263. Winding effects on brane/antibrane pairs
Save This Post as PDFThis is a guest blog post by my old friend and co-author Niko Jokela (who is nowadays working for Technion). Dmitry.
Here I want to write a guest post by the kind invitation of Dmitry. He asked me to give some flavor on our recent paper: “Winding effects on brane/anti-brane pairs” (0901.0281) – coauthored with Matti Jarvinen and Sean Nowling.
In the paper we are interested in constructing an effective action for a brane/anti-brane pair in compact space. In compact spacetimes you can have strings wrapping the compact direction and hence you feel stringy effects coming from winding modes. In particular, the string can wrap the compact direction infinitely many times, so a priori it is not always clear how to consistently truncate the physics to a fewer, a finite, number of degrees of freedom. This is where, as we show in our paper, one can find non-trivial scaling solutions and thus usual intuitions may break down.
Before considering the brane/anti-brane system, let us focus on a simpler system which introduces us to the main subtlety in trying to deal with an infinite number of degrees of freedom. To this end, consider having two classical particles living in one dimension, on a circle, that are connected with a spring. Locally (and without winding), the potential resembles the one on a real line: 
. However, the spring may connect the two particles with few wounds around the circle. It can wind around the circle an infinitely many times. Taking into account such winding modes is easily acquired by switching to the covering space (real line) and implementing an infinite order shift orbifold – a fancy word for the method of images – and identifying fields under shifts 
. In doing so, one ends up with a potential
.
Since the sum runs from 
to 
, it is apparent that one needs to decide how to make sense of the resulting theory. One needs to regularize the theory. Notice that this provides a clean example, where the renormalization takes place already at the classical system and is totally decoupled from the perturbation theory.
In the paper, we will discuss in detail how to regularize the theory. As in familiar from field theory examples, we shall introduce a cut-off parameter, 
, to render the effective action finite. Next we will introduce renormalized quantities through wavefunction and coupling constant counterterms. As we remove the cut-off, the divergences must be absorbed into the definition of the counterterms as functions of . As in other cases, it is not sufficient to simply absorb the divergences. We must also satisfy renormalization conditions. For us it is most natural to impose conditions as the radius 
to ensure that the correct non-compact theory is obtained. However, this procedure is very subtle and I need to relegate the detailed discussion into the paper. I will point out though, that one cannot simply introduce polynomial counterterms (as one usually does) because one needs to respect the shift symmetry, 
. After the dust settles, one finds a finite effective action by essentially replacing the potential 
with 
, which is natural for point particles on a circle.
Ok, so how does this translate into the brane picture? A string is roughly a collection of harmonic oscillators. In a 
system the lowest lying (non-oscillating) modes of an open string strecthing between the branes, the NS sector bosons have masses 
, where 
indicates the winding number. We call these modes winding tachyons 
even though you would expect a “tachyonic” mass only at small separation 
for zero winding tachyon 
.
The simplest thing to proceed with is to consider the background closest to the non-compact case, to consider only the zero winding tachyon. Naively, this seems like to be the only choice. However, then the T-dual system (
wrapping a circle) could only have a homogeneous tachyon decay. From worldsheet calculations we know that the inhomogeneous decay is relevant. We shall therefore consider a tachyon background with higher winding modes turned on (the T-dual profile is highly inhomogeneous), which amounts to identifying all tachyons:
.
Now the problem of regularizing the theory boils down to the earlier harmonic oscillator example and one again essentially replaces the term 
in the linearized effective action of the brane/anti-brane pair.
We are therefore able to develop an effective field theory description of the brane/anti-brane configuration in compact space such that we reduced the number of fields to a finite number in a manner which remembers the compact nature of the problem. Although our line of reasoning might sound counter-intuitive, the new effective action should be used whenever it has a lower energy than that of the single tachyon background. In the paper, we explicitly showed that this is the case in the Sakai-Sugimoto model of chiral symmetry breaking in holographic QCD.
