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Haagerup symmetries of the E8E_8 theory and conformal nets at central charge eight

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

Contributed by Cédric ARHANCET ↗

Version 1 / Nov 03, 2026 / CC BY 4.0

Abstract

A Haagerup algebra is a positive strongly rational vertex operator algebra UU of central charge eight with $\Rep(U)$ braided equivalent to the Drinfeld center Z(H3)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 UU has exactly three holomorphic extensions, each isomorphic to VE8V_{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 H2\cal H_2 and H3\cal H_3 are the two lattice extensions of VE6⊕A2V_{E_6 \oplus A_2}. Along a strategy outlined by Huang, we prove a recognition theorem for H2\cal H_2 and H3\cal H_3 among unitary fusion categories of rank six with three invertible objects, a noncommutative fusion ring and a pointed part with trivial associator. 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⊕A2S \subset V_{E_6 \oplus 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 AS\cal A_S is completely rational with $\Rep(\cal A_S) \cong Z(\cal H_3)$, that the Haagerup subfactor arises from AS\cal A_S, and that AS\cal A_S solves a problem of Bischoff on generalized orbifolds. Under the same hypothesis, using a unitary structure on VE8⊕E8V_{E_8 \oplus E_8} found by exact computer calculations, we prove that SS is completely unitary and that Rep(AS)≅Rep(S)\mathrm{Rep}(\cal A_S) \cong \mathrm{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+.

Provenance statement

The author acknowledges the use of AI tools for language polishing, LaTeX editing, exploratory mathematical discussions and feedback, for drafting parts of the arguments, and for writing and running the computer programs of Section \ref{subsec:complete-unitarity} and the certificate \texttt{omega\_certificate.json}. The author takes full responsibility for the content of the manuscript.

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  1. v1Submitted by Cédric ARHANCETInitial depositCurrentNov 02, 2026