Non-Relativistic Chern-Simons Theories and Three-Dimensional Horava-Lifshitz Gravity

TL;DR

This work constructs a trio of non-relativistic Chern–Simons theories based on extended Bargmann, Newton–Hooke, and Schrödinger algebras and shows how they reproduce 3D Hořava–Lifshitz gravity in both its projectable and non-projectable forms via Newton–Cartan geometry. The Bargmann– and Newton–Hooke–based CS actions map to projectable HL gravity (λ=1), while the Schrödinger-based CS action yields twistless torsional NC geometry corresponding to non-projectable HL gravity and hosts a z=2 Lifshitz vacuum. A key contribution is the explicit link between non-relativistic algebras with non-degenerate metrics and HL gravity, including derivations of cosmological solutions and Lifshitz geometries within a CS framework. The results open a minimal bulk-action route to Lifshitz holography, suggest boundary anomaly structures, and invite extensions to other algebras and matter couplings to explore richer HL dynamics.

Abstract

We show that certain three-dimensional Horava-Lifshitz gravity theories can be written as Chern-Simons gauge theories on various non-relativistic algebras. The algebras are specific extensions of the Bargmann, Newton-Hooke and Schroedinger algebra each of which has the Galilean algebra as a subalgebra. To show this we employ the fact that Horava-Lifshitz gravity corresponds to dynamical Newton-Cartan geometry. In particular, the extended Bargmann (Newton-Hooke) Chern-Simons theory corresponds to projectable Horava-Lifshitz gravity with a local U(1) gauge symmetry without (with) a cosmological constant. Moreover we identify an extended Schroedinger algebra containing 3 extra generators that are central with respect to the subalgebra of Galilean boosts, momenta and rotations, for which the Chern-Simons theory gives rise to a novel version of non-projectable conformal Horava-Lifshitz gravity that we refer to as Schroedinger gravity. This theory has a z=2 Lifshitz geometry as a vacuum solution and thus provides a new framework to study Lifshitz holography.