Wilson loops and chiral correlators on squashed spheres

Abstract

Abstract:We study chiral deformations of N = 2 and N = 4 supersymmetric gauge theories obtained by turning on τ J tr Φ J interactions with Φ the N = 2 superfield. Using localization, we compute the deformed gauge theory partition function Z( τ |q) and the expectation value of circular Wilson loops W on a squashed four-sphere. In the case of the deformed 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 N = 4 theory. Using the AGT correspondence, the τ J -deformations are related to the insertions of commuting integrals of motion in the four-point CFT correlator and chiral correlators are expressed as τ -derivatives of the gauge theory partition function on a finite Ω-background. In the so called Nekrasov-Shatashvili limit, the entire ring of chiral relations is extracted from the -deformed Seiberg-Witten curve. As a byproduct of our analysis we show that SU(2) gauge theories on rational Ω-backgrounds are dual to CFT minimal models.

Chiral trace relations in Ω-deformed $\mathscr{N}$ = 2 theories

“…The basic observation in understanding the duality at this level is the strong evidence that the chiral ring of the gauge theory retains the structure of integrals of motion, so chiral trace relations should emerge in a very natural way in this setup. This is clear in particular from the results of [30,31] for the SU (2) N = 2 theory with N f = 4 fundamentals hypermultiplets. As stated by AGT duality, this theory is related to a Liouville CFT on the 4-punctured sphere, with the instanton partition function being identified with the 4-point conformal block.…”

“…2 The authors of [30,31] found a simple relation between the vacuum expectation values of chiral trace operators and the CFT correlation function with charge insertions: where V α are primary fields in this extended algebra with conformal dimension Δ α = α(Q − α). Guided by these results, in [25] we searched for similar relations in the SU (2) N = 2 * and pure gauge theories.…”

On the large R-charge $$ \mathcal{N} $$ = 2 chiral correlators and the Toda equation

“…7 To be clear, this means the correction O(g 4 YM ) with respect to the tree level value. 8 See also [55] for a discussion in terms of Alday-Gaiotto-Tachikawa duality.…”

Wilson loop and its correlators in the limit of large coupling constant

“…Let us find now a connected correlator between a WL in a symmetric representation and a chiral primary operator. This correlator can be written via matrix model integrals [5].…”