Quaternionic toric manifolds

In the present paper we introduce and study a new notion of toric manifold in the quaternionic setting. We develop a construction with which, starting from appropriate m-dimensional Delzant poly-topes, we obtain manifolds of real dimension 4 m, acted on by m copies of the group Sp(1) of unit quaternions. These manifolds, are quaternionic regular in the sense of [11] and can be endowed with a 4-plectic structure and a generalized moment map. Convexity properties of the image of the moment map are studied. Quaternionic toric manifolds appear to be a large enough class of examples where one can test and study new results in quaternionic geometry.