Haagerup symmetries of the $E_8$ theory and conformal nets at central charge eight

A Haagerup algebra is a positive strongly rational vertex operator algebra $U$ of central charge eight with $\mathrm{Rep}(U)$ braided equivalent to the Drinfeld center $Z(\cal H_3)$ of the Haagerup fusion category. Such an algebra has been proposed by Jiao, with a computer-assisted proof, and announced by Riesen and by Huang. We show that $U$ has exactly three holomorphic extensions, each isomorphic to $V_{E_8}$, and that the categories of modules over the corresponding Lagrangian algebras are the three Haagerup fusion categories of Grossman and Snyder. The extensions attached to $\cal H_2$ and $\cal H_3$ are the two lattice extensions of $V_{E_6 \oplus A_2}$. Along a strategy outlined by Huang, we prove a recognition theorem for $\cal H_2$ and $\cal H_3$ among unitary fusion categories of rank six with three invertible objects. For regular unitary vertex operator algebras with Tener's bounded localized vertex operators, we prove that the fusion product of two unitary modules is unitarizable when one of them integrates to the conformal net. The explicit unitary algebra $S \subset V_{E_6 \oplus A_2}$ of our previous work, isomorphic to Jiao's algebra, is strongly local. Granting that Jiao's algebra is a Haagerup algebra, we deduce that the conformal net $\cal A_S$ is completely rational with $\mathrm{Rep}(\cal A_S) \cong Z(\cal H_3)$, that the Haagerup subfactor arises from $\cal A_S$, and that $\cal A_S$ solves a problem of Bischoff on generalized orbifolds. Under the same hypothesis, using a unitary structure on $V_{E_8 \oplus E_8}$ found by exact computer calculations, we prove that $S$ is completely unitary and that $\mathrm{Rep}(\cal A_S) \cong \mathrm{Rep}(S)$ through the correspondence given by smeared vertex operators. The diagonal construction and the passage to the twin algebra also show that every Haagerup algebra is completely unitary and satisfies Condition I+.