The Moonshine Module for Conway's Group

TL;DR

This paper constructs a moonshine module for Conway's group using a super vertex operator algebra, demonstrating that its trace functions are principal moduli with specific Fourier properties, extending moonshine phenomena.

Contribution

It introduces a new moonshine module for Conway's group, characterizes it uniquely, and proves a case of generalized moonshine relating trace functions to genus zero groups.

Findings

Trace functions are normalized principal moduli.

The module admits a unique canonically-twisted module.

Proves a case of generalized moonshine for Conway's group.

Abstract

We exhibit an action of Conway's group---the automorphism group of the Leech lattice---on a distinguished super vertex operator algebra, and we prove that the associated graded trace functions are normalized principal moduli, all having vanishing constant terms in their Fourier expansion. Thus we construct the natural analogue of the Frenkel--Lepowsky--Meurman moonshine module for Conway's group. The super vertex operator algebra we consider admits a natural characterization, in direct analogy with that conjectured to hold for the moonshine module vertex operator algebra. It also admits a unique canonically-twisted module, and the action of the Conway group naturally extends. We prove a special case of generalized moonshine for the Conway group, by showing that the graded trace functions arising from its action on the canonically-twisted module are constant in the case of Leech lattice automorphisms with fixed points, and are principal moduli for genus zero groups otherwise.