Super Carrollian and Super Galilean Field Theories

The exploration of scalar field theories that exhibit Carroll and Galilei symmetries has attracted a lot of attention. In this paper, we generalize these studies to fermionic field theories and construct consistent electric and magnetic descriptions of Carrollian and Galilean spin 1 2 fermions. We showcase various methods that offer complementary perspectives into the limiting process of the underlying relativistic theories. Moreover, we extend our study to N = 1 off-shell supersymmetric field theories in four dimensions. By introducing suitable Grassmann-analyticity conditions, we formulate the corresponding super Carrollian and super Galilean theories. These theories combine the established Carroll/Galilei scalars with the Carroll/Galilei fermions and a range of auxiliary fields into supermultiplets.


Introduction
The concept of nonrelativistic geometries 1 is not a new one, as it was introduced around a century ago by Cartan in his development of Newton-Cartan gravity [1] based on a torsionless non-Lorentzian geometry.Nevertheless, recently there has been a significant interest in the exploration and study of non-Lorentzian theories driven by a multitude of applications in a variety of physical systems (for reviews see [2][3][4][5]).These theories abide by a principle of relativity characterized by symmetries that emerge through the breaking of Lorentz invariance.This breaking, leads to many possible kinematic algebras 2 , but the main two structures are the (i) Carrollian and (ii) Galilean algebras.Both of these algebras are well-known contractions of the relativistic Poincaré algebra [10][11][12] associated with small or large characteristic velocities compared with the speed of light or equivalently with the c → 0 or c → ∞ limits respectively.
There are several motivations that encourage the study of these non-Lorentzian theories and their application to various physical systems.Nevertheless, our inspiration stems from developments in (a) flat space holography and (b) nonrelativistic/Carrollian string theory.It is a fact that the Carrollian conformal algebra is isomorphic to the BMS algebra in one dimension higher [13] and therefore relevant for the celestial approach to flat space holography [14][15][16][17][18][19][20].Moreover, Carrollian string theories [21] as well as nonrelativistic string theories [22][23][24] and tensionless string theories [25][26][27][28][29][30] are known examples of solvable string theories and provide a unitary and ultraviolet complete framework that connects different corners of string theory.Naturally, this creates a resurgence of interest in non-Lorentzian geometries and quantum field theories that arise from such string theories.
At the heart of defining nonrelativistic superstring theory lies nonrelativistic supersymmetry.The non-Lorentzian supersymmetry that emerges within this context is considerably less developed (see early works [31][32][33][34]) and not much progress has been made in the direction of nonrelativistic superspace and nonrelativistic supermultiplets.The purpose of this paper is to construct Carrollian and Galilean supersymmetric theories of matter.The bosonic parts of these theories have been explored in detail in [4].It has been shown that in addition to the naturally defined (1) electric (time derivative dominated) Carroll-invariant scalar theory and (2) magnetic (space derivative dominated) Galilei-invariant scalar theory, one can construct by application of the "seed " Lagrangian method alternative magnetic Carroll-invariant and electric Galilei-invariant scalar theories.This method is based on the fact that Carrollian and Galilean boosts are nilpotent operators.These alternative descriptions can be also understood as the appropriate (Carrollian or Galilean) limits of a dual relativistic theory [24], where the divergent quadratic terms have been tamed by the introduction of a Lagrange multiplier.
In this paper we introduce the spin 1/2 fermionic counterparts to the previously mentioned scalar theories [2,35,36].Through a range of methodologies, including Hamiltonian formulation, field expansion, and seed Lagrangian, we develop Carroll and Galilei fermionic field theories that exhibit invariance under their respective boost transformations.Similar to the scalar theories, each fermionic theory has two formulations -one electric and one magnetic.Carrollian fermions have been introduced in [37,38], however in these works the authors absorb the Inönü-Wigner contraction parameter in the background spacetime metric.Consequently, at the appropriate contraction limit there is a significant alteration in the structure of the Clifford algebra and the structure of spinors.In contrast, we absorb the contraction parameter by appropriate coordinate or field rescalings that do not impact the γ matrices.This approach proves advantageous as it enables the exploration of the supersymmetric extensions of these Carroll and Galilei theories.This is the focus of the second part of the paper where we embed the known Carroll/Galilei scalars together with their corresponding fermions within non-Lorentzian supermultiplets.There are two types of supersymmetries that can be imposed, contingent on whether the supersymmetry algebra closes under temporal translations (C-supersymmetry) or spatial translations (G-supersymmetry).For each type of supersymmetry, we develop the appropriate superspace description and use it in order to construct manifestly supersymmetric theories of matter that remain invariant under Carrollian or Galilean boosts.By introducing the notion of a C-chiral superfield, we are able to describe in superspace a C-Wess-Zumino model.This model provides the C-supersymmetric extension of electric Carroll matter, meaning that the space and time component description of the theory includes two electric Carroll scalars and an electric Carroll fermion.Similarly, we introduce a G-chiral superfield and use it to construct the G-Wess-Zumino model, which serves as the G-supersymmetric extension of magnetic Galilei matter which includes two magnetic Galilei scalars and one magnetic Galilei fermion.Furthermore, employing the seed Lagrangian approach, we utilize the G-Wess-Zumino and C-Wess-Zumino models as seeds to derive (1) a magnetic Carroll theory with G-supersymmetry and (2) an electric Galilei theory with C-supersymmetry.
