Field theory on Newton-Cartan backgrounds and symmetries of the Lifshitz vacuum

Holography for Lifshitz space-times corresponds to dual field theories on a fixed torsional Newton-Cartan (TNC) background. We examine the coupling of non-relativistic field theories to TNC backgrounds and uncover a novel mechanism by which a global U(1) can become local. This involves the TNC vector M-mu which sources a particle number current, and which for flat NC space-time satisfies M-mu = partial derivative M-mu with a Schrodinger symmetry realized on M. We discuss various toy model field theories on flat NC space-time for which the new mechanism leads to extra global space-time symmetries beyond the generic Lifshitz symmetry, allowing for an enhancement to Schrodinger symmetry. On the holographic side, the source M also appears in the Lifshitz vacuum with exactly the same properties as for flat NC space-time. In particular, the bulk diffeomorphisms that preserve the boundary conditions realize a Schrodinger algebra on M, allowing for a conserved particle number current. Finally, we present a probe action for a complex scalar field on the Lifshitz vacuum, which exhibits Schrodinger invariance in the same manner as seen in the field theory models.