Comments on Higher Loop Integrability in the $su(1|1)$ Sector of $\cal N$=4 $SYM$: Lessons From the $su(2)$ Sector

Abstract

An analysis of two loop integrability in the $su(1|1)$ sector of $\cal{N}$=4$SYM$ is presented from the point of view of Yangian symmetries. The analysis is carried out in the scaling limit of the dilatation operator which is shown to have a manifest $su(1|1)$ invariance. After embedding the scaling limit of the dilatation operator in a general (Inozemtsev like) integrable long ranged supersymmetric spin chain, the perturbative Yangian symmetry of the two loop dilatation operator is also made evident. The explicit formulae for the two loop gauge theory transfer matrix and Yangian charges are presented. Comparisons with recent results for the effective Hamiltonians for fast moving strings in the same sector are also carried out. Apart from this, a review of the corresponding results in the $su(2)$ sector obtained by Beisert, Dippel, Serban and Staudacher is also presented.