Turning rotating D-branes and BHs inside out their photon-halo

We extend our investigation on Couch-Torrence conformal inversions of BHs and D-branes in various directions. We analyse asymptotically flat rotating charged BHs in $D=4$, in particular extremal rotating BHs in STU supergavity, and find invariance for special choices of the charges. Due to the dependence of the critical radii on the impact parameter(s), the relation between the scattering angle for geodesics outside the photon-halo and the in-spiralling angle for geodesics inside the photon-halo is modified by the inclusion of a boundary term. We also consider rotating BHs in $D=5$ and rotating D3-branes and find invariance under generalised Couch-Torrence inversions for special choices of the angular momenta. Alas we don't find any similar symmetry for smooth horizonless geometries. Moreover, relying on the surprising connection between classical BH perturbation theory and quantum Seiberg-Witten curves for ${\cal N}=2$ SYM theories, we study scalar wave equations in these backgrounds and identify the near super-radiant modes produced in near-extremal BH mergers. Finally, we study scalar fluctuations around Kerr-Newman BHs in AdS$_4$ and find stringent conditions for generalised Couch-Torrence symmetry that are relaxed in the extremal case or by allowing a rescaling of the wave function.