Groups of Lie Type, Vertex Algebras, and Modular Moonshine

Abstract

We use recent work on integral forms in vertex operator algebras to construct vertex algebras over general commutative rings and Chevalley groups acting on them as vertex algebra automorphisms. In this way, we get series of vertex algebras over fields whose automorphism groups are essentially those Chevalley groups (actually, an exact statement depends on the field and involves upwards extensions of these groups by outer diagonal and graph automorphisms). In particular, given a prime power $q$, we realize each finite simple group which is a Chevalley or Steinberg variations over $\mathbb{F}_q$ as most of'' the full automorphism group of a vertex algebra over $\mathbb{F}_q$. These finite simple groups are \[ A_n(q), B_n(q), C_n(q), D_n(q), E_6(q), E_7(q), E_8(q), F_4(q), G_2(q) \] \[ \text{and } ^{2}A_n(q), ^{2}D_n(q), ^{3}D_4(q), ^{2}E_6(q), \] where $q$ is a prime power. Also, we define certain reduced VAs. In characteristics 2 and 3, there are exceptionally large automorphism groups. A covering algebra idea of Frohardt and Griess for Lie algebras is applied to the vertex algebra situation. We use integral form and covering procedures for vertex algebras to complete the modular moonshine program of Borcherds and Ryba for proving an embedding of the sporadic group $F_3$ of order $2^{15}3^{10}5^3 7^2 13{\cdot }19{\cdot} 31$ in $E_8(3)$.