A dissertation submitted to the Faculty of Science, University of the
Witwatersrand, Johannesburg, in fulfilment of the requirements for the
degree of Doctor of Philosophy. March 2013. / In the past decade, the gauge/gravity duality has been extensively explored
in the large N limit. In particular, the spectrum of anomalous dimensions have
been compared with the energy spectrum of the dual string theory showing
remarkable agreement. In this limit, for operators with a bare dimension of
order 1, planar diagrams give the leading contribution to the anomalous dimension.
To obtain the anomalous dimensions, one needs to diagonalize the
dilatation operator. One of the methods used to accomplish this is integrability.
This allows an exact computation of the spectrum of the anomalous
dimensions. There is by now a great deal of evidence that N = 4 supersymmetric
Yang-Mills (SYM) theory and N = 6 superconformal Chern Simons
(ABJ(M)) theory are integrable in the planar limit.
In this thesis we probe the gauge gravity duality at finite N using novel tools
developed from the representation theory of symmetric and unitary groups.
We start by studying the action of the nonplanar dilatation operator of N = 4
SYM theory and ABJ(M) theory. The gauge invariant operators we consider
are the restricted Schur polynomials. In the case of N = 4 SYM theory, we
obtain the spectrum of the anomalous dimension beyond the SU(2) sector
at one loop, and in the SU(2) sector at two loops. In both cases, we obtain
the spectrum at arbitrary (finite) N. We then obtain the spectrum of
anomalous dimensions in the SU(2) sector of ABJ(M) theory at two loops.
The class of gauge invariant operators we consider have classical dimension
of order O(N). In both theories, the spectrum of the anomalous dimensions
reduces to a set of decoupled harmonic oscillators at large N. This indicates,
for the first time, that N = 4 SYM theory and ABJ(M) theory exhibit nonplanar
integrability. We expect to recover non-perturbative quantum gravity
effects, from the gauge/gravity duality, when N is finite. The non-planar integrability
we discover here may play an important role in finite N studies of the
gauge/gravity duality, and hence may play an important role in understanding
non-perturbative string stringy physics. In addition, we study various classes
of correlators in ABJ(M) theory. In this context, we derive extremal n-point
correlators in ABJ(M) theory and we probe the giant graviton dynamics in
these theories.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:wits/oai:wiredspace.wits.ac.za:10539/12913 |
| Date | 29 July 2013 |
| Creators | Mohammed, Badr Awad Elseid |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | application/pdf |
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