Confined dynamical systems with Carroll or Galilei symmetries

We introduce a general method to construct classes of dynamical systems invariant under generalizations of the Carroll and of the Galilei groups. The method consists in starting from a space-time in $D+1$ dimensions and partitioning it in two parts, the first Minkowskian and the second Euclidean. Tha action consist of two terms that are separately invariant the Minkwoskian and Euclidean partitioning. One of those contains a system of lagrangian multiplies that confine the system to a subspace. The other term defines the dynamics of the system. The total lagrangian is invariant under the Carroll or the Galilei groups with zero central charge.

The Galilei and Carroll groups can be obtained via a convenient contraction of the Poincaré group [19]. In order to obtain the Bargmann group [20], that is the Galilei group with a central charge, it is necessary to extend the Poincaré group by a U (1) factor, whereas in the case of three dimensions one needs to consider the contraction of Poincaré ⊗U (1) ⊗ U (1) [21], since in this case the Galilei group has two central extensions [22]. Otherwise one gets the Galilei algebra with zero central charge. It should be noted that the "non-relativistic" limit is not unique [23][24][25][26] when we have in mind extended objects. In other words, there is not a unique contraction of the Poincaré group. In some cases these new algebras could contain non-central extensions [27,28]. From now on, unless differently specified, for Galilei and Carroll algebras, we will mean the algebras with zero central or non-central charges.
The aim of this paper is to introduce a general method to construct classes of dynamical systems invariant under generalizations of the Carroll and of the Galilei groups, [16,23,[26][27][28][29][30]. These generalizations are obtained through various contractions of the Poincaré group in (1, D) space-time dimensions. The general structure of the contracted algebras is that both contain the direct product of a Poincaré algebra in lower dimensions times an Euclidean algebra of rotations and translations in the complementary spatial dimensions. The contracted algebras contain also generalized boosts that rotate the generators of the lower Poincare with the ones of the Euclidean group. These boosts are a generalization of the ones of Galilei and Carroll groups. We will write the generators of the the Poincaré group, ISO(1, D), M µν and P µ µ, ν = 0, 1, · · · , D, as (M αβ , M ab , P α , P a , M αb ≡ B αb ), with α, β = 0, 1, · · · , k − 1 and a, b = k, · · · , D.
The contractions will be defined dividing the components of the momentum P µ , in two sets P α and P a . The contractions we will consider consist, in the Carroll case, in rescaling the momenta the coordinates of the Euclidean space and their derivatives 3 , such to conserve the momentum also in that sector. We shall see that the lagrangian multipliers of the time derivatives of the euclidean coordinates can be thought as the canonical momenta of the Euclidean space variables. The variation induced by the boosts in the action describing the dynamical system is then compensated by a corresponding variation of the lagrange multipliers. On the other hand, the equations of motion resulting from the variation of the lagrange multipliers confine the system to live in the Minkowski subspace, since the typical implication is that the time derivatives of the euclidean variables x a vanish confining the system in the Minkowski subspace and to not propagate in the euclidean part. For this reason we will call the part of the action relative to the euclidean subspace as S confining . Notice that in this procedure we do not need to scale the parameters appearing in our lagrangian. We will discuss in a detailed way the construction of discrete and continuous dynamical systems following what outlined before.
The same considerations can be made for dynamical systems invariant under the Galilei algebra. In fact, this time we will introduce an invariant action under the Euclidean group, E(D + 1 − k), depending on the variables x a only. In this case the Galilei boosts map the variables x a into the x α 's. and we can get a model invariant under the full Galilei group, by the addition of a confining action consisting of lagrange multipliers times convenient combinations of the x α 's. Note that the mass-shell constraints of our models depend on momenta and coordinates on the longitudinal or on the transverse variables, but not depending on both. Several of the previous constructed models do depend on all the variables, see for example [26,29,32], and therefore can not be obtained from our procedure.
Strictly speaking, the Galilei invariant models obtained by this procedure are not dynamical in space-time, since the action of these systems does not contain the time variable. This means that the dynamics happens only at a given time. In this sense, these models describe a sort of generalized instantons. The evolution in euclidean space can be obtained in terms of an euclidean coordinate and the associated hamiltonian. In general, it is possible to obtain interesting results. For instance, considering the dual of the 1-contraction of Carroll type (the Carroll particle), that is a D-contraction of Galilei type, it is possible to obtain, in configuration space, a model equivalent to the one, obtained in [33] as the limit of a relativistic tachyon, in the phase space, for c → ∞. This model has been called a Galilean massless particle [34], referring to the invariance under the Galilei algebra with zero central charge. This model describes an euclidean non-relativistic instanton.
The organization of the paper is as follows: in Section 2 we discuss the k contractions at the level of the algebra, whereas in Section 3 the same analysis is performed in configuration space. In Section 4 we present, in general, the construction of dynamical models invariant under Carroll and Galilei, in the discrete and in the continuous case, as, for instance, extended objects. In Section 5 we will discuss various explicit examples of our procedure, ranging from the Carroll particle to massless particles up to a Carroll string In the last Section we present our conclusions and discuss possible extensions of the present work.

