The intermediate vertex subalgebras of the lattice vertex operator algebras

TL;DR

This paper introduces intermediate vertex subalgebras of lattice vertex operator algebras, generalizing principal subspaces, and explores their bases, graded dimensions, and modular properties of their modules.

Contribution

It defines and analyzes intermediate vertex subalgebras, providing bases, graded dimensions, and demonstrating modular differential equations for their modules.

Findings

Bases and graded dimensions of intermediate subalgebras are established.

Characters of modules satisfy modular differential equations.

Results relate to the structure between E7 and E8 lattice VOAs.

Abstract

A notion of intermediate vertex subalgebras of lattice vertex operator algebras is introduced, as a generalization of the notion of principal subspaces. Bases and the graded dimensions of such subalgebras are given.As an application, it is shown that the characters of some modules of an intermediate vertex subalgebra between $E_7$ and $E_8$ lattice vertex operator algebras satisfy some modular differential equations.This result is an analogue of the result concerning the "hole" of the Deligne dimension formulas and the intermediate Lie algebra between the simple Lie algebras $E_7$ and $E_8$.