Quantum symmetries and nimreps for an exotic exceptional modular invariant of F4 at level 6

Abstract

We revisit an old non-diagonal modular invariant of $F_4$ at level $6$, discovered in 1989. At the time of its discovery, it was noticed that it could not be obtained from a conformal embedding of affine Lie algebras. Later, it was shown that it can be obtained from an extension of a vertex operator algebra which is not of Lie type. To this modular invariant, one can attach a full CFT defined (or formulated) as a module category $\mathcal M$ for the fusion category $\mathcal A = \mathcal C(F_4,6)$. More recently, it was recognized that $\mathcal M$ is associated with an étale algebra of $\mathcal A$. In other words, the resulting quantum module is a quantum subgroup (it is also said to have self-fusion, to be of type I, or to be flat). We study the dual $\mathcal O$ of $\mathcal A$ with respect to $\mathcal M$. In particular, we describe the algebra of quantum symmetries (we obtain the chiral generators and display the Ocneanu graphs for the fundamental irreps of $\mathcal A$). The chiral graphs are not connected, each one containing two pairs of isomorphic components. The vertex sets of these components are identified with the simple objects, up to equivalence, of two module categories $\mathcal M$ and $\mathcal M^\prime$ sharing the same modular invariant. $\mathcal M$ enjoys self-fusion, whereas $\mathcal M^\prime$ does not. The restriction of the action of the Verlinde algebra $A$ to the associated abelian groups $M$ and $M^\prime$ defines two families of matrices with non-negative integer entries (nimreps) and their adjacency graphs. As a by-product, we obtain the matrices describing the defect lines of the theory, the integer and rational fusion polynomials attached to $A$ and its modules, and the relevant numerical invariants, quantum dimensions, quantum masses, and induction-restriction rules.