Apollonian Packings and Kac-Moody Root Systems

TL;DR

This paper explores the deep mathematical connection between Apollonian circle packings and Kac-Moody root systems, introducing a generating function linked to automorphic forms and revealing geometric structures within the Tits cone.

Contribution

It establishes a novel relationship between Apollonian packings and Kac-Moody root systems, expressing the packing generating function via automorphic forms and analyzing its convergence domain.

Findings

Expressed the generating function in terms of Jacobi theta functions and Siegel modular forms.

Identified the Tits cone as the domain of convergence, reflecting the packing's geometric complexity.

Linked the symmetry group of packings to Weyl-Kac characters and automorphic forms.

Abstract

We study Apollonian circle packings in relation to a certain rank 4 indefinite Kac-Moody root system $\Phi$. We introduce the generating function $Z(\mathbf{s})$ of a packing, an exponential series in four variables with an Apollonian symmetry group, which relates to Weyl-Kac characters of $\Phi$. By exploiting the presence of affine and Lorentzian hyperbolic root subsystems of $\Phi$, with automorphic Weyl denominators, we express $Z(\mathbf{s})$ in terms of Jacobi theta functions and the Siegel modular form $\Delta_5$. We also show that the domain of convergence of $Z(\mathbf{s})$ is the Tits cone of $\Phi$, and discover that this domain inherits the intricate geometric structure of Apollonian packings.