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 par
tition 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.
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 obtain
ed, via a saddle point analysis, from an equation which can be viewed as a non commutative version of the standard Seiberg and Witten curve.
We compute the partition functions of D(-1)D7 systems describing the multi-instanton dynamics of SO(N) gauge theories in eight dimensions. This is the simplest instance of the so called exotic instantons. In analogy with the Seiberg-Witten theory in
four space-time dimensions, the prepotential and correlators in the chiral ring are derived via localization formulas and found to satisfy relations of the Matone type. Exotic prepotentials of SO(N) gauge theories with N=2 supersymmetries in four-dimensions are also discussed.
We study the non-perturbative dynamics of an unoriented Z_5-quiver theory of GUT kind with gauge group U(5) and chiral matter. At strong coupling the non-perturbative dynamics is described in terms of set of baryon/meson variables satisfying a quantu
m deformed constraint. We compute the effective superpotential of the theory and show that it admits a line of supersymmetric vacua and a phase where supersymmetry is dynamically broken via gaugino condensation.
