$\Z_3$-orbifold construction of the Moonshine vertex operator algebra and some maximal $3$-local subgroups of the Monster

Abstract

In this article, we describe some maximal 3-local subgroups of the Monster simple group using vertex operator algebras (VOA). We first study the holomorphic vertex operator algebra obtained by applying the orbifold construction to the Leech lattice vertex operator algebra and a lift of a fixed-point free isometry of order 3 of the Leech lattice. We also consider some of its special subVOAs and study their stabilizer subgroups using the symmetries of the subVOAs. It turns out that these stabilizer subgroups are 3-local subgroups of its full automorphism group. As one of our main results, we show that its full automorphism group is isomorphic to the Monster simple group by using a 3-local characterization and that the holomorphic VOA is isomorphic to the Moonshine VOA. This approach allows us to obtain relatively explicit descriptions of two maximal 3-local subgroups of the shape 3 1+12 .2. Suz :2 and 3 8 .Ω − (8, 3).2 in the Monster simple group.

A Self-Dual Integral Form of the Moonshine Module

“…The claims about existence and uniqueness of lifts of automorphisms to V Λ follow from Section 4.2 of [32], and a short version of the argument is Proposition 2.1 in [9]. The identification of prime order cyclic orbifolds with V follows from the main construction of [27] for p = 2, Theorem 1.1 of [10] for p = 3, and Theorem 4.4 in [1] for p = 5, 7, 13. The identification of order p 1 p 2 cyclic orbifolds with V Λ follows from Theorem 4.1 in [1] for p 1 p 2 ∈ {6, 10, 14, 26}, and for the others, the result follows from essentially the same argument: it suffices to show that the weight 1 subspace of the irreducible twisted module V Λ (g p 1 p 2 ) has dimension 24 (p 1 −1)(p 2 −1) , and one can do this by manipulating the frame shape.…”