Branelike vertex operators, defining backgrounds with ghost–matter mixing in Neveu–Schwarz–Ramond superstring theory, play an important role in a world-sheet formulation of D branes and M theory, being creation operators for extended objects in the second quantized formalism. We show that the dilaton beta function in ghost–matter mixing backgrounds becomes stochastic. The renormalization group (RG) equations in ghost–matter mixing backgrounds lead to non-Markovian Fokker–Planck equations whose solutions describe superstrings in curved spacetimes with branelike metrics. We show that the Feigenbaum universality constant δ = 4.669 ..., describing transitions from order to chaos in a huge variety of dynamical systems, appears analytically in these RG equations. We find that the appearance of this constant is related to the scaling of relative spacetime curvatures at fixed points of the RG flow. In this picture, the fixed points correspond to the period doubling of Feigenbaum iterational schemes.
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