,9/16)(7/16,15/16),(3/16,11/16)(3/16,11/16),
       (3/16,11/16)(5/16,13/16),(3/16,11/16)(7/16,15/16),(5/16,13/16)(5/16,13/16),(5/16,13/16)(7/16,15/16),(7/16,15/16)(7/16,15/16),

each pair giving two different h_W's and hence two different MFCs.]

    (h_X,h_Y,h_Z,h_S,h_T,h_U,h_V,h_W)   =(1/2,0,1/2,1/16,1/16,9/16,9/16,1/8),(1/2,0,1/2,1/16,3/16,11/16,9/16,1/4),
           (1/2,0,1/2,1/16,5/16,13/16,9/16,3/8),(1/2,0,1/2,1/16,7/16,15/16,9/16,1/2),
           (1/2,0,1/2,1/16,9/16,1/16,9/16,5/8),(1/2,0,1/2,1/16,11/16,3/16,9/16,3/4),
           (1/2,0,1/2,1/16,13/16,5/16,9/16,7/8),(1/2,0,1/2,1/16,15/16,7/16,9/16,0),
           (1/2,0,1/2,3/16,3/16,11/16,11/16,3/8),(1/2,0,1/2,3/16,5/16,13/16,11/16,1/2),
           (1/2,0,1/2,3/16,7/16,15/16,11/16,5/8),(1/2,0,1/2,3/16,11/16,3/16,11/16,7/8),
           (1/2,0,1/2,3/16,13/16,5/16,11/16,0),(1/2,0,1/2,3/16,15/16,7/16,11/16,1/8),
           (1/2,0,1/2,5/16,5/16,13/16,13/16,5/8),(1/2,0,1/2,5/16,7/16,15/16,13/16,3/4),
           (1/2,0,1/2,5/16,13/16,5/16,13/16,1/8),(1/2,0,1/2,5/16,15/16,7/16,13/16,1/4),
           (1/2,0,1/2,7/16,7/16,15/16,15/16,7/8),(1/2,0,1/2,7/16,15/16,7/16,15/16,3/8)  (1).

Including the two signs of categorical dimensions, we have 40 unitary MFCs.



  
(d_S,d_T)=(√(2),-√(2)). In this case, we have smaller symmetries; just exchange of orders in two outer and inner pairs, (h_S,h_V),(h_T,h_U). Different MFCs are given by 32 conformal dimensions

    (h_X,h_Y,h_Z,h_S,h_T,h_U,h_V,h_W)   =(1/2,0,1/2,1/16,1/16,9/16,9/16,1/8),(1/2,0,1/2,1/16,3/16,11/16,9/16,1/4),
           (1/2,0,1/2,1/16,5/16,13/16,9/16,3/8),(1/2,0,1/2,1/16,7/16,15/16,9/16,1/2),
           (1/2,0,1/2,1/16,9/16,1/16,9/16,5/8),(1/2,0,1/2,1/16,11/16,3/16,9/16,3/4),
           (1/2,0,1/2,1/16,13/16,5/16,9/16,7/8),(1/2,0,1/2,1/16,15/16,7/16,9/16,0),
           (1/2,0,1/2,3/16,1/16,9/16,11/16,1/4),(1/2,0,1/2,3/16,3/16,11/16,11/16,3/8),
           (1/2,0,1/2,3/16,5/16,13/16,11/16,1/2),(1/2,0,1/2,3/16,7/16,15/16,11/16,5/8),
           (1/2,0,1/2,3/16,9/16,1/16,11/16,3/4),(1/2,0,1/2,3/16,11/16,3/16,11/16,7/8),
           (1/2,0,1/2,3/16,13/16,5/16,11/16,0),(1/2,0,1/2,3/16,15/16,7/16,11/16,1/8),
           (1/2,0,1/2,5/16,1/16,9/16,13/16,3/8),(1/2,0,1/2,5/16,3/16,11/16,13/16,1/2),
           (1/2,0,1/2,5/16,5/16,13/16,13/16,5/8),(1/2,0,1/2,5/16,7/16,15/16,13/16,3/4),
           (1/2,0,1/2,5/16,9/16,1/16,13/16,7/8),(1/2,0,1/2,5/16,11/16,3/16,13/16,0),
           (1/2,0,1/2,5/16,13/16,5/16,13/16,1/8),(1/2,0,1/2,5/16,15/16,7/16,13/16,1/4),
           (1/2,0,1/2,7/16,1/16,9/16,15/16,1/2),(1/2,0,1/2,7/16,3/16,11/16,15/16,5/8),
           (1/2,0,1/2,7/16,5/16,13/16,15/16,3/4),(1/2,0,1/2,7/16,7/16,15/16,15/16,7/8),
           (1/2,0,1/2,7/16,9/16,1/16,15/16,0),(1/2,0,1/2,7/16,11/16,3/16,15/16,1/8),
           (1/2,0,1/2,7/16,13/16,5/16,15/16,1/4),(1/2,0,1/2,7/16,15/16,7/16,15/16,3/8)  (1).

With two signs of categorical dimensions, there are 64 MFCs.



  
(d_S,d_T)=(-√(2),-√(2)). In this case, symmetries are the same as (d_S,d_T)=(√(2),√(2)). Thus, the same 20 conformal dimensions give 40 MFCs.

Therefore, there are

    40+64+40=144

MFCs, among which those 40 in the first case are unitary. The S-matrices are given by

    S=[       1     d_X  d_Xd_Z     d_Z     d_S     d_T  d_Zd_T  d_Xd_S  d_Sd_T;     d_X       1     d_Z  d_Xd_Z  d_Xd_S    -d_T -d_Zd_T     d_S -d_Sd_T;  d_Xd_Z     d_Z       1     d_X -d_Xd_S -d_Zd_T    -d_T    -d_S  d_Sd_T;     d_Z  d_Xd_Z     d_X       1    -d_S  d_Zd_T     d_T -d_Xd_S -d_Sd_T;     d_S  d_Xd_S -d_Xd_S    -d_S       0  d_Sd_T -d_Sd_T       0       0;     d_T    -d_T -d_Zd_T  d_Zd_T  d_Sd_T       0       0 -d_Sd_T       0;  d_Zd_T -d_Zd_T    -d_T     d_T -d_Sd_T       0       0  d_Sd_T       0;  d_Xd_S     d_S    -d_S -d_Xd_S       0 -d_Sd_T  d_Sd_T       0       0;  d_Sd_T -d_Sd_T  d_Sd_T -d_Sd_T       0       0       0       0       0 ].

They have additive central charges

    c( B)=c()+c()  (8)

where

    c()=1/2   (h_S,T=1/16),
    3/2   (h_S,T=3/16),
    5/2   (h_S,T=5/16),
    7/2   (h_S,T=7/16),
    -7/2   (h_S,T=9/16),
    -5/2   (h_S,T=11/16),
    -3/2   (h_S,T=13/16),
    -1/2   (h_S,T=15/16).  (8)


In order to classify connected étale algebras, we set an ansatz

    A≅1⊕ n_XX⊕ n_YY⊕ n_ZZ⊕ n_SS⊕ n_TT⊕ n_UU⊕ n_VV⊕ n_WW

with n_j∈ N. It has

    _ B(A)=1+n_X+n_Y+n_Z+√(2)(n_S+n_T+n_U+n_V)+2n_W.

For this to obey (<ref>), the natural numbers can take only 50 values. The sets contain the one with all the n_j's be zero. It is the trivial connected étale algebra A≅1 giving B_A^0≃ B_A≃ B. All other candidates with nontrivial simple objects except Y or W fail to be commutative because they have nontrivial conformal dimensions. Thus, we are left with nontrivial candidates just with Y,W. Setting n_X,n_Z,n_S,n_T,n_U,n_V to zero, apart from the trivial one, we find five natural numbers

    (n_Y,n_W)=(1,0),(0,1),(2,0),(1,1),(3,0).

Some of them are ruled out by studying Frobenius-Perron dimensions. The two candidates (n_Y,n_W)=(0,1),(2,0) have _ B(A)=3, and demands ( B_A^0)=16/9, but there is no MFC with such Frobenius-Perron dimension. Thus, the two candidates are ruled out. Therefore, we are left with three candidates

    (n_Y,n_W)=(1,0),(1,1),(3,0).

We study these one after another.



  
A≅1⊕ Y. This is a Z/2 Z algebra, and it is also commutative because it has (d_Y,h_Y)=(1,0) (mod 1 for h_Y) and c_Y,Y≅ id_1 <cit.>. It further turns out separable (hence étale). Let us check this point by identifying B_A.

First, _ B(A)=2 demands

    ( B_A^0)=4,  ( B_A)=8.

The MFC B_A^0 is identified as

    B_A^0≃_ Z/2 Z^-1⊠_ Z/2 Z^-1   (h_W=2n+1/4 with n∈ N  (1)),
       (h_W=0  (1/2)),
    _ Z/4 Z^α   (h_W=2n+1/8 with n∈ N  (1)).

This can be seen in two ways. First, let us perform anyon condensation. This `identifies' 1,Y, and hence pairs (X,Z),(S,V),(T,U), and W splits into two. Since S and V have different conformal dimensions, the resulting simple object in B_A is confined. Similarly, the simple object from T,U is confined. The simple object coming from X,Z is deconfined, and is an object of B_A^0 with conformal dimension 1/2. Finally, W splits into two invertible simple objects with conformal dimensions h_W mod 1. Thus, if h_W=n/8 (mod 1) with an odd natural number n, the four simple objects form Z/4 Z MFC, while if h_W=n/8 (mod 1) with an even natural number n, the four simple objects form Z/2 Z× Z/2 Z MFC. More precisely, if h_W=n/4 (mod 1) with n=1,3, it is _ Z/2 Z^-1⊠_ Z/2 Z^-1, and if h_W=n/4 (mod 1) with n=0,2, then it is . This gives the identifications.

Another way to see the identification is to compute the (additive) central charge c( B). The Ising MFC has central charge 2n+1/2 (mod 8) with an integer n=-4,-3,…,3. Let us denote the central charges of the two factors 2n_1+1/2,2n_2+1/2, respectively. Then, the Deligne tensor product ⊠ has (additive) central charge

    c( B)=n_1+n_2+1  (8).

Therefore, the MFC B_A^0 has (additive) central charge

    c( B_A^0)=0   (n_1+n_2∈2 Z+1),
    1   (n_1+n_2∈2 Z)..  (2)

An MFC with =4 and this (additive) central charge is either Z/2 Z× Z/2 Z MFC or Z/4 Z MFC, respectively.[They have additive central charges

    c(_ Z/2 Z^-1⊠_ Z/2 Z^-1)=0   (h_x≠ h_y),
    2   (h_x=1/4=h_y),
    -2   (h_x=3/4=h_y),  (8)


    c()=0   (h_x,h_y,h_z)=(1/2,0,0),
    4   (h_x,h_y,h_z)=(1/2,1/2,1/2),  (8)

or

    c(_ Z/4 Z^α)=1   (h_ Z/4 Z=1/8),
    3   (h_ Z/4 Z=3/8),
    -3   (h_ Z/4 Z=5/8),
    -1   (h_ Z/4 Z=7/8),  (8)

respectively. Here, conformal dimensions mean those of two nontrivial simple objects in two factors of Z/2 Z MFC (h_x,h_y), those of three nontrivial simple objects (), or those of Z/4 Z simple objects (Z/4 Z MFC).] (One may naively think  MFCs also match the Frobenius-Perron dimensions, but they are ruled out because they cannot match the additive central charges.) More precise matching is given as follows.



