Gauged linear sigma models for noncompact Calabi-Yau varieties

We study gauged linear sigma models for noncompact Calabi-Yau manifolds described as a line bundle on a hypersurface in a projective space. This gauge theory has a unique phase if the Fayet-Iliopoulos parameter is positive, while there exist two distinct phases if the parameter is negative. We find four massless effective theories in the infrared limit, which are related to each other under the Calabi-Yau/Landau-Ginzburg correspondence and the topology change. In the T-dual theory, on the other hand, we obtain two types of exact massless effective theories: one is the sigma model on a newly obtained Calabi-Yau geometry as a mirror dual, while the other is given by a Landau-Ginzburg theory with a negative power term, indicating N=2 superconformal field theory on SL(2,R)/U(1). We argue that the effective theories in the original gauged linear sigma model are exactly realized as N=2 Liouville theories coupled to well-defined Landau-Ginzburg minimal models