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Haagerup vertex operator algebras at central charge eight

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Contributed by Cédric ARHANCET ↗

Version 1 / Nov 02, 2026 / CC BY-SA 4.0

Abstract

Evans and Gannon showed that a positive vertex operator algebra of central charge eight whose representation category is the Drinfeld center Z(H3)Z(\cal 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 UU has six currents, that U1≅sl2⊕sl2U_1 \cong \mathfrak{sl}_2 \oplus \mathfrak{sl}_2 embeds into VE6⊕A2V_{E_6 \oplus A_2} in one of two explicit ways, with levels (1,39)(1,39) or (3,13)(3,13), and we determine the lowest weight spaces of its modules and U2U_2 as U1U_1-modules. For the levels (3,13)(3,13), U2U_2 is one of four Galois conjugate subspaces, and the algebra it generates is unitary, C2C_2-cofinite and isomorphic to the algebra constructed by Jiao; for the levels (1,39)(1,39), UU contains the algebra announced by Riesen. The associated lattice of UU is that of the form x2+xy+10y2x^2+xy+10y^2, resp.\ 3x2+3xy+4y23x^2+3xy+4y^2, the two classes of the principal genus of discriminant −39-39. Exchanging them in the decomposition of UU over its Heisenberg commutant gives a Haagerup algebra U♮U^\natural of the other levels, and U↦U♮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(H3)Z(\cal 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 VE6⊕A2V_{E_6 \oplus A_2}, and genera of vertex operator algebras.

Provenance statement

The author acknowledges the use of AI tools for language polishing, exploratory mathematical discussions, the writing of the verification scripts and feedback during the development and preparation of this manuscript. The author takes full responsibility for the mathematical content of the manuscript.

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Version history

  1. v1Submitted by Cédric ARHANCETInitial depositCurrentOct 31, 2026