  
Even c( B). When n_1+n_2∈2 Z+1, the ambient category has even additive central charge. It is matched as follows:

    c( B)=0   ( B_A^0≃_ Z/2 Z^-1⊠_ Z/2 Z^-1 with h_x≠ h_y, or  with (h_x,h_y,h_z)=(1/2,0,0)),
    2   (_ Z/2 Z^-1⊠_ Z/2 Z^-1 with h_x=1/4=h_y),
    4   ( with (h_x,h_y,h_z)=(1/2,1/2,1/2)),
    6   (_ Z/2 Z^-1⊠_ Z/2 Z^-1 with h_x=3/4=h_y).




  
Odd c( B). When n_1+n_2∈2 Z, the central charge of the ambient category can only be matched by Z/4 Z MFCs:

    c( B)=1   (_ Z/4 Z^α with h_ Z/4 Z=1/8),
    3   (_ Z/4 Z^α with h_ Z/4 Z=3/8),
    5   (_ Z/4 Z^α with h_ Z/4 Z=5/8),
    7   (_ Z/4 Z^α with h_ Z/4 Z=7/8).  (8)


Together with the invariance of topological twist (<ref>), we can uniquely identify B_A^0.

Having specified the category B_A^0 of dyslectic right A-modules, let us next figure out the category B_A of right A-modules. In view of anyon condensation, it has two more simple objects with Frobenius-Perron dimensions √(2). Thus, we learn

    ( B_A)=8,  ( B_A)=6.

With this information, we search for six-dimensional NIM-reps. Indeed, we find a solution

    n_1=1_6=n_Y,      n_X=[ 0 1 0 0 0 0; 1 0 0 0 0 0; 0 0 1 0 0 0; 0 0 0 1 0 0; 0 0 0 0 0 1; 0 0 0 0 1 0 ]=n_Z,   n_S=[ 0 0 1 0 0 0; 0 0 1 0 0 0; 1 1 0 0 0 0; 0 0 0 0 1 1; 0 0 0 1 0 0; 0 0 0 1 0 0 ]=n_V,
       n_T=[ 0 0 0 1 0 0; 0 0 0 1 0 0; 0 0 0 0 1 1; 1 1 0 0 0 0; 0 0 1 0 0 0; 0 0 1 0 0 0 ]=n_U,   n_W=[ 0 0 0 0 1 1; 0 0 0 0 1 1; 0 0 0 2 0 0; 0 0 2 0 0 0; 1 1 0 0 0 0; 1 1 0 0 0 0 ].

The NIM-rep gives a multiplication table

From this, we can identify

    m_1≅1⊕ Y,   m_2≅ X⊕ Z,   m_3≅ S⊕ V,   m_4≅ T⊕ U,   m_5≅ W≅ m_6.

In B_A, they have quantum dimensions

    d_ B_A(m_1)=1=d_ B_A(m_2),   d_ B_A(m_3)=±√(2),   d_ B_A(m_4)=±√(2),   d_ B_A(m_5)=±1=d_ B_A(m_6).

Working out the monoidal products ⊗_A, we find
for B_A^0≃_ Z/2 Z× Z/2 Z^α (i.e., h_W=0,1/4,1/2,3/4 mod 1), and
for B_A^0≃_ Z/4 Z^α (i.e., h_W=1/8,3/8,5/8,7/8 mod 1). These give identifications[The identification with the notation in AnyonWiki is given by

    m_1≅1,   m_2≅4,   m_3≅5,   m_4≅6,   m_5≅2,   m_6≅3,

or its permutations (56),(23), and

    m_1≅1,   m_2≅2,   m_3≅5,   m_4≅6,   m_5≅3,   m_6≅4,

or its permutations (34),(56), respectively.]

    B_A≃ C(FR^6,0_1)   (c( B)=0 mod 2),
     C(FR^6,0_2)   (c( B)=1 mod 2).

This shows B_A is semisimple, hence A is separable and étale.



  
A≅1⊕ Y⊕ W. When h_W=0 (mod 1), W can give commutative algebra. It has _ B(A)=4, and demands

    ( B_A^0)=1,  ( B_A)=4.

Computing b_j⊗ A, we find candidate simple objects

    1⊕ Y⊕ W,   X⊕ Z⊕ W,   S⊕ T⊕ U⊕ V.

They have Frobenius-Perron dimensions

    1,  1,  √(2),

and their contributions to ( B_A) match, 1^2+1^2+√(2)^2=4. This suggests B_A have rank three. Indeed, we find a three-dimensional NIM-rep

    n_1=1_3=n_Y,   n_X=[ 0 1 0; 1 0 0; 0 0 1 ]=n_Z,   n_S=n_T=n_U=n_V=[ 0 0 1; 0 0 1; 1 1 0 ],   n_W=[ 1 1 0; 1 1 0; 0 0 2 ].

The solution gives identifications

    m_1≅1⊕ Y⊕ W,   m_2≅ X⊕ Z⊕ W,   m_3≅ S⊕ T⊕ U⊕ V.

By computing quantum dimensions (<ref>), we find the candidate can be separable only when d_S=d_T. Then, in B_A, they have

    d_ B_A(m_1)=1=d_ B_A(m_2),   d_ B_A(m_3)=±√(2).

Working out the monoidal products ⊗_A, we find

We arrive the identification

    B_A≃.

Since this is semisimple, A is separable and étale. We found[The existence of the Lagrangian algebra can also be understood from the lemma 3 because when h_W=0 (mod 1), our ambient MFC is a Drinfeld center B≃ Z().]

    A≅1⊕ Y⊕ W  (d_S,d_T,h_S,h_T)=   (√(2),√(2),1/16,15/16),(√(2),√(2),3/16,13/16),
       (-√(2),-√(2),1/16,15/16),(-√(2),-√(2),3/16,13/16).
      (1 for h)




  
A≅1⊕3Y. It has _ B(A)=4, and demands

    ( B_A^0)=1,  ( B_A)=4.

Calculating b_j⊗ A, we find candidate simple objects (assuming the smallest Frobenius-Perron dimensions)

    1⊕3Y,   X⊕3Z,  3⊕ Y,  3X⊕ Z,   S⊕3V,   T⊕3U,  3T⊕ U,  3S⊕ V,  2W.

In a putative B_A, they have Frobenius-Perron dimensions

    1,  1,  1,  1,  √(2),  √(2),  √(2),  √(2),  1.

Their contributions to ( B_A) exceed four, and the candidate is ruled out.

We conclude

All the 144 MFCs B≃⊠'s fail to be completely anisotropic.

Let us comment on the consistency of our results. The nontrivial connected étale algebra leads to the category of dyslelctic right A-modules, _ Z/4 Z^α,_ Z/2 Z^-1⊠_ Z/2 Z^-1 (with h_x=h_y), or  with (h_x,h_y,h_z)=(1/2,1/2,1/2) or (h_x,h_y,h_z)=(1/2,0,0). The first three are completely anisotropic, and the “cascades” of conformal embeddings terminate there. On the other hand, for the last B_A^0, it can admit nontrivial connected étale algebras. More precisely, the number of additional nontrivial connected étale algebras are given by

    0   (d_W,h_W)=(-2,0),
    2   (d_W,h_W)=(2,0).

The additional connected étale algebras lead to . One of the two paths is given by composition with itself, 1⊕ Y. The other path

    ⊠→

is given by the additional connected étale algebra 1⊕ Y⊕ W. This appears exactly when (d_W,h_W)=(2,0). See (<ref>).



  §.§.§ B≃_ Z/3 Z^1⊠ psu(2)_5

The MFCs have nine simple objects {1,X,Y,Z,S,T,U,V,W} obeying monoidal products

(One can identify _ Z/3 Z^1={1,X,Y},psu(2)_5={1,Z,U}, and S≅ Y⊗ Z,T≅ X⊗ Z,V≅ X⊗ U,W≅ Y⊗ U.) Thus, they have

    _ B(1)=_ B(X)   =_ B(Y)=1,  _ B(Z)=_ B(S)=_ B(T)=sin2π/7/sinπ/7,
       _ B(U)=_ B(V)=_ B(W)=sin3π/7/sinπ/7,

and

    ( B)=21/4sin^2π/7≈27.9.

Their quantum dimensions d_j's are solutions of the same multiplication rules d_id_j=∑_k=1^9N_ij^kd_k. There are three solutions

    (d_X,d_Y,d_Z,d_S,d_T,d_U,d_V,d_W)   =(1,1,sinπ/7/cosπ/14,sinπ/7/cosπ/14,sinπ/7/cosπ/14,-sin2π/7/cosπ/14,-sin2π/7/cosπ/14,-sin2π/7/cosπ/14),
           (1,1,-sin3π/7/cos3π/14,-sin3π/7/cos3π/14,-sin3π/7/cos3π/14,sinπ/7/cos3π/14,sinπ/7/cos3π/14,sinπ/7/cos3π/14),
           (1,1,sin2π/7/sinπ/7,sin2π/7/sinπ/7,sin2π/7/sinπ/7,sin3π/7/sinπ/7,sin3π/7/sinπ/7,sin3π/7/sinπ/7)

with categorical dimensions

    D^2( B)=21/4cos^2π/14(≈5.5),  21/4cos^23π/14(≈8.6),  21/4sin^2π/7,

respectively. They have four conformal dimensions each:

    (h_X,h_Y,h_Z,h_S,h_T,h_U,h_V,h_W)
       =(1/3,1/3,3/7,16/21,16/21,1/7,10/21,10/21),(1/3,1/3,4/7,19/21,19/21,6/7,4/21,4/21),(2/3,2/3,3/7,2/21,2/21,1/7,17/21,17/21),(2/3,2/3,4/7,5/21,5/21,6/7,11/21,11/21)   (1st),
    (1/3,1/3,2/7,13/21,13/21,3/7,16/21,16/21),(1/3,1/3,5/7,1/21,1/21,4/7,19/21,19/21),(2/3,2/3,2/7,20/21,20/21,3/7,2/21,2/21),(2/3,2/3,5/7,8/21,8/21,4/7,5/21,5/21)   (2nd),
    (1/3,1/3,1/7,10/21,10/21,5/7,1/21,1/21),(1/3,1/3,6/7,4/21,4/21,2/7,13/21,13/21),(2/3,2/3,1/7,17/21,17/21,5/7,8/21,8/21),(2/3,2/3,6/7,11/21,11/21,2/7,20/21,20/21)   (3rd).
       (1)

