We uncover novel features in the spectrum of BPS and near-BPS states in asymptotically AdS3 × S3 × S3 × S1 spacetimes. This follows from a careful analysis of semiclassical and quantum black holes in this theory, which have peculiarities due to the nonlinear large \( \mathcal{N} \) = 4 superconformal symmetry. Notably, we find that the S3 × S3 angular momentum spectrum of BPS states in the Ramond sector exhibits discrete jumps as a function of the ratio between the radii of the two three-spheres. This phenomenon is a quantum gravity effect for which no microscopic derivation is currently known. In addition, we construct a family of non-extremal supersymmetric black holes that contribute to a supersymmetric index yet possess a temperature-dependent free energy. Analogous results apply to six-dimensional black holes with AdS2 × S2 × S2 near-horizon geometries constructed in M-theory compactifications.
