We construct level one dominant representations of the affine Kac-Moody algebra $mathfrak{gl}_k$ 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 $mathfrak{gl}_k$, which proves the AGT correspondence for pure N=2 U(1) gauge theory on Xk. 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 $mathfrak{gl}_k≃mathfrak{h}oplus$mathfrak{sl}_k$, these vertex operators decompose as products of bosonic exponentials associated to the Heisenberg algebra h and primary fields of $mathfrak{sl}_k$. We use these operators to prove the AGT correspondence for N=2 superconformal abelian quiver gauge theories on $X_k$.
AGT relations for abelian quiver gauge theories on ALE spaces
Sala F.;
2016-01-01
Abstract
We construct level one dominant representations of the affine Kac-Moody algebra $mathfrak{gl}_k$ 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 $mathfrak{gl}_k$, which proves the AGT correspondence for pure N=2 U(1) gauge theory on Xk. 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 $mathfrak{gl}_k≃mathfrak{h}oplus$mathfrak{sl}_k$, these vertex operators decompose as products of bosonic exponentials associated to the Heisenberg algebra h and primary fields of $mathfrak{sl}_k$. We use these operators to prove the AGT correspondence for N=2 superconformal abelian quiver gauge theories on $X_k$.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.