The S-matrices are given by

    S=[             1           d_X           d_Y           d_Z        d_Yd_Z        d_Xd_Z           d_U        d_Xd_U        d_Yd_U;           d_X     e^∓2π i/3     e^±2π i/3        d_Xd_Z  e^±2π i/3d_Z  e^∓2π i/3d_Z        d_Xd_U  e^∓2π i/3d_U  e^±2π i/3d_U;           d_Y     e^±2π i/3     e^∓2π i/3        d_Yd_Z  e^∓2π i/3d_Z  e^±2π i/3d_Z        d_Yd_U  e^±2π i/3d_U  e^∓2π i/3d_U;           d_Z        d_Xd_Z        d_Yd_Z          -d_U       -d_Yd_U       -d_Xd_U             1           d_X           d_Y;        d_Yd_Z  e^±2π i/3d_Z  e^∓2π i/3d_Z       -d_Yd_U -e^∓2π i/3d_U -e^±2π i/3d_U           d_Y     e^±2π i/3     e^∓2π i/3;        d_Xd_Z  e^∓2π i/3d_Z  e^±2π i/3d_Z       -d_Xd_U -e^±2π i/3d_U -e^∓2π i/3d_U           d_X     e^∓2π i/3     e^±2π i/3;           d_U        d_Xd_U        d_Yd_U             1           d_Y           d_X          -d_Z       -d_Xd_Z       -d_Yd_Z;        d_Xd_U  e^∓2π i/3d_U  e^±2π i/3d_U           d_X     e^±2π i/3     e^∓2π i/3       -d_Xd_Z -e^∓2π i/3d_Z -e^±2π i/3d_Z;        d_Yd_U  e^±2π i/3d_U  e^∓2π i/3d_U           d_Y     e^∓2π i/3     e^±2π i/3       -d_Yd_Z -e^±2π i/3d_Z -e^∓2π i/3d_Z ].

(All signs are correlated. In other words, one S-matrix is given by choosing upper signs, and the other is its complex conjugate.) There are

    3(quantum dimensions)×4(conformal dimensions)×2(categorical dimensions)=24

MFCs, among which those eight with the third quantum dimensions give unitary MFCs. We classify connected étale algebras in all 24 MFCs simultaneously.

An ansatz

    A≅1⊕ n_XX⊕ n_YY⊕ n_ZZ⊕ n_SS⊕ n_TT⊕ n_UU⊕ n_VV⊕ n_WW

with n_j∈ N has

    _ B(A)=1+n_X+n_Y+sin2π/7/sinπ/7(n_Z+n_S+n_T)+sin3π/7/sinπ/7(n_U+n_V+n_W).

For this to obey (<ref>), the natural number can take only 66 values. The sets contain the one with all n_j's be zero. It is the trivial connected étale algebra A≅1 giving B_A^0≃ B_A≃ B. The other 65 sets contain simple object(s) with nontrivial conformal dimensions, and they all fail to be commutative.

We conclude

All the 24 MFCs B≃_ Z/3 Z^1⊠ psu(2)_5's are completely anisotropic.



  §.§.§ B≃⊠ psu(2)_5

The MFCs have nine simple objects {1,X,Y,Z,S,T,U,V,W} obeying monoidal products

(One can identify ={1,X,Y},psu(2)_5={1,Z,T}, and S≅ X⊗ Z,U≅ X⊗ T,V≅ Y⊗ Z,W≅ Y⊗ T.) Thus, they have

    _ B(1)=1=_ B(X),  _ B(Y)=√(2),  _ B(Z)=sin2π/7/sinπ/7=_ B(S),
    _ B(T)=sin3π/7/sinπ/7=_ B(U),  _ B(V)=√(2)sin2π/7/sinπ/7,  _ B(W)=√(2)sin3π/7/sinπ/7,

and

    ( B)=7/sin^2π/7≈37.2.

Their quantum dimensions d_j's are solutions of the same multiplication rules d_id_j=∑_k=1^9N_ij^kd_k. There are six (nonzero) solutions

    (d_X,d_Y,d_Z,d_S,d_T,d_U,d_V,d_W)   =(1,-√(2),sinπ/7/cosπ/14,sinπ/7/cosπ/14,-sin2π/7/cosπ/14,-sin2π/7/cosπ/14,-√(2)sinπ/7/cosπ/14,√(2)sin2π/7/cosπ/14),
           (1,√(2),sinπ/7/cosπ/14,sinπ/7/cosπ/14,-sin2π/7/cosπ/14,-sin2π/7/cosπ/14,√(2)sinπ/7/cosπ/14,-√(2)sin2π/7/cosπ/14),
           (1,-√(2),-sin3π/7/cos3π/14,-sin3π/7/cos3π/14,sinπ/7/cos3π/14,sinπ/7/cos3π/14,√(2)sin3π/7/cos3π/14,-√(2)sinπ/7/cos3π/14),
           (1,√(2),-sin3π/7/cos3π/14,-sin3π/7/cos3π/14,sinπ/7/cos3π/14,sinπ/7/cos3π/14,-√(2)sin3π/7/cos3π/14,√(2)sinπ/7/cos3π/14),
           (1,-√(2),sin2π/7/sinπ/7,sin2π/7/sinπ/7,sin3π/7/sinπ/7,sin3π/7/sinπ/7,-√(2)sin2π/7/sinπ/7,-√(2)sin3π/7/cosπ/7),
           (1,√(2),sin2π/7/sinπ/7,sin2π/7/sinπ/7,sin3π/7/sinπ/7,sin3π/7/sinπ/7,√(2)sin2π/7/sinπ/7,√(2)sin3π/7/cosπ/7)

with categorical dimensions

    D^2( B)=7/cos^2π/14(≈7.4),  7/cos^23π/14(≈11.5),  7/sin^2π/7,

respectively for each pair. Each pair has 16 conformal dimensions

    (h_X,h_Y,h_Z,h_S,h_T,h_U,h_V,h_W)=(1/2,1/16,3/7,13/14,1/7,9/14,55/112,23/112),(1/2,1/16,4/7,1/14,6/7,5/14,71/112,103/112),
    (1/2,3/16,3/7,13/14,1/7,9/14,69/112,37/112),(1/2,3/16,4/7,1/14,6/7,5/14,85/112,5/112),
    
        (1/2,5/16,3/7,13/14,1/7,9/14,83/112,51/112),(1/2,5/16,4/7,1/14,6/7,5/14,99/112,19/112),
    (1/2,7/16,3/7,13/14,1/7,9/14,97/112,65/112),(1/2,7/16,4/7,1/14,6/7,5/14,1/112,33/112),
    
        (1/2,9/16,3/7,13/14,1/7,9/14,111/112,79/112),(1/2,9/16,4/7,1/14,6/7,5/14,15/112,47/112),
    (1/2,11/16,3/7,13/14,1/7,9/14,13/112,93/112),(1/2,11/16,4/7,1/14,6/7,5/14,29/112,61/112),
    (1/2,13/16,3/7,13/14,1/7,9/14,27/112,107/112),(1/2,13/16,4/7,1/14,6/7,5/14,43/112,75/112),
    (1/2,15/16,3/7,13/14,1/7,9/14,41/112,9/112),(1/2,15/16,4/7,1/14,6/7,5/14,57/112,89/112),   (1st&2nd)
    
    
        (1/2,1/16,2/7,11/14,3/7,13/14,39/112,55/112),(1/2,1/16,5/7,3/14,4/7,1/14,87/112,71/112),
    (1/2,3/16,2/7,11/14,3/7,13/14,53/112,69/112),(1/2,3/16,5/7,3/14,4/7,1/14,101/112,85/112),
    
        (1/2,5/16,2/7,11/14,3/7,13/14,67/112,83/112),(1/2,5/16,5/7,3/14,4/7,1/14,3/112,99/112),
    (1/2,7/16,2/7,11/14,3/7,13/14,81/112,97/112),(1/2,7/16,5/7,3/14,4/7,1/14,17/112,1/112),
    
        (1/2,9/16,2/7,11/14,3/7,13/14,95/112,111/112),(1/2,9/16,5/7,3/14,4/7,1/14,31/112,15/112),
    (1/2,11/16,2/7,11/14,3/7,13/14,109/112,13/112),(1/2,11/16,5/7,3/14,4/7,1/14,45/112,29/112),
    
        (1/2,13/16,2/7,11/14,3/7,13/14,11/112,27/112),(1/2,13/16,5/7,3/14,4/7,1/14,59/112,43/112),
    (1/2,15/16,2/7,11/14,3/7,13/14,25/112,41/112),(1/2,15/16,5/7,3/14,4/7,1/14,73/112,57/112),   (3rd&4th)
    
    
        (1/2,1/16,1/7,9/14,5/7,3/14,23/112,87/112),(1/2,1/16,6/7,5/14,2/7,11/14,103/112,39/112),
    (1/2,3/16,1/7,9/14,5/7,3/14,37/112,101/112),(1/2,3/16,6/7,5/14,2/7,11/14,5/112,53/112),
    
        (1/2,5/16,1/7,9/14,5/7,3/14,51/112,3/112),(1/2,5/16,6/7,5/14,2/7,11/14,19/112,67/112),
    (1/2,7/16,1/7,9/14,5/7,3/14,65/112,17/112),(1/2,7/16,6/7,5/14,2/7,11/14,33/112,81/112),
    
        (1/2,9/16,1/7,9/14,5/7,3/14,79/112,31/112),(1/2,9/16,6/7,5/14,2/7,11/14,47/112,95/112),
    (1/2,11/16,1/7,9/14,5/7,3/14,93/112,45/112),(1/2,11/16,6/7,5/14,2/7,11/14,61/112,109/112),
    
        (1/2,13/16,1/7,9/14,5/7,3/14,107/112,59/112),(1/2,13/16,6/7,5/14,2/7,11/14,75/112,11/112),
    (1/2,15/16,1/7,9/14,5/7,3/14,9/112,73/112),(1/2,15/16,6/7,5/14,2/7,11/14,89/112,25/112).   (5th&6th).  (1)

The S-matrices are given by

    S=[       1     d_X     d_Y     d_Z  d_Xd_Z     d_T  d_Xd_T  d_Yd_Z  d_Yd_T;     d_X       1    -d_Y  d_Xd_Z     d_Z  d_Xd_T     d_T -d_Yd_Z -d_Yd_T;     d_Y    -d_Y       0  d_Yd_Z -d_Yd_Z  d_Yd_T -d_Yd_T       0       0;     d_Z  d_Xd_Z  d_Yd_Z    -d_T -d_Xd_T       1     d_X -d_Yd_T     d_Y;  d_Xd_Z     d_Z -d_Yd_Z -d_Xd_T    -d_T     d_X       1  d_Yd_T    -d_Y;     d_T  d_Xd_T  d_Yd_T       1     d_X    -d_Z -d_Xd_Z     d_Y -d_Yd_Z;  d_Xd_T     d_T -d_Yd_T     d_X       1 -d_Xd_Z    -d_Z    -d_Y  d_Yd_Z;  d_Yd_Z -d_Yd_Z       0 -d_Yd_T  d_Yd_T     d_Y    -d_Y       0       0;  d_Yd_T -d_Yd_T       0     d_Y    -d_Y -d_Yd_Z  d_Yd_Z       0       0 ].

