Brane Supersymmetry Breaking su Orbifold Z

String theory provides a consistent framework to quantize gravity and unify the four fundamental interactions. However some of its features such as the presence of ten spacetime dimensions and supersymmetry, although interesting, seem to be incompatible with Nature, as we observe it nowadays. Following an idea first proposed by Kaluza in the early twenties, one can think of these extra dimensions as highly curved and wrapped on small scales so to be undetectable at current energies. String theory then constrains the shape of the internal manifold and its topological properties determine the couplings of the four-dimensional effective action. This geometrical procedure, known with the name of compactification, results very difficult if the internal manifold is a smooth curved space. Thus, in order to have an explicit CFT description of the compactified theory, we are led to consider compactification on singular spaces, called Orbifolds, which simplify the task. In particular, in this thesis I shall consider compactification on the so-called Z orbifold, a peculiar singular space obtained from a six-dimensional torus by suitable identifications, with 27 points fixed under the action of the symmetry group Z3. Similar problems concern supersymmetry. Although being a very interesting and fascinating idea, which might offer a natural solution to the hierarchy problem and the unification of gauge coupling constants, it has not been observed so far and hence it must at best be broken. For these reasons, in this thesis I have addressed the problem of compactification and supersymmetry breaking in string theory. In particular, I have built a four-dimensional string vacuum configuration with broken supersymmetry, by implementing the so-called Brane Supersymmetry Breaking (BSB) mechanism in the Z-orbifold compactification. Various ways to break supersymmetry have been investigated during the years. In BSB models, the breaking is achieved by introducing a particular combination of antibranes and orientifold planes in spacetime and thus affects only the open-string sector, while leaving the closed-string one untouched. Moreover, the BSB mechanism breaks supersymmetry directly at the string scale, and thus is clearly compatible with experimental results. Henceforth, One might wonder whether it is consistent or not to couple a non-supersymmetric open-string sector to a closed one which is invariant under local supersymmetry. The answer is yes: the consistency is actually guaranteed by the very nature of the BSB mechanism, which is accompanied by a dilation tadpole and always provides a neutral (gauge singlet) fermion. This plays the role of the goldstino in a Volkov-Akulov theory, the analogous of the Goldstone boson for bosonic symmetries in conventional Field Theory. In such a theory, the particle spectrum consists only of a neutral massless Weyl fermion, the so-called goldstino, and hence supersymmetry is realized non-linearly, without requiring the presence of bosonic superpartners. Therefore, in the last part of the thesis, I have addressed the problem of non-linear realizations of supersymmetry and presented two different but equivalent points of view, the first one introduced by Akulov and Volkov in the early seventies, the second one proposed by Roček in the late seventies and extended by Komargodski and Seiberg in recent years. The discussion is incomplete but provides prompts for future prosecution of this work.