Supersymmetry of AdS and flat backgrounds in M-theory

We give a systematic description of all warped $AdS_n$ and ${\mathbb{R}}^{n-1,1}$ backgrounds of M-theory and identify the a priori number of supersymmetries that these backgrounds preserve. In particular, we show that $AdS_n$ backgrounds preserve $N= 2^{[{n\over2}]} k$ for $n\leq4$ and $N= 2^{[{n\over2}]+1} k$ for $4<n\leq 7$ supersymmetries while ${\mathbb{R}}^{n-1,1}$ backgrounds preserve $N= 2^{[{n\over2}]} k$ for $n\leq4$ and $N= 2^{[{n+1\over2}]} k$ for $4<n\leq7$, supersymmetries. Furthermore for $AdS_n$ backgrounds that satisfy the requirements for the maximum principle to hold, we show that the Killing spinors can be identified with the zero modes of Dirac-like operators on $M^{11-n}$ coupled to fluxes thus establishing a new class of Lichnerowicz type theorems. We also demonstrate that the Killing spinors of generic warped $AdS_n$ backgrounds do not factorize into products of Killing spinors on $AdS_n$ and Killing spinors on the transverse space.