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

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.