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We prove a non-existence theorem for smooth AdS5 solutions with connected,compact without boundary internal space that preserve strictly 24supersymmetries. In particular, we show that D=11 supergravity does not admitsuch solutions, and that all such solutions of IIB supergravity are locallyisometric to the AdS_5 * S^5 maximally supersymmetric background. Furthermore,we prove that (massive) IIA supergravity also does not admit such solutions,provided that the homogeneity conjecture for massive IIA supergravity is valid.In the context of AdS/CFT these results imply that if strictly N=3superconformal theories in 4-dimensions exist, their gravitational dualbackgrounds are either singular or their internal spaces are not compact.

AdS5 Backgrounds with 24 Supersymmetries