Relations between closed string amplitudes at higher-order tree level and open string amplitudes

KLT relations almost factorize closed string amplitudes on S2 by two open string tree amplitudes which correspond to the left- and the right-moving sectors. In this paper, we investigate string amplitudes on D2 and RP2. We find that KLT factorization relations do not hold in these two cases. The relations between closed and open string amplitudes have new forms. On D2 and RP2, the left- and the right-moving sectors are connected into a single sector. Then an amplitude with closed strings on D2 or RP2 can be given by one open string tree amplitude except for a phase factor. The relations depends on the topologies of the world-sheets. Under T-duality, the relations on D2 and RP2 give the amplitudes between closed strings scattering from D-brane and O-plane respectively by open string partial amplitudes. In the low energy limits of these two cases, the factorization relations for graviton amplitudes do not hold. The amplitudes for gravitons must be given by the new relations instead.