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Non-linear Realizations, Goldstone bosons of broken Lorentz rotations and effective actions for p-branes

We consider the non-linear realizations of the Poincare group for p-branes with local subgroup SO(1,p)*SO(D-(p+1)). The Nambu-Goto p-brane action is constructed using the Maurer Cartan forms of the unbroken translations. We perform a throughout phase space analysis of the action and show that it leads to the canonical action of a p-brane. We also construct some higher order derivative terms of the effective p-brane action using the MC forms of the broken Lorentz transformations.


Introduction
Recently there has been a renewal of interest to construct the effective string theory action [1] (see also [2]) using non-linear Lorentz invariance. The lowest order term in derivatives is the well known Nambu-Goto (NG) action. In [3], and references therein, it has been examined the next order Lorentz invariant corrections to the NG action in the static gauge 1 . In that approach it is required the invariance under the broken Lorentz transformations of coordinates of the string, which are the Goldstone bosons associated with the broken translations. The broken Lorentz transformations rotate the longitudinal and transverse directions of the string. In this paper we consider p-brane diffeomorphism (Diff) invariant actions in the non-linear realization (NLR) approach [6], see also [7]. In our case they contain the Goldstone bosons of both broken and unbroken translations and Lorentz rotations. The MC forms, pullbacked to the world volume, are used to build these Lagrangians with the geometric tools of the wedge product, Hodge operator, and covariant differentiation. These actions are functionals of both of the embedding variables x and the Lorentz variables Φ.
The general action will be where the volume form e[x, Φ] is constructed by using the Maurer Cartan (MC) forms associated to unbroken translations. With L = 1, it leads to the NG action for the embedding variables, see for example [8] [9][10] [11], once the Inverse Higgs (IH) mechanism [12], or equivalently the equations of motion (EOM) of the Lorentz variables are used 2 . In this lowest order case in derivatives, the Lorentz variables are non-dynamical.
To construct more general actions with L = 1 we can use the MC forms associated to the broken and the unbroken Lorentz transformations. In this case the Lorentz variables are dynamical. Note that the form of the invariant scalar Lagrangian has no additional restrictions, and there is an infinite arbitrariness of choices. For p-brane actions we should also consider WZ terms L W Z that are constructed from closed invariant p+2 forms, Ω p+2 = dL W Z .
If we want to obtain a Lagrangian depending on the embedding coordinates x's alone, we should express the Lorentz variables in terms of these coordinates. A standard procedure is to impose an invariant set of constraints, so called the IH mechanism (see (1.7)) and to construct the corresponding MC forms on the world volume associated to the broken Lorentz transformations and to the unbroken rotation (to build covariant derivatives) and use them to construct the polynomial and non-polynomial invariants of the coefficients of these forms along the lines of, for example [7]. This procedure give all possible local terms in the effective action.
In this paper we explore a possible different method to express the Lorentz variables in terms of geometrical quantities by solving the equations of motion perturbatively. A starting Lagrangian is called the seed Lagrangian in the following sense: since we want to obtain a Lagrangian depending on the embedding coordinates x's alone, we should express the Lorentz variables in terms of these variables. This can be done by solving perturbatively the equations of motion of the Lorentz variables and substitute these perturbative solutions into the seed Lagrangian. Thus from a seed Lagrangian we obtain a perturbative series of Lagrangians in terms of the embedding variables. Let us insist however that there is an infinite freedom of choosing the seed Lagrangian. Obviously we could limit ourselves to polynomial seed Lagrangians, organized in number of derivatives, as is done in effective theories.
