Umbral Moonshine and the Niemeier Lattices

TL;DR

This paper establishes a deep connection between umbral moonshine and Niemeier lattices, associating finite groups and mock modular forms to each lattice, extending previous work and including Mathieu moonshine as a special case.

Contribution

It constructs a framework linking Niemeier lattices, finite groups, and mock modular forms, proposing the umbral moonshine conjecture and extending prior moonshine phenomena.

Findings

Identification of mock modular forms associated with Niemeier lattices

Extension of Mathieu moonshine to all Niemeier lattices

Discovery of a correspondence between genus zero groups and Niemeier lattices

Abstract

In this paper we relate umbral moonshine to the Niemeier lattices: the 23 even unimodular positive-definite lattices of rank 24 with non-trivial root systems. To each Niemeier lattice we attach a finite group by considering a naturally defined quotient of the lattice automorphism group, and for each conjugacy class of each of these groups we identify a vector-valued mock modular form whose components coincide with mock theta functions of Ramanujan in many cases. This leads to the umbral moonshine conjecture, stating that an infinite-dimensional module is assigned to each of the Niemeier lattices in such a way that the associated graded trace functions are mock modular forms of a distinguished nature. These constructions and conjectures extend those of our earlier paper, and in particular include the Mathieu moonshine observed by Eguchi-Ooguri-Tachikawa as a special case. Our analysis also highlights a correspondence between genus zero groups and Niemeier lattices. As a part of this relation we recognise the Coxeter numbers of Niemeier root systems with a type A component as exactly those levels for which the corresponding classical modular curve has genus zero.