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

## TL;DR

This paper explores connections between vertex algebras, mock theta functions, and umbral moonshine, providing new modules for specific cases.

## Contribution

The paper introduces new modules for umbral moonshine at lambencies 8, 12, and 16 using vertex algebra trace functions.

## Key findings

- Indefinite theta functions of signature (1,1) are linked to vertex algebra trace functions.
- McKay–Thompson series for umbral moonshine at lambencies 8, 12, and 16 are expressed using these trace functions.
- The results establish new connections between mock theta functions and Appell–Lerch sums.

## 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 \documentclass[12pt]{minimal}
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\begin{document}$$\ell =8,12,16$$\end{document}ℓ=8,12,16 in terms of trace functions of vertex algebras modules, and thereby provide the modules for these instances of umbral moonshine.