We first illustrate it here in the case of a particle (see details in the section 3). We consider a coset G/H = (Poincare)/SO(D −1) and the coset element g = g 0 U, where g 0 = e ix 0 P 0 e ix a ′ P a ′ is the coset representing the Minkowski space and U = e iθ 0 b ′ J 0 b ′ is a general Lorentz rotation, x a ′ (a ′ = 1, ...D − 1) are the Goldstone bosons of broken translations, x 0 is the Goldstone boson of the unbroken time translation 3 and U is parametrized by the Goldstone bosons of the broken Lorentz transformations. The MC forms are given by, (a = 0, ..., D − 1) 4 , where Φ a b (θ) is a Lorentz boost parametrized by θ 0 b ′ . The action 5 of a free massive particle is the SO(D−1) invariant one form L 0 associated to the unbroken translation P 0 , This action is invariant under Diff, as is the case for every action built from the pullback of MC forms and with the tools mentioned above. If we compute the momenta p a = ∂L ∂ẋ a = −mΦ a 0 (1.4) therefore Φ a 0 (θ) are functions of the momenta of the particle. This fact implies that the Goldstone bosons of broken rotations θ 0 b ′ are actually phase space degrees of freedom of the particle. Now if we regard Φ a 0 as independent degrees of freedom we can rewrite the action as (1.5) where γ is the Lagrange multiplier to constrain Φ a 0 to be a time-like unit Lorentz vector. It is the canonical action of the relativistic free particle, as can be extracted from [15] [16]. Therefore the NLR action that include the Goldstone bosons of the translations and the Goldstone bosons associated to the broken boost is the canonical action of the massive relativistic particle.
If we use the equations of motion of Φ a 0 and γ, we obtain Note that Φ a a ′ is normal to the worldline and is equivalent to the vanishing the MC form L a ′ associated to the broken translations which is known as the inverse Higgs (IH) mechanism [12].
We can construct corrections to the NG action of the particle by considering the pullback of the MC forms associated to the broken Lorentz transformations 6 . We use the standard tools (wedge product, Hodge operator, and covariant differentiation) available with the MC forms. In the case of the particle we define the Hodge operator by assigning L 0 = dτ e as the volume form which implies * dτ = −e −1 . Since we want corrections to the NG action in the spirit of effective theories, we first consider the term with first order derivatives of the Lorentz variables. In particular the polynomial one, which is quadratic in the velocities, whereas the coefficient is chosen to be small. This is made by choosing, among the SO(D − 1) invariant one forms, the structure L 0a ′ * L 0a ′ , which depends at most on the velocitiesΦ ab . Adding this term to the action (1.5) we obtain 8) or better, with a redefinition of γ, where the constant parameters β is dimensionless and the term with β is considered as a perturbation of the free particle action 7 . Notice that in this Lagrangian the Φ b 0 variables are independent dynamical variables. We have been unable to solve dynamically to all orders in β the equation of motion of Φ b 0 in terms of x a , γ.
We do not use other possible Lagrangian candidates, like L a ′ * L 0a ′ or L a ′ * L a ′ , since they vanish at lowest order due to (1.7). We could also add invariant terms with an increasing number of derivatives of Φ like for instance (D * L 0a ′ ) * (D * L 0a ′ ), whereD is the covariant differential, as in effective theories. We will focus in this paper the lowest order in derivatives, and we will use this action (1.9) as a seed to produce a set of geometrical invariant actions. In fact in order to make contact with an effective Lagrangian written in terms of the geometric quantities of the world-line in Minkowski space we use the procedure of solving the equations of motion of Φ b 0 perturbatively in β, in terms of the velocities, accelerations,... of the particle. At lowest order in β, Φ b 0 is given by (1.6) and the geometrical action becomes 8 , in terms of proper time s, In [17] [18] the MC forms associated to broken Lorentz rotations were used to construct relativistic gauge fixed (super) particle action in two dimensions with extrinsic curvature. 7 We could have also added terms with higher derivatives in Φ. We will not consider those terms in this paper. 8 If we only consider the term of order β the dynamics of that Lagrangian is not equivalent to the starting Lagrangian (1.9). The equivalence would only be achieved with the full perturbative series.
where κ 1 is the first curvature of the world-line, κ 2 1 = (A) 2 , A a = d 2 x a ds 2 . Note that we do not make use of the EOM of the coordinates x, unlike what is customarily done in the standard procedure for effective field theories [2] [3]. At first order in β the solution for Φ 0 b from (1.9) departs from the IH mechanism, where U a = dxa ds and Σ a = is the relativistic jerk vector, see for example [19] [20]. Note that there is a rationale for the presence of the relativistic jerk vector, since it is orthogonal to the velocity, and this orthogonality is already imposed on Φ a 0 in (1.11) due to the Lorentz constraints η ab Φ a 0 Φ b 0 + 1 = 0. By substituting (1.11) into (1.9) we can consider the expansion of the action up to order β 2 in terms of the embedding variables, is the projector orthogonal to accelerations and velocities (s is proper time). We should notice that the combination of curvatures κ 2 1 κ 2 2 + ( dκ 1 ds ) 2 appearing at order β 2 of the effective Lagrangian (1.12) is the Σ 2 of the relativistic jerk vector Σ a .
