Rootless pairs of $EE_8$-lattices

TL;DR

This paper classifies pairs of $EE_8$-lattices with specific properties, revealing their embeddings in the Leech lattice and connections to vertex operator algebras and moonshine phenomena.

Contribution

It provides a new classification of rootless, integral $EE_8$-lattice pairs with dihedral symmetry of order at most 12, linking lattice theory to algebraic and moonshine structures.

Findings

Most pairs embed in the Leech lattice.

Classification of pairs with dihedral group order ≤ 12.

Connections to vertex operator algebra theory.

Abstract

We describe a classification of pairs $M, N$ of lattices isometric to $EE_8:=\sqrt 2 E_8$ such that the lattice $M + N$ is integral and rootless and such that the dihedral group associated to them has order at most 12. It turns out that most of these pairs may be embedded in the Leech lattice. Complete proofs will appear in another article. This theory of integral lattices has connections to vertex operator algebra theory and moonshine.