Supersymmetry reduction of N-extended supergravities in four dimensions

We consider the possible consistent truncation of N-extended supergravities to lower N' theories. The truncation, unlike the case of N-extended rigid theories, is non trivial and only in some cases it is sufficient just to delete the extra N−N' gravitino multiplets. We explore different cases (starting with N = 8 down to N' ≥ 2) where the reduction implies restrictions on the matter sector. We perform a detailed analysis of the interesting case N = 2N = 1. This analysis finds applications in different contexts of superstring and M-theory dynamics.