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

TL;DR

This paper compares Calabi-Yau threefolds constructed via toric hypersurfaces and elliptic fibrations, showing that high Hodge number examples from the Kreuzer-Skarke database can be realized through Weierstrass models, revealing deep structural correspondences.

Contribution

It demonstrates that all high Hodge number Calabi-Yau threefolds in the Kreuzer-Skarke database can be explicitly constructed as elliptic fibrations with Weierstrass models, establishing a detailed correspondence with toric methods.

Findings

High Hodge number pairs are realizable via Weierstrass models.

Structural correspondence between toric tops and Tate form tunings.

Existence of exotic constructions with non-toric bases and large Hodge shifts.

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 $h^{2, 1}\geq 240$ 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.