Newton–Cartan (super)gravity as a non-relativistic limit

The method used in this paper can be viewed as an extension of the contraction of a relativistic spacetime symmetry algebra to a non-relativistic one. In particular, its aim is to mimic the algebra contraction to obtain an irreducible multiplet of fields that represents the contracted non-relativistic algebra starting from an irreducible multiplet of the parent relativistic algebra.

Recall that when performing a standard Inönü–Wigner contraction of a symmetry algebra one first redefines the generators of the algebra, by taking linear combinations of the original generators with coefficients that involve a contraction parameter ω. The contracted algebra is then obtained by calculating commutators of the redefined generators, re-expressing the result in terms of them and taking $\omega \to \infty$ in the end. An Inönü–Wigner contraction performed in this way does not change the number of generators. Moreover, when considering finite ω the algebra of redefined generators is equivalent to the original one.

When extending this algebra contraction to an irreducible multiplet of fields that represents the parent relativistic algebra, it is useful to divide the fields in three categories. A first category consists of independent fields that can be viewed as gauge-fields that are associated to certain generators of the algebra. For instance, the vielbein of general relativity roughly plays the role of the gauge-field of local translations [31]. A second category comprises gauge-fields that are not independent, but that instead depend on other fields in the multiplet. This is the way in which the spin-connection of general relativity can be viewed, namely as a dependent gauge-field for local Lorentz transformations. Finally, the last category contains independent fields that cannot be interpreted as gauge-fields of the underlying spacetime symmetry algebra. This is for instance the case when considering off-shell supergravity multiplets, where typically auxiliary fields are necessary to guarantee the closure of the commutator algebra and, in the relativistic case, to ensure that the number of bosonic and fermionic degrees of freedom match. For simplicity, we will call all such additional fields, that are not gauge-fields corresponding to a generator, auxiliary fields. The fields that can be viewed as gauge-fields corresponding to the generators of the algebra are in general subject to constraints. Some of these constraints are called 'conventional' and merely serve to express the dependent gauge-fields in terms of the independent ones. These constraints are identically satisfied, once the explicit expressions for the dependent fields are plugged in. There will in general also be a second type of 'un-conventional' constraints, that are not identically satisfied. This second type of constraints will play a crucial role in ensuring consistency of the limiting procedure.

The first step of the limiting procedure consists of extending the redefinition of the algebra generators, with the contraction parameter ω, to all fields and symmetry parameters of the relativistic multiplet. Let us therefore denote the parent algebra generators collectively by T_A, the parent symmetry parameters by ${\xi }^{A}$ and the parent fields that correspond to gauge-fields by ${A}_{\mu }^{A}.$ The redefinition of the generators T_A to generators ${\tilde{T}}_{A},$ that involves ω and defines the contraction, can then be extended to redefinitions of ${\xi }^{A}$ to ${\tilde{\xi }}^{A}$ and of ${A}_{\mu }^{A}$ to ${\tilde{A}}_{\mu }^{A}$ such that ⁴

$$\begin{eqnarray}&&{\xi }^{A}{T}_{A}={\tilde{\xi }}^{A}{\tilde{T}}_{A},\qquad {A}_{\mu }^{A}{T}_{A}={\tilde{A}}_{\mu }^{A}\tilde{{T}_{A}}.\end{eqnarray} \tag{ 1 }$$

This defines the tilded parameters and fields in terms of the original ones and the contraction parameter ω. For finite ω, this redefinition merely corresponds to a field redefinition and the redefined multiplet is equivalent to the original one. Note that (1) involves both independent and dependent gauge-fields. For the dependent fields, one should take special care that the redefinition suggested by (1) is consistent with the one obtained by performing the redefinitions in the explicit expressions of the dependent fields in terms of the independent ones and that their $\omega \to \infty$ limit is well-defined. This amounts to a non-trivial consistency check, that we will discuss further in the next paragraph. Equation (1) does not determine the redefinitions of the auxiliary fields. These are found by examining all transformation rules in terms of redefined gauge-parameters and gauge-fields and by requiring that no term in the transformation rules diverges when taking the limit $\omega \to \infty .$ As we will see in specific examples, this can typically be achieved by suitably rescaling the auxiliary fields with the contraction parameter ω.

Once all redefinitions have been determined, one can apply them in the transformation rules of all independent fields and in the un-conventional constraints. Their non-relativistic versions are then obtained by sending $\omega \to \infty .$ For properly chosen redefinitions, no divergences are encountered here. One does however need to check whether other quantities are finite in this limit. In particular, one needs to examine the expressions of the dependent fields in terms of the independent ones and check whether one obtains a finite result, consistent with the redefinition implied by (1), in the limit $\omega \to \infty .$ Typically, terms that are proportional to positive powers of ω, and hence blow up in the $\omega \to \infty$ limit, do show up in the expressions for the dependent gauge-fields. One can however use the un-conventional constraints, written in terms of redefined fields, to replace these by terms that are finite or vanishing in the $\omega \to \infty$ limit. In this way, the relativistic dependent gauge-fields have a well-defined $\omega \to \infty$ limit, consistent with (1) and lead to the correct finite expressions for the non-relativistic dependent gauge fields.

Once all non-relativistic transformation rules, dependent gauge-fields and constraints have been found in this way, one needs to check whether the constraints form a consistent set. This involves varying all non-trivial constraints found so far under all symmetry transformations and checking that they form a closed set.

Finally, we mention that the limiting procedure can lead to the elimination of a number of auxiliary fields. This is due to the fact that we are interested in obtaining an irreducible multiplet. Loosely speaking, the non-relativistic theory can have less equations of motion than the relativistic one. The number of auxiliary fields that are needed to realize the non-relativistic algebra can therefore differ from the number that is needed to realize the relativistic algebra. This explains why some of the auxiliary fields can be eliminated in the limiting process.

We can summarize the procedure in the following way:

• I.  
  We first write the relativistic gauge-fields in terms of new redefined ones, using a contraction parameter ω. This field redefinition is dictated by the same redefinition that one performs on the generators to define the contraction of the algebra. The new fields will become the proper non-relativistic fields after we have taken the $\omega \to \infty$ limit. At this point the scaling of the auxiliary fields is still arbitrary.
• II.  
  Using the above redefinitions and taking the limit $\omega \to \infty$ we can derive a first set of non-relativistic constraints by taking the limit $\omega \to \infty$ of the relativistic un-conventional ones.
• III.  
  In a third step we derive the transformation rules of all fields. Requiring that no terms diverge in the limit $\omega \to \infty$ fixes the scalings of the auxiliary fields. At this point, we can check the limit of the dependent gauge-fields, such as, e.g., the spin-connection field. Requiring that they have a well-defined limit may involve the use of the un-conventional constraints, written in terms of the redefined fields, in order to replace divergent terms by terms with a proper limit.
• IV.  
  In this step we check whether the constraints found in step II form a closed set under the different symmetry transformations or whether we are forced to introduce additional constraints. An example where many new constraints are found by continuous variation under supersymmetry is given by the chain of constraints in equations (61)–(63).
• V.  
  The number of auxiliary fields that are needed in the non-relativistic case may be less then the number that is needed in the relativistic case. In such cases, in order to obtain an irreducible multiplet, we eliminate the redundant auxiliary fields. This occurs, for instance, in the example of section 3.

In the next subsections, we will illustrate the above limiting procedure by applying it to re-derive various results on Newton–Cartan geometry and (super)gravity that have been obtained in the literature using other methods.

In this section we illustrate the limiting procedure that we just described, to obtain the vielbein description of Newton–Cartan gravity of [23] from the vielbein formulation of general relativity. We will pay specific attention to how the transformation rules of the Newton–Cartan fields arise, to how the limiting procedure leads to the correct constraints that these fields have to obey and to how the correct dependent gauge-fields are obtained. First, we summarize in section 2.2.1 our starting point, namely the kinematics of general relativity as a gauging of the Poincaré algebra. Then, using those formulas, we deduce Newton–Cartan geometry and gravity in section 2.2.2.

2.2.1. The kinematics of general relativity

It is well-known that the vielbein formulation of general relativity can be viewed as a gauging of the Poincaré algebra [31]. Here we will briefly summarize this gauging procedure. Note that this leads to the kinematics of general relativity. In order to obtain the dynamics, one has to supplement the formulas collected here with the Einstein equations. If one is however only interested in geometrical aspects and not so much in dynamics, one need not do so. This section will serve as the starting point of our limiting procedure, that will lead to the formulation of Newton–Cartan gravity as obtained via a gauging of the Bargmann algebra in [23].

