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

TL;DR

This paper demonstrates that certain three-dimensional Horava-Lifshitz gravity theories can be formulated as Chern-Simons gauge theories based on non-relativistic algebras, revealing new connections and solutions relevant for Lifshitz holography.

Contribution

It introduces a novel formulation of Horava-Lifshitz gravity as Chern-Simons theories on extended non-relativistic algebras, including a new Schroedinger gravity with Lifshitz vacuum solutions.

Findings

Chern-Simons formulations for various Horava-Lifshitz theories

Identification of Schroedinger gravity with Lifshitz vacuum

Extension of non-relativistic algebras for gravity models

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.