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

TL;DR

The paper analyzes the spectrum of BPS and near-BPS black holes in asymptotically AdS$_3\times S^3\times S^3\times S^1$ with large $\mathcal{N}=4$ symmetry, using the large $\mathcal{N}=4$ Schwarzian to resolve near-extremal dynamics.A key result is that the Ramond-sector BPS states exhibit discrete jumps in their $(j_+,j_-)$ spins as the flux ratio $\alpha=Q_5^-/Q_5^+$ is varied, governed by a nonlinear BPS bound and a precise degeneracy formula $N_{j_+j_-u}$ with ground-state energy $E_{\rm BPS}$.The authors also construct non-extremal BPS black holes whose gravitational path integral yields a temperature-dependent index (elliptic genus), and they verify these gravity results against the Schwarzian theory, including explicit Killing spinor analyses.These results extend to six-dimensional settings with AdS$_2$ throats and suggest a broader applicability of the Schwarzian formalism to BPS spectra in theories with large $\mathcal{N}=4$ symmetry, with implications for proposed dual CFT descriptions.

Abstract

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.

Figures (2)

• Figure 1: The BPS spectrum to the leading order in the semiclassical regime for a large ${\cal N}=4$ theory is displayed. Each gray background point corresponds to a BPS multiplet labeled by spins $(j_+,j_-)\in \frac{1}{2}\mathbb{Z}_{\ge 0}\times \frac{1}{2}\mathbb{Z}_{\ge 0}$. (a) In the NS sector, the BPS multiplets are restricted to have $j_+=j_-$. (b) In the R sector, the allowed spins of BPS multiplets lie within the set defined in \ref{['eqn:BPSset']}. Colored dots represent the BPS multiplets for different values of $Q_5^-$ and $Q_5^+$, with increasing ratio from bottom to top: $Q_5^- / Q_5^+=0.2,\,0.6,\,1,\,2,\,8$, respectively; the straight lines, meanwhile, represent the expectations from classical gravity: $j_-\approx Q_5^- / Q_5^+ \,j_+$.
• Figure 2: Left: Case with $R_- \ll R_+$, or equivalently $Q_5^- \ll Q_5^+$. To leading order in $G_N$, BPS states rotate mostly on the larger sphere $S_+^3$. Right: Case with comparable $R_+ \sim R_-$ or equivalently $Q_5^- \sim Q_5^+$. To leading order, BPS states distribute their angular momentum mostly equally.