On the coupling of Galilean-invariant field theories to curved spacetime

TL;DR

We address how to covariantly couple Galilean-invariant quantum field theories to curved spacetime by promoting Newton-Cartan geometry augmented with a Milne (one-form shift) symmetry. The core construction identifies the Newton-Cartan data $(n_ u,h^{ u ho},v^ u,A_ u)$ as the background, ensuring invariance under diffeomorphisms, $U(1)$ gauge transformations, and Milne boosts; this yields Ward identities tying particle-number, momentum, and energy currents. A key check is that this structure arises naturally from null reductions of Lorentzian manifolds and matches boundary data for Schrödinger holography, with extensions to Weyl (Schrödinger) invariance for $z=2$ theories. The framework provides a covariant, Milne-invariant route to derive currents, Ward identities, and, potentially, low-energy effective actions and hydrodynamics for Galilean-invariant systems, with clear connections to holography and conformal structure. Overall, the work clarifies how to consistently couple NR field theories to spacetime and sets the stage for further exploration of anomalies, topological terms, and hydrodynamic responses in NC geometry.

Abstract

We consider the problem of coupling Galilean-invariant quantum field theories to a fixed spacetime. We propose that to do so, one couples to Newton-Cartan geometry and in addition imposes a one-form shift symmetry. This additional symmetry imposes invariance under Galilean boosts, and its Ward identity equates particle number and momentum currents. We show that Newton-Cartan geometry subject to the shift symmetry arises in null reductions of Lorentzian manifolds, and so our proposal is realized for theories which are holographically dual to quantum gravity on Schrödinger spacetimes. We use this null reduction to efficiently form tensorial invariants under the boost and particle number symmetries. We also explore the coupling of Schrödinger-invariant field theories to spacetime, which we argue necessitates the Newton-Cartan analogue of Weyl invariance.