NUT-charged black holes in matter-coupled N=2, D=4 gauged supergravity

Using the results of arXiv:0804.0009, where all timelike supersymmetric backgrounds of N=2, D=4 matter-coupled supergravity with Fayet-Iliopoulos gauging were classified, we construct genuine nut-charged BPS black holes in AdS_4 with nonconstant moduli. The calculations are exemplified for the SU(1,1)/U(1) model with prepotential F=-iX^0X^1. The resulting supersymmetric black holes have a hyperbolic horizon and carry two electric, two magnetic and one nut charge, which are however not all independent, but are given in terms of three free parameters. We find that turning on a nut charge lifts the flat directions in the effective black hole potential, such that the horizon values of the scalars are completely fixed by the charges. We also oxidize the solutions to eleven dimensions, and find that they generalize the geometry found in hep-th/0105250 corresponding to membranes wrapping holomorphic curves in a Calabi-Yau five-fold. Finally, a class of nut-charged Nernst branes is constructed as well, but these have curvature singularities at the horizon.