The kinematics of general relativity in a $(d+1)$-dimensional spacetime is described by the vielbein ${E}_{\mu }{}^{A}$ and spin-connection ${{\rm{\Omega }}}_{\mu }{}^{{AB}},$ $A,B=0,\ldots ,d,$ that can be viewed as gauge-fields associated to translations ${\hat{P}}_{A}$ and Lorentz rotations M_AB. Under diffeomorphisms (with parameter ${\xi }^{\mu }$) and Lorentz rotations (with parameter ${\lambda }^{{AB}}$) these fields transform as

$$\begin{eqnarray}&&\delta {E}_{\mu }{}^{A}={\xi }^{\rho }{\partial }_{\rho }{E}_{\mu }{}^{A}+{E}_{\rho }{}^{A}{\partial }_{\mu }{\xi }^{\rho }+{\lambda }^{A}{}_{B}{E}_{\mu }{}^{B},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\delta {{\rm{\Omega }}}_{\mu }{}^{{AB}}={\xi }^{\rho }{\partial }_{\rho }{{\rm{\Omega }}}_{\mu }{}^{{AB}}+{{\rm{\Omega }}}_{\rho }{}^{{AB}}{\partial }_{\mu }{\xi }^{\rho }+{\partial }_{\mu }{\lambda }^{{AB}}+2\;{\lambda }^{[A}{}_{C}\;{{\rm{\Omega }}}_{\mu }{}^{{CB}]}.\end{eqnarray} \tag{ 3 }$$

One may define gauge covariant curvatures, dictated by the structure constants of the Poincaré algebra, as follows:

$$\begin{eqnarray}&&{R}_{\mu \nu }{}^{A}(E)=2\;{\partial }_{[\mu }{E}_{\nu ]}{}^{A}-2\;{{\rm{\Omega }}}_{[\mu }{}^{A}{}_{B}\;{E}_{\nu ]}{}^{B},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{R}_{\mu \nu }{}^{{AB}}({\rm{\Omega }})=2\;{\partial }_{[\mu }{{\rm{\Omega }}}_{\nu ]}{}^{{AB}}-2\;{{\rm{\Omega }}}_{[\mu }{}^{A}{}_{C}\;{{\rm{\Omega }}}_{\nu ]}{}^{{CB}}.\end{eqnarray} \tag{ 5 }$$

The spin-connection ${{\rm{\Omega }}}_{\mu }{}^{{AB}}$ is not an independent field; rather it is given in terms of ${E}_{\mu }{}^{A}$ by solving the torsion constraint

$$\begin{eqnarray}&&{R}_{\mu \nu }{}^{A}(E)=0.\end{eqnarray} \tag{ 6 }$$

This constraint also allows one to replace local translations by general coordinate transformations. The solution of (6) is given by

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{\mu }{}^{{AB}}(E)=-2\;{E}^{\rho [A}{\partial }_{[\mu }{E}_{\rho ]}{}^{B]}+{E}_{\mu C}{E}^{\rho A}{E}^{\nu B}{\partial }_{[\rho }{E}_{\nu ]}{}^{C}.\end{eqnarray} \tag{ 7 }$$

Note that imposing the torsion constraint (6) also implies that the Riemann curvature tensor (5) identically satisfies the Bianchi identity

$$\begin{eqnarray}&&{R}_{[\mu \nu \rho ]}{}^{B}\left({\rm{\Omega }}(E)\right)={R}_{[\mu \nu }{}^{{AB}}\left({\rm{\Omega }}(E)\right){E}_{\rho ]A}=0.\end{eqnarray} \tag{ 8 }$$

The above defines the kinematics of general relativity. The dynamics can be obtained by imposing equations of motion, i.e. putting the theory on-shell. In general relativity this is done by imposing the Einstein equations

$$\begin{eqnarray}&&{E}^{\mu }{}_{A}\;{R}_{\mu \nu }{}^{{AB}}\left({\rm{\Omega }}(E)\right)=0.\end{eqnarray} \tag{ 9 }$$

The limiting procedure by which we will obtain non-relativistic geometry and gravity is an extension of contractions of relativistic symmetry algebras. As explained in [23], torsionless Newton–Cartan geometry and gravity is linked to the Bargmann algebra. In order to obtain the Bargmann algebra from an Inönü–Wigner contraction, one should start from a direct sum of the Poincaré algebra with an abelian factor with generator ${\mathcal{Z}}.$ In this way, one ensures that the algebra before contraction has the same number of generators as the Bargmann algebra. The abelian factor is represented by an abelian gauge-field M_μ, that transforms under diffeomorphisms and abelian gauge transformations (with parameter Λ) as follows:

$$\begin{eqnarray}&&\delta {M}_{\mu }={\xi }^{\rho }{\partial }_{\rho }{M}_{\mu }+{M}_{\rho }\;{\partial }_{\mu }{\xi }^{\rho }+{\partial }_{\mu }{\rm{\Lambda }}.\end{eqnarray} \tag{ 10 }$$

The curvature of M_μ is given by

$$\begin{eqnarray}&&{F}_{\mu \nu }(M)=2\;{\partial }_{[\mu }{M}_{\nu ]}.\end{eqnarray} \tag{ 11 }$$

In order to take the non-relativistic limit we will need to impose constraints on this curvature. For example, if we consider the dynamics of general relativity, we do not want to add extra degrees of freedom, apart from the ones contained in the vielbein. In this case, we will thus set the curvature ${F}_{\mu \nu }(M)$ to zero so that M_μ corresponds to a pure gauge-field. Another example where we will constrain ${F}_{\mu \nu }(M)$ to be zero, will appear when we discuss on-shell supersymmetry. There, this constraint will ensure that the equality of bosonic and fermionic degrees of freedom in the relativistic multiplet is not upset, after M_μ is added by hand to an existing on-shell multiplet. Even if we are only interested in the kinematics we cannot allow for a completely arbitrary ${F}_{\mu \nu }(M).$ As we will see in the following we have to constrain the spatial projection of ${F}_{\mu \nu }(M)$ to be zero in order to take the non-relativistic limit in a consistent manner. In particular, this will be crucial to obtain finite expressions for the non-relativistic dependent spin-connections as limits of the relativistic one.

In the following section, we will apply the limiting procedure to the formulas collected above.

2.2.2. Newton–Cartan gravity from relativistic gravity

In this first example, we will extend the contraction that gives the Bargmann algebra from the Poincaré algebra, to the vielbein and spin-connection of general relativity, along the lines of section 2.1. As explained above, the contraction and its extension involve redefining generators and fields using a contraction parameter ω. We will be careful in distinguishing quantities that are merely redefined relativistic ones, for which ω is finite, from non-relativistic ones, that are obtained in the limit $\omega \to \infty .$ In particular, we will denote the former ones with a tilde, whereas for the latter the tilde will be dropped.

Let us first briefly recall the Inönü–Wigner contraction of the Poincaré algebra to the Bargmann algebra. The contraction is best described by starting from a direct sum of the Poincaré algebra (with translation generators ${\hat{P}}_{A}$ and Lorentz generators M_AB) with an abelian factor (with generator ${\mathcal{Z}}$)⁵ . Starting from the Poincaré algebra

$$\begin{eqnarray}&&\left[{\hat{P}}_{A},{M}_{{BC}}\right]=2\;{\eta }_{A[B}\;{\hat{P}}_{C]},\left[{M}_{{AB}},{M}_{{CD}}\right]=4\;{\eta }_{[A[C}\;{M}_{D]B]},\end{eqnarray} \tag{ 12 }$$

supplemented with the generator ${\mathcal{Z}},$ we redefine the generators using a contraction parameter ω as follows⁶

$$\begin{eqnarray}&&{\hat{P}}_{0}\to \displaystyle \frac{1}{2\omega }\;\tilde{H}+\omega \;\tilde{Z},\quad {\mathcal{Z}}\to \displaystyle \frac{1}{2\omega }\;\tilde{H}-\omega \;\tilde{Z},\quad {M}_{a0}\to \omega \;{\tilde{G}}_{a},\end{eqnarray} \tag{ 13 }$$

where we have split the spacetime index A into a time-like 0-index and spatial $a,b$-indices. Note that the spatial translations ${\hat{P}}_{a}$ and spatial rotations M_ab are left untouched, i.e. ${\hat{P}}_{a}={\tilde{P}}_{a}$ and ${M}_{{ab}}={\tilde{M}}_{{ab}}.$ We will in the following denote ${\tilde{M}}_{{ab}}={\tilde{J}}_{{ab}}$ to conform to earlier literature. Calculating commutators of $\tilde{H},$ ${\tilde{P}}_{a},$ ${\tilde{G}}_{a},$ ${\tilde{J}}_{{ab}},$ $\tilde{Z},$ re-expressing the result in terms of the same generators and taking $\omega \to \infty ,$ we obtain the Bargmann algebra with the non-vanishing commutators

$$\begin{eqnarray}\left[{P}_{a},{J}_{{bc}}\right] & = & 2\;{\delta }_{a[b}\;{P}_{c]},\quad \left[{J}_{{ab}},{J}_{{cd}}\right]=4\;{\delta }_{[a[c}\;{J}_{d]b]},\\ \left[{G}_{a},{J}_{{bc}}\right] & = & 2\;{\delta }_{a[b}\;{G}_{c]},\quad \left[H,{G}_{a}\right]={P}_{a},\quad \left[{P}_{a},{G}_{b}\right]={\delta }_{{ab}}\;Z.\end{eqnarray} \tag{ 14 }$$

