BHK mirror symmetry as introduced by Berglund–Hübsch and Marc Krawitz between Landau–Ginzburg (LG) models has been the topic of much study in recent years. An LG model is determined by a potential function and a group of symmetries. BHK mirror symmetry is only valid when the group of symmetries is comprised of the so-called diagonal symmetries. Recently, an extension to BHK mirror symmetry to include nonabelian symmetry groups has been conjectured. In this article, we provide a mirror map at the level of state spaces between the LG A-model state space and the LG B-model state space for the mirror model predicted by the BHK mirror symmetry extension for nonabelian LG models. We introduce two technical conditions, the Diagonal Scaling Condition, and the Equivariant Φ condition, under which a bi-degree preserving isomorphism of state spaces (the mirror map) is guaranteed to exist, and we prove that the condition is always satisfied if the permutation part of the group is cyclic of prime order.

中文翻译：

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具有非阿贝尔群的 Landau-Ginzburg 模型的镜像映射

Berglund-Hübsch 和 Marc Krawitz 在 Landau-Ginzburg (LG) 模型之间引入的 BHK 镜像对称性一直是近年来研究的热点。 LG 模型由势函数和一组对称性决定。 BHK 镜像对称仅在对称群由所谓的对角对称组成时才有效。最近，人们推测 BHK 镜像对称性的扩展包括非阿贝尔对称群。在本文中，我们为非交换 LG 模型的 BHK 镜像对称扩展预测的镜像模型提供了 LG A 模型状态空间和 LG B 模型状态空间之间的状态空间级别的镜像映射。我们引入了两个技术条件，即对角缩放条件和等变 Φ 条件，在这两个条件下，保证状态空间（镜像映射）的二度保持同构存在，并且我们证明，如果排列总是满足该条件该群的一部分是素数循环的。