5)
We see that the boosts behave like vectors under both groups. Finally, the commutator among boosts is given by We will consider two types of contractions, rescaling in one case the momenta P α and in the other case the momenta P a . In both cases the boosts will be rescaled. The first contraction is called a Carroll contraction [1,16,29,31] and it is obtained by the following rescaling, in the limit of ω → ∞ M αβ = M αβ ,P α = 1 ω P α ,M ab = M ab ,P a = P a , Notice that the commutators of M αβ and M ab with the momenta do not change, since they are linear in the momenta. Therefore we will be interested only in the commutators of the new generatorsP α , P a andB αa . Using eqs. (2.5), (2.6) and (2.7) we find We will denote the contracted algebra by C k (1 + D). Since we rescale the first k momenta P α we will call this contraction a "k-contraction" of Carroll type. The other contraction that will be considered will be obtained by rescaling the D + 1 − k momenta P a and this will be called a D + 1 − k-contraction of Galilei type:M αβ = M αβ ,P α = P α ,M ab = M ab ,P a = 1 ω P a , Again, the interesting commutators are This scaling is suggested by the non-relativistic limit of relativistic branes [23-25, 27, 30, 36]. The algebra obtained by this contraction will be denoted by G D+1−k (1 + D). The two algebras C k (1 + D) and G D+1−k (1 + D) are dual one to the other, they go one into the other by exchanging the role of the momenta P α and P a and of the metric tensors. It should be noticed that this duality relating the Carroll and the Galilei contractions corresponds to quite different physical situations (see also in the following). In the case of the 1-contraction a similar duality has been discussed in [31] although it is not quite the same.

k-contractions in configuration space
In this Section we will study the realization of the previous abstract algebras in a flat Minkowski space-time, M (1, D), of dimensions D + 1, with coordinates and metric x µ , µ = 0, 1, · · · , D, η µν = (−, +, · · · , +). The coordinates of the two subspaces will be chosen to agree with what done in the previous Section 4 x α ∈ M (1, k − 1), α = 0, 1, · · · , k − 1 (3.3) and x a ∈ E(D + 1 − k), a = k, k + 1, · · · , D. Therefore, as we did previously, we will define two commuting subgroups of the Poincaré group in D + 1 dimensions, ISO(1, D). These are the Poincaré group in k dimensions, ISO(1, k − 1) and the roto-translations group in D +1−k dimensions ,ISO(D +1−k). Furthermore, there are the generators that mix together the two subgroups, that is the "boosts", B αa . We will consider the Poincaré group ISO(1, D) generated by the following vector fields which satisfy the algebra (2.1).
According to the split of the space-time illustrated before, the previous generators are split in (M αβ , P α ) ∈ ISO(1, k − 1), (M ab , P a ) ∈ ISO(D + 1 − k) (3.6) and the "boosts" The contractions in configuration space are dual to the ones defined for the momenta (see, for example, [29] for Carroll and [27,28] for Galilei) as in the limit ω → ∞. These two types of contractions are not equivalent.The terminology used here derives from the case of 1-contractions. In that case the two contractions lead respectively to the Carroll and to the Galilei group with vanishing central charge (see, for instance, [19]). Therefore, we have a total of 2D possible contractions 5 .
On the other hand, the duality property considered in the previous Section at the level of the exchange of the momenta for the two contracted groups, here is expressed in terms of the exchange of the manifolds M (1, k − 1) and E(D + 1 − k) (see Fig. 1). In fact, the case in which all the variables are rescaled (k = D+1), we get the original algebra. This is because the Lorentz generators do not change under rescaling of the coordinates, being homogeneous in the coordinates themselves. As for the momenta, since the commutation relations are linear in the momenta, they are left invariant by any common rescaling of the momenta. From this observation it follows that the contractions give the same result as the ones following from (3.8).
Let us now consider explicitly the two cases:

