The Non-Relativistic Superparticle in a Curved Background

Using a component formulation, we construct the supersymmetric action for a superparticle in a three-dimensional Newton-Cartan supergravity background and clarify its symmetries. Our construction proceeds by first constructing the superparticle in a flat background. Next, by boosting up the background symmetries, we replace in a first step the flat background by a Galilean supergravity background. In a second step the Galilean supergravity background is replaced by a Newton-Cartan supergravity background. We extend our results by adding a supersymmetric cosmological constant and compare the non-relativistic superparticle with the relativistic kappa-symmetric three-dimensional superparticle.


Introduction
Over the past few years there has been an increased interest in extending the holographic principle to non-relativistic theories. The main motivation is to widen the applications of the conjectured anti-de Sitter/conformal field theory (AdS/CFT) correspondence and to test if it also holds away from its original relativistic setting. There are reasons to expect that non-relativistic holography has applications in effective descriptions of strongly correlated condensed matter systems [1][2][3][4][5]; for a review, see e.g. [6]. Usually, one only considers a nonrelativistic setting at the boundary but one might also consider non-relativistic models both in the bulk and at the boundary [7]. In the context of the AdS/CFT correspondence nonrelativistic theories also occur as boundary geometries. For example, the authors of [8,9] have found that gravity theories supporting Lifshitz geometries lead to a boundary geometry that is characterized by a so-called torsional Newton-Cartan geometry. 1 More recently, non-relativistic gravity and Newton-Cartan structures, see e.g. [10,11], have also been used as a geometric way of realizing non-relativistic symmetries of effective field theories [12].
Originally, before all these developments took place, non-relativistic superstring theories and superbranes were studied as special points in the parameter space of M-theory with non-relativistic symmetries [13,14]. Non-relativistic strings were also thought as a possible soluble sector within string theory or M-theory [15,16]. This latter expectation was based on a similar experience with the pp-wave/BMN limit [17].
The goal of this paper is to make a first step in improving our knowledge on nonrelativistic gravity and particle/string/brane theories that underly a holography with a non-relativistic setting at both the boundary and the bulk. In this work we will consider supersymmetric theories and we will restrict to particles, or more precisely superparticles [18,19], only.
Non-relativistic N = 2 massive superparticles in ten dimensions in a flat background have already been studied in [15]. In this paper we wish to extend this analysis and consider superparticles in a curved non-relativistic supergravity background. It should be stressed that, independent of the relation with non-relativistic holography, there is not much literature on non-relativistic supersymmetry; however, see e.g. [20][21][22][23][24][25][26]. This in itself provides ample reason to investigate this topic.
Part of this paper consists of a review of known results on non-relativistic superparticles in a flat background. To the best of our knowledge all non-relativistic superparticle actions 1 The torsion occurs due to the fact that the boundary geometry is invariant under an anisotropic scaling symmetry.
with background fields, with or without a cosmological constant, are new.
In the construction of massive (super-)particle actions an important role is played by symmetries, both global and local ones. In particular, it is well-known that relativistic massive superparticles have an infinitely-reducible gauge symmetry, called κ-symmetry [27,28], that eliminates half of the fermions. In the non-relativistic setting this κ-symmetry corresponds to a fermionic gauge shift symmetry, i.e. a Stückelberg symmetry [15].
When discussing the symmetries of (super-)particles in a curved background it is important to distinguish between 'proper' and 'sigma model' symmetries [29,30]. In the case of proper symmetries the background (super-)gravity fields only transform through their dependence on the embedding coordinates of the (super-)particle. On the other hand, in the case of sigma model symmetries the background fields have their own transformation rules.
We will clarify how these sigma model transformations are related to the transformations of the (super-)gravity fields when viewed as the components of a (super-)gravity multiplet defined in the target space.
To explain our construction of non-relativistic superparticles in a curved background, we will first consider the bosonic case and take as our starting point a single massive nonrelativistic particle in a flat background. The action of such a particle is invariant under the (global) Galilei symmetries, hence the name 'Galilean' particle. We next partially gauge the spatial target space translations of the Galilean particle such that the constant parameter of a spatial translation is promoted to an arbitrary function of time. The resulting extended symmetries are sometimes called the 'acceleration-extended' Galilean symmetries.
