Ladder Symmetries of Higher Dimensional Black Holes

TL;DR

The paper extends the 4D ladder symmetry explanation for static black hole Love numbers to higher dimensions by introducing the effective level \hat{\\ell}=\\ell/(D-3) and constructing first-order ladder operators D^{\\pm}_{\\hat{\\ell}} that connect solutions at neighboring multipole orders. A ground-state condition with κ_{\\hat{\\ell}}=0 yields a decoupled, first-order equation at the horizon that, when combined with horizon regularity and purely growing behavior at infinity, forces Love numbers to vanish for all ladder-connected levels; this recovers the known 4D results and explains selective vanishing patterns in higher dimensions. Applying the framework to Schwarzschild--Tangherlini in spins 0, 1, 2 and to 5D Myers--Perry (spin-0 perturbations) reveals precise, dimension- and spin-dependent sets of hat{\\ell} values for which Love numbers vanish, and shows that in 5D Myers--Perry vanishing Love numbers require tuning of the black hole spins or related quantum numbers. The work also connects ladder constructions to horizontal ladder symmetry and Wronskian methods, suggesting a deeper geometric origin and outlining directions for future exploration in worldline EFT and higher-dimensional compact objects.

Abstract

We compute the ladder operators for static tidal perturbations to higher-dimensional black holes. These operators map between solutions of the relevant equation of motion at different multipole orders. We focus on spin 0, 1 and 2 perturbations to the Schwarzschild-Tangherlini black hole and on spin 0 perturbations to the 5D Myers-Perry black hole. The ladder structure, used in conjunction with the existence of special ground state solutions, explains why the Love numbers of these higher-dimensional black holes vanish for specific combinations of the multipole moment and number of spacetime dimensions. This generalizes previous work on a ladder symmetry explanation for the vanishing of 4D black hole static Love numbers to higher dimensions.