Fibonacci-type orbifold data in Ising modular categories

An orbifold datum is a collection $\mathbb{A}$ of algebraic data in a modular fusion category $\mathcal{C}$. It allows one to define a new modular fusion category $\mathcal{C}_{\mathbb{A}}$ in a construction that is a generalisation of taking the Drinfeld centre of a fusion category. Under certain simplifying assumptions we characterise orbifold data $\mathbb{A}$ in terms of scalars satisfying polynomial equations and give an explicit expression which computes the number of isomorphism classes of simple objects in $\mathcal{C}_{\mathbb{A}}$. In Ising-type modular categories we find new examples of orbifold data which - in an appropriate sense - exhibit Fibonacci fusion rules. The corresponding orbifold modular categories have 11 simple objects, and for a certain choice of parameters one obtains the modular category for $sl(2)$ at level 10. This construction inverts the extension of the latter category by the $E_6$ commutative algebra.