Scale-Separated AdS Vacua

• Scale-separated AdS vacua are solutions where the KK scale is parametrically smaller than the AdS radius, ensuring decoupled low-energy dynamics.
• They are realized via flux compactifications in Type II, M-theory, and heterotic frameworks using orientifolds or gravitational instantons for moduli stabilization.
• These vacua address challenges from no-go theorems and Swampland constraints, offering testbeds for holography and advances in quantum gravity.

A scale-separated AdS vacuum is an anti-de Sitter (AdS) solution in supergravity or string theory for which the Kaluza-Klein (KK) scale—set by the internal compactification manifold—is parametrically smaller than the AdS radius. This leads to the existence of a large hierarchy, $\lambda = L_{\text{KK}}/L_{\text{AdS}} \ll 1$, enabling a low-energy effective field theory decoupled from the higher-dimensional KK physics. The construction and explicit realization of scale-separated AdS vacua have been longstanding problems at the intersection of string phenomenology, flux compactifications, and Swampland constraints.

1. Definition, Criteria, and Significance

A vacuum exhibits scale separation if, for a family of compactifications labeled by an unbounded parameter (typically a large flux quantum $N$), the ratio

$$\lambda \equiv \frac{L_{\text{KK}}}{L_{\text{AdS}}} \to 0~\text{as}~N \to \infty$$

while preserving large internal volume ($\mathrm{Vol}_\text{int} \gg 1$) and weak string coupling ($g_s \ll 1$). This property guarantees that the effective lower-dimensional AdS theory is under parametric control, i.e., it is decoupled from higher-dimensional corrections and threshold effects (Coudarchet, 2023).

Parametric scale separation is essential for realizing controlled AdS vacua where the KK tower lies above the AdS scale, which may then admit a well-defined holographic dual and a reliable effective field-theory expansion. In absence of scale separation, as in standard Freund-Rubin compactifications, the KK and AdS scales are tied: $L_{\text{KK}} \sim L_{\text{AdS}}$.

2. Mechanisms and Prototype Constructions

Type II Flux Compactifications

Most known classical scale-separated AdS vacuum constructions utilize O-plane orientifold sources and RR/NSNS fluxes. A prominent class arises in massive type IIA on Calabi-Yau or $SU(3)$-structure manifolds with O6-planes and unconstrained $F_4$ flux quanta. In the DeWolfe-Giryavets-Kachru-Taylor (DGKT)-type models, moduli stabilization is achieved at tree level, and scale separation is realized by taking $F_4 \sim N \to \infty$, producing

$$g_s \sim N^{-3/4}, \quad \mathrm{Vol}_{6} \sim N^{3/2}, \quad \frac{L_{\text{AdS}}}{L_{\text{KK}}}\sim N^{1/2}\to\infty$$

(Shiu et al., 2022, Coudarchet, 2023, Tringas et al., 21 Apr 2025). Similar results hold for compactifications on nilmanifolds or twisted tori with metric fluxes (Tringas, 2023, Carrasco et al., 2023).

In type IIB, analogous vacua can be found with O5/O7-planes and RR/NS fluxes on rigid $SU(2)$-structure manifolds, where at least two unbounded flux quanta allow for large volume and small $g_s$, and again a parametric hierarchy emerges in the supergravity limit (Petrini et al., 2013).

M-Theory and Heterotic Compactifications

In M-theory, scale-separated AdS$_4$ Freund-Rubin vacua with weak $G_2$-holonomy arise by lifting massless IIA solutions with O6 sources. Here, the parametric separation stems from a decoupling between the Ricci curvature (set by the AdS parameter) and the first Laplacian eigenvalue on the internal $G_2$-manifold as the flux parameter grows, i.e., $L_{KK}^{-2}\gg |R_{\text{AdS}}|$ for large $n$ (Hemelryck, 29 Aug 2024).

A recent breakthrough has been the construction of truly scale-separated AdS$_3$ vacua in heterotic string theory, compactifying on $G_2$-structure manifolds with $H$-flux and smeared gravitational instantons. Moduli are stabilized, all fluxes quantized, and the only required negative-charge sources entering the Bianchi identity are gravitational rather than orientifold, evading previous no-go theorems (Tringas et al., 11 Nov 2025). In the heterotic AdS$_3$ vacua, parametric control and scale separation are achieved as $N \to \infty$: $$g_s \sim N^{5/2}/h_7^{1/2} \ll 1, \quad \mathrm{Vol} \sim N^{5/2}, \quad \frac{L_{\text{AdS}}}{L_{\text{KK}}} \sim N \to \infty$$ (Tringas et al., 11 Nov 2025).

3. No-go Theorems, Swampland Constraints, and the Role of O-planes

The existence of parametric scale separation is highly nontrivial, tightly constrained by no-go theorems. The Maldacena-Núñez theorem and its extensions show that, in classical two-derivative supergravity without negative tension sources or large dilaton gradients, vacua with $L_{\text{KK}} \ll L_{\text{AdS}}$ cannot exist; the curvature ratio is strictly $O(1)$ (Gautason et al., 2015, Coudarchet, 2023).

