Five-dimensional SU(2) AGT conjecture and recursive formula of deformed Gaiotto state

TL;DR

The paper advances the five-dimensional AGT program by proposing a recursive formula for the inner product $\langle G|G\rangle$ of the deformed Gaiotto state and proving that the five-dimensional Nekrasov partition function $Z(\Lambda,Q,q,t)$ obeys the same recursion. Using the integral representation of $Z$ and a careful pole-residue analysis, it shows that all poles in $Q$ are simple and that their residues match the recursion terms, thereby reducing Awata–Yamada's conjecture to the conjectured inner-product recursion. This work extends the 4D AGT recursive approach to the 5D setting and links deformed Virasoro representation theory to the K-theoretic Nekrasov partition function. The methodology combines Whittaker-type states, a Deformed Virasoro algebra framework, and contour integration to establish a structurally parallel recursion between CFT objects and gauge-theory partition functions, with potential implications for higher-dimensional AGT correspondences.

Abstract

This note deals with the five-dimensional pure SU(2) AGT conjecture proposed by Awata and Yamada. We give a conjecture on a recursive formula for the inner product of the deformed Gaiotto state. We also show that the K-theoretic pure SU(2) Nekrasov partition function satisfies the same recursion relation. Therefore the five-dimensional AGT conjecture is reduced to our conjectural recursive formula.

Theorems & Definitions (16)

• Definition 2.1
• Lemma 2.2
• proof
• Conjecture 2.3
• Remark 2.4
• Proposition 3.1
• Example 3.2
• proof : Proof of Proposition \ref{['prp:int']}
• Remark 3.3
• Theorem 4.1