D ec 2 01 9 N-extended Chern-Simons Carrollian supergravities in 2 + 1 spacetime dimensions

In this work we present the ultra-relativistic N -extended AdS Chern-Simons supergravity theories in three spacetime dimensions invariant under N -extended AdS Carroll superalgebras. We first consider the (2, 0) and (1, 1) cases; subsequently, we generalize our analysis to N = (N , 0), with N even, and to N = (p, q), with p = q, such that, in particular, N = p+ q is even. The N -extended AdS Carroll superalgebras are obtained through the Carrollian (i.e., ultrarelativistic) contraction applied to an so(2) extension of osp(2|2)⊗ sp(2), to osp(2|1)⊗ osp(2, 1), to an so(N ) extension of osp(2|N ) ⊗ sp(2), and to the direct sum of an so(p) ⊕ so(q) algebra and osp(2|p)⊗ osp(2, q), respectively. We also analyze the flat limit (l→ ∞, being l the length parameter) of the aforementioned N -extended Chern-Simons AdS Carroll supergravities, in which we recover the ultra-relativistic N -extended (flat) Chern-Simons supergravity theories invariant under N -extended super-Carroll algebras. The flat limit is applied at the level of the superalgebras, Chern-Simons actions, supersymmetry transformation laws, and field equations.


Introduction
(CS) action requires to enlarge the AdS superalgebra, considering, in particular, a direct sum of an so(p) ⊕ so(q) algebra and the (p, q) AdS superalgebra osp(2|p) ⊗ osp(2, q) [39][40][41]; this is related to the fact that, as it was proven in [39], the semi-direct extension of the so(p) ⊕ so(q) automorphism algebra by the (p, q) Poincaré superalgebra allows to produce a non-degenerate invariant tensor which is used to construct a well-defined three-dimensional CS (p, q) Poincaré supergravity theory (in particular, when either p or q is greater than one, it is not possible to obtain a non-degenerate invariant tensor without considering this extension).
It is well assumed that a three-dimensional (super)gravity theory can be described by a CS action as a gauge theory, offering an interesting toy model to approach higher-dimensional theories [42][43][44][45][46][47][48][49][50]. In the last decades, diverse three-dimensional supergravity models have been studied, and, in this context, there has also been a growing interest to extend AdS and Poincaré supergravity theories to other symmetries (see [38,51] and references therein).
In the present work, we apply the method of [35] with the improvements of [34] to develop in a systematic way the ultra-relativistic N -extended AdS CS supergravity theories in three (that is 2 + 1) spacetime dimensions invariant under N -extended AdS Carroll superalgebras. In particular, we will distinguish between the two N -extended cases N = (N , 0) and N = (p, q), generalizing the results presented at the algebraic level in [13] and also developing the associated CS supergravity theories in three dimensions. More specifically, we start by considering the (2, 0) and (1, 1) cases, and then generalize our analysis to N = (N , 0), with N even, and to N = (p, q), that is N = p + q, with p = q. 2 The N -extended AdS Carroll superalgebras are obtained through the Carrollian (i.e., ultra-relativistic) contraction applied to an so(2) extension of osp(2|2)⊗sp (2), to osp(2|1)⊗osp (2,1), to an so(N ) extension of osp(2|N ) ⊗ sp(2) (with N even), and to the direct sum of an so(p) ⊕ so(q) algebra and osp(2|p) ⊗ osp(2, q) (with p = q), respectively. Let us mention that the N = (N , 0) case (and thus also the N = (2, 0) one) will be more subtle, since it will require the definitions of new supersymmetry generators in order to properly study the Carroll limit, on the same lines of what was done in [13] (see also [52], which deals with non-relativistic superalgebras, and references therein). The ultra-relativistic N -extended AdS Carroll supergravity actions are constructedà la CS, by exploiting the non-vanishing components of the corresponding invariant tensor. The aforementioned actions are all based on a non-degenerate, invariant bilinear form (i.e., an invariant metric). Our result was also an open problem suggested in Ref. [13], and it represents the Nextended generalization of [34].
Subsequently, we study the flat limit (ℓ → ∞, being ℓ the length parameter) of the aforesaid N -extended CS AdS Carroll supergravities, in which we recover the ultra-relativistic N -extended (flat) CS supergravity theories invariant under N -extended super-Carroll algebras. The flat limit is applied at the level of the superalgebras, CS actions, supersymmetry transformation laws, and field equations.
