Rademacher sums, moonshine and gravity

TL;DR

This paper generalizes Rademacher sums to construct bases for automorphic integrals, linking moonshine, monstrous Lie algebras, and quantum gravity, and proposing conjectures on their interrelations.

Contribution

It introduces a new framework using Rademacher sums to analyze automorphic integrals and connects moonshine phenomena with quantum gravity and Lie algebra structures.

Findings

Characterization of monstrous moonshine groups via Rademacher sums

Connections established between Rademacher sums and monstrous Lie algebras

Conjectures relating moonshine, quantum gravity, and algebraic structures

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 all groups commensurable with the modular group. We use these Rademacher sums to illuminate various aspects of the structure of the spaces of automorphic integrals, including the actions of Hecke operators. We obtain a new characterization of the discrete groups of monstrous moonshine in terms of Rademacher sums, and we develop connections between Rademacher sums and a family of monstrous Lie algebras recently introduced by Carnahan. Our constructions suggest conjectures relating monstrous moonshine to a distinguished family of 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.