Duality in Supergravity Theories

We present a unified treatment in superspace of the two dual formulations of $D=10$, $N=1$ {\it pure} supergravity based on a strictly super-geometrical framework: the only fundamental objects are the super Riemann curvature and torsion, and the related Bianchi identities are sufficient to set the theory on shell; there is no need to introduce, from the beginning, closed three- or seven-superforms. This formulation extends also to {\it non minimal} models. Moreover, in this framework the algebraic analogy between pure super Yang--Mills theories and pure supergravity in $D=10$ is manifest. As an additional outcome in the present formulation the supersymmetric partner of the ABBJ-Lorentz anomaly in pure $D=10$ supergravity can be computed in complete analogy to the ABBJ-gauge anomaly in supersymmetric Yang--Mills theories in ten dimensions. In the same framework we attack the issue of duality in $N=1$, $D=11$ supergravity showing in detail that duality holds at the kinematical level in superspace while it is broken by the dynamics. We discuss also possible extensions of this theory which could be related to quantum corrections of the eleven dimensional membrane.