In\"on\"u-Wigner Contraction and $D=2+1$ Supergravity

TL;DR

This paper generalizes Inönü-Wigner contraction for Lie superalgebras, enabling explicit derivation of Chern-Simons supergravity actions, including new limits leading to Maxwell supergravity in three dimensions.

Contribution

It introduces a novel contraction method that rescales generators and constants, facilitating the derivation of contracted supergravity theories and new supergravity models in three dimensions.

Findings

Derived explicit Chern-Simons supergravity actions for contracted superalgebras.

Showed how to obtain Poincaré and Maxwell supergravity limits in 2+1 dimensions.

Constructed a new (2,0) Maxwell supergravity theory as a limit of (2,0) AdS-Lorentz supergravity.

Abstract

We present a generalization of the standard In\"on\"u-Wigner contraction by rescaling not only the generators of a Lie superalgebra but also the arbitrary constants appearing in the components of the invariant tensor. The procedure presented here allows to obtain explicitly the Chern-Simons supergravity action of a contracted superalgebra. In particular we show that the Poincar\'e limit can be performed to a $D=2+1$ $\left(p,q\right) $ $% AdS$ Chern-Simons supergravity in presence of the exotic form. We also construct a new three-dimensional $\left(2,0\right) $ Maxwell Chern-Simons supergravity theory as a particular limit of $\left(2,0\right) $ $AdS$ -Lorentz supergravity theory. The generalization for $\mathcal{N}=p+q$ gravitini is also considered.