Yangian Symmetry for the Action of the N = 4 Supersymmetric Yang–Mills Theory

Abstract

A Yangian algebra based on the superalgebra psu(2, 2|4), Y(psu(2, 2|4)), has been identified as the symmetry algebra governing the behavior of numerous observables of the N = 4 supersymmetric Yang–Mills theory in the planar limit. It appeared in the study of the scattering amplitudes as the dual superconformal symmetry, it was shown to be a symmetry of the smooth Wilson loops, and made appearance in the investigation of the spectrum of the theory. In this thesis we show that the Y(psu(2, 2|4)) is also a symmetry of the classical action of the N = 4 supersymmetric Yang–Mills theory. In this way we relate the discoveries made on the level of various observables to the usual notion of the symmetry of the theory as the invariance of its action. This is achieved first by understanding the behavior of the equations of motion of the theory under the action of the Yangian generators. This leads us to propose the notion of a weak and a strong invariance of the equations of motion. The strong invariance under the Yangian symmetry is subsequently argued to be equivalent to the invariance of the action. We then use it to derive how the Yangian generators are to be applied to the action of the theory in order to demonstrate that it is indeed left invariant. We show that the symmetry can be preserved also in the presence of gauge-fixing terms. To this end we introduce new types of Yangian-like generators, which combine local and global transformations. We then derive the identity satisfied by the action of the N = 4 supersymmetric Yang–Mills theory with the addition of the BRST gauge-fixing terms. Finally we propose Slavnov–Taylor identities for the Yangian symmetry. These are the identities satisfied by the correlation functions of the fundamental fields of the theory. We demonstrate that our proposed identities hold in numerous examples due to invariance of the action, also taking into account the effects of gauge-fixing. We also argue why the symmetry should hold in the quantum theory.