A precision test of averaging in AdS/CFT

TL;DR

This work resolves aspects of the factorization paradox by showing that Euclidean AdS wormholes can be stabilized as constrained saddles when the total boundary energy is fixed, yielding a computable microcanonical spectral form factor that encodes non-factorizing physics consistent with averaging. By analyzing a precision test with detuned couplings, the authors derive a universal exponential decay rate $e^{-\pi X^2 T}$ for the two-replica form factor and demonstrate exact agreement between bulk wormhole calculations and one-replica chaos predictions in both AdS$_3$ and AdS$_5\times S^5$ settings, thereby providing a precise test of averaging in AdS/CFT. The paper further extends the analysis to correlations across theories with different $N$, showing exponentially damped inter-replica signals, and discusses the interpretation of bulk gravity as a mesoscopic description that naturally smears over energy scales rather than couplings. Overall, the results support a view in which gravitational EFT captures averaged, energy-window physics of chaotic black-hole spectra while preserving factorization on the full boundary theory, with broad implications for how holographic duality encodes ensemble-like behavior. $$Z_{\rm wormhole}(\beta_1,\beta_2)=\int_{E_0}^\infty dE\, e^{-(\beta_1+\beta_2)E} f(E)(1+O(G))$$ and related constrained saddles provide the computational backbone for these insights.

Abstract

We reconsider the role of wormholes in the AdS/CFT correspondence. We focus on Euclidean wormholes that connect two asymptotically AdS or hyperbolic regions with $\mathbb{S}^1\times \mathbb{S}^{d-1}$ boundary. There is no solution to Einstein's equations of this sort, as the wormholes possess a modulus that runs to infinity. To find on-shell wormholes we must stabilize this modulus, which we can do by fixing the total energy on the two boundaries. Such a wormhole gives the saddle point approximation to a non-standard problem in quantum gravity, where we fix two asymptotic boundaries and constrain the common energy. Crucially the dual quantity does not factorize even when the bulk is dual to a single CFT, on account of the fixed energy constraint. From this quantity we extract a smeared version of the microcanonical spectral form factor. For a chaotic theory this quantity is self-averaging, i.e. well-approximated by averaging over energy windows, or over coupling constants. We go on to give a precision test involving the microcanonical spectral form factor where the two replicas have slightly different coupling constants. In chaotic theories this form factor is known to smoothly decay at a rate universally predicted in terms of one replica physics, provided that there is an average either over a window or over couplings. We compute the expected decay rate for holographic theories, and the form factor from a wormhole, and the two exactly agree for a wide range of two-derivative effective field theories in AdS. This gives a precision test of averaging in AdS/CFT. Our results interpret a number of confusing facts about wormholes and factorization in AdS and suggest that we should regard gravitational effective field theory as a mesoscopic description, analogous to semiclassical mesoscopic descriptions of quantum chaotic systems.