The paper is structured as follows: In Section 2, we delve into the contractions of the 4D, N = 1 super-Poincaré algebra leading to the super-Carroll and super-Galilei algebras.Moreover, using the group manifold approach we (a.) define suitable bosonic and fermionic superspace coordinates, (b.) derive their transformations under supersymmetry and boosts and (c.) define the differential operators that realize the action of supersymmetry and boost generators on the space of (super)fields as well as the necessary superspace spinorial derivatives.In Section 3, we develop electric and magnetic spin 1/2 fermionic theories that are invariant under Carroll boosts.Moving on to Section 4, we repeat the process for fermionic theories invariant under Galilei boosts.Sections 5 and 6 discuss the supersymmetric extension of electric and magnetic Carroll theories as well as electric and magnetic Galilei theories.In Appendices B and C, we offer a comprehensive review of all established methodologies used to construct the known electric and magnetic descriptions of Carroll and Galilei scalars.Also, for completeness, we discuss in Appendix D the formal off-shell supersymmetric extension of the "simplest" theory for magnetic Carroll fields as defined in [39].
2 Super-Carroll and Super-Galilei Algebras We start with the 4D, N = 1 super-Poincaré algebra generators J mn , P m , Q α , Q α and introduce a contraction parameter c by defining new generators for some exponents s 0 , r 0 , s 1 , r 1 , k.Notice that the exponents associated with time components are in general different than those associated with spatial components and thus allow for Lorentz symmetry breaking.Also the exponents of Q α and Q α are the same due to Hermiticity.The nonvanishing parts of the algebra of these generators are: In general, depending on the values of the various exponents there can be three types of terms: (i) c independent terms, (ii) c n , n > 0 terms and (iii) 1 c n , n > 0 terms.In order for this algebra to be well defined when we take the limit of the contraction parameter to zero (Carroll) all terms must be of type (i) or type (ii).Similarly, when we take the limit of the contraction parameter to infinity (Galilei) all terms must be of type (i) or type (iii).Moreover, we require the survival of the algebra of spatial rotations (s 1 = 0) and we explore algebras that deviate from the starting algebra (s 0 ̸ = 0).With all that in mind, we find two distinct, nontrivial consistent algebras that correspond to the following choices: In order for the generators of these algebras to behave properly in the appropriate c limit one should choose r 0 = 1 because the Hamiltonian must exist in both cases, and s 0 = 1 for Carroll or s 0 = −1 for Galilei.These restrictions, combined with (2.3), determine all the remaining exponents uniquely, namely, r 1 = 0 for both cases, and k = 1/2 for Carroll or k = 0 for Galilei.
As a result, we find the super-Carroll and super-Galilei algebras.The super-Carroll algebra is generated by: which have the following nonzero (anti)commutators This algebra match precisely the N =1 subalgebra of the Carroll superconformal algebra constructed in [5] and also agrees with the flat limit of the AdS super-Carroll algebra discussed in [40].
The super-Galilei algebra is generated by: which have the following nonzero commutators For each one of these algebras we can introduce an appropriate coordinate system {t, x i , θ α , θ α} via the coset manifold approach.These coordinates parametrize a choice of a representative Ω(t, x i , θ α , θ α) for every left coset generated by the orbits of right multiplication with the subgroup elements generated by spatial rotations and boosts.
A particular, but not often practical, representation of Ω(t, x i , θ α , θ α) is the exponential map (2.9) We can use this representation to deduce the transformation of these coordinates under the action of the above symmetry generators as a result of a motion on the coset space Ω(t, x i , θ α , θ α) → Ω(t ′ , x ′i , θ ′α , θ′ α) produced by left group multiplication of any group element g with the coset representative Ω: where h(t, x i , θ α , θ α, g) is a subgroup element generated by particular spatial rotations and boosts, which depend on the coordinates and the group element g: We find that under Carrollian supersymmetry (δ S ) and Carrollian boosts (δ B ) the coordinates transformations are: Additionally, under Galilean supersymmetry (δ S ) and Galilean boosts (δ B ) the coordinates transform as follows: A useful observation is that both Carrollian and Galilean boost transformations of the coordinates are nilpotent: Galilean: Moreover, we define fields and superfields on the appropriate (super)space and time manifolds.The transformation of these (super)fields is defined based on the particular irreducible representation of the fields and the above coordinate transformations.The infinitesimal Carrollian/Galilean supersymmetry and boost transformations3 of superfields Φ(t, x i , θ α , θ α) are: where the corresponding Q α , Q α, K i differential operators are defined as follows: It is straightforward to check that these differential operators together with time and space translations, P 0 = − i ∂ ∂t and P i = − i ∂ ∂x i , satisfy the super-Carroll (2.6) and super-Galilei (2.8) algebras respectively.
Later in the paper we will consider theories with manifest Carrollian or Galilean supersymmetry.For that purpose, it is convenient to construct the appropriate superspace covariant derivatives.Their defining property is to impose constraints that remain invariant under the corresponding supersymmetry transformations.Therefore, we have the Carrollian superspace covariant derivatives and the Galilean superspace covariant derivatives By comparing equations (2.6) and (2.8), one can observe that the distinction between the two superalgebras lies in the (anti)commutators of the corresponding generators K i , P i , P 0 , Q α , Q α.Nevertheless, the nature of the disparity suggests the existence of a formal mapping between the two structures, achieved by interchanging temporal and spatial directions (t ←→ ⃗ x) or more specifically by interchanging indices 0 and i (0 ←→ i).This formal relation, which becomes exact in 1+1 dimensions, was initially introduced in [41] and remains valid for the supersymmetric extensions of these algebras, as it maps σ 0 P 0 ←→ σ i P i .Furthermore, this duality, rooted in a particle perspective that discerns between time and spatial directions, can be extended to a broader p-brane viewpoint [42].In this broader context, time is part of the longitudinal directions, which are treated distinctively from the transverse directions to the brane.In D = d + 1 dimensions one can introduce an index A = 0, 1, . . ., p which captures longitudinal directions and corresponds to a p-brane where index i = p + 1, . . ., d refers to the remaining transverse directions which correspond to a (D − p − 2)-brane.The two superalgebras can be immediately generalized in the following manner: Ai , P p -brane super-Galilei algebra: Ai , P (2.20) It is now clear that there is a formal relation between these superalgebras generated by the interchanges of indices A and i: A ←→ i.In this relation P A ←→ P i and γ A P A ←→ γ i P i .

