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The homogeneity theorem for supergravity backgrounds

We prove the strong homogeneity conjecture for eleven- and ten-dimensional (Poincaré) supergravity backgrounds. In other words, we show that any backgrounds of 11-dimensional, type I/heterotic or type II supergravity theories preserving a fraction greater than one half of the supersymmetry of the underlying theory are necessarily locally homogeneous. Moreover we show that the homogeneity is due precisely to the supersymmetry, so that at every point of the spacetime one can find a frame for the tangent space made out of Killing vectors constructed out of the Killing spinors.