# Ghost-Matter Mixing and Feigenbaum Universality in String Theory

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Referee this paper: arXiv:hep-th/0212137 by Ian I. Kogan, Dimitri Polyakov

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requested Dec 19, 2018

paper authored Dec 12, 2002 to hep-th

## 1 Review

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I stumbled on this paper while looking for occurrences of Feigenbaum's constant δ (from chaos theory) in renormalization group studies of field theory. Here the log of the constant allegedly shows up as a coefficient in the beta function of the dilaton in certain curved backgrounds.

This is part of a series of papers by coauthor Dmitri Polyakov which seeks to introduce branes to the RNS formalism for string theory, by introducing new operators for "brane-like states". These operators mix ghost and matter fields, are picture-dependent, and lead to space-time beta functions with a "stochastic" component deriving from dependence on worldsheet variables. The philosophy is explained further in a follow-up paper by Polyakov.

In that follow-up paper, Polyakov writes that a factor 1/(log δ) provides a "universal" normalization of the stochastic contribution to these beta functions, which is due specifically to the "brane-like" operators. In the original paper. Polyakov and coauthor Ian Kogan write, if I am reading it correctly, that the stochastic RG equations (for the specific case they study) have a series of fixed points that correspond to increasingly large curvatures, that this is somehow a cascade of period doublings (which in chaos theory involves a control parameter being altered by increments which get smaller by factors of Feigenbaum's δ), and that the onset of chaos corresponds to the appearance of a curvature singularity.

Polyakov has written many papers about adding brane operators to RNS but they haven't received much attention. Kogan died suddenly while his collaboration with Polyakov was ongoing - the follow-up paper, by Polyakov alone, was published in a three-volume memorial to his work.

So it seems these claims haven't received much scrutiny. The "evidence" that the normalization factor is genuinely related to δ seems to be a mix of numerical and intuitive, there's certainly no theorem here. I also don't see what corresponds to periodic behavior in the alleged period-doubling cascade. (Indeed, Sergei Gukov's "RG Flows and Bifurcations" classifies period doubling as a kind of bifurcation that "we do not expect to see... in RG flows".) It could be that these were preliminary guesses by Polyakov and Kogan as to what they were seeing, that are simply wrong.

Polyakov's larger theoretical framework is interesting, and his conception of curvature singularities as corresponding to a chaotic fluid phase of string theory, might be compared to the "fluid/gravity correspondence", black holes as fast scramblers, quantum chaos in M(atrix) black holes, and so on.

reviewed Dec 19, 2018 by (1,650 points)

Remarkable analyze! However, notice that Log(4.669201...) =~ 1.54098... and not 1.534... The equation 20 gives sigma exactly equal to 1 + (27995 / 52488) =~ 1,533360006...

@mitchellporter : don't you modify the review for the bad value? I hope it is not taboo to report such errors. If I'm not wrong ( how it could be? just check the equation 20, it converges to a rational which is not what is claimed ), this article is merely unrelevant. Or else please tell why I'ld be wrong...

Despite the claim of universality, on page 9 they report slightly differing values of σ for other kinds of curved backgrounds, and one of those values is much closer to the logarithm as you have calculated it. This was all to be explored in a paper - their reference 8 - that was never written.

I think the claim that σ has something to do with 1/(log δ) remains not proved and not disproved. There could be a connection that is not exact because of extra physical effects that modify σ. But they also don't give good enough reasons to substantiate the connection with δ. Just because σ is involved with a kind of universality, and δ is also involved with a kind of universality, doesn't mean it's the same kind! The numerical relation could just be a misleading coincidence.

So thanks for your comments, because they help to clarify that in order to really know whether the connection is real or fictitious, someone will have to understand what kind of universality is behind the similar σ values - is it somehow Feigenbaum "approach to chaos", or is it something else entirely.

Finding close invariants ( or constants ) has not a specific meaning by itself. For the reader, with the equation (20), it was easy to derive that the constant is transcendantal but it is not the first Feigenbaum one. Anyway, the hope to get an exact mathematical connection with the current bifurcation theory vanishes. A connection related to a transcendantal number would be so rich. TY.

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