Squashed Toric Sigma Models and Mock Modular Forms

We study a class of two-dimensional \({\mathcal{N}=(2,2)}\) 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 \({\mathcal{N}=(2,2)}\) \({SL(2,\mathbb{R}) / U(1)}\) cigar coset.