math.QA — Quantum Algebra

An order-five Haagerup--Izumi vertex operator algebra at central charge eight

Contributed by Tzu Chen Huang

We construct an explicit unitary strongly rational vertex operator algebra $U$ of central charge eight inside $V_{A_4\oplus A_4}$ and identify $Rep(U)$ with the Drinfeld center of the positive order-five Haagerup--Izumi category. The construction adjoins one algebraic weight-two field to a diagonal affine $\mathfrak{sl}_2$ at level forty and the ambient conformal vector. Exact finite lattice identities imply strong generation and $C_2$-cofiniteness. A finite integral order in the actual Zhu algebra descends semisimplicity from a $1590$-dimensional upper presentation over $\mathbb F_{10069}$. To determine the modules, we combine the good-prime block bound with lattice character growth and a Wronskian inequality. The resulting full branching has an irreducible congruence complement of dimension at most fifteen; its exact classification forces all fourteen denominator-$29$ sectors. Finally, the actual holomorphic $E_8$ extension determines the boundary fusion category and its associator. The proof is computer-assisted, with exact arithmetic certificates separated from exploratory numerical evidence.