T-dualities and scale-separated AdS$_3$ in massless IIA on $(X_6 \times S^1)/\mathbb{Z}_2$

Abstract: Motivated by the question of whether scale-separated AdS$_3$ flux vacua arising from G$_2$ compactifications admit an uplift to eleven-dimensional supergravity, we construct scale-separated AdS$_3$ flux vacua in massless type IIA with only O6-planes. We first present new scale-separated solutions in massive type IIA on a G$_2$ holonomy toroidal orbifold with four smeared O6-planes, analyze their properties, and then perform a double T-duality to obtain the corresponding massless backgrounds. In the dual frame, the internal space is locally given by a six-dimensional quotient space $X_6$ with an $\mathrm{SU}(3)$ structure of Iwasawa type times an untwisted circle $S^1$, while globally it is further modded out by a non-trivial $\mathbb{Z}_2$ quotient inherited from the G$_2$ orbifold action. Finally, we use T-duality to derive the corresponding superpotential in massless type IIA and identify parametrically classical, scale-separated families of solutions, as well as a family with parametrically large radii, scale separation, and strong coupling, thus allowing for an uplift to eleven-dimensional supergravity.

• The paper demonstrates that scale-separated AdS3 vacua can be constructed in massless IIA using double T-dualities from a G2 orbifold with four O6-planes.
• It employs an explicit superpotential and precise flux configurations to fully stabilize moduli and achieve clear separation between the AdS curvature and KK scales.
• The analysis reveals a clean 11D geometric uplift without non-geometric fluxes, offering a controlled framework to test swampland and holographic constraints.

T-Dualities and Scale-Separated AdS$_3$ Vacua in Massless IIA on $(X_6\times S^1)/\mathbb{Z}_2$

Motivation and Context

The investigation of scale separation in AdS flux vacua addresses whether string compactifications can yield effective lower-dimensional theories where the AdS curvature scale is parametrically separated from the Kaluza-Klein (KK) scale. Such constructions are crucial for decoupling higher-dimensional modes and for realizing controllable effective supergravity descriptions. There are extensive constraints from the swampland program and related holographic consistency conditions, and in many explicit cases, parametric scale separation has been achieved only in the presence of O-planes and typically invokes smeared source approximations.

Most established constructions of scale-separated vacua, e.g., the DGKT scenario in massive IIA on Calabi-Yau or G$_2$ orientifolds, utilize massive IIA backgrounds containing Romans mass and intersecting O6/O2-planes. However, the presence of Romans mass complicates geometric uplifts to 11D supergravity and may introduce non-geometric fluxes under dualities, while intersecting orientifold sources produce localized singularities challenging to resolve globally.

This work is driven by the question of whether scale-separated AdS$_3$ flux vacua, particularly those arising from G$_2$ compactifications, can be realized in massless IIA with only net O6-plane contributions, without non-geometric fluxes and with a clean 11D geometric uplift. The paper develops explicit orientifold configurations enabling this and systematizes the application of double T-dualities to relate massive IIA vacua to massless IIA ones on spaces of the form $(X_6 \times S^1)/\mathbb{Z}_2$.

Massive IIA Setup: G$_2$ Holonomy Orbifolds with Four O6-Planes

The starting point is a G$_2$ orbifold modeled on a $T^7/(\mathbb{Z}_2^3)$ quotient, following Joyce's construction, supplemented by orientifold involutions that produce a specific configuration of (smeared) O6-planes. The essential insight is that previous “maximal” O6-plane setups yield nontrivial 4-form tadpoles upon dualization and inevitably introduce O4-planes and non-geometric fluxes in the T-dual frame. To circumvent this, the author constructs a “minimal” O6 configuration with exactly four O6-planes: $$\mathrm{O6}_\alpha, \mathrm{O6}_{\alpha\beta}, \mathrm{O6}_{\alpha\gamma}, \mathrm{O6}_{\alpha\beta\gamma}$$.

The flux Ansatz in massive IIA is engineered such that its T-dual image will avoid non-geometric fluxes. The explicit G$(X_6\times S^1)/\mathbb{Z}_2$0 structure and basis of forms are presented, with the internal metric diagonal and parameterized by seven radii, which are related to the G$(X_6\times S^1)/\mathbb{Z}_2$1 moduli. Tadpole cancellation is ensured: in this setup, the net O2-plane charge can be cancelled against explicit D2-branes, and the fluxes can be freely adjusted to achieve stabilization.

