Abstract
We show that an extended 3D Schrödinger algebra introduced in [1] can be reformulated as a 3D Poincaré 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 Schrödinger 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) × SU(1, 2) Chern-Simons theory with a non principal embedding of SL(2, ℝ) into SU(1, 2). The non-relativisic Schrödinger gravity of [1] and its extended Poincaré 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 Schrödinger algebra as its member with l = 1/2, and construct ChernSimons higher-spin gravities based on these algebras.
Highlights
A main activity has been in studying relativistic gravity models in AdS3 and Minkovski space backgrounds
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
One of the aims of this note is to show that the CS theory based on the extended Schrodinger algebra can be reinterpreted in terms of a relativistic CS theory
Summary
The structure of these relations resembles the form of a centrally extended N = 2, D = 3 Poincare superalgebra, but with the commuting (bosonic) spinor generators Zαi instead of anti-commuting ones. To make this similarity more explicit, let us rewrite the extended Schrodinger algebra in a manifestly D = 2 + 1 Lorentz invariant form. The generator I of the 2d Galilean rotations is traded for the SO(2) R-symmetry generator, the generators of the 1d conformal algebra become that of the SO(1, 2) rotations, while the Galilean translations and boosts form the doublet of SL(2, R) spinors according to (2.6) and (2.8)
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