Comparing elliptic and toric hypersurface Calabi-Yau threefolds at large Hodge numbers

Abstract

We compare the sets of Calabi-Yau threefolds with large Hodge numbers that are constructed using toric hypersurface methods with those can be constructed as elliptic fibrations using Weierstrass model techniques motivated by F-theory. There is a close correspondence between the structure of "tops" in the toric polytope construction and Tate form tunings of Weierstrass models for elliptic fibrations. We find that all of the Hodge number pairs (h 1,1 , h 2,1 ) with h 1,1 or h 2,1 ≥ 240 that are associated with threefolds in the Kreuzer-Skarke database can be realized explicitly by generic or tuned Weierstrass/Tate models for elliptic fibrations over complex base surfaces. This includes a relatively small number of somewhat exotic constructions, including elliptic fibrations over non-toric bases, models with new Tate tunings that can give rise to exotic matter in the 6D F-theory picture, tunings of gauge groups over non-toric curves, tunings with very large Hodge number shifts and associated nonabelian gauge groups, and tuned Mordell-Weil sections associated with U(1) factors in the corresponding 6D theory. arXiv:1805.05907v6 [hep-th]

Fibers add flavor. Part I. Classification of 5d SCFTs, flavor symmetries and BPS states

“…Examples are the conformal matter theories [45], which have a particularly nice characterization in terms of non-flat resolutions of the (noncompact) elliptically fibered Calabi-Yau threefold. In the context of F-theory, such fibrations have been systematically studied in [46] for Kodaira fibers, with examples in codimension two and three appearing in [13,29,[47][48][49][50][51][52][53][54][55][56][57]. Unlike the more commonly studied resolutions of minimal collisions of elliptic singularities [58][59][60][61][62], which result in complex one-dimensional fibers, non-minimal singularities require insertions of complex surfaces, S i , into the fiber in order to resolve the singularity.…”

Mirror symmetry and elliptic Calabi-Yau manifolds

“…This rank-preserving tuning between SU (2) and G2 connects two phases of the same Calabi-Yau geometry. 11 Additional vertices from tops for all generic fibrations over Hirzebruch surfaces can be looked up in Table 11 in [15].…”

Fibration structure in toric hypersurface Calabi-Yau threefolds

“…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 condition stated in [25] is that the base B of such a fibration should be identified by constructing the 2D toric variety from the set of primitive rays in the image of ∇ under the projection that takes ∇ 2 → 0. Indeed, in many cases, such as the "standard stacking" polytopes corresponding to many generic and (Tate) tuned Weierstrass models over a given base, one can use the structure of the polytopes and tops to determine the base and additional tuned Kodaira singularity types to directly construct a Weierstrass model for an elliptic fibration over the given base, thus identifying a Calabi-Yau threefold that is elliptically fibered and has the requisite Hodge numbers associated with the polytope, circumventing the explicit construction of fans compatible with a toric morphism from ∇ to ∇ 2 [22]. Particularly for more complicated fibrations with general fibers and twists, however, there is no systematic methodology for implementing such a direct construction; and in any case, it is desirable to know in general which triangulations of ∇ are compatible with the fibration structure, and whether in fact such triangulations always exist.…”

“…It was shown in [20] that many polytopes in the Kreuzer-Skarke (KS) database [21] have a structure compatible with a K3 fibration. 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.…”