In this thesis, we discuss some aspects of the Aharony, Bergman, Jafferis & Maldacena (ABJM) theory. In particular, encouraged by the recent construction of fuzzy sphere solutions in the ABJM theory, we re-analyze the latter from the perspective of a Matrix-like model. In particular, we argue that a vortex solution exhibits properties of a supergraviton, while a kink represents a 2-brane. Other solutions are also consistent with the Matrix-type interpretation. We study vortex scattering and compare with graviton scattering in the massive ABJM background, however our results are inconclusive. We speculate on how to extend our results to construct a Matrix theory of ABJM. We also present an embedding of the 3-dimensional relativistic Landau-Ginzburg model for condensed matter systems in an N = 6, U(N) × U(N) Chern-Simons-matter theory (the ABJM model) by consistently truncating the latter to an abelian effective field theory encoding the collective dynamics of O(N) of the O(N²) modes. In fact, depending on the VEV on one of the ABJM scalars, a mass deformation parameter μ and the Chern-Simons level number k, our abelianization prescription allows us to interpolate between the abelian Higgs model with its usual multi-vortex solutions and a φ⁴ theory. We sketch a simple condensed matter model that reproduces all the salient features of the abelianization. In this context, the abelianization can be interpreted as giving a dimensional reduction from four dimensions. Finally we present ansätze that reduce the mass-deformed ABJM model to gauged Abelian scalar theories, using the fuzzy sphere matrices Gα. One such reduction gives a Toda system, for which we find a new type of nonabelian vortex. Another gives the standard Abelian-Higgs model, thereby allowing us to embed all the usual (multi-)vortex solutions of the latter into the ABJM model. By turning off the mass deformation at the level of the reduced model, we can also continuously deform to the massive φ⁴ theory in the massless ABJM case. In this way we can embed the Landau-Ginzburg model into the AdS/CFT correspondence as a consistent truncation of ABJM. In this context, the mass deformation parameter μ and a field VEV <φ> act as g and gc respectively, leading to a well-motivated AdS/CMT construction from string theory. To further this particular point, we propose a simple model for the condensed matter field theory that leads to an approximate description for the ABJM abelianization. Finally, we also find some BPS solutions to the mass-deformed ABJM model with a spacetime interpretation as an M2-brane ending on a spherical M5-brane.
Reference:
Mohammed, A. 2014. Some aspects of the mass deformed ABJM theory. University of Cape Town.
Mohammed, A. (2014). Some aspects of the mass deformed ABJM theory. (Thesis). University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics. Retrieved from http://hdl.handle.net/11427/9298
Mohammed, Asadig. "Some aspects of the mass deformed ABJM theory." Thesis., University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 2014. http://hdl.handle.net/11427/9298
Mohammed A. Some aspects of the mass deformed ABJM theory. [Thesis]. University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 2014 [cited yyyy month dd]. Available from: http://hdl.handle.net/11427/9298