Carroll-type
Re-expressing the old variables in terms of the new ones, (3.8), we get and, in the limit ω → ∞ We see that this contraction coincides with the one of the previous Section. Therefore the commutation relations of the vector fields are the same obtained for the abstract generators of C k (1 + D).
Let us now consider the Galilei case:

Galilei-type
Re-expressing the old variables in terms of the new ones, (3.8), we get and For the commutation relations, the considerations made in the Carroll case can be repeated here. In any case, for completeness we repeat here the relevant commutators for the two cases:

Building Dynamical Models
We will consider here the case of dynamical systems described by discrete variables and then by continuous variables. The first case corresponds to having a certain number of interacting point-like objects. The second case corresponds to having extended objects as, for instance, branes. We will show how to construct models invariant either under Carroll, or under Galilei. However, we will not consider here the case of field theories.

Discrete models
Let us start from the Carroll type of symmetry and consider a k-contraction. We suppose to have an action, describing N interacting particles, invariant under a a linear realization Poincaré group in k dimensions: where the index i = 1, · · · , N describes the type of particle. From the invariance under translations, Introducing the canonical momenta the equations of motion areṗ implying the conservation of the total momentuṁ Now, let us consider the other part of the total space-time M (1, D), that is the euclidean space E(D + 1 − k). We will consider our point particles living only in the space M (1, k − 1), therefore we will introduce a set of constraints, confining the particles to stay in M (1, k − 1). Furthermore, if we want to implement the Carroll type of symmetry we have to require that this confinement action is invariant under ISO(D + 1 − k). In particular, we have to require invariance under translations. This can be realized introducing the center of mass coordinates in the euclidean sector 6 In order to get the momentum conservation the confining action must depend only on the relative coordinates Notice that we have the identity The equations of motion for the variables x a i arė 6 Notice that any linear combination N i a i x a i , with N i a i = 1, could make the job implying the conservation of the total momentum P a : In this way we enforce the translation invariance in E(D + 1 − k). Furthermore, assuming that all the lagrange multipliers transform like vectors under the rotation group SO(D + 1 − k), the confining action satisfies all our requirements. The equations of motion resulting from this action arė We will now assume as total action S = S P L + S confining . (4.13) Since the two actions S P L and the confining one have separated variables, all the relations we have derived so far, continue to hold for the total action. Notice that the role of S confining is just to constrain the particles living in M (1, k − 1) to not escape from this space, they are "confined". We will show now that the total action has more symmetries than We have only to check the invariance under the boosts which are given by (compare with eq. (3.12)) (4.14) A boost with parameters v α a will generate the transformation and δx a i = 0, (4. 16) where we have used the Poisson brackets Let us now evaluate the variation of the action S under a Carroll boost. We have Notice that from eq. (4.5), due to the translational invariance of S P L , it follows implying that also the variation of the term proportional tox a is zero. Therefore the variations of the lagrange multipliers λ i a and µ i a are consistent with the translational invariance of the confining action, as it follows from (4.7). This is not accidental: the boost invariance of S and the translation invariance of S P L and S confining are strictly related. In fact, since from Noether's theorem follows that continuous symmetries imply constants of motion, we see that if boosts and translational invariance in E(D + 1 − k) are satisfied, then S P L must be translational invariant, as it follows from the commutation relations 21) and the fact that the commutator of two constants of motion is a constant of motion.
We can check directly that the boosts are conserved quantities using the equations of motion (4.12), (4.4) and (4.2) Our S confining action leads to the following primary constraints and π µ i a = 0., since the action does not contain the time derivatives of the lagrange multipliers. Of course, other constraints could arise from S P L The first constraint is consistent with the variation of the lagrange multipliers under a boost. In fact The constraints (4.23) are second class, in fact To evaluate the canonical lagrangian in the reduced phase space, where we have eliminated λ i a , we first notice that the terms λ i aẋ a i do not contribute to the hamiltonian, being homogeneous of first degree in the time derivative. Therefore where H P L is the hamiltonian evaluated from L P L . If the lagrangian implies some constraints as in the case of gauge invariance, for example invariance under diffeomorphisms, one needs to use the Dirac hamiltonian. This action is boost invariant under the transformations with The Galilei case, dual to the previous one, can be discussed exactly along the same lines. This time the action for the point-like particles is defined in the euclidean space dual to the Minkowski space of the Carroll case: and the confining term defined in M (1, k − 1) Under boost we have the transformations We will call the Carroll and the Galilei cases dual one to the other. A simple way to get this result, is to start with an action Poincaré invariant in the total space M (1, D), say S T and define the actions in the two subspaces as In other words, the point-like actions for the two cases are obtained by restricting the action S T to the two respective subspaces. Of course, this is is a quite general way of proceeding in order to get dual models. That is starting from an invariant action in the total space and restricting it to the two subspaces.

