Scale-separated vacua with extended supersymmetry

Abstract: We propose the first examples of scale-separated vacua with extended supersymmetry. They arise as circle compactifications of four-dimensional vacua of massive type IIA supergravity with scale separation, upon introducing additional fluxes and sources. We provide both the ten-dimensional solutions and the three-dimensional effective descriptions in terms of Kähler potential and superpotential. The conformal dimensions of the putative dual two-dimensional field theory appear not to be integers. The superpotential for the additional fluxes of one of our models was guessed by ChatGPT and, to the best of our knowledge, it does not appear in existing literature. Should these vacua be solutions of string theory, they would allow to address the open problem of scale separation from the vantage point of extended supersymmetry.

• The paper presents the first explicit AdS3 vacua with extended N=2 supersymmetry and genuine scale separation, achieved via circle compactification of DGKT and CFI models.
• It introduces new S1 flux components that stabilize additional moduli, ensuring a controlled hierarchy between the AdS length and the Kaluza–Klein scale.
• Key findings include complete moduli stabilization and non-integer conformal dimensions in the dual CFT, challenging existing swampland conjectures.

Scale-Separated Vacua with Extended Supersymmetry

Introduction and Motivation

The construction of string vacua exhibiting genuine scale separation—the hierarchy between the AdS length and the compactification (Kaluza-Klein) scale—remains of fundamental importance for addressing the emergence of lower-dimensional physics from higher-dimensional string theory. Previous explicit examples of scale-separated vacua are sparse and are predominantly constrained to AdS solutions with minimal supersymmetry. Furthermore, strong swampland conjectures and various bottom-up and holographic arguments have cast doubt on the existence of such vacua with higher supersymmetry. No explicit construction of scale-separated vacua with extended supersymmetry has been presented prior to this work.

The paper "Scale-separated vacua with extended supersymmetry" (2604.26755) constructs and analyzes—for the first time—explicit examples of three-dimensional AdS string vacua with parametric scale separation and $\mathcal{N}=2$ supersymmetry, originating from circle reductions of four-dimensional massive type IIA flux vacua (DGKT and CFI models) with suitable fluxes and source content. These models bridge significant gaps in the landscape of flux compactifications and string vacua, allowing new perspectives on familiar theoretical questions.

Construction of Models

Circle Compactification of DGKT and CFI

The starting point of the analysis involves the well-known DGKT [DeWolfe:2005uu] and CFI [Camara:2005pr] constructions, which yield four-dimensional $\mathcal{N}=1$ AdS vacua with all moduli stabilized, realized via toroidal orbifolds of Calabi–Yau threefolds equipped with orientifold $O6$-planes, fluxes, and sources.

The models in this paper are obtained via circle compactification to three dimensions, with an internal space $X_7 = (T^6/\Gamma) \times S^1$ (with $\Gamma = \mathbb{Z}_3^2$ for DGKT and $\mathbb{Z}_2^2$ for CFI), with appropriate modifications of fluxes and the inclusion of additional source contributions.

Additional Fluxes and Sources

A key technical innovation is the introduction of new flux components along the $S^1$, leading to the necessary stabilization of the would-be modulus associated with the extra circle, which would otherwise obstruct scale separation by pinning the AdS curvature to the Kaluza–Klein scale. In both models, the four-form flux $F_4$ is augmented by components corresponding to four-cycles involving the new $S^1$, while $H_3$ is simultaneously generalized to maintain compatibility with orbifold symmetry and orientifold parity constraints. The Bianchi identities and D-brane/O-plane tadpole cancellations are managed by inserting D6-branes and ensuring mutually supersymmetric configurations.

Ten-Dimensional Supergravity and Moduli Stabilization

The analysis pursues explicit solutions to the ten-dimensional (massive type IIA) supergravity equations, implementing the Killing spinor equations, Bianchi identities, and flux quantization conditions. For both the DGKT and CFI-based models, the authors construct the internal metric, harmonic bases of forms, and explicit flux ansätze, confirming algebraically that for large flux quantum numbers, all geometric moduli and the string coupling are stabilized at parametrically large volume and weak coupling. The essential scaling limit is

$\mathcal{N}=1$0

which yields scale separation that becomes arbitrarily sharp as $\mathcal{N}=1$1. The analysis is validated for both untwisted and, qualitatively, for twisted moduli via scaling.