Carroll fermions
Naturally, one would like to consider field theories that respect the above superalgebras.Focusing on the bosonic subalgebras (Carroll or Galilei), several methodologies have been developed in order to explore non-Lorentzian models.Prototypical examples are the electric or magnetic Carroll (Galilei) boost invariant scalar field theories constructed by various approaches such as the Hamiltonian formulation [2], the field expansion approach [35], the Lagrange multiplier method [35], and the seed Lagrangian method [36] (a review of each framework is provided in appendix B).Motivated by these techniques, except the Lagrange multiplier method which fails for fermions due to the nonquadratic kinetic term, we employ them to construct fermionic theories that exhibit Carroll (Galilei) boost invariance.

Hamiltonian method
The method of Hamiltonian formulation was first used to build Carroll scalar field theories [2] (for a brief review see B.1).This method depends on having two independent variables in the formulation, the field and its conjugate momentum, that enable us to take different limits.We apply this method to construct Carroll Dirac Lagrangians.For this purpose, we start from the Lagrangian density of the massless Dirac field in four-dimensional Minkowski spacetime, which involves the speed of light c, as follows (up to total derivatives): where ψ := ψ † iγ 0 is the Dirac adjoint.The canonical momenta conjugate to ψ and ψ are and, as a result, the Hamiltonian density becomes We can now derive the action of the massless Dirac field in the Hamiltonian formulation by4 Since this formulation includes independent variables, it is possible to take the Carroll limit c → 0 in two different ways.First, we may take the limit explicitly without rescaling the fields.In this case, the action (3.4) does not involve any explicit c dependence, so we simply replace the fields π ψ and π ψ by auxiliary fields π η and πη , respectively, to indicate that they lose their canonical relation (3.2) with ψ and ψ after taking the limit.By doing this, we obtain the so-called "magnetic" Carroll Dirac action in the Hamiltonian formulation as follows: Alternatively, we can take the limit after rescaling the fields in (3.4), i.e.
π′ ψ and ψ → √ c ψ′ ) preserving the canonical structure.This effectively eliminates c from the canonical relations (3.2) and ensures their well-definedness in the limit.Therefore, by rescaling the fields in the action (3.4), taking the limit c → 0, and dropping the prime, we find the so-called "electric" Carroll Dirac action in the Hamiltonian formulation

Field expansion method
By expanding the field around c = 0, the field expansion method was initially employed to build Carroll scalar Lagrangians [35] (see B.2 for a short overview).The expansion results in extra field that appears in the Lagrangian and leads to obtaining electric and magnetic Carroll invariant Lagrangians at the same time.We use this method for spin 1/2 fermions.For this purpose, let us consider again the Lagrangian density of the massless Dirac field in four dimensions We then perform an expansion around c = 0 in the Dirac field5 for some β, and introduce L eC and L mC through the Lagrangian density By substituting the field expansion (3.8) into the Lagrangian (3.7) and comparing with (3.9), we can extract both the electric and magnetic Carroll Dirac Lagrangians, respectively, as follows: We next explore the transformation properties of these Lagrangians under Carroll boost.To do this, we use the fact that the spinor field ψ transforms like a scalar (B.13) under Carroll boosts.Indeed, as we know, when subjected to a Lorentz boost, the commutator between the Lorentz boost generator J 0i and a spinor field ψ takes the form [ J 0i , ψ ] ∝ σ 0i ψ.By introducing the Carroll boost generator (2.5), K (C) i = c J 0i , this commutation relation will take an overall factor of c on the right-hand side 6 .As we consider the Carroll limit c → 0, the commutator simplifies to [ K (C) i , ψ ] = 0, demonstrating that the Carroll spinor field transforms as a scalar under the Carroll boost.As a result of this observation, we find that the electric (3.10) and magnetic (3.11) Carroll Dirac Lagrangians remain invariant (up to total derivatives) when subjected to the following Carroll boost transformations where b i are the Carroll boost parameters.The next subsection will provide another approach to obtain these transformations.
The equations of motion for the electric (3.10) and magnetic (3.11) Lagrangians read as follows: eC − e.o.m : mC − e.o.m : Similar to the bosonic case (B.16), a matrix equation can be used to represent two magnetic Carroll Dirac field equations (3.14) more compactly In this form, the magnetic Carroll Dirac Lagrangian density (3.11) can be expressed by Finally, we note that the electric Carroll Lagrangian (3.10) is equivalent to the one in the Hamiltonian formulation (3.6) by imposing the conditions π ψ = − 1 2 ψ γ 0 and π ψ = 1 2 γ 0 ψ, which follow from rescaling the fields in (3.2).Moreover, the magnetic Carroll Lagrangian (3.11) would be equivalent to (3.5) by defining π η := − η γ 0 and πη := γ 0 η, which arises from the fact of being Hermitian.

Seed Lagrangian method
The seed Lagrangian method is an alternative approach of constructing Carroll invariant theories.It was introduced in [36] as a method to construct new non-Lorentzian scalar theories and it relies on the nilpotence of the Carrollian and Galilean boost transformations (2.14).We use this method as an alternative approach to constructing the magnetic Carroll Dirac Lagrangian (for a review of magnetic scalar Lagrangian refer to B.4).In this case the seed Lagrangian is the magnetic Galilei Dirac Lagrangian (4.9) (discussed in the next section): (3.17) Obviously, this theory is not invariant under Carrollian boosts, specifically its transformation under δ where we used the fact that (i) the integral's measure is invariant under infinitesimal boost transformations, (ii) the spinor field ψ(t, x) transforms like a scalar under boosts7 and (iii) the spatial derivative of ψ transforms Now observe that the quantity ∂ t ψ is invariant under Carrollian boost transformations: Hence the nilpotence of δ B is realized in this theory in the following manner: A consequence of this, is that the nonzero transformation (3.18) of the seed Lagrangian can be compensated by the addition of a second term in the Lagrangian which involves a fermionic Lagrange multiplier η By choosing the transformation of the Lagrange multiplier η under Carrollian boosts appropriately we can construct a Carrollian invariant theory for the Dirac field where the spatial derivatives dominate (magnetic) The Lagrange multiplier η of the seed Lagrangian method we just described can be understood as the η fermionic field in the field expansion method (3.8).

Galilei fermions
In this section, we use the methods that have been employed to build Carroll scalar fields [2,35,36] to develop fermionic theories that respect Galilei boost symmetry 8 .We exclude the Lagrange multiplier method, which fails for fermions, and use other methods to derive electric and magnetic Galilei fermionic field Lagrangians.We refer the reader to appendix C for a review of the Galilei scalar field theories obtained by different techniques.

Hamiltonian method
As we explained in the Galilei scalar part C.1, the standard Hamiltonian formulation method for Carroll Lagrangians [2] cannot be used to derive electric and magnetic Galilei Lagrangians.Instead, we adopt a novel approach called the "Hamiltonian-like" formulation, which involves introducing new conjugate momenta for the fields.To illustrate this, we consider the massless Dirac field in four-dimensional Minkowski spacetime, whose Lagrangian density is given by (3.1).We then define ψi ≡ ∂ i ψ and ψi ≡ ∂ i ψ, and introduce the canonical momenta associated with the fields ψ and ψ as follows: Thus, the Hamiltonian-like density, which is different from the Hamiltonian density (3.3), becomes As a result, the action of the massless Dirac field in the Hamiltonian-like formulation reads9 We have reached a point where we can apply the Galilei limit c → ∞ to (4.3) in two distinct ways.First, we take the limit directly without rescaling the fields, which also avoids any singularity in (4.1).This implies that (4.3) results in the magnetic Galilei Dirac action We can also apply the Galilei limit c → ∞ to (4.3) after rescaling the fields by a factor of √ c.That is, we replace Π i ψ by 1 √ c Π i ψ and ψ by √ c ψ, and similarly for Πi ψ and ψ.The rescaling eliminates the factor of 1 c from (4.3), but it also violates the canonical relations (4.1) in the limit.Therefore, we have to replace Π i ψ and Πi ψ by new auxiliary fields Π i η and Πi η respectively after taking the limit.By doing this, we obtain the electric Galilei Dirac action

