| Donaldson's correspondence between instantons on R4 and vector bundles (or, allowing for degenerations, torsion-free sheaves) on the complex projective plane is the starting point for the application of powerful algebraic-geometric techniques to gauge theory. It is quite natural to ask if these techniques can be extended to other spaces. In these lectures I would like to study the case where R4 is replaced by a minimal resolution of a singularity of type Ak (ALE space). It turns out that the role of the complex projective plane is then played by a suitable stacky compactification of the ALE space. Instantons on the ALE space, with prescribed holonomy at infinity, are in a one-to-one correspondence with framed bundles on the stack, with the holonomy data encoded in the framing. In dimension 2, the moduli functor for framed sheaves on projective stacks is representable, and is represented by a scheme, for which it is possible to compute the obstruction to smoothness. The resulting partition functions show nice factorization properties with respect to the toric structure of the ALE spaces and it is possible to prove Nekrasov's conjecture for them; in the vanishing limit of the equivariant parameters they define suitable pre-potentials. |