Skip to content
DraconiticResearch
Search Sources API
201,818 studies31 sourcesUpdated 42 min ago
PreprintarXiv licencearXiv

Haagerup symmetries of the E8 theory and conformal nets at central charge eight

  • Cédric Arhancet
Open at sourcePDFDOICite

Abstract

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(cal 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 cal H2 and cal H3 are the two lattice extensions of VE6 ⊕ A2. Along a strategy outlined by Huang, we prove a recognition theorem for cal H2 and cal 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 cal AS is completely rational with Rep(cal AS) ≅ Z(cal H3), that the Haagerup subfactor arises from cal AS, and that cal 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(cal 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+.

Related

  1. Haagerup conformal field theory at central charge eight
    Tzu-Chen Huang
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryMathematical physicsarXiv
  2. Haagerup-Izumi modular data from hyperbolic words: S-fold TQFTs, central charge, torsion and the Bethe vacua
    David E Evans
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryAnalysis & PDEsarXiv
  3. Haagerup vertex operator algebras at central charge eight
    Cédric Arhancet
    arXivPreprintarXiv licenceFull textAlgebra & number theoryMathematical physicsarXiv
  4. Central Haagerup Tensor Products and Completely Bounded Maps under Strong Morita Equivalence
    Ilja Gogić
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  5. Scaling Symmetries and Conformal Relative Equilibria on Poisson Manifolds, with Applications to Lie-Poisson Systems
    Manuele Santoprete
    arXivPreprintCC BY 4.0Full textDynamical systemsGeometry & topologyarXiv

Cite

BibTeXRIS
BibTeX
@misc{arhancet2026haagerup,
  title = {{Haagerup symmetries of the $E_8$ theory and conformal nets at central charge eight}},
  author = {Cédric Arhancet},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.03560},
  eprint = {2611.03560},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.03560},
}
RIS
TY  - UNPB
TI  - Haagerup symmetries of the E_8 theory and conformal nets at central charge eight
AU  - Cédric Arhancet
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2611.03560
UR  - https://arxiv.org/abs/2611.03560
AB  - 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(cal H_3) 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 V_E_8, 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 cal H_2 and cal H_3 are the two lattice extensions of V_E_6 ⊕ A_2. Along a strategy outlined by Huang, we prove a recognition theorem for cal H_2 and cal H_3 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 ⊂ V_E_6 ⊕ A_2 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 cal A_S is completely rational with Rep(cal A_S) ≅ Z(cal H_3), that the Haagerup subfactor arises from cal A_S, and that cal A_S solves a problem of Bischoff on generalized orbifolds. Under the same hypothesis, using a unitary structure on V_E_8 ⊕ E_8 found by exact computer calculations, we prove that S is completely unitary and that Rep(cal A_S) ≅ 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+.
ER  -