Calabi-Yau threefolds of type K (II): mirror symmetry (Q311073)

A Calabi-Yau threefold is called a type K if it admits en étale Galois covering by the product of a \(K3\) surface and an elliptic curve. In the previous article of the authors [``Calabi-Yau threefolds of type K (I) Classification'', Preprint, \url{arXiv:1409.7601}], the full classification of Calabi-Yau threefolds of type K was obtained. There are eight classes of Calabi-Yau threefolds of type K, up to deformation, with Hodge numbers varying from \(3\) to \(11\). The equivalence class is uniquely determined by the Galois group \(G\) of the minimal splitting covering, which is isomorphic to one of the following groups: NEWLINE\[NEWLINEC_2, C_2^2, C_2^3, D_6, D_8, D_{10}, D_{12}, C_2\times D_8,NEWLINE\]NEWLINE where \(C_2\) is a cyclic group and \(D_{\ell}\) is a dihedral group.NEWLINENEWLINEThis paper continues the investigating of Calabi-Yau threefolds of type K from a viewpoint of mirror symmetry. Let \(X\) be a Calabi-Yau threefold of type \(K\) and \(\pi: S\times E\to X\) be its minimal splitting covering, where \(S\) is a \(K3\) surface and \(E\) is an elliptic curve. The geometry of \(X\) is equivalent to the \(G\)-equivariant geometry of the covering space \(S\times E\), where \(G=\text{{Gal}}(\pi)\) is the Galois group of the covering \(\pi\). The Galois group \(G\) is a semi-direct product of \(H:=\text{{Ker}}(G\to \mathrm{GL}(H^{2,0}(S)))\) and the cyclic group \(C_2\). Let \(M_G:=H^2(S,{\mathbb{Z}})^G\) and \(N_G:=H^2(S,{\mathbb{Z}})^H_{C_2}\).NEWLINENEWLINEThe first main result is formulated in the following theorem, which asserts the existence of lattice isomorphisms from which the mirror symmetry of Calabi-Yau threefolds is derived. \smallskip {Theorem 1}: There exists a lattice isomorphism \(U\oplus M_G\cong N_G\) over the rational numbers \({\mathbb{Q}}\).NEWLINENEWLINEIn particular, \(X\) is a self-mirror symmetric.NEWLINENEWLINEThere are several consequences of this fundamental duality on, e.g., the Yukawa couplings, special Lagrangian fibrations.NEWLINENEWLINEThe second main result is the explicit computation of the Brauer groups of Calabi-Yau threefolds \(X\) of type K. Let \(\text{{Br}}(X)\) denote the Brauer group of \(X\).NEWLINENEWLINE{Theorem 2.}: Let \(X\) be a Calabi-Yau threefold of type K. Then \(\text{{Br}}(X)\cong {\mathbb{Z}}_2^{\oplus m}\), where \(m\) ranges from \(1\) to \(3\) depending on the eight possible Galois group \(G\) of the minimal splitting covering. In this case, \(H_1(X,{\mathbb{Z}})\cong \text{{Br}}(X)\otimes{\mathbb{Z}}_2^{\oplus 2}.\)NEWLINENEWLINEAs an application, it is shown that any derived equivalent Calabi-Yau threefolds of type K have isomorphic Galois groups of the minimal splitting coverings.