Asymptotic symmetries of Schr\"odinger spacetimes

TL;DR

This paper investigates the asymptotic symmetries of Schr"odinger spacetimes, revealing that their symmetry algebra is finite-dimensional and does not extend to the infinite-dimensional Schr"odinger-Virasoro algebra, contrary to expectations.

Contribution

It demonstrates that the asymptotic symmetry algebra of Schr"odinger spacetimes is only isomorphic to the exact background symmetry, challenging previous assumptions about infinite-dimensional extensions.

Findings

Asymptotic symmetry algebra is finite-dimensional

Infinite-dimensional extension is not correctly represented by Dirac brackets

Analysis extends to Lifshitz spacetimes

Abstract

We discuss the asymptotic symmetry algebra of the Schrodinger-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 Schrodinger algebra possesses an infinite-dimensional extension, the Schrodinger-Virasoro algebra. However, we show that the asymptotic symmetry algebra of Schrodinger 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 Schrodinger 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.