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Scanning the skeleton of the 4D F-theory landscape. (English) Zbl 1384.83066
Summary: Using a one-way Monte Carlo algorithm from several different starting points, we get an approximation to the distribution of toric threefold bases that can be used in four-dimensional F-theory compactification. We separate the threefold bases into “resolvable” ones where the Weierstrass polynomials \((f,g)\) can vanish to order (4,6) or higher on codimension-two loci and the “good” bases where these (4,6) loci are not allowed. A simple estimate suggests that the number of distinct resolvable base geometries exceeds \(10^{3000}\), with over \(10^{250}\) “good” bases, though the actual numbers are likely much larger. We find that the good bases are concentrated at specific “end points” with special isolated values of \(h^{1,1}\) that are bigger than 1,000. These end point bases give Calabi-Yau fourfolds with specific Hodge numbers mirror to elliptic fibrations over simple threefolds. The non-Higgsable gauge groups on the end point bases are almost entirely made of products of \(E_8\), \(F_4\), \(G_2\) and SU(2). Nonetheless, we find a large class of good bases with a single non-Higgsable SU(3). Moreover, by randomly contracting the end point bases, we find many resolvable bases with \(h^{1,1}(B) \sim 50--200\) that cannot be contracted to another smooth threefold base.

MSC:
83E30 String and superstring theories in gravitational theory
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
32Q25 Calabi-Yau theory (complex-analytic aspects)
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
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