Continuous models
We would like to consider the dynamics of extended objects embedded in a confined region of the spacetime. We will show that the space-time symmetries of these models are precisely the ones deriving from the k-contractions. Let us begin considering the Carroll-type of contractions [1,16,29,31].
An extended object (for instance a brane) is mathematically described by mappings of a manifold (world-sheet) to a target space. Let us start considering the target space as a flat space-time in D + 1 dimensions. Then, suppose to confine the extended object in the M (1, k − 1) Minkowski subspace of the space-time. Assume that the world-sheet is described by the coordinates (τ, σ i ), with i = 1, 2, · · · , m ≤ k − 1. We will assume for this extended object an action invariant under a a linear representation of ISO(1, k − 1) with x α the coordinates of the target space M (1, k − 1), V the volume which defines the system and x α ,i = ∂x α /∂σ i . Since we want to confine the extended object inside this space, we will add to this action a term keeping into account this condition where x a are the coordinates of the euclidean target space, and the λ's and the µ's are lagrange multipliers. The aim of the confining term is to make vanish all the possible motions or vibrations of the extended object that could end in the space E(D + 1 − k). The total action is given by The confining term is invariant under the euclidean group ISO(D + 1 − k), assuming that the all lagrange multipliers transform as vectors under the rotation group SO(D + 1 − k). Let us define the following quantities where P α is the momentum density. The equations of motion are from which we have that the total momentum, Therefore the action is invariant by assuming the following transformation law for the lagrange multiplier and in this way we have shown that this construction leads automatically to models invariant under the Carroll type of contracted groups considered in Section 2. An analogous procedure can be made for the Galilei type.
Other features of these models that can be discussed before specifying S EO We have P a = δL confining δẋ a = λ a . for any dynamical variable A. Therefore in the reduced phase space, the momenta P a and the lagrange multipliers λ a can be identified. Notice that the same boosts generating the transformation (4.44) would generate the following variations for the momenta: which is consistent with the variation of λ a . The construction of the action in the reduced phase space goes as in the point-like case. Analogous models can be considered for the Galilei case. In this circumstance, we would take an action describing an extended object inside the euclidean space E(D + 1 − k) with a confining term Then, one can repeat the same considerations made in the Carroll case. However, there is a very deep physical difference between the two cases, since the Carroll models are formulated in a Minkowski space-time, whereas the Galilei case is formulated in an euclidean space. As a consequence the Galilei case resembles the description of an instanton. Notice that the Carroll case corresponding to a kcontraction is dual to the D + 1 − k-contraction of Galilei type. However, when we use this duality to describe extended objects there are some conditions to be respected. In fact, we will assume that the dimensions of the world-sheet of the extended object, m + 1, are smaller or equal to the ones of the target space. Therefore, to have dual objects we have to require m + 1 ≤ k , in the Carroll case, and m + 1 ≤ D + 1 − k. Therefore to have duality we need the condition (see Fig. 3) Also in the continuous case we can proceed as in the point-like one, that is starting from an invariant action describing an extended object in M (1, D) Minkowski space. Then, the action for Carroll is obtained by restricting this action to the space M (1, k − 1), that is by putting to zero the variables x a . Analogously, in the Galilei case, one takes the restriction to E(D + k − 1), that is by putting to zero the variables x α . Of course, in order to get dual models, the condition (4.51) must be satisfied.

