Fibonacci categories

The Fibonacci category is the simplest nonpointed rank-two fusion category. Tensor powers of its nontrivial simple object have Fibonacci multiplicities, and its positive unitary realization is a standard anyon model for topological quantum computation. Its other algebraic realization is the nonunitary Yang–Lee Galois conjugate Rowell et al. (2009), p. 4 and §5.3.2.

The constructor fibonacci_category(K) uses labels $(\mathbb 1,\tau)$ and the rule $\tau\otimes\tau=\mathbb 1\oplus\tau$. An optional second positional argument selects the $a$-th root $r$ of $x^2-x-1$ returned by OSCAR: fibonacci_category(K,a). The default is $a=1$, and the constructor sets $b=-r$. The field $K$ must contain the selected root. A complex realization also requires a field embedding.

The only nonidentity associator block is

\[\label{eq:fibonacci-package-associator} A_{\tau,\tau,\tau}^{\tau}= \begin{pmatrix}b&b\\1&-b\end{pmatrix},\qquad b^2+b=1.\]

This is the actual stored row-coordinate matrix, not the commonly displayed symmetric gauge. The overload fibonacci_category(a) uses the same root selector over $\mathbb Q(\sqrt5)$, with fibonacci_category() again taking $a=1$. The constructor retains the skeletal model's all-one pivotal components; for both algebraic roots these pass the package's exact pivotal check.

using TensorCategories, Oscar
C = fibonacci_category()
A = C.ass[2,2,2,2]
b = A[1,1]
@assert b^2 + b == 1
@assert A == matrix(base_ring(C),[b b; 1 -b])
@assert A^2 == identity_matrix(base_ring(C),2)
@assert pentagon_axiom(C)
@assert !is_braided(C)
A

[-ϕ   -ϕ]
[ 1    ϕ]

The two algebraic roots give the two familiar rank-two categorifications after choosing an embedding. To identify the positive Fibonacci realization versus its nonunitary Galois conjugate, inspect the root and pivotal dimensions under that embedding.

For the positive root $b$, put $D=\operatorname{diag}(1,1/\sqrt b)$. Then

\[\label{eq:fibonacci-symmetric-associator} D^{-1}A_{\tau,\tau,\tau}^{\tau}D =\begin{pmatrix}b&\sqrt b\\ \sqrt b&-b\end{pmatrix}.\]

The resulting symmetric matrix is precisely the positive unitary Fibonacci gauge displayed in Rowell et al. (2009), §5.3.2, after $b=\varphi^{-1}$. Thus the implementation agrees with the underlying fusion category in that reference by an explicit change of fusion basis.

The constructor supplies no braiding. In particular, it does not implement the $R$-symbols of the complete modular category displayed in that reference.