3. Electronic Theses and Dissertations (ETDs) - All submissions
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Item Symmetry classifications on a curved geometry(2020) Mathebula, AgreementIn this thesis, we consider one-parameter point transformations that leave a differential equation invariant. In particular, we show that Noether symmetry classifications of any diagonal metric may be simplified by geometric criteria. We describe the Klein-Gordon equation for some general spaces and deal with the corresponding Killing algebra. Moreover, our investigation consists of several metrics, their Lie algebras, the point generators of the Klein-Gordon equation and their associated potential functions. Finally, we study a class of ecological diffusive equations and determine higher-order symmetries of non-linear diffusion equationsItem Probing space-time geometry using young diagrams(2018) Nkumane, LwaziQuantum field theories and theories of gravity play an essential role in understanding nature. A dramatic recent development has been the discovery that quantum field theories are equivalent or dual to theories of quantum gravity on negatively curved spacetime. This duality goes under the name of the AdS/CFT correspondence. Sometimes the computation of certain observables in field theory are more difficult than the computation of the same observables in the theory of gravity and the opposite is also true. This makes the correspondence a powerful tool, that might provide an approach to strong coupling dynamics. We explore the AdS/CFT correspondence between type IIB string theory on asymptotically AdS5×S5 backgrounds as our gravity theory and N = 4 super Yang-Mills as our conformal field theory. We study BPS operators with bare dimension of order N2 in the field theory and identify them with BPS geometries on the gravity side of the correspondence. The dynamics of 1/2 BPS geometries are identified with gauge invariant operators constructed using a single field in the field theory, while the dynamics of 1/4 BPS geometries are identified with gauge invariant operators constructed using two fields. We find a sector of the two matrix model defined by the SU(2) sector in the field theory, that can be reduced to eigenvalue dynamics. The BPS operators in this sector are associated to solutions on the gravity side of the correspondence. We also identify the gauge invariant operators with bare dimension of order N, constructed using three fields, with 1/8 BPS giant graviton states. We count these gauge invariant operators constructed using three fields in the field theory and show that the counting of these operators is in agreement with the number of giant graviton states. We also demonstrate a correspondence between correlation functions of the field theory and the overlaps of the giant graviton wave functions. By working in terms of the eigenvalues we have managed to go from the matrix, which contains O(N2) degrees of freedom, to the eigenvalues which are O(N) degrees of freedom. Thus our work points to a significant simplification of the dynamics, something that deserves to be understood better. Another concrete result that we have achieved, is a proposal for some of the operators that are dual to the 1/4 BPS geometries. This is a genuine two matrix problem so it represents a novel extension of the understanding achieved by LLM of the 1/2 BPS geometries, constructed using a single matrix. The observables dual to new geometries have a bare dimension of O(N2). We have also considered operators with a bare dimension O(N), which are dual to 1/4 BPS giant gravitons. In this case too, we demonstrate that the eigenvalue description is useful. Almost all of the studies of the large N limit of CFT have focused on the planar limit. Here, since the operator dimensions scale as we take N →∞, we are considering large N but non-planar limits of the CFT. In these limits non-planar diagrams are not suppressed and the problem is considerably more difficult. The fact that we are able to explore this limit is concrete evidence for the power of the eigenvalue description and it suggests that a systematic treatment of large N but non-planar limits is possible.Item Emergent spacetime(2017) Mathaba, KagisoIn this dissertation we explore the connection between entanglement and geometry. Recent work in the AdS/CFT correspondence has uncovered fascinating connections between quantum information and geometry, suggesting that entanglement in the CFT results in the emergence of spacetime in the bulk . We work in the 1/2 BPS sector of the duality between N = 4 super Yang Mills on R x S3 and IIB string theory on AdS5 S5. We aim to test this connection by calculating the Renyi entropies in the presence of 1/2 BPS operators heavy enough to deform the background geometry. This allows us to calculate the entanglement of these operators via the replica trick. The Ryu-Takayanagi formula relates this calculation to a minimal surface in the dual supergravity geometry, thus allowing us to observe how the boundary entanglement affects the bulk spacetime. We build a