Examples
In this Section we will start from the description of a massive relativistic particle in the full space-time M (1, D) described by the invariant Diff action We will consider various examples. In particular we will study, in the case of 1-contraction of Carroll type, a massive particle and its Galilei dual, obtaining the Carroll particle studied in [16]. Its dual corresponds to the so called Galileian massless particle [33,34]. Notice that, here, massless refers to the fact that the corresponding representation of the Galilei algebra is with zero central charge. We will consider also a relativistic massless particle described by the action

The Carroll massive particle
We will consider the Carroll type 1-contraction [1], using the action (5.1). The Minkowski space-time reduces to a one dimensional space described by the variable x 0 . The total action is given by the restriction of (5.1) at this space, plus the confining term This action is invariant under diffeomorphisms in τ . The momenta are given by There is one first-class constraint φ = p 2 0 − M 2 = 0, (5.5) and 2D second-class constraints with Poisson brackets Introducing Dirac brackets, one can eliminate the lagrange multipliers λ in favour of the momenta p. The action (5.3) is invariant under the Carroll boost transformations (see eq. (3.12)) with In the reduced phase space, these transformations become These transformations are generated by corresponding to the vector fields of eq. (3.12). The action in the reduced phase space is This action is invariant under the boost transformations given in (5.9) and coincides with the phase space action studied in [16], where it has been obtained from the relativistic action (5.1) in the limit of zero light-velocity. From our point of view, the limit c → 0 is equivalent to the Carroll 1-contraction of eq. (3.8). Notice that the physical description of this particle corresponds to a massive particle in its rest-frame. In fact, from the variation of λ , we get˙ x = 0.

The Galilei particle
The D-contracted Galilei case corresponds to the model described in the literature as the Galileian massless particle [33][34][35]. The action is obtained by the restriction of the action (5.1) to the euclidean space E(D) and adding the confining part. We get 7 S = dτ M ẋ aẋb η ab + λẋ 0 = dτ M ˙ x 2 + λẋ 0 ≡ dτ L, η ab = δ ab . (5.12) This action is invariant under diffeomorphisms in τ , therefore the canonical hamiltonian vanishes identically. The momenta are given by Therefore there are three constraints since the time derivative of λ does not appear in the action. We see that the first constraint is first-class, whereas the other two are second-class. In fact, their Poisson bracket is not zero Therefore, introducing the Dirac brackets in the reduced phase space, p 0 and λ can be identified. This action is invariant under the boost transformations of Galilei type generated by the Galilei boosts (see eq. (3.17)), with parameters v a . B a0 = x 0 p a , (5.16) In the reduced phase space, these transformations become 18) and the action is is given by This action coincides with the one studied in [33][34][35]. The particle described by this model can be seen as a tachyon in the standard frame of its velocity, that is the frame whereẋ 0 = 0, as it follows varying λ in the action. The model can also be obtained by the contraction in eq. (3.9), in the limit ω → ∞ for a relativistic tachyon. This is equivalent to consider the limit c → ∞. Therefore, this particle exists at a single instant of time at any point of space. Since this model can be also obtained via Wick rotation on the action of a massive relativistic particle, technically it describes an instanton. Notice also that the conditioṅ x 0 = 0 implies that the physical velocity d x/dx 0 → ∞.

