Nowadays, the understanding of the global aspects of the rich class of the models which populate the world of theoretical physics is one of the most fascinating problems which requires hard effort and common work between physicists of different areas. By global aspects are meant all the features which, in some way, are related to the global properties of the considered model; as it happens, for example, in the two-dimensional conformal theories on a Riemann surface, in the recently proposed topological gauge theories in four-dimensions and in the three-dimensional Chern-Simons model. The aim of this thesis is to show how these global aspects can be explored by using the methods and the techniques of the quantum field theory. We begin by considering a two-dimensional bosonic sigma model defined on a generic Riemannian space M. Here the global aspect is represented by the Friedan operator [1] which characterizes the transition functions between different charts and which can be used to define global quantities on M. From a field theory viewpoint the Friedan operator corresponds to a non-linear symmetry which, by the help of the BRS technique, can be translated into a Slavnov-Taylor identity.
This identity will be used to discuss the ultraviolet stability of the model and to identify the parameters which affect the metric tensor. As a second example, we consider the recently proposed topological omega-model [7]. Here the
idea is to build a model in which the observables are of a global topological nature; for instance, in the case of the topological IT-model the observables are given by the De Rham cohomology of the manifold on which the model is defined.
In a field theory realization, the observables of the topological omega-model can be recovered through the so-called basic cohomology [10], which is a kind of restriction of the usual BRS cohomology and which can be defined by combining the original BRS symmetry [7] with the Friedan symmetry.
The third example considered is the Chern-Simons model in three dimensions. The Chern-Simons action, being a three-form, is metric independent and it is not only the starting point for computing topological quantities of a three manifolds, but it is also expected to be strongly related with the two-dimensional conformal models.
However, also at the perturbative level [14], the Chern-Simons model has numerous aspects which render it interesting. This is due to the expected finiteness [15] of the perturbative expansion and to its rich class of symmetries. We will see, indeed, that these symmetries form a supersymmetry algebra and can be used to characterize the perturbative expansion of the model. Also if a complete proof of the finiteness of the Chern-Simons model has not yet been established, it is interesting to note that these symmetries, being anomaly-free, impose quite strong conditions on the possible counterterms.
