Wilson Loops and Chiral Correlators on Squashed Sphere

Understand

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 $Z(\vec\tau|q)$ 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.