2d Sigma Models on Non-compact Calabi-Yau and $\mathcal{N}=2$ Liouville Theory

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