In order to achieve this partial gauging the particle must move in a curved Galilean background that is characterized by the Newton potential Φ [31]. 2 We will call this particle a 'Curved Galilean' particle. Finally, we perform a full gauging of the Galilean symmetries such that the parameter of spatial translations becomes an arbitrary function of the spacetime coordinates and the full symmetries of the particle action are the general coordinate transformations. Actually, to perform this gauging it is necessary to extend the Galilei symmetries with an additional central charge transformation 3 [32,35,36]. The background is now promoted to a Newton-Cartan gravity background that is characterized by a spacelike Vielbein e µ a , a timelike Vielbein τ µ and a central charge gauge field m µ . We will call the 2 There is also a different way of gauging the spatial translations where the background is given by a vector field rather than a scalar potential [32,33]. 3 In sections 3 and 4, where we discuss supersymmetric particle actions, we will restrict ourselves to d = 3.
In that case, it is known that the Bargmann algebra admits a second central charge, see e.g. [34]. We will however not consider it in this paper.
corresponding particle a 'Newton-Cartan' (NC) particle. In table 1 we have summarized   the three different situations described above. background fields particle Λ = 0 symmetries Λ = 0 particle Λ = 0 symmetries Λ = 0 In this work we will extend the superparticle actions considered so far in two directions.
First of all, we will deform the different backgrounds with a cosmological constant Λ. For a flat 4 background the Galilei symmetries are then deformed to so-called Newton-Hooke (NH) symmetries [38,39] 5 . We will call the corresponding particle a 'Newton-Hooke' particle.
The cosmological extensions of the Curved Galilean and NC particle are called the Curved NH and NC NH particle, respectively. It turns out that the symmetries corresponding to these particle actions are the same as in the absence of a cosmological constant, see The second extension we will consider is the one from the bosonic particle to the superparticle. This requires a supersymmetric extension of the gravity backgrounds in the first place. Since non-relativistic supergravity multiplets to our knowledge have only been explicitly constructed in three dimensions, we will only consider superparticles in a threedimensional (3D) background. A supersymmetric version of the 3D Galilean and NC back- 4 Note that after adding the cosmological constant space is flat while space-time is curved, see e.g. [37]. 5 For earlier works on NH symmetries including their supersymmetric extension, see e.g. [  This figure displays the different backgrounds used in this paper and the sections where they are discussed.
For each background, the first section discusses the bosonic case while the second section treats the supersymmetric case. The arrows indicate the direction of gauge-fixing. From left to right we turn on a cosmological constant Λ.
grounds was recently constructed by gauging the Galilei, or better Bargmann, superalgebra [47]. We will make full use of the construction of [47] which, in particular, explains how to switch between different backgrounds, with different symmetries, by partial gauging or partial gauge-fixing. Our aim will be to investigate the action of a 3D superparticle first in a flat background and, next, in a Galilean and NC supergravity background with and without a cosmological constant. To indicate the different cases we will use the same nomenclature as in the bosonic case but with the word particle replaced by superparticle.
This work is organized as follows. In section 2 we go through the known descriptions of the bosonic particle to introduce our notation and discuss the different gaugings and gauge-fixings in a simple setting. After discussing the Λ = 0 case in some detail we repeat the analysis for a non-zero cosmological constant. Sections 3 and 4 are devoted to the supersymmetrization of the theories discussed in section 2 without and with a non-zero cosmological constant, respectively. Section 5 is reserved for some comments on the role of κ-symmetry in the non-relativistic case. A few technicalities, that are used in the main text, are relegated to three appendices. Finally, we have given an alternative table of contents of this paper in figure 1 which summarizes in which sections we discuss the different backgrounds that we will consider.
6 This section will serve as an introduction as well as a means to familiarize the reader with our notation. Moreover, it is a useful warm-up exercise for the rest of the paper when we discuss the superparticle. In subsection 2.1 we start by discussing the bosonic Galilean particle in a flat spacetime without gravitational dynamics. Next, in subsection 2.2 we will partially gauge the Galilei symmetries to allow for arbitrary time-dependent translations.
This introduces the Curved Galilean particle which has gravitational dynamics described by a Newton potential, see [48]. In subsection 2.3 we will perform a further gauging and obtain the Newton-Cartan (NC) particle action which is invariant under general coordinate transformations. Finally, in subsection 2.4, we repeat the analysis of the previous three subsections in the presence of a cosmological constant.

