The case for D-Branes in closed strings: Boundary States

In a recent post about the "brane forest" I said that although in type II strings, which are closed strings, there are d-branes, and play a very important role, I had not seen an explicit construction of them.

After all in open string theory there is an easy way to see the appearance of d-branes, they are hyperplanes at which the extremes of open string can end. But, clearly, the same image badly could work in closed string theory because closed strings can leave a d-brane (these is the stringy inspiration for the wrapped brane sceneries ala Randall-Sundrum). It has not been totally trivial to find a minimally satisfactory answer (for my taste) of the question. Before going to it I´ll review some heuristic arguments.

The most common one is based in the analysis of d-branes interactions. These can be understood in different ways. One is by the mediation of open strings joining the two d-branes. Considering a one-loop diagram of these strings is equivalent to consider a tree level diagram of a closed string going between these two d-branes. Ultimately these is related to the fact that poles of open strings diagrams of open strings would correspond to closed strings and that, maybe, open strings alone would not be an self-consistent theory and would need a closed string sector.

Well, anyway, these is as far as many articles and books who review d-brane theory go in these picture. They present some other arguments about the existence of d-branes in closed strings, but before commenting on some of them It is time to present the way in which these picture can be achieved in a math formalism which goes beyond wordy arguments. The trick is the use of a an artifact of conformal field theories known as "boundary states" (for the sake of truth I must say that the book on d-branes of Clifford Jonshon actually mentions it, but is far from presenting it in any clear way). I we go to the original literature we can see that the original paper of Polchinsky on the subject http://arxiv.org/pdf/hep-th/9510017 uses these formalism. For people not familiar with conformal field theories, and particularly with the boundary state formalism I leave a few links:

http://wildcard.ph.utexas.edu/~shaji/papers/misc2.pdf (easy to understand intro to boundary states, it requires previous knowledge of CFT)

http://arxiv.org/pdf/hep-th/0011109 (General introduction to CFT´s, including boundary states, it is more formal/rigorous than the previous. As an advise for casual reader to comment that what in the first paper is called "method of images" i these is called "Stocky conditions").

IF someone doesn’t want to read these papers he can get a "quick version" in the original paper of Polchinsky or, for example in http://arxiv.org/PS_cache/hep-th/pdf/9707/9707068v1.pdf (additional papers could be http://arxiv.org/PS_cache/hep-th/pdf/9510/9510161v1.pdf or http://arxiv.org/PS_cache/hep-th/pdf/9510/9510135v2.pdf).

The argument is as follows, you can consider a diagram of closed strings and impose that the "out state" is not given by a closed string but an BRST invariant operator |B> that inserts
a boundary on the world-sheet and enforces on it the appropriate boundary conditions.

I’ll not go further in the details, and refer the reader to the literature cited before, specially these paper. Just to mention that boundary states utility in string theory is not only to introduce D-branes but are useful for many other purposes (for example in the CFT treatment compactifications). In CFT terms a boundary states has a role very related to it’s name, it imposes conditions in the boundary of the world-sheet. Just for completitude I´ll mention that boundary state formalism is a very "natural" way to introduce D-branes in the string field theory version of strings theories. I hope to write some entries in SFT so I´ll give them the details.

Well, these is the most rigorous way to introduce d-branes in closed strings. But in most of the literature it is not made, why? Probably because it is supposed that CFT is reserved to the "advanced" reader, or maybe because supposedly the other ways are in some why "more physical". For example in CFT you can get a more rigorous way to achieve effective actions for certain approximation of situations of string theories. Aa very related example, the Born-Infield action of the d-brane) but the picture of the fields associated to the extremes of the open string dictating the physic of the branes is more appealing.