There are

    6(quantum dimensions)×16(conformal dimensions)×2(categorical dimensions)=192

MFCs, among which those 32 with the sixth quantum dimensions are unitary. We classify connected étale algebras in all 192 MFCs simultaneously.

An ansatz

    A≅1⊕ n_XX⊕ n_YY⊕ n_ZZ⊕ n_SS⊕ n_TT⊕ n_UU⊕ n_VV⊕ n_WW

with n_j∈ N has

    _ B(A)=1+n_X+√(2)n_Y+sin2π/7/sinπ/7(n_Z+n_S)+sin3π/7/sinπ/7(n_T+n_U)+√(2)sin2π/7/sinπ/7n_V+√(2)sin3π/7/sinπ/7n_W.

For this to obey (<ref>), the natural number can take only 75 values. The sets contain the one with all n_j's be zero. It is the trivial connected étale algebra A≅1 giving B_A^0≃ B_A≃ B. The other 74 candidates contain nontrivial simple object(s) b_j≇1 with nontrivial conformal dimensions. Thus, they all fail to be commutative.

We conclude

All the 192 MFCs B≃⊠ psu(2)_5's are completely anisotropic.



  §.§.§ B≃ so(11)_2

The MFCs have nine simple objects {1,X,Y,Z,S,T,U,V,W} obeying monoidal products

Thus, they have

    _ B(1)=1=_ B(X),  _ B(Y)=   _ B(Z)=_ B(S)=_ B(T)=_ B(U)=2,
    _ B(V)=   √(11)=_ B(W),

and

    ( B)=44.

Their quantum dimensions d_j's are solutions of the same multiplication rules d_id_j=∑_k=1^9N_ij^kd_k. There are two (nonzero) solutions

    (d_X,d_Y,d_Z,d_S,d_T,d_U,d_V,d_W)=(1,2,2,2,2,2,-√(11),-√(11)),(1,2,2,2,2,2,√(11),√(11))

with the same categorical dimension

    D^2( B)=44.

They have four conformal dimensions[Naively one finds 40 consistent conformal dimensions, however, the other 36 are equivalent to one of the four in the main text under permutations (VW) or (YZUST) of simple objects.]

    (h_X,h_Y,h_Z,h_S,h_T,h_U,h_V,h_W)   =(0,1/11,9/11,3/11,5/11,4/11,1/8,5/8),(0,1/11,9/11,3/11,5/11,4/11,3/8,7/8),
           (0,10/11,2/11,8/11,6/11,7/11,1/8,5/8),(0,10/11,2/11,8/11,6/11,7/11,3/8,7/8)  (1).

The S-matrices are given by

    S=[     1   d_X   d_Y   d_Z   d_S   d_T   d_U   d_V   d_W;   d_X   d_X   d_Y   d_Z   d_S   d_T   d_U  -d_V  -d_W;   d_Y   d_Y   s_1   s_2   s_3   s_4   s_5     0     0;   d_Z   d_Z   s_2   s_5   s_4   s_1   s_3     0     0;   d_S   d_S   s_3   s_4   s_2   s_5   s_1     0     0;   d_T   d_T   s_4   s_1   s_5   s_3   s_2     0     0;   d_U   d_U   s_5   s_3   s_1   s_2   s_4     0     0;   d_V  -d_V     0     0     0     0     0 ± d_V ∓ d_W;   d_W  -d_W     0     0     0     0     0 ∓ d_W ± d_W ]

with

    s_1=4sin3π/22,   s_2=-4cosπ/11,   s_3=4cos2π/11,   s_4=-4sinπ/22,   s_5=-4sin5π/22.

They have additive central charges

    c( B)=2   (1st&2nd h),
    -2   (3rd&4th h).  (8)

There are

    2(quantum dimensions)×4(conformal dimensions)×2(categorical dimensions)=16

MFCs, among which those eight with the second quantum dimensions are unitary. We classify connected étale algebras in all 16 MFCs simultaneously.

An ansatz

    A≅1⊕ n_XX⊕ n_YY⊕ n_ZZ⊕ n_SS⊕ n_TT⊕ n_UU⊕ n_VV⊕ n_WW

with n_j∈ N has

    _ B(A)=1+n_X+2(n_Y+n_Z+n_S+n_T+n_U)+√(11)(n_V+n_W).

For this to obey (<ref>), the natural numbers can take only 72 values. The sets contain the one with all n_j's be zero. It is the trivial connected étale algebra giving B_A^0≃ B_A≃ B. Other 32 candidates without X all fail to be commutative because nontrivial simple object(s) entering them have nontrivial conformal dimensions. Thus, we are left with five nontrivial candidates with n_X=1,2,3,4,5. Since X has (d_X,h_X)=(1,0) (mod 1 for h_X), it has c_X,X≅ id_1 <cit.>, and can give commutative algebras. An obstruction is the Frobenius-Perron dimensions. The Frobenius-Perron dimensions 2,3,4,5,6 of the candidates demand ( B_A^0)=11,44/9,11/4,44/25,11/9, but there are no MFCs with such Frobenius-Perron dimensions for all of them but the first. Therefore, the only nontrivial candidate is

    A≅1⊕ X.

This is just a Z/2 Z algebra, and it is commutative. It further turns out separable. Let us check this point by identifying B_A.

Its Frobenius-Perron dimension =2 demands

    ( B_A^0)=11,  ( B_A)=22.

It turns out

    B_A^0≃_ Z/11 Z^1,   B_A≃TY( Z/11 Z).

This identification also matches central charges because

    c(_ Z/11 Z^1)=2   (h_X,h_Y,h_Z,h_Q,h_R,h_S,h_T,h_U,h_V,h_W)=(1/11,4/11,9/11,5/11,3/11,3/11,5/11,9/11,4/11,1/11),
    -2   (h_X,h_Y,h_Z,h_Q,h_R,h_S,h_T,h_U,h_V,h_W)=(10/11,7/11,2/11,7/11,8/11,8/11,6/11,2/11,7/11,10/11).  (8)

One of the easiest ways to see the identifications is to perform anyon condensation. Since we `identify' 1 and X, V and W are also identified. The other simple objects `split' into two each. Since V and W have different conformal dimensions, they are confined. Thus, the category of dyslectic right A-modules consists of 11 invertible simple objects, and it is identified as a Z/11 Z MFC. In the category B_A of right A-modules, V⊕ W has Frobenius-Perron dimension √(11). Thus, the category is identified as a Z/11 Z Tambara-Yamagami category TY( Z/11 Z).[More rigorously, we should search for NIM-reps. Indeed, we find a 12-dimensional NIM-rep

    n_1=1_12=n_X,   n_Y=[ 0 1 1 0 0 0 0 0 0 0 0 0; 1 0 0 0 0 0 0 0 0 1 0 0; 1 0 0 0 0 0 0 0 0 0 1 0; 0 0 0 0 0 0 0 1 0 1 0 0; 0 0 0 0 0 0 0 0 1 0 1 0; 0 0 0 0 0 0 1 1 0 0 0 0; 0 0 0 0 0 1 0 0 1 0 0 0; 0 0 0 1 0 1 0 0 0 0 0 0; 0 0 0 0 1 0 1 0 0 0 0 0; 0 1 0 1 0 0 0 0 0 0 0 0; 0 0 1 0 1 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 2 ],   n_Z=[ 0 0 0 1 1 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 1 0 0 1 0; 0 0 0 0 0 0 0 0 1 1 0 0; 1 0 0 0 0 0 1 0 0 0 0 0; 1 0 0 0 0 1 0 0 0 0 0 0; 0 0 0 0 1 0 0 0 0 1 0 0; 0 0 0 1 0 0 0 0 0 0 1 0; 0 1 0 0 0 0 0 0 1 0 0 0; 0 0 1 0 0 0 0 1 0 0 0 0; 0 0 1 0 0 1 0 0 0 0 0 0; 0 1 0 0 0 0 1 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 2 ],
       n_S=[ 0 0 0 0 0 1 1 0 0 0 0 0; 0 0 0 0 0 0 1 0 1 0 0 0; 0 0 0 0 0 1 0 1 0 0 0 0; 0 0 0 0 1 0 0 0 0 0 1 0; 0 0 0 1 0 0 0 0 0 1 0 0; 1 0 1 0 0 0 0 0 0 0 0 0; 1 1 0 0 0 0 0 0 0 0 0 0; 0 0 1 0 0 0 0 0 0 0 1 0; 0 1 0 0 0 0 0 0 0 1 0 0; 0 0 0 0 1 0 0 0 1 0 0 0; 0 0 0 1 0 0 0 1 0 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 2 ],   n_T=[ 0 0 0 0 0 0 0 1 1 0 0 0; 0 0 0 0 1 1 0 0 0 0 0 0; 0 0 0 1 0 0 1 0 0 0 0 0; 0 0 1 0 0 0 0 0 1 0 0 0; 0 1 0 0 0 0 0 1 0 0 0 0; 0 1 0 0 0 0 0 0 0 0 1 0; 0 0 1 0 0 0 0 0 0 1 0 0; 1 0 0 0 1 0 0 0 0 0 0 0; 1 0 0 1 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 1 0 0 0 1 0; 0 0 0 0 0 1 0 0 0 1 0 0; 0 0 0 0 0 0 0 0 0 0 0 2 ],
       n_U=[ 0 0 0 0 0 0 0 0 0 1 1 0; 0 0 1 1 0 0 0 0 0 0 0 0; 0 1 0 0 1 0 0 0 0 0 0 0; 0 1 0 0 0 1 0 0 0 0 0 0; 0 0 1 0 0 0 1 0 0 0 0 0; 0 0 0 1 0 0 0 0 1 0 0 0; 0 0 0 0 1 0 0 1 0 0 0 0; 0 0 0 0 0 0 1 0 0 1 0 0; 0 0 0 0 0 1 0 0 0 0 1 0; 1 0 0 0 0 0 0 1 0 0 0 0; 1 0 0 0 0 0 0 0 1 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 2 ],   n_V=[ 0 0 0 0 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 0 0 0 0 1; 1 1 1 1 1 1 1 1 1 1 1 0 ]=n_W.

The NIM-rep gives identifications

    m_1≅1⊕ X,   m_2≅   Y≅ m_3,   m_4≅ Z≅ m_5,
    
        m_6≅ S≅ m_7,   m_8≅ T≅ m_9,      m_10≅ U≅ m_11,   m_12≅ V⊕ W.

