  To clarify the mathematical structure of the RG-derived holographic dual
field theory, we rewrite the string-theory based conventionally utilized dual
holographic effective field theory based on the ADM decomposition of the metric
tensor. This comparison leads us to claim that the RG-derived emergent
holographic dual field theory takes into account higher-derivative curvature
terms with gauge fixing in the string-theory based conventionally utilized
Einstein-Klein-Gordon theory, giving rise to the RG flow of the metric tensor
beyond the AdS (anti-de Sitter space) geometry. Furthermore, we compare the
Hamilton-Jacobi equation for the effective IR on-shell action of the
string-theory based conventionally utilized dual holographic effective theory
with that of the RG-based holographic dual field theory. It turns out that the
effective IR on-shell action of the string-theory based dual holography can be
identified with the IR boundary effective action of the RG-based emergent
holographic dual description, where the Wilsonian RG-transformation procedure
may be regarded as an inverse process of the holographic renormalization. This
demonstration leads us to propose an effective dual holographic field theory
with the diffeomorphism invariance and higher derivative curvature terms, where
the IR boundary condition is newly introduced to clarify the deep connection
between UV microscopic and IR macroscopic degrees of freedom.
