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Cubic Closed String Field Theory on a Double Layer

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.