The SU(3) spin chain sigma model and string theory

TL;DR

This work derives the continuum non-linear sigma model for the ferromagnetic SU(3) spin chain, linking the one-loop anomalous dimensions of ${\cal N}=4$ SYM operators built from the three complex scalars to a sigma-model description. It then shows that, in a large angular momentum regime, this spin-chain sigma model exactly matches the string sigma model for strings rotating on $S^5$ within $AdS_5\times S^5$, reproducing the energy and fluctuation spectra of circular three-spin solutions. By extending prior SU(2) analyses to SU(3), the paper provides a concrete bridge between integrable spin chains and semiclassical string dynamics, while noting open issues for higher-loop corrections and enlarged sectors such as $SU(2|3)$. Overall, it reinforces the view that continuum spin-chain descriptions encode key aspects of string theory in the AdS/CFT correspondence and offers a framework for exploring integrability across both sides of the duality.

Abstract

The ferromagnetic integrable SU(3) spin chain provides the one loop anomalous dimension of single trace operators involving the three complex scalars of N=4 supersymmetric Yang-Mills. We construct the non-linear sigma model describing the continuum limit of the SU(3) spin chain. We find that this sigma model corresponds to a string moving with large angular momentum in the five-sphere in AdS_5xS^5. The energy and spectrum of fluctuations for rotating circular strings with angular momenta along three orthogonal directions of the five-sphere is reproduced as a particular case from the spin chain sigma model.