Torsional Newton-Cartan Geometry and the Schrödinger Algebra

TL;DR

This work demonstrates that twistless torsional Newton–Cartan (TTNC) geometry emerges by gauging the Schrödinger algebra with dynamical exponent $z$ and applying curvature constraints that align diffeomorphisms with time and space translations. The authors show that for $z=2$ the TTNC boundary geometry mirrors Lifshitz holography data with a bulk massive vector, while for $z\neq2$ an extra degree of freedom $b_0$ appears, which can be removed by introducing a Stückelberg scalar $\chi$ and a supplementary special conformal symmetry, yielding holographically consistent TTNC; they further extend the formalism to generic torsional Newton–Cartan geometry by relaxing hypersurface orthogonality of the timelike vielbein. The second part of the paper promotes the central charge to a Stückelberg symmetry and constructs a vielbein formalism that accommodates the scalar $\chi$, resulting in a TTNC description that matches holographic expectations for Lifshitz spacetimes and clarifies how to recover full TNC geometry when torsion is allowed. Overall, the results establish a direct link between local Schrödinger invariance and boundary non-relativistic geometry in Lifshitz holography, providing a robust framework for holographic dictionaries on TTNC and TNC backgrounds.

Abstract

We show that by gauging the Schrödinger algebra with critical exponent $z$ and imposing suitable curvature constraints, that make diffeomorphisms equivalent to time and space translations, one obtains a geometric structure known as (twistless) torsional Newton-Cartan geometry (TTNC). This is a version of torsional Newton-Cartan geometry (TNC) in which the timelike vielbein $τ_μ$ must be hypersurface orthogonal. For $z=2$ this version of TTNC geometry is very closely related to the one appearing in holographic duals of $z=2$ Lifshitz space-times based on Einstein gravity coupled to massive vector fields in the bulk. For $z\neq 2$ there is however an extra degree of freedom $b_0$ that does not appear in the holographic setup. We show that the result of the gauging procedure can be extended to include a Stückelberg scalar $χ$ that shifts under the particle number generator of the Schrödinger algebra, as well as an extra special conformal symmetry that allows one to gauge away $b_0$. The resulting version of TTNC geometry is the one that appears in the holographic setup. This shows that Schrödinger symmetries play a crucial role in holography for Lifshitz space-times and that in fact the entire boundary geometry is dictated by local Schrödinger invariance. Finally we show how to extend the formalism to generic torsional Newton-Cartan geometries by relaxing the hypersurface orthogonality condition for the timelike vielbein $τ_μ$.