Conformal field theories in higher genus Riemann surfaces

Two dimensional conformal field theories have received a lot of attention due to their relevance in string theory, as well as the fact that, in statistical mechanics, they describe systems at second order phase transitions. The motivation to study conformal field theories in higher genus Riemann surfaces is both physical (in connection with the Polyakov perturbative scheme for string theory, or with models in statistical mechanics having particular boundary conditions) and mathematical: the higher genus shares the same features as the genus zero case but the complications arising from the non-trivial topology are introduced. This thesis is devoted to the study of the conformal field theories in higher genus with a particular concern on two aspects, namely in establishing a prescription to compute the correlation functions of these theories, and in elaborating an operatorial framework for them. For what concerns the former aspect we recall that one of the key properties of the OFT is the factorizability of the correlation functions in chiral and antichiral blocks. Our aim is to give a prescription to compute such chiral blocks. We have to remember that in OFT such as the minimal models the correlation functions are fully constrained by the conformal invariance. To solve the partial differential equation so implied is however, from a practical point of view, extremely hard. At genus zero the Coulomb gas method elaborated in [1] gives an integral representation of the PDE. The basic idea underlying this method is the existence of a certain free field theory whose correlation functions, suitably constrained, have to be interpreted as correlation functions for the minimal models. The free field theory considered is that of a free bosonic field <P with a background charge. However such a theory is not easily treatable in higher genus. Motivated by the duality property between bosons and fermions in two dimension, we will introduce a new system, a real weight chiral anticommuting b- c system, which gives rise to a fermionized version of the Coulomb gas and can be formulated in a straightforward way on any genus Riemann surface. This system allows to compute chiral blocks for any Conformal Field Theory which admit a representation in terms of free fields (not only minimal models then, but also WZW models for instance). The b - c system allows to compute, in a very easy way, also quantities which are relevant for string theories, like spin-field correlation functions and their generalizations. Regarding this, one should remark that the b - c system theory is a well-defined theory interesting in itself, not only as a recipe to compute correlation functions for given OFT. Besides knowing the correlation functions for the Conformal Field Theories, one should also be able to identify their operatorial content. In eh. 4 .we will generalize the construction of the chiral vertex operator to higher genus Riemann surfaces. We will work in the Krichever-Novikov framework [2] which has the striking advantage that it mimics to higher genus the procedure already carried out on the sphere. Morever, the global data concerning the Riemann surfaces are present from the very beginning. An interesting question concerns the possibility that in higher genus the Krichever-Novikov algebra, which should replace in higher genus the Virasoro algebra as a classifying tool, could be more refined than the Virasoro one. The material is so organized: in eh. 1 a discussion on conformal field theories is carried out, emphasizing particularly the aspects which will be relevant later. In eh. 2 the real weight b - c system is introduced on the sphere and a detailed presentation about how the information about the vertex operators of the theory under consideration, as well as the procedure to get primary field correlation functions, is given. In eh. 3 the generalization of such b - c systems to higher genus is performed. In eh. 4, besides the construction of the vertex operator, a detailed analysis of the higher genus oscillators is presented.