Haagerup vertex operator algebras at central charge eight

@misc{arhancet2026haagerup,
  title = {{Haagerup vertex operator algebras at central charge eight}},
  author = {Cédric Arhancet},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01618},
  eprint = {2611.01618},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01618},
}

TY  - UNPB
TI  - Haagerup vertex operator algebras at central charge eight
AU  - Cédric Arhancet
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01618
UR  - https://arxiv.org/abs/2611.01618
AB  - Evans and Gannon showed that a positive vertex operator algebra of central charge eight whose representation category is the Drinfeld center Z(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 ≅ sl_2 ⊕ sl_2 embeds into V_E_6 ⊕ 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 ↦ 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(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 ⊕ A_2, and genera of vertex operator algebras.
ER  -