Love numbers and Love symmetries for $p$-form and gravitational perturbations of higher-dimensional spherically symmetric black holes

The static Love numbers of four-dimensional asymptotically flat, isolated, general-relativistic black holes are known to be identically vanishing. The Love symmetry proposal suggests that such vanishings are addressed by selection rules following from the emergence of an enhanced $\text{SL}(2,\mathbb{R})$ ("Love") symmetry in the near-zone region; more specifically, it is the fact that the black hole perturbations belong to a highest-weight representation of this near-zone $\text{SL}(2,\mathbb{R})$ symmetry, rather than the existence of the Love symmetry itself, that outputs the vanishings of the corresponding Love numbers. In higher spacetime dimensions, some towers of magic zeroes with regards to the black hole response problem have also been reported for scalar, electromagnetic and gravitational perturbations of the Schwarzschild-Tangherlini black hole. Here, we extend these results by supplementing with $p$-form perturbations of the Schwarzschild-Tangherlini black hole. We furthermore analytically extract the static Love numbers and the leading order dissipation numbers associated with spin-$0$ scalar and spin-$2$ tensor-type tidal perturbations of the higher-dimensional Reissner-Nordstr\"om black hole. We find that Love symmetries exist and that the vanishings of the static Love numbers are captured by representation theory arguments even for these higher spin perturbations of the higher-dimensional spherically symmetric black holes of General Relativity. Interestingly, these near-zone $\text{SL}(2,\mathbb{R})$ structures acquire extensions to Witt algebras. Our setup allows to also study the $p$-form response problem of a static spherically symmetric black hole in a generic theory of gravity. We perform explicit computations for some black holes in the presence of string-theoretic corrections and investigate under what geometric conditions Love symmetries emerge in the near-zone.