AGT relations for abelian quiver gauge theories on ALE spaces

Abstract

We construct level one dominant representations of the affine Kac-Moody algebra ${\widehat{\mathfrak{gl}}}_k$ 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 ${\mathbb{C}}^2/{\mathbb{Z}}_k$. We show that the direct sum of the fundamental classes of these moduli spaces is a Whittaker vector for ${\widehat{\mathfrak{gl}}}_k$, which proves the AGT correspondence for pure $\mathcal{N}=2$ $U(1)$ 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 ${\widehat{\mathfrak{gl}}}_k\simeq \mathfrak{h}\oplus {\widehat{\mathfrak{sl}}}_k$, these vertex operators decompose as products of bosonic exponentials associated to the Heisenberg algebra $\mathfrak{h}$ and primary fields of ${\widehat{\mathfrak{sl}}}_k$. We use these operators to prove the AGT correspondence for $\mathcal{N}=2$ superconformal abelian quiver gauge theories on $X_k$.

58 pages; v2: typos corrected, reference added; v3: Introduction expanded, minor corrections and clarifying remarks added throughout, references added and updated; Final version published in Journal of Geometry and Physics