Holography for Schrodinger backgrounds

TL;DR

This work establishes a holographic framework for Schrödinger backgrounds by showing that Schrödinger spacetimes arise as irrelevant deformations of relativistic CFTs that are exactly marginal with respect to the Schrödinger group. It analyzes two bulk realizations—3D topologically massive gravity and a massive vector model—identifying deforming operators X and X_v that couple to extrinsic curvature or a null vector field, and demonstrates, via conformal perturbation theory and linearized bulk solutions, that these deformations yield momentum-dependent operator dimensions Δ_s(k_v) and consistent Schrödinger two-point functions. A key finding is that holographic renormalization in this setting requires non-local counterterms in the lightlike direction v, reflecting the non-locality of the dual Schrödinger-invariant field theory. The results unify field-theoretic and gravitational perspectives, showing that the dual theory is a non-local Schrödinger-invariant deformation of a CFT, with a null-dipole structure suggested by the bulk analysis. The work also outlines a refined dilatation operator framework and highlights important open questions about the precise holographic dictionary for TMG, the full role of the vielbein, and the deeper null-dipole interpretation of these holographic systems.

Abstract

We discuss holography for Schrodinger solutions of both topologically massive gravity in three dimensions and massive vector theories in (d+1) dimensions. In both cases the dual field theory can be viewed as a d-dimensional conformal field theory (two dimensional in the case of TMG) deformed by certain operators that respect the Schrodinger symmetry. These operators are irrelevant from the viewpoint of the relativistic conformal group but they are exactly marginal with respect to the non-relativistic conformal group. The spectrum of linear fluctuations around the background solutions corresponds to operators that are labeled by their scaling dimension and the lightcone momentum k_v. We set up the holographic dictionary and compute 2-point functions of these operators both holographically and in field theory using conformal perturbation theory and find agreement. The counterterms needed for holographic renormalization are non-local in the v lightcone direction.