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 \Phi. We introduce the generating function Z(\mathbf {s}) of a packing, an exponential series in four variables with an Apollonian symmetry group, which is a symmetric function for \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. [ABSTRACT FROM AUTHOR]