Nonrelativistic Chern-Simons theories and three-dimensional Horava-Lifshitz gravity

We show that certain three-dimensional Horava-Lifshitz gravity theories can be written as Chern-Simons gauge theories on various nonrelativistic algebras. The algebras are specific extensions of the Bargmann, Newton-Hooke and Schrodinger algebras 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 Schrodinger algebra containing three 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 nonprojectable conformal Horava-Lifshitz gravity that we refer to as Chern-Simons Schrdinger gravity. This theory has a z = 2 Lifshitz geometry as a vacuum solution and thus provides a new framework to study Lifshitz holography.