\(\mathcal{N}\)-extended Chern-Simons Carrollian supergravities in 2 + 1 spacetime dimensions. (English) Zbl 1435.83191
Summary: 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 \(\mathfrak{so}\left(\mathcal{N}\right)\) 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 \(( \mathcal{l} \rightarrow \infty\), being \(\mathcal{l}\) 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.
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