WHAT IS SPECIAL KAHLER GEOMETRY

The scalars in vector multiplets of N = 2 supersymmetric theories in f our dimensions exhibit 'special Kahler geometry', related to duality s ymmetries, due to their coupling to the vectors. In the literature the re is some confusion on the definition of special geometry. We show eq uivalences of some definitions and give examples which show that earli er definitions are not equivalent, and are not sufficient to restrict the Kahler metric to one that occurs in N = 2 supersymmetry. We treat the rigid as well as the local supersymmetry case. The connection is m ade to moduli spaces of Riemann surfaces and Calabi-Yau 3-folds. The c onditions for the existence of a prepotential translate to a condition on the choice of canonical basis of cycles. (C) 1997 Elsevier Science B.V. PACS: 02.40.-k; 04.65.+e.