The covariant path integral formalism for theories of open and closed strings isused to study the first order of string perturbation theory beyond tree level for theclosed-string states, in which the string world sheet has the topology of the disk or thereal projective plane. We find that scattering amplitudes (in flat spacetime) confirmthese surfaces' contribution to the low-energy effective action for the bosonic stringtheory, as derived by another method, demanding consistency of string propagation inbackground gravitational and dilaton fields (the "sigma model approach"). However,we are not able to obtain results consistent with this effective action by demandingthat amplitudes in a curved background be finite; this is an unresolved puzzle. Decouplingof spurious tachyon states from the superstring S-matrix is discussed, andfiniteness of amplitudes for the disk plus projective plane is demonstrated for a largeclass of external states, when the gauge group is SO(32).
