Classical/quantum integrability in AdS/CFT

Vladimir A. Kazakov, Andrei Marshakov, Joseph A. Minahan, Konstantin Zarembo

Research output: Contribution to journalArticlepeer-review

408 Citations (Scopus)


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 × 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.

Original languageEnglish
Pages (from-to)499-553
Number of pages55
JournalJournal of High Energy Physics
Issue number5
Publication statusPublished - 1 May 2004
Externally publishedYes


  • 1/N Expansion
  • AdS-CFT and dS-CFT Correspondence
  • Bethe Ansatz
  • Integrable Equations in Physics


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