Harmonic forms on asymptotically AdS metrics

In this paper we study the rotationally invariant harmonic cohomology of a 2-parameter family of Einstein metrics $g$ which admits a cohomogeneity one action of $SU (2) \times U (1)$ and has AdS asymptotics. Depending on the values of the parameters, $g$ is either of NUT type, if the fixed-point locus of the $U (1)$ action is 0-dimensional, or of bolt type, if it is 2-dimensional. We find that if $g$ is of NUT type then the space of $SU (2)$-invariant harmonic 2-forms is 3-dimensional and consists entirely of self-dual forms; if $g$ is of bolt type it is 4-dimensional. In both cases we explicitly determine a basis. The pair $(g,F)$ for $F$ a self-dual harmonic 2-form is also a solution of the bosonic sector of $4D$ supergravity. We determine for which choices it is a supersymmetric solution and the amount of preserved supersymmetry.