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Apollonian Packings and Kac-Moody Root Systems

We study Apollonian circle packings in relation to a certain rank 4 indefinite Kac-Moody root system $Φ$. 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 $Φ$. By exploiting the presence of affine and Lorentzian hyperbolic root subsystems of $Φ$, with automorphic Weyl denominators, we express $Z(\mathbf{s})$ in terms of Jacobi theta functions and the Siegel modular form $Δ_5$. We also show that the domain of convergence of $Z(\mathbf{s})$ is the Tits cone of $Φ$, and discover that this domain inherits the intricate geometric structure of Apollonian packings.