Extensão supersimétrica do modelo BF bisimensional e a quantização de laços - Bautista, Luis Ivan Morales (2012)
One of the main challenges in theoretical physics over the last fifty years has been to reconcile Quantum Mechanics with General Relativity into a theory of Quantum Gravity. Theory that has not yet been found, in a concrete way, due to its complexity, specially when we deal with gravity-matter systems, and lack of technologies that may give us experimental evidences. But, there are many theoretical models which try to explain this theory, among of them we have Loop Quantum Gravity. In order to understand and simplify the difficulties of of Loop Quantum Gravity theory in 3 +1 dimensions, we study models in lower dimensions. Starting from a topological BF model, discussed in this thesis gravity-matter systems of two-dimensional space-time, by means of supersymmetric extensions N = 1. We discuss two models: 1.) In the first model, the gauge group of the theory is given by the super-(anti-) de Sitter, S(A)dS, supergroup, that is a supersymmetric extension N = 1 of the (A)dS gauge group, which have three bosonic generator and two fermionic generators. 2.) In the second model, we couple topological matter, being guided by the existence of a rigid supersymmetry (especifically we study the Euclidean gravity with positive cosmological constant), where the fields content is of the theory is expressed in terms of superfields, with the gauge group being a "supersymmetrization"of SU(2). In this particular case we quantize the model by extending techniques well used in Loop Quantum Gravity. In both cases, we discuss the canonical structure of the model, we show that the Hamiltonian of the theory is completely constrained, we also construct gauge invariant quantities (Dirac observables)