The gauge-invariant N=2 supersymmetric σ-model with general scalar potential

We construct the supersymmetric σ-model, in six dimensions, for an arbitrary hyper-Kähler manifold, and its minimal coupling to super-Yang-Mills theory. Non-trivial reduction to five or four dimensions yields the corresponding five- or four-dimensional N = 2 supersymmetric model with general scalar potential. We discuss briefly the coupling to supergravity in six dimensions and we give the on-shell supergravity torsion constraints.