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

TL;DR

This paper explores the coupling of non-relativistic field theories to torsional Newton-Cartan backgrounds, revealing a mechanism for localizing a global U(1) symmetry and demonstrating how this leads to enhanced Schrödinger symmetry in both field theories and holographic models.

Contribution

It uncovers a novel mechanism by which a global U(1) symmetry becomes local through TNC backgrounds, leading to symmetry enhancement in Lifshitz holography.

Findings

A new mechanism for U(1) localization via TNC vector $M_$.

Enhanced Schrf6dinger symmetry in toy models and holographic Lifshitz vacua.

A probe scalar field action exhibiting Schrf6dinger invariance.

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 $M_\mu$ which sources a particle number current, and which for flat NC space-time satisfies $M_{\mu}=\partial_{\mu}M$ 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.