Localization on $Ad{S}_3\times S^2$ I: the 4d/5d connection in off-shell Euclidean supergravity

We begin to develop the formalism of localization for the functional integral of supergravity on $Ad{S}_3\times S^2$. We show how the condition of supersymmetry in the Euclidean ${\mathbb{H}}^3\mathrm{/}\mathbb{Z}\times S^2$ geometry naturally leads to a twist of the S${}^2$ around the time direction of $Ad{S}_3$. The twist gives us a five-dimensional Euclidean supergravity background dual to the elliptic genus of $(0,4)$ SCFT${}_2$ at the semiclassical level. On this background we set up the off-shell BPS equations for one of the Killing spinors, such that the functional integral of five-dimensional Euclidean supergravity on ${\mathbb{H}}^3\mathrm{/}\mathbb{Z}\times S^2$ localizes to its space of solutions. We obtain a class of solutions to these equations by lifting known off-shell BPS solutions of 4-dimensional supergravity on $Ad{S}_2\times S^2$. In order to do this consistently, we construct and use a Euclidean version of the off-shell 4d/5d lift of arxiv:1112.5371, which could be of independent interest.