Counting AdS Vacua

TL;DR

This paper investigates the count of AdS vacua with a UV cutoff, proposing a power-law decay as the cutoff approaches zero, and explores implications for the distribution of vacua and potential dS analogs.

Contribution

It provides evidence that the number of AdS vacua scales as a power law with the UV cutoff and highlights the role of moduli space volume in this count.

Findings

Number of AdS vacua scales as a power law with the UV cutoff.

Large moduli spaces contribute minimally to the vacuum count.

Speculation on the number of quasi-dS vacua scaling with the cosmological constant.

Abstract

We study the 'number' $\mathfrak{N}(\mu)$ of AdS vacua with a UV cut off $ \mu$. It has been proposed that this number is finite. We find evidence that $\mathfrak{N}(\mu)\sim a \ \mu^{-b}$ as $\mu \rightarrow 0$ for some constants $a$ and $b$ of $O(1)$ in Planck units that may depend on dimension and the number of supercharges. For this result to hold it is crucial to integrate over the volume of massless and tachyonic directions of AdS which corresponds to the volume of the space of marginal and relevant deformations of the dual CFT. We are led to the surprising prediction that theories with large number of light moduli contribute very little to the volume measure among all theories. We also speculate about the dS case leading to the number of quasi-dS vacua of the order of $\Lambda^{-\alpha}$ for some $O(1)$ parameter $\alpha$.