A note on the holographic consistency of DGKT-type vacua with $h^{2,1}=0$

Technical Summary: A note on the holographic consistency of DGKT-type vacua with $h^{2,1} = 0$

Problem Statement
The existence of scale-separated Anti-de Sitter (AdS) vacua in String Theory remains a debated topic. While classical constructions like the DGKT scenario (massive Type IIA with O6-planes) achieve parametric separation between the AdS radius ($R_{AdS}$) and the Kaluza-Klein scale ($R_{KK}$), their consistency as effective field theories is scrutinized. Recent work [46] established a holographic consistency condition for such vacua: in the putative dual Conformal Field Theory (CFT), certain "extremal" three-point functions of scalar moduli must vanish. An extremal arrangement is defined by conformal dimensions satisfying $\Delta_i = \Delta_j + \Delta_k$.

Previous analysis [46] confirmed this vanishing condition for the simplest DGKT vacuum based on the $T^6/(\mathbb{Z}_3 \times \mathbb{Z}_3)$ orbifold. However, subsequent findings [47] indicated that for compactifications with complex structure moduli ($h^{2,1} \neq 0$), this constraint is typically violated due to the proliferation of extremal arrangements. The central problem addressed in this work is whether the holographic constraint persists in more complex DGKT-type vacua that lack complex structure deformations ($h^{2,1} = 0$) but possess reduced symmetry and more intricate triple-intersection numbers compared to the original orbifold.

Methodology
The authors analyze the DGKT construction on toroidal orientifolds classified in [49], specifically focusing on models where $h^{2,1} = 0$. The methodology involves:

1. Effective Field Theory (EFT) Setup: The 4D $\mathcal{N}=1$ supergravity description is utilized, where the Kähler moduli $t_a = b_a + i v_a$ and the axio-dilaton $N = \xi + iz$ are stabilized by fluxes ($F_0, F_4, F_6, H_3$). The scalar potential is derived from the superpotential $W$ and Kähler potential $K$.
2. Spectrum Analysis: The authors identify the conformal dimensions of the scalar moduli in the large flux limit. For $h^{2,1}=0$, the spectrum consists of integer dimensions: saxions with $\Delta_s = \{10, 6, \dots\}$ and axions with $\Delta_a = \{11, 5, \dots\}$. This leads to two specific extremal triplets: $(10, 5, 5)$ and $(11, 6, 5)$.
3. Constraint Verification: The holographic constraint requires the effective cubic coupling $c'_{ijk}$ to vanish for these extremal triplets. This coupling is a linear combination of the scalar potential cubic terms ($c_{ijk}$) and derivative interaction terms ($d_{ijk}$):
   $$c'_{ijk} = c_{ijk} + \frac{m_i^2 - m_j^2 - m_k^2}{2} d_{ijk}$$
   Due to 3D $\mathcal{N}=1$ supersymmetry relations (detailed in Appendix B), verifying the vanishing of the coupling for the $(10, 5, 5)$ triplet is sufficient to imply the vanishing for the $(11, 6, 5)$ triplet.
4. Explicit Computation: The authors explicitly compute the vacuum expectation values, kinetic matrices, and mass matrices for three distinct classes of models (II/IV, III, and V) from the classification in [49]. This involves diagonalizing non-trivial kinetic matrices (which are not diagonal in the original coordinates for models III and V) and transforming the cubic couplings into the mass eigenstate basis.

Key Contributions and Results
The paper provides explicit verification of the holographic consistency condition for DGKT vacua beyond the original $T^6/(\mathbb{Z}_3 \times \mathbb{Z}_3)$ example:

• Models II and IV: These models share the same untwisted sector as the original DGKT orbifold but with identified Kähler moduli. The authors confirm that the cancellation $c'_{\phi_0 a_i a_j} = 0$ holds, consistent with the simpler case.
• Model III: This model features a non-diagonal kinetic matrix and lower symmetry. Despite the increased computational complexity, the authors demonstrate that the effective cubic coupling vanishes exactly.
• Model V: This is the most complex example considered, involving six Kähler moduli and a highly non-trivial volume function. Through explicit diagonalization and basis transformation, the authors show that the cancellation persists.

In all examined cases, the effective coupling for the $(10, 5, 5)$ arrangement takes the form:
$$c'_{\phi_0 a_i a_j} = 0$$
The individual contributions $c_{ijk}$ and $d_{ijk}$ exhibit a clear pattern: the diagonal entries for the $h^{1,1}_- - 1$ volume-preserving deformations are proportional to $(-100/\sqrt{13}, 4/\sqrt{13})$, while the entry corresponding to the overall volume mixing with the dilaton is proportional to $(200/(13\sqrt{13}), -8/(13\sqrt{13}))$. These specific ratios ensure the exact cancellation required by the holographic constraint.

Significance and Claims
The authors claim that their results extend the evidence for the holographic consistency of DGKT-type vacua to a broader class of geometries with $h^{2,1} = 0$. The persistence of the cancellation across models with varying topological data (different triple intersection numbers and reduced symmetries) suggests that this property is not an artifact of the high symmetry of the original $T^6/(\mathbb{Z}_3 \times \mathbb{Z}_3)$ orbifold.

The paper posits that these cancellations likely hold more generally for arbitrary Calabi-Yau orientifolds with $h^{2,1} = 0$, drawing a parallel to the universality of the low-lying spectrum observed in [37]. The authors explicitly state that the detailed analysis of these explicit examples provides the necessary insight to generalize this conclusion to arbitrary Calabi-Yau compactifications, a task they intend to address in future work [48]. Conversely, they reiterate that for $h^{2,1} \neq 0$, violations of the constraint are expected, aligning with previous findings [47].

The work does not propose new experimental applications or modify the DGKT construction itself; rather, it serves as a consistency check within the AdS/CFT framework, supporting the viability of these classical supergravity solutions as duals to consistent CFTs with a parametric gap.