Open strings from closed strings

Question

This issue comes up in Shiraz's lecture here on 29th October 2008.

I understand that he is saying that one can think of closed string theory as having two minima and that the ground state in the non-global minima is kind of a condensate of closed strings and open strings emerge as massless fluctuations about them. It seems that he is trying to say that open string theory cannot exist on its own and there is no sense in which there can be a low energy limit/effective action of open strings which is made up of only open strings. (also he is alluding to an interpretation of D-branes as a condensate of closed strings and closed strings as a soliton of open strings and thats what is in this second minima)

So I guess there is an interpretation that if enough energy is provided to closed strings then they will on their own produce D-branes and open-strings? (by moving to this second minima from the global minima?)

• I am quite sure I said a million things wrong in the above paragraph and Shiraz doesn't provide references which derive these ideas which lets one go back and forth between closed and open string interpretation.

  I would like to know as to what are are corresponding completely correct facts which Shiraz is alluding to and if someone can tell me reference which derive that (hopefully from scratch!)

• He comes to this issue in trying to interprete how the appearance of D-branes in the T-dual picture of compactified Bosonic closed strings is "kind of" a breaking of translation invariance. And that this translation invariance breaking can be seen to be rooted in the interpretation of closed strings breaking up into open strings.

  It would be great if someone can help explain this open/closed relationship to T-duality and hopefully give pedagogic references for that.

Answer 1

This question has many aspects.

First, there is the fact that theories without closed strings are inconsistent. If we allow an open string to split into two open strings (line intervals) and vice versa (Hermiticity), the same interaction may also merge the end points of an open string and produce a closed string.

Alternatively, one-loop diagrams in a theory of open strings may also be read in the "intermediate closed string channel", thus proving that there are intermediate closed strings and there must be physical closed strings, too.

In type II-based string theories, it is also possible to produce D-branes out of closed strings only. D-branes automatically allow strings to terminate on them, so one produces the potential for open strings or the open strings themselves, too.

The reason is that a D-brane is physically the same object as a black $p$-brane which is a higher-dimensional generalization of a black hole. Just like one can prove that under certain circumstances, black holes will be formed (the singularity theorems by Penrose and Hawking), there will also be black $p$-branes formed out of closed string fields (the closed string Ramond-Ramond electromagnetic-like fields are around the charged black $p$-branes found in type II string theories).

It is possible to show that the minimum-energy configurations with the same charge must look like D-branes at weak coupling, and their excitations are therefore described by open strings attached to these D-branes.

Now, the D-branes localized in some circular dimensions are T-dual to D-branes that are wrapped around these directions. In this T-duality map (equivalence), the location of the localized D-brane is mapped to the value of the Wilson line of the wrapped brane. Choosing one of the positions of the D-branes from the list of a priori allowed many positions breaks the translational invariance, obviously. This is equivalent to a nonzero vacuum expectation of the Wilson loop around the circle.

Shiraz in particular has repeatedly studied a similar phenomenon in AdS/CFT in which this extra circle is the Euclideanized time in thermal calculations. The Wilson loops around this thermal circle are essentially called the Polyakov loops, at least in some non-Abelian generalizations, and these Polyakov loops can get a vev. This is a description of the deconfinement transition in the gauge theory.

It seems to me that Shiraz has been talking about many independent topics and each of them is discussed in different references and many of them are rather standard parts of the string theory basic education and textbooks (whose rest is sort of needed to understand these particular aspects, too, so it doesn't make much sense to try to isolate these aspects from the rest).