Is there a tower of charges to be discovered?

TL;DR

This paper explores higher loop integrability in a q-deformed su(2) sector of N=4 SYM, constructing a generalized spin chain model and examining commuting charges to establish integrability.

Contribution

It introduces a generalized long-range spin chain model based on Temperley-Lieb algebra for the q-deformed N=4 SYM and derives explicit local charges at one and two loops.

Findings

Constructed a Temperley-Lieb algebra-based spin chain model.

Derived explicit expressions for local charges at one and two loops.

Identified constraints for integrability at three loops.

Abstract

We investigate higher loop integrability for a q-deformation of the su(2)-sector of N=4 SYM theory. First we construct a generalisation of the long range spin chain, which for the lowest orders describes the non-deformed dilatation operator. This generalised model is built up from Temperley-Lieb algebra generators and describes the deformed theory to at least two loops. When constructing the model we have demanded the existence of one commuting charge, which puts strong constraints on the parameters to three loop orders. We also write down the five first charges for this model at two loops order. Our main goal is to obtain an explicit expression for an infinite number of commuting charges, all commuting with the dilatation operator. This would imply integrability. As a step towards this goal we present in this paper an expression for a generic local charge of the one-loop dilatation operator, which happens to be a generator of the Temperley-Lieb algebra.