AGT correspondence: Ding–Iohara algebra at roots of unity and Lepowsky–Wilson construction

It was recently conjectured that the AGT correspondence between the $U\left(r\right)$—instanton counting on $R^4/Z_p$ and the two-dimensional field theories with the conformal symmetry algebra $A(r,p)$ can be considered as a root of unity limit of its K-theoretic analogue. From this point of view, the algebra $A(r,p)$ 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 $A(1,p)$ symmetry in this limit. We also found that the vertex operators in the special basis have factorized AFLT form. (paper)