Three-Dimensional Effective Supergravity

$\mathcal{N}=1$2 AdS Vacua and Superpotentials

Dimensional reduction further produces effective $\mathcal{N}=1$3 supergravity models in three dimensions. The authors extract explicit Kähler potentials and superpotentials, constructed by lifting the known four-dimensional effective field theories and incorporating the effects of the new fluxes. In particular, for DGKT, the nontrivial contributions to the three-dimensional superpotential can be deduced from symmetry and matching conditions; in the CFI case, the form of the superpotential was obtained using an LLM (ChatGPT), exhibiting structure not previously derived in the literature.

Strong claims are made here: the explicit Kähler and superpotential yield complete classical moduli stabilization, with full agreement between the critical points of the effective three-dimensional theory and the ten-dimensional solutions.

Absence of Partial Supersymmetry Breaking

The possibility that the effective theory could realize extended supersymmetry only partially—by virtue of D-term $\mathcal{N}=1$4 F-term structure—was investigated. Arguments based on the structure of three-dimensional $\mathcal{N}=1$5 supergravity, the structure of the possible isometries, and the explicit forms of the superpotentials preclude any avenue towards partial SUSY breaking: the vacua preserve exactly $\mathcal{N}=1$6 supersymmetry. This is particularly relevant given the intricate relation between scalar potentials and the presence/absence of R-symmetry gaugings and D-terms.

Conformal Dimensions in the Holographic Dual

For both models, the analysis of operator conformal dimensions in the putative dual two-dimensional CFTs reveals a significant result: all computed conformal dimensions are non-integer, in contrast with the four-dimensional DGKT and CFI ancestors. The occurrence of non-integer dimensions demonstrates that neither scale separation nor extended supersymmetry necessarily enforce the integrality condition, deepening the complexity of the AdS/CFT mapping in these vacua.

Implications, Limitations, and Future Work

Significance for the Swampland and String Landscape

These constructions provide explicit counterexamples to the prevailing expectation that scale-separated vacua with extended supersymmetry are either non-existent or belong to the swampland. The models allow for parametric control over moduli stabilization and the UV/IR hierarchy, broadening the landscape of possible phenomenological and holographic studies.

Limitations

Smeared brane and O-plane approximations are employed throughout, as is common in the majority of the literature, but recent progress in localized source backreaction [Junghans:2020acz, Marchesano:2020qvg, Junghans:2023yue] may ultimately require refinement or even modification of the ten-dimensional setups. Moreover, the full inclusion and analysis of the twisted sector, especially in the CFI-based model, is only addressed heuristically.

AI Assistance in Theoretical Developments

Notably, the nontrivial structure of the new superpotential terms for the CFI-based circle compactification was obtained using a LLM (ChatGPT) upon feeding the context and reviewing failed naive analytic attempts. This signals the potential for AI tools to infer nontrivial analytic structures in high-dimensional moduli spaces and effective theories, contingent on researcher supervision and physical consistency requirements. However, this does not in itself constitute a systematic framework for automated model building, and the critical assessment and verification by experts remain indispensable.

Conclusion

This paper provides the first explicit constructions of scale-separated AdS$\mathcal{N}=1$7 vacua with extended ($\mathcal{N}=1$8) supersymmetry in string theory, achieved via circle reductions of four-dimensional type IIA compactifications with new fluxes and sources. The solutions stabilize all moduli at weak coupling and large radius with parametrically tunable separation of scales. The study demonstrates the absence of partial supersymmetry breaking, the emergence of non-integer conformal operator dimensions in the dual CFTs, and highlights the role of computational tools in discovering nontrivial effective superpotentials. This advances the landscape of viable flux compactifications and opens new avenues in the interface between supergravity, swampland constraints, and the application of AI to formal theory.