Apollonian packings and Kac-Moody root systems

Abstract

We study Apollonian circle packings using the properties of a certain rank 4 indefinite Kac-Moody root system Φ \Phi . We introduce the generating function Z ( s ) Z(\mathbf {s}) of a packing, an exponential series in four variables with an Apollonian symmetry group, which is a symmetric function for Φ \Phi . By exploiting the presence of affine and Lorentzian hyperbolic root subsystems of Φ \Phi , with automorphic Weyl denominators, we express Z ( s ) Z(\mathbf {s}) in terms of Jacobi theta functions and the Siegel modular form Δ 5 \Delta _5 . We also show that the domain of convergence of Z ( s ) Z(\mathbf {s}) is the Tits cone of Φ \Phi , and discover that this domain inherits the intricate geometric structure of Apollonian packings.