Field expansion method
The field expansion method for Carrollian fields [35] uses c = 0 as the expansion point.For Galilean fields, we choose c = ∞ as the expansion point.We applied this method to Galilei scalar fields in C.2 and reproduced Galilei scalar Lagrangians in this manner.Now we use this method for fermions.To do this, let us take into account the Lagrangian density of the massless Dirac field in four dimensions where Ψ := Ψ † iγ 0 is the Dirac adjoint.We can then make an expansion around c = ∞ in the Dirac field10 for some β, and introduce L mG and L eG via the Lagrangian density When we use (4.7) for the Lagrangian (4.6) and compare it with (4.8), we can read both the magnetic and electric Galilei Dirac Lagrangians respectively We will now investigate the transformation properties of these Lagrangians under Galilei boost.We use the fact that the spinor field ψ transforms like a scalar (C.11) under Galilei boost.We know that a Lorentz boost on a spinor field ψ makes the commutator [ J 0i , ψ ] ∝ σ 0i ψ.We define the Galilei boost generator (2.7) as Then the commutator gets a factor of 1 c on the right-hand side11 .In the Galilei limit c → ∞, the commutator becomes [ K (G) i , ψ ] = 0.This illustrates that the Galilei spinor field transforms as a scalar under Galilei boost.Because of this fact, we find that the magnetic (4.9) and electric (4.10) Galilei Dirac Lagrangians are invariant (up to total derivatives) under the following Galilei boost transformations: where b i are the parameters of the Galilei boost.We will obtain these transformations in a different way in the next subsection.
The magnetic Galilei Lagrangian (4.9) gives the equation of motion γ i ∂ i ψ = 0.The electric Galilei Lagrangian (4.10) has the equation of motion in a compact form as In this form, the electric Galilei Dirac Lagrangian density (4.10) can be expressed as We remark that by imposing the canonical relations (4.1), the magnetic Galilei Dirac Lagrangian (4.9) is the same as the one in the Hamiltonian formulation (4.4).Furthermore, by defining Π i η := − η γ i and Πi η := γ i η the electric Galilei Dirac Lagrangian (4.10) would be the same as (4.5).

Seed Lagrangian method
The electric formulation of a Galilei boost invariant fermion can be also deduced via the seed Lagrangian method.In this case the seed Lagrangian is that of an electric Carroll fermion This theory is not invariant under Galilean boost, however its transformation under δ B takes the following form: This is special because the transformation is the product of two factors and one of them (∼ ∂ i ψ) is Galilean boost invariant, hence the deformation of the theory can be absorbed by adding an appropriate compensating term.Equation (4.15) can be easily verified by the transformation of the fermion and its derivatives under Galilean boosts: In this theory, the nilpotence of δ B is realized in the following manner: To restore G-boost invariance, we must add the following compensating terms where λ i is a collection of compensating fields whose transformation under Galilean boosts are We conclude that the electric description of a Galilei boost invariant fermion can take the form We note that the latter is equivalent to the electric Galilei Lagrangian in the Hamiltonian formulation (4.5), by introducing Π i η := − λi γ 0 and Πi η := γ 0 λ i .Moreover, it can be connected to the one in the field expansion method (4.10), by defining λ i := − γ 0 γ i η.

Carroll Boost Invariant Supersymmetric Theories
A natural question to ask is whether the various formulations (electric or magnetic) of Carroll or Galilei fermions described in sections 3 and 4 can be combined with the corresponding formulations of Carroll or Galilei scalars [2,35,36] (see appendices B and C for a review) to define supersymmetric theories.As established in section 2, there are two types of supersymmetries.We can have either (i ) C-supersymmetry or (ii ) G-supersymmetry In what follows, we demonstrate the construction of supersymmetric theories in which the Carrollian fermion can be incorporated into either a C-supermultiplet or a G-supermultiplet 12 , contingent upon whether we utilize its electric or magnetic description respectively.Similarly, magnetic Galilean fermions are embedded naturally in G-supermultiplets, but electric Galilean fermions are members of C-supersymmetric multiplets.
The description of these theories is given in the appropriate superspace which makes the corresponding supersymmetries manifest.Using the C/G-supersymmetric covariant derivatives (2.17) and (2.18), one can define irreducible representations of matter multiplets (including spin 0 and spin 1/2 fields) by imposing constraints on scalar superfields.In our description we will consider C-Chiral and G-Chiral superfields that are defined as follows: The superspace and component Lagrangian descriptions of these matter multiplets are respectively: (5.3b) We will refer to (5.3a) as the C-Wess-Zumino model and, similarly, we will call (5.3b) the G-Wess-Zumino model.It is evident that the C-Wess-Zumino model will be relevant for "electric" descriptions, whereas the G-Wess-Zumino model will be appropriate for "magnetic" descriptions.
An interesting observation is that Carrollian supersymmetry (5.1a) is equivalent to the quantum mechanical supersymmetry algebra, therefore one can use the techniques of Adinkras [45][46][47][48] in order to classify and construct representations of C-supersymmetry beyond the C-chiral (5.2a) one.
In this section, we study the transformation properties of the C-Wess-Zumino and G-Wess-Zumino models under Carrollian boosts, generated by K we find that Carrollian boosts preserve the C-chiral condition (5.2a), but not the G-chiral condition (5.2b).However, notice that the deviation from G-chirality (θ α (σ i ) α α ∂ t ) is Carroll boost invariant.Therefore, we can construct two Carroll boost invariant supersymmetric theories.The first one will have C-supersymmetry and it will be given precisely by the C-Wess-Zumino model, while the second one will have G-supersymmetry and it will be described by the G-Wess-Zumino model with the addition of a Lagrange multiplier superfield that imposes appropriate constraints 13 on the prepotential Λ (G) of the G-chiral multiplet.

Supersymmetric extension of electric Carroll theory
By solving the C-chiral constraint (5.2a), we can express Φ (C) in terms of an unconstrained scalar superfield, Λ (C) : Under Carrollian boosts, the unconstrained prepotential Λ (C) transforms as expected: and because the C-covariant derivative also remains invariant (5.4), the C-chiral superfield transforms as follows: (5.7) Therefore the C-Wess-Zumino model (5.3a) is invariant under Carrollian boost and provides the C-supersymmetric extension of the electric Carroll scalar.The component action of this theory in a two component spinor description is given in (5.3a) and in four-component spinors the action takes the following form: This action includes the following terms: (i ) two real electric Carroll scalar fields (B.20) ϕ R and ϕ I which are the real and imaginary parts of ϕ, (ii ) two real auxiliary scalar fields F R and F I which are the real and imaginary parts of F , they have no dynamics and are required by off-shell C-supersymmetry invariance, and (iii ) an electric Carroll Majorana spinor ψ, which is governed by 13 Using (5.4) one can guess that the appropriate constraint should be the vanishing of the time derivative of the G-chiral superfield and that of its prepotential Λ (G) .

