Apollonian packings and Kac-Moody root systems

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