Haagerup vertex operator algebras at central charge eight

Abstract

Evans and Gannon showed that a positive vertex operator algebra of central charge eight whose representation category is the Drinfeld center $Z(\mathcal{H}_3)$ of the Haagerup fusion category has one of two character vectors, with six or nineteen currents. Call such an algebra, if strongly rational, a Haagerup algebra. We prove that a Haagerup algebra $U$ has six currents, that $U_1 \cong \mathfrak{sl}_2 \oplus \mathfrak{sl}_2$ embeds into $V_{E_6 \oplus A_2}$ in one of two explicit ways, with levels $(1,39)$ or $(3,13)$, and we determine the lowest weight spaces of its modules and $U_2$ as $U_1$-modules. For the levels $(3,13)$, $U_2$ is one of four Galois conjugate subspaces, and the algebra it generates is unitary, $C_2$-cofinite and isomorphic to the algebra constructed by Jiao; for the levels $(1,39)$, $U$ contains the algebra announced by Riesen. The associated lattice of $U$ is that of the form $x^2+xy+10y^2$, resp. $3x^2+3xy+4y^2$, the two classes of the principal genus of discriminant $-39$. Exchanging them in the decomposition of $U$ over its Heisenberg commutant gives a Haagerup algebra $U^\natural$ of the other levels, and $U \mapsto U^\natural$ is a bijection between the isomorphism classes of the two cases. Consequently, if either explicit algebra of Jiao or Riesen is positive and strongly rational with representation category braided equivalent to $Z(\mathcal{H}_3)$, there are exactly two Haagerup algebras up to isomorphism, forming a single hyperbolic genus. The explicit algebras of Jiao and Riesen have rational forms. The proofs combine exact trace relations given by vector-valued modular forms, exact computations in a model of $V_{E_6 \oplus A_2}$, and genera of vertex operator algebras.