The Umbral Moonshine Module for the Unique Unimodular Niemeier Root System

TL;DR

This paper constructs the umbral moonshine module for the unique Niemeier lattice with its root sublattice using super vertex operator algebra modules, providing explicit formulas for associated mock modular forms.

Contribution

It introduces a novel construction of the umbral moonshine module for a specific Niemeier lattice, linking super vertex operator algebras to mock modular forms.

Findings

Explicit expressions for vector-valued mock modular forms are provided.

Four of Ramanujan's fifth order mock theta functions appear as components.

The construction clarifies the connection between lattice automorphisms and moonshine phenomena.

Abstract

We use canonically-twisted modules for a certain super vertex operator algebra to construct the umbral moonshine module for the unique Niemeier lattice that coincides with its root sublattice. In particular, we give explicit expressions for the vector-valued mock modular forms attached to automorphisms of this lattice by umbral moonshine. We also characterize the vector-valued mock modular forms arising, in which four of Ramanujan's fifth order mock theta functions appear as components.