The Galilean Particle
We consider a bosonic particle moving in a flat d-dimensional background with embedding coordinates {t(τ ) , x i (τ )} , i = 1, . . . , d − 1, with τ the evolution parameter. The action describing the dynamics of this bosonic particle is given by The dot indicates a differentiation with respect to the evolution parameter τ . The equations of motion for t and x i corresponding to the action (2.1) are not independent. The first can be derived from the latter which is given by d dτ This implies the existence of a gauge transformation. In fact, the action (2.1) is invariant under worldline reparameterizations with parameter ρ(τ ): 6 together with the Galilei symmetries Here (ζ , λ i j , v i , a i ) parametrize a (constant) time translation, spatial rotation, boost transformation and space translation, respectively. Note that the Lagrangian (2.1) is invariant under boosts only up to a total τ -derivative. This has the consequence that the algebra of Nöther charges is given by a centrally extended Galilei algebra which is called the Bargmann algebra [49,50]. The transformations (2.4) are a representation of the Galilean algebra where Z is the generator of central charge transformations. Physically, the occurrence of the central charge transformations is related to the fact that at the non-relativistic level the mass of a particle is conserved.

The Curved Galilean Particle
We next consider the Curved Galilean particle, i.e. a bosonic particle in a Galilean gravity background described by a Newton potential Φ(t, x). The action for the Curved Galilean particle can be derived by gauging the spatial translations (2.4) to allow for arbitrary timedependent boost parameters ξ i (t). This extension of the Galilean symmetries is called the 'acceleration-extended Galilei symmetries' or the Milne symmetries [51]. As explained in [48] this gauging procedure leads to the following action: The acceleration-extended symmetries are not a proper symmetry of the action (2.8). Instead, the Newton potential should be viewed as a background field and the accelerationextended symmetries as sigma model symmetries. In particular, the transformation rule of the background field, that we will denote by the symbol δ bg , lacks the transport terms that are present in the transformation rule associated to a proper symmetry, denoted in this paper by δ pr : 7 (2.10) Using this we find that the action (2.8) is invariant under the acceleration-extended symmetries (2.9) provided the Newton potential Φ transforms as follows: The second term with the arbitrary function σ(t) represents a standard ambiguity in any potential describing a force and gives a boundary term in the action (2.8). An important qualification of the background field is that it has to obey the constraint We will refer to this constraint as the equation of motion of the background field.
We can re-obtain the particle in flat space by setting This reduces the action and transformation rules of the Curved Galilean particle to the one of the Galilean particle given in the previous subsection.

The Newton-Cartan Particle
We now wish to extend the Curved Galilean particle to a NC particle, i.e. a particle moving in a NC gravity background and invariant under general coordinate transformations. To describe a NC background we need a temporal Vielbein τ µ and a spatial Vielbein e µ a , a = 1, . . . , d − 1,. We furthermore need to introduce a central charge gauge field m µ . The action for the NC particle was already derived in [52], see also [48], and is given by This action is invariant under the worldline reparameterizations (2.3) and under the gauged Galilei symmetries, i.e. general coordinate transformations. The embedding coordinates 7 Assuming that x µ → x µ + δx µ a transport term is given by −δx µ ∂µ so that the second term in (2.10) cancels the transport term present in the proper transformation rule represented by the first term.
transform under these general coordinate transformations, with parameters ξ µ (x ν ), in the standard way: The transformation rules of the background fields τ µ , e µ a and m µ follow from the known proper transformation rules by omitting the transport term, see eq. (2.10): (2.16) Here, λ a b , λ a and σ are the parameters of a local spatial rotation, boost transformation and central charge transformation, respectively.
The proper transformation rules of the background fields τ µ , e µ a and m µ can be obtained by gauging the Bargmann algebra, see e.g. [36]. Since the NC background is the most general background one must be able to obtain the transformations of the Curved Galilean and flat backgrounds discussed in the two previous subsections by gauge-fixing some of the general coordinate transformations. This is discussed in detail in [47]. For the convenience of the reader, we list in table 2 the gauge-fixing conditions that need to be imposed on the NC background fields, and the compensating gauge transformations that come along with it, that bring us to the Curved Galilean background in terms of the Newton potential Φ.

Adding a Cosmological Constant (bosonic)
We now extend the analysis and consider particles in a background with a cosmological constant Λ. We only consider the case of a negative cosmological constant. One way to obtain the action and transformation rules is by using the method of non-linear realizations [53,54] applied to the non-relativistic contraction of the (A)dS algebra which is the socalled Newton-Hooke algebra. In the supersymmetric case this is done in appendix B. The corresponding Newton-Hooke superalgebra is given in appendix A. As we will see below, in the bosonic case an easier way of getting a cosmological extension is to take the nonrelativistic limit, as described e.g. in [48], of the relativistic particle action in an (A)dS background. Below we will discuss the cosmological extension of the Galilean, Curved Galilean and NC particle, one after the other. Following the nomenclature given in table 1, this will lead to the NH, Curved NH and NC NH particle, respectively. gauge condition compensating transformations Table 2 This follows from the gauge conditions made at that point.

