VEV of $Q$-operator in $U(1)$ linear quiver 5d gauge theories

TL;DR

This paper studies the expectation values of Baxter's Q operator in 5d U(1) linear quiver gauge theories, revealing a specific Young diagram contribution restriction and deriving explicit formulas relating to the Q operator and its 4d limit.

Contribution

It establishes a connection between the Q operator VEVs and restricted Young diagram contributions in 5d gauge theories, providing explicit formulas and the 4d limit expressions.

Findings

Restricted Young diagram contributions to the partition function.

Explicit formulas for Q operator VEVs in terms of q-deformed Appel functions.

Derivation of 4d limit expressions for the Q operator VEVs.

Abstract

Linear quiver ${\cal N}=1$ 5d gauge theory in $\Omega$ background is considered. It is shown that under certain restrictions on the VEV's of the adjoint scalar field corresponding to the first node, only the array of Young diagrams, such that the first diagram is a single column and the others are empty, contribute to the partition function. Furthermore it is proved that this partition function in a simple way is related to the expectation values of Baxter's $Q$ operator (at specific discrete values of the spectral parameter) in the gauge theory with the special node removed. Using known expression of the partition function in the $U(1)$ quiver, Baxter's T-Q difference equations are established and explicit expressions for the VEV of the $Q$ operator in terms of generalized q-deformed Appel's functions is found. Finally the corresponding expressions for the 4d limit are derived.