On the commuting charges for the highest dimension SU(2) operators in planar ${\cal N}=4$ SYM

TL;DR

This work analyzes the commuting charges of the highest-dimension SU(2) operator in planar ${\cal N}=4$ SYM within the non-wrapping regime, linking gauge theory data to integrable structures via Bethe root densities. By deriving a linear integral equation for the Bethe root density and a coupled linear system for the charges, the authors uncover the strong-coupling scaling of the charges with $Q_r \sim g^{-(r-1/2)}$ and show how the density localizes around $u\approx\pm 2g$, providing a direct bridge to semiclassical string predictions. The special case $\kappa=2$ yields exact leading terms, and a finite-length non-linear integral equation is presented to include wrapping corrections up to order $g^{2L-2}$, with explicit expressions for the finite-size charges. Overall, the paper advances the understanding of strong-coupling behavior in the AdS/CFT integrable structure of the SU(2) sector and furnishes a framework to compute commuting charges and finite-size effects from first principles.

Abstract

We consider the highest anomalous dimension operator in the SU(2) sector of planar ${\cal N}=4$ SYM at all-loop, though neglecting wrapping contributions. In any case, the latter enter the loop expansion only after a precise length-depending order. In the thermodynamic limit we write both a linear integral equation for the Bethe root density and a linear system obeyed by the commuting charges. Consequently, we determine the leading strong coupling contribution to the density and from this an approximation to the leading and sub-leading terms of any charge $Q_r$: it scales as $λ^{1/4-r/2}$, which generalises the Gubser-Klebanov-Polyakov energy law. In the end, we briefly extend these considerations to finite lengths and 'excited' operators by using the idea of a non-linear integral equation.