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

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 Chern-Simons higher-spin gravities based on these algebras.

Non-relativistic symmetries in three space-time dimensions and the Nappi-Witten algebra

“…Therefore, even though this algebra can be brought to the Maxwellian exotic Bargmann algebra given in (3.6) and (3.13) by a redefinition of the generators, they are physically different. The Chern-Simons Lagrangian invariant under (3.18) defines a non-relativistic limit of the Hietarinta Chern-Simons gravity studied in [31,47], pretty much in the same way as the Chern-Simons form (3.15) describes a NR limit of Maxwell gravity 4 [32]. The expanded invariant tensor (2.13) can be read off from (3.14) by interchanging P a ↔ Z a , H ↔ Z, M ↔ T , and the Chern-Simons form (2.15) is constructed using…”

“…In 2+1 dimensions, Newton-Cartan gravity can be formulated as a Chern-Simons theory invariant under the Bargmann algebra [26,27] (see also [28] for a general classification of non-relativistic limits of three-dimensional Einstein gravity). This can be generalized to the case of the Newton-Hooke and Schrödinger symmetries, whose gravity theories lead to different generalization of Horava-Lifshitz gravity in 2+1 dimensions [29,30] (for a relativistic reinterpretation of the three-dimensional Schrödinger symmetry and the corresponding Chern-Simons theory, see [31]). Moreover, the Bargmann-invariant Chern-Simons theory can be non-centrally extended to include a covariantly constant background field [32], leading to Maxwellian Exotic Bargmann gravity.…”

“…In doing this, we will recover several symmetries known in the literature such as the Extended Bargmann, Extended Newton-Hooke, Carroll, and Maxwellian Exotic Bargmann algebras together with their corresponding Chern-Simons actions. Furthermore, this procedure allows one to obtain novel symmetries such as the NR limit of the semi-simple extension of the Poincaré algebra in 2+1 dimensions [45,46], the NR limit of the bosonic Hietarinta algebra studied in [31,47], and NR versions of the generalized Poincaré and generalized AdS symmetries introduced in [40,41].…”

Three-dimensional Maxwellian extended Bargmann supergravity

“…The NR theories have received a renewed interest since they play an important role to approach condensed matter systems [8][9][10][11][12][13][14][15] and NR effective field theories [16][17][18][19]. It seems then natural to extend NR gravity theories [20][21][22][23][24][25][26][27][28][29][30][31][32][33] to the presence of supersymmetry. In particular, NR supergravity models can be seen as a starting point to approach supersymmetric field theories on curved backgrounds by means of localization [34,35].…”

On the supersymmetric extension of asymptotic symmetries in three spacetime dimensions

“…where e a is the vielbein, ω a corresponds to the spin connection, σ a is the gravitational Maxwell gauge field, R a = dω a + 1 2 ǫ abc ω b ω c is the Lorentz curvature and T a = D ω e a is the torsion two-form. More recent results have been presented in [73,74] in the dual version of the Maxwell algebra known as Hietarinta algebra [75]. The Hietarinta symmetry appears by interchanging the role of the Maxwell gravitational generator Z a with the momentum generator P a .…”