Vertex operator algebras, extended $E_8$ diagram, and McKay’s observation on the Monster simple group

Abstract:

We study McKay’s observation on the Monster simple group, which relates the $2A$-involutions of the Monster simple group to the extended $E_8$ diagram, using the theory of vertex operator algebras (VOAs). We first consider the sublattices $L$ of the $E_8$ lattice obtained by removing one node from the extended $E_8$ diagram at each time. We then construct a certain coset (or commutant) subalgebra $U$ associated with $L$ in the lattice VOA $V_{\sqrt {2}E_8}$. There are two natural conformal vectors of central charge $1/2$ in $U$ such that their inner product is exactly the value predicted by Conway (1985). The Griess algebra of $U$ coincides with the algebra described in his Table 3. There is a canonical automorphism of $U$ of order $|E_8/L|$. Such an automorphism can be extended to the Leech lattice VOA $V_\Lambda$, and it is in fact a product of two Miyamoto involutions. In the sequel (2005) to this article, the properties of $U$ will be discussed in detail. It is expected that if $U$ is actually contained in the Moonshine VOA $V^\natural$, the product of two Miyamoto involutions is in the desired conjugacy class of the Monster simple group.