Note that we could consider within our approach more general, non-polynomial, actions, like (1. 13) In this case the relation to orderβ among the Lorentz variables and the geometrical quantities is given As long as the seed Lagrangian contains the term with first time derivatives of the Lorentz variables, the first correction term to the NG will be a functional of the first extrinsic curvature.
In this paper we will see that the previous results for the particle are extended to the case of a p-brane and show that: 1. The Goldstone bosons of broken Lorentz rotations are functions of the phase space degrees of freedom of the p-brane.
2. The NLR action constructed from the MC forms of unbroken translations leads in a natural way to the canonical form of NG action. Since the Goldstone bosons of the broken rotations are non-dynamical we can eliminate them through their own equations of motion and one gets the world-volume NG action [10].
3. We can also construct a WZ term for a p-brane in dimensions p + 2 using only the MC forms of the translations 9 .
4. As in (1.5) it is often useful to introduce also Goldstone bosons associated to the unbroken Lorentz rotations as independent degrees of freedom without using explicit parametrization of the Lorentz transformation. In this case the Poincare invariant p-brane action will be invariant under local SO(1, p)×SO(D −(p+1)) rotations as well as world-volume diffeomorphism (Diff).

5.
We also consider terms of the effective action of a p-brane with non-linearly realized Poincare symmetry by adding invariant forms associated with the broken Lorentz transformations. In these cases actions, like the particle case, will contain the first derivatives of the Goldstone bosons of broken rotations and the Φ's become dynamical. The Lagrangian with both the embedding variables and the Lorentz variables is used as a seed to produce corrections, in terms of the embedding variables, to the NG Lagrangian. The procedure consist in solving for the Lorentz variables by using the equations of motion (EOM) iteratively and substituting them back into the Lagrangian so that it will produce an effective action in terms of geometrical quantities, like the intrinsic curvature, extrinsic curvature and the higher curvatures of the world-volume.
The organization of the paper is as follows. In section 2 we will construct the action of a tensionfull p-brane with the lowest order derivatives using the non-linear realization approach with local subgroup SO(1, p)×SO(D−(p+1)). We will perform the Hamiltonian-Dirac analysis that leads to the canonical action of the NG action. In section 3 we will construct leading order -and beyond-corrections to the NG action. Finally we will present some conclusions and outlook. There are two appendices with technical details.
Note added: when this paper was in a process of submission it has appeared a paper by Gliozzi and Meineri [24] which proposes a different procedure to construct some higher order derivative corrections to the Nambu Goto action. Their results agree with ours.
2 Nonlinear realization with local subgroup SO(1, p) × SO(D − (p + 1)) In this section we will construct the p-brane action using the NLR approach [11] by Callan, Coleman, Wess and Zumino [6]. However we will not follow the method in terms of coset representatives. In the present case it is more useful to reformulate the theory of non-linear realizations in terms of the group elements 10 of G themselves rather than those of the coset G/H. We consider group elements g of G and take the symmetries of the non-linearly realized theory given by where g 0 ∈ G is a rigid transformation while the second independent local transformation h(x) is an arbitrary space-time dependent transformation that belongs to H.
In this more general way the theory contains gauge degrees of freedom that can be 10 For the case of internal symmetries see [25]. fixed using the local H transformations. We will refer to such a formulation as a non-linear realization of a group G with local subgroup H. The global G transformations of the coset coordinates x's are determined up to the local H transformations and are in general non-linear. One could from the beginning use these local H transformations to fix the gauge freedom and so work only with coset representatives. This is equivalent to the original approach and it requires the H-compensating transformations for the global g 0 transformations. One can also work in a half way house where only some of the fields are removed.