The remain of the paper is organized as follows: In Section 2, we first introduce a new N = (2, 0) AdS Carroll superalgebra, which is obtained as the ultra-relativistic contraction of an so(2) extension of osp(2|2) ⊗ sp (2). Here, the so(2) extension is necessary in order to end up with an invariant non-degenerate inner product in the ultra-relativistic limit, providing a well-defined CS action. In fact, this allows us to subsequently develop the three-dimensional CS supergravity action invariant under the (2, 0) AdS Carroll superalgebra, which we call (2, 0) CS AdS Carroll supergravity in 2 + 1 dimensions. In Section 3, we repeat the same analysis for the (1, 1) case, ending up with the CS supergravity action invariant under the (1, 1) AdS Carroll superalgebra. Subsequently, we generalize our study to the cases of N = (N , 0), with N even, and N = (p, q), with p = q, respectively in Section 4 and 5. In Section 6, we discuss the flat limit ℓ → ∞ of the N -extended CS AdS Carroll supergravities introduced in the previous part of the work. Section 7 contains some final comments and remarks.
2 (2, 0) AdS Carroll supergravity in 2 + 1 dimensions In this section, we first introduce a new N = (2, 0) AdS Carroll superalgebra, which is obtained as the ultra-relativistic contraction of an so(2) extension of osp(2|2) ⊗ sp (2). The so(2) extension of osp(2|2) ⊗ sp (2) is needed in order to end up with an invariant non-degenerate inner product in the ultra-relativistic limit, namely with a well-defined invariant tensor (this is reminescent of what was done in [39] in the case of relativistic theories), in such a way to be able to construct a well-defined CS action. Indeed, this allows us to subsequently develop the three-dimensional CS supergravity action invariant under this N = (2, 0) AdS Carroll superalgebra, which we call (2, 0) CS AdS Carroll supergravity.
Let us mention, here, that a N = (2, 0) AdS Carroll superalgebra has been first introduced in [13]. Nevertheless, due to the degeneracy of the invariant tensor for that superalgebra, one could not construct a well-defined CS action in that case. On the other hand, as we will see in the following, our N = (2, 0) AdS Carroll superalgebra will be different from the one presented in [13], allowing, in particular, the formulation of a three-dimensional CS action in the supergravity context.
where ℓ is a length parameter, C denotes the charge conjugation matrix, and Γ A and Γ AB represent the Dirac matrices in three dimensions. The generatorsJ AB ,P A ,Z ij , andQ i α have a dual description in terms of 1-form fields,ω AB (spin connection),Ṽ A (vielbein),z ij (1-form field dual to the generatorZ ij ), andψ α i (gravitinos), respectively. Here, we consider the ultra-relativistic contraction of an so(2) extension of osp(2|2) ⊗ sp (2), involving an extra generatorS ij = ǫ ijS . This will allow the formulation of a well-defined ultrarelativistic CS action, based on a non-degenerate invariant tensor, which would not be possible by considering the N = (2, 0) superalgebra of [13]. In particular, we extend (2.1) by adding the extrã S generator and we perform, on the same lines of [39], the redefinitioñ T ≡Z − ℓS , (2.2) to eliminateZ in favour ofT (this redefinition is particularly convenient for discussing the flat limit, see also [39]). Consequently, we rewrite the (anti)commutation relations (2.1) as follows (we adopt dimensionful generators from the very beginning, on the same lines of [38]): 3 Note that, in the flat limit ℓ → ∞,S becomes the central element of the N = (2, 0) Poincaré superalgebra extended with the extra so(2) generatorT (see [39]). The non-vanishing components of an invariant tensor for the superalgebra (2.3), which will be useful in the sequel, are given by where α 0 and α 1 are arbitrary constants and ǫ ABC is the Levi-Civita symbol in three dimensions. 3 We use the metric ηAB with the signature (−, +, +).
To take the Carrollian (i.e., ultra-relativistic) contraction of the superalgebra (2.3), we decompose the indices as A → (0, a) , a = 1, 2 . (2.5) This induces the following decomposition of the generators: Furthermore, we define new supersymmetry charges bỹ Then, we rescale the generators with a parameter σ as follows: Taking the limit σ → ∞ 4 and removing the tilde symbol also on the generators that we have not rescaled, we end up with a new N = (2, 0) AdS Carroll superalgebra (differing from the one of [13], due to the presence of the generator S), which fulfills the following non-trivial (anti)commutation relations: (2.10) We will now construct a CS supergravity action in three dimensions invariant under the N = (2, 0) AdS Carroll superalgebra (2.10): The (2, 0) AdS Carroll CS supergravity action.