The Newton-Hooke Particle
We consider a negative cosmological constant Λ < 0 with the AdS radius given by In global coordinates the metric of an AdS spacetime is given by By taking the non-relativistic limit of a relativistic particle in such an AdS background we arrive at the action The other Galilei transformations are not deformed. The symmetry algebra is the NH algebra, which consists of the Galilei algebra (2.5) plus the additional commutator (2.20) In the limit R → ∞ the action, transformation rules and the algebra reduce to those of the Galilean particle.

The Curved Newton-Hooke Particle
We can partially gauge the symmetries of the NH particle in the same way as we did in the case of the Galilean particle, see subsection 2.2, by introducing a gravitational potential φ.
The resulting action is given by It is invariant under the acceleration-extended symmetries (2.9) but the transformation of the background field changes: The explicit 1/R terms that appear in the action (2.21) and the transformation rule (2.22) are due to our choice of the gravitational potential φ. Namely, we demand that the equation Setting the gravitational potential φ to zero leads us back to the NH particle. The differential equation that follows from eq. (2.22) then fixes the time-dependence of ξ i (t) to be the one given in eq. (2.19).

The Newton-Cartan Newton-Hooke Particle
A cosmological extension of the NC particle discussed in subsection 2.3 can be obtained along the same lines as we discussed for the Curved NH particle above. The following action is obtained: This action is invariant under the gauged Galilei symmetries (which are equivalent to the gauged NH symmetries). The transformations of the background fields τ µ and e µ a are given 9 Note that we could simplify the formulas (2.21) and (2.22) by defining a new potential (2.25) By imposing the gauge-fixing conditions enlisted in table 2 one obtains the action and symmetries of the Curved NH particle, see eqs. (2.21) and (2.22).
This finishes our discussion of the bosonic particle in different backgrounds. In the remaining part of this work we will discuss the supersymmetric extension of these different particles and backgrounds.

The Non-relativistic Superparticle
In the same way that the non-relativistic bosonic particle is based upon the Galilei algebra, or its centrally extended version, the Bargmann algebra, the action and transformation rules of the non-relativistic 3D N = 2 superparticle are based upon the supersymmetric extension of the Galilei or Bargmann algebra. It turns out that we need two supersymmetries since one of the supersymmetries is, like the time translations in the bosonic case, a Stückelberg symmetry.
The 3D N = 2 Galilei superalgebra is given by the bosonic commutation relations (2.5) plus the additional relations [25,26] J ab , where the Majorana spinors Q ± α are the generators corresponding to the two supersymmetries. 10 In the case of the 3D N = 2 Bargmann superalgebra there is a central extension given by eq. (2.7) and by the following equation: The 3D N = 2 supergravity background has been obtained in [47] by gauging the algebra above. This result allows us to discuss the non-relativistic superparticle for Λ = 0, i.e. the supersymmetric version of the second column of table 1. In the three subsections below we will discuss the Galilean, the Curved Galilean and the NC superparticle, respectively. 10 We use a Majorana representation in which the charge conjugation matrix is given by C = iγ 0 and all γ-matrices are real, i.e. γ µ = (iσ2, σ1, σ3).
In the relativistic case the superparticle has an additional so-called κ-symmetry. In the non-relativistic analog this becomes a Stückelberg symmetry [15]. For the purpose of this section we will fix κ-symmetry to avoid unnecessary cluttering of our formulas. Some remarks about restoring κ-symmetry are separately given in section 5.

The Galilean Superparticle
The Galilean superparticle was already discussed in [15]. In terms of the bosonic and fermionic embedding coordinates {t, x i , θ − } the action is given by This action is invariant under the following Galilei symmetries: The same action is invariant under two supersymmetries with constant parameters ǫ + and ǫ − : One may verify that the above set of transformation rules (3.4) and (3.5) closes off-shell.
Note that the transformation with parameter ǫ + is realized linearly. Instead, the one with parameter ǫ − is realized non-linearly, i.e. it is a broken symmetry. This implies that the superparticle corresponds to a 1/2 BPS state.

