Nonlinear realisation approach to extended supergravity theories in three dimensions

TL;DR

The paper addresses extending 3D ${\cal N}$-extended supergravity to AdS and Poincaré settings within a nonlinear realisation framework. It introduces a novel local ${\cal N}$-extended supersymmetry and constructs Stückelberg-type extensions of $(p,q)$ AdS and ${\cal N}$-extended Poincaré supergravity, along with a Stückelberg reformulation of the supersymmetric Lorentz Chern-Simons action. The key contributions include invariance of these extended actions under two local supersymmetries, the recovery of standard AdS/Poincaré supergravity in unitary gauge, and a detailed analysis of topological terms and their compatibility with the nonlinear realisation; the work also clarifies limitations for ${\cal N}>1$ in describing topologically massive supergravity. The findings offer a unified viewpoint on spontaneous SUSY breaking in 3D gravity and highlight where off-shell methods or conformal extensions may be needed for higher ${\cal N}$ theories, informing future studies of extended supergravities and their topological couplings.

Abstract

We elaborate on the nonlinear realisation approach to spontaneously broken supergravity in three dimensions presented in arXiv:2304.09506. Using this approach we provide a novel derivation of $\mathcal{N}$-extended supergravity, with and without a cosmological term. It corresponds to a Stückelberg-type extension of the following theories: (i) the $(p,q)$ anti-de Sitter (AdS) supergravity theories with $p+q=\mathcal{N}$, $p\geq q\geq 0$ proposed by Achúcarro and Townsend; and (ii) the $\mathcal{N}$-extended Poincaré supergravity of Marcus and Schwarz. We also apply the approach to obtain a Stückelberg reformulation of the supersymmetric Lorentz Chern-Simons action for arbitrary $\mathcal{N}$. In our construction, the pure supergravity actions (Poincaré and AdS) share the invariance under two different local $\mathcal{N}$-extended supersymmetries. One of them acts on the Goldstini, while the other supersymmetry leaves the Goldstini inert. The $\mathcal{N}$-extended supersymmetric Lorentz Chern-Simons action proposed in our setting shares the former supersymmetry, but differs in the one that leaves the Goldstini inert. The supersymmetry that acts on the Goldstini can be used to gauge them away, and then the resulting actions coincide with that given in the literature.