Since m_1,m_2,…,m_11 have quantum dimensions

    d_ B_A(m_j)=d_ B(m_j)/d_ B(A)=+1,

and conformal dimensions (1/11,9/11,3/11,5/11,4/11) or its conjugate, B_A^0 is identified as a Z/11 Z MFC. The non-invertible simple object m_12 has d_ B_A(m_12)=±√(11) and obeys monoidal products

    m_j⊗_A m_12≅ m_12≅ m_12⊗_Am_j (j=1,2,…,11),   m_12⊗_Am_12≅⊕_j=1^11m_j.

This shows B_A≃TY( Z/11 Z).] Since TY( Z/11 Z) is semisimple, A is separable and étale. Note that this example also has

    ( B_A)>( B).


We conclude

All the 16 MFCs B≃ so(11)_2's fail to be completely anisotropic.



  §.§.§ B≃ su(2)_8

The MFCs have nine simple objects {1,X,Y,Z,S,T,U,V,W} obeying monoidal products

Thus, they have

    _ B(1)=1=_ B(X),  _ B(Y)=√(5+√(5)/2)=   _ B(Z),  _ B(S)=3+√(5)/2=_ B(T),
    _ B(U)=√(5+2√(5))=
        _ B(V),     _ B(W)=1+√(5),

and

    ( B)=30+10√(5)≈52.4.

Their quantum dimensions d_j's are solutions of the same multiplication rules d_id_j=∑_k=1^9N_ij^kd_k. There are four (nonzero) solutions

    (d_X,d_Y,d_Z,d_S,d_T,d_U,d_V,d_W)   =(1,-√(5-√(5)/2),-√(5-√(5)/2),3-√(5)/2,3-√(5)/2,√(5-2√(5)),√(5-2√(5)),1-√(5)),
           (1,√(5-√(5)/2),√(5-√(5)/2),3-√(5)/2,3-√(5)/2,-√(5-2√(5)),-√(5-2√(5)),1-√(5)),
           (1,-√(5+√(5)/2),-√(5+√(5)/2),3+√(5)/2,3+√(5)/2,-√(5+2√(5)),-√(5+2√(5)),1+√(5)),
           (1,√(5+√(5)/2),√(5+√(5)/2),3+√(5)/2,3+√(5)/2,√(5+2√(5)),√(5+2√(5)),1+√(5)),

with categorical dimensions

    D^2( B)=30-10√(5)(≈7.6),  30+10√(5),

respectively for each pair. They have four conformal dimensions[Naively, one finds eight consistent conformal dimensions, but the other four are equivalent to one in the main text under permutations (YZ)(UV) of simple objects.]

    (h_X,h_Y,h_Z,h_S,h_T,h_U,h_V,h_W)   =(0,1/40,21/40,2/5,2/5,1/8,5/8,1/5),(0,9/40,29/40,3/5,3/5,1/8,5/8,4/5),
           (0,11/40,31/40,2/5,2/5,3/8,7/8,1/5),(0,19/40,39/40,3/5,3/5,3/8,7/8,4/5)  (1)

for the first two quantum dimensions and

    (h_X,h_Y,h_Z,h_S,h_T,h_U,h_V,h_W)   =(0,3/40,23/40,1/5,1/5,3/8,7/8,3/5),(0,7/40,27/40,4/5,4/5,7/8,3/8,2/5),
           (0,13/40,33/40,1/5,1/5,5/8,1/8,3/5),(0,17/40,37/40,4/5,4/5,1/8,5/8,2/5)  (1)

for the last two quantum dimensions. The S-matrices are given by

    S=[     1   d_X   d_Y   d_Z   d_S   d_T   d_U   d_V   d_W;   d_X   d_X  -d_Y  -d_Z   d_S   d_T  -d_U  -d_V   d_W;   d_Y  -d_Y ± d_U ∓ d_U  -d_U   d_U ± d_Y ∓ d_Y     0;   d_Z  -d_Z ∓ d_U ± d_U  -d_U   d_U ∓ d_Y ± d_Y     0;   d_S   d_S  -d_U  -d_U     1     1   d_Y   d_Y  -d_W;   d_T   d_T   d_U   d_U     1     1  -d_Y  -d_Y  -d_W;   d_U  -d_U ± d_Y ∓ d_Y   d_Y  -d_Y ± d_U ∓ d_U     0;   d_V  -d_V ∓ d_Y ± d_Y   d_Y  -d_Y ∓ d_U ± d_U     0;   d_W   d_W     0     0  -d_W  -d_W     0     0   d_W ].

They have additive central charges

    c( B)=4/5   (1st&3rd h),
    -4/5   (2nd&4th h),   (1st&2nd quantum dimensions)
    12/5   (1st&3rd h),
    -12/5   (2nd&4th h).   (3rd&4th quantum dimensions)  (8)

There are

    4(quantum dimensions)×4(conformal dimensions)×2(categorical dimensions)=32

MFCs, among which those eight with the fourth quantum dimensions are unitary. We classify connected étale algebras in all 32 MFCs simultaneously.

An ansatz

    A≅1⊕ n_XX⊕ n_YY⊕ n_ZZ⊕ n_SS⊕ n_TT⊕ n_UU⊕ n_VV⊕ n_WW

with n_j∈ N has

    _ B(A)=1+n_X+√(5+√(5)/2)(n_Y+n_Z)+3+√(5)/2(n_S+n_T)+√(5+2√(5))(n_U+n_V)+(1+√(5))n_W.

For this to obey (<ref>), the natural numbers can take only 85 values. The sets contain the one with all n_j's be zero. It is the trivial connected étale algebra A≅1 giving B_A^0≃ B_A≃ B. Other candidates without X contain simple object(s) with nontrivial conformal dimensions, and fail to be commutative. Thus, we are left with candidates 1⊕ n_XX with n_X=1,2,3,4,5,6. Since X has (d_X,h_X)=(1,0), it has trivial braiding c_X,X≅ id_1 <cit.>, and the candidates can be commutative. The obstruction is the Frobenius-Perron dimensions. The Frobenius-Perron dimensions 2,3,4,5,6,7 of the candidates demand ( B_A^0)=15+5√(5)/2,30+10√(5)/9,30+10√(5)/16,30+10√(5)/25,30+10√(5)/36,30+10√(5)/49, but there is no MFC with such Frobenius-Perron dimensions for n_X>1. Therefore, the only candidate is A≅1⊕ X. This is indeed a commutative Z/2 Z algebra, and it turns out separable. To see this, we identify B_A.

Its Frobenius-Perron dimension demands

    ( B_A^0)=15+5√(5)/2,  ( B_A)=15+5√(5).

From the Frobenius-Perron dimension, the MFC B_A^0 is identified as

    B_A^0≃⊠.

This can also be seen via anyon condensation. Condensation of A≅1⊕ X `identifies' a pair (1,X), and hence (Y,Z),(S,T),(U,V), and it `splits' W into two simple objects. Since Y and Z have different conformal dimensions, the resulting simple object in B_A is confined. Similarly for the simple object descending from U,V. Thus, the deconfined phase B_A^0 consists of four simple objects with quantum dimensions 1,3±√(5)/2,1±√(5)/2,1±√(5)/2. This is identified as ⊠ where two Fibonacci objects have the same quantum and conformal dimensions. The identification also matches central charges. To describe the matching more precisely, let us denote two Fibonacci objects x,y. The central charges are matched as

    c( B)=c( B_A^0)=±4/5   (d_W,h_W,d_x,d_y,h_x,h_y)=(1-√(5),±1/5,1-√(5)/2,1-√(5)/2,±1/5,±1/5),
    ±12/5   (d_W,h_W,d_x,d_y,h_x,h_y)=(1+√(5),±2/5,1+√(5)/2,1+√(5)/2,±2/5,±2/5),  (1 for h)

where signs are correlated. The category B_A of right A-modules have two more simple objects with quantum dimensions ±√(5±√(5)/2) and √(5±2√(5)). Such a rank six fusion category is not listed in the AnyonWiki, but we can figure out their properties.

More rigorously, we should find NIM-reps. Indeed, we find a six-dimensional solution

    n_1=1_6=n_X,      n_Y=[ 0 1 0 0 0 0; 1 0 1 0 0 0; 0 1 0 1 0 0; 0 0 1 0 1 1; 0 0 0 1 0 0; 0 0 0 1 0 0 ]=n_Z,   n_S=[ 0 0 1 0 0 0; 0 1 0 1 0 0; 1 0 1 0 1 1; 0 1 0 2 0 0; 0 0 1 0 0 1; 0 0 1 0 1 0 ]=n_T,
       n_U=[ 0 0 0 1 0 0; 0 0 1 0 1 1; 0 1 0 2 0 0; 1 0 2 0 1 1; 0 1 0 1 0 0; 0 1 0 1 0 0 ]=n_V,   n_W=[ 0 0 0 0 1 1; 0 0 0 2 0 0; 0 0 2 0 1 1; 0 2 0 2 0 0; 1 0 1 0 1 0; 1 0 1 0 0 1 ].

This gives a multiplication table

We obtain identifications

    m_1≅1⊕ X,   m_2≅ Y⊕ Z,   m_3≅ S⊕ T,   m_4≅ U⊕ V,   m_5≅ W≅ m_6.

They have

    d_ B_A(m_1)=1,   d_ B_A(m_2)=d_Y,   d_ B_A(m_3)=d_S,   d_ B_A(m_4)=d_U,   dd_ B_A(m_5)=d_W/2=d_ B_A(m_6).

They obey monoidal products

Note that the fusion ring has multiplicity two. Thus, it is natural that the ring is not listed in <cit.> and AnyonWiki. Luckily, however, we know this is a fusion category as follows. Let us choose the specific su(2)_8 MFC describing su(2)_8 WZW model. (The MFC is also denoted as C(A_1,8).) It is known <cit.> that C(A_1,8) has two connected étale algebras, 1,1⊕ X. The second thus gives a fusion category C(A_1,8)_A. The fusion category corresponds to the D_6 Dynkin diagram. The correspondence especially implies ( C(A_1,8)_A)=6. Therefore, we learn the fusion ring with multiplicity two gives rank six fusion category. Let us collectively denote the fusion categories with the fusion ring as C(D_6).

We now relax the assumption on conformal dimensions and consider all su(2)_8 MFCs. Our analysis above showed A≅1⊕ X is a connected commutative algebra giving a fusion category B_A≃ C(D_6). Since C(D_6) is semisimple, 1⊕ X is separable, hence étale.

We conclude

All the 32 MFCs B≃ su(2)_8's fail to be completely anisotropic.



