Rademacher sums, Moonshine and Gravity

In 1939 Rademacher demonstrated how to express Klein’s modular invariant as a sum over elements of the modular group. In this article we generalize Rademacher’s approach so as to construct bases for the spaces of automorphic integrals of arbitrary even integer weight, for an arbitrary group commensurable with the modular group. Our methods provide explicit expressions for the Fourier expansions of the Rademacher sums we construct at arbitrary cusps, and illuminate various aspects of the structure of the spaces of automorphic integrals, including the actions of Hecke operators. We give a moduli interpretation for a class of groups commensurable with the modular group which includes all those that are associated to the Monster via Monstrous Moonshine. We show that the behavior of the Rademacher sums attached to these groups allows us to characterize exactly those groups that correspond to elements of the Monster. In particular, the genus zero property of the groups of Monstrous Moonshine is encoded naturally in the properties of the corresponding Rademacher sums. Just as Klein’s modular invariant gives the graded dimension of the Moonshine Module, the exponential generating function of the Rademacher sums associated to the modular group furnishes the bi-graded dimension of the Verma module for the Monster Lie algebra. This result generalizes naturally to all the groups of Monstrous Moonshine, and recovers a certain family of Monstrous Lie algebras recently introduced by Carnahan. Our constructions suggest conjectures relating Monstrous Moonshine to a family of distinguished chiral three dimensional quantum gravities, and relating Monstrous Lie algebras and their Verma modules to the second quantization of this family of chiral three dimensional quantum gravities. Harvard University, Department of Mathematics, One Oxford Street, Cambridge, MA 02138, USA. Yale University, Department of Mathematics, 10 Hillhouse Avenue, New Haven, CT 06520, USA. The research of I.F. was supported by NSF grant DMS-0457444.