In order to derive Newton–Cartan gravity, as obtained via the gauging of (14) in [23], we extend the contraction (13) to the vielbein ${E}_{\mu }{}^{A},$ spin-connection ${{\rm{\Omega }}}_{\mu }{}^{{AB}}(E)$ and abelian gauge-field M_μ of the previous section. In particular, we make the following redefinition for the relativistic vielbein:

$$\begin{eqnarray}&&{E}_{\mu }{}^{A}={\delta }_{0}^{A}\left(\omega \;{\tilde{\tau }}_{\mu }+\displaystyle \frac{1}{2\omega }\;{\tilde{m}}_{\mu }\right)+{\delta }_{a}^{A}\;{\tilde{e}}_{\mu }{}^{a},\end{eqnarray} \tag{ 15 }$$

where ${\tilde{\tau }}_{\mu },$ ${\tilde{e}}_{\mu }{}^{a}$ and ${\tilde{m}}_{\mu }$ will be identified, in the limit $\omega \to \infty$ (where we will drop the tilde), as the independent gauge-fields of Newton–Cartan geometry and gravity. It is convenient to define fields ${\tilde{e}}^{\mu }{}_{a},$ ${\tilde{\tau }}^{\mu }$ as follows:

$$\begin{eqnarray}&&{\tilde{e}}^{\mu }{}_{a}{\tilde{e}}_{\mu }{}^{b}={\delta }_{a}^{b},{\tilde{\tau }}^{\mu }{\tilde{\tau }}_{\mu }=1,{\tilde{\tau }}^{\mu }{\tilde{e}}_{\mu }{}^{a}={\tilde{\tau }}_{\mu }{\tilde{e}}^{\mu }{}_{a}=0,{\tilde{e}}^{\rho }{}_{a}{\tilde{e}}_{\mu }{}^{a}={\delta }_{\mu }^{\rho }-{\tilde{\tau }}_{\mu }{\tilde{\tau }}^{\rho }.\end{eqnarray} \tag{ 16 }$$

In the limit $\omega \to \infty ,$ the fields ${\tau }_{\mu },{\tau }^{\mu },{e}_{\mu }{}^{a}$ and ${e}^{\mu }{}_{a}$ can be used to define two separate non-degenerate Galilei-invariant metrics, one in the time direction and one in the three spatial directions. These will not be needed in the present discussion. Using the defining relations (16) the following expansion of the relativistic inverse vielbein is obtained

$$\begin{eqnarray}&&{E}^{\mu }{}_{A}={\delta }_{A}^{a}\left[{\tilde{e}}^{\mu }{}_{a}+{\mathcal{O}}\left(\displaystyle \frac{1}{{\omega }^{2}}\right)\right]+\displaystyle \frac{1}{\omega }\;{\delta }_{A}^{0}\left[{\tilde{\tau }}^{\mu }+{\mathcal{O}}\left(\displaystyle \frac{1}{{\omega }^{2}}\right)\right].\end{eqnarray} \tag{ 17 }$$

Note that in this expression, we have only explicitly given the terms of leading order in ω. There are in principle an infinite number of corrections of lower order in ω, that we have denoted by ${\mathcal{O}}\left((,1/{\omega }^{2},)\right)$ and that will not be needed in the following (as they will not contribute in the $\omega \to \infty$ limit).

The abelian gauge-field M_μ is redefined as follows

$$\begin{eqnarray}&&{M}_{\mu }=\omega \;{\tilde{\tau }}_{\mu }-\displaystyle \frac{1}{2\omega }\;{\tilde{m}}_{\mu }.\end{eqnarray} \tag{ 18 }$$

The relativistic spin-connection ${{\rm{\Omega }}}_{\mu }{}^{{AB}}(E)$ is a dependent field, determined by the torsion constraint (6). As already mentioned under equation (11), we now have to impose a constraint on the curvature of the relativistic gauge-field M_μ in order to lower the powers of ω in certain terms in the expression for ${{\rm{\Omega }}}_{\mu }{}^{{AB}}(E),$ see equation (25) below, such that the limit can be taken. In particular we mentioned the following choices

$$\begin{eqnarray}&&\mathrm{dynamical}\;:\quad {F}_{\mu \nu }(M)=0,\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&\mathrm{kinematical}\;:\quad {\tilde{e}}^{\mu }{}_{a}{\tilde{e}}^{\nu }{}_{b}\;{F}_{\mu \nu }(M)={\tilde{F}}_{{ab}}(M)=0.\end{eqnarray} \tag{ 20 }$$

The last choice is the least restrictive one and is sufficient to take the $\omega \to \infty$ limit in a consistent manner, if one is only interested in geometrical and kinematical aspects. The first choice represents a stronger condition and should be adopted when one is also interested in taking the non-relativistic limit of dynamical aspects of general relativity. In particular, this constraint implies that M_μ is pure gauge and does not represent extra degrees of freedom in the parent relativistic theory.

We can then define spin- and boost-connections ${\tilde{\omega }}_{\mu }{}^{{ab}}(\tilde{e},\tilde{\tau },\tilde{m}),$ ${\tilde{\omega }}_{\mu }{}^{a}(\tilde{e},\tilde{\tau },\tilde{m}),$ that will be identified with the non-relativistic ones when $\omega \to \infty ,$ as the coefficients of the terms of leading order in an ω-expansion of ${{\rm{\Omega }}}_{\mu }{}^{{AB}}(E):$

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{\mu }{}^{{ab}}(E)={\tilde{\omega }}_{\mu }{}^{{ab}}(\tilde{e},\tilde{\tau },\tilde{m})+{\mathcal{O}}\left(\displaystyle \frac{1}{{\omega }^{2}}\right),\end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{\mu }{}^{0a}(E)=\displaystyle \frac{1}{\omega }\;{\tilde{\omega }}_{\mu }{}^{a}(\tilde{e},\tilde{\tau },\tilde{m})+{\mathcal{O}}\left(\displaystyle \frac{1}{{\omega }^{3}}\right),\end{eqnarray} \tag{ 22 }$$

where

$$\begin{eqnarray}&&{\tilde{\omega }}_{\mu }{}^{{ab}}(\tilde{e},\tilde{\tau },\tilde{m})=-2\;{\tilde{e}}^{\nu [a}{\partial }_{[\mu }{\tilde{e}}_{\nu ]}{}^{b]}+{\tilde{e}}_{\mu }{}^{c}{\tilde{e}}^{\rho a}{\tilde{e}}^{\nu b}{\partial }_{[\rho }{\tilde{e}}_{\nu ]}{}^{c}-{\tilde{\tau }}_{\mu }{\tilde{e}}^{\rho a}{\tilde{e}}^{\nu b}{\partial }_{[\rho }{\tilde{m}}_{\nu ]},\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&{\tilde{\omega }}_{\mu }{}^{a}(\tilde{e},\tilde{\tau },\tilde{m})={\tilde{\tau }}^{\nu }{\partial }_{[\mu }{\tilde{e}}_{\nu ]}{}^{a}+{\tilde{e}}_{\mu }{}^{b}{\tilde{e}}^{\rho a}{\tilde{\tau }}^{\nu }{\partial }_{[\rho }{\tilde{e}}_{\nu ]}{}^{b}+{\tilde{e}}^{\nu a}{\partial }_{[\mu }{\tilde{m}}_{\nu ]}-{\tilde{\tau }}_{\mu }{\tilde{e}}^{\rho a}{\tilde{\tau }}^{\nu }{\partial }_{[\rho }{\tilde{m}}_{\nu ]}.\end{eqnarray} \tag{ 24 }$$

Note that to obtain these formulas, one only needs the kinematical constraint (20) in the form

$$\begin{eqnarray}&&\omega \;{\tilde{e}}^{\mu }{}_{a}{\tilde{e}}^{\nu }{}_{b}\;{\partial }_{[\mu }{\tilde{\tau }}_{\nu ]}=\displaystyle \frac{1}{2\omega }\;{\tilde{e}}^{\mu }{}_{a}{\tilde{e}}^{\nu }{}_{b}\;{\partial }_{[\mu }{\tilde{m}}_{\nu ]},\end{eqnarray} \tag{ 25 }$$

to replace terms that diverge in the $\omega \to \infty$ limit by terms that have the correct leading ω-order as indicated in the expansions (21) and (22). Since the stronger constraint (19) implies (25), it will achieve the same goal. The subleading terms in (21) and (22) are due to the fact that the relativistic spin-connection ${{\rm{\Omega }}}_{\mu }{}^{{AB}}(E)$ depends on the inverse vielbein ${E}^{\mu }{}_{A}.$

The rationale behind the redefinitions (15)–(22) is that they leave the sum of the products of the gauge-fields with their respective generators invariant, up to subleading terms in ω, that stem from the dependent spin-connection via (21) and (22). One thus has:

$$\begin{eqnarray}&&{\hat{P}}_{A}\;{E}_{\mu }{}^{A}+{\mathcal{Z}}\;{M}_{\mu }+{M}_{{AB}}\;{{\rm{\Omega }}}_{\mu }{}^{{AB}}(E)\\ &&\quad ={\tilde{P}}_{a}\;{\tilde{e}}_{\mu }{}^{a}+\tilde{H}\;{\tilde{\tau }}_{\mu }+\tilde{Z}\;{\tilde{m}}_{\mu }+{\tilde{J}}_{{ab}}\;{\tilde{\omega }}_{\mu }{}^{{ab}}(\tilde{e},\tilde{\tau },\tilde{m})\\ &&\quad -2\;{\tilde{G}}_{a}\;{\tilde{\omega }}_{\mu }{}^{a}(\tilde{e},\tilde{\tau },\tilde{m})+{\mathcal{O}}\left(\displaystyle \frac{1}{{\omega }^{2}}\right).\end{eqnarray} \tag{ 26 }$$

