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.

Generalized Mathieu Moonshine

“…The key idea was that the multiplicative lift of the modular J-function could be interpreted as the denominator formula for a specific GKM, now commonly called the Monster Lie algebra, whose root system carries an action of the Monster group M. The generalized moonshine ideas of Norton [14] suggest that similar GKMs should exist for each class [g] ∈ M [20,21,54]. In the context of Mathieu Moonshine the role of generalized Kac-Moody algebras has so far not been understood.…”

Fricke Lie algebras and the genus zero property in Moonshine

“…The first step was done in the landmark paper [van Ekeren, et al, 2015], steps 2,3, and 5 were done in [Carnahan ≥2017], step 4 was done in [Carnahan 2012], and step 6 was done in [Carnahan 2010]. We shall describe the last step in more detail.…”

“…in the sense of [Dong-Lepowsky 1993]) formed from a sum of irreducible twisted V -modules, constructed in [van Ekeren, et al, 2015] with minor adjustments in [Carnahan ≥2017] to account for anomalous weights. Our functor is the following composite: …”

Mixed Rademacher and BPS black holes

“…Although the constant term c 0 might need more care for the proof of convergence, it can be recovered by analyzing the behavior of the Kloosterman sum at n = 0 [3]. The method was later developed for modular forms of various weights, modular groups and multiplier systems in several works like [5,[20][21][22][23][24][59][60][61][62], to quote just a few. 10 Progress in the context of harmonic Maass forms has also been achieved in [64][65][66].…”

“…Adding a certain non-holomorphic integral of the shadow to the form restores modularity at the expense of holomorphicity. When the shadow is a cusp form, the Rademacher series can be applied to recover the Fourier coefficients of mock Jacobi forms [5,[20][21][22][23][24]. However, the mock Jacobi forms arising in the counting problem of 1/4-BPS states in Type IIB string theory on K3 × T 2 have mixed mock components, and their shadows are not cusp forms.…”