Cone Vertex Algebras, Mock Theta Functions, and Umbral Moonshine Modules

TL;DR

This paper constructs vertex algebra modules from lattice cones to express indefinite theta functions, linking them to mock theta functions and providing modules for specific cases of umbral moonshine.

Contribution

It introduces a new family of indefinite theta functions related to vertex algebra trace functions, connecting mock theta functions and umbral moonshine modules.

Findings

Expressed McKay-Thompson series in terms of vertex algebra trace functions

Constructed modules for umbral moonshine at lambency 8, 12, 16

Linked indefinite theta functions with mock theta functions

Abstract

We describe a family of indefinite theta functions of signature $(1,1)$ that can be expressed in terms of trace functions of vertex algebras built from cones in lattices. The family of indefinite theta functions considered has interesting connections with mock theta functions and Appell-Lerch sums. We use these relations to write the McKay-Thompson series of umbral moonshine at lambency $\ell=8,12,16$ in terms of trace functions of vertex algebras modules, and thereby provide the modules for these instances of umbral moonshine.