Test of the Anti–de Sitter-Space/Conformal-Field-Theory Correspondence Using High-Spin Operators

In two remarkable recent papers the planar perturbative expansion was proposed for the universal function of the coupling appearing in the dimensions of high-spin operators of the $\mathcal{N}=4$ super Yang-Mills theory. We study numerically the integral equation derived by Beisert, Eden, and Staudacher, which resums the perturbative series. In a confirmation of the anti–de Sitter-space/conformal-field-theory (AdS/CFT) correspondence, we find a smooth function whose two leading terms at strong coupling match the results obtained for the semiclassical folded string spinning in ${\mathrm{AdS}}_5$. We also make a numerical prediction for the third term in the strong coupling series.

Figure 1

Plot of the solutions of the integral equations: $f_m(g)$ for the Eden and Staudacher kernel [18] $K^{(m)}$ (upper curve, red), $f_0(g)$ for the kernel $K^{(m)}+K^{(c)}$ (middle curve, green), and $f(g)$ for the BES kernel $K^{(m)}+2K^{(c)}$ (lower curve, blue). Notice the different asymptotic behaviors. The inset shows the three functions in the crossover region $0<g<1$.