## ABSTRACT

Michael Singer proposed a definition for a four-dimensional conformal field theory. In this theory the role of compact Riemann surfaces, which occur in standard conformal field theory, is played instead by compact flat twistor spaces. It is therefore tempting to ask questions about these twistor spaces which are, in some way, natural extensions from Riemann surfaces. The properties of compact Riemann surfaces are well known and have been extensively documented and so there is an immensely rich source of possible questions that can be asked about flat twistor spaces. This chapter deals with exploring the number of linearly independent meromorphic functions on a compact Riemann surface with distinct points. The question is answered by converting the question into one involving global data. The problem then becomes one of determining the dimension of the cohomology group.