We proceed by taking the limit $\omega \to \infty$ and derive the kinematics of Newton–Cartan gravity. For example, dropping the tildes on all fields we see that the expressions (23) and (24) coincide with the expressions of [23], which were obtained by setting

$$\begin{eqnarray}{R}_{\mu \nu }{}^{a}(P) & = & 2\;{\partial }_{[\mu }{e}_{\nu ]}{}^{a}-2\;{\omega }_{[\mu }{}^{{ab}}{e}_{\nu ]}{}^{b}-2\;{\omega }_{[\mu }{}^{a}{\tau }_{\nu ]}=0,\\ {R}_{\mu \nu }(Z) & = & 2\;{\partial }_{[\mu }{m}_{\nu ]}-2\;{\omega }_{[\mu }{}^{a}{e}_{\nu ]}{}^{a}=0.\end{eqnarray} \tag{ 27 }$$

These constraints are thus satisfied identically. Further constraints can be derived from the relativistic Bianchi identity (8) and the constraints on the curvature of the relativistic gauge-field M_μ, see equation (19) or (20). Using the inverse vielbein (17) and the expansion

$$\begin{eqnarray}&&{R}_{\mu \nu }{}^{{AB}}({\rm{\Omega }})={\delta }_{a}^{A}{\delta }_{b}^{B}\;{\tilde{R}}_{\mu \nu }{}^{{ab}}(\tilde{J})-\displaystyle \frac{1}{\omega }\;{\delta }_{a}^{A}{\delta }_{0}^{B}\;{\tilde{R}}_{\mu \nu }{}^{a}(\tilde{G})+\displaystyle \frac{1}{\omega }\;{\delta }_{0}^{A}{\delta }_{b}^{B}\;{\tilde{R}}_{\mu \nu }{}^{b}(\tilde{G}),\end{eqnarray} \tag{ 28 }$$

in (8) we obtain the non-relativistic Bianchi identities

$$\begin{eqnarray}&&{R}_{[\mu \nu }{}^{a}(G){e}_{\rho ]}{}^{a}=0,\quad {R}_{[\mu \nu }{}^{{ab}}(J){e}_{\rho ]}{}^{b}+{R}_{[\mu \nu }{}^{a}(G){\tau }_{\rho ]}=0,\end{eqnarray} \tag{ 29 }$$

where the curvatures of the spin- and boost-connection gauge-fields are given by

$$\begin{eqnarray}{R}_{\mu \nu }{}^{{ab}}(J) & = & 2\;{\partial }_{[\mu }{\omega }_{\nu ]}{}^{{ab}}-2\;{\omega }_{[\mu }{}^{a}{}_{c}\;{\omega }_{\nu ]}{}^{{cb}},\\ {R}_{\mu \nu }{}^{a}(G) & = & 2\;{\partial }_{[\mu }{\omega }_{\nu ]}{}^{a}-2\;{\omega }_{[\mu }{}^{{ab}}\;{\omega }_{\nu ]}{}^{b}.\end{eqnarray} \tag{ 30 }$$

We consider the implications of equations (19) and (20) separately. In the first case (19), we simply find

$$\begin{eqnarray}&&{R}_{\mu \nu }(H)=2\;{\partial }_{[\mu }{\tau }_{\nu ]}=0.\end{eqnarray} \tag{ 31 }$$

In the kinematical case (20), we find the less restrictive condition

$$\begin{eqnarray}&&{R}_{{ab}}(H)=2\;{e}^{\mu }{}_{a}{e}^{\nu }{}_{b}\;{\partial }_{[\mu }{\tau }_{\nu ]}=0.\end{eqnarray} \tag{ 32 }$$

Interestingly, this constraint is equivalent to writing

$$\begin{eqnarray}&&{\partial }_{[\mu }{\tau }_{\nu ]}={b}_{[\mu }{\tau }_{\nu ]},\end{eqnarray} \tag{ 33 }$$

where b_μ is completely arbitrary. This resembles the constraint found in the twistless torsional Newton–Cartan geometry of [25].

In either of the two cases, we can show that no further constraints are obtained by applying symmetry transformations on the constraints (31) and (32). To do so we need to derive the transformation rules of the Newton–Cartan fields ${\tau }_{\mu },$ ${e}_{\mu }{}^{a}$ and m_μ. This can be done by applying the relativistic transformation rules (2) and (10) to the decompositions (15) and (18). For this purpose, we first express the new fields in terms of the old ones, i.e.

$$\begin{eqnarray}&&{\tilde{\tau }}_{\mu }=\displaystyle \frac{1}{2\omega }\left({E}_{\mu }{}^{0}+{M}_{\mu }\right),{\tilde{m}}_{\mu }=\omega \left({E}_{\mu }{}^{0}-{M}_{\mu }\right).\end{eqnarray} \tag{ 34 }$$

Having done this, it is straightforward to obtain

$$\begin{eqnarray}\delta {\tau }_{\mu } & = & 0,\\ \delta {e}_{\mu }{}^{a} & = & {\lambda }^{a}{}_{b}\;{e}_{\mu }{}^{b}+{\lambda }^{a}{\tau }_{\mu },\\ \delta {m}_{\mu } & = & {\partial }_{\mu }\sigma +{\lambda }_{a}\;{e}_{\mu }{}^{a},\end{eqnarray} \tag{ 35 }$$

where we defined

$$\begin{eqnarray}&&{\lambda }^{a}=\omega \;{\lambda }^{a}{}_{0},{\rm{\Lambda }}=-\displaystyle \frac{\sigma }{\omega },\end{eqnarray} \tag{ 36 }$$

in agreement with equations (22) and (18). All fields transform under diffeomorphisms in the usual way. The transformations of the spin-connections can be found as well:

$$\begin{eqnarray}\delta {\omega }_{\mu }{}^{{ab}}(e,\tau ,m) & = & {\partial }_{\mu }{\lambda }^{{ab}}+2\;{\lambda }^{[a}{}_{c}\;{\omega }_{\mu }{}^{{cb}]},\\ \delta {\omega }_{\mu }{}^{a}(e,\tau ,m) & = & {\partial }_{\mu }{\lambda }^{a}+{\lambda }^{a}{}_{b}\;{\omega }_{\mu }{}^{b}-{\omega }_{\mu }{}^{a}{}_{c}\;{\lambda }^{c}.\end{eqnarray} \tag{ 37 }$$

The transformations (35)–(37) together with the constraints (31) and (29) make up the Newton–Cartan theory of gravity as described in [23].

At this point, we may impose equations of motion on the Newton–Cartan gauge-fields, in addition to the constraint (31) or (32), for example by performing the limiting procedure on the Einstein equations. One can show that this leads to the equation of motion presented in [23].

In this section we will extend the previous example to the three-dimensional Newton–Cartan supergravity theory constructed in [24]. The reason that we work in three-dimensions is that this is the only dimension in which an example of an on-shell Newton–Cartan supergravity theory is known so far. Since the discussion mainly parallels the previous section, we will skip most intermediate steps, where the contraction parameter ω is finite, and we will mostly focus on the results obtained in the $\omega \to \infty$ limit. Here and in the following, we will therefore no longer resort to the notation using tildes, to denote quantities at finite ω.

The underlying gauge algebra in this case is the ${\mathcal{N}}=2$ Bargmann superalgebra. This superalgebra can be obtained by contracting the ${\mathcal{N}}=2$ Poincaré superalgebra, with central extension ${\mathcal{Z}},$ that is given by

$$\begin{eqnarray}\left[{M}_{{AB}},{\hat{P}}_{C}\right] & = & -2\;{\eta }_{C[A}{\hat{P}}_{B]},\quad \left[{M}_{{AB}},{M}_{{CD}}\right]=4\;{\eta }_{[A[C}{M}_{D]B]},\\ \left[{M}_{{AB}},{Q}^{i}\right] & = & -\displaystyle \frac{1}{2}\;{\gamma }_{{AB}}{Q}^{i},\quad \left\{{Q}^{i},{Q}^{j}\right\}=-{\gamma }^{A}{C}^{-1}\;{\hat{P}}_{A}\;{\delta }^{{ij}}+{C}^{-1}\;{\mathcal{Z}}\;{\epsilon }^{{ij}}.\end{eqnarray} \tag{ 38 }$$

Here, the supercharges ${Q}^{i}\ (i=1,2)$ are two-component Majorana spinors. For the gamma-matrices we choose a real basis, i.e. ${\gamma }^{A}=({\rm{i}}{\sigma }_{2},{\sigma }_{1},{\sigma }_{3})$ and the charge conjugation matrix is taken to be $C={\rm{i}}{\gamma }^{0}.$ In order to define the Inönü–Wigner contraction, we first define the combinations

$$\begin{eqnarray}&&{Q}_{\pm }=\displaystyle \frac{1}{\sqrt{2}}\left({Q}_{1}\pm {\gamma }_{0}{Q}_{2}\right),\end{eqnarray} \tag{ 39 }$$

and split the three-dimensional flat indices A, B into time-like and space-like indices $\{0,a\}.$ As before, we set ${M}_{{ab}}={J}_{{ab}}$ for the purely spatial rotations. The motivation for choosing the combinations of the relativistic spinors as given in equation (39), stems from the non-relativistic algebra (and later on from the transformation rules of the gravitini). It leads to particularly simple transformations of the spinors under boosts. Before making the contraction we perform the following redefinition of the generators:

$$\begin{eqnarray}{Q}_{-} & \to & \sqrt{\omega }\;{Q}_{-},\quad {Q}_{+}\to \displaystyle \frac{1}{\sqrt{\omega }}\;{Q}_{+},\quad {M}_{a0}\to \omega \;{G}_{a},\\ {\mathcal{Z}} & \to & -\omega Z+\displaystyle \frac{1}{2\omega }\;H,\quad {\hat{P}}_{0}\to \omega Z+\displaystyle \frac{1}{2\omega }\;H.\end{eqnarray} \tag{ 40 }$$