The Curved Galilean Superparticle
We will now extend the Galilean superparticle to a Curved Galilean superparticle thereby replacing the flat background by a Galilean supergravity background. This corresponds to extending the bosonic particle in a Galilean gravity background, discussed in section 2.2, to the supersymmetric case.
Our starting point is the superparticle action in a flat background, see eq. respectively.
The Galilean supergravity multiplet, that we will use to perform the partial gauging of the transformations (3.4) and (3.5), was introduced in [47]. Alongside the Newton potential Φ(x), it also contains a fermionic background field Ψ(x). The equations of motion for these two background fields are: There is a slight subtlety regarding this Galilean supergravity multiplet, stemming from the fact that Ψ(x) is the superpartner of the Newton force Φ i ≡ ∂ i Φ(x) and not of the Newton potential itself. The transformation rules of the Newton force are, however, compatible with the integrability condition ∂ [i Φ j] = 0, so that they can be integrated to transformation rules of the Newton potential Φ(x). This is done via the introduction of a fermionic prepotential χ(x), that will be called the 'Newtino potential', defined via in [47] and is related to Φ(x) via the Cauchy-Riemann equations for Φ + i Ξ: The dual Newton potential was introduced in [47] in order to write down the supersymmetry transformation rule for χ. Since this is a transformation rule for both real and imaginary parts of χ 1 + i χ 2 , it is natural to expect that it involves also both real and imaginary parts of Φ + i Ξ and this is indeed the case.
We find that the action of the Curved Galilean superparticle in terms of the Galilean supergravity background fields Φ and Ψ is given by One may verify that the action (3.11) is invariant under the transformations (3.6) and (3.7) provided that the background fields transform under the bosonic symmetries as and under the fermionic symmetries as 14) The only invariance that is non-trivial to show is the one under the linear ǫ + -transformations. Varying the action (3.11) under ǫ + -transformations one is left with the following terms: The first two terms combine into a total τ -derivative, since χ is a function of x i and t and therefore d dτ The second term vanishes upon using the equation of motion for the background field χ.
To calculate the commutator algebra it is important to keep in mind that the background fields do not transform as fundamental fields but, instead, according to background fields, see eq. (2.10). This explains the 'wrong' sign transport term in the ǫ + -transformation and the absence of transport terms for all other symmetries. It also has the consequence that partial derivatives do not commute with background variations: Another subtlety when calculating the commutator algebra is related to the fact that the parameters ξ i and ǫ − are functions of the time t but that t itself is a scalar function t(τ ) of the world-line parameter τ . This implies that when we calculate commutators we have to vary the t inside the parameters. Keeping the above subtleties in mind we find that the commutation relations close off-shell on the embedding coordinates and the background fields.
Imposing the gauge-fixing conditions we recover the Galilean superparticle with the flat spacetime transformation rules (3.4) and (3.5). Imposing the additional gauge-fixing condition we find agreement with the algebra obtained in [47].

The Newton-Cartan Superparticle
We wish to extend the result of the previous subsection to arbitrary frames corresponding to a superparticle in a Newton-Cartan supergravity background. Due to the complexity of the calculations we only give the result up to quartic fermions in the action. We find that using this approximation the action is given by where the Lorentz-covariant derivative D τ is defined as To lowest order in the fermions the action (3.20) is invariant under the following bosonic and fermionic symmetries of the embedding coordinates: and In the following we refrain from explicitly denoting the local x µ -dependence of the parameters.
The transformation rules of the background fields follow from the supergravity result given in [47] and application of the identity (2.10). We find that the bosonic transformation rules are given by δ pr e µ a = λ a b e µ b + λ a τ µ , δ pr ψ µ+ = 1 4 λ ab γ ab ψ µ+ , δ pr ω µ ab = ∂ µ λ ab , δ pr ψ µ− = 1 4 λ ab γ ab ψ µ− − 1 2 λ a γ a0 ψ µ+ , δ pr ω µ a = ∂ µ λ a − λ b ω µ ab + λ ab ω µb . (3.25) The variation of ω µ ab is only zero on-shell, i.e. upon using the equations of motion of the background fields. The explicit form of these equations of motion are given in [47]. They also follow by taking the R → ∞ limit of the formulas in appendix C. In the same manner we can write the variation of ω µ a as whereψ µν− is the covariant curvature of ψ µ− , see [47] or appendix C with R → ∞. One

