Special Kähler Geometry

Abstract

Special (Kähler) geometry [2] is, by definition, the geometry of vectormultiplet scalars in N = 2 supergravity. However, one would like to define this geometry only referring to these scalars, not to any other fields. Therefore one needs to know the most general way of coupling vectormultiplets to supergravity. Originally, supergravity actions for vectormultiplets were constructed using a holomorphic ‘prepotential’. As turned out later, duality transformations can lead to actions for which a prepotential does not exist [4]. In [1] a formulation (‘definition’) of special geometry was given which is manifestly invariant under duality transformations. It was proved that this formulation is equivalent to the original one, in the sense that it is always possible to perform a duality transformation such that a prepotential exists in the dual formulation of the theory. Moreover, it describes all presently known examples of special geometry. All constraints imposed in this definition have a nice physical interpretation (related to duality invariance and positivity of the kinetic energy), except for one constraint in the special case of only one vectormultiplet. This exception suggests that one could try to construct a more general supergravity theory for one vectormultiplet, one that could not be encoded in a holomorphic prepotential.