AGT relations for abelian quiver gauge theories on ALE spaces

Abstract

We construct level one dominant representations of the affine Kac–Moody algebra g l ̂ k on the equivariant cohomology groups of moduli spaces of rank one framed sheaves on the orbifold compactification of the minimal resolution X k of the A k − 1 toric singularity C 2 / Z k . We show that the direct sum of the fundamental classes of these moduli spaces is a Whittaker vector for g l ̂ k , which proves the AGT correspondence for pure N = 2 U ( 1 ) gauge theory on X k . 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 g l ̂ k ≃ h ⊕ s l ̂ k , these vertex operators decompose as products of bosonic exponentials associated to the Heisenberg algebra h and primary fields of s l ̂ k . We use these operators to prove the AGT correspondence for N = 2 superconformal abelian quiver gauge theories on X k .