Adding a Cosmological Constant
In this section we are going to describe the superparticle in the presence of a cosmological constant. Like in the bosonic case we can derive the action by taking the non-relativistic limit of a superparticle in AdS space, see e.g. [16] for an example in 10 dimensions.
In the presence of a cosmological constant the relativistic AdS superalgebra in three dimensions is not unique. Instead, in the case of N supersymmetries, one always finds N different versions, often referred to as (p, q) AdS superalgebras [55]. In appendix A we give both the (1, 1) and (2, 0) N = 2 AdS superalgebras. As explained in appendix A, the Newton-Hooke superalgebra that we will use below is obtained by contracting the N = (2, 0) AdS superalgebra. 11 For the bosonic case these conditions are given in table 2.

The Newton-Hooke Superparticle
We find that the action of the NH superparticle takes the form A realization of the Newton-Hooke superalgebra, whose explicit form is given in eq. (A.6), on the embedding coordinates is given by the following bosonic transformation rules supplemented with the following fermionic transformations: Here, the time-dependence of the parameters ξ i (t) and ǫ ± (t) is given by

The Curved Newton-Hooke Superparticle
The Curved Newton-Hooke superparticle is obtained in analogy to the Curved Galilean superparticle, see subsection 3.2. This means that we gauge the spatial translations and one of the supersymmetries, the broken one, such that their parameters become arbitrary time-dependent functions. This introduces the Newton potential φ and its supersymmetric partner ψ. However, based upon our experience in the bosonic case, we expect that either the transformation rules of the NH background fields φ and ψ differ from the Galilean background fields Φ and Ψ, or their equations of motion change, see the comment in footnote 9. Either way, we cannot make use of the results of [47]. 12 Instead, we should first derive the transformations rules of Newton-Hooke supergravity as a 1/R modification of Galilean supergravity. Once we have obtained the NH supergravity transformation rules we can use eq. (2.10) to get the transformation rules of the NH background fields.

Newton-Hooke supergravity
We find that the following bosonic and fermionic transformation rules provide a realization of the acceleration-extended NH algebra: and These transformation rules constitute the NH supergravity extension of the Galilean supergravity result given in [47]. The subscript t on ǫ +,t indicates that it is a function of time.
However, unlike ξ i (t) or ǫ − (t), the time-dependence of ǫ +,t is a very specific one, namely the one given in eq.

The superparticle action
Now that we have constructed the Newton-Hooke supergravity theory, it is easy to construct an action for the Curved NH superparticle where the NH supergravity fields occur as background fields. We find that this action is given by The transformation rules of the NH background fields follow from the NH supergravity rules given in eqs. (4.5)-(4.7) after applying the relation (2.10).
As far as we know, for Λ = 0 there is no redefinition akin to (2.23) that would enable us to deduce the transformation rules of the Curved Newton-Hooke background fields φ and ψ from the Curved Galilean ones given in eqs. (3.13) and (3.14), respectively. 12 The equations of motion are needed to close the supersymmetry algebra.

The Newton-Cartan Newton-Hooke Superparticle
To obtain the NC NH superparticle action we first need to derive the transformation rules of NC NH supergravity, viewed as a 1/R modification of the NC supergravity theory constructed in [47]. We find the following result.

Newton-Cartan Newton-Hooke supergravity
The bosonic transformation rules of the background fields are given by (3.24) and (2.25).
With respect to supersymmetry, the only 1/R modifications occur in the transformation rules of m µ , ψ µ+ and ψ µ− : The supersymmetry rules of τ µ , e µ a are un-deformed and coincide with the ones of NC supergravity, see (3.25). The above modifications induce the following 1/R modifications in the transformation rules of the dependent fields ω µ ab and ω µ a : (4.13) The superparticle action Having constructed the NC NH supergravity theory, it is straightforward to construct the action for the NC NH superparticle. We find that the action, up to quartic fermions, is given by (4.14) One can show that, upon using the equations of motion of the background fields, see appendix C, this action is indeed invariant up to quartic fermions. This concludes our description of the NC NH superparticle action.
It is well-known that the relativistic superparticle is invariant under an additional infinitelyreducible fermionic symmetry called κ-symmetry [27,28]. In the case of strings and branes, this κ-symmetry is needed to obtain the correct counting of degrees of freedom. It is known from the work of [15] that in the non-relativistic case the κ-symmetry is just a Stückelberg symmetry that acts as a shift on one of the fermionic coordinates. It can easily be gauge- Below we just give the results thereby focusing our attention on the κ-symmetry aspects.
For the full details we refer the reader to appendix B.
The κ-symmetric Galilean superparticle The κ-symmetric version of the flat Galilean superparticle action (3.3) depends on an additional fermionic coordinate θ + . The action is given by where the line-elements π 0 and π i are defined as The κ-transformations that leave the action (5.1) invariant are given by The commutator algebra of all symmetry transformations including local κ-transformations and worldline reparameterizations closes on-shell.
We see from eq. (5.3) that the κ-transformation acts as a simple Stückelberg shift symmetry on the embedding coordinate θ + . Therefore, one could just fix this symmetry by imposing the gauge condition which leads us back to the formulas of the Galilean superparticle used in subsection 3.1.
The κ-symmetric Newton-Hooke superparticle An analogous construction leads to the κ-symmetric Newton-Hooke superparticle. We find the following action with the line-elements given by The κ-transformations read as follows: Like in the case of the κ-symmetric Galilean superparticle discussed above the κ-symmetry can be gauge-fixed by imposing the gauge condition θ + = 0.

