Some Applications of Hyperbolic Geometry in String Perturbation Theory

In this thesis, we explore some applications of recent developments in the hyperbolic geometry of Riemann surfaces and moduli spaces thereof in string theory. First we show how a proper decomposition of the moduli space of hyperbolic surfaces can be achieved using the hyperbolic parameters. The decomposition is appropriate to define off-shell amplitudes in bosonic-string, heterotic-string and type-II superstring theories. Since the off-shell amplitudes in bosonic-string theory are dependent on the choice of local coordinates around the punctures, we associate local coordinates around the punctures in various regions of the moduli space. The next ingredient to define the off-shell amplitudes is to provide a method to integrate the off-shell string measure over the moduli space of hyperbolic surfaces. We next show how the integrals appearing in the definition of bosonic-string, heterotic-string and type-II superstring amplitudes can be computed by lifting them to appropriate covering spaces of the moduli space. In heterotic-string and typeII superstring theories, we also need to provide a proper distribution of picture-changing operators. We provide such a distribution. Finally, we illustrate the whole construction in few examples. We then describe the construction of a consistent string field theory using the tools from hyperbolic geometry