N-extended Chern-Simons Carrollian supergravities in 2+1 spacetime dimensions

Abstract

In this work we present the ultra-relativistic $\mathcal{N}$-extended AdS Chern-Simons supergravity theories in three spacetime dimensions invariant under $\mathcal{N}$-extended AdS Carroll superalgebras. We first consider the $(2,0)$ and $(1,1)$ cases; subsequently, we generalize our analysis to $\mathcal{N}=(\mathcal{N},0)$, with $\mathcal{N}$ even, and to $\mathcal{N}=(p,q)$, with $p,q>0$. The $\mathcal{N}$-extended AdS Carroll superalgebras are obtained through the Carrollian (i.e., ultra-relativistic) contraction applied to an $so(2)$ extension of $\mathfrak{osp}(2|2)\otimes \mathfrak{sp}(2)$, to $\mathfrak{osp}(2|1)\otimes \mathfrak{osp}(2,1)$, to an $\mathfrak{so}(\mathcal{N})$ extension of $\mathfrak{osp}(2|\mathcal{N})\otimes \mathfrak{sp}(2)$, and to the direct sum of an $\mathfrak{so}(p) \oplus \mathfrak{so}(q)$ algebra and $\mathfrak{osp}(2|p)\otimes \mathfrak{osp}(2,q)$, respectively. We also analyze the flat limit ($\ell \rightarrow \infty$, being $\ell$ the length parameter) of the aforementioned $\mathcal{N}$-extended Chern-Simons AdS Carroll supergravities, in which we recover the ultra-relativistic $\mathcal{N}$-extended (flat) Chern-Simons supergravity theories invariant under $\mathcal{N}$-extended super-Carroll algebras. The flat limit is applied at the level of the superalgebras, Chern-Simons actions, supersymmetry transformation laws, and field equations.