Circle Compactification of DGKT & CFI

• The paper introduces a circle compactification method that reduces 4D flux compactifications to 3D AdS vacua with extended N=2 supersymmetry.
• The approach employs additional fluxes threading the circle to achieve moduli stabilization at large volume and weak coupling with a clear KK-AdS scale separation.
• The construction provides explicit Kähler and superpotential formulations that enable detailed exploration of non-integer conformal dimensions in the dual CFT.

The circle compactification of DGKT (DeWolfe–Girsy–Kachru–Taylor) and CFI (Cribiori–Farakos–Iatrakis) refers to a concrete procedure for constructing scale-separated three-dimensional AdS vacua with extended $\mathcal{N}=2$ supersymmetry in massive type IIA supergravity. These vacua arise by reducing known four-dimensional flux compactifications on a further $S^1$, combined with the introduction of new fluxes threading the circle. This approach provides fully stabilized solutions with all moduli fixed at large volume, weak coupling, and a parametrically large hierarchy between the Kaluza–Klein (KK) and AdS scales, thereby achieving genuine scale separation. The construction applies both to the original DGKT model, based on a $T^6/\mathbb{Z}_3^2$ orientifold, and to the toroidal $\mathbb{Z}_2 \times \mathbb{Z}_2$-orientifold sector known as CFI, each extended with additional circle fluxes that result in three-dimensional supergravity descriptions featuring explicit Kähler and superpotential terms (Cribiori et al., 29 Apr 2026).

1. Ten-Dimensional Starting Point and Compactification Ansatz

The initial setup considers type IIA supergravity on a 7-torus constructed as $T^6 / \mathbb{Z}_3^2 \times S^1$, where the $T^6$ is realized as three complex planes, each quotiented by $\mathbb{Z}_3^2$, and $S^1$ parametrizes the circle direction. The ten-dimensional string frame metric takes the form

$$ds_{10}^2 = e^{2A(\hat{y})} ds_{AdS_3}^2 + ds_7^2, \qquad ds_7^2 = \sum_{i=1}^3 r_i^2 (d\hat{y}^i{}^2 + d\hat{y}^{i+3}{}^2) + r_7^2 d\hat{y}^7{}^2,$$

with a (constant) warp factor $A$ and radii $S^1$0, $S^1$1 for the $S^1$2 and $S^1$3 respectively. The construction commonly assumes an isotropic Ansatz $S^1$4, leading to moduli $S^1$5 and $S^1$6, so that $S^1$7. Fluxes of the RR and NSNS sectors are introduced, with $S^1$8 (Romans mass), and $S^1$9 and $T^6/\mathbb{Z}_3^2$0 expanded on the harmonic forms $T^6/\mathbb{Z}_3^2$1 and their Hodge duals $T^6/\mathbb{Z}_3^2$2 on $T^6/\mathbb{Z}_3^2$3. Additional fluxes $T^6/\mathbb{Z}_3^2$4 and $T^6/\mathbb{Z}_3^2$5 thread the extra $T^6/\mathbb{Z}_3^2$6 factor, subject to the constraint $T^6/\mathbb{Z}_3^2$7 to avoid D2/O2 tadpoles.

Local sources include O6-planes and D6-branes wrapping appropriate 3-cycles to cancel the $T^6/\mathbb{Z}_3^2$8-tadpole. In the smeared approximation, the source currents are $T^6/\mathbb{Z}_3^2$9 and $\mathbb{Z}_2 \times \mathbb{Z}_2$0 with $\mathbb{Z}_2 \times \mathbb{Z}_2$1, $\mathbb{Z}_2 \times \mathbb{Z}_2$2. The algebraic constraints ensuring tadpole cancellation and supersymmetry for fluxes and sources are

$\mathbb{Z}_2 \times \mathbb{Z}_2$3

with further sign constraints $\mathbb{Z}_2 \times \mathbb{Z}_2$4 for the supersymmetric branch.

