Umbral moonshine

Abstract

We describe surprising relationships between automorphic forms of various kinds, imaginary quadratic number fields and a certain system of six finite groups that are parameterised naturally by the divisors of twelve. The Mathieu group correspondence recently discovered by Eguchi-Ooguri-Tachikawa is recovered as a special case. We introduce a notion of extremal Jacobi form and prove that it characterises the Jacobi forms arising by establishing a connection to critical values of Dirichlet series attached to modular forms of weight two. These extremal Jacobi forms are closely related to certain vector-valued mock modular forms studied recently by Dabholkar-Murthy-Zagier in connection with the physics of quantum black holes in string theory. In a manner similar to monstrous moonshine the automorphic forms we identify constitute evidence for the existence of infinite-dimensional graded modules for the six groups in our system. We formulate an umbral moonshine conjecture that is in direct analogy with the monstrous moonshine conjecture of Conway-Norton. Curiously, we find a number of Ramanujan's mock theta functions appearing as McKay-Thompson series. A new feature not apparent in the monstrous case is a property which allows us to predict the fields of definition of certain homogeneous submodules for the groups involved. For four of the groups in our system we find analogues of both the classical McKay correspondence and McKay's monstrous Dynkin diagram observation manifesting simultaneously and compatibly. *

Calabi-Yau manifolds and sporadic groups

“…This Jacobi form for χ CY 6 = 8 and the corresponding expansion in terms of N = 4 characters appeared in [21] (see eq. (2.49)), and it is related to 2.AGL 3 (2) via the umbral moonshine conjecture (proven in [41]).…”

“…However, we point out that a particular extremal Jacobi form that plays a role in umbral moonshine [21] arises as elliptic genus of the product of two CY 3-folds. We again study this further by looking at twined functions.…”

“…The symmetry groups of the K3 manifold at different points in moduli space have found a beautiful string theory explanation in [20], where the authors study type IIA string theory on K3×T 3 . The resulting three dimensional theory does not only have M 24 symmetry at one point in moduli space but it actually has all umbral symmetry groups [21,22] as symmetry groups at certain points in its moduli space. These symmetries are partially broken upon taking the decompactification limit for the T 3 .…”

“…Nevertheless we can check under what conditions the extremal Jacobi forms that appear in umbral moonshine [21,22] can arise as elliptic genera of Calabi-Yau manifolds or products thereof.…”

“…In particular the product of any two CY 3-folds has χ 0 = χ 1 = 0. 7 The other interesting Jacobi forms in [21] have weight zero and index 4, 6 and 12. Let us look first at the case of d = 8 that leads to a Jacobi form with index 4.…”