Quaternionic and hyper-Kähler metrics from generalized sigma models

The problem of finding new metrics of interest, in the context of SUGRA, is reduced to two stages: first, solving a generalized BPS sigma model with full quaternionic structure proposed by the authors and, second, constructing the hyper-Kähler metric, or suitable deformations of this condition, taking advantage of the correspondence between the quaternionic left-regular potential and the hyper-Kähler metric of the target space. As illustration, new solutions are obtained using generalized Q-sigma model for Wess–Zumino type superpotentials. Explicit solutions analog to the Berger?s sphere and Abraham–Townsend type are given and generalizations of 4-dimensional quaternionic metrics, product of complex ones, are shown and discussed.