Finally, let me discuss a little bit about a possible application of our work that we didn’t end up putting in our paper. The applications that I have in mind are brane/anti-brane inflationary scenarios. In these cases, the inflaton field is the separation mode l between the brane and the anti-brane. What if you considered inflation in a compact space? Then, the argument goes that one should replace the term 
in the Lagrangian with 
. No doubt, inflation would proceed much differently in this case.
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I am sorry to say but this article makes no sense to me. Are the authors trying to revolutionize string theory on a circle? Are they really convinced that this is not a basic textbook material, e.g. in chapter 8 of Polchinski’s book?
Could they please repeat what they found wrong with (or going beyond) this chapter? I have read this whole text and portions of the preprint but I have no idea what the answer could be – except that they seem to oversimplify even the already-simple picture from the textbook.
Also, I have no idea what they mean by regularizing the theory infinitely many winding modes. The winding modes are real and their infinite degeneracy doesn’t really generate infinite contributions to physical quantities we care about – such as cross sections – and string-theoretical calculations of loops automatically resum all of the winding modes at the same moment. Also, there is a T-dual description which wraps the T-dual branes on the T-dual circle, and reinterprets the winding modes as KK momentum modes.
Of course, when one wants to translate the background as a string field theory, one may truncate the winding modes above/at a critical winding number but it’s not really necessary for most calculations. The authors seem to think that just because there are infinitely many winding modes, there must be a problem. But this “Zeno” argument is a fallacy and no such a problem exists.
Moreover, only a finite number of the winding modes are tachyonic, as the blog article seems to realize (because the winding adds multiples of a basic quantum to the squared mass) but the preprint is remarkably confusing about this point because it talks about infinitely many winding modes.
When all these confusing exercises involving a popular, high-school-level oversimplification of the basic textbook material from chapter 8 of Polchinski are completed, it is being claimed that these exercises are important for AdS/QCD. Huh!?
That’s quite a leap. I must have missed something here. Are the authors claiming that Sakai and Sugimoto don’t understand something elementary about D-branes located on a circle? Do they really believe so?
Well, I am puzzled.
Dear Lubos,
Thank you for the interest in our work. I feel, however, that you have missed the point of our paper. I understand why you want to try to understand the system using the more traditional way. However, notice that Polchinski Ch. 8 does not discuss tachyon condensation on a circle and yes we think that there is more to learn about
system in compact space.
The most common approach is to truncate the effective tachyon Lagrangian to only include zero winding modes. This is usually justified by imagining that the tachyons condense one at a time. In such cases, it is true that as the branes approach one another, the zero winding tachyon’s mass becomes negative before any of the other tachyons.
Here we are suggesting that you could also consider a background where all the winding tachyons are turned on. Classically, you would never think about considering such a background: the EOM for any individual tachyon
, would tell us that the vev vanishes. Only the zero winding tachyon (at small separation) could condense. However, we identified a different way to condense the tachyons which only exists when there are infinitely many tachyons.
I did not discuss the justification for turning all and identifying all the winding tachyons in the blog post too much in detail, but it is discussed in the preprint. By switching to the T-dual system one finds that such a identification is not only natural, but also necessary for inhomogeneous condensation.
Whether or not you like the approach we are proposing, I think that the ultimate decider is still the energy. What we showed is that our background is
preferred over the simplest background one might choose. The obvious
background one might use would be to only condense the “lightest” tachyon. Having found one additional background, we fully expect there to be others.
In regard to Sakai-Sugimoto model, for any non-antipodal embedding of the
-branes the tachyons are in the game and should be taken into account. Furthermore, in the original case all the quarks are massless, a situation which can be improved by incorporating the tachyon physics. This is not something new, see for example Bergman-Seki-Sonnenschein.
I hope this clarified the overall picture.
Best wishes,
Niko
The concept of brane and anti-brane
have continued to evolve with several
‘authors’ giving their special ’spin’
to the concepts. To avoid so much clutter,
we need to consolidate the various nuances
and to post a solid definition for the
brane and the anti-brane