Classical/quantum integrability in AdS/CFT

TL;DR

This paper explores the integrability of the AdS/CFT correspondence, linking operator dimensions in N=4 super Yang-Mills to string energies, and develops a unified approach to compute anomalous dimensions matching string theory results.

Contribution

It introduces a unified method for solving long wavelength Bethe equations and classical string solutions in the SU(2) sector, connecting gauge theory and string theory predictions.

Findings

Computed anomalous dimensions up to two loops.

Demonstrated agreement with string theory predictions.

Developed a general solution using complex curves with meromorphic differentials.

Abstract

We discuss the AdS/CFT duality from the perspective of integrable systems and establish a direct relationship between the dimension of single trace local operators composed of two types of scalar fields in N=4 super Yang-Mills and the energy of their dual semiclassical string states in AdS(5) X S(5). The anomalous dimensions can be computed using a set of Bethe equations, which for ``long'' operators reduces to a Riemann-Hilbert problem. We develop a unified approach to the long wavelength Bethe equations, the classical ferromagnet and the classical string solutions in the SU(2) sector and present a general solution, governed by complex curves endowed with meromorphic differentials with integer periods. Using this solution we compute the anomalous dimensions of these long operators up to two loops and demonstrate that they agree with string-theory predictions.