Discussion
In this paper we have constructed the superparticle actions describing the dynamics of a supersymmetric particle in a 3D Curved Galilean and Newton-Cartan supergravity background. Furthermore, we constructed the actions for a superparticle moving in the cosmological extension of these backgrounds by including a cosmological constant. Due to the computational complexity we gave the action in the Newton-Cartan case only up to terms quartic in the fermions. The Newton-Cartan background is characterized by more fields and corresponds to more symmetries than the Galilean background. One can switch between the two backgrounds either by a partial gauging of symmetries (from Galilean to Newton-Cartan) or by gauge-fixing some of the symmetries (from Newton-Cartan to Galilean). An important role in the construction is played by symmetries. At several occasions we stressed that, as far as the background fields are concerned, one should use the background transformations and not the proper transformations. The latter are used in the definition of the supergravity multiplet. The relation between the two kind of transformations is given in eq. (2.10).
A noteworthy feature is that the proof of invariance of the superparticle action requires that the background fields satisfy their equations of motion. This is reminiscent to what happens with the fermionic κ-symmetry in the relativistic case. We showed in two particular cases that the non-relativistic superparticle also allows a κ-symmetric formulation but that in the non-relativistic case the κ-symmetry is of a simple Stückelberg type [15]. Although being rather trivial, we expect that the formulation with κ-symmetry is indispensable for a reformulation of our results in terms of a non-relativistic superspace and superfields, see e.g. [56]. Such a superspace formulation would be useful to construct the superparticle actions in the Newton-Cartan background to all orders in the fermions.
There are several interesting directions in which one could extend our results. First of all, our efforts in this paper were limited to the three-dimensional case. Clearly, it would be desirable to construct the four-dimensional analog of our results. In order to do this, one should first be able to construct the Galilean and Newton-Cartan supergravity multiplets in four spacetime dimensions. So far, this has not yet been achieved. Another generalization of our results would be to go from superparticles to superstrings or even super p-branes. This would require taking a 'stringy' generalization of the non-relativistic limits we have been considering here, see e.g. [48]. The case of a non-relativistic superstring in a flat background was already considered in [16]. In the case of a non-relativistic curved background one could apply holography and study the corresponding non-relativistic supersymmetric boundary theory.
In the case of a (super-)particle propagating in three spacetime dimensions the Galilei algebra contains a second central charge that could be included as well. This leads to the notion of a non-commutative non-relativistic (super-)particle where the embedding coordinates are non-commutative with respect to the Dirac brackets [57,58]. Finally, we have found that some of our superparticles are 1/2 BPS, corroborating recent results for relativistic superparticles [59]. It would be interesting to verify whether the statement that "all superparticles are BPS" [59] applies to non-relativistic superparticles as well.

