AGT relations for abelian quiver gauge theories on ALE spaces

We construct level one dominant representations of the affine Kac–Moody algebra gl̂k on the equivariant cohomology groups of moduli spaces of rank one framed sheaves on the orbifold compactification of the minimal resolution XkXk of the Ak−1Ak−1 toric singularity C2/ZkC2/Zk. We show that the direct sum of the fundamental classes of these moduli spaces is a Whittaker vector for gl̂k, which proves the AGT correspondence for pure N=2U(1) gauge theory on XkXk. We consider Carlsson–Okounkov type Ext-bundles over products of the moduli spaces and use their Euler classes to define vertex operators. Under the decomposition gl̂k≃h⊕sl̂k, these vertex operators decompose as products of bosonic exponentials associated to the Heisenberg algebra hh and primary fields of sl̂k. We use these operators to prove the AGT correspondence for N=2N=2 superconformal abelian quiver gauge theories on XkXk.