Nuclear Physics B255 (1985) 569-608 © North-Holland Publishing Company LAGRANGIANS OF N = 2 SUPERGRAVITY-MATrER SYSTEMS B. de WIT Institute for Theoretical Physics, University of Utrecht, The Netherlands P.G. LAUWERS Pl~vsikalisches lnstitut, Universit~t Bonn, Bonn, FRG A. VAN PROEYEN* Institute for Theoretical Pl~vsics, K. U. Leuven, Belgium Received 27 December 1984 We present explicit expressions for general actions of vector and scalar multiplets coupled to N = 2 supergravity. We outline their construction which is based on the superconformal tensor calculus. The vector multiplets may be associated with a gauge group G which may also act on the scalar multiplets. The latter are naturally described in terms of quaternions; in the simplest case their kinetic terms define a nonlinear sigma model of a quaternionic projective space. We give an extension of the vector multiplet action which is not obtained from a chiral superspace density, and contains a Chern-Simons-type term. Transformation rules are given and the conditions for supersymmetry breaking are defined. 1. Introduction In the last few years realistic models have been constructed based on N = 1 supergravity coupled to matter (for a review with an extensive list of references see [1]). These models can be used to investigate the implications of supergravity for elementary particle phenomenology. Although their construction represents a signifi- cant step towards a more unified description of the fundamental interactions, it is clear that a more complete unification can be achieved in extended supergravity. This framework is also more compelling in view of its better prospects for giving rise to a consistent quantum theory of gravity coupled to matter. To explore the restrictions imposed by extended supergravity one may first consider the simplest case: N = 2 supergravity coupled to N = 2 matter multiplets. In this paper we deal with general locally supersymmetric actions of N = 2 vector multiplets and scalar multiplets (hypermultiplets) which transform under an optional * Bevoegdverklaard navorser NFWO, Belgium. 569