Quantum Symmetries for Exceptional $\mathrm{SU}(4)$ Modular Invariants Associated with Conformal Embeddings

Three exceptional modular invariants of $\mathrm{SU}(4)$ exist at levels 4, 6 and 8. They can be obtained from appropriate conformal embeddings and the corresponding graphs have self-fusion. From these embeddings, or from their associated modular invariants, we determine the algebras of quantum symmetries, obtain their generators, and, as a by-product, recover the known graphs $\mathcal E_4$, $\mathcal E_6$ and $\mathcal E_8$ describing exceptional quantum subgroups of type $\mathrm{SU}(4)$. We also obtain characteristic numbers (quantum cardinalities, dimensions) for each of them and for their associated quantum groupoïds.