On AGT-W Conjecture and q-Deformed W-Algebra

TL;DR

This work extends the AGT-W correspondence to five dimensions by linking the Nekrasov partition function for 5d SU(N) gauge theories to the scalar products of Gaiotto-Whittaker vectors in the q-deformed W_N algebra. The authors formulate the conjecture Z^{SU(N)}_{YM} = ⟨0,...,0,Λ^N|Λ^N,0,...,0⟩ within the q-W_N framework and provide explicit, nontrivial checks at low instanton numbers, including SU(3) YM and SU(4) YM one-instanton tests, using level-k Kac-Shapovalov matrices. They develop the necessary algebraic machinery via the q-Miura realization and Whittaker conditions, and illustrate how higher-rank cases follow the same pattern, thereby offering a coherent 5d uplift of the AGT-W program. The results suggest a deep link between 5d Nekrasov functions and representations of q-deformed W_N, with potential implications for 4d limits, SQCD with flavors, and broader gauge-group generalizations, pending further proofs. The work thus provides a concrete, testable bridge between 5d gauge theories and q-deformed conformal algebras, with significant theoretical and mathematical resonance.

Abstract

We propose an extension of the Alday-Gaiotto-Tachikawa-Wyllard conjecture to 5d SU(N) gauge theories. A Nekrasov partition function then coincides with the scalar product of the corresponding Gaiotto-Whittaker vectors of the q-deformed W_N algebra.