Looking at supersymmetric black holes for a very long time

TL;DR

This work analyzes extremal supersymmetric black holes by focusing on the zero-energy boundary dynamics governed by the N=2 super-Schwarzian, solved exactly via a supersymmetric Liouville description and an equivalent propagator formalism. By computing two- and higher-point functions for BPS and neutral operators, the authors reveal long-time constants and explicit integral expressions for n-point correlators, linking boundary correlators to bulk AdS2 physics and wormhole geometries. The paper also connects to N=2 SYK, demonstrates consistency with numerical results, and uncovers rich structure such as matter-carrying supersymmetric wormholes, a matter Casimir operator, and entanglement features, contributing to a deeper understanding of ground-state microstate statistics and emergent time in holography. A parallel discussion of the non-supersymmetric JT gravity case highlights universal long-time behavior and clarifies the role of connected versus disconnected bulk diagrams in the low-energy limit.

Abstract

We study correlation functions for extremal supersymmetric black holes. It is necessary to take into account the strongly coupled nature of the boundary supergraviton mode. We consider the case with ${\cal N}=2$ supercharges which is the minimal amount of supersymmetry needed to give a large ground state degeneracy, separated from the continuum. Using the exact solution for this theory we derive formulas for the two point function and we also give integral expressions for any $n$-point correlator. These correlators are time independent at large times and approach constant values that depend on the masses and couplings of the bulk theory. We also explain that in the non-supersymmetric case, the correlators develop a universal time dependence at long times. This paper is the longer companion paper of arXiv:2207.00407.

Figures (19)

• Figure 1: Various correlators. (a) Two point function at finite boundary euclidean times, $u$, $u'$. (b) Two point function in the zero energy limit, with $u=u'=\infty$. (c) A general long time correlator.
• Figure 2: (a) The construction of an empty supersymmetric wormhole arising after the evolution over a long Euclidean time. (b) By adding an operator during the Euclidean time evolution we produce extra matter. (c) We could add several single particle operators to produce a multiparticle state in the wormhole. The length of the green line represents the size of the wormhole.
• Figure 3: (a) Computation of the two point function of two operators separated by boundary time $u$ on one side and boundary time $u'$ on the other. (b) As $u, u' \to \infty$ we get the zero energy correlator which is a constant. In both cases $\ell$ denotes a renormalized distance between the two boundary points.
• Figure 4: Lowest energy fermion even continuum states as a function of the $R$ charge $J$. Note that for $J>0$ the lowest energy continuum state is $H_{s=0,j}$, with $J=j$. There is another state in the same multiplet, $L_{s=0,j}$ which has $J=j-1$. On the other hand, for negative $J$, the lowest energy state is $L_{s=0,j+1}$ with $J = j <0$.
• Figure 5: Neutral two point function for finite $u$ and $u' \to \infty$ in the charge $j=0$ sector. We chose $\Delta = 1/8$. Note that the correlator behaves as $u^{-2\Delta}$ for $u \ll 1$ and it becomes constant for large $u$.