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Supersymmetric branes on curved spaces and fluxes

We discuss general supersymmetric brane configurations in flux backgrounds of string and M-theory and derive a necessary condition for the worldvolume theory to be supersymmetric on a given curved manifold. This condition resembles very much the conditions found from coupling a supersymmetric field theory to off-shell supergravity but can be derived in any dimension and for up to sixteen supercharges. Apart from the topological twist, all couplings appearing in the supersymmetry condition are linked to fluxes in the bulk. We explicitly derive the condition for D3-, M2- and M5-branes, in which case the results are also useful for constructing holographic duals to the corresponding field theories. In $N=1$ setups we compare the supersymmetry conditions to those that arise by coupling the field theory to off-shell supergravity. We find that the couplings of both old and new minimal supergravity are simultaneously realized, indicating that off-shell supergravity should be coupled via the S-multiplet of 16/16 supergravity in order to describe all supersymmetric brane theories on curved spaces.