Integrability in String theory

Abstract

The AdS/CFT correspondence is duality between string theory negatively curved background of anti de Sitter space-time (AdS) in D-dimensions and conformal eld theory (CFT), living on D-1 dimensions on the boundary of this space. One of the methods used to study The AdS/CFT correspondence is the integrability. Integrability is a very useful property that allows many physical observables, such as spectrum of anomalous dimensions and scattering amplitudes can be computed e ciently. In this thesis we describe how to calculate integrability in the planar limit at weak coupling, using the method of spin-chain together with the Bethe ansatz. Furthermore we study the action of non-planar dilatation operator of N = 4 SYM theory in the sector SU(2). The gauge invariant operators, we considered are the restricted Schur polynomials with two matrix elds Z and Y . In the case of N = 4 SYM theory, we obtain the spectrum of the anomalous dimension the SU (2) sector at one loop as well as two loops in the SU (2) sector. We discuss integrability of the action at one loop and two loops dilatation operator in the SU(2) sector on restricted Schur polynomials of order O(N) at large N non-planar limit.