Working with fusion data

We now construct a category from its fusion rules and associators, read $F$- and $R$-symbols as coefficients of morphisms, and extract them from a concrete model. All matrices use the conventions fixed in the preceding chapter.

Building Ising explicitly

The Ising fusion rules, with simples $(\mathbb 1,\chi,X)$ in this order, are

\[\label{eq:ising-fusion-ring} \chi^2=\mathbb 1,\qquad \chi X=X\chi=X,\qquad X^2=\mathbb 1+\chi.\]

Over $K=\mathbb Q(\sqrt 2)$, choose $s\in K$ with $s^2=2$. The nonidentity associator blocks are $A^{\chi X\chi}_X=[-1]$, $A^{X\chi X}_{\chi}=[-1]$, and

\[\label{eq:ising-associator} A^{XXX}_X=\frac1s \begin{pmatrix}1&1\\1&-1\end{pmatrix}.\]

The rows and columns of the last block correspond to the intermediate simples $\mathbb 1,\chi$. These data give the constructor ising_category(K,s); see Tambara–Yamagami and Ising for the general formulas.

To enter the data, initialize the fusion array, specify the unit at index 1, and replace the three nonidentity associator blocks. The constructor six_j_category initially supplies identity associator blocks and all-one pivotal components. These are initial values rather than certified structures, so the example checks the completed associator and pivotal data explicitly.

using TensorCategories, Oscar
K, s = quadratic_field(2)
N = zeros(Int,3,3,3)
N[1,1,1] = N[1,2,2] = N[2,1,2] = 1
N[1,3,3] = N[3,1,3] = N[2,3,3] = N[3,2,3] = 1
N[2,2,1] = N[3,3,1] = N[3,3,2] = 1
C = six_j_category(K, N, ["1", "chi", "X"])
set_one!(C, 1; check=true)
set_associator!(C,2,3,2,3, matrix(K,1,1,[-1]))
set_associator!(C,3,2,3,2, matrix(K,1,1,[-1]))
set_associator!(C,3,3,3,3, inv(s)*matrix(K,[1 1; 1 -1]))
@assert pentagon_axiom(C)
@assert is_pivotal(C; check=true) && is_spherical(C; check=true)
D = ising_category(K,s)
@assert C.ass == D.ass
C[3] ⊗ C[3]

1 ⊕ chi

For general fusion rules the initialized identity blocks need not satisfy the pentagon, and the all-one components need not define a pivotal structure. Calling set_tensor_product! again resets the associator blocks to identities.

Either root of $s^2=2$ gives monoidal data over $K$. No braiding has been specified in this construction.

Reading an associator matrix

Let $t$ be the nonunit simple of the Fibonacci category. Since $t\otimes t\cong\mathbb 1\oplus t$, each triple-product projection space with target $t$ has two basis vectors, ordered by the intermediate simples $\mathbb 1,t$. We construct these projections explicitly and apply the defining equation

\[\label{eq:associator-row-coordinate} R_v\circ\alpha_{t,t,t}=\sum_u A_{u,v}L_u.\]

using TensorCategories, Oscar
C = fibonacci_category()
t = C[2]
S = simples(C)
L = [q ∘ (p ⊗ id(t)) for e in S
     for p in basis(Hom(t⊗t,e)) for q in basis(Hom(e⊗t,t))]
R = [q ∘ (id(t) ⊗ p) for f in S
     for p in basis(Hom(t⊗t,f)) for q in basis(Hom(t⊗f,t))]
A = C.ass[2,2,2,2]
@assert A != transpose(A)
for v in eachindex(R)
    @assert R[v] ∘ associator(t,t,t) ==
            sum(A[u,v]*L[u] for u in eachindex(L))
end
A

[-ϕ   -ϕ]
[ 1    ϕ]

Fusion multiplicities and braiding

The rank-four $\mathrm{SU}(3)_3$ subcategory has simples $(\mathbb 1,8,10,\overline{10})$. The rule $8\otimes8\cong\mathbb 1\oplus2\cdot8\oplus10\oplus\overline{10}$ gives two basis vectors in $\operatorname{Hom}(8\otimes8,8)$. For the triple product with target $8$, the intermediate simple $8$ contributes four paths, one for each pair of binary basis indices. Together with the paths through $\mathbb 1,10,\overline{10}$, this gives a seven-dimensional block.

In the code below, x = C[2] represents the simple $8$. We form the projection bases and read both the associator and the braiding in these coordinates:

using TensorCategories, Oscar
C = TensorCategories.su_3_3_subcategory()
x = C[2]
S = simples(C)
@assert int_dim(Hom(x⊗x,x)) == 2
L = [q ∘ (p ⊗ id(x)) for e in S
     for p in basis(Hom(x⊗x,e)) for q in basis(Hom(e⊗x,x))]
R = [q ∘ (id(x) ⊗ p) for f in S
     for p in basis(Hom(x⊗x,f)) for q in basis(Hom(x⊗f,x))]
A = C.ass[2,2,2,2]
@assert size(A) == (7,7)
@assert A[3,6] == 1//2 && A[6,3] == -1//2
for v in eachindex(R)
    @assert R[v] ∘ associator(x,x,x) ==
            sum(A[u,v]*L[u] for u in eachindex(L))
end
P = basis(Hom(x⊗x,x))
B = C.braiding[2,2,2]
for j in eachindex(P)
    @assert P[j] ∘ braiding(x,x) ==
            sum(B[i,j]*P[i] for i in eachindex(P))
end
@assert TensorCategories.dict_to_associator(F_symbols(C)) == C.ass

Here the left channel order is $(\mathbb 1),(8,1,1),(8,1,2),(8,2,1),(8,2,2),(10),(\overline{10})$, where the two indices attached to $8$ specify the inner and outer projection bases, respectively.

Extracting from an existing model

The category of $C_2$-graded vector spaces is already available as a concrete model. Its skeletal realization is obtained by choosing its two simple graded lines and computing the associator in decomposition bases:

using TensorCategories, Oscar
G = cyclic_group(2)
V = graded_vector_spaces(QQ,G)
S = skeletonize(V)
@assert length(simples(S)) == 2
@assert pentagon_axiom(S)
multiplication_table(S)

2×2×2 Array{Int64, 3}:
[:, :, 1] =
 1  0
 0  1

[:, :, 2] =
 0  1
 1  0

This generic conversion applies to a split fusion category whose implementation supports decomposition and the necessary Hom-space operations. The resulting symbols use the Hom bases chosen by the input model. The concrete model remains available for computations on graded vectors and matrices.

The related function six_j_symbols(V) computes only the array of associator blocks in such chosen bases. The generic skeletonize(V) packages these blocks with the fusion rules and unit, transports a braiding when one is present, and attempts to transport the pivotal or spherical structure. If that transport is not available, the skeletal initializer's coefficients $P_i=1$ remain in place; use is_pivotal(C; check=true) before treating them as a pivotal structure.

For GradedVectorSpaces there is also a specialized direct conversion six_j_category(V). It reads the group multiplication, cocycle associator, and stored pivotal scalars directly, but currently omits the braiding; use skeletonize(V) when the braiding must be transported. Finally, F_symbols(S) converts the blocks of an existing SixJCategory to one of the documented dictionary layouts.

Continue with Numerical fusion categories, which applies the same categorical formulas over arbitrary-precision ball fields.