Wormholes and black hole microstates in AdS/CFT

TL;DR

This work argues that Euclidean wormholes provide a nonperturbative, coarse-grained description of the energy level statistics of black hole microstates in AdS/CFT. By developing constrained instanton techniques and a lambda-solution perspective, the authors derive integral representations for wormhole amplitudes across Einstein gravity and IIB supergravity on AdS5 x S5, identifying macroscopic saddles related to the double-cone geometry and connecting them to smeared two-point functions of the density of states. The results yield evidence for level repulsion in black hole microstate spectra and reveal a factorization paradox: wormhole contributions imply connected two-boundary amplitudes in theories believed to be single-CFT duals. A thorough stability analysis shows perturbative robustness of these wormhole saddles in higher dimensions and within string theory, and the late-time Lorentzian continuation indicates that brane-nucleation instabilities do not destroy the ramp behavior in the spectral form factor, supporting a nuanced ensemble-like interpretation within AdS/CFT.

Abstract

It has long been known that the coarse-grained approximation to the black hole density of states can be computed using classical Euclidean gravity. In this work we argue for another entry in the dictionary between Euclidean gravity and black hole physics, namely that Euclidean wormholes describe a coarse-grained approximation to the energy level statistics of black hole microstates. To do so we use the method of constrained instantons to obtain an integral representation of wormhole amplitudes in Einstein gravity and in full-fledged AdS/CFT. These amplitudes are non-perturbative corrections to the two-boundary problem in AdS quantum gravity. The full amplitude is likely UV sensitive, dominated by small wormholes, but we show it admits an integral transformation with a macroscopic, weakly curved saddle-point approximation. The saddle is the "double cone" geometry of Saad, Shenker, and Stanford, with fixed moduli. In the boundary description this saddle appears to dominate a smeared version of the connected two-point function of the black hole density of states, and suggests level repulsion in the microstate spectrum. Using these methods we further study Euclidean wormholes in pure Einstein gravity and in IIB supergravity on Euclidean AdS$_5\times\mathbb{S}^5$. We address the perturbative stability of these backgrounds and study brane nucleation instabilities in 10d supergravity. In particular, brane nucleation instabilities of the Euclidean wormholes are lifted by the analytic continuation required to obtain the Lorentzian spectral form factor from gravity. Our results indicate a factorization paradox in AdS/CFT.

Figures (6)

• Figure 1: A log-log plot of the spectral form factor $Z(\beta + i T, \beta - i T)$ (normalized by $Z(\beta, \beta)$) as a function of $T$ for the Gaussian Unitary Ensemble (GUE). We have considered the GUE for $500 \times 500$ matrices with the real and imaginary parts of each matrix element sampled from a Gaussian distribution with variance $1/500$. The black curve is averaged over $10000$ samples and the light blue curve is a single sample.
• Figure 2: Depiction of a double-scaled density of states.
• Figure 3: The critical value of $b$ as a function of $\frac{\beta_1\beta_2}{(\beta_1+\beta_2)^2}$. The shaded region indicates wormholes with $b<b_c$, which are stable against the nucleation of 3-brane pairs.
• Figure 4: 3-brane embeddings in the wormhole with torus cross-section and $\beta_1 = \beta_2$ (upon rescaling $\tau \to \frac{\tau}{\beta b}$). The innermost solid loop is the trajectory of the $D3$-brane instanton, with $x= 1$. The dashed loop indicates the trajectory for a toy model of a 3-brane with a mass-to-charge ratio $1.15$, the outermost loop is the trajectory for the toy model with mass-to-charge ratio $x=1.19967$, just below the critical value of $x_0 = 1.19967864..$. The red line is the trajectory when $x=1.19968$, just above the critical value.
• Figure 5: A visual summary of the two brane instantons found in this Section. The circle is the Euclidean time direction, and we have suppressed the other directions. The blue/red pair wrapping the time circle are the $D3-\overline{D3}$ brane pair at constant $\rho$ from Subsection \ref{['S:basicBrane']}. For the $\mathbb{S}^1\times\mathbb{S}^3$ wormhole, this arrangement is a saddle of probe 3-branes and, if $b>b_c$, placing the branes there lowers the wormhole action. To the left of the $\overline{D3}$-brane, and to the right of the $D3$-brane, there are $N$ units of 5-form flux, while in between there are $N-1$. The green circle is the instanton from Subsection \ref{['S:otherBrane']}. This configuration leads to an instanton correction to the wormhole amplitude, again for $b>b_c$. If we analytically continued $\tau$ to real time then this instanton leads to a dynamical instability of the wormhole to the production of a 3-brane pair.