(2, 0) AdS Carroll supergravity action
The general form of a three-dimensional CS action is given by where k = 1/(4G) is the CS level of the theory (and for gravitational theories it is related to the gravitational constant G), A corresponds to the gauge connection 1-form, . . . denotes the invariant tensor, and the integral is over a three-dimensional manifold M. 5 The CS action (2.11) can also be rewritten as 12) in terms of the curvature 2-form F = dA In the case of the N = (2, 0) AdS Carroll superalgebra (2.10), the connection 1-form reads 6 where ω ab , k a , V a , h, t, s, ψ + , and ψ − are the 1-form fields dual to the generators J ab , K a , P a , H, T , S, Q + , and Q − , respectively. The corresponding curvature 2-form F is with Now, in order to construct a CS action (that is an action of the form (2.11)) invariant under the N = (2, 0) super-AdS Carroll group, we require the connection 1-form given in (2.13) and the corresponding non-vanishing components of the invariant tensor. 5 In the sequel, we will omit the wedge product "∧" between differential forms. 6 Here and in the following, for simplicity, we will omit the spinor index α.
Concerning the invariant tensor, that is the fundamental ingredient for the construction of a CS action, we now apply the method of [35], which consists in rescaling not only the generators but also the coefficients appearing in the invariant tensor before applying a contraction, in order to end up with a non-trivial invariant tensor for the contracted (super)algebra on which the desired CS theory will be based. Specifically, we consider the non-vanishing components of the invariant tensor for the so(2) extension of osp(2|2)⊗ sp(2) (see (2.3)) given in (2.4), we decompose the indices as in (2.5) and consider the new supersymmetry charges (2.8), and then we rescale not only the generators in compliance with (2.9) but also the coefficients appearing in (2.4) as follows: (2. 16) In this way, taking the limit σ → ∞, we end up with the following non-vanishing components of an invariant tensor for the N = (2, 0) AdS Carroll superalgebra: This bilinear form is non-degenerate if α 1 = 0. Thus, using the connection 1-form in (2.13) and the non-vanishing components of the invariant tensor given in (2.17) in the general expression (2.11) for a three-dimensional CS action, we can finally write the (2, 0) AdS Carroll CS supergravity action in three spacetime dimensions invariant under (2.10), which reads as follows: written in terms of the curvatures appearing in (2.15). We can see that in (2.18) we have two different sectors, one proportional to α 0 and the other proportional to α 1 . Observe that the term proportional to α 0 corresponds to the exotic Lagrangian involving the Lorentz contribution, a torsional piece, and a contribution from the 1-form field t, while it does not contain any contribution from the 1-form fields ψ + and ψ − . Let us mention that the CS action (2.18) can also be rewritten up to boundary terms as The CS action (2.18), characterized by two coupling constants α 0 and α 1 , is invariant by construction under the N = (2, 0) AdS Carroll superalgebra (2.10). In particular, the local gauge transformations δ λ A = dλ + [A, λ] with gauge parameter are given by Restricting ourselves to supersymmetry, we have The equations of motion obtained from the variation of the action (2.18) with respect to the fields ω ab , k a , V a , h, t, s, ψ + , and ψ − are, respectively, up to boundary terms, and we can see that when α 1 = 0 they reduce to the vanishing of the (2, 0) super-AdS Carroll curvature 2-forms, namely Here, we can also observe that α 1 = 0 is a sufficient condition to recover (2.24), meaning that one could consistently set α 0 = 0, which corresponds to the vanishing of the exotic term in the CS action (2.18).

(1, 1) AdS Carroll supergravity in 2 + 1 dimensions
In this section, we repeat the analysis done in Section 2 in the (1, 1) case. To this aim, we first review the derivation of the N = (1, 1) AdS Carroll superalgebra introduced in [13], which is obtained as the ultra-relativistic contraction of osp(2|1) ⊗ osp(2, 1). Then, we write the nonvanishing components of the invariant tensor of the N = (1, 1) AdS Carroll superalgebra (obtained as the Carrollian contraction of the non-vanishing components of the invariant tensor for osp(2|1) ⊗ osp(2, 1)). This allows us to construct a three-dimensional CS supergravity action invariant under the N = (1, 1) AdS Carroll superalgebra, which we call the (1, 1) AdS Carroll CS supergravity action.
The superalgebra osp(2|1) ⊗ osp(2, 1) is generated by the set {J AB ,P A ,Q + α ,Q − α } obeying the following (anti)commutation relations: Note that by taking the flat limit ℓ → ∞ of (3.1) one recovers the N = (1, 1) Poincaré superalgebra. The non-vanishing components of an invariant tensor for the superalgebra (3.1) are being α 0 and α 1 arbitrary independent constants. Now, to take the Carrollian contraction of the superalgebra (3.1), we decompose the indices A, B, . . . = 0, 1, 2 as in (2.5), which induces the decomposition (2.6), together with (2.7). Then, we rescale the generators with a parameter σ as Subsequently, taking the limit σ → ∞ (and removing the tilde symbol also on the generators that we have not rescaled), we end up with the N = (1, 1) AdS Carroll superalgebra introduced in Ref. [13], whose (anti)commutation relations read In the sequel we will construct a CS action in three-dimension invariant under the N = (1, 1) AdS Carroll superalgebra (3.4).
