Newton-Cartan Supergravity

TL;DR

The paper constructs a supersymmetric extension of three-dimensional Newton-Cartan gravity by gauging the $\mathcal{N}=2$ super-Bargmann algebra. Due to the universal Newtonian time, only one supersymmetry is gauged, while the other acts as a fermionic Stueckelberg symmetry, resulting in a Galilean-frame theory governed by a Newton potential and its dual, with the latter required for consistent SUSY transformations. Via full gauging one obtains Newton-Cartan supergravity; via partial gauge fixing one obtains Galilean supergravity, where the Newton potential and its fermionic partner, together with a dual potential, realize a holomorphic structure $\Phi+i\Xi$ that constrains the dynamics through a harmonic condition $\Delta\Phi=0$. The work highlights a holomorphic pair of potentials and a Stueckelberg mechanism in non-relativistic supersymmetry, discusses the four-dimensional extension and its additional field content, and outlines directions for higher-dimensional generalizations and representation theory of the super-Bargmann algebra.

Abstract

We construct a supersymmetric extension of three-dimensional Newton-Cartan gravity by gauging a super-Bargmann algebra. In order to obtain a non-trivial supersymmetric extension of the Bargmann algebra one needs at least two supersymmetries leading to a N=2 super-Bargmann algebra. Due to the fact that there is a universal Newtonian time, only one of the two supersymmetries can be gauged. The other supersymmetry is realized as a fermionic Stueckelberg symmetry and only survives as a global supersymmetry. We explicitly show how, in the frame of a Galilean observer, the system reduces to a supersymmetric extension of the Newton potential. The corresponding supersymmetry rules can only be defined, provided we also introduce a `dual Newton potential'. We comment on the four-dimensional case.