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Haagerup vertex operator algebras at central charge eight

A mix of human-written and AI-generated textHuman understanding: some partsmath.QA — Quantum Algebra

Contributed by Cédric ARHANCET

Version 2 / 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. Together with U1U_1 it generates the unitary C2C_2-cofinite algebra SS, which is isomorphic to the algebra constructed by Jiao. For the levels (1,39)(1,39), UU contains the explicit algebra of 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. The classification gives exactly two isomorphism classes, forming a single hyperbolic genus, as soon as either explicit candidate is known to be a Haagerup algebra. This property is proved for SS in [ArhIII], where the twin S♮S^\natural is also identified, with the algebra of Huang and Riesen, using the results of Dong, Jiao and Xu [DJX26]. 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.

Author comments: 75 pages

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. v1 · 75 pagesSubmitted by Cédric ARHANCETInitial depositSupersededOct 31, 2026
  2. v2 · 75 pagesSubmitted by Cédric ARHANCET75 pagesCurrentNov 06, 2026