To this aim, we introduce the connection 1-form A associated with (3.4), that is being ω ab , k a , V a , h, ψ + , and ψ − the 1-form fields respectively dual to the generators J ab , K a , P a , H, Q + , and Q − obeying the (anti)commutation relations given in (3.4), and the related curvature 2-form F , which reads Now, we move to the explicit construction of a CS action invariant under the N = (1, 1) super-AdS Carroll group, on the same lines of what we have previously done in Section 2 for N = (2, 0). Thus, we consider the non-vanishing components of the (relativistic) invariant tensor given in (3.2), we decompose the indices as in (2.5), and we rescale not only the generators in compliance with (3.3) but also the coefficients appearing in (3.2) as in (2.16). Then, taking the ultra-relativistic limit σ → ∞, we get the following non-vanishing components of an invariant tensor for the N = (1, 1) AdS Carroll superalgebra: This invariant tensor is non-degenerate if α 1 = 0. Substituting the connection 1-form in (3.5) and the non-vanishing components of the invariant tensor (3.8) in the general expression (2.11) for a three-dimensional CS action, we end up with the (1, 1) AdS Carroll CS supergravity action in 2 + 1 spacetime dimensions, that is which can also be rewritten omitting boundary terms as follows: (3.10) The action (3.9) has been written in terms of the curvatures appearing in (3.7), it involves two different sectors, respectively proportional to α 0 (which corresponds to the exotic Lagrangian) and to α 1 , and it is invariant by construction under the N = (1, 1) AdS Carroll superalgebra (3.4). The local gauge transformations and, restricting ourselves to supersymmetry, we are left with the following transformation rules: The equations of motion obtained from the variation of the action (3.9) with respect to the 1-form fields ω ab , k a , V a , h, ψ + , and ψ − are respectively; for α 1 = 0, they reduce to the vanishing of the (1, 0) super-AdS Carroll curvature 2-forms, namely Analogously to what happened in the (2, 0) case discussed in Section 2, we can see that α 1 = 0 is a sufficient condition to recover (3.15), which means that one could consistently set α 0 = 0, making the exotic term in the CS action (3.9) disappear.

(N , 0) AdS Carroll supergravity theories in 2 + 1 dimensions
Now, we generalize our analysis to the (N , 0) case, with N even. First, we present the derivation of the N = (N , 0) AdS Carroll superalgebra as the Carrollian contraction of an so(N ) extension of osp(2|N ) ⊗ sp (2). This also provides us with a non-degenerate invariant tensor in the ultra-relativistic limit. Then, we can subsequently formulate a well-defined three-dimensional CS supergravity action invariant under the aforesaid N = (N , 0) AdS Carroll superalgebra.

N = (N , 0) AdS Carroll superalgebra
Let us first take the direct sum of osp(2|N ) ⊗ sp(2) and an so(N ) algebra (we consider N even), that is reminiscent of what was done in Ref. [39]. In this case, the non-trivial (anti)commutation relations are with A, B, . . . = 0, 1, 2, i, j, . . . = 1, . . . , N (where we have considered N = 2x, x = 1, . . . , N 2 ), and whereZ ij = −Z ji ,S ij = −S ij . Then, we do the following redefinition (on the same lines of [39]): which is a generalization of the one performed in Section 2. Thus, we can now rewrite the (anti)commutation relations (4.1) as Observe that, taking the limit ℓ → ∞ of (4.3), we get the N = (N , 0) Poincaré superalgebra involving a semi-direct so(N ) extension (with N even). The non-vanishing components of an invariant tensor for (4.3), which will be useful in the sequel, are given by being α 0 and α 1 arbitrary independent constants.
Furthermore, from the commutation relations involving the generators T ij and S ij , we get the following non-vanishing ones (recall that we haveŨ ′λµ = −Ũ µλ andṼ ′λµ = −Ṽ µλ ): and Now, let us rescale the generators with a parameter σ as 13) where we have also removed the tilde symbol on the generators. Taking the limit σ → ∞ (and removing the tilde symbol also on the generators that we have not rescaled), we end up with the N = (N , 0) AdS Carroll superalgebra (with N even), whose non-trivial (anti)commutation relations read as follows (recall the definitions (4.8), (4.10), since we have expressed the anticommutation relations in terms of the combinations given in that expressions, together with the fact thatŨ ′λµ = −Ũ µλ andṼ ′λµ = −Ṽ µλ , and the (anti)commutation relations (4.9), (4.11), and (4.12)): Notice that if we restrict ourselves to the special case N = (2, 0), that is x = 1, after some algebraic calculations, exploiting the definitions (4.6), (4.8), (4.10), and the symmetry properties (4.7), we exactly reproduce the N = (2, 0) AdS Carroll superalgebra obtained in Section 2, given by (2.10). 7 In the sequel, we will construct a CS action in 2 + 1 dimensions invariant under (4.14).