For the case of relativistic p-branes we consider the NLR of G = ISO(D − 1, 1) with the local subgroup SO(1, p) × SO(D − (p + 1)). The generators 11 P a , J a b , J a ′ b ′ generate the unbroken translations, local longitudinal and transverse rotations via the right transformations [14]. In the non-linear realization the reparametrization invariance corresponds to the invariance under unbroken translations with additional local Lorentz rotations for a general p-brane, see [11]. The group elements are parametrized by where g 0 = e ix a Pa e ix a ′ P a ′ is the coset representing the Minkowski space and U is a general element of the Lorentz group. In (2.2) we have introduced SO(1, p) × SO(D − (p + 1)) degrees of freedom in addition to the coset elements of SO(1,p)×SO(D−(p+1)) used for example in [10]. The MC form is given by where the vielbein and spin connection forms in the flat D dimensional target space are 4) and are satisfying the MC equations Here Φ b a is a finite Lorentz transformation parametrized by θ a b ′ , θ a b , θ a ′ b ′ . The explicit form of the finite Lorentz transformation Φ in terms of the Goldstone fields θ is not simple. Also, the local SO(1, p) × SO(D − (p + 1)) transformations of θ are involved.
Like in the case of the particle (1.5), it is more convenient to consider the D 2 elements of Φ a b of the Lorentz transformation themselves as new variables restricted by the orthonormality constraint and with the implicit choice of det Φ = 1, are the ones associated to the local SO(D − (p + 1)). So the number of non-gauge degrees of freedom for the Φ variables will be (p + 1)(D − (p + 1)) which agrees with the counting of Goldstone bosons θ a b ′ associated to broken Lorentz rotations of the coset [10].

NG p-brane Lagrangian
The action of relativistic p-branes with lowest order of derivatives is constructed from the pullback of the MC forms (2.4) on the world-volume, with dimensionless coordinates (ξ 0 , ξ m ), m = 1, ..., p. It must be invariant under world-volume diffeomorphism associated to the longitudinal unbroken translation invariance and local longitudinal and transverse Lorentz rotations. The Lagrangian density, for a generic dimension, is given by with L 0 is the world-volume density of the p + 1-brane 12 where κ is the tension of the p-brane with dimension [κ] = m p+1 and e = det(e i a ) with (p + 1)-bein given by which is the pullback of the L a in (2.4). L 1 is the Lagrange multiplier term for the orthonormality of the D-beins Φ a c , (2.6), We can write the Lagrangian (2.8) as 13 with Not all of the Euler-Lagrange (EL) equations of motion are independent but there are p + 1 Noether identities for diffeomorphism invariance, (2.14) 12 We often omit the ∧ symbol for the wedge products. 13 A similar action in terms of Lorentz harmonics was considered in [26] [27].
The Noether identities associated with the local SO(p, relations among these equations of motion. The presence of gauge symmetries and their associated Noether identities results in the appearance of the first class constraints in the Hamiltonian formalism. In addition, since the variables Φ b a , γ ba are non-dynamical (for the Lagrangian (2.8)), the EL equations will produce Lagrange constraints (relations among coordinates and velocities). As we will see they will produce Hamiltonian secondary constraints and relations among the arbitrary functions appearing in the Hamiltonian. Now we will make contact with ordinary NG action written in terms of the coordinates of the p-brane. In fact, if we introduce the inverse vielbein e b i , e b i e i a = δ a b , the IH mechanism L a ′ = 0 like the particle case (1.7) and the orthonormality condition we have also Taking the determinant of (2.17) we have that the Lagrangian (2.12) becomes which is the ordinary NG action.

WZ Lagrangian density
Apart from the invariant Lagrangian density (2.8) valid for any dimension we can construct a WZ term for a p-brane in dimensions D = p + 2 using only the MC forms of the translations 15 L a . In fact is a closed invariant (p + 2) form. Since det Φ = 1 we can write it as We can either add L W Z on the NG Lagrangian [21] [22] or consider the p + 1 form L W Z solely as a particle model Lagrangian. The WZ term of the p-brane action represents the coupling of the p-brane to a (p+1) form Abelian gauge potential given by A p+1 = ǫ a 0 ···a p+1 x a 0 ∧ dx a 1 ∧ · · · dx a p+1 , this coupling breaks parity invariance for odd p.
