Haagerup symmetries of the E₈ 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 Rep(U) braided equivalent to the Drinfeld center Z( H₃) 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 VE₈, 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 H₂ and H₃ are the two lattice extensions of VE₆ ⊕ A₂. Along a strategy outlined by Huang, we prove a recognition theorem for H₂ and H₃ 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 ⊂ VE₆ ⊕ A₂ 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 AS is completely rational with Rep( AS) Z( H₃), that the Haagerup subfactor arises from AS, and that AS solves a problem of Bischoff on generalized orbifolds. Under the same hypothesis, using a unitary structure on VE₈ ⊕ E₈ found by exact computer calculations, we prove that S is completely unitary and that Rep( AS) 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+.