Wilson Loops and Chiral Correlators on Squashed Sphere

We study chiral deformations of $\mathcal{N}=2$ and $\mathcal{N}=4$ supersymmetric gauge theories obtained by turning on ${\tau }_J\mathrm{t}\mathrm{r}{\mathrm{\Phi }}^J$ interactions with $\mathrm{\Phi }$ the $\mathcal{N}=2$ superfield. Using localization, we compute the deformed gauge theory partition function $Z(\vec{\tau }\mathrm{\mid }q)$ and the expectation value of circular Wilson loops $W$ on a squashed four-sphere. In the case of the deformed $\mathcal{N}=4$ 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 $\mathcal{N}=4$ theory. Using the AGT correspondence, the ${\tau }_J$-deformations are related to the insertions of commuting integrals of motion in the four-point CFT correlator and chiral correlators are expressed as $\tau$-derivatives of the gauge theory partition function on a finite $\mathrm{\Omega }$-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 $\mathrm{\Omega }$-backgrounds are dual to CFT minimal models.