Stringa in Spazi di Kasner

In this thesis we discuss how bosonic strings behave in a time-dependent background with a Big Crunch/Big Bang type singularity. More precisely, as background we consider a generalisation of the so called Kasner spacetime, which, under certain conditions, is an exact solution of Einstein's vacuum field equations, thus a suitable choice for the target space where the string lives. Now, it is worth noticing that Bosonic String Theory in Kasner background, since the latter is generically non flat, as a two dimensional quantum field theory on the world-sheet of the string, is non trivial, i.e., it is an interacting quantum field theory, even if the conformal gauge is chosen; moreover, since the Kasner metric is time-dependent, the energy of the system is not conserved, which is quite an unusual situation to face. Now, the main goal of this thesis is to understand if (Bosonic) String Theory provides a satisfactory description of Big Crunch/Big Bang type singularities, as it is expected from a promising candidate for a theory of quantum gravity. To this end, our intent is computing closed and open string tree level and one-loop amplitudes. Given technical difficulties and subtleties, however, we were not able to quantise the bosonic string and discuss String Perturbation Theory in Kasner background, thus, as for now, we cannot state if String Theory in this specific background is well defined or not. We long to answer this point in future works.