A Holographic Constraint on Scale Separation

We propose a new consistency condition for the compatibility of a gravitational effective field theory in AdS with a dual holographic description in terms of a family of large-$N$ CFTs. Using large-$N$ factorization of correlation functions combined with a properly defined notion of single- and multi-particle operators, we argue that the cubic scalar bulk couplings for fields dual to operators with extremal arrangements of the conformal dimensions, i.e. $Δ_i=Δ_j+Δ_k$, should vanish. We apply this criterion to the 4d $\mathcal{N}=1$ effective supergravity theory describing the simplest DGKT AdS$_4$ vacua in type IIA string theory and show that it is non-trivially satisfied. In addition, we calculate explicitly all non-vanishing three-point correlation functions of low-lying scalar operators in the putative 3d CFTs dual to these AdS$_4$ string theory backgrounds.

💡 Research Summary

This paper addresses a fundamental challenge in the AdS/CFT correspondence: determining the criteria for a gravitational effective field theory (EFT) in Anti-de Sitter (AdS) space to be compatible with a dual holographic description in terms of a large-$N$ Conformal Field Theory (CFT). The authors propose a novel consistency condition based on the principle of large-$N$ factorization of correlation functions.

The core of the research lies in analyzing the behavior of scalar operators in the boundary CFT. The authors focus on the “extremal” case, where the conformal dimensions of the operators satisfy the relation $\Delta_i = \Delta_j + \Delta_k$. By examining the properties of single- and multi-particle operators and their correlation functions in the large-$N$ limit, they demonstrate that for the theory to maintain the required factorization property—where multi-particle correlation functions decompose into products of single-particle functions—the cubic scalar couplings in the bulk gravity theory must vanish whenever this extremal condition is met. If these couplings were non-zero, it would lead to a violation of the fundamental structural consistency of the dual CFT.

To validate this theoretical proposal, the authors apply the derived criterion to a concrete and physically significant model: the 4D $\mathcal{N}=1$ effective supergravity theory describing the DGKT AdS$_4$ vacua in Type IIA string theory. This is a non-trivial test, as the DGKT vacua involve complex interactions within a string-theoretic framework. The results show that the proposed holographic constraint is non-trivially satisfied, reinforcing the validity of the DGKT construction and the proposed consistency condition.

Beyond the verification of the constraint, the paper provides a significant computational contribution by explicitly calculating all non-vanishing three-point correlation functions for the low-lying scalar operators in the putative 3D CFTs dual to these backgrounds. This provides essential data for researchers studying the dynamics of these specific 3D conformal field theories.

In conclusion, this work contributes to the broader “Swampland Program” by providing a precise, calculable tool to distinguish between consistent gravitational theories (the Landscape) and those that cannot be UV-completed (the Swampland). By establishing a new holographic constraint on scale separation, the authors have provided a powerful mathematical filter for evaluating the holographic legitimacy of gravitational effective field theories.