Fibrations in CICY threefolds

Abstract

In this work we systematically enumerate genus one fibrations in the class of 7, 890 Calabi-Yau manifolds defined as complete intersections in products of projective spaces, the so-called CICY threefolds. This survey is independent of the description of the manifolds and improves upon past approaches that probed only a particular algebraic form of the threefolds (i.e. searches for "obvious" genus one fibrations as in [1,2]). We also study K3-fibrations and nested fibration structures. That is, K3 fibrations with potentially many distinct elliptic fibrations. To accomplish this survey a number of new geometric tools are developed including a determination of the full topology of all CICY threefolds, including triple intersection numbers. In 2, 946 cases this involves finding a new "favorable" description of the manifold in which all divisors descend from a simple ambient space. Our results consist of a survey of obvious fibrations for all CICY threefolds and a complete classification of all genus one fibrations for 4, 957 "Kähler favorable" CICYs whose Kähler cones descend from a simple ambient space. Within the CICY dataset, we find 139, 597 obvious genus one fibrations, 30, 974 obvious K3 fibrations and 208, 987 nested combinations. For the Kähler favorable geometries we find a complete classification of 377, 559 genus one fibrations. For one manifold with Hodge numbers (19, 19) we find an explicit description of an infinite number of distinct genus-one fibrations extending previous results for this particular geometry that have appeared in the literature. The data associated to this scan is available here [3].

F-theory on quotients of elliptic Calabi-Yau threefolds

“…In this section we discuss an example of a non-prime quotient that reduces the numbers of tensors and also the number of Abelian gauge group factors. The geometry is realized as a complete intersection CY threefold [21,24,48] given by the configuration matrix.…”

“…The Schoen manifold is well-known to be an exceptional point in the landscape of Calabi-Yau manifolds. As a fiber product of two rational elliptic surfaces it has a range of remarkable features, including a vast number of freely acting discrete symmetries and in fact, an infinite number of inequivalent genus-one fibrations [48,56,57]. For generic points in its complex structure moduli space, the Schoen manifold has a non-trivial, rank 8 Mordell-Weil group, the highest rank explicitly known for a Calabi-Yau threefold [58].…”

“…These include [49] for CICY threefolds and [51] for toric hypersurfaces. An analysis of the former has been undertaken to determine which symmetries are consistent with fibration structures [21] (based on the tools and classification in [48,[62][63][64]). Although quotients by non-Abelian discrete groups are known for CY threefolds.…”

Rigid limit for hypermultiplets and five-dimensional gauge theories

“…It appears first in the list of CICYs studied in [77], which simultaneously has h 1,1 = 3 and the property of being Kähler favourable (model 5299 in this database). The latter property means that its Kähler cone descends from the one of the ambient projective space.…”

Fibration structure in toric hypersurface Calabi-Yau threefolds

“…Some of the largest known sets of Calabi-Yau threefolds are the 7,890 complete intersection Calabi-Yau (CICY) threefolds [14], more generalized complete intersection Calabi-Yaus [15], and the Calabi-Yau threefolds constructed as hypersurfaces in toric varieties associated with the 473.8 million reflexive 4D polytopes [16]. It was shown in [17,18] that 99.3% of the CICY threefolds have an "obvious" genus one or elliptic fibration, and that many of these threefolds admit many such fibrations; similar results hold for discrete quotients of the CICY threefolds [19]. It was shown in [20] that many polytopes in the Kreuzer-Skarke (KS) database [21] have a structure compatible with a K3 fibration.…”