Topological stars and scalar wave equation: Exact resummation of the renormalized angular momentum in the eikonal limit

We show that for a topological star the renormalized angular momentum parameter, $\ensuremath{\nu}$, appearing in the Mano-Suzuki-Takasugi-type or in the quantum-Seiberg-Witten-type approaches of the perturbation equations, (1) has a direct link with the geodesic radial action computed along the null orbits of the background and (2) admits an exact resummation in terms of hypergeometric functions, generalizing previous results valid in the Schwarzschild case; see [M. M. Ivanov, Y. Z. Li, J. Parra-Martinez, and Z. Zhou, Resummation of universal tails in gravitational waveforms, arXiv:2504.07862.].