Counting AdS Vacua

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 $\mu \to 0$ 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 ${\mathrm{\Lambda }}^{-\alpha }$ for some $O(1)$ parameter $\alpha$.