  It will be shown analytically that the light sector of the identity block of
a mixed heavy-light correlator in the large central charge limit is given by a
correlation function of light operators on an effective background geometry.
This geometry is generated by the presence of the heavy operators. It is shown
that this background geometry is a solution to the Liouville equation of motion
sourced by corresponding heavy vertex operators and subsequently that the light
sector of the identity block matches the Liouville correlation function in the
semi-classical limit. This method effectively captures the spirit of Einstein
gravity as a theory of dynamical geometry in AdS/CFT. The reason being that
Liouville theory is closely related to semi-classical asymptotically AdS$_3$
gravity.
