The statistical mechanics of near-BPS black holes

TL;DR

This work shows that near-BPS black holes in supergravity have a temperature-dependent partition function governed by an exact N=4 super-Schwarzian boundary theory, arising from a PSU(1,1|2) BF description of the near-horizon region. Unlike non-supersymmetric cases, the spectrum exhibits a mass gap E_gap ~ 1/(8Φ_r) and a large degeneracy e^{S0} at extremality, with extremal states organized into bosonic multiplets and a precise density-of-states formula that matches string/M-theory expectations. The analysis is carried out in two distinct settings: 4D N=2 ungauged supergravity and 3D AdS3 with (4,4) supersymmetry, both reducing to the same N=4 JT/Schwarzian boundary dynamics and yielding a universal near-extremal spectrum shaped by the broken superconformal symmetry. The results reconcile gravitational thermodynamics with microstate counting by showing that supersymmetry imposes a gap and exact extremal degeneracy, providing a robust bridge between gravitational path integrals and string-theoretic accounts of black hole microstates. Implications extend to holography, SYK-like models with extended SUSY, and potential generalizations to gauged supergravity and higher-dimensional near-BPS systems.

Abstract

Due to the failure of thermodynamics for low temperature near-extremal black holes, it has long been conjectured that a "thermodynamic mass gap" exists between an extremal black hole and the lightest near-extremal state. For non-supersymmetric near-extremal black holes in Einstein gravity, with an AdS$_2$ throat, no such gap was found. Rather, at that energy scale, the spectrum exhibits a continuum of states, up to non-perturbative corrections. In this paper, we compute the partition function of near-BPS black holes in supergravity where the emergent, broken, symmetry is $PSU(1,1|2)$. To reliably compute this partition function, we show that the gravitational path integral can be reduced to that of a $\mathcal N=4$ supersymmetric extension of the Schwarzian theory, which we define and exactly quantize. In contrast to the non-supersymmetric case, we find that black holes in supergravity have a mass gap and a large extremal black hole degeneracy consistent with the Bekenstein-Hawking area. Our results verify several string theory conjectures, concerning the scale of the mass gap and the counting of extremal micro-states.

Figures (5)

• Figure 1: The near extremal 4D black hole. A similar picture applies to near-extremal rotating BTZ black hole in AdS$_3$.
• Figure 2: Schematic shape of the black hole spectrum at fixed $SU(2)$ charge $J$ as a function of energy above extremality $E$. In 4D $J$ is angular momentum while in $AdS_3$ it is the $SU(2)$ charge (one of the two rotations along extra $S^3$ in string theory) that breaks supersymmetry. We show the semiclassical answer (red dashed) and the solution including quantum effects (purple). (a) Answer for Einstein gravity. We see there is no gap at scale $E\sim 1/\Phi_r$ and the extremal entropy goes to zero. (b) Answer for supergravity (either $\mathcal{N}=2$ in 4D or $\mathcal{N}=(4,4)$ in 3D). We find a gap at the scale $E_{gap}=\frac{1}{8\Phi_r}$ and a number $e^{S_0}$ of extremal states, consistent with string theory expectations. (c) Einstein gravity spectrum for $J\neq 0$. (d) Supergravity spectrum for $J\neq 0$, the jumps indicate contributions from different supermultiplets $\mathbf{J}, \mathbf{J+1/2}$ and $\mathbf{J+1}$.
• Figure 3: Left: Density of supermultiplets labeled by the highest spin $\mathbf{J}$. We show $\mathbf{0}$, which is simply a delta function at $E=0$; $\mathbf{1/2}$ which is continuous but starts at $E_{\rm gap}\equiv E_0(1/2)$; and $\mathbf{1}$ which is also continuous starting at $E_0(1)$. Right: Degeneracy for all states with $J=0$. These come from $\mathbf{0}$, the delta function at $E=0$, $\mathbf{1/2}$, starting at $E_{\rm gap}$, and $\mathbf{1}$, starting at $E_0(1)$. All other supermultiplets do not have a $J=0$ component.
• Figure 4: Plot of the entropy and energy dependence on temperature for near-BPS black holes in $\mathcal{N}=2$ ungauged supergravity. The black curves represent results obtained from the naive semiclassical computation while the red curves account for quantum corrections. The inset figures zoom in on the temperature range smaller than $E_\text{gap}$ where quantum corrections become important. Both the entropy and energy approach zero temperature as $\# + \# e^{-E_\text{gap}/T}$ while in the semiclassical analysis, they approach zero temperature as $\# T$ and $\# T^2$, respectively.
• Figure 5: Density of supermultiplets as a function of energy $E$ and charge $Q$. Left: Odd $\widehat{q}$ and no anomaly. The delta function at $E=0$ involves charges in the range $|Q|<1/2$. The supermultiplet with the lowest gap has $Q_0=1/2\pm 1/(2\widehat{q})$ with $E_{\rm gap}=E_0(Q_0)$. Other supermultiplets labeled by $Q$ start at higher energies as shown. Right: Odd $\widehat{q}$ and anomaly. The delta function at $E=0$ involves charges in the range $|Q|<1/2$. The supermultiplet with $Q=1/2$ has no gap. Other supermultiplets have a gap, as shown.