$$ \mathcal{N} $$-extended Chern-Simons Carrollian supergravities in 2 + 1 spacetime dimensions

Abstract

Abstract In this work we present the ultra-relativistic $$ \mathcal{N} $$ N -extended AdS Chern-Simons supergravity theories in three spacetime dimensions invariant under $$ \mathcal{N} $$ N -extended AdS Carroll superalgebras. We first consider the (2, 0) and (1, 1) cases; subsequently, we generalize our analysis to $$ \mathcal{N} $$ N = ($$ \mathcal{N} $$ N , 0), with $$ \mathcal{N} $$ N even, and to $$ \mathcal{N} $$ N = (p, q), with p, q > 0. The $$ \mathcal{N} $$ N -extended AdS Carroll superalgebras are obtained through the Carrollian (i.e., ultra-relativistic) contraction applied to an so(2) extension of $$ \mathfrak{osp} $$ osp (2|2) ⊗ $$ \mathfrak{sp} $$ sp (2), to $$ \mathfrak{osp} $$ osp (2|1) ⊗ $$ \mathfrak{osp} $$ osp (2, 1), to an $$ \mathfrak{so}\left(\mathcal{N}\right) $$ so N extension of $$ \mathfrak{osp} $$ osp (2|$$ \mathcal{N} $$ N ) ⊗ $$ \mathfrak{sp} $$ sp (2), and to the direct sum of an $$ \mathfrak{so} $$ so (p) ⊕ $$ \mathfrak{so} $$ so (q) algebra and $$ \mathfrak{osp} $$ osp (2|p) ⊗ $$ \mathfrak{osp} $$ osp (2, q), respectively. We also analyze the flat limit (ℓ → ∞, being ℓ the length parameter) of the aforementioned $$ \mathcal{N} $$ N -extended Chern-Simons AdS Carroll supergravities, in which we recover the ultra-relativistic $$ \mathcal{N} $$ N -extended (flat) Chern-Simons supergravity theories invariant under $$ \mathcal{N} $$ 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.