All Weight Systems for Calabi–Yau Fourfolds from Reflexive Polyhedra

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.

Section: Discussionmentioning

confidence: 99%

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

Huang

Taylor

2019

“…On the other hand, complete intersection Calabi-Yau d-folds of codimension m are related to nef-partitions of d + m-dimensional reflexive polytopes [61]. For recent progress on the classification of five-dimensional reflexive polytopes see [132].…”

Section: Eliptically Fibered Calabi-yau As Toric Hypersurfacesmentioning

confidence: 99%

“…For a given reflexive polytope one can systematically search for the toric fibrations of the corresponding toric variety. This has been used by [134] and [132] to scan for fibrations in the complete Kreuzer-Skarke list.…”

Section: Jhep11(2019)170mentioning

confidence: 99%

Topological strings on genus one fibered Calabi-Yau 3-folds and string dualities

Cota¹,

Klemm

Schimannek

2019

121

We calculate the generating functions of BPS indices using their modular properties in Type II and M-theory compactifications on compact genus one fibered CY 3-folds with singular fibers and additional rational sections or just N-sections, in order to study string dualities in four and five dimensions as well as rigid limits in which gravity decouples. The generating functions are Jacobi-forms of Γ 1 (N) with the complexified fiber volume as modular parameter. The string coupling λ, or the ± parameters in the rigid limit, as well as the masses of charged hypermultiplets and non-Abelian gauge bosons are elliptic parameters. To understand this structure, we show that specific auto-equivalences act on the category of topological B-branes on these geometries and generate an action of Γ 1 (N) on the stringy Kähler moduli space. We argue that these actions can always be expressed in terms of the generic Seidel-Thomas twist with respect to the 6-brane together with shifts of the B-field and are thus monodromies. This implies the elliptic transformation law that is satisfied by the generating functions. We use Higgs transitions in F-theory to extend the ansatz for the modular bootstrap to genus one fibrations with N-sections and boundary conditions fix the all genus generating functions for small base degrees completely. This allows us to study in depth a wide range of new, non-perturbative theories, which are Type II theory duals to the CHL Z N orbifolds of the heterotic string on K3 × T 2. In particular, we compare the BPS degeneracies in the large base limit to the perturbative heterotic one-loop amplitude with R 2 + F 2g−2 + insertions for many new Type II geometries. In the rigid limit we can refine the ansatz and obtain the elliptic genus of superconformal theories in 5d.

“…The largest h 1,1 for a Calabi-Yau threefold associated with a non-fibered 4D polytope comes from the case with Hodge numbers (140, 62), as previously identified in [23]. The next-largest values of h 1,1 come from polytopes associated with Calabi-Yau threefolds having Hodge numbers It is very interesting that the polytopes corresponding with large h 1,1 that do not have 2D reflexive subpolytopes have Hodge numbers h 2,1 in the narrow range 60-63; the first exception as h 1,1 decreases has the Hodge numbers (95, 55), and for all h 1,1 > 61 the non-fibered cases have h 2,1 in the range 48-65. It would be interesting to understand better whether this family of polytopes has some common structure associated with the lack of an elliptic or genus one fibration for the corresponding toric varieties.…”

Section: Fiber Analysis Of All Reflexive 4d Polytopesmentioning

confidence: 63%

“…In [54] it was shown that the fraction of CICY Calabi-Yau fourfolds that has an obvious elliptic or genus one fibration (99.95%) is even larger than the fraction of CICY Calabi-Yau threefolds with this property (99.3%). There is no complete analysis of reflexive 5D polytopes, though there are some partial results in this direction [55] and some analysis of the fibration structure of these polytopes was carried out in [26]. In fact, direct construction of toric threefold bases that support elliptic and genus one toric hypersurface Calabi-Yau fourfolds shows that the number of such bases alone is extraordinarily large, on the order of 10 3000 [48,56,57].…”

Section: Discussionmentioning

confidence: 99%

Fibration structure in toric hypersurface Calabi-Yau threefolds

Huang

Taylor

2020

We find through a systematic analysis that all but 29,223 of the 473.8 million 4D reflexive polytopes found by Kreuzer and Skarke have a 2D reflexive subpolytope. Such a subpolytope is generally associated with the presence of an elliptic or genus one fibration in the corresponding birational equivalence class of Calabi-Yau threefolds. This extends the growing body of evidence that most Calabi-Yau threefolds have an elliptically fibered phase. arXiv:1907.09482v2 [hep-th]