Three-dimensional (higher-spin) gravities with extended Schr\"odinger and $l$-conformal Galilean symmetries

We show that an extended $3D$ Schr\"odinger algebra introduced in [1] can be reformulated as a $3D$ Poincar\'e 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 Schr\"odinger algebra studied in [1] can be reinterpreted as a relativistic Chern-Simons theory. The latter can be obtained by a contraction of the $SU(1,2)\times SU(1,2)$ Chern-Simons theory with a non principal embedding of $SL(2,\mathbb R)$ into $SU(1,2)$. The non-relativisic Schr\"odinger gravity of [1] and its extended Poincar\'e 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 Schr\"odinger algebra as its member with $l=1/2$, and construct Chern-Simons higher-spin gravities based on these algebras.