Fibonacci-type orbifold data in Ising modular categories

TL;DR

This paper explores orbifold data in Ising modular categories, revealing Fibonacci-like fusion rules, explicit classifications, and connections to well-known quantum groups, expanding the understanding of modular fusion categories.

Contribution

It characterizes orbifold data via polynomial equations and constructs new orbifold modular categories with Fibonacci fusion rules, including examples related to $sl(2)$ at level 10.

Findings

New orbifold data with Fibonacci fusion rules in Ising categories

Explicit formulas for simple object counts in orbifold categories

Construction of orbifold categories related to $sl(2)$ level 10

Abstract

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.