2. Reduction to Three Dimensions: Light Moduli and Effective Data

Compactification on the $\mathbb{Z}_2 \times \mathbb{Z}_2$5 yields a three-dimensional $\mathbb{Z}_2 \times \mathbb{Z}_2$6 supergravity theory containing the following moduli:

• The dilaton–volume modulus $\mathbb{Z}_2 \times \mathbb{Z}_2$7, with $\mathbb{Z}_2 \times \mathbb{Z}_2$8,
• Three Kähler moduli $\mathbb{Z}_2 \times \mathbb{Z}_2$9 representing 2-cycles of $T^6 / \mathbb{Z}_3^2 \times S^1$0, $T^6 / \mathbb{Z}_3^2 \times S^1$1,
• The circle modulus $T^6 / \mathbb{Z}_3^2 \times S^1$2, $T^6 / \mathbb{Z}_3^2 \times S^1$3, in addition to their axionic superpartners.

The three-dimensional Kähler potential is

$T^6 / \mathbb{Z}_3^2 \times S^1$4

while the superpotential receives contributions both from the inherited DGKT piece,

$T^6 / \mathbb{Z}_3^2 \times S^1$5

and from the circle-flux corrections,

$T^6 / \mathbb{Z}_3^2 \times S^1$6

where the sum is over cyclic permutations of $T^6 / \mathbb{Z}_3^2 \times S^1$7 and the isotropic sector $T^6 / \mathbb{Z}_3^2 \times S^1$8 simplifies the structure. The total superpotential is $T^6 / \mathbb{Z}_3^2 \times S^1$9.

3. Scalar Potential, Moduli Stabilization, and Scale Separation

The F-term scalar potential in 3d $T^6$0 supergravity takes the form

$T^6$1

Expanding in the isotropic case and setting axions to zero, the potential becomes a function of $T^6$2, and flux parameters. The explicit form includes contributions depending on fluxes $T^6$3 and enforces the D2/O2-tadpole constraint via the $T^6$4 combination.

Supersymmetric AdS$T^6$5 vacua are found by solving $T^6$6, leading to stabilized moduli: $T^6$7 and AdS scale $T^6$8. For large flux $T^6$9, this configuration guarantees large internal volume $\mathbb{Z}_3^2$0, weak coupling $\mathbb{Z}_3^2$1, and a scale-separated regime $\mathbb{Z}_3^2$2 as $\mathbb{Z}_3^2$3.

4. Dual CFT Data and Conformal Dimensions

The stabilized AdS$\mathbb{Z}_3^2$4 vacuum allows computation of the spectrum of untwisted real scalar masses, translated to conformal dimensions of putative dual two-dimensional operators via $\mathbb{Z}_3^2$5. For the DGKT–circle model, the resulting conformal dimensions are

$\mathbb{Z}_3^2$6

none of which are integers. This non-integrality distinguishes the theory from standard scenarios where protected multiplets would yield integer conformal dimensions.

5. CFI Extension and Generalization

The CFI sector is constructed by starting from the toroidal $\mathbb{Z}_3^2$7 orientifold (with Hodge number $\mathbb{Z}_3^2$8), with the four-dimensional superpotential

$\mathbb{Z}_3^2$9

and circle-flux corrections incorporated upon compactification. Specifically, in three dimensions the corrected superpotential is

$S^1$0

with the same Kähler potential as before augmented by $S^1$1. The resulting F-term scalar potential reproduces the ten-dimensional uplift and, as in the DGKT case, yields a fully stabilized AdS$S^1$2 vacuum with identical flux scaling and non-integer conformal dimensions (in this case, 16 values).

6. Significance and Outlook

The construction of circle-compactified DGKT and CFI vacua provides first explicit examples of scale-separated solutions with extended supersymmetry and all moduli stabilized at large volume and weak coupling. The existence of these solutions with AdS$S^1$3 geometry and non-integer conformal dimensions for the dual CFT operators presents new avenues for addressing the challenge of scale separation in string theory. The explicit determination of the Kähler potential, superpotential, and F-term potential in three dimensions facilitates further exploration of holographic correspondences, moduli stabilization, and flux compactification mechanisms in settings with extended supersymmetry (Cribiori et al., 29 Apr 2026).