51 constructions of the Moonshine module

Abstract

We show using Borcherds products that for any fixed-point free automorphism of the Leech lattice satisfying a "no massless states" condition, the corresponding cyclic orbifold of the Leech lattice vertex operator algebra is isomorphic to the Monster vertex operator algebra. This induces an "orbifold duality" bijection between algebraic conjugacy classes of fixed-point free automorphisms of the Leech lattice satisfying this condition and algebraic conjugacy classes of non-Fricke elements in the Monster. We use the duality to show that non-Fricke Monstrous Lie algebras are Borcherds-Kac-Moody Lie algebras, and prove a refinement of Norton's Generalized Moonshine conjecture: the ambiguous constants relating generalized moonshine Hauptmoduln under conjugation and modular transformations are necessarily roots of unity. We also describe a class of rank 2 Borcherds-Kac-Moody Lie algebras attached to the Conway group.

Characterizing Moonshine Functions by Vertex-Operator-Algebraic Conditions

“…Proof . By our assumption that T g k is a non-Fricke monstrous function, the proof of Theorem 4.5 of [4] implies V /g k is isomorphic to V Λ . We note that the statement of the theorem assumes V ∼ = V , but the proof only uses the completely replicable function T g k and its replicates and SL 2 (Z)-transforms, so it works under the assumption that V is only moonshine-like.…”

“…In [4], the first author proved a conjecture in [19], giving an orbifold-duality correspondence between non-Fricke automorphisms of V and fixed-point free automorphisms of the Leech lattice satisfying a "no massless states" condition. Proposition 4.1.…”

“…2. In [4], we find that cyclic orbifold duality gives a correspondence between non-Fricke elements of the monster and fixed-point free automorphisms of the Leech lattice satisfying a "no massless states" condition. This was conjectured by Tuite in [19].…”

“…More recent developments give us an additional condition: In [19], a large amount of computational and physical evidence is given for a conjectured cyclic orbifold duality between non-Fricke automorphisms of V and fixed-point free automorphisms of the Leech lattice satisfying a "no massless states" condition. Using the cyclic orbifold theory established in [12], the conjecture was proved in [4], and it implies that any completely replicable function with a non-Fricke monstrous replicate must be monstrous in order to be a trace function. We note that this implies any counterexample to the uniqueness of V has no non-Fricke automorphisms.…”

Duality defect of the monster CFT

“…• Our investigation of Z 2A has a natural generalization to Z pA with p a prime number dividing the order of the Monster group. For each such p, the Monster CFT is known to be self-dual under the Z pA orbifold [46,29,30,48,49]. 18 , which implies that the Monster CFT must have a duality defect belonging to a Z p Tamabara-Yamagami category.…”