A dissertation presented
to
The Faculty of Science
University of the Witwatersrand
Johannesburg
in ful lment of the requirements
for the degree of
Master of Science
June 2015 / At nite N, the number of restricted Schur polynomials is greater than, or equal to the
number of generalized restricted Schur polynomials. In this dissertation we study this dis-
crepancy and explain its origin. We conclude that, for quiver gauge theories, in general, the
generalized restricted Shur polynomials correctly account for the complete set of nite N
constraints and they provide a basis, while the restricted Schur polynomials only account
for a subset of the nite N constraints and are thus overcomplete. We identify several
situations in which the restricted Schur polynomials do in fact account for the complete set
of nite N constraints. In these situations the restricted Schur polynomials and the gen-
eralized restricted Schur polynomials both provide good bases for the quiver gauge theory.
Further, we demonstrate situations in which the generalized restricted Schur polynomials
reduce to the restricted Schur polynomials and use these results to study the anomalous
dimensions for scalar operators in ABJM theory in the SU(2) sector. The operators we
consider have a classical dimension that grows as N in the large N limit. Consequently,
the large N limit is not captured by summing planar diagrams { non-planar contributions
have to be included. We nd that the mixing matrix at two-loop order is diagonalized
using a double coset ansatz, reducing it to the Hamiltonian of a set of decoupled oscilla-
tors. The spectrum of anomalous dimensions, when interpreted in the dual gravity theory,
shows that the energy of the
uctuations of the corresponding giant graviton is dependent
on the size of the giant. The rst subleading corrections to the large N limit are also
considered. These subleading corrections to the dilatation operator do not commute with
the leading terms, indicating that integrability probably does not survive beyond the large
N limit.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:wits/oai:wiredspace.wits.ac.za:10539/19367 |
| Date | 22 January 2016 |
| Creators | Kreyfelt, Rocky |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | application/pdf |
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