The statistical mechanics of near-BPS black holes

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.