Four dimensional supersymmetrization of

We review our recent works on the supersymmetrization of the leading string correction (the R 4 term) to N = 1, 2 supergravity theories in four dimensions. We show that, in the ”old minimal ” formulations of these theories, when going on-shell in the presence of this correction, the auxiliary fields which come from multiplets with physical fields cannot be eliminated, but those ones that come from compensating multiplets without any physical fields can be eliminated. We conjecture similar results for other versions of these theories