I am trying to understand precisely the interactions in open/closed string theories from several (interconnected) aspects:

  1. relation between the "intuitive" picture (by that I mean spatial sections of the string, as one could mimick with ropes) vs the worldsheet description (as a Riemann surface);
  2. what is the minimal set of interactions;
  3. what changes for non-oriented strings.

I will first describe the general setup, and then ask the various questions. Note that I check in the various books on string theory and none of them describe these aspects completely.


To fix the ideas, here are the figures I will refer to below:

String interactions

The first approach to string interactions ist just to consider ropes and see how one can join and split them (what I call the intuitive picture). The simplest interactions are those at most two strings splitting/joining at a single point in at most two strings. This is the only ones which have a chance of being fundamental (i.e. no decomposable in other processes): other interactions with more strings or interaction points look so only in one specific Lorentz frame, by boosting in another frame the process would appear to be composed of several more elementary processes. Then this set can be decomposed in two classes of processes (I just describe one direction for the interaction, the converse being also valid):

  1. Figure 1: the string splits at one point (figures 1b and 1c) or the string pinches and the loop becomes independent (figures 1a and 1d);
  2. Figure 2: the string intersects itself or another string.

These diagrams appear in question #22475. Some of them are described in Polchinski vol. 1 (p. 79). The worldsheet diagrams of figure 1 are given in figure 3 and the coupling constant are respectively $g_s$, $\sqrt{g_s}$, $\sqrt{g_s}$ and $g_s$.

Part 1: Open-open-closed diagram

All interactions of Figure 1 looked first fundamental to me because I do not see any of them as a composition of the other ones. This view was reinforced while reading Green-Schwarz-Witten vol. 1 (GSW1), where this diagram is introducted in sec. 1.5.6 (p. 53) to describe the aspects of unitarity and the relation between the coupling constants. The way it is done let me think first that the diagram is fundamental.

On the other hand the diagram 1d has a worldsheet (figure 3d) which can be seen as gluing 3b and 3c by lengthening the open-closed region on the right (the diagram are topologically equivalent). Thus this diagram would not be fundamental. This can be sustained by string field theory where one needs only 3b and 3c to describe the interactions of open strings, cf hep-th/9202015 (I am aware that worldsheet and SFT interactions are not in one-to-one mapping in general, but here I am concerned with open strings, for which this is the case for Witten's cubic theory).

So here is my question: if diagram 1d is not fundamental, how can I see it as a composition of 1b and 1c? I really mean in terms of ropes, and not of worldsheet. Is this a question of Lorentz frame?

I tried to look at this last point by decomposing the process 1d (see figure below), but then it looks very baroque – and wrong: due to the high-number of vertices the overall coupling constant would be too big ($g_s^3$ against $g_s$; edit: in fact there is another simpler diagram, but still of order $g_s^2$). What am I missing here?

enter image description here

Part 2: Diagrams with intersection

A colleague indicated me that the diagrams of Figure 2 are possible only in unoriented theories. The reason is that there is a twist in the string which allows it to self-intersect or which happens when the strings reconnect differently. Such a twist would make the worldsheet non-orientable.

Nonetheless I am confused because it seems to me that I could construct such a process using the worldsheets of Figure 3 and then obtain a graph which is globally orientable but non-planar (see figure below).

Hence my question: can we have non-planar graphs in oriented string theories? Are the graphs of Figure 2 possible in oriented or unoriented string theories? If some graphs exist only in unoriented strings, are they part of the minimal set of interactions? Finally do these diagrams look fundamental only due to the frame chosen for drawing the pictures?

I was in particularly confused by Polchsinki vol. 1 (p. 79) because he gives just a subset of the interactions, and moreover all the strings have an arrow, letting me think that they are oriented. Other books discuss unoriented strings only through the worldsheet, using twists or cross-caps, but this does not help me to understand the intuitive picture and make contact with the above questions.

enter image description here


I tried to be as explicit as possible, but I am not sure I have managed to really convey my confusion, so: I am looking towards as many information as possible to describe the diagrams below and explain their differences, in particular in the oriented vs unoriented cases.


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