Extended D = 3 Bargmann supergravity from a Lie algebra expansion

J.A De Azcárraga,D Gútiez,J.M Izquierdo

doi:10.1016/j.nuclphysb.2019.114706

Abstract

In this paper we show how the method of Lie algebra expansions may be used to obtain, in a simple way, both the extended Bargmann Lie superalgebra and the Chern-Simons action associated to it in three dimensions, starting from D=3, N=2 superPoincaré and its corresponding Chern-Simons supergravity.

Highlights

In recent years, the supersymmetric version of Newtonian Gravity, i.e. Newtonian supergravity, has received some attention in the context of a non-relativistic version of the AdS/CFT correspondence
In [3] this was done in D = 3, 4 and in the absence of fermions by starting from the centrally extended Galilei algebra or Bargmann algebra, and imposing certain conditions on the curvatures
The supersymmetric case was studied [5,6] in D = 3 for a superalgebra that contains two fermionic generators and such that the bosonic part is the Bargmann algebra

Summary

Introduction

The supersymmetric version of Newtonian Gravity, i.e. Newtonian supergravity, has received some attention in the context of a non-relativistic version of the AdS/CFT correspondence (see, for instance, [1,2]). The supersymmetric case was studied [5,6] in D = 3 for a superalgebra that contains two fermionic generators and such that the bosonic part is the Bargmann algebra In this way, the D = 3 NC supergravity was obtained. We point out that the Galilean superalgebra and the CS action mentioned above may be found alternatively by using the method of Lie algebra expansions, which has its origin in the work of [9] and was formulated and studied in general in [10,11] (see [12,13] for other applications and [14] for a generalization involving semigroups). We show here that, starting from D = 3, N = 2 Poincaré supergravity and a CS action associated to it, the method of expansions applied to the algebra leads to the extended Bargmann superalgebra of [7]. We will comment on the possible future applications of the method in the context of Galilean gravity and supergravity

Lie Algebra expansions

Expanded CS actions

Poincaré algebra and space-time splitting

Dual version of the algebra

Expansion of the action

Outlook