Landscape of Narain CFTs

TL;DR

This paper analyzes the AdS$_{3}$/CFT$_{2}$ dual of an ensemble of Narain CFTs under Swampland constraints, focusing on AdS distance and finiteness to bound the boundary landscape and bulk couplings. It identifies gauge and gravitational anomalies in the bulk and demonstrates two complementary anomaly-cancellation mechanisms: a stringy boundary WZW construction and fermionic worldsheet degrees of freedom, which together restore consistency for both standard and generalized Narain theories. By combining these mechanisms with unitarity and AdS distance arguments, the authors derive explicit finiteness bounds, including $1/4 \\lesssim k^{G} \\lesssim 166$ and $3/2 \\lesssim c \\lesssim 10^{3}$, and relate them to bounds on the AdS radius $l_{AdS_3}$ and the central charges. The results map the Narain landscape to a finite region controlled by the CS level and bulk geometry, offering a concrete framework to connect Swampland constraints with holographic CFT ensembles and bootstrap insights.

Abstract

In this work, we investigate the AdS$_{3}$ gravitational bulk dual to an ensemble of Narain CFTs and their generalisations to establish bounds consistent with the Swampland program. Focusing on the AdS distance and finiteness conjectures, we show that the central charge of Narain CFTs forming the ensemble must be finite. Combining anomaly and unitary requirements, we derive an upper bound on the rank of the abelian U(1) gauge symmetries that can consistently couple to the AdS$_{3}$ gravity. We give explicit realisations of these constraints by determining the range of the Chern-Simons level $k^{G}$ corresponding to a bounded AdS$_{3}$ radius$.$ Accordingly, the Narain landscape is finite with a number of admissible CFTs constrained as $3/2\lesssim c\lesssim 10^{3}.$

Figures (1)

• Figure 1: Narain Landscape region (in yellow) represents the viable space of consistent theories with central charge $c$ constrained by the cutoff as $\frac{3}{2}\left( G_{N}\Lambda _{cut-off}\right) ^{-1}$ (blue line). The red and green lines indicate the minimal and maximal values of c determined by the allowed range of the AdS radius $G_{N}\lesssim l_{AdS_{3}}\lesssim 600G_{N}$. The purple line denotes the central charge $c_{g}$ of the current algebra and bounds the gray shaded area excluded by unitarity. For each c, many values of $c_{g}=\frac{p+q-2}{2}$ are possible with a minimum value of $c_{g_{\min }}=1$ corresponding to a Narain CFT with $p=q=2$.