Supersymmetric AdS$(X_6\times S^1)/\mathbb{Z}_2$2 vacua are obtained by extremizing an explicit superpotential expressed in terms of the G$(X_6\times S^1)/\mathbb{Z}_2$3 moduli. The scalar potential exhibits parametric regimes where all moduli are stabilized at large volume and weak coupling, with parametric scale separation between the AdS and KK scales. The analysis distinguishes different solution branches, isolating those where all internal lengths and the string coupling become parametrically large or small.

Double T-Duality and the Massless IIA Frame

A double T-duality is performed along two toroidal directions, mapping the original G$(X_6\times S^1)/\mathbb{Z}_2$4 orbifold to a massless IIA compactification on a twisted six-dimensional space $(X_6\times S^1)/\mathbb{Z}_2$5 times an untwisted $(X_6\times S^1)/\mathbb{Z}_2$6, modulo a remnant $(X_6\times S^1)/\mathbb{Z}_2$7 quotient. Under duality, the NSNS fluxes become metric fluxes generating a twisted Iwasawa-type nilmanifold structure on $(X_6\times S^1)/\mathbb{Z}_2$8, and the RR fluxes are appropriately mapped, resulting in explicit $(X_6\times S^1)/\mathbb{Z}_2$9 fluxes.

Detailed analysis of the orientifold involutions in the new frame shows that only O6-planes remain, with O2/O4 charges trivialized by brane combinations. No non-geometric fluxes are present, in contrast with previous constructions. The author provides a complete description of the twisted geometry, including Maurer-Cartan equations, the decomposition of G$_2$0 forms into SU(3) structure data, and the identification of left-invariant forms and torsion classes.

The explicit form of the superpotential is constructed in the dual frame, both in terms of the radii and dilaton and in differential-form language using the SU(3) structure. All contributions—metric flux, RR fluxes—are matched under duality.

Moduli Stabilization and Scale-Separated Vacua in Massless IIA

A systematic extremization of the massless IIA superpotential yields families of AdS$_2$1 vacua with all moduli stabilized. The analysis identifies several distinct parametric solution regimes, including:

• Vacua with parametric scale separation, all string-frame radii large, and weak coupling.
• Vacua where all three-cycle volumes are large but some radii shrink, yet the KK scale remains separated and the effective theory under control.
• Notably, a parametric strong-coupling regime with large internal radii and scale separation. This is not accessible in the massive frame but arises naturally after dualization due to the changed dependence of the dilaton on the flux quanta.

The paper emphasizes that in these solutions, due to the absence of non-geometric fluxes and the exclusive presence of net O6-planes, an uplift to a well-defined geometric eleven-dimensional background is feasible, which is essential for matching the vacua to 11D supergravity or M-theory and to further test swampland or holographic constraints.

Geometric and Physical Insights

The analytic structure of these compactifications is elucidated, with the Iwasawa nilmanifold underlying $_2$2 supporting a half-flat SU(3) structure. The author computes all intrinsic SU(3) torsion classes, showing the structure is half-flat (nonvanishing $_2$3), as expected for T-duals of G$_2$4 spaces with flux.

The geometric construction clarifies the precise mapping of orientifold actions, the decomposition of cohomology under $_2$5 quotients, and the consistency of smeared sources in the limit where backreaction is negligible. All fluxes are shown to possess the correct orientifold parities.

Implications and Future Directions

This work demonstrates that scale-separated AdS$_2$6 vacua with only O6-planes and no non-geometric fluxes can be constructed in massless type IIA, with a full set of stabilized moduli and with strong-coupling, large-volume regimes available. The existence of such vacua has implications for the possibility of consistent string/M-theory backgrounds with clean geometric M-theory uplifts, providing key test cases for the DGKT scaffold and generalized swampland/holographic constraints on AdS vacua.

The approach can be generalized to study other orientifold and flux configurations or extended to higher-dimensional AdS vacua. The massless IIA frame avoids complications associated with localized source singularities and non-geometric fluxes, thus facilitating precise matchings to CFT duals or testing of conjectured scale separation bounds.

Future directions include explicit construction and analysis of the corresponding eleven-dimensional uplifts, extension to cases with less supersymmetry or nontrivial spin(7) structures, and clarification of the dual CFT operator spectra and their moduli dependence.

Conclusion

This paper provides a systematic construction of scale-separated AdS$_2$7 vacua in massless IIA supergravity on a $_2$8 geometry, obtained via double T-duality from a massive IIA G$_2$9 setup with four O6-planes. The results establish the existence of vacua with stabilized moduli, large internal dimensions, parametric scale separation, and controllable couplings—even realizing parametrically strong coupling appropriate for a geometric uplift to M-theory. This framework enables a deeper assessment of AdS scale separation in string theory and presents a tangible platform for addressing swampland, holographic, and moduli stabilization paradigms in controlled flux backgrounds (2603.26615).