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Register a new userWilson Loops and Chiral Correlators on Squashed Sphere
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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 $Z(vectau|q)$ 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.
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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 $Z(vectau|q)$ 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.
We study the correlator of concentric circular Wilson loops for arbitrary radii, spatial and internal space separations. For real values of the parameters specifying the dual string configuration, a typical Gross-Ooguri phase transition is observed. In addition, we explore some analytic continuation of a parameter $gamma$ that characterizes the internal space separation. This enables a ladder limit in which ladder resummation and string theory computations precisely agree in the strong coupling limit. Finally, we find a critical value of $gamma$ for which the correlator is supersymmetric and ladder diagrams can be exactly resummed for any value of the coupling constant.
By considering a Gaussian truncation of ${cal N}=4$ super Yang-Mills, we derive a set of Dyson equations that account for the ladder diagram contribution to connected correlators of circular Wilson loops. We consider different numbers of loops, with different relative orientations. We show that the Dyson equations admit a spectral representation in terms of eigenfunctions of a Schrodinger problem, whose classical limit describes the strong coupling limit of the ladder resummation. We also verify that in supersymmetric cases the exact solution to the Dyson equations reproduces known matrix model results.
We continue our study of the correlators of a recently discovered family of BPS Wilson loops in N=4 supersymmetric U(N) Yang-Mills theory. We perform explicit computations at weak coupling by means of analytical and numerical methods finding agreement with the exact formula derived from localization. In particular we check the localization prediction at order g^6 for different BPS latitude configurations, the N=4 perturbative expansion reproducing the expected results within a relative error of 10^(-4). On the strong coupling side we present a supergravity evaluation of the 1/8 BPS correlator in the limit of large separation, taking into account the exchange of all relevant modes between the string world-sheets. While reproducing the correct geometrical dependence, we find that the associated coefficient does not match the localization result.
We study at quantum level correlators of supersymmetric Wilson loops with contours lying on Hopf fibers of $S^3$. In $mathcal{N}=4$ SYM theory the strong coupling analysis can be performed using the AdS/CFT correspondence and a connected classical string surface, linking two different fibers, is presented. More precisely, the string solution describes oppositely oriented fibers with the same scalar coupling and depends on an angular parameter, interpolating between a non-BPS configuration and a BPS one. The system can be thought as an alternative deformation of the ordinary antiparallel lines giving the static quark-antiquark potential, that is indeed correctly reproduced, at weak and strong coupling, as the fibers approach one another.
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