Supersymmetric extension of magnetic Carroll theory
Similar to the C-chiral superfield, the G-chiral superfield can also be expressed in terms of an unconstrained prepotential Λ (G) by solving constraint (5.2b): Φ which is an unconstrained scalar superfield, transforms under Carrollian boost in the usual manner: However, as indicated by (5.4), the G-covariant derivatives do transform under Carrollian boosts, therefore the transformation of Φ (G) is: (5.12) which obviously breaks the G-chirality of Φ (G) .Therefore, the Carroll boost invariant theory cannot be expressed purely in terms of superfield Φ (G) and the bare prepotential Λ (G) must participate.
Moreover, the Carroll boost transformation of D(G) α Λ (G) is: hence we get the following sequence which realizes the nilpotent property of δ It becomes obvious that the deformation of the G-Wess-Zumino model under Carroll boost has a special form, it is written as the product of two factors with one of them being Carroll boost invariant.This type of deformation can be compensated by the introduction of an appropriate compensator superfield Σ.Specifically, we find that the following action is invariant under Carroll boosts and G-supersymmetry: where Σ (G) is a G-complex linear superfield [ D(G) ] 2 Σ (G) = 0 and under Carrollian boosts transforms: which respects the complex linear nature of Σ (G) .Action (5.14) can be written equivalently as: where G) , and satisfies the identity D(G) α , and it is defined modulo the redundancy Ξ Using (5.15), the transformation of Ξ α under Carroll boosts is β . (5.18) The corresponding component action is: The terms in the first line are the G-Wess-Zumino terms (5.3b) consisting of: (i ) the magnetic Galilei Lagrangian (C.17) for two real scalar fields π R and π I which are the real and imaginary parts of π, (ii ) two real auxiliary scalar fields, G R and G I , the real and imaginary parts of G, which have no dynamics but are required by G-supersymmetry, and (iii ) a magnetic Galilei Lagrangian (4.9) for Majorana field λ.The terms in the second line, are the Lagrange multiplier terms that impose appropriate constraints and make the theory invariant under Carroll boosts.The first two terms are the constraint terms for the two scalars π R , π I , as expected by (B.26), the following two terms are the analog terms for the auxiliary fields G R , G I , and the last two terms are the constraint terms for the fermion λ as found in (3.11).Finally, the terms in the third and forth lines are required by the supersymmetry transformation of the terms in the second line.Specifically, the component fields that appear in the above action are organized in the following supermultiplets Γ (G) and Ξ (G) in the following manner: Each multiplet carries 12 bosonic and 12 fermionic off-shell degrees of freedom.The supersymmetry transformations of these fields are easily derived from their corresponding supermultiplets and they generalise the transformations (6.5) of the smaller G-Chiral multiplet {π R , π I , G R , G I , λ}.The closure of these transformations is consistent with G-supersymmetry: signals that the Galilean boosts preserve the G-chiral condition (5.2b), but not C-chirality (5.2a).Hence, the supersymmetric extension of the magnetic Galilei scalar theory will be provided by the G-Wess-Zumino model (5.3b) which is manifestly invariant under G-supersymmetry, whereas the supersymmetric extension of the electric Galilei scalar theory will be given by the C-Wess-Zumino model (5.3a) which is manifestly C-supersymmetric.

Supersymmetric extension of magnetic Galilei theory
Repeating the same type of arguments as in section 5.1 and using (6.1) we conclude that the G-chiral superfield Φ (G) is invariant under Galilean boosts: Hence, the G-Wess-Zumino model is invariant under Galilean boost as well as G-supersymmetry: 3) The component action of this theory in four component description is The action includes the following terms: (i ) the Lagrangians of two real magnetic Galilei scalar fields (C.17) ϕ R and ϕ I , (ii ) two real auxiliary scalar fields G R and G I which have no dynamics and are required by off-shell G-supersymmetry, and (iii ) a magnetic Galilei Majorana field λ, which is described by the Lagrangian (4.9).The explicit G-supersymmetry transformations of all the fields are δπ R = ε λ , δπ I = ε i γ 5 λ , (6.5a) where ε is an arbitrary constant Majorana spinor object that parametrizes the supersymmetry transformations.It is straightforward to check that the algebra of the above transformations indeed closes off-shell to the expected G-supersymmetry, for all fields: Systems that respect this type of supersymmetry have been studied in [49].

Supersymmetric extension of electric Galilei theory
Using the seed Lagrangian approach [36], it was shown that an electric description of Galilean scalar can be obtained (see C.4 for review) by considering the electric Carrollian description and adding an appropriate Lagrange multiplier term that enforces Galilei boost invariance and imposes the appropriate constraints.The supersymmetric extension of that theory is straightforward.The electric Carrollian scalar acquires a C-supersymmetric partner, the electric Carroll fermion (3.10).
Together with the appropriate auxiliary fields they make the C-Wess-Zumino model (5.3a) described by a C-chiral superfield Φ (C) .Similarly, the bosonic and fermionic Lagrange multipliers that appear in the electric description of the Galilean bosons (C.22) and fermions (4.20) respectively, will be organized into a second C-supermultiplet.
Using (6.1), we calculate the transformation of the prepotential Λ (C) and its derivatives These transformations realize the nilpotency of the Galilean boost transformation δ Applying the seed Lagrangian approach to superspace, we start with the C-Wess-Zumino model S C) and transform it under Galilean boosts: . (6.9) Once again the deformation of the theory is factorized and one of the factors ( invariant, as seen in (6.8).Therefore, the deformation can be absorbed by introducing a collection of compensator superfields Σ (C) i with appropriately chosen transformations under Galilean boosts.The following action is invariant under Galilei boosts and C-supersymmetry: The action (6.10) can be expressed in terms of the variables Γ(C) α and Ξ(C) i α defined as: where Ξ(C) i α is defined up modulo the redundancy Ξ The component action of this theory can be extracted from the above superspace action: The terms on the first line are the C-Wess-Zumino terms (5.3a) consisting of: (i ) the electric Carroll Lagrangians (B.20) for two real electric Carroll scalar fields ϕ R and ϕ I which are the real and imaginary parts of ϕ, (ii ) two real auxiliary scalar fields, F R and F I , the real and imaginary parts of F , which have no dynamics but are required by C-supersymmetry, and (iii ) an electric Carroll Lagrangian (3.10) for Majorana field ψ.The terms on the second line, are the Lagrange multiplier terms that impose appropriate constraints and make the theory invariant under Galilei boosts.The first two terms are the constraint terms for the two scalars ϕ R , ϕ I , as expected by [36] (see (C.22)), the following two terms are the analog terms for the auxiliary fields F R , F I , and the last two terms are the constraint terms for the fermion ψ as found in (4.20).Finally, the third and fourth line terms are required by the supersymmetry transformations of the second line terms.The component fields are organized into four supermultiplets Γ (C) and Ξ (C) i , i = 1, 2, 3 as follows: Each one of these supermultiplets has 12 bosonic and 12 fermionic d.o.f. and one can easily check that their supersymmetry transformations close off-shell to the C-supersymmetry algebra As mentioned previously supersymmetric Carrollian multiplets can be understood as supersymmetric quantum mechanical systems.In this case, the detailed analysis of the quantum mechanical complex linear supermultiplet, which corresponds to the Carrollian multiplet Σ (C) can be found in [50][51][52].