14 Our convention is

NLR action versus canonical action
In this subsection we will see how the Goldstone bosons of the broken Lorentz generators are functions of the phase space variables of the p-brane. This result generalizes (1.4) of the particle, sketched in the introduction to the p-brane case. We will see that the NLR action (2.8) leads in a natural way to the canonical action of a pbrane. The definition of all the canonical momenta from the action (2.8) 16 gives primary constraint equations (See the details of the Hamiltonian formalism in the appendix B) Note that the combination ee b 0 does not depend on the velocities 17ẋa . The Hamiltonian is a sum of primary constraints, where λ a , Λ a b and λ γ ab are arbitrary functions of ξ at this moment. Using the Hamilton's equations they are related to the velocities, The Hamilton's equations for momenta reproduce the EL equations. The consistency condition that the primary constraints (2.21)-(2.23) remain zero under time evolution gives the secondary constraints and conditions on λ a and Λ, where ee b m , (m = 1, ..., p) are linear functions of λ a =ẋ a . Eq.(2.26) is the orthonormality of Φ and Eq.(2.27) fixes the γ ab . Eq.(2.28) and Eq.(2.29) coincide with the vanishing of the MC forms L b ′ in (2.4) associated to broken translations, which is known as the inverse Higgs mechanism (IH) [12].
The stability of the secondary constraints does not produce any further constraints but only conditions on the multipliers, where g mℓ is the inverse of the spatial world-sheet induced metric g mℓ = e m a e ℓa and g is the determinant of g mℓ . There remain arbitrary anti-symmetric Hamiltonian multipliers Ω ab , Ω a ′ b ′ andλ a ≡ λ a Φ a a corresponding to local SO(p + 1), SO(D − (p + 1)) and Diff p+1 gauge invariances.
The Hamiltonian is written as a linear combinations of constraints with the independent arbitrary multipliers as where we have used the fact that products of two constraints vanish as strong equations. The constraints appearing here are the first class combinations of the constraints 18 and the arbitrary multipliersλ ⊥ ,λ m ,Ω ba ,Ω b ′ a ′ are given in (B.61) and (B.73). The other independent constraints are the second class constraints that are used to reduce the phase space variables. In order to find them explicitly we rewrite the variables Φ a b in terms of new variables as where B 1 and B 2 are symmetric matrices defined by In terms of these variables the orthonormality constraints (2.26) are written as It means φ a b is a group element of SO(p, 1) and φ a ′ b ′ belongs to SO(D − (p + 1)). Using the local invariances we can take φ a In this gauge the second class constraints areφ a b ′ = ϕ a b ′ and ϕ a b ′ are expressed in terms of p a and ∂ m x a as 19 where the replacement ∂ 0 x a → p a is to be done in the right hand side of the previous equations. Therefore the Goldstone bosons ϕ a b ′ associated to Lorentz broken rotations are written as functions of the phase space variables of the p-brane. Now all second class constraints are used to reduce the phase space to that of the space-time coordinates and momenta of a p-brane (x a , p a ). The action of p-brane obtained from the nonlinear realization in the reduced space leads to the canonical action of a Dirac-Nambu-Goto p-brane in configuration space. The canonical Lagrangian becomes In summary the action (2.8), which depends on the coordinates of the p-brane and the Goldstone bosons associated to the broken Lorentz rotations, leads in a natural way to the canonical action of a p-brane.

Corrections to the NG Lagrangian
In section 2 we have constructed in generic dimensions an invariant action for p-brane in terms of the MC forms of unbroken translations. We have also constructed the WZ Lagrangian (quasi invariant Lagrangian) in dimension p + 2 for a p-brane using the MC forms of all translations. In this section we will see that we can use the MC forms of the broken Lorentz rotations to discuss possible terms of the effective action of a p-brane 20 . In order to construct these terms we will need to use the worldvolume geometry, in particular the Hodge star operator on the MC forms. According to the dimensionality of the couplings, the candidate Lagrangians built up with the MC forms will be added as corrections to the lowest order Lagrangian (2.8). As was mentioned in the introduction we make a perturbative dynamical determination of the variables associated with the broken Lorentz rotations to obtain higher order derivative corrections to the Nambu-Goto Lagrangian instead of the minimal use of the IH condition [12]. In the spirit of the effective theories, the coefficients of these terms will be small compared to the NG term. After considering general properties of the p-brane we will consider in some detail the case of the particle and the string.