Using these redefinitions, the supersymmetric extension of the Bargmann algebra is then obtained in the limit $\omega \to \infty ,$ in a similar way as discussed in the previous subsection. In particular, we find the following non-vanishing commutation relations:

$$\begin{eqnarray}\left[{J}_{{ab}},{P}_{c}\right] & = & -2\;{\delta }_{c[a}{P}_{b]},\quad \left[{J}_{{ab}},{G}_{c}\right]=-2\;{\delta }_{c[a}{G}_{b]},\\ \left[{G}_{a},H\right] & = & -{P}_{a},\qquad \left[{G}_{a},{P}_{b}\right]=-{\delta }_{{ab}}\;Z,\\ \left[{J}_{{ab}},{Q}_{\pm }\right] & = & -\frac{1}{2}\;{\gamma }_{{ab}}{Q}_{\pm },\quad \left[{G}_{a},{Q}_{+}\right]=-\frac{1}{2}\;{\gamma }_{a0}{Q}_{-},\\ \left\{{Q}_{+},{Q}_{+}\right\} & = & -{\gamma }^{0}{C}^{-1}\;H,\quad \left\{{Q}_{+},{Q}_{-}\right\}=-{\gamma }^{a}{C}^{-1}\;{P}_{a},\\ \left\{{Q}_{-},{Q}_{-}\right\} & = & -2\;{\gamma }^{0}{C}^{-1}\;Z.\end{eqnarray} \tag{ 41 }$$

The bosonic part of the algebra corresponds to the Bargmann algebra, see equation (14). Note that, since we are working in three-dimensions, the spatial rotations are abelian.

We now wish to extend this contraction to the fields of the on-shell, relativistic ${\mathcal{N}}=2$ supergravity multiplet, whose supersymmetry transformation rules (with parameter ${\eta }_{i}$) are given by

$$\begin{eqnarray}&&\delta {E}_{\mu }{}^{A}=\displaystyle \frac{1}{2}\;{\delta }^{{ij}}\;{\bar{\eta }}_{i}\;{\gamma }^{A}{{\rm{\Psi }}}_{\mu j},\end{eqnarray} \tag{ 42 }$$

$$\begin{eqnarray}&&\delta {{\rm{\Psi }}}_{\mu i}={D}_{\mu }{\eta }_{i}={\partial }_{\mu }{\eta }_{i}-\displaystyle \frac{1}{4}\;{{\rm{\Omega }}}_{\mu }{}^{{AB}}(E,{{\rm{\Psi }}}_{i}){\gamma }_{{AB}}{\eta }_{i},\end{eqnarray} \tag{ 43 }$$

where D_μ is the Lorentz-covariant derivative and the dependent spin-connection ${{\rm{\Omega }}}_{\mu }{}^{{AB}}(E,{{\rm{\Psi }}}_{i})$ is given by the supersymmetric analog of (7), i.e.

$$\begin{eqnarray}{{\rm{\Omega }}}_{\mu }{}^{{AB}}(E,{{\rm{\Psi }}}_{i}) & = & -2\;{E}^{\rho [A}\left({\partial }_{[\mu }{E}_{\rho ]}{}^{B]}-\displaystyle \frac{1}{4}\;{\delta }^{{ij}}\;{\bar{{\rm{\Psi }}}}_{[\mu i}{\gamma }^{B]}{{\rm{\Psi }}}_{\nu ]j}\right)\\ & & +{E}_{\mu C}{E}^{\rho A}{E}^{\nu B}\left({\partial }_{[\rho }{E}_{\nu ]}{}^{C}-\displaystyle \frac{1}{4}\;{\delta }^{{ij}}\;{\bar{{\rm{\Psi }}}}_{[\rho i}{\gamma }^{C}{{\rm{\Psi }}}_{\nu ]j}\right).\end{eqnarray} \tag{ 44 }$$

From this expression one derives that the supersymmetry transformation of the (dependent) spin-connection is given by

$$\begin{eqnarray}&&\delta {{\rm{\Omega }}}_{\mu }{}^{{AB}}(E,{{\rm{\Psi }}}_{i})=-\displaystyle \frac{1}{2}\;{\delta }^{{ij}}\;{E}^{\rho [A}\;{\bar{\eta }}_{i}\;{\gamma }^{B]}{\hat{{\rm{\Psi }}}}_{\mu \rho j}+\displaystyle \frac{1}{4}\;{\delta }^{{ij}}\;{E}_{\mu C}{E}^{\rho A}{E}^{\nu B}\;{\bar{\eta }}_{i}\;{\gamma }^{C}\;{\hat{{\rm{\Psi }}}}_{\rho \nu j}.\end{eqnarray} \tag{ 45 }$$

Note that this transformation rule is zero on-shell, i.e. it vanishes upon using the fermionic equations of motion

$$\begin{eqnarray}&&{\hat{{\rm{\Psi }}}}_{\mu \nu i}=2\;{D}_{[\mu }{{\rm{\Psi }}}_{\nu ]i}=0.\end{eqnarray} \tag{ 46 }$$

One may verify that the supersymmetry algebra on the fields (42) and (43) closes on-shell.

As in the previous subsection, we will introduce a field M_μ, associated to the central charge transformation ${\mathcal{Z}}$ of the ${\mathcal{N}}=2$ algebra. Its transformation rule under supersymmetry is determined by the Poincaré superalgebra (38)

$$\begin{eqnarray}&&\delta {M}_{\mu }=\displaystyle \frac{1}{2}\;{\varepsilon }^{{ij}}\;{\bar{\eta }}_{i}{{\rm{\Psi }}}_{\mu j}.\end{eqnarray} \tag{ 47 }$$

This field is ordinarily not introduced in the supergravity multiplet. In order not to upset the on-shell counting of bosonic and fermionic degrees of freedom, we are thus obliged to set the supercovariant curvature of M_μ to zero, i.e.

$$\begin{eqnarray}&&{\hat{F}}_{\mu \nu }(M)=2\;{\partial }_{[\mu }{M}_{\nu ]}-\displaystyle \frac{1}{2}\;{\varepsilon }^{{ij}}\;{\bar{{\rm{\Psi }}}}_{[\mu i}{{\rm{\Psi }}}_{\nu ]j}=0,\end{eqnarray} \tag{ 48 }$$

so that this field corresponds to a pure gauge degree of freedom. Note that this constraint also implies that the commutator of two supersymmetry transformations acting on M_μ closes to a general coordinate transformation and a central charge transformation. Moreover, since this constraint is the supercovariant version of (19), it will allow us to obtain finite expressions for the non-relativistic spin-connections from the relativistic one. Starting from expression (48), the full set of relativistic equations of motion is obtained by the following chain of supersymmetry transformations

$$\begin{eqnarray}&&{\hat{F}}_{\mu \nu }(M)=0\quad \to \quad {\hat{{\rm{\Psi }}}}_{\mu \nu i}=0\quad \to \quad {\hat{R}}_{\mu \nu }{}^{{AB}}({\rm{\Omega }})=0.\end{eqnarray} \tag{ 49 }$$

This concludes the summary of our relativistic starting point. Let us now extend the algebra contraction to the fields of this on-shell supergravity multiplet. For the bosonic fields, this entails the redefinitions involving ω that were introduced in the previous section. The redefinitions of the gravitini follow from the way we contract the generators of the three-dimensional ${\mathcal{N}}=2$ Poincaré superalgebra to get the Bargmann superalgebra, see the definitions (39). Hence, we define new spinors

$$\begin{eqnarray}&&{{\rm{\Psi }}}_{\pm }=\displaystyle \frac{1}{\sqrt{2}}\left({{\rm{\Psi }}}_{1}\pm {\gamma }_{0}{{\rm{\Psi }}}_{2}\right),\end{eqnarray} \tag{ 50 }$$

and similarly for the parameters η. We then introduce the scalings:

$$\begin{eqnarray}{{\rm{\Psi }}}_{\mu +} & = & \sqrt{\omega }\;{\psi }_{\mu +},\quad {\eta }_{+}=\sqrt{\omega }\;{\epsilon }_{+},\\ {{\rm{\Psi }}}_{\mu -} & = & \displaystyle \frac{1}{\sqrt{\omega }}\;{\psi }_{\mu -},\quad {\eta }_{-}=\displaystyle \frac{1}{\sqrt{\omega }}\;{\epsilon }_{-}.\end{eqnarray} \tag{ 51 }$$