A light-like particle of Galilei type
Let us consider the Poincaré group in D + 1 dimensions. If we introduce light cone variables, we can write the generators as P − , P + , P a , , M −+ = M 01 , M ab , B +a , B −a , a, b = 2, 3, · · · , D, (5.20) where We can see that we have two subalgebras G ± : (M 01 , P ± , M ab , P a , B ± a ). (5.22) These subalgebras have the property that leave invariant a null direction n µ , explicitly Λ µ ν n ν = λn µ . G + leaves invariant the direction n µ + = δ µ + ,whereas G − leaves invariant the direction n µ − = δ µ − . In the case of D + 1 = 4, this algebras are known as ISim(2) ± . They are symmetries of the Very Special Relativity [37] in place of the Poincaré group.
The relevant non vanishing commutation relations are In order to construct a a massless particle model we consider a Galilean invariant model, it is useful to consider a k = 2 contraction of the Poincaré group. Introducing the light-cone coordinates the generators of the contracted algebra in configuration space (see Section 3) are The transformation under the boosts B ± a are δx a = −v a ± x ± , δx ± = 0. (5.26) In other words the main feature of these contracted algebras is that they leave invariant one of the two branches of the light-cone, x ± = 0, respectively. Notice that the generator M 01 acts as a dilation operator on the light-cone variables x ± . It follows that it leaves invariant one of the planes x ± = constant. This suggests to consider a massless particle in Minkowski space M (1, D) given in (5.2) in the light-cone coordinates Following the lines for a 1-contraction of Galilei type, we will consider two possible actions for describing the particle in the E(D − 1) euclidean space These actions are invariant under the two subalgebras separately. We can check the invariance under the boosts B ± a . In fact, we have Let us notice that the two actions S ± are invariant under the transformations generated by M 01 . In fact, the corresponding transformations of x ± are δx ± = ∓sx ± , (5.30) where s is the infinitesimal parameter. The invariance follows assuming Also this transformation is compatible with the identification of λ ± with p ± as it follows from the commutation relations given in eqs. (5.23).
For simplicity let us now consider S + . The same considerations will hold for S − . The canonical momenta are, using light cone variables, we have the constraints p + = λ + , , p − = 0, p 2 = 0 → p = 0, π λ+ = 0.. (5.32) We have D+1 first class constraints and two second class constraints, with 2(D+2) degrees of freedom. Then, the model is described by 2 d.o.f,, p + and its conjugated coordinate. The Dirac hamiltonian is given by

A light-like particle of Carroll type
We will consider now a light-like particle within a 2-contraction of Carroll type. The action, in the Minkowski part of the space-time, will depend on the two variables x 0 and x 1 . We start with the massless action in eq. (5.2) restricted to the plane (x 0 , x 1 ) and we add the confining term: The canonical momenta are As in the previous examples, in the reduced phase space (after using the second class constraints), we can identify p a with λ a . We will assume the λ a 's transforming like vectors under the generators of SO(D − 1)). The model is invariant under translations, and under the transformations generated by M 01 , since this generates a Lorentz boost in the direction D and it leaves invariant the quadratic form in the action. Furthermore, we have invariance under the two type of boosts B 0a and B 1a : and The rest of the discussion goes as in the other examples. In this case the Carroll particle moves in the plane (x 0 , x 1 ) at the speed of light.
In the case of the 2-contraction we have reported only the case of a massless particle.. We could as well to start with the action for a massive particle in M (1, D). In this case we would obtain for the Galilei particle the mass-shell conditions that is a tachyon in D + 1 dimensions, whereas for the Carroll case with considerations completely analogous to the ones discussed for the massless case.

