Fibrations in non-simply connected Calabi-Yau quotients

Abstract

In this work we study genus one fibrations in Calabi-Yau three-folds with a non-trivial first fundamental group. The manifolds under consideration are constructed as smooth quotients of complete intersection Calabi-Yau three-folds (CICYs) by a freely acting, discrete automorphism. By probing the compatibility of symmetries with genus one fibrations (that is, discrete group actions which preserve a local decomposition of the manifold into fiber and base) we find fibrations that are inherited from fibrations on the covering spaces. Of the 7,890 CICY three-folds, 195 exhibit known discrete symmetries, leading to a total of 1,695 quotient manifolds. By scanning over 20,700 fiber/symmetry pairs on the covering spaces we find 17,161 fibrations on the quotient Calabi-Yau manifolds. It is found that the vast majority of the non-simply connected manifolds studied exhibit multiple different genus one fibrations -echoing a similar ubiquity of such structures that has been observed in other data sets. The results are available at [1]. The possible base manifolds are all singular and are catalogued. These Calabi-Yau fibrations generically exhibit multiple fibers and are of interest in F-theory as backgrounds leading to theories with superconformal loci and discretely charged matter.

Mirror symmetry and elliptic Calabi-Yau manifolds

“…The largest known set of Calabi-Yau threefolds are constructed from the class of over 400 million reflexive 4D polytopes found by Kreuzer and Skarke [5,6], and exhibit this mirror symmetry structure. More recently, an increasing body of evidence [7][8][9][10][11][12][13][14][15][16] suggests that a large fraction of known Calabi-Yau threefolds have the property that they can be described as genus one or elliptic fibrations over a complex two-dimensional base surface. We recently showed that this is true of all but at most 4 Calabi-Yau threefolds in the Kreuzer-Skarke database having one or the other Hodge number h 2,1 , h 1,1 at least 140, and that at small h 1,1 the fraction of polytopes in the Kreuzer-Skarke database that lack an obvious elliptic or genus one fibration decreases roughly as 0.1 × 2 5−h 1,1 .…”

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.…”

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.…”

“…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.…”

“…To construct quotients of elliptically fibered manifolds it is necessary to systematically understand how the discrete symmetry acts on the fibers/sections. To this end, we are aided by previous explorations such as [7,21,22] which produce discrete symmetries of elliptic CY threefolds by demanding that the discrete action maps sections into one another in a fibrationpreserving manner. As we will review in Section 2, this can be accomplished in some cases by demanding that the fibers are of a form to support Mordell-Weil (MW) Torsion.…”