Open Access

Umbral moonshine and the Niemeier lattices

Research in the Mathematical Sciences20141:3

DOI: 10.1186/2197-9847-1-3

Received: 21 January 2014

Accepted: 3 February 2014

Published: 17 June 2014


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 and 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.

AMS subject classification

11F22; 11F37; 11F46; 11F50; 20C34; 20C35


Mock modular form Niemeier lattice Umbral moonshine

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Authors’ Affiliations

Korteweg-de Vries Institute of Mathematics, University of Amsterdam
Department of Mathematics, Applied Mathematics and Statistics, Case Western Reserve University
Enrico Fermi Institute and Department of Physics, University of Chicago


© Cheng et al.; licensee Springer. 2014

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