The following non-relativistic supersymmetry transformation rules then follow

$$\begin{eqnarray}\delta {\tau }_{\mu } & = & \displaystyle \frac{1}{2}\;{\bar{\epsilon }}_{+}{\gamma }^{0}{\psi }_{\mu +},\\ \delta {e}_{\mu }{}^{a} & = & \displaystyle \frac{1}{2}\;{\bar{\epsilon }}_{+}{\gamma }^{a}{\psi }_{\mu -}+\displaystyle \frac{1}{2}\;{\bar{\epsilon }}_{-}{\gamma }^{a}{\psi }_{\mu +},\\ \delta {m}_{\mu } & = & {\bar{\epsilon }}_{-}{\gamma }^{0}{\psi }_{\mu -},\end{eqnarray} \tag{ 52 }$$

as well as

$$\begin{eqnarray}\delta {\psi }_{\mu +} & = & {\partial }_{\mu }{\epsilon }_{+}-\displaystyle \frac{1}{4}\;{\omega }_{\mu }{}^{{ab}}{\gamma }_{{ab}}{\epsilon }_{+},\\ \delta {\psi }_{\mu -} & = & {\partial }_{\mu }{\epsilon }_{-}-\displaystyle \frac{1}{4}\;{\omega }_{\mu }{}^{{ab}}{\gamma }_{{ab}}{\epsilon }_{-}+\displaystyle \frac{1}{2}\;{\omega }_{\mu }{}^{a}{\gamma }_{a0}{\epsilon }_{+}.\end{eqnarray} \tag{ 53 }$$

The transformation rules of the spinors with respect to the non-relativistic bosonic symmetries stem from the relativistic rule $\delta {{\rm{\Psi }}}_{\mu }=1/4\;{\lambda }^{{AB}}{\gamma }_{{AB}}{{\rm{\Psi }}}_{\mu }$ and are found to be

$$\begin{eqnarray}\delta {\psi }_{\mu +} & = & \displaystyle \frac{1}{4}\;{\lambda }^{{ab}}{\gamma }_{{ab}}{\psi }_{\mu +},\\ \delta {\psi }_{\mu -} & = & \displaystyle \frac{1}{4}\;{\lambda }^{{ab}}{\gamma }_{{ab}}{\psi }_{\mu -}-\displaystyle \frac{1}{2}\;{\lambda }^{a}{\gamma }_{a0}{\psi }_{\mu +}.\end{eqnarray} \tag{ 54 }$$

It is understood that the spin-connections ${\omega }_{\mu }{}^{a},{\omega }_{\mu }{}^{{ab}}$ in (53) are dependent, i.e. ${\omega }_{\mu }{}^{a}={\omega }_{\mu }{}^{a}(e,\tau ,m,{\psi }_{\pm })$ and ${\omega }_{\mu }{}^{{ab}}={\omega }_{\mu }{}^{{ab}}(e,\tau ,m,{\psi }_{\pm }).$ The expressions for these non-relativistic spin-connections can be obtained from the relativistic expressions given in equations (44) and (48). We find

$$\begin{eqnarray}{\omega }_{\mu }{}^{{ab}}(e,\tau ,m,{\psi }_{\pm }) & = & -2\;{e}^{\nu [a}\left({\partial }_{[\mu }{e}_{\nu ]}{}^{b]}-\displaystyle \frac{1}{2}\;{\bar{\psi }}_{[\mu +}{\gamma }^{b]}{\psi }_{\nu ]-}\right)\\ & & +{e}_{\mu }{}^{c}{e}^{\rho a}{e}^{\nu b}\left({\partial }_{[\rho }{e}_{\nu ]}{}^{c}-\displaystyle \frac{1}{2}\;{\bar{\psi }}_{[\rho +}{\gamma }^{c}{\psi }_{\nu ]-}\right)\\ & & -{\tau }_{\mu }{e}^{\rho a}{e}^{\nu b}\left({\partial }_{[\rho }{m}_{\nu ]}-\displaystyle \frac{1}{2}\;{\bar{\psi }}_{[\rho -}{\gamma }^{0}{\psi }_{\nu ]-}\right),\end{eqnarray} \tag{ 55 }$$

$$\begin{eqnarray}{\omega }_{\mu }{}^{a}(e,\tau ,m,{\psi }_{\pm }) & = & {\tau }^{\nu }\left({\partial }_{[\mu }{e}_{\nu ]}{}^{a}-\displaystyle \frac{1}{2}\;{\bar{\psi }}_{[\mu +}{\gamma }^{a}{\psi }_{\nu ]-}\right)\\ & & +{e}_{\mu b}{e}^{\rho a}{\tau }^{\nu }\left({\partial }_{[\rho }{e}_{\nu ]}{}^{b}-\displaystyle \frac{1}{2}\;{\bar{\psi }}_{[\rho +}{\gamma }^{b}{\psi }_{\nu ]-}\right)\\ & & +{e}^{\nu a}\left({\partial }_{[\mu }{m}_{\nu ]}-\displaystyle \frac{1}{2}\;{\bar{\psi }}_{[\mu -}{\gamma }^{0}{\psi }_{\nu ]-}\right)\\ & & -{\tau }_{\mu }{e}^{\rho a}{\tau }^{\nu }\left({\partial }_{[\rho }{m}_{\nu ]}-\displaystyle \frac{1}{2}\;{\bar{\psi }}_{[\rho -}{\gamma }^{0}{\psi }_{\nu ]-}\right).\end{eqnarray} \tag{ 56 }$$

In order to obtain these expression, we have mimicked the discussion around equations (21)–(25). This time however, we have used equation (48) to replace terms that diverge in the $\omega \to \infty$ limit, by terms with the expected ω-order. Like in the bosonic case, the above expressions for the spin-connections identically solve the supercovariant curvature constraints

$$\begin{eqnarray}{\hat{R}}_{\mu \nu }{}^{a}(P) & = & {R}_{\mu \nu }{}^{a}(P)-{\bar{\psi }}_{[\mu +}{\gamma }^{a}{\psi }_{\nu ]-}=0,\\ {\hat{R}}_{\mu \nu }(Z) & = & {R}_{\mu \nu }(Z)-{\bar{\psi }}_{[\mu -}{\gamma }^{0}{\psi }_{\nu ]-}=0.\end{eqnarray} \tag{ 57 }$$

The so-called conventional constraints (57) are identically fulfilled, so we need not worry about variations thereof. They can be used to determine the transformations of the spin- and boost-connections (55) and (56).

The $\omega \to \infty$ limit of (48) leads to the further constraint

$$\begin{eqnarray}&&{\hat{R}}_{\mu \nu }(H)={R}_{\mu \nu }(H)-\displaystyle \frac{1}{2}\;{\bar{\psi }}_{[\mu +}{\gamma }^{0}{\psi }_{\nu ]+}=0.\end{eqnarray} \tag{ 58 }$$

This leads to further conditions upon variation under supersymmetry. To check this we need to know the supersymmetry variations of the spin-connections. Using the transformation rules (52) and (53) in the expressions (55), (56) we find

$$\begin{eqnarray}{\delta }_{Q}{\omega }_{\mu }{}^{{ab}}(e,\tau ,m,{\psi }_{\pm }) & = & \displaystyle \frac{1}{2}\;{\bar{\epsilon }}_{+}{\gamma }^{[b}{\hat{\psi }}^{a]}{}_{\mu -}+\displaystyle \frac{1}{4}\;{e}_{\mu c}\;{\bar{\epsilon }}_{+}{\gamma }^{c}{\hat{\psi }}^{{ab}}{}_{-}-\displaystyle \frac{1}{2}\;{\tau }_{\mu }\;{\bar{\epsilon }}_{-}{\gamma }^{0}{\hat{\psi }}^{{ab}}{}_{-}\\ & & +\displaystyle \frac{1}{2}\;{\bar{\epsilon }}_{-}{\gamma }^{[b}{\hat{\psi }}^{a]}{}_{\mu +}+\displaystyle \frac{1}{4}\;{e}_{\mu c}\;{\bar{\epsilon }}_{-}{\gamma }^{c}{\hat{\psi }}^{{ab}}{}_{+},\end{eqnarray} \tag{ 59 }$$

$$\begin{eqnarray}{\delta }_{Q}{\omega }_{\mu }{}^{a}(e,\tau ,m,{\psi }_{\pm }) & = & \displaystyle \frac{1}{2}\;{\bar{\epsilon }}_{-}{\gamma }^{0}{\hat{\psi }}_{\mu }{}^{a}{}_{-}+\displaystyle \frac{1}{2}\;{\tau }_{\mu }\;{\bar{\epsilon }}_{-}{\gamma }^{0}{\hat{\psi }}_{0}{}^{a}{}_{-}+\displaystyle \frac{1}{4}\;{e}_{\mu b}\;{\bar{\epsilon }}_{+}{\gamma }^{b}{\hat{\psi }}^{a}{}_{0-}+\displaystyle \frac{1}{4}\;{\bar{\epsilon }}_{+}{\gamma }^{a}{\hat{\psi }}_{\mu 0-}\\ & & +\displaystyle \frac{1}{4}\;{e}_{\mu b}\;{\bar{\epsilon }}_{-}{\gamma }^{b}{\hat{\psi }}^{a}{}_{0+}+\displaystyle \frac{1}{4}\;{\bar{\epsilon }}_{-}{\gamma }^{a}{\hat{\psi }}_{\mu 0+}.\end{eqnarray} \tag{ 60 }$$

