No Ensemble Averaging Below the Black Hole Threshold

TL;DR

The authors address the puzzle of ensemble averaging in AdS/CFT by isolating fixed-energy, sub-threshold observables that should be insensitive to ensemble averages. They develop a geometric framework in AdS$_3$/CFT$_2$ using the renormalized volume $V_R$ of hyperbolic 3-manifolds to determine when bulk saddles can contribute to sub-threshold amplitudes, proving that only connected-boundary (Schottky) geometries contribute to these observables and that compressible cycles drive $V_R\to-\infty$. Their mathematical analysis connects Teichmüller theory, Weil-Petersson geometry, and the convex core/bending lamination to show when bulk geometries can yield fixed-energy propagation without ensemble averaging. They argue that ensemble averaging in gravity arises from chaotic black hole physics and the lack of a large-$N$ limit for black hole sectors, effectively averaging over nearby $N$ in a random-matrix sense. The work clarifies which CADB amplitudes can be reconciled with a single CFT description and why certain observables remain robust against ensemble effects.

Abstract

In the AdS/CFT correspondence, amplitudes associated to connected bulk manifolds with disconnected boundaries have presented a longstanding mystery. A possible interpretation is that they reflect the effects of averaging over an ensemble of boundary theories. But in examples in dimension $D\geq 3$, an appropriate ensemble of boundary theories does not exist. Here we sharpen the puzzle by identifying a class of "sub-threshold" observables that we claim do not show effects of ensemble averaging. These are amplitudes that do not involve black hole states. To support our claim, we explore the example of $D=3$, and show that connected solutions of Einstein's equations with disconnected boundary never contribute to sub-threshold observables. To demonstrate this requires some novel results about the renormalized volume of a hyperbolic three-manifold, which we prove using modern methods in hyperbolic geometry. Why then do any observables show apparent ensemble averaging? We propose that this reflects the chaotic nature of black hole physics and the fact that the Hilbert space describing a black hole does not have a large $N$ limit.

Figures (5)

• Figure 1: A three-holed sphere, with its boundaries labeled by fixed energy states $i,j,k$.
• Figure 2: A genus two surface $M$ with three nonintersecting and homotopically independent one-cycles labeled ${\sf A}$, ${\sf B}$, and ${\sf C}$. $M$ is the conformal boundary of a hyperbolic three-manifold $X$. If $X$ is such that $V_R(X)\to -\infty$ when ${\sf A}$ is pinched, then the contribution of $X$ to $Z(M)$ has a part that describes propagation through ${\sf A}$ of a fixed energy state. In general, however, black hole states are propagating through ${\sf B}$ and ${\sf C}$ and this amplitude is subject to ensemble averaging. If we stipulate that $V_R(X)\to -\infty$ when any of ${\sf A}$, ${\sf B}$, or ${\sf C}$ is pinched, we get an amplitude that has a contribution that describes fixed energy states propagating through each of ${\sf A}$, ${\sf B}$, and ${\sf C}$, and interacting via two three-holed spheres, one to the left of ${\sf A}$ and one to the right. A contribution of this type, according to our conjecture, should not be subject to ensemble averaging. So this behavior of $V_R(X)$ should be possible only if the conformal boundary of $X$ is connected.
• Figure 3: A surface $M$ of genus $g=3$, with $3g-3=6$ independent circles marked. Surgery on $\sf A$ produces a surface $M_1$ of genus $g_1=g-1=2$. In passing to $M_1$, $\sf A$ disappears, $\sf B$ becomes nullhomotopic, and $\sf C$ and $\sf D$ become homotopic to each other, leaving three independent circles on $M_1$. Surgery on $\sf B$ leaves a surface $M_1'$ of genus $g_1'=1$ and a surface $M_1"$ of genus $g_1"=2$. Of the original 6 independent circles on $M$, $\sf B$ disappears in the surgery, $\sf A$ remains on $M_1'$, and $\sf C$ and $\sf D$ become homotopic. We remain with $3g_1"-3=3$ independent circles on $M_2'$.
• Figure 4: Planes almost orthogonal to thin tubes cannot intersect.
• Figure 5: A pair of geodesic rays $\alpha,\beta$ normal to a convex subset $K$ cannot intersect, because the sum of the angles of triangle $xx'y$ would be greater than $\pi$.

Theorems & Definitions (10)

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• proof : Proof of Theorem \ref{['tm:compare2']}
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• proof : Proof of Theorem \ref{['tm:asymptotics']}