  §.§.§ B≃ psu(2)_5⊠ psu(2)_5

The MFCs have nine simple objects {1,X,Y,Z,S,T,U,V,W} obeying monoidal products

(One can identify psu(2)_5={1,X,S},{1,Y,Z}, and T≅ X⊗ Y,U≅ Y⊗ S,V≅ X⊗ Z,W≅ Z⊗ S.) Thus, they have

    _ B(1)=1,  _ B(X)=sin2π/7/sinπ/7=_ B(Y),  _ B(Z)=sin3π/7/sinπ/7=_ B(S),
    _ B(T)=(sin2π/7/sinπ/7)^2,  _ B(U)=sin2π/7sin3π/7/sin^2π/7=
        _ B(V),  _ B(W)=(sin3π/7/sinπ/7)^2,

and

    ( B)=49/16sin^4π/7≈86.4.

Their quantum dimensions d_j's are solutions of the same multiplication rules d_id_j=∑_k=1^9N_ij^kd_k. There are nine solutions

    (d_X,d_Y,d_Z,d_S,d_T,d_U,d_V,d_W)
       =(sinπ/7/cosπ/14,sinπ/7/cosπ/14,-sin2π/7/cosπ/14,-sin2π/7/cosπ/14,(sinπ/7/cosπ/14)^2,-sinπ/7sin2π/7/cos^2π/14,-sinπ/7sin2π/7/cos^2π/14,(sin2π/7/cosπ/14)^2,),
           (-sin3π/7/cos3π/14,sinπ/7/cosπ/14,-sin2π/7/cosπ/14,sinπ/7/cos3π/14,-sinπ/7/cos3π/14,sin^2π/7/cosπ/14cos3π/14,1,-sinπ/7sin2π/7/cosπ/14cos3π/14),
           (sinπ/7/cosπ/14,-sin3π/7/cos3π/14,sinπ/7/cos3π/14,-sin2π/7/cosπ/14,-sinπ/7/cos3π/14,1,sin^2π/7/cosπ/14cos3π/14,-sinπ/7sin2π/7/cosπ/14cos3π/14),
           (-sin3π/7/cos3π/14,-sin3π/7/cos3π/14,sinπ/7/cos3π/14,sinπ/7/cos3π/14,(sin3π/7/cos3π/14)^2,-sinπ/7sin3π/7/cos^23π/14,-sinπ/7sin3π/7/cos^23π/14,(sinπ/7/cos3π/14)^2),
           (sinπ/7/cosπ/14,sin2π/7/sinπ/7,sin3π/7/sinπ/7,-sin2π/7/cosπ/14,sinπ/7sin2π/7/cosπ/14sinπ/7,-sin^22π/7/cosπ/14sinπ/7,1,-sin2π/7sin3π/7/cosπ/14sinπ/7),
           (sin2π/7/sinπ/7,sinπ/7/cosπ/14,-sin2π/7/cosπ/14,sin3π/7/sinπ/7,sin2π/7sinπ/7/sinπ/7cosπ/14,1,-sin^22π/7/sinπ/7cosπ/14,-sin2π/7sin3π/7/sinπ/7cosπ/14),
           (-sin3π/7/cos3π/14,sin2π/7/sinπ/7,sin3π/7/sinπ/7,sinπ/7/cos3π/14,-sin3π/7sin2π/7/cos3π/14sinπ/7,1,-sin^23π/7/cos3π/14sinπ/7,sin3π/7/cos3π/14),
           (sin2π/7/sinπ/7,-sin3π/7/cos3π/14,sinπ/7/cos3π/14,sin3π/7/sinπ/7,-sin3π/7sin2π/7/cos3π/14sinπ/7,-sin^23π/7/cos3π/14sinπ/7,1,sin3π/7/cos3π/14),
           (sin2π/7/sinπ/7,sin2π/7/sinπ/7,sin3π/7/sinπ/7,sin3π/7/sinπ/7,(sin2π/7/sinπ/7)^2,sin2π/7sin3π/7/sin^2π/7,sin2π/7sin3π/7/sin^2π/7,(sin3π/7/sinπ/7)^2)

with categorical dimensions

    D^2( B)=49/16cos^2π/14(≈3.4)   (1st),
    49/16cos^2π/14cos^23π/14(≈5.3)   (2nd&3rd),
    49/16cos^43π/14(≈8.2)   (4th),
    49/16cos^2π/14sin^2π/7(≈17.1)   (5th&6th),
    49/16cos^23π/14sin^2π/7(≈26.6)   (7th&8th),
    49/16sin^4π/7   (9th),

respectively. Since the two quantum dimensions giving the same categorical dimension are equivalent under permutations (XY)(ZS)(UV) of simple objects, we choose the first quantum dimensions from each pair and fix the order of two factors to remove redundancies. They have conformal dimensions

    (h_X,h_Y,h_Z,h_S,h_T,h_U,h_V,h_W)
       =(4/7,3/7,1/7,6/7,0,2/7,5/7,0),(4/7,4/7,6/7,6/7,1/7,3/7,3/7,5/7),(3/7,3/7,1/7,1/7,6/7,4/7,4/7,2/7)   (1st),
    (5/7,3/7,1/7,4/7,1/7,0,6/7,5/7),(5/7,4/7,6/7,4/7,2/7,1/7,4/7,3/7),(2/7,3/7,1/7,3/7,5/7,6/7,3/7,4/7),(2/7,4/7,6/7,3/7,6/7,0,1/7,2/7)   (2nd),
    (2/7,5/7,4/7,3/7,0,1/7,6/7,0),(5/7,5/7,4/7,4/7,3/7,2/7,2/7,1/7),(2/7,2/7,3/7,3/7,4/7,5/7,5/7,6/7)   (4th),
    (3/7,6/7,2/7,1/7,2/7,0,5/7,3/7),(4/7,6/7,2/7,6/7,3/7,5/7,6/7,1/7),(3/7,1/7,5/7,1/7,4/7,2/7,1/7,6/7),(4/7,1/7,5/7,6/7,5/7,0,2/7,4/7)   (5th),
    (2/7,6/7,2/7,3/7,1/7,2/7,4/7,5/7),(2/7,1/7,5/7,3/7,3/7,4/7,0,1/7),(5/7,6/7,2/7,4/7,4/7,3/7,0,6/7),(5/7,1/7,5/7,4/7,6/7,5/7,3/7,2/7)   (7th),
    (6/7,1/7,5/7,2/7,0,3/7,4/7,0),(1/7,1/7,5/7,5/7,2/7,6/7,6/7,3/7),(6/7,6/7,2/7,2/7,5/7,1/7,1/7,4/7)   (9th).  (1)

The S-matrices are given by

    S=[       1     d_X     d_Y     d_Z     d_S  d_Xd_Y  d_Yd_S  d_Xd_Z  d_Zd_S;     d_X    -d_S  d_Xd_Y  d_Xd_Z       1 -d_Yd_S     d_Y -d_Zd_S     d_Z;     d_Y  d_Xd_Y    -d_Z       1  d_Yd_S -d_Xd_Z -d_Zd_S     d_X     d_S;     d_Z  d_Xd_Z       1    -d_Y  d_Zd_S     d_X     d_S -d_Xd_Y -d_Yd_S;     d_S       1  d_Yd_S  d_Yd_S    -d_X     d_Y -d_Xd_Y     d_Z -d_Xd_Z;  d_Xd_Y -d_Yd_S -d_Xd_Z     d_X     d_Y  d_Zd_S    -d_Z    -d_S       1;  d_Yd_S     d_Y -d_Zd_S     d_S -d_Xd_Y    -d_Z  d_Xd_Z       1    -d_X;  d_Xd_Z -d_Zd_S     d_X -d_Xd_Y     d_Z    -d_S       1  d_Yd_S    -d_Y;  d_Zd_S     d_Z     d_S -d_Yd_S -d_Xd_Z       1    -d_X    -d_Y  d_Xd_Y ].

They have additive central charges

    c( B)=c(psu(2)_5)+c(psu(2)_5)  (8)

where

    c(psu(2)_5)=4/7   (h_X,Y,h_S,Z)=(3/7,1/7),
    -4/7   (h_X,Y,h_S,Z)=(4/7,6/7),   (d_X,Y,d_S,Z)=(sinπ/7/cosπ/14,-sin2π/7/cosπ/14),
    12/7   (h_X,Y,h_S,Z)=(2/7,3/7),
    -12/7   (h_X,Y,h_S,Z)=(5/7,4/7),   (d_X,Y,d_S,Z)=(-sin3π/7/cos3π/14,sinπ/7/cos3π/14),
    -8/7   (h_X,Y,h_S,Z)=(1/7,5/7),
    8/7   (h_X,Y,h_S,Z)=(6/7,2/7),   (d_X,Y,d_S,Z)=(sin2π/7/sinπ/7,sin3π/7/sinπ/7).  (8)

Taking the two signs of categorical dimensions, each class of quantum dimensions has 6,8,6,8,8,6 MFCs, respectively. Therefore, there are

    6+8+6+8+8+6=42

MFCs, among which those six with the ninth quantum dimensions are unitary. We classify connected étale algebras in all 42 MFCs simultaneously.

An ansatz

    A≅1⊕ n_XX⊕ n_YY⊕ n_ZZ⊕ n_SS⊕ n_TT⊕ n_UU⊕ n_VV⊕ n_WW

with n_j∈ N has

    _ B(A)=1+sin2π/7/sinπ/7(n_X+n_Y)+sin3π/7/sinπ/7(n_Z+n_S)+(sin2π/7/sinπ/7)^2n_T+sin2π/7sin3π/7/sin^2π/7(n_U+n_V)+(sin3π/7/sinπ/7)^2n_W.

For this to obey (<ref>), the natural numbers can take only 110 values. The sets contain the one with all n_j's be zero. It is the trivial connected étale algebra A≅1 giving B_A^0≃ B_A≃ B_A. Other candidates with X,Y,Z,S contain nontrivial simple object(s) with nontrivial conformal dimensions, and they all fail to be commutative. We are left with just T,U,V,W. Setting n_X,n_Y,n_Z,n_S to zero, apart from the trivial one, we find 11 sets

    (n_T,n_U,n_V,n_W)   =(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1),
           (2,0,0,0),(1,1,0,0),(1,0,1,0),(1,0,0,1),
           (0,2,0,0),(0,1,1,0),(0,0,2,0).

All but the eighth are ruled out because they demand

    ( B_A^0)≈4.8, 3.4, 3.4, 2.3, 1.5, 1.3, 1.3, 1, 1.04, 1.04, 1.04,

respectively. There is no MFC with these Frobenius-Perron dimensions except ( B_A^0)=1. Thus, we are only left with

    A≅1⊕ T⊕ W.

This candidate can be commutative only for the first, fourth, and ninth quantum dimensions and the first conformal dimensions in each quantum dimension. Namely, when the two factors have the same quantum dimensions and opposite conformal dimensions. In other words, when the ambient MFC is a Drinfeld center, B≃ Z(psu(2)_5). In this case, from the lemma 3, we know there exists a Lagrangian algebra. From our analysis above, the only candidate is A≅1⊕ T⊕ W. Therefore, we learn it is connected étale without any computation. In order to see the consistency, let us identify the category of right A-modules.

