T-dualities and scale-separated AdS$_3$ in type I (2509.12801v1)

Abstract: We perform three T-dualities on previously found, classical $\mathcal{N}=1$ scale-separated AdS$_3$ solutions of massive type IIA supergravity. These solutions arose from a compactification on a toroidal $G_2$-holonomy space with smeared O2/D2 and O6/D6 sources. The T-dual backgrounds are classical $\mathcal{N}=1$ AdS$_3$ solutions of type IIB supergravity with O5/D5 and O9/D9 sources (type I) compactified on a space with $G_2$-structure and non-vanishing Ricci scalar. We generalize the original solutions in IIA in the T-dual picture and present on the type IIB side fully classical solutions with parametric control, scale separation, and integer conformal dimensions for the dual operators in the corresponding CFT. We also obtain strongly coupled solutions with the same properties. These are S-dual to parametrically controlled classical solutions of the heterotic SO(32) string theory.

• The paper develops a systematic construction of scale-separated AdS₃ vacua in type I string theory via T-dualities from massive IIA compactifications.
• It employs detailed flux mappings and tadpole cancellation mechanisms to achieve moduli stabilization and integer conformal dimensions in the dual CFT.
• The study identifies solution families with controlled scale separation and parametrically large fluxes, hinting at broader implications for the string landscape.

T-dualities and Scale-Separated AdS$_3$ Vacua in Type I String Theory

Introduction and Motivation

The paper investigates the construction of scale-separated AdS$_3$ vacua in type I string theory via T-dualities applied to previously established massive type IIA supergravity solutions. The original IIA backgrounds are compactifications on toroidal $G_2$-holonomy spaces with smeared O2/D2 and O6/D6 sources, yielding classical $\mathcal{N}=1$ AdS$_3$ vacua with parametric scale separation and moduli stabilization. The central aim is to understand how these properties translate under three T-dualities to type IIB backgrounds with O5/D5 and O9/D9 sources, and to analyze the resulting moduli stabilization, scale separation, and conformal dimensions in the dual CFT.

Geometric and Flux Setup

The starting point is a toroidal orbifold $T^7/(\mathbb{Z}_2^3)$ with $G_2$-holonomy, characterized by invariant three- and four-forms $\Phi$ and $\Psi$. The moduli $s^i$ parametrize the sizes of the internal three-cycles, and the volume is given by $\text{vol}(X) = (\prod_{i=1}^7 s^i)^{1/3}$. The orientifold projection introduces O2/O6-planes, with the orbifold group acting to produce a web of wrapped and transverse directions for the sources.

Upon three T-dualities, the $H_3$ flux in IIA is mapped to metric fluxes in IIB, resulting in a $G_2$-structure manifold with non-vanishing Ricci scalar and torsion classes $W_1$ and $W_{27}$. The IIB setup includes O5/O9-planes, with the O9-plane charge canceled by 32 D9-branes, yielding a type I background.

Effective 3D Supergravity and Moduli Stabilization

Dimensional reduction yields a 3D $\mathcal{N}=1$ supergravity theory with eight scalar fields: the internal volume modulus $\upsilon$, the dilaton $\phi$, and six independent shape moduli $\tilde{s}^a$. The scalar potential is derived from a real superpotential $P(\varphi^I)$, with explicit dependence on the fluxes and moduli. The IIA superpotential includes $F_0$, $F_4$, and $H_3$ fluxes, while the IIB superpotential after T-duality involves $F_3$, $F_7$, and metric fluxes.

The analysis distinguishes between fluxes that are unbounded (parametrically large, controlling scale separation and moduli stabilization) and those that are bounded by tadpole cancellation conditions. In particular, the mechanism for tadpole cancellation—whether by fluxes or by branes—directly impacts the conformal dimensions of dual CFT operators.

Scale Separation and Kaluza-Klein Spectrum

Scale separation is quantified by the ratio $\langle V \rangle / m_{\text{KK}}^2$, with $m_{\text{KK}}^2$ estimated from the internal radii. The construction ensures that for parametrically large unbounded fluxes, the AdS scale is decoupled from the KK scale, and higher-order $\alpha'$ corrections are suppressed for large string-frame radii.

Explicit Vacua and T-dual Constructions

Minimal IIA Construction

The minimal IIA flux configuration involves $F_0 = m_0$, $F_4$ threading selected four-cycles, and $H_3$ threading selected three-cycles. Tadpole cancellation for the O2-plane is achieved by D2-branes, leaving $F_4$ fluxes unconstrained. The O6-plane tadpole is canceled by $H_3 F_0$ contributions and D6-branes. This setup yields integer conformal dimensions for dual CFT operators.

Non-minimal IIA Construction

A more general $F_4$ flux threads all seven four-cycles, giving masses to all scalar fields. Tadpole cancellation involves both fluxes and branes, and integer conformal dimensions are recovered only in the parametric limit where certain fluxes dominate.

Type I T-dual Vacua

Three T-dualities map the IIA backgrounds to type IIB with $G_2$-structure, O5/O9-planes, and $F_3$, $F_7$ fluxes. The unbounded IIA fluxes become closed, unbounded IIB fluxes, while bounded fluxes remain bounded and ensure tadpole cancellation. The explicit flux mapping is detailed, and the resulting moduli stabilization and scale separation are analyzed.

Families of Solutions and Parametric Control

The paper identifies three families of solutions in the type I setup:

• Classical, weakly coupled, scale-separated vacua: All cycles large, string coupling small, scale separation achieved for $1 \ll N^2 \ll \mathcal{G} \ll N^4$.
• T-dual of IIA with shrinking cycles: Weak coupling, one cycle parametrically small, scale separation maintained.
• Strongly coupled, scale-separated vacua: $F_7$ flux dominates, corresponding to S-dual heterotic solutions.

In all cases, scale separation is controlled by unbounded $F_3$ fluxes, and moduli are stabilized by fluxes. The explicit parametric dependence of the moduli, string coupling, and vacuum energy on the fluxes is provided.

Mass Spectrum and Conformal Dimensions

The mass spectrum of the scalar fields is computed from the Hessian of the scalar potential. In the minimal construction, five massive and three massless scalars are found, with conformal dimensions $\Delta = \{8,4,4,4,4,2,2,2\}$, all integers. In the non-minimal case, all moduli are stabilized, but the conformal dimensions become flux-dependent and generically non-integer, except in the parametric limit.

Implications and Future Directions

The results demonstrate that scale-separated AdS$_3$ vacua with parametric control and moduli stabilization can be constructed in type I string theory via T-dualities from massive IIA backgrounds. The mechanism of tadpole cancellation and the structure of the fluxes are crucial for achieving integer conformal dimensions in the dual CFT. The existence of strongly coupled solutions S-dual to heterotic backgrounds broadens the landscape of scale-separated vacua.

Potential future directions include exploring T-dualities along different cycles, utilizing nilmanifold classifications to generate vacua with distinct Betti numbers, and uplifting the constructions to M-theory. These avenues may yield further insight into the universality and robustness of scale separation, moduli stabilization, and holographic properties in string compactifications.

Conclusion

This work provides a systematic construction of scale-separated AdS$_3$ vacua in type I string theory via T-dualities from massive IIA backgrounds, elucidating the interplay between fluxes, tadpole cancellation, moduli stabilization, and holographic conformal dimensions. The explicit parametric control and identification of families of solutions contribute to the broader understanding of flux compactifications and their holographic duals, with implications for the Swampland program and the structure of the string landscape.