  We study a class of two-dimensional N=(2,2) sigma models called squashed
toric sigma models, using their Gauged Linear Sigma Models (GLSM) description.
These models are obtained by gauging the global U(1) symmetries of toric GLSMs
and introducing a set of corresponding compensator superfields. The geometry of
the resulting vacuum manifold is a deformation of the corresponding toric
manifold in which the torus fibration maintains a constant size in the interior
of the manifold, thus producing a neck-like region. We compute the elliptic
genus of these models, using localization, in the case when the unsquashed
vacuum manifolds obey the Calabi-Yau condition. The elliptic genera have a
non-holomorphic dependence on the modular parameter $\tau$ coming from the
continuum produced by the neck. In the simplest case corresponding to squashed
$\mathbb{C}/\mathbb{Z}_{2}$ the elliptic genus is a mixed mock Jacobi form
which coincides with the elliptic genus of the N=(2,2) SL(2,R)/U(1) cigar
coset.
