Classical/quantum integrability in non-compact sector of AdS/CFT

TL;DR

This work analyzes the non-compact $SL(2,\mathbb{R})$ sector of AdS/CFT, comparing integrable structures in ${\cal N}=4$ SYM and the classical $AdS_3\times S^1$ sigma-model. It formulates and solves a unified Riemann-Hilbert problem for finite-gap (classical) solutions and shows that at one loop the gauge-theory scaling limit matches the classical limit of the string Bethe equations, establishing a precise gauge-string correspondence in this non-compact sector. The authors develop the general scaling-limit solution of the Bethe equations via hyperelliptic Riemann surfaces, provide explicit one-cut (circular string) and two-pole (sigma-model) solutions, and demonstrate explicit agreement with perturbative gauge theory. They discuss the implications for a full quantum integrable description of the AdS sigma-model, potential two-loop discrepancies, and the need for discrete SL(2) Bethe equations to complete the integrable picture of the AdS/CFT correspondence.

Abstract

We discuss non-compact SL(2,R) sectors in N=4 SYM and in AdS string theory and compare their integrable structures. We formulate and solve the Riemann-Hilbert problem for the finite gap solutions of the classical sigma model and show that at one loop it is identical to the classical limit of Bethe equations of the spin (-1/2) chain for the dilatation operator of SYM.

Figures (1)

• Figure 1: The Riemann surface $\Sigma$ for $K=2$.