(N , 0) AdS Carroll supergravity
We can now move to the formulation of a three-dimensional CS supergravity action invariant under the superalgebra (4.14). We call this action (N , 0) AdS Carroll CS supergravity action (where, in our analysis, N is even).
To this aim, let us start by introducing the connection 1-form A associated with the superalgebra (4.14), that is where ω ab , k a , V a , h, t λµ , t ′λµ , u λµ , s λµ , s ′λµ , v λµ , ψ + λ , and ψ − λ are the 1-form fields dual to the generators J ab , K a , P a , H, T λµ , T ′ λµ , U λµ , S λµ , S ′ λµ , V λµ , Q + λ , and Q − λ , respectively. The corresponding curvature 2-form F is F = 1 2 R ab J ab + K a K a + R a P a + HH with where we have used and where we have In order to develop a CS action invariant under (4.14), we have to consider the non-vanishing components of the invariant tensor in (4.4), decompose the indices as in (2.5), exploit (4.5), (4.6), (4.7), and rescale not only the generators in compliance with (4.13) but also the coefficients appearing in (4.4) as in (2.16). Consequently, in the ultra-relativistic limit σ → ∞ we get the following non-vanishing components of an invariant tensor for (4.14): (4.20) The invariant tensor for (4.14) above is non-degenerate if α 1 = 0. Then, substituting the connection 1-form in (4.15) and the non-zero components of the invariant tensor (4.20) into (2.11), we end up with the three-dimensional (N , 0) AdS Carroll CS supergravity action, which reads where we have also exploited (4.18). The action (4.21) has been written in terms of the curvatures appearing in (4.17) and it involves two coupling constants, that are α 0 and α 1 . Up to boundary terms, (4.21) can be rewritten as (4.22) The contribution proportional to α 0 corresponds to the exotic Lagrangian, and, in the present case, it involves, besides the Lorentz and torsional terms, also pieces including the 1-form fields t λµ , t ′λµ , and u λµ . On the other hand, the contribution proportional to α 1 also includes terms involving the 1-form fields s λµ , s ′λµ , and v λµ , plus the spinor 1-form fields ψ + λ and ψ − λ . The CS action (4.21) is invariant by construction under (4.14), and the local gauge transforma- are given by where we have also used the properties and definitions (4.25) Restricting ourselves to supersymmetry, we get the following supersymmetry transformation laws: (4.26) Finally, one can prove that from the variation of the action (4.21) with respect to the 1-form fields ω ab , k a , V a , h, t λµ , t ′λµ , u λµ , s λµ , s ′λµ , v λµ , ψ + λ , and ψ − λ , we get, respectively, the equations of motion δω ab : α 0 R ab + α 1 ǫ ab H = 0 , δk a : α 1 R a = 0 ,  written up to boundary contributions. We observe that, for α 1 = 0, the equations (4.27) reduce precisely to the vanishing of the (N , 0) super-AdS Carroll curvature 2-forms given in (4.16), that is to say (4. 28) Also in this case, we notice that α 1 = 0 is a sufficient condition to recover (4.28), which means that we can consistently impose α 0 = 0 in the CS action (4.21), making the exotic term disappear.
Let us also mention that, restricting ourselves to the special case N = (2, 0), that is x = 1, after some algebraic calculations, we exactly reproduce the results of Section 2, that is to say, in fact, the (2, 0) AdS Carroll supergravity theory.

(p, q) AdS Carroll supergravity theories in 2 + 1 dimensions
We finally extend our analysis to the (p, q) case, with p = q. We first derive the N = (p, q) AdS Carroll superalgebra as the Carrollian contraction of the direct sum of an so(p) ⊕ so(q) algebra and osp(2|p) ⊗ osp(2, q). This allows us to end up with a non-degenerate invariant tensor in the ultra-relativistic limit, and to consequently construct the three-dimensional CS supergravity theory invariant under the aforesaid N = (p, q) AdS Carroll superalgebra.
