One loop effective actions in Kerr-(A)dS Black Holes

We compute new exact analytic expressions for one-loop scalar effective actions in Kerr (A)dS black hole (BH) backgrounds in four and five dimensions. These are computed by the connection coefficients of the Heun equation via a generalization of the Gelfand-Yaglom formalism to second-order linear ODEs with regular singularities. The expressions we find are in terms of Nekrasov-Shatashvili special functions, making explicit the analytic properties of the one-loop effective actions with respect to the gravitational parameters and the precise contributions of the quasi-normal modes. The latter arise via an associated integrable system. In particular, we prove asymptotic formulae for large angular momenta in terms of hypergeometric functions and give a precise mathematical meaning to Rindler-like region contributions. Moreover we identify the leading terms in the large distance expansion as the point particle approximation of the BH and their finite size corrections as encoding the BH tidal response. We also discuss the exact properties of the thermal version of the BH effective actions by providing a proof of the DHS formula and explicitly computing it for new relevant cases. Although we focus on the real scalar field in dS-Kerr and (A)dS-Schwarzschild in four and five dimensions, similar formulae can be given for higher spin matter and radiation fields in more general gravitational backgrounds.