math.QA — Quantum Algebra

The Haagerup VOA at Central Charge Eight

Contributed by Andrew Riesen

We give an explicit construction of a vertex operator algebra $U$ of central charge eight inside $V_{E_6\oplus A_2}$, and prove the ribbon equivalence $\Rep(U)\simeq Z(\HH_3)$, where $\HH_3$ is a Haagerup fusion category. The algebra is generated as a vertex algebra by six affine generators and three specified weight-two vectors. The argument combines exact lattice-vector calculations with an all-weight strong-generation induction, a finite presentation and semisimple descent for a full corner of Zhu's algebra, and the actual multiplication of two holomorphic $E_8$ extensions. The categorical identification uses splitting maps and a comparison with the Evans--Gannon reconstruction, not modular data alone.