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# Integrability and the AdS/CFT correspondence

The AdS/CFT correspondence is a fascinating duality relating string theory in Anti de Sitter backgrounds and non-gravitational gauge theories. It provides a nonperturbative formulation of a quantum theory of gravity and might provide us a new framework to understand the strong coupling phenomena of gauge theories, such as confinement and the existence of a mass gap in QCD. On the other hand, since we are dealing with a weak/strong duality, the check of this correspondence, that is the comparison of the results obtained from either side of the duality, is very hard. Fortunately, the appearance of integrable structures in N = 4 supersymmetric Yang-Mills theory in the planar limit and in free type IIB superstrings in AdS_5xS^5 leads to the belief that these theories might be indeed exactly solvable [1].

This opened a new exciting venue of investigation in which the goal is not to check the AdS/CFT correspondence but rather to use it as an inspirational guide in the process of completely solving these two integrable theories. The impressive outcome of this endeavor is a set of asymptotic Bethe equations which are supposed to interpolate between the perturbative gauge theory and the classical string regimes [2]. The solutions to these equations yield the planar anomalous dimensions of very large single trace operators in SYM, or, on the string side, the energy of the corresponding dual string states with large angular momentum, to at any values of the t’Hooft coupling l.

In the future it would be extremely interesting to go beyond this limitation and compute the exact anomalous dimensions of any, even small, single trace operator (such as the Konishi multiplet) in the planar limit of N = 4 SYM. The general solution to this problem will most likely involve the understanding of the covariant string quantization in AdS_5xS^5 and will probably comprise the introduction of an extra level of physical particles in the known asymptotic Bethe equations.

[1] J. A. Minahan and K. Zarembo, “The Bethe-ansatz for N = 4 super Yang-

Mills,” JHEP 0303, 013 (2003) [arXiv:hep-th/0212208]. I. Bena, J. Polchinski

and R. Roiban, “Hidden symmetries of the AdS(5) x S**5 superstring,” Phys.

Rev. D 69 (2004) 046002 [arXiv:hep-th/0305116].

[2] N. Beisert and M. Staudacher, “Long-range PSU(2,2—4) Bethe ansaetze for

gauge theory and strings,” Nucl. Phys. B 727, 1 (2005) [arXiv:hep-th/0504190].

N. Beisert, R. Hernandez and E. Lopez, “A crossing-symmetric phase for

AdS(5) x S**5 strings,” JHEP 0611, 070 (2006) [arXiv:hep-th/0609044]. N. Beisert,

B. Eden and M. Staudacher, “Transcendentality and crossing,” J. Stat. Mech.

0701, P021 (2007) [arXiv:hep-th/0610251].

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