263. Winding effects on brane/antibrane pairs

This 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

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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:

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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.