Linear sigma models for the R8/ZkR8/Zk orbifold

We construct N=4N=4 gauged linear sigma models in two dimensions whose Higgs branch has a R8/ZkR8/Zk orbifold singularity or its generalization. Our linear sigma models have either ALF or ALE type hyper-Kähler 8-manifolds as their Higgs branch. For the ALE case, the matter content of our model is specified by a quiver diagram which is a union of two A-type extended Dynkin diagrams overlapping at one link.