  Squashed toric sigma models are a class of sigma models whose target space is
a toric manifold in which the torus fibration is squashed away from the fixed
points so as to produce a neck-like region. The elliptic genera of squashed
toric-Calabi-Yau manifolds are known to obey the modular transformation
property of holomorphic Jacobi forms, but have an explicit non-holomorphic
dependence on the modular parameter. The elliptic genus of the simplest
one-dimensional example is known to be a mixed mock Jacobi form, but the
precise automorphic nature for the general case remained to be understood. We
show that these elliptic genera fall precisely into a class of functions called
higher-depth mock modular forms that have been formulated recently in terms of
indefinite theta series. We also compute a generalization of the elliptic
genera of these models corresponding to an additional set of charges
corresponding to the toric symmetries. Finally we speculate on some relations
of the elliptic genera of squashed toric models with the Vafa-Witten partition
functions of $\mathcal{N}=4$ SYM theory on $\mathbb{CP}^2$.
