2D sigma models on noncompact Calabi-Yau manifolds and 𝒩 = 2 Liouville Theory

We consider a class of two dimensional conformal 𝒩 = 2 supersymmetric 𝑈⁡(1) gauge linear sigma models with 𝑁 fields of charges +1 and 𝑁 fields of charges −1, whose Higgs branches are noncompact toric Calabi-Yau manifolds of complex dimension 2⁢𝑁 −1. We show, starting from large-𝑁 approximation, that the Coulomb branch of these models, which opens up at strong coupling, is described by 𝒩 = 2 Liouville theory and then extrapolate it to exact equivalence demanding the central charge of the Liouville theory to be ^𝑐 = 2⁢𝑁 −1. Next, we concentrate on mostly physically attractive 𝑁 =2 and 𝑁 ≥3 cases and find there a perfect agreement of the set of complex moduli on the Calabi-Yau side with the marginal deformations in 𝒩 =2 Liouville theory, supporting proposed exact equivalence.