Wormholes and Spectral Statistics in the Narain Ensemble

Abstract

We study the spectral statistics of primary operators in the recently formulated ensemble average of Narain's family of free boson conformal field theories, which provides an explicit (though exotic) example of an averaged holographic duality. In particular we study moments of the partition function by explicit computation of higher-degree Eisenstein series. This describes the analog of wormhole contributions coming from a sum of over geometries in the dual theory of "U(1) gravity" in AdS${}_3$. We give an exact formula for the two-point correlation function of the density of primary states. We compute the spectral form factor and show that the wormhole sum reproduces precisely the late time plateau behaviour related to the discreteness of the spectrum. The spectral form factor does not exhibit a linear ramp.

39 pages + appendices. v2: references added, typos corrected

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• Adding Flavor to the Narain Ensemble
• Alpha states demystified: Towards microscopic models of AdS${}_2$ holography
• Non-rational Narain CFTs from codes over $F_4$
• Calabi-Yau CFTs and Random Matrices
• A note on ensemble holography for rational tori
• Microstructure in matrix elements
• Gravity without averaging
• (Half) Wormholes under Irrelevant Deformation
• Baby universes, ensemble averages and factorizations with matters
• Spacing Statistics of Energy Spectra: Random Matrices, Black Hole Thermalization, and Echoes
• A universal sum over topologies in 3d gravity
• Factorization and global symmetries in holography