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

Abstract

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

Keywords

• Chern-Simons Theories
• Gauge Symmetry
• Space-Time Symmetries
• Supergravity Models