The non-vanishing components of an invariant tensor for the superalgebra (5.3), that will be useful in the following study, are given by where α 0 and α 1 are arbitrary constants. In order to take the ultra-relativistic contraction of the superalgebra (5.3), we decompose, as usual, the indices A, B, . . . = 0, 1, 2 as in (2.5), which induces the decomposition (2.6), together with (2.7). After that, we rescale the generators with a parameter σ as (5.5) Then, taking the limit σ → ∞ (and removing the tilde symbol also on the generators that we have not rescaled), we end up with the N = (p, q) AdS Carroll superalgebra whose non-trivial (anti)commutation relations read Notice that if we restrict ourselves to the special case N = (1, 1), that is p = q = 1, we exactly reproduce the N = (1, 1) AdS Carroll superalgebra obtained in Section 3, namely (3.4).
In the following, we will construct a three-dimensional CS action invariant under (5.6).

(p, q) AdS Carroll supergravity
We will now construct a three-dimensional CS supergravity action invariant under the superalgebra (5.6) just introduced. We call this action (p, q) AdS Carroll CS supergravity action (recall that, in our case, p = q, such that N = p + q is even).
To this aim, let us first introduce the connection 1-form A associated with (5.6), namely being ω ab , k a , V a , h, t ij , t IJ , s ij , s IJ , ψ i , and ψ I the 1-form fields respectively dual to the generators J ab , K a , P a , H, T ij , T IJ , S ij , S IJ , Q i , and Q I (obeying the (anti)commutation relations given in (5.6)), and the related curvature 2-form F , that is We can now move to the explicit construction of a CS action invariant under (5.6). To this aim, consider the non-vanishing components of the invariant tensor given in (5.4), decompose the indices as in (2.5), and rescale not only the generators in compliance with (3.3) but also the coefficients appearing in (5.4) as in (2.16). Consequently, the Carroll limit σ → ∞ leads to the following non-vanishing components of an invariant tensor for the superalgebra (5.6): (5.10) The invariant tensor whose components are given in (5.10) is non-degenerate when α 1 = 0. Then, substituting the connection 1-form in (5.7) and the non-zero components of the invariant tensor (5.10) into the general expression (2.11), we end up with the (p, q) AdS Carroll CS supergravity action in three dimensions, that is which is written in terms of the curvatures appearing in (5.9) and it involves two coupling constants, that are α 0 and α 1 . Up to boundary terms, the action (5.11) can be reworked as follows: As usual, the contribution proportional to α 0 corresponds to the exotic Lagrangian, and we can see that it involves, besides the Lorentz and torsional terms, also pieces including the 1-form fields t ij and t IJ . However, it does not contain terms involving ψ i and ψ I . On the other hand, the contribution proportional to α 1 also includes pieces involving the 1-form fields s ij , s IJ , ψ i , and ψ I . The action (5.11) is invariant by construction under (5.6), and the local gauge transformations are given by (5.14) Thus, the restriction to supersymmetry transformations gives us Finally, the equations of motion obtained from the variation of (5.11) with respect to the 1-form fields ω ab , k a , V a , h, t ij , t IJ , s ij , s IJ , ψ i , and ψ I are, respectively (up to boundary contributions), and, for α 1 = 0, they reduce precisely to the vanishing of the (p, q) super-AdS Carroll curvature 2-forms given in (5.8), that is to say We observe that, as usual, α 1 = 0 is a sufficient condition to recover (5.17), meaning that α 0 can be consistently set to zero, making the exotic term in the CS action (5.11) disappear. Let us finally mention that, if we restrict ourselves to the case p = q = 1, we exactly reproduce the results of Section 3, that is to say, as properly expected, the (1, 1) AdS Carroll supergravity theory.
6.1 (2, 0) Carroll supergravity from the ℓ → ∞ limit In the limit ℓ → ∞, the (anti)commutation relations of the N = (2, 0) AdS Carroll superalgebra (2.10) reduce to the following non-vanishing ones: On the other hand, by applying the ℓ → ∞ limit to the three-dimensional CS action (2.18), we end up with which is written in terms of the super-Carroll curvatures appearing in (6.2). The latter must not be confused with the super-AdS Carroll ones given in (2.15), since (6.2) correspond to the flat limit of (2.15). Here we signal that we have done a little abuse of notation. The action (6.3) can also be derived by using the following non-vanishing components of the invariant tensor: which are obtained by taking the limit ℓ → ∞ of (2.17), and the connection 1-form for the N = (2, 0) (flat) Carroll superalgebra (6.1) in the general expression (2.11). Notice that the exotic term, which is the one proportional to α 0 in (6.3), now reduces purely to the so-called Lorentz Lagrangian. The CS action (6.3) is invariant by construction under the super-Carroll group associated with (6.1). In particular, concerning the flat limit of the gauge transformations (2.21), we get the local gauge transformations The restriction to supersymmetry transformations reads and we can see that, when α 1 = 0, they exactly reduce to the vanishing of the curvature 2-forms given in (6.2). We can also observe that, in analogy with the AdS case of Section 2, also in the flat limit α 1 = 0 results to be a sufficient condition to recover the vanishing of the curvature 2-forms (6.2) obtained in the flat limit, which means that one could consistently set α 0 = 0 and thus neglect the exotic term (i.e., the Lorentz Lagrangian) in the CS action (6.3). 8 6.2 (1, 1) Carroll supergravity from the ℓ → ∞ limit The limit ℓ → ∞ performed on the (anti)commutation relations of the N = (1, 0) AdS Carroll superalgebra (3.4) leads to the following non-vanishing ones: These are the (anti)commutation relations of the N = (1, 1), D = 3 super-Carroll algebra (see [13], where (6.8) corresponds to the superalgebra obtained in the R → ∞ limit of the N = (1, 1) AdS-Carroll superalgebra of Section C.4 of the same paper). It could be also obtained by considering the ultra-relativistic contraction of the N = (1, 1), D = 3 Poincaré superalgebra.