A The Newton-Hooke Superalgebra
The NH (super-)algebra can be derived as a contraction of the AdS (super-)algebra. In the case of two supersymmetries there are two independent versions of the latter one, the so-called N = (1, 1) and N = (2, 0) algebras. In the main text we use only the N = (2, 0) algebra for reasons we will explain below.
We proceed by discussing the contraction of the 3D N = (2, 0) AdS algebra. The basic commutators are given by (A = 0, 1, 2) To show that the above algebra corresponds to the N = (2, 0) AdS algebra it is conve-nient to define the new generators In terms of these new generators we obtain the following (anti-)commutation relations: while the charges Q i do not transform under J − A . This identifies the algebra as the N = (2, 0) AdS algebra.
To make the non-relativistic contraction we define new supersymmetry charges by and rescale the generators with a parameter ω as follows: We also set x = 1/(2ωR). Taking the limit ω → ∞ and dropping the tildes on the Q ± we get the following 3D N = (2, 0) Newton-Hooke superalgebra: The central extension Z that leads to the Bargmann version of the NH superalgebra occurs in the following (anti-)commutation relations: The reason why we do not use the N = (1, 1) AdS algebra for the contraction is essentially the same reason as why we are interested in N = 2 rather than N = 1 algebras.
The authors of [47] gauged the N = 2 Galilei superalgebra in order to obtain commutator relations that yield {Q, Q} ∼ H and {Q, Q} ∼ / P , that is, the commutator of two supersymmetries gives timeand space-translations. For N = 1 one can have only one of them.
The N = (1, 1) AdS algebra is equal to the direct product OSp(1|2)⊗OSp(1|2). Taking the non-relativistic contraction thereof amounts to taking the simultaneous contractions of two independent N = 1 algebras. However, we already argued that this cannot lead to a superalgebra of the desired form.

B Non-linear Realizations
In this appendix we obain the action and transformation rules for the flat Galilean and Newton-Hooke superparticles by using the method of non-linear realizations [53,54]. 13 We will derive in the first subsection the κ-symmetric action and transformation rules of the Galilean superparticle, see e.g. [62,63]. In the second subsection we will do the same for the NH superparticle. The normalizations of the (to-be) embedding coordinates that occur in this appendix differ from those in the main text.

B.1 The Kappa-symmetric Galilean Superparticle
The starting point is the N = 2 Bargmann superalgebra given in section 3. We derive the transformation rules for the coordinates (t, x i , s, θ α − , θ α + , k i ) using the coset This leads to the Maurer-Cartan form with the L's given by It is convenient to define the line-elements which are related to the Maurer-Cartan form via the pull-backs The action of the Galilean superparticle is given by the pull-back of all L's that are invariant under rotations, hence Here we replaced the Goldstone field k i by its equation of motion k i = −π i /π 0 . This procedure is know as inverse Higgs mechanism [64], see also [65]. The bosonic transformations of the embedding coordinates are and the supersymmetry transformations are To derive an action that is invariant under κ-transformations we need to find a fermionic gauge-transformation that leaves L H and/or L Z invariant. 14 The variation of L H and L Z under gauge-transformations is given by For κ-transformations we find, using the explicit expressions for L + and L − , It follows that to obtain a κ-symmetric action we need to take the pull-back of either L H or L Z , with [δz + ] = 0 or [δz − ] = 0, respectively. We focus here on the second case, i.e. we choose a = 0. Then the action and κ-symmetry rules are given by where κ is an arbitrary (local) parameter. Using this we find the following κ-transformations of the coordinates: 14) The corresponding κ-symmetric action is given by To compare to the action and transformations rules given in the main text one needs to make the following redefinitions 16) and rescale all spinors by 1/ √ 2.

B.2 The Kappa-symmetric Newton-Hooke Superparticle
The previous subsection was a warming up exercise for the derivation of the NH superparticle which is a deformation of the Galilean superparticle. Here, we give only the main results. Starting from the NH superalgebra given in appendix A we choose the same coset as in the previous subsection, see eq. (B.1), and find the following expressions for the L's: As before we use L i P and L H to define line-elements using the definition (B.5). These will be useful when we write the action.
Using the Maurer-Cartan form one can derive the transformation rules of the Goldstone fields that realize the NH superalgebra (A.6). We find that the bosonic transformation rules are given by while all others fields are invariant (except s). This transformation leaves the line-elements invariant. For the ǫ + -transformations we find the following rules δt = 1 2ǭ + (t)γ 0 θ + , δx i =ǭ + (t)γ i θ − 1 − 1 4Rθ + θ + , In this appendix we collect a few formulas that are needed to show that the NC NH superparticle action (4.14) is invariant under the bosonic and fermionic symmetries given in The curvaturesR µν a (P ) andR µν (Z) do not change w.r.t. the NC case discussed in [47].
Therefore one can still use these, by settingR µν a (P ) =R µν (Z) = 0, to solve for ω µ ab and ω µ a in terms of the other fields. Furthermore one can impose the additional constraintŝ The additional constraints (C.2) and the equations of motion (C.3) are needed to proof invariance of the NC NH superparticle action (4.14).