Gluing of Branched Surfaces by Sewing of Fermionic String Vertices

We glue together two branched spheres by sewing of two Ramond (dual) two-fermion string vertices and present a rigorous analytic derivation of the closed expression for the four-fermion string vertex. This method treats all oscillator levels collectively and the obtained answer verifies that the closed form of the four vertex previously argued for on the basis of explicit results restricted to the first two oscillator levels is the correct one. [email protected] [email protected] 1 There are by now several different methods available for calculating correlation functions in string theory and conformal field theory (CFT). When applied to correlation functions involving only untwisted fields, the method of operator sewing of string (Reggeon) vertices is relatively straightforward and can be used to produce closed expressions at arbitrary genus and number of external legs; for a recent review see [1] and references therein. The sewn answers are expressable in closed form in terms of geometrical quantities given in the Schottky representation of the Riemann surface in question. However when applied to twisted fields, i.e. fields with noninteger power expansions in z or z̄ (e.g the Ramond (R) fields of the Neveu-Schwarz-Ramond (NSR) string), this method has so far been far less successful. Here the derivation of a closed geometrical form of the sewn expressions is a step that still remains to be done. In fact even for such a fundamental and conceptually simple object as the vertex for four external twisted fermions, i.e. the four Ramond vertex, the closed geometric form is surprisingly difficult to derive by sewing, and it is only recently that the old result for the scattering of massless fermions in the NSR string [2, 3, 4, 5, 6, 7] has been extended (in the matter sector) to the full sewn vertex [8]. (Some results along these lines have also been obtained recently for Z3 twisted fermions [9].) This development also made it clear that the sewn vertex can be cast into a unique closed geometrical form. That is, it was demonstrated in [8] that by computing explicitly the matrix expressions for a large number of terms in the exponent of the vertex at oscillator level zero and one, one checks easily that they are all generated by the closed form of the four Ramond vertex already argued for in [10]. In CFT the difficulties with twisted fields can be sidestepped by resorting to various other means of computing correlations functions, see for instance [11, 12]. As long as one is studying CFT the situation is in principle quite satisfactory. In string theory, on the other hand, the method of sewing together fundamental vertices to produce the different terms in the perturbation expansion plays a more significant role. This fact becomes particularly clear in the context of string field theory when discussed as for instance in [13, 14, 15]. Consequently, since sewing involving vertices containing twisted fields is a poorly developed subject, our understanding of (open/closed) gauge invariant interacting NSR superstring field theory (in the twisted sector) is not as detailed and explicit as one would like. Comparing to the situation for the bosonic 2 string field theory where things are rather well under control, a similar level of understanding of superstring field theory seems significantly more difficult to obtain. In particular, following [13, 14] it is possible to show that sewing (at tree level) does produce results which are connected to the Riemann surface (via the corresponding propagator) that one naively expects to have generated. This goes under the name of the Generalized Gluing and Resmoothing Theorem (GGRT) when the transformations involved in the sewing are general conformal transformations. As far as the Ramond sector of the superstring is concerned no results along these lines have yet been obtained. However, restricting the transformations to projective ones the explicit results of ref. [8] provides a strong indication for the validity of the theorem also in the twisted case (at least in the restricted sense). In this paper we will improve the situation involving twisted fermion fields by presenting an analytic proof of the fact that the two expressions for the four Ramond vertex, i.e. the one obtained by sewing and the corresponding closed geometrical form, discussed in detail in [8], are equal (both will be given explicitly below). This proof is valid for all oscillator levels. The important new feature of our approach is that after the initial sewing one can recollect all Ramond modes into transported Ramond fields and carry through the proof keeping the fields intact. This then means that the fermion fields play a far less significant role in the proof and one can focus entirely on the issue of the equivalence (under a double integral) of the propagator of the sewn surface and its representation obtained in the actual sewing. We now turn to a brief review of the origin of these two forms of the four Ramond vertex. The matrix form of the expression for the four Ramond vertex obtained from sewing is easily derived by inserting a NS completeness relation between two Ramond emission vertices [2], or equivalently between two dual Ramond vertices previously derived in [16]. Following the latter reference, the four Ramond vertex is obtained by computing a correlation function in an auxiliary Hilbert space as follows: ŴR1,R2(V1, V2) = aux〈0|ŴR1(V1)ŴR2(V2)|0〉aux (1) where, in terms of the complex fermions defined in [8] (the exact definition of which will not