Abstract
Internal intervals spanned by finite ranges of a conformal coordinate z and terminating at a pair of singularities are a common feature of many string compactifications with broken supersymmetry. The squared masses emerging in lower-dimensional Minkowski spaces are then eigenvalues of Schrödinger-like operators, whose potentials have double poles at the ends of the intervals. For one-component systems, the possible self-adjoint extensions of Schrödinger operators are described by points in AdS3 × S1, and those corresponding to independent boundary conditions at the ends of the intervals by points on the boundary of AdS3. The perturbative stability of compactifications to Minkowski space time depends, in general, on these choices of self-adjoint extensions. We apply this setup to the orientifold vacua driven by the “tadpole potential” \( V=T\ {e}^{\frac{3}{2}\phi } \) and find, in nine dimensions, a massive scalar spectrum, a unique choice of boundary conditions with stable tensor modes and a massless graviton, and a wide range of choices leading to massless and/or massive vector modes.
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Acknowledgments
We are grateful to C. Bachas, G. Dall’Agata, E. Dudas, S. Raucci 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 grateful to Dr. M. Nardelli, who kindly retrieved some mathematical literature.
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Mourad, J., Sagnotti, A. Non-supersymmetric vacua and self-adjoint extensions. J. High Energ. Phys. 2023, 41 (2023). https://doi.org/10.1007/JHEP08(2023)041
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DOI: https://doi.org/10.1007/JHEP08(2023)041