Supersymmetric gauge theories onve-manifolds

We construct rigid supersymmetric gauge theories on Riemannianve- manifolds. We follow a holographic approach, realizing the manifold as the conformal boundary of a six-dimensional bulk supergravity solution. This leads to a systematic clas- sication ofve-dimensional supersymmetric backgrounds with gravity duals. We show that the background metric is furnished with a conformal Killing vector, which generates a transversely holomorphic foliation with a transverse Hermitian structure. Moreover, we prove that any such metric denes a supersymmetric background. Finally, we construct supersymmetric Lagrangians for gauge theories coupled to arbitrary matter on such back- grounds.