Semiclassical relativistic strings in S^5 and long coherent operators in N=4 SYM theory

TL;DR

This work derives a universal semiclassical framework linking fast-moving strings on S^5 to long coherent operators in N=4 SYM via a phase-space sigma-model reduction. By isolating a fast angular coordinate and performing controlled expansions in ${\tilde{\lambda}}=\lambda/L^2$, the authors show that both the string action and the SO(6) spin-chain action converge to the same Landau-Lifshitz-type sigma model on the coset $G_{2,6}$. They provide explicit mappings between string solutions (including pulsating and rotating configurations) and locally BPS coherent SYM operators, and obtain exact energy–action relations using action-angle variables, along with higher-order corrections via canonical perturbation theory. The results unify rotating and pulsating dynamics within a single coherent framework and suggest extensions to two-loop corrections and broader subsectors of the AdS_5 × S^5 correspondence.

Abstract

We consider the low energy effective action corresponding to the 1-loop, planar, dilatation operator in the scalar sector of N=4 SU(N) SYM theory. For a general class of non-holomorphic ``long'' operators, of bare dimension L>>1, it is a sigma model action with 8-dimensional target space and agrees with a limit of the phase-space string sigma model action describing generic fast-moving strings in the S^5 part of AdS_5 x S^5. The limit of the string action is taken in a way that allows for a systematic expansion to higher orders in the effective coupling $λ/L^2$. This extends previous work on rigid rotating strings in S^5 (dual to operators in the SU(3) sector of the dilatation operator) to the case when string oscillations or pulsations in S^5 are allowed. We establish a map between the profile of the leading order string solution and the structure of the corresponding coherent, ``locally BPS'', SYM scalar operator. As an application, we explicitly determine the form of the non-holomorphic operators dual to the pulsating strings. Using action--angle variables, we also directly compute the energy of pulsating solutions, simplifying previous treatments.