  Semi-holography has been proposed as an effective nonperturbative framework
which can combine perturbative and nonperturbative effects consistently for
theories like QCD. It is postulated that the strongly coupled nonperturbative
sector has a holographic dual in the form of a classical gravity theory in the
large N limit, and the perturbative fields determine the gravitational boundary
conditions. In this work, we pursue a fundamental derivation of this framework
particularly showing how perturbative physics by itself can determine the
holographic dual of the infrared, and also the interactions between the
perturbative and the holographic sectors. We firstly demonstrate that the
interactions between the two sectors can be constrained through the existence
of a conserved local energy-momentum tensor for the full system up to hard-soft
coupling constants. As an illustration, we set up a bi-holographic toy theory
where both the UV and IR sectors are strongly coupled and holographic with
distinct classical gravity duals. In this construction, the requirement that an
appropriate gluing can cure the singularities (geodetic incompletenesses) of
the respective geometries leads us to determine the parameters of the IR theory
and the hard-soft couplings in terms of those of the UV theory. The high energy
scale behaviour of the hard-soft couplings is state-independent but their
runnings turn out to be state-dependent. We discuss how our approach can be
adapted to the construction of the semi-holographic framework for QCD.