Now we readily derive that under supersymmetry transformations, with parameters ${\epsilon }_{+}$ and ${\epsilon }_{-},$ the following set of constraints is generated:

$$\begin{eqnarray}&&\overset{{Q}_{-}}{\longrightarrow }\quad {\hat{\psi }}_{{ab}-}=0,\end{eqnarray} \tag{ 61 }$$

$$\begin{eqnarray}&&{\hat{R}}_{\mu \nu }(H)=0\quad \overset{{Q}_{+}}{\longrightarrow }\quad {\hat{\psi }}_{\mu \nu +}=0,\quad \overset{{Q}_{+}}{\longrightarrow }\quad {R}_{\mu \nu }{}^{{ab}}(J)=0.\phantom{\rule{6em}{0ex}}\end{eqnarray} \tag{ 62 }$$

$$\begin{eqnarray}&&\overset{{Q}_{+}}{\longrightarrow }\quad {\gamma }^{a}{\hat{\psi }}_{a0-}=0\quad \overset{{Q}_{+}}{\longrightarrow }\quad {\hat{R}}_{0a}{}^{a}(G)=0\end{eqnarray} \tag{ 63 }$$

Note that the variation of the ${\hat{\psi }}_{\mu \nu +}=0$ constraint leads to three different constraints. Of these three constraints only the variation of the ${\gamma }^{a}{\hat{\psi }}_{a0-}=0$ leads to one more constraint. Using the last constraint given in equation (62) the non-relativistic Bianchi identities reduce to

$$\begin{eqnarray}&&{\hat{R}}_{{ab}}{}^{c}(G)=0,\quad {\hat{R}}_{0[a}{}^{b]}(G)=0.\end{eqnarray} \tag{ 64 }$$

These identities are e.g. needed to show that the variation of the constraint given in equation (61) does not lead to further constraints. We did not check the variation of the last constraint in equation (63). The calculation is quite involved and has also not been carried out in [24]. We will, however, show in section 3 that the full set of constraints (61)–(63) can be derived from an off-shell version of this multiplet where we have checked the consistency of the whole set of constraints.

At this point we have finished the derivation of the three-dimensional on-shell Newton–Cartan supergravity constructed in [24], i.e. we obtained all constraints and transformation rules. The terminology 'on-shell' stems from the fact that the constraints given in equation (63) both can be interpreted as equations of motion for Newton–Cartan supergravity: the first constraint is necessary to obtain closure of the supersymmetry algebra while the bosonic part of the second constraint is precisely the equation of motion of the bosonic Newton–Cartan gravity theory. Note, however, that to call some constraints 'equation of motion' and others not is slightly ambiguous when talking about Newton–Cartan (super)gravity, due to the absence of an action principle that can be used to derive these equations of motion. In section 3, we will construct a different, 'off-shell' version of three-dimensional Newton–Cartan supergravity, that includes an auxiliary scalar field in the supermultiplet. The terminology 'off-shell' will be justified in the sense that the first constraint given in equation (63) will no longer be needed for closure of the supersymmetry algebra. Both constraints given in equation (63) will in fact not appear at all. Equations of motion can thus be identified in a pragmatic way as those constraints that can be removed by adding auxiliary degrees of freedom to a non-relativistic supermultiplet.

Let us stress/repeat some important aspects of this second example. In the case at hand we can draw the commuting diagram, given in figure 1. The left column represents a chain of relativistic constraints while the right column contains a similar chain of non-relativistic constraints.

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The diagram shows all non-relativistic constraints that are obtained by a $\omega \to \infty$ limit of the relativistic ones. However, in this way we do not obtain the full set of non-relativistic constraints. This is due to the fact that in the left column we have included both supersymmetries but in the right column we have only included the variation under ${Q}_{+}$ transformations. Further constraints follow from the variation under the ${Q}_{-}$ transformations, but those non-relativistic constraints are not obtained as limits of relativistic constraints.

We note that in the limit $\omega \to \infty$ the two relativistic constraints given in the second row of the left column, namely those containing the two gravitino curvatures, lead to just the single non-relativistic constraint given in the second row of the right column. This is in line with the fact that the constraint ${\hat{R}}_{\mu \nu }(H)=0$ only varies under one of the two non-relativistic supersymmetries and hence its variation under supersymmetry only leads to one of the non-relativistic gravitino curvatures. This observation is of vital importance to understand the off-shell case treated in section 3. There we are also going to impose the constraint ${\hat{F}}_{\mu \nu }(M)=0,$ but since its non-relativistic limit does not necessarily lead to the non-relativistic equations of motion, imposing this constraint does not force us to go to the non-relativistic on-shell multiplet of the current section.

We finish this section with a third illustration of the non-relativistic limiting procedure in which we consider a superparticle in a curved background.

In this third example we apply the limiting procedure to a superparticle moving in a curved background. To be concrete, we use it to derive the action and transformation rules of the non-relativistic superparticle in a curved background, put forward in [30]. The non-relativistic superparticle in a flat background was already discussed in [35–37]. We note that the limit that was taken in [34] to derive the non-relativistic superparticle in a flat background can be understood as a special case of the analysis in this section.

It is illustrative to first discuss the bosonic particle. To derive the action of a non-relativistic bosonic point-particle in an arbitrary Newton–Cartan background we start from the relativistic action

$$\begin{eqnarray}&&{S}_{\mathrm{rel}}=-M\int {\rm{d}}\lambda \left(\sqrt{-{\eta }_{{AB}}({\dot{x}}^{\mu }{E}_{\mu }{}^{A})({\dot{x}}^{\nu }{E}_{\nu }{}^{B})}-{\dot{x}}^{\mu }{M}_{\mu }\right).\end{eqnarray} \tag{ 65 }$$

All dots refer to derivatives w.r.t. the worldline parameter λ, i.e. ${\dot{x}}^{\mu }={\rm{d}}{x}^{\mu }/{\rm{d}}\lambda .$ We use mostly plus signature and we also added a 'charge' term ${\dot{x}}^{\mu }{M}_{\mu }.$ Here, we impose that the curvature of the abelian gauge-field M_μ vanishes, implying that it can locally be written as ${M}_{\mu }={\partial }_{\mu }{\rm{\Gamma }}$ and the second term in (65) corresponds to a total derivative. Using the expressions (15) and (18) in the relativistic action (65) and taking $M=\omega \;m,$ we obtain, in the limit $\omega \to \infty ,$ the following non-relativistic action:

$$\begin{eqnarray}&&{S}_{\mathrm{nr}}=m\int {\rm{d}}\lambda \left[\displaystyle \frac{{\delta }_{{ab}}({\dot{x}}^{\mu }{e}_{\mu }{}^{a})({\dot{x}}^{\nu }{e}_{\nu }{}^{b})}{2{\tau }_{\rho }{\dot{x}}^{\rho }}-{m}_{\mu }{\dot{x}}^{\mu }\right].\end{eqnarray} \tag{ 66 }$$

This action agrees with the action, calculated by other means, in e.g. [4, 28, 38]. Note that one of the reasons to add the term ${\dot{x}}^{\mu }{M}_{\mu }$ is to cancel a divergent (total derivative) term that otherwise would arise in the limit $\omega \to \infty ,$ see also [32]. In contrast, the combination ${\dot{x}}^{\mu }{m}_{\mu }$ in the non-relativistic action is not a total derivative term. This non-relativistic term does not only follow from the relativistic ${\dot{x}}^{\mu }{M}_{\mu }$ term, but it also receives contributions from the kinematic term $\sqrt{-{\dot{x}}^{2}}.$

We now generalize the discussion of the non-relativistic bosonic particle to the non-relativistic superparticle. The relativistic superparticle in a curved background is most conveniently written using superspace techniques, see [39]. Since so far a non-relativistic superspace description is lacking, we will refrain from using superspace notation and simplify the discussion and notation by considering only the terms in the action that are at most quadratic in the fermions. Thus, the supersymmetric analog of (65) takes the form

$$\begin{eqnarray}&&{S}_{\mathrm{rel}}=-M\int {\rm{d}}\lambda \left[\sqrt{-{\eta }_{{AB}}\;{{\rm{\Pi }}}^{A}{{\rm{\Pi }}}^{B}}-\displaystyle \frac{1}{4}\;{\varepsilon }^{{ij}}\;{\bar{\theta }}_{i}{D}_{\lambda }{\theta }_{j}-{\dot{x}}^{\mu }\left({M}_{\mu }-\displaystyle \frac{1}{2}\;{\varepsilon }^{{ij}}\;{\bar{\theta }}_{i}\;{{\rm{\Psi }}}_{\mu j}\right)\right].\end{eqnarray} \tag{ 67 }$$

The background fields ${E}_{\mu }{}^{A},$ M_μ and ${{\rm{\Psi }}}_{\mu i}$ are those of the three-dimensional on-shell theory discussed in section 2.3 and the embedding coordinates are ${x}^{\mu }$ and ${\theta }_{i}.$ The supersymmetric line-element ${{\rm{\Pi }}}^{A}$ is defined as

$$\begin{eqnarray}&&{{\rm{\Pi }}}^{A}={\dot{x}}^{\mu }\left({E}_{\mu }{}^{A}-\displaystyle \frac{1}{2}\;{\delta }^{{ij}}\;{\bar{\theta }}_{i}\;{\gamma }^{A}{{\rm{\Psi }}}_{\mu j}\right)+\displaystyle \frac{1}{4}\;{\delta }^{{ij}}\;{\bar{\theta }}_{i}\;{\gamma }^{A}{D}_{\lambda }{\theta }_{j},\end{eqnarray} \tag{ 68 }$$

where the derivative D_λ is covariantized w.r.t. Lorentz transformations, i.e.

$$\begin{eqnarray}&&{D}_{\lambda }\theta =\dot{\theta }-\displaystyle \frac{1}{4}\;{\dot{x}}^{\mu }\;{{\rm{\Omega }}}_{\mu }{}^{{AB}}(E,{{\rm{\Psi }}}_{i}){\gamma }_{{AB}}\theta .\end{eqnarray} \tag{ 69 }$$

As we are only interested in terms up to second order in fermions expressions like ${\eta }_{{AB}}{{\rm{\Pi }}}^{A}{{\rm{\Pi }}}^{B}$ are understood to contain only such terms and all terms quartic in fermions are discarded.

