  We develop a new approach to the study of supersymmetric gauge theories on
ALE spaces using the theory of framed sheaves on root toric stacks, which
illuminates relations with gauge theories on $\mathbb{R}^4$ and with
two-dimensional conformal field theory. We construct a stacky compactification
of the minimal resolution $X_k$ of the $A_{k-1}$ toric singularity
$\mathbb{C}^2/\mathbb{Z}_k$, which is a projective toric orbifold
$\mathscr{X}_k$ such that $\mathscr{X}_k\setminus X_k$ is a
$\mathbb{Z}_k$-gerbe. We construct moduli spaces of torsion free sheaves on
$\mathscr{X}_k$ which are framed along the compactification gerbe. We prove
that this moduli space is a smooth quasi-projective variety, compute its
dimension, and classify its fixed points under the natural induced toric
action. We use this construction to compute the partition functions and
correlators of chiral BPS operators for $\mathcal{N}=2$ quiver gauge theories
on $X_k$ with nontrivial holonomies at infinity. The partition functions are
computed with and without couplings to bifundamental matter hypermultiplets and
expressed in terms of toric blowup formulas, which relate them to the
corresponding Nekrasov partition functions on the affine toric subsets of
$X_k$. We compare our new partition functions with previous computations,
explore their connections to the representation theory of affine Lie algebras,
and find new constraints on fractional instanton charges in the coupling to
fundamental matter. We show that the partition functions in the low energy
limit are characterised by the Seiberg-Witten curves, and in some cases also by
suitable blowup equations involving Riemann theta-functions on the
Seiberg-Witten curve with characteristics related to the nontrivial holonomies.
