Duality Rotations in Nonlinear Electrodynamics and in Extended Supergravity

Abstract

We review the general theory of duality rotations which, in four dimensions, exchange electric with magnetic fields. Necessary and sufficient conditions in order for a theory to have duality symmetry are established. A nontrivial example is Born-Infeld theory with n abelian gauge fields and with Sp(2n,R) self-duality. We then review duality symmetry in supergravity theories. In the case of N=2 supergravity duality rotations are in general not a symmetry of the theory but a key ingredient in order to formulate the theory itself. This is due to the beautiful relation between the geometry of special Kaehler manifolds and duality rotations.Comment: Invited contribution to Rivista del Nuovo Cimento in occasion of the 2005 Enrico Fermi Prize of the Italian Physical Society. 96 pages, corrected typo