The holography principle relates the quantum gravity in the bulk, described by closed string, the gauge theory, described by open string on the boundary with certain asymptotic conditions. Thus, it is important to understand intimate relations between open string theory and closed string theory:
In the present work 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.