Rigid Supersymmetry from Conformal Supergravity in Five Dimensions

TL;DR

The paper constructs the most general rigid supersymmetric backgrounds for 5d N=2 conformal supergravity on Euclidean manifolds, showing that the existence of a conformal Killing vector is both necessary and sufficient for supersymmetry. It provides a gravitino-dilatino analysis that renders the solution in terms of intrinsic torsions, a CKV v, a triplet Δ^{ij}, and a horizontal vector W, with THF structures arising when the SU(2) curvature abelianizes. A key finding is that turning on a standard Yang–Mills term requires the CKV to be Killing, while backgrounds with a CKV allow a position-dependent YM coupling via a background vector multiplet scalar; explicit examples illustrate these constraints on spaces like R^5, R×S^4, Sasaki–Einstein manifolds, and S^5. The results clarify the geometric structure of rigid supersymmetric five-dimensional theories and inform localization and partition-function computations on these curved backgrounds.

Abstract

We study the rigid limit of 5d conformal supergravity with minimal supersymmetry on Riemannian manifolds. The necessary and sufficient condition for the existence of a solution is the existence of a conformal Killing vector. Whenever a certain $SU(2)$ curvature becomes abelian the backgrounds define a transversally holomorphic foliation. Subsequently we turn to the question under which circumstances these backgrounds admit a kinetic Yang-Mills term in the action of a vector multiplet. Here we find that the conformal Killing vector has to be Killing. We supplement the discussion with various appendices.