An exact slope for AdS/CFT

Abstract

We present a conjecture for the small spin limit of the minimal scaling dimension of Wilson operators in the sl(2) sector of the planar N=4 Super-Yang-Mills theory. The expression is given in closed form as a function of the 't Hooft coupling and twist of the operator. The formula should stand as a prediction of the Asymptotic Bethe Ansatz equations for the spectrum of scaling dimensions and evidence is given at both weak and strong coupling that it should be exact. In particular, agreement is found with established one-loop spectroscopy of string energies at strong coupling.

Figures (2)

• Figure 1: Plot of the slope $\alpha_{J}$ as a function of the coupling $\sqrt{\lambda}$ for $J = 2, \ldots , 5$. The upper (blue) line stands for $J=2$. The slope decreases monotonically as a function of the twist $J$ at fixed coupling
• Figure 2: Sketch of the (expected) complex spin plane for minimal scaling dimension. The central grey dot marks the origin -- it is not the locus of a singularity at finite coupling. The closest singularity lies on the negative $S$-axis at a position $S_{0} < 0$. Extra singularities are depicted in the upper and lower half plane -- they are expected to lie at a distance $\sim \sqrt{\lambda}$ from the origin. Presumably all singularities collide at $S = -J +1$ at weak coupling.