Wilson Loops and Chiral Correlators on Squashed Sphere

TL;DR

This paper explores deformations of supersymmetric gauge theories on a squashed sphere, deriving exact formulas for partition functions and Wilson loops, and connecting these to conformal field theory via the AGT correspondence.

Contribution

It provides explicit formulas for deformed gauge theory partition functions and Wilson loops, and relates these to integrable models and conformal field theories through novel dualities.

Findings

Exact formulas for $Z$ and $W$ in deformed ${ m N}=4$ theory.

Connection between $ au_J$-deformations and integrals of motion in CFT.

Duality between $SU(2)$ gauge theories on rational $oldsymbol{ extOmega}$-backgrounds and minimal models.

Abstract

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. 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.