The action (67) is invariant under the following supersymmetry transformations of the embedding coordinates

$$\begin{eqnarray}&&\delta {x}^{\mu }=-\displaystyle \frac{1}{4}\;{\delta }^{{ij}}\;{\bar{\eta }}_{i}\;{\gamma }^{A}{\theta }_{j}\;{E}^{\mu }{}_{A},\delta {\theta }_{i}={\eta }_{i}.\end{eqnarray} \tag{ 70 }$$

These transformations should be accompanied by the following σ-model transformations [40, 41] of the background fields, as explained e.g. in [4, 30]:

$$\begin{eqnarray}\delta {E}_{\mu }{}^{A} & = & \displaystyle \frac{1}{2}\;{\delta }^{{ij}}\;{\bar{\eta }}_{i}\;{\gamma }^{A}{{\rm{\Psi }}}_{\mu j}-\displaystyle \frac{1}{4}\;{\delta }^{{ij}}\;{\bar{\eta }}_{i}\;{\gamma }^{B}\;{\theta }_{j}\;{E}^{\rho }{}_{B}{\partial }_{\rho }{E}_{\mu }{}^{A},\quad \delta {{\rm{\Psi }}}_{\mu i}={D}_{\mu }{\eta }_{i},\\ \delta {M}_{\mu } & = & \displaystyle \frac{1}{2}\;{\varepsilon }^{{ij}}\;{\bar{\eta }}_{i}\;{{\rm{\Psi }}}_{\mu j}-\displaystyle \frac{1}{4}\;{\delta }^{{ij}}\;{\bar{\eta }}_{i}\;{\gamma }^{B}\;{\theta }_{j}\;{E}^{\rho }{}_{B}{\partial }_{\rho }{M}_{\mu }.\end{eqnarray} \tag{ 71 }$$

The action (67) is also left invariant by the κ-transformations

$$\begin{eqnarray}&&{\delta }_{\kappa }{x}^{\mu }=-\displaystyle \frac{1}{4}\;{\delta }^{{ij}}\;{\bar{\theta }}_{i}\;{\gamma }^{A}{\delta }_{\kappa }{\theta }_{j}\;{E}^{\mu }{}_{A},\quad {\delta }_{\kappa }{\theta }_{1}=\kappa ,\quad {\delta }_{\kappa }{\theta }_{2}=-\displaystyle \frac{{{\rm{\Pi }}}^{A}{\gamma }_{A}}{\sqrt{-{{\rm{\Pi }}}^{2}}}\;\kappa .\end{eqnarray} \tag{ 72 }$$

In this case all background fields transform under κ-symmetry only through their dependence on the embedding coordinates. To show invariance under supersymmetry and κ-symmetry, one needs to use the equations of motion of the background fields.

With all these preliminaries at hand it is now straightforward to apply the limiting procedure to the relativistic superparticle action (67). This yields the following result:

$$\begin{eqnarray}{S}_{\mathrm{nr}} & = & \displaystyle \frac{m}{2}\displaystyle \int {\rm{d}}\lambda \left[\displaystyle \frac{{\hat{\pi }}^{a}{\hat{\pi }}^{b}{\delta }_{{ab}}}{{\hat{\pi }}^{0}}-2\;{\dot{x}}^{\mu }\left({m}_{\mu }-{\bar{\theta }}_{-}{\gamma }^{0}{\psi }_{\mu -}\right)\right.\\ & & -\left.{\bar{\theta }}_{-}{\gamma }^{0}\hat{D}{\theta }_{-}-\displaystyle \frac{1}{2}\;{\dot{x}}^{\mu }{\omega }_{\mu }{}^{a}\;{\bar{\theta }}_{+}{\gamma }_{a}{\theta }_{-}\right],\end{eqnarray} \tag{ 73 }$$

where we have defined the following supersymmetric line elements

$$\begin{eqnarray}&&{\hat{\pi }}^{0}={\dot{x}}^{\mu }\left({\tau }_{\mu }-\displaystyle \frac{1}{2}\;{\bar{\theta }}_{+}{\gamma }^{0}{\psi }_{\mu +}\right)+\displaystyle \frac{1}{4}\;{\bar{\theta }}_{+}{\gamma }^{0}\hat{D}{\theta }_{+},\end{eqnarray} \tag{ 74 }$$

$$\begin{eqnarray}{\hat{\pi }}^{a} & = & {\dot{x}}^{\mu }\left({e}_{\mu }{}^{a}-\displaystyle \frac{1}{2}\;{\bar{\theta }}_{+}{\gamma }^{a}{\psi }_{\mu -}-\displaystyle \frac{1}{2}\;{\bar{\theta }}_{-}{\gamma }^{a}{\psi }_{\mu +}\right)+\displaystyle \frac{1}{4}\;{\bar{\theta }}_{+}{\gamma }^{a}\hat{D}{\theta }_{-}+\displaystyle \frac{1}{4}\;{\bar{\theta }}_{-}{\gamma }^{a}\hat{D}{\theta }_{+}\\ & & +\displaystyle \frac{1}{8}\;{\bar{\theta }}_{+}{\gamma }^{a}{\gamma }_{b0}{\theta }_{+}\;{\dot{x}}^{\mu }{\omega }_{\mu }{}^{b}.\end{eqnarray} \tag{ 75 }$$

Note that the supercovariant derivative $\hat{D}$ is covariant w.r.t. spatial rotations, not boosts. The boost-connection ${\omega }_{\mu }^{a}$ that appears in equations (73) and (75) is the dependent boost-connection (56). For notational simplicity we do not denote below its dependence on the other fields. The transformations of the embedding coordinates under κ-symmetry are given by

$$\begin{eqnarray}\delta t & = & -\displaystyle \frac{1}{4}\;{\bar{\theta }}_{+}{\gamma }^{0}\kappa ,\quad \delta {\theta }_{+}=\kappa ,\\ \delta {x}^{i} & = & -\displaystyle \frac{1}{4}\;{\bar{\theta }}_{-}{\gamma }^{i}\kappa -\displaystyle \frac{1}{8}\displaystyle \frac{{\hat{\pi }}^{k}}{{\hat{\pi }}^{0}}\;{\bar{\theta }}_{+}{\gamma }^{0i}{\gamma }_{k}\kappa ,\quad \delta {\theta }_{-}=-\displaystyle \frac{{\hat{\pi }}^{i}}{2{\hat{\pi }}^{0}}\;{\gamma }_{i0}\kappa .\end{eqnarray} \tag{ 76 }$$

This reproduces precisely, to second order in fermions, the κ-symmetric non-relativistic superparticle in a curved background as presented in [30].

Fixing kappa-symmetry by setting ${\theta }_{+}=0$ we obtain the result of [30]. When we gauge-fix the Newton–Cartan background to a Galilean background with a Newton potential Φ, the one described in [24], the action (73) reduces to

$$\begin{eqnarray}&&{S}_{\mathrm{nr}}=\displaystyle \frac{m}{2}\int {\rm{d}}\lambda \left[\displaystyle \frac{{\pi }_{{\rm{\Phi }}}^{i}{\pi }_{{\rm{\Phi }}}^{i}}{{\pi }^{0}}-2\dot{t}\left({\rm{\Phi }}-{\bar{\theta }}_{-}{\gamma }^{0}{\rm{\Psi }}\right)-{\bar{\theta }}_{-}{\gamma }^{0}{\dot{\theta }}_{-}+\displaystyle \frac{\dot{t}}{2}\;{\partial }_{i}{\rm{\Phi }}\;{\bar{\theta }}_{+}{\gamma }^{i}{\theta }_{-}\right],\end{eqnarray} \tag{ 77 }$$

with the 'super-Galilean' line-elements given by

$$\begin{eqnarray}&&{\pi }^{0}=\dot{t}+\displaystyle \frac{1}{4}\;{\bar{\theta }}_{+}{\gamma }^{0}{\dot{\theta }}_{+},\end{eqnarray} \tag{ 78 }$$

$$\begin{eqnarray}&&{\pi }_{{\rm{\Phi }}}^{i}={\dot{x}}^{i}-\displaystyle \frac{1}{2}\;\dot{t}\;{\bar{\theta }}_{+}{\gamma }^{i}{\rm{\Psi }}+\displaystyle \frac{1}{4}\;{\bar{\theta }}_{+}{\gamma }^{i}{\dot{\theta }}_{-}+\displaystyle \frac{1}{4}\;{\bar{\theta }}_{-}{\gamma }^{i}{\dot{\theta }}_{+}-\displaystyle \frac{1}{8}\;\dot{t}\;{\partial }_{j}{\rm{\Phi }}\;{\bar{\theta }}_{+}{\gamma }^{i}{\gamma }_{j0}{\theta }_{+}.\end{eqnarray} \tag{ 79 }$$

This finishes our discussion of the superparticle in a non-relativistic curved background.