Non-Relativistic Chern-Simons Supergravity with Torsion

TL;DR

This paper develops a three-dimensional non-relativistic Chern--Simons supergravity theory with torsion, using a systematic algebraic expansion to ensure consistency and explore various models.

Contribution

It introduces a novel non-relativistic supergravity formulation based on an $ ext{N}=2$ extension and semigroup expansion, overcoming previous construction challenges.

Findings

The model includes parameters $(p,q)$ interpolating between different supergravity theories.

The approach ensures algebra closure and a non-degenerate invariant bilinear form.

It unifies various non-relativistic supergravity models with torsion.

Abstract

In this work, we construct a three-dimensional non-relativistic Chern--Simons supergravity theory with both curvature and torsion within the Mielke--Baekler framework. We show that a consistent non-relativistic supergravity formulation requires starting from a $\mathcal{N}=2$ supersymmetric extension of the Mielke--Baekler algebra and implementing a non-relativistic expansion via the semigroup expansion method, rather than a naive contraction. This procedure allows one to overcome the usual difficulties of non-relativistic supergravity constructions, ensuring closure of the superalgebra and the existence of a non-degenerate invariant bilinear form. The resulting model is characterized by two parameters $(p,q)$, which interpolate between different non-relativistic supergravity theories, including the extended Bargmann, Newton--Hooke, and torsional models. Our results provide a unified framework for non-relativistic supergravity with torsion and open new avenues for exploring supersymmetric extensions of Galilean and Carrollian gravity theories.