Taking ℓ → ∞, the 2-form curvatures (3.7) reduce to R ab = dω ab = R ab , 9) and the ℓ → ∞ limit of the CS action (3.9) leads us to the following three-dimensional one: written in terms of the curvatures appearing in (6.9) (again, we are doing a little abuse of notation). The action (6.10) can also be derived by using the connection 1-form for the N = (1, 1) (flat) Carroll superalgebra (6.8) together with the non-vanishing components of the invariant tensor J ab J cd = α 0 (δ ad δ bc − δ ac δ bd ) , (6.11) in the general expression for a three-dimensional CS action (2.11). Analogously to what happened in the (2, 0) flat theory, also in the current case the exotic term, proportional to α 0 , now reduces purely to the Lorentz Lagrangian. By construction, the CS action (6.10) is invariant under the (1, 1) super-Carroll group, that is associated with the superalgebra given in (6.8). In particular, taking the flat limit of the gauge transformations (3.12), we get the following local gauge transformations: 12) and restricting ourselves to supersymmetry transformations, we are left with Concluding, the equations of motion for the action (6.10) (flat limit of the equations of motion given in (3.14)) read as follows: (6.14) When α 1 = 0, the eqs. (6.14) exactly reduce to the vanishing of the curvature 2-forms in (6.9) (α 1 = 0 is a sufficient condition to recover the vanishing of the curvature 2-forms (6.9), meaning that one could consistently set α 0 = 0, omitting the exotic term, that is the Lorentz Lagrangian, in the CS action (6.10)).
Let us observe that, restricting ourselves to the purely bosonic theory, we end up with the N = (N , 0) Carroll gravity theories (with N even) in three dimensions, invariant under the N = (N , 0) Carroll algebra. At the purely bosonic level, the fields t λµ , t ′λµ , u λµ , s λµ , s ′λµ , and v λµ , and the corresponding terms in the action, can also be consistently discarded by performing an IW contraction.
On the other hand, considering the special case N = (2, 0), that is x = 1, after some algebraic calculations, we can prove that the (2, 0) theory in the flat limit previously discussed in this section is exactly reproduced.
6.4 (p, q) Carroll supergravity theories from the ℓ → ∞ limit Applying the flat limit ℓ → ∞ to the (anti)commutation relations given in (5.6), we get the following non-vanishing ones: These are the (anti)commutation relations of the N = (p, q), D = 3 super-Carroll algebra (with p = q), and we could also have obtained it by applying the Carroll contraction to the semi-direct extension of the so(p)⊕so(q) automorphism algebra by the N = (p, q), D = 3 Poincaré superalgebra (see Ref. [39]).
the limit ℓ → ∞ of (5.10) gives us the following non-vanishing components: J ab J cd = α 0 (δ ad δ bc − δ ac δ bd ) , The CS action (6.24) is invariant by construction under the super-Carroll group associated with (6.22), and, in particular, concerning the ℓ → ∞ limit of the local gauge transformations (5.14), we get Concluding, the equations of motion for the action (6.24) (flat limit of the equations of motion given in (5.16)) read as follows:  When α 1 = 0, the eqs. (6.28) exactly reduce to the vanishing of the curvature 2-forms given in (6.23) (α 1 = 0 is a sufficient condition to recover the vanishing of the curvature 2-forms (6.23), and the coefficient α 0 can also be consistently set to zero, making the exotic term disappear from the action (6.24)).
Restricting ourselves to the purely bosonic theory, we end up with the N = (p, q) Carroll gravity theories (with p = q) in three dimensions, invariant under the N = (p, q) Carroll algebra. At the purely bosonic level, the fields t ij , t IJ , s ij , and s IJ , and the corresponding terms in the action, can also be consistently discarded by performing an IW contraction.
