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

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.