Type I superstring theory is unoriented, and it seems that it needs to be so in order to exist.
Now, we always have open-closed duality, that connects at least the ultraviolet sector of a theory with the infrared sector of another, so in principle we should have some oriented open strings coming from it, by duality with a closed oriented theory.
¿Is some of these D-brane examples an oriented theory?
¿Is there some canonical, well known, example of a consistent oriented theory with open strings?
No. There is no theory of open, oriented strings. Any string theory must contain closed strings, while the open strings are optional. If there is a string theory which contains oriented open strings, then it has the problem that the oriented open strings cannot couple to the oriented closed strings. Why?
This is my understanding of the explanation given by the by Thomas Mohaupt in Lecture notes "Introduction to String theory":
In the closed string spectrum, there is an algebra and and an algebra which lead to different string theories. Both have 32 supercharges. In each of these,, there are 2 gravitinoes, dilationoes, 1 in the Ramond Neveu-Schwarz Sector and 1 more in the Neveu-Schwarz Ramond Sector. These 2 gravitinos need 2 different supercurrents to couple to. But the N=1 supersymmetric algebra with only 16 supercharges clearly cannot allow this!
Thus, the open oriented strings would not couple with the closed oriented strings. The solution is to have open unoriented strings instead. This along with the unoriented closed strings IS the Type I string theory. The "only unoriented closed strings" theory is also inconsistent because of other reasons.
What about the strings between D-branes? Are they unoriented too, even if they have Chan-Paton labels? - arivero
@arivero: They are unoriented. Orientation is a property of the 2d worldsheet, not the 1d string. Unoriented just means that crosscaps and the like can appear in the worldsheet. – user1504