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

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 $V^{♮}$ are also discussed. We show that there is no Ising vector of $\sigma$-type in $V^{♮}$. Moreover, we compute the centralizer $C_{\aut V^\natural}(z, τ_e)$ 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 $E_8$ .

22 pages