Loops, Twists and String Vertices: The Sewing Approach to String Perturbation Theory

String theory provides the most promising candidate for a unified theory of all fundamental forces and a consistent quantum mechanical theory of gravity. String theory is, at present, defined through its perturbation expansion. There exists various methods for treating the string diagrams one of which is through the sewing of Reggeon vertices. We investigate the sewing procedure for both untwisted and twisted strings. Ordinary untwisted strings satisfy periodic boundary conditions whereas the twisted strings pick up non-zero phases when taken around the origin in complex z- plane. Apart from the Ramond sector of the superstring, twisted strings are also relevant in connection with string propagation on orbifolds and the construction of realistic string models. For untwisted strings, it is well understood how to perform the sewing and it is possible to derive completely general answers for string scattering amplitudes. In Papers I and IV, we show that the construction of loop vertices is simplified if we take the basic building blocks to be dual four-Reggeon vertices instead of the usual Sciuto- Della Selva-Saito three vertices. For twisted strings, such as the Ramond string, the situation is much more complicated. For vertices involving four external twisted strings, answers have previously only been derived through sewing for external ground states. In Paper II, we extend these results and give an algorithm by means of which the four vertex can be calculated for arbitrary external states. This is used to find the closed form of the vertex for scattering of four Ramond fermions. In Paper V, we use this algorithm, and the dual vertex for twisted scalars constructed in Paper I, to investigate the four-Reggeon vertex for Z2- twisted scalar fields. In Paper III, we generalize the sewing procedure to fields with higher twists; in this case Z3-twisted fermions