---
title: Circle Compactification of DGKT & CFI
url: https://www.emergentmind.com/topics/circle-compactification-of-dgkt-and-cfi
type: topic
---

# Circle Compactification of DGKT & CFI

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 [2604.26755].

## 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 $r_i$, $r_7$ for the $T^6$ and $S^1$ respectively. The construction commonly assumes an isotropic Ansatz $r_1 = r_2 = r_3 \equiv r$, leading to moduli $s \equiv r^3$ and $t \equiv r^2 r_7$, so that $\mathrm{Vol}_7 = s^{4/3} t$. Fluxes of the RR and NSNS sectors are introduced, with $F_0 = m$ (Romans mass), and $H_3$ and $F_4$ expanded on the harmonic forms $\Phi_i$ and their Hodge duals $\Psi_i$ on $T^6 \times S^1$. Additional fluxes $h'$ and $f'$ thread the extra $S^1$ factor, subject to the constraint $F_4 \wedge H_3 = 0$ to avoid D2/O2 tadpoles.

Local sources include O6-planes and D6-branes wrapping appropriate 3-cycles to cancel the $F_2$-tadpole. In the smeared approximation, the source currents are $J_{O6} = Q_{O6}(\Phi_1 + \dots + \Phi_4)$ and $J_{D6} = Q_{D6}(\Phi_5 + \Phi_6 + \Phi_7)$ with $Q_{O6} < 0$, $Q_{D6}>0$. The algebraic constraints ensuring tadpole cancellation and supersymmetry for fluxes and sources are
\[
4f' h + 3f h' = 0, \qquad h m + Q_{O6} = 0, \qquad h' m + Q_{D6} = 0,
\]
with further sign constraints $fh > 0,\, f'h' > 0,\, hm > 0,\, h'm < 0$ for the supersymmetric branch.

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

Compactification on the $S^1$ yields a three-dimensional $\mathcal{N}=2$ supergravity theory containing the following moduli:
- The dilaton–volume modulus $S$, with $\operatorname{Re} S = e^{-\phi} \sqrt{\mathrm{Vol}_6}$,
- Three Kähler moduli $T_i$ representing 2-cycles of $T^6$, $\operatorname{Re} T_i = r_i r_{i+3} =: t_i / r_7$,
- The circle modulus $R$, $\operatorname{Re} R = (r_7 \operatorname{Re} S)^2$,
in addition to their axionic superpartners. 

The three-dimensional Kähler potential is
\[
K = -\log(R + \overline{R}) - \sum_{i=1}^3 \log(T_i + \overline{T}_i) - 4 \log(S + \overline{S}),
\]
while the superpotential receives contributions both from the inherited DGKT piece,
\[
W_{\mathrm{DGKT}} = 2i h S - \frac{i}{2} f \sum_{i=1}^3 T_i + 2i m T_1 T_2 T_3,
\]
and from the circle-flux corrections,
\[
W_{cf} = +\frac{i h'}{2} S^2 \sum_{\text{cyc}} \sqrt{\frac{T_j T_k}{R T_i}} - 2i f' S \sqrt{\frac{T_1 T_2 T_3}{R}},
\]
where the sum is over cyclic permutations of $\{i,j,k\} = \{1,2,3\}$ and the isotropic sector $T_i \equiv T$ simplifies the structure. The total superpotential is $W = W_{\mathrm{DGKT}} + W_{cf}$.

## 3. Scalar Potential, Moduli Stabilization, and Scale Separation

The F-term scalar potential in 3d $\mathcal{N}=2$ supergravity takes the form
\[
V = e^K (K^{I \bar J} D_I W\, \overline{D_J W} - 4 |W|^2), \qquad I,J \in \{S, T_i, R\}.
\]
Expanding in the isotropic case and setting axions to zero, the potential becomes a function of $s,t,g_s = e^\phi$, and flux parameters. The explicit form includes contributions depending on fluxes $m, h, h', f, f'$ and enforces the D2/O2-tadpole constraint via the $4 f' h + 3 f h' = 0$ combination.

Supersymmetric AdS$_3$ vacua are found by solving $D_I W = 0$, leading to stabilized moduli:
\[
s = \frac{7^{3/4}}{2\sqrt{2}} \left( \frac{f}{m} \right)^{3/4}, \qquad
t = 2\sqrt{2}\,7^{3/4} \frac{f'}{m} \left( \frac{m}{f} \right)^{1/4}, \qquad
g_s = \frac{2\sqrt{2}}{3} 7^{1/4} \frac{h}{m} \left( \frac{m}{f} \right)^{3/4},
\]
and AdS scale $L_{AdS}^{-1} = \mu = - \frac{7^{1/4}}{3\sqrt{2}} h \left( \frac{m}{f} \right)^{3/4}$. For large flux $N \equiv f \simeq f'$, this configuration guarantees large internal volume $\mathrm{Vol}_7 \sim N^{7/4}$, weak coupling $g_s \sim N^{-3/4}$, and a scale-separated regime $L_{KK} / L_{AdS} \sim N^{-1/2} \ll 1$ as $N \to \infty$.

## 4. Dual CFT Data and Conformal Dimensions

The stabilized AdS$_3$ vacuum allows computation of the spectrum of untwisted real scalar masses, translated to conformal dimensions of putative dual two-dimensional operators via $\Delta = 1 + \sqrt{1 + m^2 L_{AdS}^2}$. For the DGKT–circle model, the resulting conformal dimensions are
\[
\Delta \in \{3.57,\,3.57,\,4.57,\,4.57,\,1.91,\,3.18,\,8.94,\,2.91,\,4.18,\,7.94\},
\]
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 $\mathbb{Z}_2 \times \mathbb{Z}_2$ orientifold (with Hodge number $h^{2,1} = 3$), with the four-dimensional superpotential
\[
W_{CFI,4d} = \frac{i}{2}h\,(S+U_1+U_2+U_3) - \frac{i}{2} f\, \sum T_i + 2i m T_1 T_2 T_3,
\]
and circle-flux corrections incorporated upon compactification. Specifically, in three dimensions the corrected superpotential is
\[
W_{cf}' = i\frac{h'}{2}\sqrt{S\,U_1U_2U_3} \sum_{\rm cyc} \sqrt{\frac{T_jT_k}{R\,T_i}}
\;-\;\frac{i\,f'}{2}\sqrt{S\,U_1U_2U_3} \left(\frac{1}{S}+\sum_{i=1}^3\frac{1}{U_i}\right) \sqrt{\frac{T_1T_2T_3}{R}},
\]
with the same Kähler potential as before augmented by $\sum \log(U_i+\overline{U}_i)$. The resulting F-term scalar potential reproduces the ten-dimensional uplift and, as in the DGKT case, yields a fully stabilized AdS$_3$ 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$_3$ 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 [2604.26755].

Source: https://www.emergentmind.com/topics/circle-compactification-of-dgkt-and-cfi