Superstring scattering on the real projective plane

Superstring scattering from orientifold planes requires considering string amplitudes on world-sheets with crosscaps with the lowest order case (in string coupling constant) having the topology of the real projective plane. While amplitudes on the latter have been formulated for the trivial one- and two-point cases in this work we go beyond these cases thereby solving various technicalities. The latter include reducing the complex world-sheet integration of closed string insertions over the real projective plane to pure real open string integrals. As a result we find that scattering of $n$ closed strings on the real projective plane can be expressed in terms of disk amplitudes involving $2n$ open strings. In this work we explicitly work out in pure spinor formalism the case $n=3$ which can be written as a linear combination of two (gauge-invariant) six open string amplitudes. We also present the low-energy expansion of this result necessary to construct closed string couplings on orientifold planes.