On the other hand, let us finally mention that, if we now consider the particular case p = q = 1, we exactly reproduce the results previously obtained in this section for the (1, 1) theory in the flat limit.
All the studies of the flat limit presented in this section represent a new development and generalization of the previous works concerning Carroll superalgebras in three dimensions, in particular in the context of three-dimensional CS supergravity theories.

Conclusions
Motivated by the recent development of applications of Carroll symmetries (in particular, by their prominent role in the context of holography), and by the fact that, nevertheless, the study of their supersymmetric extensions in the context of supergravity models still remains poorly explored, in this paper we have presented, in a systematic fashion, the ultra-relativistic N -extended AdS CS supergravity theories in three (2 + 1) spacetime dimensions, which are invariant under N -extended AdS Carroll superalgebras, extending the results recently presented in [34] (where the construction of the three-dimensional N = 1 CS supergravity theory invariant under the so-called AdS Carroll superalgebra, ultra-relativistic contraction of the N = 1 AdS superalgebra [13], together with the study of its flat limit, has been presented for the first time). In particular, we have applied the method introduced in [35] with the improvements of [34] to construct the aforesaid ultra-relativistic N -extended AdS CS supergravity theories.
A N = (2, 0) AdS Carroll superalgebra in three dimensions was previously introduced in [13]. Nevertheless, the latter does not allow for a non-degenerate invariant tensor, meaning that one cannot construct a well-defined CS action based on this superalgebra. To overcome this point, we have considered an so(2) extension of osp(2|2) ⊗ sp(2) and performed the ultra-relativistic contraction on it, ending up with a new N = (2, 0) AdS Carroll superalgebra endowed with a nondegenerate invariant tensor. This has allowed us to develop the three-dimensional CS supergravity action invariant under this N = (2, 0) AdS Carroll superalgebra. We have called this action the (2, 0) AdS Carroll CS supergravity action. We have done an analogous analysis in the (1, 1) case, and subsequently generalized our study to N = (N , 0) (with N even) and to N = (p, q) (with p = q). In particular, after having introduced the ultra-relativistic superalgebras, we have constructed the respective CS supergravity theories in three-dimensions by exploiting the nonvanishing components of the corresponding invariant tensor. The aforementioned actions are all based on a non-degenerate, invariant bilinear form (i.e., an invariant metric), and each of them is characterized by two coupling constants and involve an exotic contribution. The results presented in this paper were also open problems suggested in Ref. [13], and they represent the N -extended generalization of [34]. Interestingly, one can observe that the CS formulation in the N -extended cases N = (N , 0) and N = (p, q) requires the presence of so(N ) and so(p) ⊕ sp(q) generators, respectively, also at the ultra-relativistic level, that is in the Carroll limit; thus, what happens at the relativistic level for three-dimensional N -extended CS Poincaré and AdS supergravity theories (see [39]), that is the need to introduce the aforementioned extra generators (together with their dual 1-form fields) in the theory in order to obtain a non-degenerate invariant tensor, has repercussions also on (and still holds at) the ultra-relativistic level.
We have also analyzed the flat limit ℓ → ∞ of the aforementioned models, in which we have recovered the ultra-relativistic N -extended (flat) CS supergravity theories invariant under N -extended super-Carroll algebras. The flat limit has been applied at the level of the superalgebras, CS actions, supersymmetry transformation laws, and field equations. Also all the studies of the flat limit presented in Section 6 represent a new development and generalization of the previous works presented in the literature concerning Carroll (super)algebras in three dimensions, in particular in the context of three-dimensional CS (super)gravity theories.
The recently discovered relations among the Carrollian world and flat holography suggest that this work might represents a starting point to go further in the analysis of supersymmetry invariance of flat supergravity in the presence of a non-trivial boundary, along the lines of [53]. Besides, now, having well-defined three-dimensional CS (super)gravity theories respectively invariant under the N -extended AdS-Carroll and Carroll (super)algebras, it would be intriguing to go beyond and study the asymptotic symmetry of these models, following, for instance, the prescription given in Ref. [54]. It would also be interesting to further extend our analysis to more general amount of supersymmetry, involving also odd N cases, and to higher-dimensional models (recently, a study exploring the Carroll limit corresponding to M2-as well as M3-branes propagating over D = 11 supergravity backgrounds in M-theory has been presented [55]), where Carrollian (super)gravity theories still remain poorly explored. Finally, all these ultra-relativistic theories constructedà la CS could have some applications in the context of Carrollian fluids (and their relations with flat holography, see Refs. [30][31][32][33]).

Acknowledgments
L.R. is grateful to L. Andrianopoli and B.L. Cerchiai for useful discussions and support.