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

Abstract

We consider the highest anomalous dimension operator in the SU(2) sector of planar $\mathcal{N}=4$ 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 ${\lambda }^{1/4-r/2}$, 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.

Latex file, 20 pages, some typos corrected, some technical details expanded and explained

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• Extended Y-system for the $Ad{S}_5/CF{T}_4$ correspondence
• A Generalized Scaling Function for AdS/CFT
• TBA and Y-system for planar $Ad{S}_4/CF{T}_3$
• On Drinfeld's second realization of the AdS/CFT su(2|2) Yangian
• Large spin corrections in $\mathcal{N}=4$ SYM sl(2): still a linear integral equation
• Yangians, S-matrices and AdS/CFT
• Strong coupling for planar $\mathcal{N}=4$ SYM theory: an all-order result
• Asymptotic Bethe Ansatz on the GKP vacuum as a defect spin chain: scattering, particles and minimal area Wilson loops
• The generalised scaling function: a note
• Resurgence of the dressing phase for ${\mathrm{AdS}}_5\times {\mathrm{S}}^5$
• The generalised scaling function: a systematic study
• Structure of the string R-matrix
• Review of AdS/CFT Integrability, Chapter III.4: Twist states and the cusp anomalous dimension
• On the logarithmic powers of $sl(2)$ SYM${}_4$
• From nesting to dressing
• TBA-like equations and Casimir effect in (non-)perturbative AdS/CFT
• The virtual scaling function of twist operators in the N=6 Chern-Simons theory
• Giant Spinons
• On the dressing phase in the SL(2) Bethe Ansatz