The geometry of Schrödinger symmetry in non-relativistic CFT

TL;DR

The paper clarifies how non-relativistic Schrödinger symmetry emerges from a Bargmann/Newton-Cartan framework and shows that NR conformal transformations correspond to ξ-preserving conformal isometries of a higher-dimensional relativistic space. It analyzes concrete gravitational and field-theoretic settings, including Siklos AdS waves, Newton-Hooke spacetimes, topologically massive gravity, and Madelung hydrodynamics, to illustrate how mass, gauge potentials, and inertial effects are encoded geometrically. A key methodological message is that conformal Bargmann transformations can be recast as isometries after an appropriate rescaling, linking NR conformal symmetry to AdS/CFT-inspired backgrounds and to oscillator-flat-space dualities. The results highlight how the Equivalence Principle is encoded in this geometric framework and demonstrate the versatility of the approach across gravitational backgrounds and non-relativistic fluids. Overall, the work provides a unified geometric account of NR Schrödinger symmetry in diverse contexts and clarifies the role of conformal extensions (e.g., Newton-Hooke) and Madelung-type reductions within this scheme.

Abstract

The non-relativistic conformal "Schroedinger" symmetry of some gravity backgrounds proposed recently in the AdS/CFT context, is explained in the "Bargmann framework". The formalism incorporates the Equivalence Principle. Newton-Hooke conformal symmetries, which are analogs of those of Schroedinger in the presence of a negative cosmological constant, are discussed in a similar way. Further examples include topologically massive gravity with negative cosmological constant and the Madelung hydrodynamical description.