Spectrum of BPS black holes in $AdS_3 imes S^3 imes S^3 imes S^1$

We uncover novel features in the spectrum of BPS and near-BPS states in asymptotically $AdS_3 \times S^3 \times S^3 \times S^1$ 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 $S^3 \times S^3$ 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 $AdS_2 \times S^2 \times S^2$ near-horizon geometries constructed in M-theory compactifications.

💡 Research Summary

This paper investigates the spectrum of supersymmetric (BPS) and near‑BPS black holes in the Type IIB background AdS₃ × S³₊ × S³₋ × S¹, a setting characterized by the large N=4 superconformal algebra Aγ and its associated supergroup D(2,1|α). The geometry is sourced by Q₁ D1‑branes intersecting Q₊ and Q₋ D5‑branes, giving rise to two three‑spheres with radii ℓ₊ and ℓ₋. The ratio α = ℓ₋/ℓ₊ = Q₋/Q₊ controls the relative size of the spheres and enters the supergroup as a continuous parameter (though quantized in string theory).

Classical black holes.
The authors first review the classical BPS black hole solutions: they carry equal energy and AdS₃ angular momentum (E = J) and carry SU(2)₊ × SU(2)₋ spins (j₊, j₋). At the classical level the BPS condition reduces to a linear relation j₋ = α j₊, analogous to the familiar condition for extremal Reissner‑Nordström black holes.

Large N=4 Schwarzian theory.
The core of the analysis is the exact solution of the large N=4 Schwarzian effective theory, previously derived in