Conclusions and outlook
In this paper, we have studied the non-Lorentzian algebras and theories associated with the Carroll and Galilei symmetries, and we have addressed the following important issues.
• We explored the construction of super-Carroll and super-Galilei algebras through the introduction of contraction parameters in the 4D, N = 1 super-Poincaré algebra.These algebras exhibit a breaking of Lorentz symmetry due to a characteristic distinction between temporal and spatial components which is generated by a difference in the associated exponents of the contraction parameter.To ensure the proper behavior of these algebras in the limits of c → 0 (Carroll) and c → ∞ (Galilei), we imposed specific conditions on these exponents.These conditions, ultimately give rise to the super-Carroll and super-Galilei algebras, which have been described in detail, including their generators (2.5), (2.7) and commutation relations (2.6), (2.8) respectively.
• Furthermore, we discussed the action of Carrollian and Galilean supersymmetries, as well as Carrollian and Galilean boosts on coordinates and (super)fields.Notably, both Carrollian and Galilean boost transformations of coordinates were shown to be nilpotent.To facilitate the formulation of theories with manifest Carrollian or Galilean supersymmetry, we introduced the corresponding superspace covariant derivatives (2.17), (2.18).These derivatives play an important role in understanding the behavior of the theory under Carrollian or Galilean boosts within a supersymmetric framework.
• From the field theoretical point of view, the scalar field theories with Carroll and Galilei symmetries have been investigated, and they exhibit electric and magnetic sectors [2,35,36].The electric sector corresponds to the theory where the time derivatives are dominant, while the magnetic sector is the theory in which the spatial derivatives are dominant.We reviewed four existing methods to construct the Carroll scalar field theories in appendix B: the Hamiltonian formulation, the field expansion approach, the Lagrange multiplier technique, and the seed Lagrangian method.For the Galilei scalar field theory, only the seed Lagrangian method has been used in the literature [36], but we extend it to the other three methods in appendix C.
• Motivated by the success of the four methods employed in the scalar field theory, our investigation naturally expanded to encompass the construction of Carroll and Galilei fermionic field theories, including Dirac, Majorana, and Weyl fermions.We used three methods, except the Lagrange multiplier method which is not suitable for fermions due to the nonquadratic kinetic term, to construct the Carroll and Galilei fermionic field theories in sections 3 and 4 respectively.Two important points that facilitate these constructions are: (i ) Carroll and Galilei fermions transform like a scalar under Carroll/Galilei boosts as a result of the algebra contraction process, and (ii ) rather than incorporating the contraction parameter within the metric as in [37,38] 16 , we opted to include it within the component coordinates; thus, we kept the Minkowski metric, the Clifford algebra, and gamma matrices unchanged.While this consideration has no significant effect in the scalar field theory, it has an important effect in the fermionic theories and allows us to use the same gamma matrices as in relativistic theory and consequently led us to consistent fermionic theories with supersymmetry.
• Moreover, we investigated the supersymmetric extension of these models.There are two types of non-Lorentzian supersymmetry that one can consider, C-supersymmetry (5.1a) and Gsupersymmetry (5.1b).We found that a natural description of simple matter theories that are manifestly invariant under C or G supersymmetry, is provided by the C-Wess-Zumino (5.3a) and G-Wess-Zumino (5.3b) models.These models combine the known descriptions of Carroll/Galilei scalars with the newly found Carroll/Galilei fermions and promote them into appropriate supermultiplets with the addition of appropriate auxiliary fields.The superspace description of these theories is provided by the definition of appropriate C-chiral superfields (5.2a) and G-chiral superfields (5.2b).In addition, we provided the component description of these models and the detailed supersymmetry transformation laws for all the fields.
• The structure of the component description of C-Wess-Zumino model makes obvious that it corresponds to the C-supersymmetrization of electric Carrollian scalar and fermionic fields.On the other hand, the G-Wess-Zumino model provides the G-supersymmetrization of magnetic Galilean scalars and fermions.Naturally, we searched for the supersymmetric extensions of the magnetic Carrollian and electric Galilean theories.By applying the seed Lagrangian methodology, presented in [36], in superspace and using G-Wess-Zumino and C-Wess-Zumino theories as seeds we found the corresponding magnetic Carroll (5.19) and electric Galilei (6.14) theories.It is interesting to emphasize that the super magnetic Carroll theory is invariant under G-supersymmetry, whereas the super electric Galilei theory is invariant under C-supersymmetry.However, it is important to emphasize that in both cases there can be supersymmetry breaking solutions where the supersymmetric auxiliary fields acquire a nontrivial vev.
In the future, we would like to investigate the C/G supersymmetric extensions of non-Lorentzian gauge theories.A lot of work has be done for non-Lorentzian (super)gravities [53][54][55][56][57] (for a review see [58] and references therein) but not much is known about non-Lorentzian higher spin theories 17 and their C/G supersymmetric generalizations.Another interesting direction is the exploration of variant representations of non-Lorentzian supersymmetries.The existence of variant supersymmetric representations is a characteristic feature of supersymmetric theories in various dimensions and it is often related to special features of target space geometry or the existence of (weak/strong) dualities.Therefore, it would be interesting to search for variant descriptions of non-Lorentzian matter or gauge theories.For the case of C-supersymmetry such a search can be assisted by the Adinkras [45][46][47][48] methodology of classifying one dimensional representations of supersymmetry algebra.Specifically, by interpreting C-supersymmetry (5.1a) not as the outcome of a Lorentz breaking contraction of the Super-Poincaré algebra, but as its one dimensional (QM) reduction, one can use Adinkras to discover such variant representations of C-supersymmetry and extended C-supersymmetries.Another interesting direction is the Carroll/Fracton correspondence, which has been studied in recent works (see, e.g., [59,60] and references therein).It would be interesting to investigate the existence of a similar type correspondence for Galilean theories.and the seed Lagrangian method [36].It is worth noting that while these methods differ in their approaches, they are fundamentally equivalent to each other.

