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Umbral moonshine and the Niemeier lattices

Research in the Mathematical Sciences20141:3

  • Received: 21 January 2014
  • Accepted: 3 February 2014
  • Published:


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, the Netherlands
Department of Mathematics, Applied Mathematics and Statistics, Case Western Reserve University, Cleveland, OH 44106, USA
Enrico Fermi Institute and Department of Physics, University of Chicago, Chicago, IL 60637, USA


© Cheng et al.; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.