Torsional Newton-Cartan Geometry from the Noether Procedure

TL;DR

The paper develops a field-theoretic Noether procedure to gauge space-time symmetries in non-relativistic settings, showing that linearized gauging of Galilean and Bargmann algebras yields torsional Newton–Cartan geometry with a minimal affine connection linear in the mass gauge field M_mu. In Bargmann theories M_mu couples to the conserved mass current, while in massless Galilean theories it couples to a topological current, and the procedure identifies a distinguished linear-in-M_mu connection that generalizes NC geometry to torsionful backgrounds. The authors provide a detailed analysis of conserved currents, their improvements, and the resulting couplings of gauge fields to currents, culminating in a minimal coupling framework that matches linearized TNC geometry. They illustrate the formalism with Galilean electrodynamics, showing a clean agreement between Noether-derived couplings and the linearized GED action on TNC backgrounds. These results offer a clear, covariant way to couple non-relativistic field theories to curved backgrounds, informing holography, condensed matter, and non-relativistic gravity research, and pave the way for nonlinear and Carrollian extensions.

Abstract

We apply the Noether procedure for gauging space-time symmetries to theories with Galilean symmetries, analyzing both massless and massive (Bargmann) realizations. It is shown that at the linearized level the Noether procedure gives rise to (linearized) torsional Newton-Cartan geometry. In the case of Bargmann theories the Newton-Cartan form $M_μ$ couples to the conserved mass current. We show that even in the case of theories with massless Galilean symmetries it is necessary to introduce the form $M_μ$ and that it couples to a topological current. Further, we show that the Noether procedure naturally gives rise to a distinguished affine (Christoffel type) connection that is linear in $M_μ$ and torsionful. As an application of these techniques we study the coupling of Galilean electrodynamics to TNC geometry at the linearized level.