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We will consider a particular deformation of N=4 Super Yang Mills theory, the
Leigh-Strassler deformation. There exists much evidence that N=4 Super Yang
Mills theory is in the planar limit a fully integrable model. In particular,
the planar dilatation operator seems to be mapped to a long range integrable
spin chain. Our main question is to which extent can this be done for the
dilatation operator of the Leigh-Strassler deformation.
After a review of the basics, we will find the parameter values for which the
dilatation operator in the full scalar field theory satisfies Reshetikhins
criteria for integrability. Then we will have a look at a particular three
state subsector of the q-deformed theory with su_q(3) symmetry which preserves
integrability. And in the end we will discuss higher loop integrability for
the q-deformed su(2) sector.