SPECIAL GEOMETRY AND SYMPLECTIC TRANSFORMATIONS

Special Kahler manifolds are defined by coupling of vector multiplets to N = 2 supergravity. The coupling in rigid supersymmetry exhibits si milar features. These models contain n vectors in rigid supersymmetry and n + 1 in supergravity, and n complex scalars. Apart from exception al cases they are defined by a holomorphic function of the scalars. Fo r supergravity this function is homogeneous of second degree in an (n + 1)-dimensional projective space. Another formulation exists which do es not start from this function, but from a symplectic (2n)- or (2n 2)-dimensional complex space. Symplectic transformations lead either t o isometries on the manifold or to symplectic reparametrizations. Fina lly we touch on the connection with special quaternionic and very spec ial real manifolds, and the classification of homogeneous special mani folds.