On Nonrelativistic Diffeomorphism Invariance

TL;DR

The paper develops a systematic framework for nonrelativistic diffeomorphism invariance on curved spaces with background gauge fields, showing how to construct invariant actions by integrating out dynamical degrees of freedom and enforcing foliation-preserving diffeomorphisms. It provides concrete two-dimensional examples that yield topological (Wen-Zee, Chern-Simons) and drift-induced terms, and extends the symmetry to multi-gauge sectors with extended gauge invariance. By deriving continuity and Ward identities, it connects the NR symmetry to transport quantities in quantum Hall systems, including the relation between Hall viscosity and the $q^2$ part of the Hall conductivity. It also discusses an HL gravity perspective and a constructive NR-invariant action program, offering multiple routes to NR covariant formulations and highlighting their physical implications for Hall physics and NR gravity. The work thus furnishes a versatile toolkit for NR diffeomorphism-invariant effective field theories with direct relevance to condensed matter and quantum gravity contexts.

Abstract

We study certain aspects of the recently proposed notion of nonrelativistic diffeomorphism invariance. In particular, we consider specific examples of invariant actions, extended gauge symmetry as well as an application to the theory of quantum Hall effect. We also discuss an alternative approach based on Horava-Lifshitz gravity.