Averaging Over Narain Moduli Space

TL;DR

The paper investigates whether averaging over the Narain moduli space of 2D CFTs with $D$ free bosons and ${ m U}(1)^{2D}$ current algebras can be dual to a bulk theory in three dimensions. Using the Siegel-Weil formula, the authors show that genus-one and higher-genus averages yield Eisenstein-series inputs $E_{D/2}( au)$ or $E_{D/2}(oldsymbol ext Omega)$ that govern the averaged partition functions, and they interpret these results via a bulk theory built from ${ m U}(1)^{2D}$ Chern-Simons dynamics, effectively identifying a bulk sum over handlebodies with the boundary average. They extend the analysis to include extended chiral algebras, disconnected boundaries, and higher-genus surfaces, revealing both the power and limitations of this bulk-averaging picture, including large-$D$ simplifications and genus-dependent divergences at finite $D$. The work highlights the potential of exotic bulk theories to encode ensemble-averaged CFT data, while underscoring challenges in achieving a fully nonperturbative, geometric bulk dual.

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.

Figures (1)

• Figure 1: If a closed Riemann surface $\Sigma$ is embedded in ${\mathbb{R}}^3$ in an arbitrary fashion, then its "interior" is, topologically, a handlebody $Y$. Such an embedding of $\Sigma$ determines a distinguished sublattice $\Gamma_0$ of the first homology $\Gamma=H_1(\Sigma,{\mathbb Z})$, spanned by one-cycles that are contractible in $Y$. In the present example, $\Sigma$ has genus 2, and $\Gamma_0$ is spanned by the two one-cycles drawn.