Wilson Loops and Chiral Correlators on Squashed Spheres

TL;DR

This paper computes deformed gauge theory partition functions and Wilson loop expectation values on squashed spheres using localization, relating them to CFT correlators via the AGT correspondence, and explores their dependence on deformations and matrix models.

Contribution

It provides exact formulas for deformed partition functions and Wilson loops in ${ m N}=4$ theories on squashed spheres, connecting gauge theory results with CFT via AGT and matrix models.

Findings

Exact formulas for $Z$ and $W$ in terms of matrix models.

Relation of deformations to insertions of integrals of motion in CFT.

Chiral correlators as derivatives of the partition function.

Abstract

After a very brief recollection of how my scientific collaboration with Ugo started, in this talk I will present some recent results obtained with localization: the deformed gauge theory partition function $Z(\vec\tau|q)$ and the expectation value of circular Wilson loops $W$ on a squashed four-sphere will be computed. The partition function is deformed by turning on $\tau_J \,{\rm tr} \, \Phi^J$ interactions with $\Phi$ the ${\cal N}=2$ superfield. For the ${\cal N}=4$ theory SUSY gauge theory exact formulae for $Z$ and $W$ in terms of an underlying $U(N)$ interacting matrix model can be derived thus replacing the free Gaussian model describing the undeformed ${\cal N}=4$ theory. These results will be then compared with those obtained with the dual CFT according to the AGT correspondence. The interactions introduced previously are in fact related to the insertions of commuting integrals of motion in the four-point CFT correlator and the chiral correlators are expressed as $\tau$-derivatives of the gauge theory partition function on a finite $\Omega$-background.