DGKT AdS₄ Vacua

• DGKT AdS₄ vacua are fully stabilized compactifications emerging from massive type IIA string theory with a no-scale scalar potential that fixes all moduli.
• They achieve parametric scale separation by tuning the F₄-flux, resulting in an AdS₄ radius vastly larger than the Kaluza–Klein scale in a controlled, large volume regime.
• Their unique mass spectrum yields integer conformal dimensions for dual operators, underpinning holographic consistency through vanishing extremal cubic couplings.

DGKT AdS$_4$ vacua refer to a family of four-dimensional anti-de Sitter (AdS$_4$) flux compactifications originally constructed in massive type IIA string theory. These vacua, associated with the work of DeWolfe, Giryavets, Kachru, and Taylor (DGKT), realize full moduli stabilization at tree level, exhibit parametric scale separation between the AdS radius and the Kaluza–Klein (KK) scale, and possess distinctive features such as integer conformal dimensions in the spectrum of dual operators. The DGKT solutions have provided a cornerstone in the study of geometric and physical properties of AdS vacua arising from string flux compactifications.

1. Geometric Setup and Vacua Construction

DGKT AdS$_4$ vacua arise from compactifications of massive type IIA supergravity on Calabi–Yau orientifolds $X_6$ with O6-plane involution. The ten-dimensional spacetime is of the form AdS$_4 \times X_6$, with the metric ansatz in string frame given by

$$ds_{10}^2 = e^{2A(y)} ds^2_{\mathrm{AdS}_4} + ds^2_{X_6}(y)\,,$$

where $A(y)$ is a warp factor and $ds^2_{X_6}$ encodes an $\mathrm{SU}(3)\times \mathrm{SU}(3)$-structure deformation of the Calabi–Yau metric (Marchesano et al., 2020). The geometry includes the following fluxes:

• RR 0-form (Romans mass): $F_0 = m$,
• RR 4-form: $F_4 = e_A \tilde{\omega}^A$ (where, crucially, $e_A$ can be taken large and is unconstrained by tadpoles),
• NSNS 3-form: $H_3 = h^K \alpha_K$,
• Localized O6-plane sources ensuring tadpole cancellation.

The background admits a unique AdS$_4$ critical point with all geometric moduli stabilized, determined by extremizing the $4\mathrm{d}$ scalar potential, which takes a no-scale form in terms of Kähler and superpotentials (Apers et al., 2022, Apers, 2022). The Kähler moduli, axio-dilaton, and any residual complex-structure moduli are frozen at tree level, and the size of the compactification space and the AdS$_4$ radius can be made arbitrarily large by dialing the unconstrained $F_4$-flux (i.e., large $e_A$).

2. Moduli Stabilization and Scale Separation

A central property of DGKT vacua is their complete moduli stabilization at the classical level, with the F-term equations solved algebraically for all geometric moduli. At the critical point: $$t^a \propto \frac{e_a}{m}\,, \quad s \sim \frac{m h}{\mathcal{K}^{1/2}}\,, \quad V_{\min} \sim -N^{-9/2}\,,$$ where $N \sim e_a$ is the large, unconstrained integer parameterizing the $F_4$ flux, $t^a$ are Kähler volumes, and $s$ is the 4d dilaton (Apers et al., 2022, Apers, 2022). The AdS$_4$ radius $L$ and the KK scale $m_{\textrm{KK}}$ scale as

$$L^2 \sim N^{9/2}\,, \quad m_{\textrm{KK}} \sim N^{-1/4}\,,$$

implying $L m_{\textrm{KK}}\sim N \to \infty$ for $N\gg 1$. The string coupling $g_s$ decreases with $N$ as $g_s \sim N^{-3/4}$ (Apers, 2022, Andriot et al., 2023). Thus, the vacua are in a regime of large volume, weak coupling, and strong scale separation, where the AdS radius can be made parametrically larger than the inverse compactification scale.

3. Ten-Dimensional Uplift and SU(3)×SU(3)-Structure

The ten-dimensional uplift of the DGKT vacua involves an $\mathrm{SU}(3)\times \mathrm{SU}(3)$-structure on $X_6$ with Majorana–Weyl internal spinors. The leading solution is “smeared” (i.e., with delocalized O6-planes), but perturbative expansions allow the inclusion of localized sources and backreacted warp factors. To first order in $\epsilon\sim g_s \sim \mu/F_0$, the internal metric, dilaton, and fluxes receive controlled deformations, but the key vacuum features (moduli stabilization, integer conformal dimensions, flux quantization) persist (Marchesano et al., 2020, Andriot et al., 2023). All four supersymmetry and Bianchi equations are satisfied up to $O(\epsilon^2)$. Warped and partially localized extensions have also been constructed, confirming the robustness of the mass spectrum (Andriot et al., 2023).

