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

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.