Wilson loops and chiral correlators on squashed spheres

We study chiral deformations of 𝒩=2 and 𝒩=4 supersymmetric gauge theories obtained by turning on τ J tr Φ J interactions with Φ the 𝒩=2 superfield. Using localization, we compute the deformed gauge theory partition function Z(.τ|q) and the expectation value of circular Wilson loops W on a squashed four-sphere. In the case of the deformed 𝒩=4 theory, exact formulas for Z and W are derived in terms of an underlying U( N ) interacting matrix model replacing the free Gaussian model describing the 𝒩=4 theory. Using the AGT correspondence, the τ J -deformations are related to the insertions of commuting integrals of motion in the four-point CFT correlator and chiral correlators are expressed as τ -derivatives of the gauge theory partition function on a finite Ω-background. In the so called Nekrasov-Shatashvili limit, the entire ring of chiral relations is extracted from the ϵ -deformed Seiberg-Witten curve. As a byproduct of our analysis we show that SU(2) gauge theories on rational Ω-backgrounds are dual to CFT minimal models.