Extended $D=3$ Bargmann supergravity from a Lie algebra expansion

TL;DR

The paper demonstrates that Lie algebra expansions provide a straightforward route to the extended Bargmann superalgebra and its three-dimensional Chern-Simons action by expanding from $D=3$, $\\mathcal{N}=2$ Poincaré supergravity. The method is applied first to the bosonic Poincaré algebra to recover the extended Bargmann (Galilei) structure, then extended to the full $D=3$, $\\mathcal{N}=2$ case to obtain the supersymmetric extension with a corresponding CS action. Key results include the explicit construction of the expanded algebra, its dual commutators, and the $\\lambda^2$-term of the expanded action, which reproduces the action studied in Bergshoeff et al. BR:16. The approach highlights a systematic, algebraic route to non-relativistic supergravity in $D=3$, with potential extensions to higher dimensions and connections to FDAs and Carroll-type theories.

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 $D=3$, $\mathcal{N}=2$ superPoincaré and its corresponding Chern-Simons supergravity.