B.1 Hamiltonian method
The Hamiltonian formulation consists of two independent variables (the field ϕ and the canonical momentum conjugate to the field π ϕ ), which allows us to take the Carroll limit in two different ways: explicit limit and the limit after rescaling the fields [2].To this end, let us consider a massless scalar field in Minkowski spacetime with the Lagrangian density The canonical momentum conjugate to the field ϕ reads and then the Hamiltonian density becomes Accordingly, the Hamiltonian action of the massless scalar field can be written as There are two ways to take the Carroll limit c → 0. First, we may directly take the limit of the visible c's in (B.4), without rescaling the fields.However, this procedure violates the canonical relation between the field and its conjugate momentum, as seen from (B.2).Therefore, after taking the limit, one should replace π ϕ by an auxiliary field π φ in (B.4) to reflect the breakdown of the canonical relation.This yields the magnetic Carroll scalar action in the Hamiltonian formulation Alternatively, we can rescale fields in (B.4), i.e. ϕ → c ϕ ′ and π ϕ → 1 c π ′ ϕ , which preserves the canonical structure.Particularly the canonical relation (B.2) becomes π ′ ϕ = φ′ .Therefore, in this way, we first perform the rescaling, take the limit c → 0, and remove the primes from the rescaled variables.This leads to the electric Carroll scalar action in the Hamiltonian formulation

B.2 Field expansion method
The field expansion technique [35] is another approach to derive Carroll scalar field Lagrangians.The Lagrangian density of the massless scalar field, with the speed of light c involved, is given by Let us now make an expansion around c = 0 in the scalar field18 for some α, and define L eC and L mC trough the Lagrangian density By applying (B.8) to (B.7) and comparing with (B.9), one can derive the electric and magnetic Carroll scalar Lagrangian densities, respectively Let us discuss the invariance of the electric and magnetic Lagrangians under a Carroll boost, which is an ultrarelativistic limit of a Lorentz boost.Under a Lorentz boost, with ⃗ β being the Lorentz boost parameter, the field Φ in (B.7) transforms as

B.3 Lagrange multiplier method
Another way to obtain Carroll scalar Lagrangians is by using the Lagrange multiplier method, as presented in [35].For this purpose, let us consider two equivalent Lagrangian densities for a massless scalar field, given by (B.1) and where χ in the latter is a Lagrange multiplier.Indeed, the variation of L ′ with respect to χ equal to zero leads to the equation of motion of χ, which is c 2 χ − ∂ t ϕ = 0. Solving for χ and substituting back into L ′ one finds L ′ = L. Accordingly, two equivalent expressions for the scalar field Lagrangian density allow us to obtain the Carroll limit, c → 0, in two different ways.First, we may directly take the limit of the visible c in the Lagrangian L ′ (B.18), without rescaling the field.This yields the magnetic Carroll scalar Lagrangian density We note that the Lagrange multiplier χ in this case can be understood as the momentum of φ, χ = π φ , in the Hamiltonian action (B.5), or equivalently as its time derivative, χ = ∂ t φ, in the Lagrangian (B.11).
Alternatively, one can rescale the field in the Lagrangian L (B.1), i.e. ϕ → c ϕ, and then take the limit c → 0. This leads to the electric Carroll scalar Lagrangian density Using the Lagrange multiplier method, we can see that the electric (B.20) and magnetic (B.19) Carroll scalar Lagrangians are invariant (up to total derivatives) under the Carroll boost transformations, which have the following form We observe that the magnetic Carroll Lagrangian (B.19) preserves its Carroll boost invariance when we set ∂ i ϕ = 0.Such a Lagrangian, called the "simplest" magnetic Lagrangian in [39], would be invariant under (B.21) by setting ∂ i ϕ = 0. We have generalized this case and its fermionic counterpart to a supersymmetric theory with C-supersymmetry, as shown in appendix D.

B.4 Seed Lagrange method
The seed Lagrangian method is an alternative approach to obtain the Carroll scalar field theories [36].Starting from the Lagrangian density of a massless scalar field (B.1), we can rescale the field as ϕ → c ϕ, and take the Carroll limit c → 0. This led to the electric Carroll scalar Lagrangian (B.20) which is invariant up to total derivatives under the Carroll boost transformation To find the magnetic Carroll scalar Lagrangian, one can use the magnetic Galilei scalar Lagrangian (discussed in the next section) However, the seed Lagrangian can be compensated by the addition of a second term in the Lagrangian which involves a Lagrange multiplier χ This implies that the magnetic Carroll scalar Lagrangian can be obtained as We then impose the symmetry of the Lagrangian (B.26) under the Carroll boost transformation (B.22).This allows us to find the transformation of χ as

C Galilei scalar field
The Galilei scalar field theories (electric and magnetic) have been formulated solely through the seed Lagrangian method [36].In this section, we not only provide a review of the seed Lagrangian method in C.4, but also motivated by [2,35], present three additional approaches for deriving the Galilei scalar Lagrangians: the Hamiltonian formulation method, the field expansion approach, and the Lagrange multiplier method.These methods have different procedures, but they all lead to the equivalent results.

C.1 Hamiltonian method
The Hamiltonian method that applies to Carroll scalars theories [2] is ineffective for the construction of Galilei scalar theories.Indeed, when we take the Galilei limit, c → ∞, the field π ϕ becomes zero according to (B.2).This leads to an indeterminate form of (0 × ∞) in the Hamiltonian (B.3) and in the middle term of the action (B.4).Therefore, to avoid this ambiguity, we establish an alternative formulation, called "Hamiltonian-like" formulation, in which the Galilei scalar field action is well-defined in the limit c → ∞.To this end, considering the Lagrangian density of a massless scalar field (B.1), we just define a new canonical momentum conjugate to the field ϕ, subject to the spatial derivatives of ϕ, which is Then, the Hamiltonian-like density, which is different from the Hamiltonian density (B.3), defines and the action of a massless scalar field in the Hamiltonian-like formulation becomes We observe that the limit c → ∞ becomes well-defined in this new formulation.Accordingly, we can first take the limit without rescaling the fields, which also does not cause any singularity in (C.1).This leads to the action of magnetic Galilei scalar field in the Hamiltonian-like formulation Alternatively, we can rescale the fields in (C.3) as follows: ϕ → c ϕ and Π i ϕ → 1 c Π i ϕ .Then, we apply the limit and replace Π i ϕ with the auxiliary field Π i φ , since the rescaling makes the canonical relation (C.1) singular in the limit.This arrives us at the action of electric Galilei scalar field in the Hamiltonian-like formulation