A model of two particles
We start we considering the relativistic two-particle model [38] where x j (τ ), (j = 1, 2) are the space-time coordinates of the two particles. V (r 2 ) is any Poincaré invariant function of the squared relative distance r 2 = (x 2 − x 1 ) 2 , m j0 's are the rest masses of the particles and m 2 j (r 2 ) = m 2 j0 − V (r 2 ) are the effective masses of the particles. The interaction breaks the individual invariance under diffeomorphism (Diff) of the action of two free particles, leaving a universal Diff invariance. The momenta are given by Following our procedure the Carroll lagrangian will be given by where r 0 = x 0 1 − x 0 2 . Like in the k = 1 Carroll particle, the presence of the second class constraints allows to eliminate the λ j in terms of the momenta p j . In the reduced space the canonical action becomes The particles do not move, although the momenta of the particles is not individually conserved. Notice that for these Carroll particles the momenta are not related to the velocities of the particles. This model is different from the two particle model of [16] where the two mass shell constraints depend on the total momenta of the particles. The Galileian counterpart is given Like in the k = 1 Galilei particle, the presence of the second class constraints allows to eliminate the λ i in terms of the momenta p 0i . The action canonical in the reduced space is The two particles described by this model can be seen as two tachyons in their standard frame of their velocity,

2-contraction for a Carroll string
We recall the string action Following what we have done in Section 4.2 we assume the following action where in the string action X α = (X 0 , X 1 ), whereas a = 2, 3, · · · , D. We get for the string lagrangian density L string The component of the canonical momentum are given by We get the primary constraints P α X α = 0, P 2 + T 2 X 2 = 0. (5.53) Let us evaluate the quantities Q σ α = ∂L/∂X The canonical action in the reduced space is given by This model is different from the Carroll string of [29]. Choosing the gauge conditions X 0 = τ, X 1 = σ, These equations show that the string is at rest and, having fixed end points, it satisfies the Dirichlet boundary conditions. The total momentum is conserved P α = π 0 dσ P α , E = P 0 = πT, P 1 = 0. (5.60) Therefore in the case of a 2-contraction of Carroll type, the string is the equivalent of the Carroll particle, that is the case of 1-contraction. For the considerations about the invariance under the boosts, one can repeat the general considerations made in Section 4.2

Conclusions and outlook
In this paper we have introduced a general method to construct models with invariant lagrangians under generalized Carroll or Galilei algebras (in the text called k-contractions). The method consists in starting from a space-time in D + 1 dimensions and partitioning it in two parts, the first Minkowskian and the second Euclidean. Then, a Carroll invariant model can be obtained by introducing a Minkowski invariant action in the first part of the space-time, whereas in the second part a system of lagrange multipliers, transforming in an appropriate way under the Euclidean group is introduced. This system is such to compensate the variations, induced by the Carroll boosts, of the action previously defined. The same procedure is done for the Galilei case, this time using an action lagrangian defined in the Euclidean sector and enlarging it with a system of lagrange multipliers living in the first part of the space-time. The main difference between Carroll and Galileian models constructed in this way, is that in the first case we have a real dynamical system, since the time coordinates are in the action, whereas in the second case the time variables appear only in the confined part, meaning that one is describing an instanton-like object. This procedure could be generalized as follows: start with a target space (in the text the original space-time), partitioned in two parts. Then, suppose that the target space supports a natural representation of some group G (in the text the realization in terms of vector fields on the space-time). Assume that the two sectors support natural representations of two groups G I and G II , which are both subgroups of G. Eventually assume that the Lie algebra of G can be decomposed as follows: Lie G = Lie G I ⊕ Lie G II ⊕ I, (6.1) where I are a set of intertwining generators mapping Lie G I into Lie G II or viceversa. Then, dynamical models describing systems, living in the target space, can be constructed following what we have done in the text. All these models would exhibit the confinement in one of the two sectors of the target space. One could think to various possible extensions. For instance, one could describe a dynamical model with a generalized Galilei invariance, starting with a space-time with two times, M (2, D − 1). Or else, starting from a euclidean space E(D + 1) one could consider statistical systems confined to dimensions lower that D + 1, but exhibiting a bigger symmetry. This approach could also be useful in theories with extra-dimensions. We hope to be able to consider in the future some of these extensions in greater detail.