We show how the Newton-Cartan formulation of Newtonian gravity can be obtained from gauging the Bargmann algebra, i.e., the centrally extended Galilean algebra. In this gauging procedure several curvature constraints are imposed. These convert the spatial (time) translational symmetries of the algebra into spatial (time) general coordinate transformations, and make the spin connection gauge fields dependent. In addition we require two independent Vielbein postulates for the temporal and spatial directions. In the final step we impose an additional curvature constraint to establish the connection with (on-shell) Newton-Cartan theory. We discuss a few extensions of our work that are relevant in the context of the AdS-CFT correspondence.

We construct a 'stringy' version of Newton-Cartan gravity in which the concept of a Galilean observer plays a central role. We present both the geodesic equations of motion for a fundamental string and the bulk equations of motion in terms of a gravitational potential which is a symmetric tensor with respect to the longitudinal directions of the string. The extension to include a nonzero cosmological constant is given. We stress the symmetries and (partial) gaugings underlying our construction. Our results provide a convenient starting point to investigate applications of the AdS/CFT correspondence based on the nonrelativistic 'stringy' Galilei algebra.

We construct the N = 1 three-dimensional supergravity theory with cosmological, Einstein-Hilbert, Lorentz Chern-Simons, and general curvature squared terms. We determine the general supersymmetric configuration, and find a family of supersymmetric adS vacua with the supersymmetric Minkowski vacuum as a limiting case. Linearizing about the Minkowski vacuum, we find three classes of unitary theories; one is the supersymmetric extension of the recently discovered 'massive 3D gravity'. Another is a 'new topologically massive supergravity' (with no Einstein-Hilbert term) that propagates a single (2, 3 2 ) helicity supermultiplet.

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. IntroductionIt is known that non-relativistic Newtonian gravity can be reformulated in a geometric way, invariant under general coordinate transformations, thus mimicking General Relativity. This reformulation is known as Newton-Cartan theory [1,2]. By (partially) gauge fixing general coordinate transformations, non-geometric formulations can be obtained. The extreme case is the one in which one gauge fixes such that one only retains the Galilei symmetries, corresponding to a description in free-falling frames, in which there is no gravitational force. A less extreme case is obtained by gauge fixing such that one not only considers free-falling frames, but also includes frames that are accelerated, with an arbitrary time-dependent acceleration, with respect to a free-falling frame. The observers in such a frame are called 'Galilean observers ' [3, 4] and the corresponding formulation of non-relativistic gravity is called 'Galilean gravity'1 . In such a frame, the gravitational force is described by the Newton potential Φ. Such frames are related to each other by the so-called 'acceleration extended' Galilei symmetries, consisting of an extension of the Galilei symmetries in which constant spatial translations become time-dependent ones 2 . In this paper, we will construct a supersymmetric version of both Newton-Cartan gravity, as well as Galilean gravity, and show how they are related via a partial gauge fixing.In a previous work we showed how four-dimensional Newton-Cartan gravity can be obtained by gauging the Bargmann algebra 3 which is a central extension of the Galilei algebra [8]. An important step in this gauging procedure is the imposition of a set of constraints on the curvatures corresponding to the algebra [9]. The purpose of these constraints is to convert the abstract time and space translations of the Bargmann algebra into general coordinate transformations. In the relativistic case, i.e. when gauging the Poincaré algebra, one imposes that the torsion, i.e. the curvature corresponding to the spacetime translations, vanishes:These constraints are called conventional constraints. The same set of constraints serves another purpose: it can be used to solve for the spin-connection fields corresponding to the...

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