Lorentz Chern-Simons terms in N=1 4D supergravity consistent with supersymmetry and string compactification

We consider the effective N=1 4D supergravity obtained from superstring compactification and we establish full supersymmetry of the Lorentz Chern-Simons terms induced by the modified D=10 supergravity lagrangian. Supersymmetry completion requires an extra coupling of the chiral multiplet S (containing an axion) to a chiral gravitational multiplet which includes both the Hirzebruch signature and the Euler characteristic densities. Then the consistency of the topological condition coming from superstring compactification and full off-shell supersymmetry is eventually proved. (orig.)