D-brane sonda in teoria di Yang-Mills

In a series of articles published between the end of 2013 and early 2014, Professor Frank Ferrari of the Université Libre de Bruxelles developed an extension of the concept of D-brane, and in particular of the concepts of background and probe D-branes, to pure gauge theories with no underlying string theory and no supersimmetry. In this framework, he defines a probe brane effective action which in principle contains all the information about the large N limit of the original gauge theory. He also proposes a method of computing such an action which involves several non trivial steps. Firstly, a particular partial gauge fixing procedure is needed, which consists of an equivariant version of the BRST cohomology. Secondly, all the quartic interaction vertices which are present in the action are removed with the introduction of a set of auxiliary fields. Finally, the action, which in general can be expressed as an expectation value of functional determinants, is further simplified in a peculiar approximation scheme, called the ¿bare bubble approximation¿. Such an approximation, which is inherently non-perturbative, is in principle capable of capturing both the weakly and the strongly coupled regimes of the theory. In one of his articles Professor Ferrari applies this method to the zero-dimensional matrix model with good results: the approximate free energy he computes matches the exact one with an error never exceeding 4% all the way from the weakly coupled to the strongly coupled regime. In this thesis I will attempt an application of this method to 2 Yang-Mills theory, firstly by introducing the proper set of auxiliary and ghost fields, then by integrating out all the unwanted background variables, thus finally obtaining an expression for the probe brane effective action.