Abstract
We recently constructed type-IIB compactifications to four dimensions depending on a single additional coordinate, where a five-form flux Φ on an internal torus leads to a constant string coupling. Supersymmetry is fully broken when the internal manifold includes a finite interval of length ℓ, which is spanned by a conformal coordinate in a finite range 0 < z < zm. Here we examine the low-lying bosonic spectra and their classical stability, paying special attention to self-adjoint boundary conditions. Special boundary conditions result in the emergence of zero modes, which are determined exactly by first-order equations. The different sectors of the spectrum can be related to Schrödinger operators on a finite interval, characterized by pairs of real constants μ and \( \overset{\sim }{\mu } \), with μ equal to 1/3 or 2/3 in all cases and different values of \( \overset{\sim }{\mu } \). The potentials behave as \( \frac{\mu^2-1/4}{z^2} \) and \( \frac{{\overset{\sim }{\mu}}^2-1/4}{{\left({z}_m-z\right)}^2} \) near the ends and can be closely approximated by exactly solvable trigonometric ones. With vanishing internal momenta, one can thus identify a wide range of boundary conditions granting perturbative stability, despite the intricacies that emerge in some sectors. For the Kaluza-Klein excitations of non-singlet vectors and scalars the Schrödinger systems couple pairs of fields, and the stability regions, which depend on the background, widen as the ratio Φ/ℓ4 decreases.
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Acknowledgments
We are grateful to C. Bachas, G. Dall’Agata, E. Dudas and A. Tomasiello for stimulating discussions. AS was supported in part by Scuola Normale, by INFN (IS GSS-Pi) and by the MIUR-PRIN contract 2017CC72MK_003. JM is grateful to Scuola Normale Superiore for the kind hospitality while this work was in progress. AS is grateful to Université de Paris Cité and DESY-Hamburg for the kind hospitality, and to the Alexander von Humboldt foundation for the kind and generous support, while this work was in progress. Finally, we are both very grateful to Dr. M. Nardelli, who kindly helped us to retrieve some mathematical literature.
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Mourad, J., Sagnotti, A. A 4D IIB flux vacuum and supersymmetry breaking. Part II. Bosonic spectrum and stability. J. High Energ. Phys. 2023, 61 (2023). https://doi.org/10.1007/JHEP11(2023)061
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DOI: https://doi.org/10.1007/JHEP11(2023)061