# Super Vertex Algebras, Meromorphic Jacobi Forms and Umbral Moonshine

## TL;DR

This paper constructs super vertex operator algebras for specific umbral moonshine cases, linking meromorphic Jacobi forms with algebraic structures and verifying their properties through explicit constructions.

## Contribution

It provides explicit constructions of super vertex operator algebras for certain umbral moonshine cases, solving parts of the meromorphic module problem.

## Key findings

- Constructed super vertex operator algebras for Coxeter numbers 7 and 13.
- Verified trace functions recover specified meromorphic Jacobi forms.
- Partial solutions for Coxeter numbers 4 and 5 using subgroup structures.

## Abstract

The vector-valued mock modular forms of umbral moonshine may be repackaged into meromorphic Jacobi forms of weight one. In this work we constructively solve two cases of the meromorphic module problem for umbral moonshine. Specifically, for the type A Niemeier root systems with Coxeter numbers seven and thirteen, we construct corresponding bigraded super vertex operator algebras, equip them with actions of the corresponding umbral groups, and verify that the resulting trace functions on canonically twisted modules recover the meromorphic Jacobi forms that are specified by umbral moonshine. We also obtain partial solutions to the meromorphic module problem for the type A Niemeier root systems with Coxeter numbers four and five, by constructing super vertex operator algebras that recover the meromorphic Jacobi forms attached to maximal subgroups of the corresponding umbral groups.