Rademacher sums, moonshine and gravity

Abstract

In 1939 Rademacher derived a conditionally convergent series expression for the elliptic modular invariant, and used this expression-the first Rademacher sum-to verify its modular invariance. By generalizing Rademacher's approach we construct bases for the spaces of automorphic integrals of arbitrary even integer weight, for groups 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 within this class the monstrous groups can be characterized just in terms of the behavior of their Rademacher sums. In particular, the genus zero property of monstrous moonshine is encoded naturally in the properties of Rademacher sums. Just as the elliptic 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 distinguished family of chiral three-dimensional (3D) quantum gravities, and relating monstrous Lie algebras and their Verma modules to the second quantization of this family of chiral 3D quantum gravities.