Lattice-integrality of certain group-invariant integral forms in vertex operator algebras

TL;DR

This paper investigates conditions under which certain group-invariant integral forms in vertex operator algebras are lattice-integral, demonstrating the existence of such forms in notable cases like the Moonshine VOA.

Contribution

It establishes criteria for lattice-integrality of integral forms in VOAs, including invariant forms under finite groups, and constructs a lattice-integral Monster-invariant form.

Findings

Lattice-integrality can be achieved under specific hypotheses.

Existence of a lattice-integral Monster-invariant form in the Moonshine VOA.

Provides a framework for understanding integral forms in VOAs.

Abstract

Certain vertex operator algebras have integral forms (integral spans of bases which are closed under the countable set of products). It is unclear when they (or integral multiples of them) are integral as lattices under the natural bilinear form on the VOA. We show that lattice-integrality may be arranged under some hypotheses, including cases of integral forms invariant by finite groups. In particular, there exists a lattice-integral Monster-invariant integral form in the Moonshine VOA.