Nonlinear realization approach to extended supergravity theories in three dimensions

We elaborate on the nonlinear realization approach to spontaneously broken supergravity in three dimensions presented by Kuzenko and Stirling []. Using this approach, we provide a novel derivation of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi mathvariant="script">N</a:mi> </a:math> -extended supergravity, with and without a cosmological term. It corresponds to a Stückelberg-type extension of the following theories: (i) the <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:mo stretchy="false">(</d:mo> <d:mi>p</d:mi> <d:mo>,</d:mo> <d:mi>q</d:mi> <d:mo stretchy="false">)</d:mo> </d:math> anti–de Sitter (AdS) supergravity theories with <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:mi>p</h:mi> <h:mo>+</h:mo> <h:mi>q</h:mi> <h:mo>=</h:mo> <h:mi mathvariant="script">N</h:mi> </h:math> , <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>p</k:mi> <k:mo>≥</k:mo> <k:mi>q</k:mi> <k:mo>≥</k:mo> <k:mn>0</k:mn> </k:math> proposed by Achúcarro and Townsend and (ii) the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi mathvariant="script">N</m:mi> </m:math> -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 <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"> <p:mi mathvariant="script">N</p:mi> </p:math> . In our construction, the pure supergravity actions (Poincaré and AdS) share the invariance under two different local <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:mi mathvariant="script">N</s:mi> </s:math> -extended supersymmetries. One of them acts on the Goldstini, while the other supersymmetry leaves the Goldstini inert. The <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"> <v:mi mathvariant="script">N</v:mi> </v:math> -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.