Love numbers of black p-branes: fine tuning, Love symmetries, and their geometrization

Abstract

We compute scalar static response coefficients (Love numbers) of non-dilatonic black p-brane solutions in higher dimensional supergravity. This calculation reveals a fine-tuning behavior similar to that of higher dimensional black holes, which we explain by “hidden” near-zone Love symmetries. In general, these symmetries act on equations for perturbations but they are not background isometries. The Love symmetry of charged p = 0 branes is described by the usual SL(2, ℝ) algebra. For p = 1 the Love symmetry has an algebraic structure SL(2, ℝ) × SL(2, ℝ). The p = 0, 1 Love symmetries reduce to isometries of the near-horizon Schwarzschild-AdSp+2 metric in the near-extremal finite temperature limit. They further reduce to the AdSp+2 isometries in the extremal zero-temperature limit. We call this process geometrization. In contrast, for the p > 1 cases, the Love symmetry is always an SL(2, ℝ), and there is no limit in which it becomes geometric. We interpret geometrization and its absence as a consequence of the local equivalence between the Schwarzschild-AdSp+2 and pure AdSp+2 spaces for p = 0, 1, which does not hold for p > 1. We also show that the static Love numbers of extremal p-branes are always zero regardless of spacetime dimensionality, which contrasts starkly with the non-extremal case. Overall, our results suggest that the Love symmetry is hidden by nature, and it can acquire a geometric meaning only if the background has an AdS2 or AdS3 limit.