Scaling solutions in context of Denef - Moore

Question

My question is based on the paper Split states, entropy enigma, holes, halos.

What are the scaling solutions discussed on page 49 of the paper ?

It is stated that the equations ${\sum_{j, i\neq j}\frac{I_{ij}}{r_{ij}} = \theta_{i}}$ always have solutions os the form $r_{ij}= \lambda I_{ij}$. why is that true?

I don't understand this as some of the I's may be negative and then a single $\lambda$ can cannot give such a solutions as the distance will be negative in such cases.

I would greatly appreciate an answer explaining the proper meaning of such solutions and what are the conditions for their existence.

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Answer 1

$I_{13}$, $I_{32}$ and $I_{21}$ in eq. (3.56) are positive, as shown in the sentence below (3.57) and also in the caption of Fig. 6.

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Comments

thanks for pointing this out. But in this case the solution should be $r_{13}, r_{21}, r_{32}= \lambda I's$ not $r_{ij}= \lambda I$ for all $i,j$. Also what will happen in the case when say $r_{21},r_{32} < 0$.

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Yes indeed. But you need to understand that Prof. Moore is almost the most rigorous person as far as string theorists are concerned. You need to learn to relax and read what the authors meant behind what is in fact written. (Un)fortunately, string theory is not math.

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Also, the discussions of scaling solutions in the paper by Denef-Moore did not satisfy everyone (including you). This led to a few related papers, e.g. http://arxiv.org/abs/0807.4556 . So, when you have a very specific question in a paper, you shouldn't just ask it here... You need to think about it yourself, and then write a paper about it.

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Thanks for reply. Well I'm aware of prof. Moore's and also of other string theorist's mathematical acumen and that's one of the reasons why I read their papers. Sometimes, I get confused, and that's because of my ignorance and things I don't pay attention to. Thanks for pointing that out.

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