Vertex operator algebras generated by two Ising vectors

TL;DR

This paper investigates the structure of vertex operator algebras generated by two Ising vectors with specific inner products, establishing isomorphisms with known algebraic structures and exploring special cases.

Contribution

It proves the isomorphism of a specific VOA generated by two Ising vectors to the 6A-algebra and analyzes several special inner product cases.

Findings

$ ext{VOA}( ext{e,f})$ is isomorphic to $ ext{6A}$-algebra $ ext{U}_{6A}$

Identifies special cases with different inner products and their algebraic properties

Provides a detailed classification of VOA generated by two Ising vectors with given inner products

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 $\langle e, f\rangle=\frac{5}{2^{10}}$. 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 $\langle e, f\rangle=\frac{1}{4}, \frac{1}{2^{5}}, 0 $ and the case $\langle e, f\rangle=\frac{1}{2^{8}}$ with $(\tau_{e}\tau_{f})^4=1$.