Well, these are the more rigorous way, but I had said that there were more considerations that support the existence of d-branes in closed string theories. In the original paper of Polchinsky, after the boundary state formalism treatment, he offers an interesting argument by referring to the Hilbert space which I reproduce here:

Periodically identify some of the dimensions in the type II string:

X^u= X^u + 2πR n=p + 1,...,9 (for a Dp-brane)

Now make the spacetime into an orbifold by further imposing

X^u= -X^u

To be precise, combine this with a world-sheet parity transformation to make an
orientifold. This is not a consistent string theory. The orientifold points are sources for the RR
elds (by the analog of the above arguments for D-branes, but
with the boundary replaced by a crosscap *), but in the compact space these
elds have
nowhere to go. One can screen this charge and obtain a consistent compacti
cation
with exactly 16 D-branes oriented. Now take R->0. The result is the
type I string

A few more comment to conclude. An interesting aspect of the boundary state formalism is that you can get a somewhat dual vision of the paper of the d-branes. Instead of considering interactions between them mediated by strings you can think on them as solitonic solutions of string theory and doing perturbative string theory around that solitonic background. For a review (previous to the introduction of d-branes) in string theory the reader could read these article, String solitons.

I, personally, haven’t still readed it, i have a lot of lectures to do, as well as many things to think about. But one thing, the last one of these post, that I wanted to comment is the following. In the viewpoint of a d-brane as an hyperplane where an open string can move semengly is obvious that a d-brane is an infinitely extended object. The full secenarie of Wrapped universes of Randall-Sundrum (or the ekitropic scenario) support these viewpoint. But as is well known d-branes are not supposed to be rigid objects because in gravity there are not such objects, and in fact, guided by the p-brane scenery of supergravity, and also by the theory of "fundamental" branes you can think of a d-branes as a generalization of an string. From these viewpoint it is not clear at all that the brane as an infinitely extended object would be the right one. In fact the 2 and f5 branes of M theroy (M-branes) are supposed to be fundamental objects (despite the fact that it is very difficult to make sense of a theory which gives an interaction picture for them) which in some limits become ordinary fundamental strings. But ok, M-branes are not d-banes, is there a picture to think about d-branees like "small" objects?.

Well, in ordinary QFT there are solitonic solutions to Yang mil theories. They can be interpreted as monopoles. And they are particles with a finite size and all that, certainly not infinitely extended objects. If-d-branes are solitones for an string it would be natural to think of them as small objects of a certain size and not, at least without further reasons to think so) infinitely extended objects. Even if you think in the hyperplane picture you have that they are hyperplanes for a single string, but, why all the strings would be constrained to the same hyperplane?. If you go to the study of d-bane interactions you see that parallel static d-branes don´t interact, but that when they form some angle they do, and, for example, break some supersymmetries. Also moving parallel d-branes interact with a potential proportional to they speed (these configurations of d-branes can, for example, be used to probe distances smaller than the string size). Books, and reviews, explain (or actually prove) these results, but they don´t mention too much about what to think about their implications. I guess that it we must think about it it could be that d-branes related to the many individual strings in the universe, interact to recombine into a single "big" d-brane, or maybe a few parallel ones, but it is conceivable that "small" d-branes with arbitrary size exist. In the literature of string theory you frequently read about d-branes wrapped around torus (or K3 spaces) when considering black hole entropy counting (i hope to write an entry abou these sometime in the future) but it is not so usual to read about actually "shaping" d-branes. Al I have readed so far is in the divulgative book of Leonard Suskind "the cosmic landscape" stating that d-branes in spherical configurations are instable. In a more formal way the d-branes book of Clifford Jonshon in the last part of the chapter in "d-brane geometry I" consider non hyperplanes configurations of d-branes, related to ALE spaces (asintotically locally eculidean), but it a very indirect treatment and it is not easy to follow the details.

Well, as the reader can easilly deduce I am not an expert in string theory (see, for example, the blogs of Lubos Motl or Jackes Distler linked in these blog if you want ton consult the expers) but I hope that maybe reading to someone who has not all the answers can be also interesting because, maybe, the reader could have arrived himself to the same doubts I have, and maybe, I could have answered some of them partially. Also I think it is interesting to discuss somewhat "old" topics and not only the last papers in arxiv (altlthought I don´t discard to discuss some, of course). In any case, fortunately, these last times I have considerable amounts of time to dedicate to the study of strings (and in less extend other approaches to QG) so I hope I will become a more reliable source of information in the near future ;-).