Framed sheaves on root stacks and supersymmetric gauge theories on ALE spaces

Understand

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.