  Using conformal field theory and localization tecniques we study the
propagation of waves in gravity backgrounds described by Schr\"odinger like
equations with Fuchsian singularities. Exact formulae for the connection
matrices relating the asymptotic behaviour of the wave functions near the
singularities are obtained in terms of braiding and fusion rules of the CFT.
The results are applied to the study of quasi normal modes, absorption cross
sections, amplification factors, echoes and tidal responses of black holes (BH)
and exotic compact objects (ECO) in four and five dimensions. In particular, we
propose a definition of dynamical Love numbers in gravity.
