The Extended Haagerup fusion categories

In this paper we construct two new fusion categories and many new subfactors related to the exceptional Extended Haagerup subfactor. The Extended Haagerup subfactor has two even parts EH1 and EH2. These fusion categories are mysterious and are the only known fusion categories which appear to be unrelated to finite groups, quantum groups, or Izumi quadratic categories. One key technique which has previously revealed hidden structure in fusion categories is to study all other fusion categories in the Morita equivalence class, and hope that one of the others is easier to understand. In this paper we show that there are exactly four categories (EH1, EH2, EH3, EH4) in the Morita equivalence class of Extended Haagerup, and that there is a unique Morita equivalence between each pair. The existence of EH3 and EH4 gives a number of interesting new subfactors. Neither EH3 nor EH4 appears to be easier to understand than the Extended Haaerup subfactor, providing further evidence that Extended Haagerup does not come from known constructions. We also find several interesting intermediate subfactor lattices related to Extended Haagerup. The method we use to construct EH3 and EH4 is interesting in its own right and gives a general computational recipe for constructing fusion categories in the Morita equivalence class of a subfactor. We show that pivotal module $\rm C^*$ categories over a given subfactor correspond exactly to realizations of that subfactor planar algebra as a planar subalgebra of a graph planar algebra. This allows us to construct EH3 and EH4 by realizing the Extended Haagerup subfactor planar algebra inside the graph planar algebras of two new graphs. This technique also answers a long-standing question of Jones: which graph planar algebras contain a given subfactor planar algebra?