Integrability of smooth Wilson loops in N = 4 $$ \mathcal{N}=4 $$ superspace

Abstract

We perform a detailed study of the Yangian symmetry of smooth supersymmetric Maldacena-Wilson loops in planar N = 4 super Yang-Mills theory. This hidden symmetry extends the global superconformal symmetry present for these observables. A gauge-covariant action of the Yangian generators on the Wilson line is established that generalizes previous constructions built upon path variations. Employing these generators the Yangian symmetry is proven for general paths in non-chiral N = 4 superspace at the first perturbative order. The bi-local piece of the level-one generators requires the use of a regulator due to divergences in the coincidence limit. We perform regularization by point splitting in detail, thereby constructing additional local and boundary contributions as regularization for all level-one Yangian generators. Moreover, the Yangian algebra at level one is checked and compatibility with local kappa-symmetry is established. Finally, the consistency of the Yangian symmetry is shown to depend on two properties: the vanishing of the dual Coxeter number of the underlying superconformal algebra and the existence of a novel superspace "G-identity" for the gauge field theory. This tightly constrains the conformal gauge theories to which integrability can possibly apply.

Yangian symmetry for the action of planar N=4 super Yang-Mills and N=6 super Chern-Simons theories

“…The non-linear term H [1] is also zero for N = 4 sYM by explicit computation. One can relate this finding to the G-identity for N = 4 sYM found in [15]. Altogether, the constituent operators of the bi-local level-one generator commute (when summed over all pairs as implied by Sweedler's notation)…”

“…However, here the corresponding worldsheet topology of the observables is a clear obstruction for Yangian invariance. Yangian invariance without any of the above restrictions has only been observed for expectation values of smooth, non-intersecting Maldacena-Wilson loops with disc topology [14,15] (even though the analysis is limited to one loop so far).…”

Yangian algebra and correlation functions in planar Gauge theories

“…However, there will be some well-prescribed terms involving K[J] and Q to compensate the remaining terms. 15 In particular, this is rather evident in a supersymmetric theory when the level-zero algebra includes supersymmetry. 16 This includes the curious statement that the unphysical fields carry no momentum, no energy and no angular momentum.…”

“…This function has been evaluated explicitly for J = P in N = 4 sYM in [15] with a very simple structure as…”

Wilson loops and integrability

“…These loop operators have been constructed and shown to be Yangian symmetric both at weak and strong coupling [81,82,83].…”