Asymptotic symmetries of Schrödinger spacetimes

TL;DR

This work analyzes the asymptotic symmetry structure of Schrödinger-invariant spacetimes and their finite-temperature gravity duals to non-relativistic CFTs. Using a covariant phase-space formalism with a spatial regulator, it shows that the physically realized asymptotic charges reproduce only the exact Schrödinger algebra $\mathfrak{sch}_z(d)$, while the proposed infinite Schrödinger–Virasoro extension does not admit a well-defined Dirac-bracket representation on the relevant phase space. The authors construct explicit charges for $z=2$ (and analyze $z=3$ as an example), demonstrating that infinite-dimensional generators fail to yield integrable, conserved charges on the regulated space, unless one modifies the bracket to account for regulator transformations. The results imply that NRCFT holography based on Schrödinger backgrounds naturally realizes finite Schrödinger symmetries rather than the full Schrödinger–Virasoro algebra, with Lifshitz cases discussed as a parallel extension.

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.