AdS$_3$ wormholes from a modular bootstrap

TL;DR

The paper uses a modular bootstrap to fix the torus×interval wormhole amplitude in AdS$_3$ gravity, treating pure gravity as an ensemble of CFTs and showing the two-boundary amplitude is uniquely determined by Virasoro symmetry, modular invariance, and a few gravity inputs, with normalization fixed by the Jackiw–Teitelboim limit. It then contrasts this gravity result with the Narain ensemble, finding that gravity exhibits eigenvalue repulsion near BTZ-threshold (ramp) consistent with chaotic spectra, whereas the Narain ensemble yields a plateau in the spectral form factor due to spectrum discreteness. By analyzing low-temperature limits and Dehn-twist sums, the authors illuminate how wormholes encode different universal statistics in these ensembles and argue that modular bootstrap can generalize to bootstrapping CFT ensembles more broadly. The work proposes a program to bootstrap ensemble averages of CFT data (partition functions on higher-genus surfaces, stress-tensor content, etc.) and discusses how such an ensemble perspective may sharpen connections between holography and random-matrix or maximum-entropy approaches in quantum gravity.

Abstract

In recent work we computed the path integral of three-dimensional gravity with negative cosmological constant on spaces which are topologically a torus times an interval. Here we employ a modular bootstrap to show that the amplitude is completely fixed by consistency conditions and a few basic inputs from gravity. This bootstrap is notably for an ensemble of CFTs, rather than for a single instance. We also compare the 3d gravity result with the Narain ensemble. The former is well-approximated at low temperature by a random matrix theory ansatz, and we conjecture that this behavior is generic for an ensemble of CFTs at large central charge with a chaotic spectrum of heavy operators.

Figures (1)

• Figure 1: The Euclidean wormhole $\mathbb{T}^2\times I$. For this configuration, the spatial and temporal circles on boundary $1$ continuously interpolate to the spatial and temporal circles on boundary $2$. Other configurations have a relative Dehn twist between the boundary tori.