Non-Relativistic Chern-Simons Supergravity with Torsion

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.