N-String Vertex in the Operatorial Formalism

The thesis is organized as follows. In chapter 2 we review the basic features of the BRST invariant operatorial formalism, giving also the expressions of the scattering amplitudes in the case of bosonic open and closed strings in interaction. In chapter 3 we discuss the properties of the N-string vertex, that is given both in the case of only closed strings and in the one of mixed strings. Projective and BRST invariance of these vertices are shown. Here it is also illustrated the technique of the conformal cut-off, very helpful in treating the zero modes in the orbital contribution to the N.;string vertex. This technique consists in writing the position operator xμ as a conformal field of weight ~c and to perform all the calculations with E which is supposed to be sent just at the end. In chapter 4 it is discussed the generalization to the Neveu-Schwarz string, giving a projective and super-projective invariant expressions for the N-string vertex. Chapter 5 is devoted to the I?ossibility of generalizing the technique of the N-string vertex to an arbitrary conformal field theory, constructing an N-point g-loop vertex for the fermionic field which gives correlation functions on arbitrary Riemann surfaces. The sewing used for such a construction is illustrated in some detail. The same procedure is then considered for free bosons checking in this way bosonization of the free fermionic theory on an arbitrary Riemann surface. In Appendix A we give definitions and properties of unitary irreducible representations of the projective group for a conformal field of arbitrary weight, which have a considerable role in the definition of an N-string vertex; in Appendix B we give some details about the Schottky description of the Riemann surface. Throughout this work we use the space-time metric TJμv = diag(-1, + 1, ... , +l), a value a'=1/2 for the Regge slope, and (super) Koba-Nielsen variables Z = (z, 0), with z = e^ir, becoming real after a Wick rotation, and 0 a Grassrnann variable.