On the prevalence of elliptic and genus one fibrations among toric hypersurface Calabi-Yau threefolds

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 h 1,1 ≥ 140 or h 2,1 ≥ 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 h 1,1 the fraction of polytopes in the KS database that do not have a genus one or elliptic fibration drops exponentially as h 1,1 increases. We also consider the different toric fiber types that arise in the polytopes of elliptic Calabi-Yau threefolds.

F-theory models with U(1) × ℤ2, ℤ4 and transitions in discrete gauge groups

“…There are situations in which a genus-one fibration has a global section and in which it does not have a global section; when a genus-one fibration does not have a global section, a discrete gauge group forms in F-theory on this fibration, as mentioned. Recent discussions of F-theory on genus-one fibrations without a global section can be found, for example, in [20,19,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45] 2 . When a genus-one fibration has a global section (in which case, the fibration is often called an elliptic fibration 3 in the F-theory literature), the U (1) gauge group forms in F-theory if the fibration has two or more independent global sections.…”

Fibration structure in toric hypersurface Calabi-Yau threefolds

“…The algorithm that we have used to identify the existence of a 2D reflexive subpolytope is a streamlined version of the algorithms discussed in [26,27,23]. The basic idea is to determine whether any pair of rays in ∇ ∩ Z 4 generate a 2D sublattice of Z 4 that intersects ∇ in a set of points that form a 2D polytope containing the origin as an interior point.…”

“…The problem of identifying 2D subpolytopes from the combinatorial data of a 4D polytope is described and discussed in some detail in [26,27,23]. We use here the notation and conventions of [22,23], to which the reader is referred for further background and references.…”

“…A systematic construction of elliptic CY threefolds at large Hodge numbers over toric base surfaces [22] showed that all Hodge number pairs in the KS database with h 1,1 ≥ 240 or h 2,1 ≥ 240 are associated with such elliptic Calabi-Yau threefolds. A systematic direct study of the fibration structure of the polytopes in the KS database was initiated in [23]; in that paper we found that all polytopes associated with Calabi-Yau threefolds having h 1,1 ≥ 150 or h 2,1 ≥ 150 have a reflexive 2D subpolytope, indicating a structure compatible with the presence of an elliptic or genus one fibration for the associated CY threefolds. In that paper we also found empirically that at small h 1,1 , the fraction of 4D polytopes lacking such a reflexive 2D subpolytope drops roughly as 2 −h 1,1 , and we gave some analytic arguments for why these results may naturally be expected.…”

“…In this paper we complete the program begun in [23], and we report on the results of a complete analysis of all 473.8 million reflexive 4D polytopes for reflexive 2D subpolytopes. The upshot of the analysis is that the Calabi-Yau threefolds that correspond to the toric hypersurface construction seem to be dominated by those that are elliptic or genus one fibered in some phase, so that in fact most known Calabi-Yau threefolds are birationally equivalent to one with an elliptic or genus one fibration.…”