  Based on the renormalization group (RG) flow of worldsheet bosonic string
theory, we construct an effective holographic dual description of the target
space theory identifying the RG scale with the emergent extra dimension. This
results in an effective dilaton-gravity theory, analogous to the low-energy
description of bosonic M-theory. We argue that this holographic dual effective
field theory is non-perturbative in the $\alpha'$ expansion. To investigate the
monotonicity of RG flow of the metric in the emergent spacetime, we consider
entropy production along the RG flow. We construct a microscopic entropy
functional based on the probability distribution function of the holographic
dual effective field theory, regarded as Gibbs or Shannon entropy. Given that
the Ricci flow represents the 1-loop RG flow equation of the target space
metric for the 2D non-linear sigma model, and motivated by Perelman's proof of
the monotonicity of Ricci flow, we propose a Perelman's entropy functional for
the holographic dual effective field theory. Furthermore, utilizing the
equivalence between the Hamilton-Jacobi equation and the local RG equation, we
show that the RG flow of holographic Perelman's entropy functional is the Weyl
anomaly. This eventually verify the monotonicity of RG flow for the emergent
target spacetime. Interestingly, we find that the microscopic entropy
production rate can be determined by integrating the rate of change of the
holographic Perelman's entropy functional over all possible metric
configurations along the flow.
