  Various studies have attempted to extend the Witten's cubic open string field
theory to the cubic closed string field theory, which may describe Einstein
gravity in the low energy sector consistently. In the present study we propose
a cubic closed string field theory, introducing a double layer to describe the
closed string world-sheet as an extension of the open string world-sheet of the
Witten's cubic open string. We mapped the closed string world-sheet onto the
complex plane, of which the lower half plane is completely covered by the
extended part of the string world-sheet. Using the Green's function on the
complex plane, evaluated the Polyakov string path integral, from which we
extracted the Neumann functions and the vertex operators. We obtained the one
and two string vertices (identities) following this procedure and finally the
cubic string vertex operator. It is notable that the obtained cubic string
scattering amplitude coincide with the cubic graviton interaction of the
Einstein gravity. Thus, the resultant cubic closed string field theory
consistently describes the Einstein gravity. We also show that the
Kawai-Lewellen-Tye (KLT) relations of the first quantized string theory may be
manifested in the cubic closed string field theory.
