Three-dimensional (higher-spin) gravities with extended Schrodinger and l-conformal Galilean symmetries

We show that an extended 3D Schrodinger algebra introduced in [1] can be reformulated as a 3D Poincare algebra extended with an SO(2) R-symmetry generator and an SO(2) doublet of bosonic spin-1/2 generators whose commutator closes on 3D translations and a central element. As such, a non-relativistic Chern-Simons theory based on the extended Schrodinger algebra studied in [1] can be reinterpreted as a relativistic Chern-Simons theory. The latter can be obtained by a contraction of the SU(1, 2) x SU(1, 2) Chern-Simons theory with a non principal embedding of SL(2, ) into SU(1, 2). The non-relativisic Schrodinger gravity of [1] and its extended Poincare gravity counterpart are obtained by choosing different asymptotic (boundary) conditions in the Chern-Simons theory. We also consider extensions of a class of so-called l-conformal Galilean algebras, which includes the Schrodinger algebra as its member with l = 1/2, and construct ChernSimons higher-spin gravities based on these algebras.