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Counting AdS Vacua

Zihni Kaan Baykara, Alessandro Tomasiello, Cumrun Vafa

TL;DR

The paper investigates the counting of AdS vacua with a UV cutoff, arguing that when massless and tachyonic (marginal and relevant) directions are integrated over, the number of vacua grows only polynomially with the cutoff, rather than exponentially. Using a blend of discrete flux data, continuous moduli-space volumes, and careful treatment of tachyonic directions (via cylindrical decomposition and, where applicable, Siegel mass-type analyses), it derives power-law bounds for maximal and lower-SUSY AdS vacua across dimensions 7, 5, and 4. The main results show explicit scaling laws: for maximal SUSY AdS vacua, $ rak N_{AdS_d}^{(Q=16)}(bc) \\sim bc^{- rac{(D-d-1)(d-2)}{D-2}}$, with corresponding CFT counts $ rak N_{CFT} \\sim bc^{ rac{2}{d-1}}$; for lower SUSY sectors there are analogous power laws with specific exponents, tempered by tachyonic volume effects. The findings imply that the landscape is not exponentially proliferative once light moduli are properly weighed, offering implications for anthropic reasoning in Λ and suggesting structured, nonpathological behavior in de Sitter extensions as well.

Abstract

We study the 'number' $\mathfrak{N}(μ)$ of AdS vacua with a UV cut off $ μ$. It has been proposed that this number is finite. We find evidence that $\mathfrak{N}(μ)\lesssim a \ μ^{-b}$ as $μ\rightarrow 0$ 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 $Λ^{-α}$ for some $O(1)$ parameter $α$.

Counting AdS Vacua

TL;DR

The paper investigates the counting of AdS vacua with a UV cutoff, arguing that when massless and tachyonic (marginal and relevant) directions are integrated over, the number of vacua grows only polynomially with the cutoff, rather than exponentially. Using a blend of discrete flux data, continuous moduli-space volumes, and careful treatment of tachyonic directions (via cylindrical decomposition and, where applicable, Siegel mass-type analyses), it derives power-law bounds for maximal and lower-SUSY AdS vacua across dimensions 7, 5, and 4. The main results show explicit scaling laws: for maximal SUSY AdS vacua, , with corresponding CFT counts ; for lower SUSY sectors there are analogous power laws with specific exponents, tempered by tachyonic volume effects. The findings imply that the landscape is not exponentially proliferative once light moduli are properly weighed, offering implications for anthropic reasoning in Λ and suggesting structured, nonpathological behavior in de Sitter extensions as well.

Abstract

We study the 'number' of AdS vacua with a UV cut off . It has been proposed that this number is finite. We find evidence that as for some constants and of 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 for some parameter .

Paper Structure

This paper contains 58 sections, 1 theorem, 243 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Minkowski Vacua
  3. Maximal SUSY AdS Vacua
  4. Scales.
  5. Towers.
  6. AdS count.
  7. CFT count.
  8. In terms of $\Lambda$.
  9. Lower SUSY AdS Vacua
  10. AdS7
  11. Discrete count
  12. Scales.
  13. Towers.
  14. Cylindrical decomposition.
  15. AdS count.
  16. ...and 43 more sections

Key Result

Theorem 1

Let $A$ be an $m\times n$ matrix, $a\in\mathbb{R}^m$, $c\in\mathbb{R}^n$, and $b\in\mathbb{R}$. Then if and only if there exists $\lambda\in\mathbb{R}^m$ with $\lambda\ge0$ such that

Figures (3)

  • Figure 1: Example of a Young diagram $Y_L=[3,3,2]$ corresponding to a nilpotent element $\mu_L\in\mathfrak{su}(8)$. Each row of the diagram represents an irreducible $SU(2)$ representation of dimension equal to the row length. The multiplicity of equal row lengths determines the unbroken subgroup of $SU(k)$ commuting with the associated embedding $\rho_{Y_L}:SU(2)\hookrightarrow SU(k)$. In this example, there are two rows of length $3$ and one row of length $2$, giving the commutant $\mathrm{Comm}_{SU(k)}(\rho_{Y_L}) = S(U(2)\times U(1)) \cong SU(2)\times U(1)$. Thus, the Young diagram encodes how the left flavor symmetry $SU(k)$ is broken along the Higgs branch of the corresponding six-dimensional $(1,0)$ SCFT.
  • Figure 2: The integration region in $(\log(k_1),\log(k_2),\log(N))$ variables for the number of theories with tower scales above the cutoff $\hat{\mu}$ obtained from \ref{['eq:AdS5-ine1']}, \ref{['eq:AdS5-ine2']}, \ref{['eq:AdS5-ine3']}, \ref{['eq:AdS5-ine4']}. For the plot we chose $\log\hat{\mu} =-10$.
  • Figure 3: The integration region in $(\log(k_1),\log(k_2),\log(N))$ variables for the number of theories with tower scales above the cutoff $\hat{\mu}$ and instatons $S_{\rm inst}\gtrsim 1$ obtained from \ref{['eq:AdS5-ine1']}, \ref{['eq:AdS5-ine3']}, \ref{['eq:AdS5-ine4']}, \ref{['eq:AdS5-inst']} with $\log\hat{\mu} =-10$. Compare with \ref{['fig:ads5-region']}.

Theorems & Definitions (1)

  • Theorem 1: Motzkin Transposition Theorem