Five-dimensional AGT Conjecture and the Deformed Virasoro Algebra

TL;DR

The paper proposes a five-dimensional analogue of the AGT correspondence for pure SU(2) gauge theory by relating the 5D instanton partition function to the inner product of a deformed Gaiotto state in the $q$-Virasoro algebra. It develops the quantum deformed Virasoro framework, constructs the deformed Gaiotto state, and provides a free-boson realization, showing an exact match with Nekrasov’s 5D instanton sum (with $k=Q^{1/2}$) up to high order. The results bridge 5D gauge theory and deformed 2D conformal symmetry via Macdonald-polynomial structures, suggesting extensions to SU(n) and matter content and linking to irregular conformal structures and Painlevé equations. The work points to concrete vertex-operator methods as avenues for a formal proof and broader applications in 5D gauge/CFT correspondences.

Abstract

We study an analog of the AGT relation in five dimensions. We conjecture that the instanton partition function of 5D N=1 pure SU(2) gauge theory coincides with the inner product of the Gaiotto-like state in the deformed Virasoro algebra. In four dimensional case, a relation between the Gaiotto construction and the theory of Braverman and Etingof is also discussed.