Asymptotic symmetries of Schrödinger spacetimes

We discuss the asymptotic symmetry algebra of the Schrödinger -invariant metrics in d+3 dimensions and its realization on finite temperature solutions of gravity coupled to matter fields. These solutions have been proposed as gravity backgrounds dual to non-relativistic CFTs with critical exponent z in d space dimensions. It is known that the Schrödinger algebra possesses an infinite-dimensional extension, the Schrödinger-Virasoro algebra. However, we show that the asymptotic symmetry algebra of Schrödinger spacetimes is only isomorphic to the exact symmetry group of the background. It is possible to construct from first principles finite and integrable charges that infinite-dimensionally extend the Schrödinger algebra but these charges are not correctly represented via a Dirac bracket. We briefly comment on the extension of our analysis to spacetimes with Lifshitz symmetry.