formula to calculate correlation functions of 1/2 BPS operators on the Riemann sheet that arises from the replica trick. This is a recursive formula based on group theory techniques. We demonstrate how the formula works for light operators and discuss how it can be generalised to include heavy operators by considering symmetric groups of higher order.Item Geophysical investigation into the geology, geometry and geochronology of the South African Pilanesberg Complex and the Pilanesberg dyke system(2016) Lee, Sally-AnneThe Mesoproterozoic Pilanesberg Complex, South Africa, is the world’s largest alkaline intrusive complex. Mapped geological field relationships suggest the Complex has circular inward dipping layers. However, it is unclear how the dipping layers extend at depth. As a result, the 3D geometry of the Pilanesberg Complex is unknown. Modelling of the Pilanesberg Complex uses 2D forward models as well as 3D forward and inversion, gravity and magnetic data models, to set limits on the 3D geometry of the Pilanesberg Complex. The 2D Bouguer gravity models and geology maps indicate that some of the Bushveld Complex Main Zone shifted to the west of the Pilanesberg Complex during emplacement. This, and a highly faulted country rock, accounts for a portion of how the host rock was able to accommodate the Pilanesberg Complex intrusion. The geometry of the Complex is explored with test gravity models where the model of outward dipping and vertically dipping cylinders are unable to match the Bouguer gravity signal over the Complex, but the inward dipping model matched the data to provide a possible solution for the geometry of the Complex. The Pilanesberg Complex geometry is modelled with 3D magnetic inversion, 3D forward gravity models and 2.5D gravity test profiles that were all constrained by the surface geology. The different models correlate so that best data fit for the Complex is represented by an overall inward dipping structure. Surface geological measurements indicate that the northern edge of the Complex dip out to the north. The 3D forward modelling was able to produce a positive solution that matched the gravity data with a northward dipping northern edge. The dipping northern edge is also observed on the University of British Columbia, UBC, 3D gravity inversion and the Euler deconvolution gravity profile solutions. The depth of the Pilanesberg Complex from 3D forward gravity modelling is estimated to be between 5 and 6 km. The Complex is suggested to have undergone block movement where the northern block and southern block are separated by the 30 km long Vlakfontein fault, which bisects the Complex from the north-east to the south-west. The image processing contact depth, Euler deconvolution solutions and the 3D Voxi inversion model suggest that the fresh bedrock is closer to surface in the north, while the southern block appears to be approximately 1km deeper than the northern block. The northern dip and block movement are explained by complicated structural events that include trap door graben settling which hinged on the northern edge as well as faulting and external block movement during a regional lateral extensional event. The Pilanesberg Complex intruded during a larger system of alkaline intrusions, known as the Pilanesberg Alkaline Province. The intrusions are associated with the Province due to their ages and chemical affinity. This Province includes two dyke swarms that radiate to the north-west and south of the Pilanesberg Complex, as well as smaller circular clinopyroxenite intrusions throughout the Bushveld Complex. The Pilanesberg dyke system and the circular clinopyroxenite intrusions are reversely magnetised with IGRF corrected values ranging between -150 to -320 nT compared to the normally magnetised 166 to 330 nT values of the Pilanesberg Complex. This suggests that a magnetic reversal occurred between the emplacement of the Pilanesberg Complex and the dyke System. The age data of the Complex and dyke Swarm suggest a magnetic reversal could have occurred between the emplacement of the Pilanesberg Complex and the Pilanesberg dyke System. The Complex is dated at 1602 ± 38 Ma and 1583 ± 10 Ma, from two white foyaite samples from the southern edge (using 40Ar/39Ar amphibole spectrum analysis). These ages are vastly different from previously reported ages, which ranged between 1200 Ma and 1450 Ma (Harmer R., 1992; Hansen et al., 2006). The error analysis has improved considerably from the published dates making the proposed dates plausible for the intrusion of the Pilanesberg Complex as the first and main intrusion of the Pilanesberg Alkaline Province. The Pilanesberg dyke System intruded much later between 1219 ± 6 Ma to 1268 ± 10 Ma for the red syenite dyke samples (using 40Ar/39Ar on feldspars spectrum analysis) and 1139 ± 18 Ma obtained for the grey syenite dyke (using 40Ar/39Ar on amphiboles inverse isochronal analysis). The dyke Swarm dates are significantly younger than the previously published ages for the dykes, which were between 1290 Ma and 1330 Ma (Van Niekerk, 1962; Emerman, 1991).