AGT correspondence, Ding-Iohara algebra at roots of unity and Lepowsky-Wilson construction

Abstract

It was recently conjectured that the AGT correspondence between the $U(r)$-instanton counting on ${\mathbb{R}}^4/{\mathbb{Z}}_p$ and the two-dimensional field theories with the conformal symmetry algebra $\mathcal{A}(r,p)$ can be considered as a root of unity limit of its K-theoretic analogue. From this point of view, the algebra $\mathcal{A}(r,p)$ and a special basis in its representation are limits of the Ding-Iohara algebra and the Macdonald polynomials respectively. In this paper we confirm this conjecture for the special case $r=1$. We uncover the implicit $\mathcal{A}(1,p)$ symmetry in this limit. We also found that the vertex operators in the special basis have factorized AFLT form.