C.2 Field expansion method
The derivation of the Galilei scalar theory can be achieved by utilizing the field expansion method analogously to the Carroll scalar theory [35].To demonstrate this, we consider the following Lagrangian density of the massless scalar field, which involves the speed of light c : If we then make an expansion around c = ∞ in the scalar field 19 for some α, and define L mG and L eG through the Lagrangian density one arrives respectively at the magnetic and electric Galilei scalar field Lagrangian densities Let us next provide the invariance of these Lagrangians under the Galilei boost.For this purpose, we apply the scalar field expansion (C.7) for (B.12) and introduce ⃗ β the Galilei boost parameter.This yields the Galilei boost transformations under which the magnetic (C.9) and the electric (C.10) Galilei scalar Lagrangians are invariant up to total derivatives.
The magnetic Galilei Lagrangian (C.9) yields the equation of motion ∂ 2 i ϕ = 0.For the electric Galilei Lagrangian (C.10), the equations of motion are ∂ 2 i ϕ = 0 and ∂ 2 i φ − ∂ 2 t ϕ = 0.These can be compactly written as The electric Galilei scalar Lagrangian (C.10), up to total derivatives, can then be expressed as

C.3 Lagrange multiplier method
The Lagrange multiplier method for Carroll scalar theory [35] included a Lagrangian (B.18) that was well-defined in the Carroll limit.However, this Lagrangian becomes singular in the Galilei limit c → ∞.To avoid this, we present and start from an alternative Lagrangian which is well-defined in the Galilei limit.To this end, we employ a Lagrange multiplier χ a and consider the following Lagrangian density for a massless real scalar field: This Lagrangian density is equivalent to the standard one after rescaling the field as ϕ → 1 c ϕ.This can be seen by eliminating the Lagrange multiplier χ a using its equation of motion, which gives χ a = c 2 ∂ a ϕ.Substituting this back into the Lagrangian density (C.14), and rescaling the field ϕ → 1 c ϕ, we obtain (C.15) which is the usual form of the Lagrangian density for a massless scalar field.Having these two Lagrangians, one can take the Galilean limit, c → ∞, explicitly.If we do this, the Lagrangian (C.(C.17) These electric and magnetic Lagrangians are identical to those obtained in [36] by using the seed Lagrangian method.Furthermore, we observe that the Lagrange multiplier χ a in this case has the interpretation of either the negative of the momentum of φ, χ a = − Π a φ , in the Hamiltonian action (C.5), or the spatial derivative of φ, χ a = ∂ a φ, in the Lagrangian (C.10).

C.4 Seed Lagrangian method
We briefly review the seed Lagrangian method, presented in [36], as another technique to derive the Galilei scalar field Lagrangians.We start from the relativistic massless scalar field theory with the Lagrangian density (C.15) and take the limit c → ∞, which corresponds to the Galilei limit.This gives us the magnetic Galilei scalar Lagrangian density (C.17

D Simplest Carroll supersymmetric theory
The theory of the electric Carroll scalar field is characterized by a straightforward Lagrangian, as represented by relation (B.20).However, the Lagrangian density for the magnetic Carroll scalar field has more terms, involving spatial derivative terms and a Lagrange multiplier term (B.19).Nonetheless, it is possible to consider a simpler Lagrangian density for the magnetic Carroll scalar field.In this simplified Lagrangian, the spatial derivatives of the field are eliminated, leaving only the Lagrange multiplier term intact.This Lagrangian, which still exhibits Carroll boost invariance, has been explored in [39] and called the "simplest" magnetic Carroll Lagrangian, given by In this section, we apply the simplest Lagrangians for Carroll fields to construct the simplest supersymmetric theory for the magnetic Carroll case with N = 1 off-shell supersymmetry.We find such an action can be given by The bosonic part of the action contains two real scalar fields, ϕ R and ϕ I , which are the real and imaginary parts of ϕ, and two Lagrange multipliers χ 1 and χ 2 .It also contains four real auxiliary scalar fields: F R and F I , which are the real and imaginary parts of F , and G R and G I , which are the real and imaginary parts of G.These auxiliary fields have no dynamics, but they are necessary for the off-shell supersymmetry.The fermionic part of the action includes a Majorana spinor field ψ and the fermionic Lagrange multiplier λ.We find that the action (D. which demonstrates that the supersymmetry algebra closes off-shell.It is important to remark that the super magnetic Carroll theory does not possess C-supersymmetry, but rather G-supersymmetry as specified by (5.21).However, when we consider the simplest Lagrangians for the Carroll fields, given by (D.1) and (D.2), we can have a C-supersymmetry (D.5) rather than a G-supersymmetry.
We could also introduce the simplest electric Galilei Lagrangians and build their supersymmetric extension, but for now, we will refrain from doing so, as the process follows a similarly straightforward method.

i
, aiming toward the construction of Carrollian boost invariant supersymmetric theories.By calculating the commutator of K

.21) 6
Galilei Boost Invariant Supersymmetric TheoriesNow, we study the transformation properties of the C-Wess-Zumino and G-Wess-Zumino models under Galilean boosts, generated by K 12) and the Lagrangian density (B.7) transforms into a total derivative.Then, one may use the scalar field expansion (B.8) for (B.12) and define ⃗ β = c ⃗ b, with ⃗ b being the Carroll boost parameter.This yields the Carroll boost transformations δ

= 1 c
⃗ b, with ⃗ b being19 Similar to the Carroll scalar case (B.8), Galilei boost invariant terms are not influenced by odd powers of c.

1 2 (
), which is invariant under the Galilean boost transformation δ(G) B ϕ = t b i ∂ i ϕ .(C.18)On the other side, to derive the electric Galilei scalar Lagrangian density, one can utilize the electric Carroll scalar Lagrangian density (B.10) as a seed Lagrangian, that isL eC = 1 2 (∂ t ϕ) 2 .(C.19)This choice of the seed Lagrangian is not invariant under the Galilean boost transformation (C.18).It transforms as δ(G) B L eC = − ϕ ∂ t b i ∂ i ϕ .(C.20)In order to obtain a Galilei boost invariant Lagrangian density, we can compensate the seed Lagrangian (C.19) by the addition of a second term in the Lagrangian which involves a Lagrange multiplierχ a L χ eC = − χ a ∂ a ϕ .(C.21)As a consequence, the electric Galilei scalar Lagrangian density can be found asL eG = L eC + L χ eC = ∂ t ϕ) 2 − χ a ∂ a ϕ .(C.22)Imposing the symmetry of this Lagrangian under the Galilei boost transformation (C.18), one can determine the transformation of χ a , which becomesδ (G) B χ a = t b i ∂ i χ a + b a ∂ t ϕ .(C.23)Therefore, the obtained electric Galilei scalar Lagrangian density (C.22) is invariant under the Galilean boost transformations (C.18) and (C.23), up to total derivatives.
L S mC = χ ∂ t ϕ .(D.1) Similarly, by eliminating the spatial derivatives of the field in (3.11), and defining λ := ∂ t η for consistency of the supersymmetric theory, we can present the simplest Carroll boost invariant magnetic Lagrangian for fermions, which is given by L S mC = λ γ 0 ψ − ψ γ 0 λ .(D.2)