Calabi–Yau threefolds of type K (II): mirror symmetry

Abstract

A Calabi-Yau threefold is called of type K if it admits anétale Galois covering by the product of a K3 surface and an elliptic curve. In our previous paper [16], based on Oguiso-Sakurai's fundamental work [24], we have provided the full classification of Calabi-Yau threefolds of type K and have studied some basic properties thereof. In the present paper, we continue the study, investigating them from the viewpoint of mirror symmetry. It is shown that mirror symmetry relies on duality of certain sublattices in the second cohomology of the K3 surface appearing in the minimal splitting covering. The duality may be thought of as a version of the lattice duality of the anti-symplectic involution on K3 surfaces discovered by Nikulin [23]. Based on the duality, we obtain several results parallel to what is known for Borcea-Voisin threefolds. Along the way, we also investigate the Brauer groups of Calabi-Yau threefolds of type K.

Special geometry and the swampland

“…The pre-potential for all Calabi-Yau of type (iii) was computed (in some convenient frame) in ref. [132] getting a purely cubic polynomial, in agreement with the prediction from fact 5 that there should exist an electro-magnetic frame with this property. Indeed, in these cases it is fairly obvious that the Weil-Petersson metric should be locally symmetric to…”

“…A 3-CY with |π 1 (X)| = ∞ has a finite unbranched cover which is either an Abelian variety A or the product of an elliptic curve E and a K3 surface K [133]. A 3-CY of the form A/Σ is said to be of A-type, while one of the form (E × K)/Σ is called of Ktype [55,132,134]. In both cases Σ is a finite group of automorphisms acting freely.…”

“…There are six deformation types of A-type 3-CY explicitly constructed in refs. [55,134]. Their Ricci-flat Kähler metric is flat, so χ(X) = 0 and c 2 (X) = 0.…”

Complex Hantzsche-Wendt manifolds

“…However they were also found independently by different authors. In particular they were listed in [9] as examples of Calabi-Yau threefolds of Type A (more detailed description, which agree with our results, can be found in [4]). We point out that there are many inequivalent definitions of Calabi-Yau manifolds in literature and that the CHW manifolds do not satisfy all of them.…”

Type II string theory on Calabi-Yau manifolds with torsion and non-Abelian discrete gauge symmetries

“…Their product admits an action of the group G = Z 2 × Z 2 , generated by the transformations; Various other quotients, by different group actions, were considered by Oguiso and Sakurai [18] in the process of studying the collection of all Calabi-Yau manifolds with an infinite fundamental group. They obtained a partial classification, which has very recently JHEP07(2017)129 been completed in [23]. All actions of the basic group G = Z 2 × Z 2 and of all its Abelian extensions were classified in [15].…”