  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 $\hat{c}=2N-1$. Next we concentrate on $N=2$ (which is
physically mostly attractive case) and $N\geq 3$ 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.
