A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. Johannesburg, 30.09.2014. / ABSTRACT
In this dissertation we study the action of the one loop dilatation operator on operators
with a classical dimension of order N. We consider the su(3) and su(2) sectors. The
operators in the su(3) sector are constructed using three complex fields X, Y and Z, while
operators in the su(2) sector are constructed from only the two complex fields Y and Z. For
the operators in these sectors non-planar diagrams contribute already at the leading order
in N and the planar and large N limits are distinct.
Although the spectrum of anomalous dimensions in su(3) has been computed for this class
of operators, previous studies have neglected certain terms which were argued to be small.
After dropping these terms diagonalizing the dilatation operator reduces to diagonalizing
a set of decoupled oscillators. In this dissertation we explicitly compute the terms which
were neglected previously and show that diagonalizing the dilatation operator still reduces
to diagonalizing a set of decoupled oscillators.
In the su(2) sector the action of the one loop and the two loop dilatation operator reduces
to a set of decoupled oscillators and factorizes into an action on the Z fields and an action on
the Y fields. Direct computation has shown that the action on the Y fields is the same at one
and two loops. In this dissertation, using the su(2) symmetry algebra as well as structural
features of field theory, we give compelling evidence that the factor in the dilatation operator
that acts on the Y s is given by the one loop expression, at any loop order.
I hereby declare that the content of this dissertation is based on my following original
works:
• R. de Mello Koch, S. Graham and W. Mabanga, “Subleading corrections to the Double
Coset Ansatz preserve integrability” (2013) [arXiv:1312.6230v1 [hep-th]]
• R. de Mello Koch, S. Graham and I. Messamah, “Higher Loop Nonplanar Anomalous
Dimensions from Symmetry” (2013) [arXiv:1312.6227v1 [hep-th]].
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:wits/oai:wiredspace.wits.ac.za:10539/16809 |
| Date | 29 January 2015 |
| Creators | Graham, Stuart |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | application/pdf |
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