Orientifold planes play a central role in bypassing these constraints. When the orientifold tension gives a leading negative contribution to the scalar potential, the scaling of the KK and AdS scales in the supergravity regime always ensures scale separation as $g_s \to 0$ or $\mathrm{Vol} \to \infty$, independent of spacetime dimension $D$ and the O$p$-plane with $p < 7$ (Tringas et al., 21 Apr 2025). Table 1 summarizes the universality of O-plane induced scale separation:

O-plane	Minimal Tadpole Flux	Parametric Scale Separation?
O2	F$_4$	Yes (AdS$_3$)
O3	F$_3$	Yes (AdS$_3$, AdS$_4$)
O5	F$_1$	Yes (AdS$_3$–$_6$)
O6	F$_0$ (Romans mass)	Yes (AdS$_3$–$_7$)

However, when the leading potential term is internal curvature (as in compactifications on coset or isotropic spaces), the hierarchy is absent: $L_{\text{KK}}\sim L_{\text{AdS}}$, i.e., no scale separation (Tringas et al., 21 Apr 2025, Tsimpis, 2012, Font et al., 2019).

4. Moduli Stabilization, Quantization, and Explicit Scale Separation

In all controlled constructions, fully stabilized compactifications require quantization of fluxes and detailed balancing of tadpoles. The stabilization mechanism typically involves extremizing a real superpotential in lower-dimensional supergravity: $$P = \int_{M_7} \left[ e^{\phi/2} H_7 - e^{-\phi/2} *_{7}\Phi \wedge H + \frac12 \Phi \wedge d\Phi \right] / (4 \,\mathrm{Vol}^{-2})$$ which fixes all geometric moduli, dilaton, and flux components jointly (Tringas et al., 11 Nov 2025). Bianchi identities are solved by matching gauge instanton and gravitational instanton charges to the induced cohomology classes (e.g., via Eguchi-Hanson ALE spaces at orbifold singularities). This structure ensures all moduli stabilization is achieved in terms of quantized data.

For the heterotic AdS$_3$ example, setting the integer fluxes $(h_7, h, N, f)$, there is a unique solution for $e^{\phi}$ and each modulus $s^i$; all shape moduli and dilaton are fixed as explicit functions of the fluxes, and the parametric region $N \to \infty$, $h_7 \gg N^5$ ensures weak coupling, large radius, and complete suppression of quantum and $\alpha'$ corrections (Tringas et al., 11 Nov 2025).

5. Regimes of Control, Swampland Distance Conjecture, and Transitions

Scale-separated AdS vacua can interpolate, through controlled moduli trajectories, between scale-separated and non-separated regimes. For instance, adding a brane modulus (e.g., a D4-brane position field) allows discrete changes in flux quanta and smooth paths through moduli space, connecting vacua with and without scale hierarchy. These interpolations facilitate checks of the (Refined) Swampland Distance Conjecture: geodesic distances in field space scale as $\Delta \sim \log N$, and the observed scaling

$m_{\rm KK}/M_{\rm AdS} \sim e^{-\alpha\Delta}$

matches expectations from the conjecture (Shiu et al., 2022, Farakos et al., 2023, Farakos et al., 2023, Coudarchet, 2023).

A subtle point is the role of discrete gauge symmetries, such as $\mathbb{Z}_k$ domain wall symmetries. In scenarios like DGKT, these underlie the scale separation (giving the $\sqrt{k}$ scaling between KK and AdS scales), while in certain AdS$_3$ vacua the necessary higher-form symmetry is absent, possibly challenging their Swampland status (Buratti et al., 2020, Apers et al., 2022).

6. Generalizations, Dualities, and Model Taxonomy

Scale-separated vacua possess T- and S-dual descriptions. Massive IIA scale-separated AdS$_3$ vacua on $G_2$-holonomy orbifolds with O2/O6 sources can be dualized to type I backgrounds (IIB with O5/D5 and O9/D9 sources) on $G_2$-structure spaces, which in turn enable a heterotic (SO(32)) dual description with preserved scale separation and identical moduli stabilization (Miao et al., 16 Sep 2025). This duality structure is crucial for exploring the broader landscape of scale-separated AdS vacua across string theories.

The structure and stabilization of the moduli space, the integer nature of the dual conformal dimensions in certain models, and the diverse realization (e.g., in type IIB, heterotic, type I, and M-theory) all point to a complex taxonomy of AdS vacua with parametric scale separation. Notably, all controlled constructions require either orientifold planes or gravitational instantons to source the necessary leading negative contribution to the potential.

7. Open Problems and Outlook

While explicit constructions of scale-separated AdS vacua now exist across string theories—including in heterotic string theory with only gravitational instantons (Tringas et al., 11 Nov 2025)—several challenges and open problems remain:

• For all models with O-plane or gravitational instanton sources, constructing fully localized (non-smeared) solutions and understanding their $g_s$ and $\alpha'$ corrections is an active area of research (Hemelryck, 29 Aug 2024, Cribiori et al., 2021).
• The holographic duals of scale-separated AdS backgrounds are not (yet) explicit, and their operator spectra may exhibit peculiar properties (e.g., integer conformal dimensions, in certain cases), raising questions about consistent large-gap AdS/CFT duals (Coudarchet, 2023).
• The precise interplay between Swampland conjectures (AdS Distance, Spin-2, Weak Gravity) and scale separation, especially in regimes lacking discrete higher-form symmetries, is under investigation (Shiu et al., 2022, Apers et al., 2022, Buratti et al., 2020).
• Extensions to dS vacua are challenging; flux quantization generally precludes controlled de Sitter critical points in these frameworks (Tringas et al., 21 Apr 2025).

Scale-separated AdS vacua remain central to the exploration of the landscape/swampland boundary, furnishing key laboratories for string, M-, and heterotic theories to test the limits of consistent quantum gravity backgrounds.