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.