Newton-Cartan supergravity with torsion and Schrödinger supergravity

TL;DR

This work constructs a torsionful $d=3$, $\mathcal{N}=2$ Newton–Cartan supergravity by gauging Schrödinger supergravity and applying a non-relativistic superconformal tensor calculus. It introduces two compensator multiplets—scalar and vector—to obtain two distinct off-shell formulations, termed old minimal and new minimal, each with explicit field content and transformation rules. Torsion arises naturally from the spatial components of the dilatation connection, and a zero-torsion truncation connects to known torsionless Newton–Cartan supergravity via specific identifications. The framework provides a solid basis for non-relativistic supersymmetric field theories on curved backgrounds and suggests avenues for matter-coupled NR supergravities and higher-dimensional generalizations.

Abstract

We derive a torsionfull version of three-dimensional N=2 Newton-Cartan supergravity using a non-relativistic notion of the superconformal tensor calculus. The "superconformal" theory that we start with is Schrödinger supergravity which we obtain by gauging the Schrödinger superalgebra. We present two non-relativistic N=2 matter multiplets that can be used as compensators in the superconformal calculus. They lead to two different off-shell formulations which, in analogy with the relativistic case, we call "old minimal" and "new minimal" Newton-Cartan supergravity. We find similarities but also point out some differences with respect to the relativistic case.