Abstract
In this paper, we explore the string theory landscape obtained from type IIB and F-theory flux compactifications. We first give a comprehensive introduction to a number of mathematical finiteness theorems, indicate how they have been obtained, and clarify their implications for the structure of the locus of flux vacua. Subsequently, in order to address finer details of the locus of flux vacua, we propose three mathematically precise conjectures on the expected number of connected components, geometric complexity, and dimensionality of the vacuum locus. With the recent breakthroughs on the tameness of Hodge theory, we believe that they are attainable to rigorous mathematical tools and can be successfully addressed in the near future. The remainder of the paper is concerned with more technical aspects of the finiteness theorems. In particular, we investigate their local implications and explain how infinite tails of disconnected vacua approaching the boundaries of the moduli space are forbidden. To make this precise, we present new results on asymptotic expansions of Hodge inner products near arbitrary boundaries of the complex structure moduli space.
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Acknowledgments
We would like to thank Michael Douglas, Damian van de Heisteeg, Aroldo Kaplan, Bruno Klingler, Erik Plauschinn, David Prieto, Lorenz Schlechter, Christian Schnell, and Mick van Vliet for useful discussions and comments. This research is supported, in part, by the Dutch Research Council (NWO) via a Start-Up grant and a Vici grant.
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Grimm, T.W., Monnee, J. Finiteness theorems and counting conjectures for the flux landscape. J. High Energ. Phys. 2024, 39 (2024). https://doi.org/10.1007/JHEP08(2024)039
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DOI: https://doi.org/10.1007/JHEP08(2024)039