John F. R. Duncan

Umbral Moonshine and the Niemeier Lattices

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 classification11F22; 11F37; 11F46; 11F50; 20C34; 20C35

Super-moonshine for Conway's largest sporadic group

We study a self-dual N=1 super vertex operator algebra and prove that the full symmetry group is Conways largest sporadic simple group. We verify a uniqueness result which is analogous to that conjectured to characterize the Moonshine vertex operator algebra. The action of the automorphism group is sufficiently transparent that one can derive explicit expressions for all the McKay-Thompson series. A corollary of the construction is that the perfect double cover of the Conway group may be characterized as a point-stabilizer in a spin module for the Spin group associated to a 24 dimensional Euclidean space.

Mathieu moonshine and \mathcal{N}=2 string compactifications

A bstractThere is a ‘Mathieu moonshine’ relating the elliptic genus of K3 to the sporadic group M24. Here, we give evidence that this moonshine extends to part of the web of dualities connecting heterotic strings compactified on K3 × T2 to type IIA strings compactified on Calabi-Yau threefolds. We demonstrate that dimensions of M24 representations govern the new supersymmetric index of the heterotic compactifications, and appear in the Gromov-Witten invariants of the dual Calabi-Yau threefolds, which are elliptic fibrations over the Hirzebruch surfaces

Mock Modular Mathieu Moonshine Modules

{{\mathbb{F}}_n}

Rademacher Sums and Rademacher Series

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Derived equivalences of K3 surfaces and twined elliptic genera

We construct super vertex operator algebras which lead to modules for moonshine relations connecting the four smaller sporadic simple Mathieu groups with distinguished mock modular forms. Starting with an orbifold of a free fermion theory, any subgroup of

Weight one Jacobi forms and umbral moonshine

\textit{Co}_0

Attractive Strings and Five-Branes, Skew-Holomorphic Jacobi Forms and Moonshine

Co0 that fixes a 3-dimensional subspace of its unique non-trivial 24-dimensional representation commutes with a certain