Wilson Loops and Chiral Correlators on Squashed Sphere

We study chiral deformations of ${cal N}=2$ and ${cal N}=4$ supersymmetric gauge theories obtained by turning on $tau_J ,{rm tr} , Phi^J$ interactions with $Phi$ the ${cal N}=2$ superfield. Using localization, we compute the deformed gauge theory partition function $Z(vectau|q)$ and the expectation value of circular Wilson loops $W$ on a squashed four-sphere. In the case of the deformed ${cal N}=4$ 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 ${cal N}=4$ 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 $Omega$-background. In the so called Nekrasov-Shatashvili limit, the entire ring of chiral relations is extracted from the $epsilon$-deformed Seiberg-Witten curve. As a byproduct of our analysis we show that $SU(2)$ gauge theories on rational $Omega$-backgrounds are dual to CFT minimal models.

Gauge theories on Omega-backgrounds from non commutative Seiberg-Witten curves

We study the dynamics of a N=2 supersymmetric SU(N) gauge theory with fundamental or adjoint matter in presence of a non trivial Omega-background along a two dimensional plane. The prepotential and chiral correlators of the gauge theory can be obtained, via a saddle point analysis, from an equation which can be viewed as a non commutative version of the standard Seiberg and Witten curve.