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 Eg diagram at each time. We then construct a certain coset (or commutant) subalgebra U associated with L in the lattice VOA V √2E8 . 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 A , 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 1 , the product of two Miyamoto involutions is in the desired conjugacy class of the Monster simple group.