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

TL;DR

This paper investigates two-loop integrability in the $su(1|1)$ sector of $ ext{N}=4$ SYM using Yangian symmetries, embedding the dilatation operator in an integrable spin chain and comparing with $su(2)$ sector results.

Contribution

It demonstrates the explicit Yangian symmetry of the two-loop dilatation operator in the $su(1|1)$ sector and provides formulas for the transfer matrix and Yangian charges.

Findings

Yangian symmetry is manifest at two loops in the $su(1|1)$ sector.

The dilatation operator can be embedded in an integrable long-range spin chain.

Comparison with $su(2)$ sector results highlights similarities and differences.

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.