On superpotentials for nonlinear sigma-models with eight supercharges

Using projective superspace techniques, we consider 4D N=2 and 5D N=1 gauged supersymmetric nonlinear sigma-models for which the hyper-Kähler target space is (an open domain of the zero section of) the cotangent bundle of a real-analytic Kähler manifold. As in the 4D N=1 case, one may gauge those holomorphic isometries of the base Kähler manifold (more precisely, their lifting to the cotangent bundle) which are generated by globally defined Killing potentials. In the U(1) case, by freezing the background vector (tropical) multiplet to a constant value of its gauge-invariant superfield strength, we demonstrate the generation of a chiral superpotential, upon elimination of the auxiliary superfields and dualisation of the complex linear multiplets into chiral ones. Our analysis uncovers a N=2 superspace origin for the results recently obtained in [J. Bagger, C. Xiong, hep-th/0601165].