Mirror Map for Landau-Ginzburg models with nonabelian groups

Abstract

BHK mirror symmetry as introduced by Berglund--H\"ubsch 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 $\Phi$ 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.