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.
