Five-dimensional electrostatic black holes in a background field

We consider $5$ dimensional electrostatic solutions to Einstein-Maxwell gravity with $2$ commuting spacelike Killing fields. Taking two distinct reductions from $5$ dimensions to a $3$ dimensional base space, we write the Einstein-Maxwell equations using some axially symmetric functions on $\mathbb{R}^3$. These equations can be viewed as arising from a harmonic map coupled to 3-dimensional gravity with the isometries of the target space of this map revealing a hidden $SL(2,\mathbb{R})$ symmetry of this sector of the theory. Depending on the choice of reduction this symmetry then gives rise to two different 1-parameter families of transformations corresponding to either charging a black hole or immersing it in a background electric field. We use these transformations to charge a static black Saturn and a static $L(n,1)$ black lens spacetime and by tuning the strength of the external field, we cure the conical singularities to give new regular solutions. Notably the electrified black lens generated is the first example of a regular black lens in Einstein-Maxwell gravity with topologically trivial asymptotics.