Umbral Moonshine
Miranda C. N. Cheng, John F. R. Duncan, Jeffrey A. Harvey

TL;DR
This paper explores deep connections between automorphic forms, finite groups, and mock modular forms, proposing an umbral moonshine conjecture that generalizes monstrous moonshine and links to quantum black hole physics.
Contribution
It introduces the concept of extremal Jacobi forms, establishes their connection to Dirichlet series, and formulates the umbral moonshine conjecture relating automorphic forms to finite groups.
Findings
Identification of automorphic forms linked to six finite groups
Discovery of Ramanujan's mock theta functions as McKay-Thompson series
Analogues of McKay correspondence for four groups
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…
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Taxonomy
TopicsAdvanced Algebra and Geometry · Advanced Mathematical Identities · Algebraic Geometry and Number Theory
