Schrödinger invariance from Lifshitz isometries in holography and field theory

We study nonrelativistic field theory coupled to a torsional Newton-Cartan (NC) geometry both directly and holographically. The latter involves gravity on asymptotically local Lifshitz space-times. We define an energy-momentum tensor and a mass current and study the relation between conserved currents and conformal Killing vectors for flat Newton-Cartan backgrounds. It is shown that flat NC space-time realizes two copies of the Lifshitz algebra that together form a Schr\"odinger algebra (without the central element). We show why the Schr\"odinger scalar model has both copies as symmetries and the Lifshitz scalar model only one. Finally we discuss the holographic dual of this phenomenon by showing that the bulk Lifshitz space-time realizes the same two copies of the Lifshitz algebra.