Choices of open/closed oriented/unoriented strings

Question

In bosonic string theory, there are various variants we could consider depending on the choice of boundary conditions to include, which give rise to different fields. Two examples are,

1. Closed oriented strings: $G_{\mu\nu}$ (graviton), $B_{\mu\nu}$ (2-form), $\Phi$ (dilaton);
2. Closed + open oriented strings:$G_{\mu\nu}$ (graviton), $B_{\mu\nu}$ (2-form), $\Phi$ (dilaton), $A_\mu$ ($U(1)$ field).

These choices are also equivalent to choosing to sum over worldsheets in defining the S-matrix, with only the specified topology, that is, for example, only closed oriented strings.

A possibility not listed in textbooks is the choice of having oriented open strings, with unoriented closed strings, that is, open and closed strings of differing type.

However, in Polchinski it is stated:

... oriented or unoriented open strings can only couple to closed strings of the same type.

Thus, two concerns arise:

• Why can closed strings only couple to closed strings of the same type of orientation?
• If the answer to the first question has to do with consistency, then ignoring this issue, if we did define a string theory with these, what fields would arise?

Answer 1

You cannot choose a sum over worldsheets that represents the coupling of oriented to unoriented strings, because there is no such things as a "partially oriented worldsheet". If the worldsheet is oriented, then so are all strings moving along it, if it isn't, then all the strings moving along it are unoriented. Orientation is a global property, you cannot choose an orientation for one part but not for another.

Since there is no worldsheet for the coupling of unoriented to oriented strings, this theory is not only inconsistent, it is impossible to write down.