Vertex operator algebras generated by two Ising vectors

Abstract

In this paper we mainly study the vertex operator algebra $\mathbb{C}\mathrm{VA}(e,f)$ generated by two Ising vectors $e$ and $f$ with $⟨e,f⟩=\frac{5}{2^{10}}$. We prove $\mathbb{C}\mathrm{VA}(e,f)$ is isomorphic to the $6A$-algebra ${\mathcal{U}}_{6A}$ constructed by Lam-Yamada-Yamauchi. We also discuss the cases $⟨e,f⟩=\frac{1}{4},\frac{1}{2^5},0$ and the case $⟨e,f⟩=\frac{1}{2^8}$ with $({\tau }_e{\tau }_f{)}^4=1$.