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 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.
Highlights
In this paper, we relate umbral moonshine to the Niemeier lattices
Taking the 23 Niemeier lattices as the starting point, in the ‘Umbral groups’ section, we identify a finite group GX - the umbral group - for each Niemeier root system X
On the other hand, using the ADE classification discussed in the ‘The Eichler-Zagier operators and an ADE classification’ section and Theorem 2, we identify a distinguished vector-valued mock modular form HX - the umbral form - for each Niemeier root system X
Summary
This relation associates one case of umbral moonshine to each of the 23 Niemeier lattices and in particular constitutes an extension of our previous work [1], incorporating 17 new instances This prescription displays interesting connections to certain interesting genus zero groups (subgroups < SL2(R) that define a genus zero quotient of the upper-half plane) and extended Dynkin diagrams via McKay’s correspondence. The dimensions of irreducible representations of M24 was pointed out in [11] This observation was later extended into a Mathieu moonshine conjecture in [12,13,14,15] by providing the corresponding twisted characters and the mock modular forms Hg(2), and was related in a more general context to the K3-compactification of superstring theory in [12]. We provide a prescription that attaches to each of the 23 Niemeier lattices a distinguished vector-valued modular form - the umbral mock modular form HX which conjecturally encodes the dimensions of the homogeneous subspaces of the
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