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

TL;DR

This paper confirms a conjecture linking the AGT correspondence at roots of unity to Ding-Iohara algebra and Macdonald polynomials, specifically for the case r=1, revealing new algebraic symmetries and simplified vertex operators.

Contribution

It demonstrates the root of unity limit of the AGT correspondence for r=1, connecting Ding-Iohara algebra, Macdonald polynomials, and uncovering new symmetries and factorized vertex operators.

Findings

Confirmed the AGT conjecture for r=1 at roots of unity.

Identified the implicit (1,p) symmetry in the limit.

Discovered factorized AFLT form of vertex operators.

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.