Non-Abelian orbifolds in string theory

Two dimensional conformal field theories (CFT) play a key role in String theory, in particular they provide a suitable description of dynamics of the string in a given space-time. In this thesis we study 2D conformal theories constructed through toroidal orbifold techniques and arising from superstring compactification on some singular limit of a Calabi-Yau manifold. Orbifolds are one of the main techniques used to construct new two dimensional conformal field theories from known ones. They are obtained by first projecting the CFT on the subsector inva-riant under some finite group of symmetries. In order to obtain a consistent new theory, one is then forced to introduce new sectors (twisted) whose analysis represent the most subtle part of the orbifold construction. In this thesis, we consider orbifolds of the form T 4/G, where T 4 is a four-dimensional torus and G is a finite non-abelian group of discrete symmetries which do not admit a geometric de-scription as isometries of T 4. Torus orbifolds T 4/G may be interpreted as singular limits of Calabi-Yau manifolds of complex dimension two (K3 surfaces). K3 surfaces are the simplest cases of Calabi-Yau manifolds: strings compactifications on K3 have been the background for the first microscopic description in string theory of the Bekenstein-Hawking formula for Black Hole entropy; they are also the framework for one of the most important examples of holographic duality in the AdS/CFT correspondence. Despite these results, generic K3 string models are difficult to describe explicitly: orbifolds T 4/G are some of the few examples where exact computations can be performed. The goal of the thesis is to analyze the main proprieties of orbifolds T 4/G, such as the spectrum, the currents algebra and boundary states, using CFT methods that do not rely on the geometri-cal action of the group G. These methods are then applied to provide the first explicit description of certain examples of T 4/G orbifolds where the group G is non-abelian and/or non-geometric. In particular, we performed explicitly the computation for the group G = 2.A5.