Holography for people with no time

TL;DR

The paper analyzes extremal supersymmetric black holes by focusing on the AdS2 near-horizon region and a single boundary graviton mode, enabling a tractable zero-energy limit in which correlators become time-independent. It shows that the dual quantum mechanics projects onto a finite set of ground states, producing a finite-length supersymmetric wormhole between two black holes, and it develops a detailed account of two- and higher-point functions within this framework. The authors connect these analytic results to ${ cal N}=2$ supersymmetric SYK numerics, confirming gap structure, ground-state degeneracy, and random-matrix features consistent with the Schwarzian effective theory. Overall, the work provides a controlled holographic setting where time emerges from a boundary theory with no time and offers new insights into black hole microstates and the role of wormholes in holography.

Abstract

We study the gravitational description of extremal supersymmetric black holes. We point out that the $AdS_2$ near horizon geometry can be used to compute interesting observables, such as correlation functions of operators. In this limit, the Hamiltonian is zero and correlation functions are time independent. We discuss some possible implications for the gravity description of black hole microstates. We also compare with numerical results in a supersymmetric version of SYK. These results can also be interpreted as providing a construction of wormholes joining two extremal black holes. This is the short version of a longer and more technical companion paper arXiv:2207.00408.

Figures (9)

• Figure 1: The density of states for the ${\cal N}=2$ superSchwarzian theory in the zero R-charge sector, from Stanford:2017thb. We see the presence of a gap, and a delta function contribution at zero which gives the extremal entropy. $E_\text{gap} = { 1 \over 32 C}$, with $C$ defined in (\ref{['ActSch']}).
• Figure 2: Diagrams for two point function. (a) The bottom propagator generates the thermofield double state $|\mathrm{TFD}(u)\rangle$ and the top one generates a similar one with $u\to u'$. The correlator involves a geodesic going between the two boundary points whose renormalized length is $\ell$ and the associated conformal dimension is $\Delta$. (b) The correlator in the $u= u'=\infty$ limit. The boundary has large fluctuations but the distance between the two operator insertions remains finite.
• Figure 3: In black, we plot the effective potential when the term involving the fermions has an effectively negative coefficient. It leads to a single normalizable state, which is shown in red.
• Figure 4: The cylinder diagram in the probe approximation. The states of the empty wormhole propagate along the cylinder.
• Figure 5: General correlators are built from $AdS$ Witten diagrams, in red, dressed by boundary graviton propagators, in blue.