At a newly identified universal class of supersymmetric type-IIB flux vacua near the large-complex-structure (LCS) point of generic Calabi-Yau orientifolds, the complete axio-dilaton/complex-structure mass spectrum is given by a simple analytic formula independent of the Calabi-Yau, but the flux superpotential satisfies W0=Vm3/2≥1/πW_0=\mathcal{V} m_{3/2}\geq 1/\sqrt{\pi}, so these vacua are compatible with the Large Volume Scenario (LVS) and not with KKLT. The open problem, stated as the primary future direction by the discovering authors and still open in the 2026 literature, is to complete moduli stabilization: include the α′\alpha' and nonperturbative corrections that stabilize the Kähler moduli within LVS and determine whether fully stabilized vacua exist in which the axio-dilaton and all complex-structure fields remain perturbatively stable — in particular the fate of the single light mode that becomes massless at the LCS point — together with the companion construction of alternative truncation schemes in which the spectrum stays analytically computable but W0W_0 can be arbitrarily small.

Background

In type-IIB string theory compactified on a Calabi-Yau orientifold, the low-energy effective supergravity contains three sets of scalar moduli: the axio-dilaton τ\tau (the string coupling plus its axionic partner), the complex-structure moduli ziz^i describing the shape of the Calabi-Yau, and the Kähler moduli TaT_a describing its size. Three-form fluxes (integer quanta of the NS and RR three-form field strengths) generate, through the Gukov-Vafa-Witten superpotential W=∫(F3−τH3)∧ΩW=\int(F_3-\tau H_3)\wedge\Omega, a potential that fixes τ\tau and the ziz^i at supersymmetric minima while leaving the Kähler moduli unfixed to leading order, a feature known as no-scale structure. Near the large-complex-structure (LCS) point of moduli space, where the imaginary parts of the ziz^i are large, the prepotential governing the geometry is well approximated by a cubic polynomial, with exponentially suppressed world-sheet instanton corrections. The two standard schemes that then fix the Kähler moduli are KKLT (Kachru, Kallosh, Linde, Trivedi), which requires an exponentially small flux superpotential W0=Vm3/2W_0=\mathcal{V} m_{3/2} (V\mathcal{V} the Calabi-Yau volume, m3/2m_{3/2} the gravitino mass), and the Large Volume Scenario (LVS) (Balasubramanian, Berglund, Conlon, Quevedo), which works at exponentially large volume with W0W_0 of order one and uses α′\alpha' (string-derivative) corrections and nonperturbative effects such as Euclidean D3-brane instantons or gaugino condensation on D7 branes wrapping blow-up divisors. Blanco-Pillado, Sousa, Urkiola and Wachter (PRD 103, 106006 (2021)) showed that for generic Calabi-Yau orientifolds at LCS there is a class of supersymmetric flux vacua, obtained by a consistent truncation of all complex-structure fields except one along a monodromy direction, at which the squared masses of the full axio-dilaton/complex-structure sector (including the truncated fields) are given, to leading order in α′\alpha' and the string coupling gsg_s and up to exponentially small instanton corrections, by a universal analytic formula depending on a single parameter ξ\xi that measures the distance to the LCS point: the modes with labels λ=0,1\lambda=0,1 (the surviving fields τ\tau and z^\hat z) have masses (1±1−2ξ3m^(ξ))2m3/22\left(1\pm\frac{\sqrt{1-2\xi}}{\sqrt{3}}\hat m(\xi)\right)^2 m_{3/2}^2 and (1±1−2ξ3 m^(ξ))2m3/22\left(1\pm\frac{\sqrt{1-2\xi}}{\sqrt{3}\,\hat m(\xi)}\right)^2 m_{3/2}^2 respectively, the paired modes with λ=2,…,h2,1\lambda=2,\ldots,h^{2,1} have masses (1±1+ξ3)2m3/22\left(1\pm\frac{1+\xi}{3}\right)^2 m_{3/2}^2 up to a common rescaling, and importantly, since the function m^(ξ)\hat m(\xi) of the source's eq. (2) satisfies m^(0)=1/3\hat m(0)=1/\sqrt{3}, a single mode, μ−12\mu^2_{-1} (the minus branch of λ=1\lambda=1), becomes massless as ξ→0\xi\to 0 at LCS while all others are of order m3/2m_{3/2}. The authors constructed an ensemble of 17,054 such vacua on the Calabi-Yau hypersurface WP[1,1,1,6,9]4\mathbb{WP}^4_{[1,1,1,6,9]} with 272 complex-structure moduli. Because the flux quantization condition implies W0=Vm3/2≥1/πW_0=\mathcal{V} m_{3/2}\geq 1/\sqrt{\pi} at these vacua, they are naturally suited to LVS but incompatible with KKLT. What is missing is the Kähler sector: including the α′\alpha' and nonperturbative corrections that break the no-scale structure of the Kähler potential also perturbs the complex-structure sector, and whether the completed vacuum remains perturbatively stable — especially regarding the light μ−1\mu_{-1} mode and the expected lifting of the degeneracy among the λ≥2\lambda\geq 2 modes — has not been established. The companion paper of the same authors (JHEP 04 (2021) 149) analysed the more general vacua with a one-field truncation on symmetric Calabi-Yaus, where W0W_0 can be arbitrarily small but the same completion question arises.

Problem Statement

The universal class of type-IIB flux vacua of Blanco-Pillado, Sousa, Urkiola and Wachter provides, at LCS and to leading order in α′\alpha' and gsg_s, the full analytic mass spectrum of the axio-dilaton and complex-structure sector at supersymmetric minima with W0=Vm3/2≥1/πW_0=\mathcal{V} m_{3/2}\geq 1/\sqrt{\pi}, but leaves the Kähler moduli unstabilized and the vacuum incomplete. The problem is to complete the stabilization of these vacua with a common objective — fully stabilized string vacua retaining analytic control over the complete moduli spectrum — consisting of: (1) include the α′\alpha' perturbative corrections and nonperturbative effects (Euclidean D3-brane instantons or gaugino condensation) that generate the Kähler-moduli potential within the Large Volume Scenario, which is the scheme compatible with the lower bound W0≥1/πW_0\geq 1/\sqrt{\pi}, and determine whether there exist fully stabilized vacua of this class in which the axio-dilaton and all complex-structure fields remain perturbatively stable — in particular establish the fate of the light μ−1\mu_{-1} mode that becomes massless as ξ→0\xi\to 0 at LCS, and how the corrected potential lifts the degeneracy of the λ≥2\lambda\geq 2 modes; and (2) as the companion direction stated by the source, construct alternative truncation schemes compatible with the more general one-field-truncation vacua of the companion paper (arXiv:2007.10381) in which the mass spectrum is still analytically computable but W0W_0 can be arbitrarily small, and assess their viability, e.g. for KKLT-type stabilization. Parts (1) and (2) are independent questions, each posed on its own, and each is resolved either by an explicit construction — a fully stabilized vacuum in a concrete Calabi-Yau orientifold with all moduli fixed, the corrected spectrum computed, and stability and parametric control demonstrated, or, for part (2), an alternative truncation scheme with an analytically computable mass spectrum and arbitrarily small W0W_0 — or by a controlled no-go result (a demonstration, within the stated EFT control regime, that no such LVS-type completion preserves complex-structure stability, or that no analytic truncation of the proposed type yields small W0W_0).

References

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