Five-dimensional AGT Relation and the Deformed beta-ensemble

TL;DR

This work extends the AGT correspondence to five dimensions by constructing a $q$-deformed beta-ensemble that automatically satisfies a $q$-W_N constraint and by relating it to the five-dimensional Nekrasov partition function for $SU(N)$ with $N_f=2N$. The authors develop a comprehensive $q$-deformed algebraic infrastructure based on the $q$-W_N algebra, including bosonic realizations, primary fields, screening currents, and singular vectors, then realize the beta-ensemble as a Macdonald-polynomial framework with a contour integral representation. They derive loop equations and a quantum spectral curve, analyze the large-$r$ saddle-point limit to obtain a reduced spectral curve, and connect the partition function to $q$-deformed Liouville correlators and $q$-hypergeometric functions, with exact matches to the 5D Nekrasov partition function in key cases (notably $N=2$). The work also clarifies the limiting behavior to four dimensions ($q o1$) and links to the classical ${ m W}_N$ and Jack polynomial structures, offering a new integrable approach to 5D gauge theories and their CFT duals.

Abstract

We discuss an analog of the AGT relation in five dimensions. We define a q-deformation of the beta-ensemble which satisfies q-W constraint. We also show a relation with the Nekrasov partition function of 5D SU(N) gauge theory with N_f=2N.