Résume | In this talk I will discuss how a new type of moonshine – the umbral moonshine – arises from the 24 even unimodular lattices in 24 dimensions. In more details, for each of the 24 Niemeier lattices we pose an umbral moonshine conjecture identifying a specific set of mock modular forms and the graded characters of a certain natural module of a finite group arising from the automorphism of the Niemeier lattice. The construction of the set of mock modular forms relies on an ADE classification of mock modular forms of a certain type, analogous to the ADE classification of modular invariant combinations of characters of $A_1^{(1)}$ affine Kac-Moody algebra by Cappelli–Itzykson–Zuber, as well as the relation between mock modular forms and meromorphic Jacobi forms studied by S. Zwegers and Dabholkar–Murthy–Zagier. If time permits I will discuss certain mysterious group theoretic properties of this moonshine, the possible relations to certain generalised Kac-Moody algebras, or/and Gromov–Witten theory of certain Calabi–Yau three-folds. This talk is mainly based on joint work with J Duncan and J Harvey. |