Averaging Over Narain Moduli Space

TL;DR

This paper computes the average of Narain's family of 2D CFTs using number theory, suggesting a possible dual description as a 3D Chern-Simons gravity theory rather than Einstein gravity.

Contribution

It introduces a method to average over Narain CFTs using the Siegel-Weil formula, proposing a new bulk dual theory in three dimensions.

Findings

Average over Narain CFTs can be computed explicitly.

The bulk dual resembles a $U(1)^{2D}$ Chern-Simons theory.

Results suggest a non-Einstein gravity dual in three dimensions.

Abstract

Recent developments involving JT gravity in two dimensions indicate that under some conditions, a gravitational path integral is dual to an average over an ensemble of boundary theories, rather than to a specific boundary theory. For an example in one dimension more, one would like to compare a random ensemble of two-dimensional CFT's to Einstein gravity in three dimensions. But this is difficult. For a simpler problem, here we average over Narain's family of two-dimensional CFT's obtained by toroidal compactification. These theories are believed to be the most general ones with their central charges and abelian current algebra symmetries, so averaging over them means picking a random CFT with those properties. The average can be computed using the Siegel-Weil formula of number theory and has some properties suggestive of a bulk dual theory that would be an exotic theory of gravity in three dimensions. The bulk dual theory would be more like $U(1)^{2D}$ Chern-Simons theory than like Einstein gravity.