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We begin to develop the formalism of localization for the functional integralof supergravity on $AdS_3 \times S^2$. We show how the condition ofsupersymmetry in the Euclidean $\mathbb{H}^3/\mathbb{Z} \times S^2$ geometrynaturally leads to a twist of the S$^2$ around the time direction of $AdS_3$.The twist gives us a five-dimensional Euclidean supergravity background dual tothe elliptic genus of $(0,4)$ SCFT$_2$ at the semiclassical level. On thisbackground we set up the off-shell BPS equations for one of the Killingspinors, such that the functional integral of five-dimensional Euclideansupergravity on $\mathbb{H}^3/\mathbb{Z} \times S^2$ localizes to its space ofsolutions. We obtain a class of solutions to these equations by lifting knownoff-shell BPS solutions of 4-dimensional supergravity on $AdS_2 \times S^2$. Inorder to do this consistently, we construct and use a Euclidean version of theoff-shell 4d/5d lift of arxiv:1112.5371, which could be of independentinterest.

Localization on $AdS_3 \times S^2$ I: the 4d/5d connection in off-shell Euclidean supergravity