Its Frobenius-Perron dimension _ B(A)=7/4sin^2π/7 requires

    ( B_A^0)=1,  ( B_A)=7/4sin^2π/7.

The only possible category of dyslectic right A-modules is the trivial MFC

    B_A^0≃.

This identification also matches central charges because the Drinfeld center has additive central charge zero.

So as to identify B_A, we follow our routine method. Calculating b_j⊗ A, we find simple objects:

    1⊕ T⊕ W,   X⊕ Y⊕ U⊕ V⊕ W,   Z⊕ S⊕ T⊕ U⊕ V⊕ W.

They have Frobenius-Perron dimensions

    1,  sin2π/7/sinπ/7,  sin3π/7/sinπ/7,

respectively, and their contributions to ( B_A) match as it should be. This suggests B_A have rank three. Indeed, we find a three-dimensional NIM-rep

    n_1=1_3,   n_X=[ 0 1 0; 1 0 1; 0 1 1 ]=n_Y,   n_Z=[ 0 0 1; 0 1 1; 1 1 1 ]=n_S,
    
        n_T=[ 1 0 1; 0 2 1; 1 1 2 ],   n_U=[ 0 1 1; 1 1 2; 1 2 2 ]=n_V,   n_W=[ 1 1 1; 1 2 2; 2 2 3 ].

The solution gives identifications

    m_1≅1⊕ T⊕ W,   m_2≅ X⊕ Y⊕ U⊕ V⊕ W,   m_3≅ Z⊕ S⊕ T⊕ U⊕ V⊕ W.

In B_A, they have quantum dimensions

    (d_ B_A(m_1),d_ B_A(m_2),d_ B_A(m_3))=(1,sinπ/7/cosπ/14,-sin2π/7/cosπ/14)   (1st quantum dimension),
    (1,-sin3π/7/cos3π/14,sinπ/7/cos3π/14)   (4th quantum dimension),
    (1,sin2π/7/sinπ/7,sin3π/7/sinπ/7)   (9th quantum dimension)..

These are nothing but the quantum dimensions of psu(2)_5. Furthermore, working out the monoidal products, we find

This also shows B_A≃ psu(2)_5. Therefore, we found a connected étale algebra

    A≅1⊕ T⊕ W  (1st or 4th or 9th quantum dimensions with 1st h's).


We conclude

Namely, those six in (<ref>) fail to be completely anisotropic, while the other 36 MFCs B≃ psu(2)_5⊠ psu(2)_5's are completely anisotropic.



  §.§.§ B≃ psu(2)_17

The MFCs have nine simple objects {1,X,Y,Z,S,T,U,V,W} obeying monoidal products

Thus, they have

    _ B(1)=1,  _ B(X)=sin2π/19/sinπ/19,  _ B(Y)=   sin3π/19/sinπ/19,  _ B(Z)=sin4π/19/sinπ/19,  _ B(S)=sin5π/19/sinπ/19,
    _ B(T)=sin6π/19/sinπ/19,  _ B(U)=sin7π/19/sinπ/19,     _ B(V)=sin8π/19/sinπ/19,  _ B(W)=sin9π/19/sinπ/19,

and

    ( B)=19/4sin^2π/19≈175.3.

Their quantum dimensions d_j's are solutions of the same multiplication rules d_id_j=∑_k=1^9N_ij^kd_k. There are nine solutions

    (d_X,d_Y,d_Z,d_S,d_T,d_U,d_V,d_W)
       =(sinπ/19/cosπ/38,-sin8π/19/cosπ/38,-sin2π/19/cosπ/38,sin7π/19/cosπ/38,sin3π/19/cosπ/38,-sin6π/19/cosπ/38,-sin4π/19/cosπ/38,sin5π/19/cosπ/38),
           (-sin3π/19/cos3π/38,-sin5π/19/cos3π/38,sin6π/19/cos3π/38,sin2π/19/cos3π/38,-sin9π/19/cos3π/38,sinπ/19/cos3π/38,sin7π/19/cos3π/38,-sin4π/19/cos3π/38),
           (sin5π/19/cos5π/38,-sin2π/19/cos5π/38,-sin9π/19/cos5π/38,-sin3π/19/cos5π/38,sin4π/19/cos5π/38,sin8π/19/cos5π/38,sinπ/19/cos5π/38,-sin6π/19/cos5π/38),
           (-sin7π/19/cos7π/38,sinπ/19/cos7π/38,sin5π/19/cos7π/38,-sin8π/19/cos7π/38,sin2π/19/cos7π/38,sin4π/19/cos7π/38,-sin9π/19/cos7π/38,sin3π/19/cos7π/38),
           (sin9π/19/cos9π/38,sin4π/19/cos9π/38,-sinπ/19/cos9π/38,-sin6π/19/cos9π/38,-sin8π/19/cos9π/38,-sin3π/19/cos9π/38,sin2π/19/cos9π/38,sin7π/19/cos9π/38),
           (-sin8π/19/cos11π/38,sin7π/19/cos11π/38,-sin3π/19/cos11π/38,-sinπ/19/cos11π/38,sin5π/19/cos11π/38,-sin9π/19/cos11π/38,sin6π/19/cos11π/38,-sin2π/19/cos11π/38),
           (sin6π/19/cos13π/38,sin9π/19/cos13π/38,sin7π/19/cos13π/38,sin4π/19/cos13π/38,sinπ/19/cos13π/38,-sin2π/19/cos13π/38,-sin5π/19/cos13π/38,-sin8π/19/cos13π/38),
           (-sin4π/19/cos15π/38,sin6π/19/cos15π/38,-sin8π/19/cos15π/38,sin9π/19/cos15π/38,-sin7π/19/cos15π/38,sin5π/19/cos15π/38,-sin3π/19/cos15π/38,sinπ/19/cos15π/38),
           (sin2π/19/sinπ/19,sin3π/19/sinπ/19,sin4π/19/sinπ/19,sin5π/19/sinπ/19,sin6π/19/sinπ/19,sin7π/19/sinπ/19,sin8π/19/sinπ/19,sin9π/19/sinπ/19)

with categorical dimensions

    D^2( B)=19/4cos^2π/38(≈4.8)   (1st),
    19/4cos^23π/38(≈5.1)   (2nd),
    19/4cos^25π/38(≈5.7)   (3rd),
    19/4cos^27π/38(≈6.8)   (4th),
    19/4cos^29π/38(≈8.8)   (5th),
    19/4cos^211π/38(≈12.6)   (6th),
    19/4cos^213π/38(≈21.0)   (7th),
    19/4cos^215π/38(≈45.1)   (8th),
    19/4sin^2π/19   (9th),

respectively. They have two conformal dimensions each:

    (h_X,h_Y,h_Z,h_S,h_T,h_U,h_V,h_W)=(2/19,18/19,10/19,16/19,17/19,13/19,4/19,9/19),(17/19,1/19,9/19,3/19,2/19,6/19,15/19,10/19)   (1st),
    (6/19,16/19,11/19,10/19,13/19,1/19,12/19,8/19),(13/19,3/19,8/19,9/19,6/19,18/19,7/19,11/19)   (2nd),
    (9/19,5/19,7/19,15/19,10/19,11/19,18/19,12/19),(10/19,14/19,12/19,4/19,9/19,8/19,1/19,7/19)   (3rd),
    (5/19,7/19,6/19,2/19,14/19,4/19,10/19,13/19),(14/19,12/19,13/19,17/19,5/19,15/19,9/19,6/19)   (4th),
    (1/19,9/19,5/19,8/19,18/19,16/19,2/19,14/19),(18/19,10/19,14/19,11/19,1/19,3/19,17/19,5/19)   (5th),
    (3/19,8/19,15/19,5/19,16/19,10/19,6/19,4/19),(16/19,11/19,4/19,14/19,3/19,9/19,13/19,15/19)   (6th),
    (7/19,6/19,16/19,18/19,12/19,17/19,14/19,3/19),(12/19,13/19,3/19,1/19,7/19,2/19,5/19,16/19)   (7th),
    (8/19,15/19,2/19,7/19,11/19,14/19,16/19,17/19),(11/19,4/19,17/19,12/19,8/19,5/19,3/19,2/19)   (8th),
    (4/19,17/19,1/19,13/19,15/19,7/19,8/19,18/19),(15/19,2/19,18/19,6/19,4/19,12/19,11/19,1/19)   (9th).  (1)

The S-matrices are given by

    S=[    1  d_X  d_Y  d_Z  d_S  d_T  d_U  d_V  d_W;  d_X -d_Z  d_T -d_V  d_W -d_U  d_S -d_Y    1;  d_Y  d_T  d_W  d_U  d_Z    1 -d_X -d_S -d_V;  d_Z -d_V  d_U -d_Y   -1  d_S -d_W  d_T -d_X;  d_S  d_W  d_Z   -1 -d_T -d_V -d_Y  d_X  d_U;  d_T -d_U    1  d_S -d_V  d_X  d_Z -d_W  d_Y;  d_U  d_S -d_X -d_W -d_Y  d_Z  d_V    1 -d_T;  d_V -d_Y -d_S  d_T  d_X -d_W    1  d_U -d_Z;  d_W    1 -d_V -d_X  d_U  d_Y -d_T -d_Z  d_S ].

There are

    9(quantum dimensions)×2(conformal dimensions)×2(categorical dimensions)=36

MFCs, among which those four with the ninth quantum dimensions are unitary. We classify connected étale algebras in all 36 MFCs simultaneously.

An ansatz

    A≅1⊕ n_XX⊕ n_YY⊕ n_ZZ⊕ n_SS⊕ n_TT⊕ n_UU⊕ n_VV⊕ n_WW

with n_j∈ N has

    _ B(A)=1+sin2π/19/sinπ/19n_X+sin3π/19/sinπ/19n_Y+sin4π/19/sinπ/19n_Z+sin5π/19/sinπ/19n_S+sin6π/19/sinπ/19n_T+sin7π/19/sinπ/19n_U+sin8π/19/sinπ/19n_V+sin9π/19/sinπ/19n_W.

For this to obey (<ref>), the natural numbers can take only 107 values. The sets contain the one with all n_j's be zero. It is the trivial connected étale algebra A≅1 giving B_A^0≃ B_A≃ B. The other 106 candidates contain nontrivial simple object(s) with nontrivial conformal dimensions, and they all fail to be commutative.

We conclude

All the 36 MFCs B≃ psu(2)_17's are completely anisotropic.



§ PHYSICAL APPLICATIONS

Our classification results have physical implications in constraining renormalization group (RG) flows. Concretely, in massive RG flows, they constrain ground state degeneracies (GSDs) and prove spontaneous symmetry breaking (SSB). In this section, we discuss the physical applications.



 §.§ Theorems

Let C be a fusion category. Two-dimensional gapped phases with C symmetry stand in bijection with C-module categories <cit.>

    {2d  C-symmetric gapped phases}≅{ C-module categories  M}.

