A Holographic Constraint on Scale Separation (2512.11031v1)

Abstract: 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.

• The paper shows that vanishing extremal cubic scalar couplings in gravitational EFTs preserve large-N factorization in AdS/CFT.
• Detailed computations in DGKT AdS4 vacua confirm that cancellations in extremal configurations are necessary for a consistent holographic dual.
• This holographic constraint offers a rigorous guideline for constructing EFTs in string theory that admit dual large-N CFTs, influencing swampland and UV completeness research.

Summary of "A Holographic Constraint on Scale Separation" (2512.11031)

Introduction and Background

The manuscript presents a rigorous investigation into the compatibility of gravitational effective field theories (EFTs) in AdS with the existence of large-$N$ holographic dual conformal field theories (CFTs). The authors propose a novel, robust holographic constraint: cubic bulk scalar couplings that are extremal with respect to conformal dimensions—i.e., $c'_{ijk}$ where $\Delta_i = \Delta_j + \Delta_k$—must vanish for a gravitational EFT in AdS to be consistent with a dual large-$N$ CFT. This assertion is motivated by the observation that extremal configurations induce divergences in the holographic three-point Witten diagrams and thus fail to yield well-defined CFT three-point functions unless the relevant bulk coupling constants vanish.

The argument leverages large-$N$ factorization and the precise identification of single- and multi-particle operators in holographic duality. The analysis generalizes previously understood constraints from the prototypical AdS$_5$/CFT$_4$ case (e.g., type IIB on AdS$_5 \times S^5$/ $\mathcal{N}=4$ SYM) and extends them to generic EFTs in AdS, including those arising in string/M-theory constructions exhibiting scale separation.

Holographic Consistency Condition

The formal derivation begins by examining scalar cubic interactions in an EFT expanded around an AdS vacuum. Employing the holographic dictionary, operators dual to bulk scalars are identified as single-particle operators (SPOs) with well-defined conformal dimensions related to bulk masses. The authors show that, in the context of large-$N$ holographic CFTs, two-point and three-point function scaling imposes strong constraints on the allowed cubic couplings. Specifically, when a configuration allows for mixing between single- and double-trace CFT operators due to coincident dimensions (the extremal case), the only way to preserve large-$N$ factorization and the SPO structure is to ensure the relevant cubic bulk couplings vanish.

Failure to observe this constraint results in anomalous scaling of CFT correlators ($1/c$ rather than $1/c^2$ for three-point functions), signaling a breakdown of the holographic correspondence in the regime of large $c$. The formalism encompasses both extremal and super-extremal cases (i.e., when $\Delta_i = \Delta_j + \Delta_k + n$ for $n \in \mathbb{Z}_{>0}$), establishing a stringent requirement for the EFT's coupling structure.

Application to DGKT AdS$_4$ Vacua

The second half of the paper tests the holographic criterion within the concrete context of DGKT-type scale-separated AdS$_4$ vacua in massive type IIA string theory, built from orientifolded orbifold compactifications such as $T^6/\mathbb{Z}_3^2$. In these models, flux choices can produce parametric scale separation and large central charge, yielding a controlled regime for EFT and holography.

Detailed calculations of the scalar and pseudoscalar operator dimensions in the effective $\mathcal{N}=1$ supergravity theory exhibit abundant extremal arrangements. The authors perform a precise expansion of the action to cubic order, compute the relevant couplings, and demonstrate explicit non-trivial cancellations: all extremal and super-extremal cubic couplings vanish after taking into account derivative and potential terms. This confirms that the EFT underlying these AdS$_4$ vacua is compatible with the proposed holographic constraint.

In addition, explicit non-extremal three-point function coefficients for low-lying scalar operators are derived. The spectrum of conformal dimensions and the detailed matching of holographic correlator scaling support the validity of the criterion in both supersymmetric and non-supersymmetric DGKT vacua.

Implications and Future Directions

The holographic constraint outlined in this work establishes a necessary consistency condition for any gravitational EFT in AdS to admit a dual large-$N$ CFT description, especially for scenarios with scale separation. The constraint is non-trivial in string compactifications, where the occurrence of extremal operator arrangements cannot be a priori excluded, as demonstrated by the DGKT flux models.

Practical implications include:

• Constraints on Model Building: Proposed scale-separated AdS vacua in string or M-theory must be scrutinized for vanishing extremal cubic couplings in their low-energy EFTs to admit consistent holographic duals.
• Guiding UV Completeness: The vanishing of the extremal couplings acts as a filter for EFTs that may be realized as consistent quantum gravity backgrounds with CFT duals.
• Impact on Swampland Program: This criterion intersects with ongoing efforts to delineate the swampland of EFTs incompatible with holography or UV completion.

Theoretically, the constraint sharpens the understanding of operator mixing and correlation function scaling in holographic dualities. It suggests extensions to other dimensions, compactification schemes, and setups with integer operator dimensions (e.g., AdS$_3$/CFT$_2$). The absence of extremal arrangements in certain constructions (e.g., KKLT/LVS) and the role of central charge tunability highlight further areas for exploration.

Several directions for future research are suggested:

• Generalization to Other Scale-Separated Vacua: The applicability of the holographic criterion to more intricate CY orientifold compactifications should be investigated, particularly given the ubiquity of extremal arrangements in these spectra.
• Study of Higher-Point Functions and Corrections: Further computation of holographic four-point functions and $1/c$ corrections to operator dimensions may reveal additional non-trivial structure and symmetry constraints.
• Explicit Construction of Dual CFTs: The search for families of 3d CFTs dual to DGKT-type vacua—potentially with large spectral gaps and exotic fusion rules—remains open.

Conclusion

This work formulates and tests a new holographic consistency condition: extremal cubic scalar couplings must vanish in gravitational EFTs in AdS to preserve large-$N$ factorization and the existence of SPOs in the dual CFT. Detailed application to DGKT AdS$_4$ flux vacua in type IIA string theory confirms this constraint via explicit calculation. The criterion strategically refines the landscape of admissible EFTs in string/M-theory compactifications and strengthens the analytic bridge between AdS gravity and CFT data, with consequences for future studies in holographic duality, scale separation, and quantum gravity model building.