On conformal supergravity and harmonic superspace

This paper describes a fully covariant approach to harmonic superspace. It is based on the conformal superspace description of conformal supergravity and involves extending the supermanifold M^4|8 by the tangent bundle of ℂP¹. The resulting superspace M^4|8×TℂP¹ can be identified in a certain gauge with the conventional harmonic superspace M^4|8×S². This approach not only makes the connection to projective superspace transparent, but simplifies calculations in harmonic superspace significantly by eliminating the need to deal directly with supergravity prepotentials. As an application of the covariant approach, we derive from harmonic superspace the full component action for the sigma model of a hyperkähler cone coupled to conformal supergravity. Further applications are also sketched.