4. Mass Spectrum and Integer Conformal Dimensions

Expanding about the unique AdS$_4$ minimum, the canonically normalized scalar fields (saxions and axions) have masses $m^2$ related to the AdS radius $L$ by $m^2 L^2 = \Delta(\Delta - 3)$. Strikingly, DGKT vacua yield integer conformal dimensions for all light moduli:

• Saxions: $\Delta = 10, 6, 6, 6$,
• Axions: $\Delta = 11, 5, 5, 5$ (Apers, 2022, Andriot et al., 2023).

The operator duals of the moduli in the would-be CFT$_3$ thus have integer scaling dimensions, a rare property among flux vacua. Polynomial spacetime-dependent shift symmetries in AdS$_4$ correspond directly to these integer values (Apers, 2022, Apers et al., 2022). Corrections to the mass spectrum, including large-$n$ deformations and partial localization, do not alter the integer conformal dimensions at leading order (Andriot et al., 2023).

5. Physical Interpretation and Holography

The $F_4$ flux in DGKT can be mapped holographically to a stack of $N$ D4-brane domain walls intersecting orthogonally, with the near-horizon limit giving rise to the AdS$_4$ throat. The central charge and free energy scale as $c \sim N^{9/2}$, distinct from standard AdS$_4$/M2-brane systems where $c \sim N^{3/2}$ (Apers, 2022, Apers et al., 3 Jun 2025). The backtracking procedure applied to DGKT recovers a strongly coupled, non-conical singularity in massive IIA, whose near-horizon CFT is conjectured to be an $\mathcal N=1$ SU($N$) gauge theory with a Chern–Simons term from the Romans mass (Apers et al., 3 Jun 2025). This is in sharp contrast to the massless-IIA (no Romans mass) case, which yields a weakly coupled conical singularity and standard M2-brane scaling.

A significant recent advance establishes a new holographic consistency criterion: extremal cubic scalar bulk couplings (where operator dimensions sum as $\Delta_k=\Delta_i+\Delta_j$) must vanish to preserve large-$N$ CFT factorization. Explicit analysis in the DGKT context confirms a nontrivial cancellation of all such couplings, establishing holographic completeness and consistency with AdS/CFT expectations (Bobev et al., 11 Dec 2025).

6. Extensions, Swampland Conditions, and Open Issues

Two major classes of extensions have been considered:

• Warped and backreacted localized models;
• Deformations in the large-$n$ expansion affecting fluxes and moduli fields (Andriot et al., 2023).

All known corrections leave the mass spectrum invariant at leading order, reinforcing the persistence of scale separation and protected properties. The space of DGKT vacua admits a positive, flux-independent metric, as derived from the off-shell quadratic variation of the effective action, supporting both the AdS Distance Conjecture and the related “Metric Positivity” swampland condition (Palti et al., 2 May 2024).

Non-supersymmetric analogues (“DGKT-like” branches) have also been constructed, some of which satisfy sharpened Weak Gravity Conjecture criteria, with superextremal membrane probes realized via wrapped D8/D6-branes (Marchesano et al., 2022). The perturbative and non-perturbative stability of these branches has been systematically addressed.

7. Summary Table: Key Properties of DGKT AdS$_4$ Vacua

Feature	Value/Scaling	Origin/Paper
Moduli stabilization	Complete at tree-level	(Apers, 2022)
Scale separation	$L_{AdS}m_{KK}\sim N \to \infty$	(Apers, 2022, Andriot et al., 2023)
String coupling	$g_s \sim N^{-3/4}$	(Apers et al., 2022)
Operator dimensions	Integers: $\{10,6,6,6;11,5,5,5\}$	(Apers, 2022, Bobev et al., 11 Dec 2025)
Central charge	$c \sim N^{9/2}$	(Apers, 2022, Apers et al., 3 Jun 2025)
Flux metric over vacua	$G_{\sigma\sigma} = 3376/27$	(Palti et al., 2 May 2024)
Extremal cubic couplings	All vanish	(Bobev et al., 11 Dec 2025)
Holographic brane picture	Strongly coupled, non-conical	(Apers et al., 3 Jun 2025)

The DGKT AdS$_4$ vacua serve as the prototypical example of scale-separated, fully stabilized string compactifications, with uniquely robust features that make them a central testing ground for swampland conjectures, holographic completeness, and the structure of AdS/CFT in minimal supersymmetric flux vacua.