A Self-Dual Integral Form of the Moonshine Module

Abstract

We construct a self-dual integral form of the moonshine vertex operator algebra, and show that it has symmetries given by the Fischer-Griess monster simple group. The existence of this form resolves the last remaining open assumption in the proof of the modular moonshine conjecture by Borcherds and Ryba. As a corollary, we find that Griess's original 196884-dimensional representation of the monster admits a positive-definite self-dual integral form with monster symmetry.In this paper, we construct self-dual R-forms of the moonshine module vertex operator algebra [27] over various commutative rings R, culminating in the universal case where R is the ring of rational integers Z. For the Z-form, we show that the vertex operator algebra has monster symmetry, and is self-dual with respect to an invariant bilinear form that respects the monster symmetry. Base change then gives us self-dual monster-symmetric vertex operator algebras over any commutative ring. Our construction yields the final step in the affirmative resolution of Ryba's modular moonshine conjecture.This paper is a contribution to the Special Issue on Moonshine and String Theory. The full collection is available at https://www.emis.de/journals/SIGMA/moonshine.html arXiv:1710.00737v4 [math.RT] 19 Apr 2019Given a homomorphism f : R → S of commutative rings, we say f is faithfully flat if S is faithfully flat as an R-module under the induced action.See [43, Section 00H9] for a brief overview of basic properties of flat and faithfully flat modules and ring maps.Definition 2.2. Let f : R → S be a homomorphism of commutative rings. A descent datum for modules with respect to f is a pair (M, φ), where M is an S-module, and φ : M ⊗ R S → S ⊗ R M is an S ⊗ R S-module isomorphism satisfying the "cocycle condition", i.e., that the following diagram commutes:

Snowmass White Paper: Moonshine

“…In the papers [140,12] Ryba and Borcherds observed some very interesting behavior in the p-modular behavior of the module V ♮ of the Monster VOA. One of their main conjectures was recently proved by Carnahan [16]. A nice overview of these developments as well as observations and speculations regarding connections between modular moonshine and generalized moonshine can be found in [19].…”

A Composite Order Generalization of Modular Moonshine

“…Ryba proposed the modular moonshine conjecture. The existence of a self-dual integral form of the monster vertex algebra was not proved when he suggested it, but Carnahan proved it [4].…”

“…The original modular moonshine conjectures of Ryba asserted the existence of vertex algebras g V over finite fields with finite group actions, such that the graded Brauer characters are genus zero modular functions. He also suggested a construction of the vertex algebra g V (see Section 3) for an element g ∈ M of prime order p and of conjugacy class pA that is the largest conjugacy class of order p, in terms of a self-dual integral form of the monster vertex algebra whose existence was proved later by Carnahan [4]. Borcherds and Ryba reinterpreted the modular moonshine conjectures in terms of Tate cohomology with coefficients in an integral form V of the monster vertex algebra.…”

On the tensor structure of modules for compact orbifold vertex operator algebras

“…In addition to the above associativity result for Y , we will need the following skew-symmetry result, which can be found in [Ca,Lemma 1.2.2] (see also the proof of [GL, Proposition 2.6]):…”