Ising vectors in the vertex operator algebra $V_{\Lambda}^+$ associated with the Leech lattice $\Lambda$

TL;DR

This paper characterizes Ising vectors in the vertex operator algebra associated with the Leech lattice, linking their structure to sublattices and exploring their properties within the moonshine VOA and related groups.

Contribution

It provides a classification of Ising vectors in $V_ ext{Leech}^+$ and analyzes their properties and symmetries within the moonshine vertex operator algebra.

Findings

Ising vectors correspond to sublattices isomorphic to $ ext{sqrt}(2)E_8$

No Ising vectors of $ ext{sigma}$-type in $V^ atural$

Computed centralizers related to Ising vectors and explained relations in the Monster group

Abstract

In this article, we study the Ising vectors in the vertex operator algebra $V_\Lambda^+$ associated with the Leech lattice $\Lambda$. The main result is a characterization of the Ising vectors in $V_\Lambda^+$. We show that for any Ising vector $e$ in $V_\Lambda^+$, there is a sublattice $E\cong \sqrt{2}E_8$ of $\Lambda$ such that $e\in V_E^+$. Some properties about their corresponding $\tau$-involutions in the moonshine vertex operator algebra $V^\natural$ are also discussed. We show that there is no Ising vector of $\sigma$-type in $V^\natural$. Moreover, we compute the centralizer $C_{\aut V^\natural}(z, \tau_e)$ for any Ising vector $e\in V_\Lambda^+$, where $z$ is a 2B element in $\aut V^\natural$ which fixes $V_\Lambda^+$. Based on this result, we also obtain an explanation for the 1A case of an observation by Glauberman-Norton (2001), which describes some mysterious relations between the centralizer of $z$ and some 2A elements commuting $z$ in the Monster and the Weyl groups of certain sublattices of the root lattice of type $E_8$ .