Field Theory on Newton-Cartan Backgrounds and Symmetries of the Lifshitz Vacuum

TL;DR

The work advances a non-relativistic holographic program by coupling field theories to torsional Newton–Cartan boundaries and analyzing Lifshitz vacua, where the TNC vector $M_$ sources a particle-number current and can be promoted to a gauge field. It shows that bulk PBH diffeomorphisms induce a local Schrödinger algebra on $M$, enabling Schrödinger enhancements of Lifshitz symmetry in flat NC space-time and revealing how a nontrivial orbit of $M$ yields global Galilean and conformal-like symmetries. The authors develop Schrödinger-invariant probe actions on Lifshitz backgrounds, relate $M$-orbit structure to symmetry breaking/enhancement, and discuss the boundary field theory implications, including conserved particle-number currents and the role of the Stückelberg sector. These results provide a coherent holographic dictionary for Lifshitz spacetimes, clarify the boundary geometry as TNC with torsion, and open avenues for non-relativistic hydrodynamics and condensed-matter applications in non-AdS holography.

Abstract

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 $M_μ$ which sources a particle number current, and which for flat NC space-time satisfies $M_μ=\partial_μM$ with a Schroedinger 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 Schroedinger 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 Schroedinger 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 Schroedinger invariance in the same manner as seen in the field theory models.