Abstract
We systematically analyze the fibration structure of toric hypersurface Calabi-Yau threefolds with large and small Hodge numbers. We show that there are only four such Calabi-Yau threefolds with h1,1 ≥ 140 or h2,1 ≥ 140 that do not have manifest elliptic or genus one fibers arising from a fibration of the associated 4D polytope. There is a genus one fibration whenever either Hodge number is 150 or greater, and an elliptic fibration when either Hodge number is 228 or greater. We find that for small h1,1 the fraction of polytopes in the KS database that do not have a genus one or elliptic fibration drops exponentially as h1,1 increases. We also consider the different toric fiber types that arise in the polytopes of elliptic Calabi-Yau threefolds.
Highlights
Calabi-Yau manifolds play a central role in string theory; these geometric spaces can describe extra dimensions of space-time in supersymmetric “compactifications” of the theory
A fairly comprehensive introductory review of the toric hypersurface construction and how elliptic fibrations are described in this context is given in the companion paper [24], in which we describe in much more detail the structure of the elliptic fibrations for CalabiYau threefolds X with very large Hodge numbers (h1,1(X) ≥ 240 or h2,1(X) ≥ 240)
Of the 495515 polytopes analyzed at large Hodge numbers, we found that only four lacked a 2D reflexive polytope fiber, and the other 495511 polytopes all lead to Calabi-Yau threefolds with a manifest genus one fiber
Summary
Calabi-Yau manifolds play a central role in string theory; these geometric spaces can describe extra dimensions of space-time in supersymmetric “compactifications” of the theory. A large class of Calabi-Yau threefolds can be described as hypersurfaces in toric varieties; these were systematically classified by Kreuzer and Skarke [2, 3] and represent most of the explicitly known Calabi-Yau threefolds at large Hodge numbers. A direct analysis of the related structure of K3 fibrations for many of the toric hypersurface constructions in the Kreuzer-Skarke database was carried out in [17], demonstrating directly the prevalence of fibrations by smaller-dimensional Calabi-Yau fibers among known Calabi-Yau threefolds. A fairly comprehensive introductory review of the toric hypersurface construction and how elliptic fibrations are described in this context is given in the companion paper [24], in which we describe in much more detail the structure of the elliptic fibrations for CalabiYau threefolds X with very large Hodge numbers (h1,1(X) ≥ 240 or h2,1(X) ≥ 240). We give only a very brief summary of the essential points
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