The correspondence in particular implies GSD of a C-symmetric gapped phase in the LHS is equal to ( M) of a C-module category in the RHS:

    GSD=( M).

This leads to constraints on GSDs:

Theorem. Let B be a multiplicity-free modular fusion category up to rank nine and A∈ B a connected étale algebra. Two-dimensional B-symmetric gapped phases described by indecomposable B_A's have

    GSD∈{7}   ( B≃ su(7)_1),
    {7}   ( B≃ su(2)_6),
    {7,8}   ( B≃ so(7)_2),
    {7}   ( B≃ psu(2)_13)

for rank seven,

    GSD∈{4,8}   ( B≃_ Z/2 Z× Z/2 Z× Z/2 Z^α≃_ Z/2 Z^-1⊠ (16 with (<ref>))),
    {8}   ( B≃_ Z/2 Z× Z/2 Z× Z/2 Z^α≃_ Z/2 Z^-1⊠ (the other 24)),
    {8}   ( B≃_ Z/2 Z× Z/4 Z^α),
    {4,8}   ( B≃ su(8)_1),
    {4,8}   ( B≃⊠_ Z/2 Z^-1⊠_ Z/2 Z^-1 (16 with (<ref>))),
    {8}   ( B≃⊠_ Z/2 Z^-1⊠_ Z/2 Z^-1 (the other 64)),
    {4,8}   ( B≃⊠ (16 with (<ref>))),
    {8}   ( B≃⊠ (the other 24)),
    {8}   ( B≃⊠_ Z/4 Z^α),
    {4,8}   ( B≃_ Z/2 Z^-1⊠⊠ (16 with (<ref>))),
    {8}   ( B≃_ Z/2 Z^-1⊠⊠ (the other 64)),
    {3,6,8,10}   ( B≃ so(9)_2 (four with (<ref>))),
    {6,8,10}   ( B≃ so(9)_2 (the other 12)),
    {3,6,8,10}   ( B≃Rep(D(D_3)) (two with (<ref>,<ref>))),
    {6,8,10}   ( B≃Rep(D(D_3)) (the other six with 1st&2nd h)),
    {8,10}   ( B≃Rep(D(D_3)) (eight with 3rd&4th h)),
    {8}   ( B≃ su(2)_7),
    {4,8}   ( B≃⊠⊠ (16 with (<ref>,<ref>))),
    {8}   ( B≃⊠⊠ (the other 24)),
    {8}   ( B≃⊠ psu(2)_7),
    {8}   ( B≃ psu(2)_15)

for rank eight, and

    GSD∈{3,9}   ( B≃ su(9)_1),
    {3,9}   ( B≃_ Z/3 Z× Z/3 Z^α (two with 1st h)),
    {9}   ( B≃_ Z/3 Z× Z/3 Z^α (two with 2nd h)),
    {9}   ( B≃_ Z/3 Z^1⊠),
    {3,6,9}   ( B≃⊠ (eight with (<ref>))),
    {6,9}   ( B≃⊠ (the other 136)),
    {9}   ( B≃_ Z/3 Z^1⊠ psu(2)_5),
    {9}   ( B≃⊠ psu(2)_5),
    {9,12}   ( B≃ so(11)_2),
    {6,9}   ( B≃ su(2)_8),
    {3,9}   ( B≃ psu(2)_5⊠ psu(2)_5 (six with (<ref>))),
    {9}   ( B≃ psu(2)_5⊠ psu(2)_5 (the other 36)),
    {9}   ( B≃ psu(2)_17)

for rank nine.

As demonstrated <cit.> for general MFC (without assumption on multiplicity), we see all of them are spontaneously broken. Here, we have the

Definition. <cit.> Let C be a fusion category and M a (left) C-module category describing a C-symmetric gapped phase. A symmetry c∈ C is called spontaneously broken if ∃ m∈ M such that c▹ m≇m. We also say C is spontaneously broken if there exists a spontaneously broken object c∈ C. A categorical symmetry C is preserved (i.e., not spontaneously broken) if all objects act trivially.

With the definition, one can show a

Lemma 3. <cit.> Let C be a fusion category and M be an indecomposable (left) C-module category. Then, ( M)>1 implies SSB of C (i.e., C is spontaneously broken.)

Therefore, we obtain a

Corollary. Let B be a nontrivial (i.e., ( B)>1) multiplicity-free modular fusion category up to rank nine and A∈ B a connected étale algebra. In two-dimensional B-symmetric gapped phases described by indecomposable B_A's, B symmetries are spontaneously broken.

Remark. As noted in <cit.>, commutativity of an algebra seems too strong; numerical computation suggests an existence of B-symmetric gapped phase described by B_A with non-commutative connected separable algebra.



 §.§ Examples

In this section, we discuss concrete examples and predict GSDs. Since unitary cases are relatively well-understood, we study less-understood non-unitary examples.

Pick a non-unitary minimal model[We basically follow the notations of <cit.>.] M(p,2p+1) with p≥2 as an ultraviolet (UV) theory. It was proved <cit.> that its relevant ϕ_5,1-deformation preserves rank (p-1) modular fusion subcategory formed by symmetry operators { L_1,1, L_1,2,…, L_1,p-1}. If the relevant deformation triggers massless renormalization group (RG) flow, it is known <cit.> that the infrared (IR) theory is another non-unitary minimal model M(p,2p-1). Its further relevant ϕ_1,2-deformation also preserves <cit.> rank (p-1) modular fusion subcategory { L_1,1, L_3,1,…, L_2p-3,1}, and IR theory can be another non-unitary minimal model M(p-1,2p-1). However, massless RG flows typically require fine-tuning. Therefore, generic relevant deformations are expected to trigger massive RG flows. The IR theories are B-symmetric gapped phases with modular B, and we can apply our classification results. Below, we study massive RG flows triggered by relevant deformations of the non-unitary minimal models.



  
M(8,15)+ϕ_1,2. The relevant deformation preserves rank seven MFC B with simple objects { L_1,1, L_3,1,…, L_13,1}. They form B≃ psu(2)_13 with identifications

    X≅ L_13,1,   Y≅ L_3,1,   Z≅ L_11,1,   U≅ L_5,1,   V≅ L_9,1,   W≅ L_7,1.

They have the first (non-unitary) quantum dimensions. Their conformal dimensions

    (h_13,1,h_3,1,h_11,1,h_5,1,h_9,1,h_7,1)=(82/5,1/15,11,6/5,20/3,17/5)

match our first conformal dimensions mod 1. Having specified the symmetry, we immediately learn from our results that the massive RG flow described by B_A has GSD=7 and B symmetry is spontaneously broken.



  
M(8,17)+ϕ_5,1. The relevant deformation preserves rank seven MFC B with simple objects { L_1,1, L_1,2,…, L_1,7}. They form B≃ su(2)_6 with identifications

    X≅ L_1,7,   Y≅ L_1,2,   Z≅ L_1,6,   U≅ L_1,5,   V≅ L_1,3,   W≅ L_1,4.

They have the third (non-unitary) quantum dimensions. Their conformal dimensions

    (h_1,7,h_1,2,h_1,6,h_1,5,h_1,3,h_1,4)=(45/2,35/32,515/32,43/4,13/4,207/32)

match our first conformal dimensions mod 1. With this knowledge on symmetry, our classification result immediately implies that the massive RG flow described by B_A has GSD=7 and B symmetry is spontaneously broken.



  
M(9,17)+ϕ_1,2. The relevant deformation preserves rank eight MFC B with simple objects { L_1,1, L_3,1,…, L_15,1}. They form B≃ psu(2)_15 with identifications

    X≅ L_15,1,   Y≅ L_3,1,   Z≅ L_13,1,   T≅ L_5,1,   U≅ L_11,1,   V≅ L_7,1,   W≅ L_9,1.

They have the first (non-unitary) quantum dimensions. Their conformal dimensions

    (h_15,1,h_3,1,h_13,1,h_5,1,h_11,1,h_7,1,h_9,1)=(385/17,1/17,276/17,20/17,185/17,57/17,112/17)

match our second conformal dimensions mod 1. Given the symmetry, we immediately learn the massive RG flow described by B_A has GSD=8 and B symmetry is spontaneously broken.



  
M(9,19)+ϕ_5,1. The relevant deformation preserves rank eight MFC B with simple objects { L_1,1, L_1,2,…, L_1,8}. They form B≃ su(2)_7 with identifications

    X≅ L_1,8,   Y≅ L_1,7,   Z≅ L_1,2,   T≅ L_1,3,   U≅ L_1,6,   V≅ L_1,5,   W≅ L_1,4.

They have the fifth (non-unitary) quantum dimensions. Their conformal dimensions

    (h_1,8,h_1,7,h_1,2,h_1,3,h_1,6,h_1,5,h_1,4)=(119/4,67/3,13/12,29/9,575/36,32/3,77/12)

match our third conformal dimensions mod 1. Having specified the symmetry, we find the massive RG flow described by B_A has GSD=8 and B symmetry is spontaneously broken.



  
M(10,19)+ϕ_1,2. The relevant deformation preserves rank nine MFC B with simple objects { L_1,1, L_3,1,…, L_17,1}. They form B≃ psu(2)_17 with identifications

    X≅ L_17,1,   Y≅ L_3,1,   Z≅ L_15,1,   S≅ L_5,1,   T≅ L_13,1,   U≅ L_7,1,   V≅ L_11,1,   W≅ L_9,1.

They have the first (non-unitary) quantum dimensions. Their conformal dimensions

    (h_17,1,h_3,1,h_15,1,h_5,1,h_13,1,h_7,1,h_11,1,h_9,1)=(568/19,1/19,427/19,22/19,306/19,63/19,205/19,124/19)

match our second conformal dimensions mod 1. With the symmetry, we immediately learn the massive RG flow described by B_A has GSD=9 and B symmetry is spontaneously broken.



  
M(10,21)+ϕ_5,1. The relevant deformation preserves rank nine MFC B with simple objects { L_1,1, L_1,2,…, L_1,9}. They form B≃ su(2)_8 with identifications

    X≅ L_1,9,   Y≅ L_1,2,   Z≅ L_1,8,   S≅ L_1,7,   T≅ L_1,3,   U≅ L_1,4,   V≅ L_1,6,   W≅ L_1,5.

They have the third (non-unitary) quantum dimensions. Their conformal dimensions

    (h_1,9,h_1,2,h_1,8,h_1,7,h_1,3,h_1,4,h_1,6,h_1,5)=(38,43/40,1183/40,111/5,16/5,51/8,127/8,53/5)

match our first conformal dimensions mod 1. With the symmetry, we find the massive RG flow described by B_A has GSD∈{6,9} and B symmetry is spontaneously broken.



§ ACKNOWLEDGMENT

We thank Gert Vercleyen for correspondences on typos in AnyonWiki. We also thank Terry Gannon for teaching us the classification results in C(B_4,2), which made us realize a mistake in the previous method.






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