T-duality and generalized complex geometry

Since a massless state with spin two was found in the spectrum of closed strings, the String theory has become the most promising field theory in trying to unify all the fundamental interactions in a unique framework. The introduction of fermionic matter in the String Theory brings to consider the supersymmetric extension of this, also called Superstrings Theory. These have critical dimension d = 10. The discrepancy with phenomenology, which provides only four dimensions, is filled by one of the most interesting theoretical aspects of Superstring Theory, that is the compactification of the 6 extra dimensions. The most used way to compactify the extra dimensions is called Kaluza-Klein reduction. It consists in taking the 6 extra dimensions as compact ones and ``small''. By assuming that the typical lenght of the compactified dimensions is of order 1/M_P - where M_P \sim 10^{19} Gev is the Planck mass - by Kaluza-Klein reduction a tower of states is obtained, each of which has mass proportional to n/R. At low energies, the only observable states are the massless ones. These are the vacua of the theory, and are intimately connected to the geometry of the 6-manifold on which the theory is compactified, also called the internal space K. The present work concerns the study of different aspects of the geometry of the internal space. Since the matter content of the theory is strictly related to the geometry of the internal space, phenomenology puts strong constraints on the geometry of K. The most accredited phenomenological models are currently those which provide for a N = 1 supersymmetric extension of the Standard model. The examples studied in the present work always assume that the compactification is done with the constraint of preserving N = 1 supersymmetry in the effective 4-dimensional theory. The vacua states of the Superstrings Theory are described by a tern of objects, namely (g, H, \phi), where g is the Riemannian metric, H is a three-form also called the Neveu-Schwarz flux, while \phi is the dilaton. Compactifications with vanishing H-flux have been intensively studied until the first half of the '90s, and they brought to the study of Calabi-Yau manifolds. In the present work we will deal only with non-vanishing H-fluxes. For non-vanishing H-flux the internal space geometry is not more \Kahler: it is called generalized \Kahler. The first part of the present work is devoted to the study of G-structures, which allow us to describe such a kind of manifolds. In particular we will see that the generalized \Kahler structures are SU(3) structures, and how they can be described also in terms of spinors on a manifold. T-duality is a non-local symmetry of the String theory. In the case of compactifications with H flux, the T-duality consists of a map T which associate to a background (g, H, \phi) its dual background (g', H', \phi'). At the level of local supergravity backgrounds, there exists a standard way to find the dual background, which is given by the Buscher rules. These consist in introducing a gauge field by gauging the non-linear sigma model defined by (g, H, \phi). The dual background can be simply obtained by integrating the gauge field out. One of the aspects of the present work is to understand under which conditions a dual background can be defined in a global manner. C. Hull has furnished general arguments to understand if the non-linear sigma model associated to a global background can be gauged in a way which defines a global dual background. It's in this context that the double field theory was born. In the present work we will explicitly study the non-physical example of the three-torus T^3. Even if this example can't be used as an actual background (its dimension is 3!) it is very useful since it allows us to highlight the mathematical details of the question. Moreover, even if a global treatment is possible in this case, we will see explicitly that the results locally agree with those given by Buscher rules. In particular we will explicitly show that the three-torus represents the simplest example in which an ungaugeable isometry can actually be gauged by using the double space technique. In particular, as it was formalized by V. Mathai, J. Evslin and P. Bouwknegt the topology of the background can change after T-duality. We will explicitly see this phenomenon in the T^3 example. The main point of the present work is however the systematic study of the Generalized Complex Geometry. It turns out to be the natural framework to describe generalized \Kahler structures. Since it provides a doubling of the degrees of freedom due to the fact that tangent space and cotangent spaces are merged togheter, it can be used to describe the doubled space in a natural way. In particular the T-duality map takes a very simple form when written in terms of generalized structures. It will be shown that Generalized Complex Geometry provides the right way to describe type II supergravity backgrounds, and in this context we will consider two explicit examples which are SU(3) structures. In particular we will study the form of the T-duality map written in terms of pure spinors for these examples, and we will see explicitly that the local form of such a map is equivalent to that prescribed by Buscher rules. Doubtless the most interesting point is to understand if such local dual supergravity backgrounds can be extended to global Superstring backgrounds. We will see explicitly that the examples considered are T-folds according to the definition given by Hull and we will study the mathematical details which descend from it. In particular we will concentrate on the generalized geometry consequences for T-folds