math.QA — Quantum Algebra

The Haagerup center at central charge eight via conformal nets

Contributed by Cédric ARHANCET

Let $S\subset V_{E_6\oplus A_2}$ be the unitary vertex operator algebra of central charge eight with currents at levels $(3,13)$ constructed in our previous work. An exact presentation of a full corner of its Zhu algebra gives strong rationality, the twelve simple modules and the lattice branching rules. Using these results, we identify the representation category of its conformal net with the Haagerup center $Z(\mathcal{H}_3)$. We construct a holomorphic diagonal extension of $\mathcal {A}_S\otimes \mathcal{A}_S$ and apply the vacuum trace class theorem of Henriques and Tener to prove trace class heat operators in every sector of $\mathcal{A}_S$. Reconstruction gives a unitary holomorphic vertex operator algebra containing $S\otimes S$ as mutual commutants. The coset theorems of Gui then give the complete unitarity of $S$ and a braided equivalence $\mathrm{Rep}(S) \cong Z(\mathcal{H}_3)$, so that $S$ is a Haagerup algebra. Using results of Dong, Jiao and Xu, we also identify the Huang--Riesen algebra with the twin of $S$ and describe their common fixed point algebra using the intersection of the two binary lattices of discriminant $-39$.

Author comments: 73 pages