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.
Highlights
We consider the problem of coupling Galilean-invariant quantum field theories to a fixed spacetime
We find that NewtonCartan geometry and the shift symmetry automatically arise in the reduction of Lorentzian manifolds in one higher dimension along a null isometry
Given a Galilean-invariant field theory, it should be coupled to a Newton-Cartan structure in such a way that the action is invariant under coordinate reparameterizations, U(1) gauge transformations, and the Milne boosts (2.12)
Summary
Consider coupling a relativistic field theory to a curved background spacetime M. One must first specify the symmetries in order to classify the potential anomalies of a field theory This is tantamount to deducing the correct and covariant couplings to a background spacetime and gauge fields, which is the very thing that is not understood. Consider a Galilean-invariant field theory, which necessarily has a conserved particle number current Jμ to which we may couple a background gauge field Aμ. We find that NewtonCartan geometry and the shift symmetry automatically arise in the reduction of Lorentzian manifolds in one higher dimension along a null isometry This is exactly the boundary geometry that appears in stringy holographic duals of Galilean-invariant field theories, and so our proposal is realized holographically. Various technical results on Newton-Cartan geometry are relegated to the Appendix
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