Three-dimensional supergravity reloaded

TL;DR

This work presents a comprehensive construction of the locally supersymmetric extension of the most general three-dimensional gravity theory, the Mielke-Baekler model, showing it can be formulated as a Chern-Simons theory for the appropriate (super)gauge group. A key technical advance is a map that shifts the spin connection by $\omega^a \rightarrow \omega^a + \beta e^a$, relating torsionful MB gravity to torsionless AdS/dS/Poincaré theories while preserving the underlying CS structure. The authors explicitly obtain the $\mathcal{N}=1$ supergravity via this map and generalize to $\mathcal{N}=p+q$ gravitini, using CS theories for $OSp(2|p)\times OSp(2|q)$. The approach clarifies the role of torsion, yields a unified CS framework, and provides a pathway to study BPS states, black holes with torsion, and asymptotic symmetries in 3D supergravity. Overall, the paper advances the understanding of how geometry and supersymmetry emerge from gauge-theoretic formulations in low dimensions, with potential implications for quantum gravity models.

Abstract

The locally supersymmetric extension of the most general gravity theory in three dimensions leading to first order field equations for the vielbein and the spin connection is constructed. Apart from the Einstein-Hilbert term with cosmological constant, the gravitational sector contains the Lorentz-Chern-Simons form and a term involving the torsion each with arbitrary couplings. The supersymmetric extension is carried out for vanishing and negative effective cosmological constant, and it is shown that the action can be written as a Chern-Simons theory for the supersymmetric extension of the Poincare and AdS groups, respectively. The construction can be simply carried out by making use of a duality map between different gravity theories discussed here, which relies on the different ways to make geometry emerge from a single gauge potential. The extension for N =p+q gravitini is also performed.