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

Abstract

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 N=4 super Yang-Mills theory. We study numerically the integral equation derived by Beisert, Eden, and Staudacher, which resumes 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 AdS5. We also make a numerical prediction for the third term in the strong coupling series.

The strong coupling limit of the scaling function from the quantum string Bethe ansatz

“…As mentioned earlier there exists a numerical prediction for the coefficient of the O(1/ √ λ) term of (89) [16]. Furthermore, a genuine string theory calculation of the same quantity seems to be under way [30].…”

“…For the moment only the leading semi-classical contribution has been derived from the BES equation by analytic means [13][14][15]. By numerical analysis of the equation both the leading [16,17] and the next to leading order term [16] can be reproduced with high accuracy. Furthermore, it is possible to predict numerically the next term in the expansion which would result from a string theory two-loop computation [16].…”

Resurgence of the cusp anomalous dimension

“…At weak coupling, it is not hard to solve (3.1) directly order by order in g 2 [11], 7 while to find the strong coupling solution is not straightforward. A direct approach to solve the BES equation at strong coupling (and at finite coupling numerically) is found in [18,19]. Below, we review the method developed in [22,35].…”

“…is Catalan's constant. These coefficients were first predicted in [18] by the numerical analysis. In [19][20][21], the leading coefficient was computed analytically, and then, in [22], the expansion was systematically computed up to 1/g 40 .…”

Gauge-String Dualities and some Applications

“…But solving (5.7) forσ(t) at values of the coupling constant g beyond the perturbative regime is not an easy task. By expanding the fluctuation density as a Neumann series of Bessel functions, 12) one can reduce the problem to an (infinite-dimensional) matrix problem, which can be consistently truncated and is thus amenable to numerical solution [53]. This was shown to give f (g) with high accuracy up to rather large values of g, and the function was found to be monotonically increasing, smooth, and in excellent agreement with the linear asymptotics predicted by string theory (5.5).…”