Aspectos de estruturas topológicas em baixas dimensões

Abstract

In this thesis, we studied the aspects of two-dimensional and three-dimensional topological structures that arise in models of classical field theory. Briefly, in the two dimensional case, we examined the kink configurations. For the three-dimensional models, we analyzed the emergence of topological magnetic vortices. In order to reach the purpose of this thesis, we started considering a multi-field theory ɸ⁶ with interaction in a flat spacetime. In this scenario, we obtained the Bogomol'nyi-Prasad Sommerfield (BPS) configurations for the fields that make up the theory. Furthermore, by adopting arguments from the configuration entropy, the most probable kink (and antikink) structures are selected. Afterward, we investigated the kink-like configurations that emerge in a two-dimensional dilaton gravity scenario. In this scenario, we inspected some aspects of these field configurations. Among them, we analyzed the stability of the matter field and the translational mode. Furthermore, we calculated the interforce produced by the structures and the scattering of these structures. To study magnetic topological vortices, we considered three different scenarios. In the first scenario, we adopted the generalized and non-canonical models described by a scalar field theory. In a moment second, multi-field theories coupled to the non-minimal O(3)-sigma model are assumed. Finally, in the third scenario, we studied the three-dimensional self-gravitating structures in Einstein's gravity. To inspect the first case, we considered a perturbative generation for scalar-vector theories and demonstrate that the dielectric permeability functions must have a non polynomial form, i.e., the profile of a logarithmic function. By using this result, we investigated the three-dimensional structures (i.e., the generalized magnetic vortices) with BPS property for a Maxwell-Gausson-like model. Afterward, we investigated the existence of three-dimensional topological structures in a model with\textit{cuscuton}-like dynamics. In this case, the topological solutions found in this scenario are not self-dual but suggest the structure's existence if some impositions on the energy of the fields are adopted. Furthermore, we showed that when spontaneous symmetry breaking does not occur, the non-canonical three-dimensional model with the contribution of the term \textit{cuscuton} will admit only non-topological solutions. By considering the O(3)-sigma model with non-minimal coupling, we constructed a multi-field theory with the non-canonical sigma field, with a scalar field and a Maxwell field modified by a dielectric function. We examine the possible vortex configurations in topological sectors of the sigma and the scalar field by mean of BPS formalism. We investigated the influence of the scalar field in the sigma field sector. Then, we build an extension of the theory considering a model constituted by the sigma field, a cuscuton-like scalar field, and Maxwell's field. In this case, we investigate BPS properties considering a theory without interaction. Furthermore, we announced the physical aspects of the topological magnetic vortices. Among these aspects, we reported the energy behavior, the profile of the magnetic fields, and the magnetic flux of each structure. Finally, by adopting Einstein's gravity, we show the influence of the solitonic matter and the gauge field in the spacetime metric functions. In this scenario, we concluded that the emergence of magnetic vortices in three-dimensional spacetime induces the appearance of a black hole.