p-brane
First we note that the volume form for p-brane is expressed in terms of Hodge operator * as where the Hodge * on q form is defined by (L b 1 ...L bq ) * (L a 1 ...L aq ) = µδ b 1 a 1 ...δ bq aq , (for b 1 < ... < b q , a 1 < ... < a q ). The NG action of a p-brane is proportional to the volume form (3.1), see (2.9). The equations of motion for the NG p-brane are expressed as The first one is obtained by the variation with respect to Φ and is known as the IH condition [12]. In the following we construct a geometric p-brane action using with the MC forms L a , L ab ′ and solve the EOM of Φ iteratively. By eliminating Φ up to a given order a geometric effective action is obtained as a function of higher derivative of the world-volume coordinates x a . As possible corrections to the NG action we here consider some invariant p + 1 forms, where is the curvature two form and is essentially the trace of the extrinsic curvature expressed as a p + 1 form transverse vector. Explicitly More correctly R and K a ′ become Riemannian and extrinsic curvatures at lowest order of β (NG p-brane) (see e.g. (3.60)). Note also that the first three quantities in (3.3) are not independent at the lowest order of string, but related by the Gauss-Codazzi equation, see for example [28]. In (3.3) we didn't consider to use the forms L a ′ because they vanish at lowest order of the EOM for Φ (3.2). They only contribute in higher order perturbations. On the other hand we do keep the extrinsic curvatures K b ′ which vanish when the EOM of x a of the NG brane is taken. We use the EOM of Φ but not the EOM of x a in constructing the effective action. This procedure is different from [2] [3] where the extrinsic curvatures are ignored in the effective action by using the EOM of x a 's at lowest order. The mass dimension of the volume form (NG term) is [µ] = m −1−p and the terms in (3.3) have dimensions (3.7) In the following we will examine the Lagrangian by adding a linear combination of these terms -we pick those that are independent at the lowest order-whose coefficients are scaled by the brane tension κ, ([κ] = m p+1 ), where the Lagrange multiplier terms are included in κ ′ (2.13) 21 and the constants β's are dimensionless.

Particle
In case of relativistic bosonic particle (p = 0) the NG particle action is proportional to the invariant one form L 0 . We define the Hodge operator on the world-line by, Since there is no two form R ab on the world-line we only consider the K 2 term in the action (3.8). Using K a ′ = L 0a ′ , the β term is considered as a perturbation of the free particle action 22 . In order to make contact with an effective Lagrangian written in terms of the geometric quantities of the world-line we solve the equations of motion of Φ perturbatively in β, in terms of the higher velocities of the particle. The EL equations of Φ are and that of γ gives δγ ; Φ 2 + 1 = 0. (3.15) Using the second we obtain (3.16) where "derivation D" is defined by D ≡ 1 e d dτ . Although it is a dynamical equation it is solved for Φ iteratively in β, It tells that the correction of e from √ −ẋ 2 starts from an O(β 2 ) term The expression of Φ a up to order β 2 is given by where U a is the velocity, we have also the space-like vectors acceleration, A a , relativistic jerk, Σ a , snap Ξ a , crackle C a [19] [20]. The following relations are verified by theses vectors

24)
where κ 1 , κ 2 are the first and second curvature of the worldline. The Lagrangian up to order β 3 is Here If we use another seed Lagrangian, for instance by introducing a non-polynomial term, (Note that the last term of (3.30) is obtained from the MC forms in the worldline by computing the square root of a scalar: L 0 ∧ ( * (L 0a ′ * L 0a ′ )) = dτ Φ2 .) then the expression for Φ a will change to and, accordingly, the Lagrangian in terms of the embedding coordinates will also change. Up to lowest order inβ it becomes, Notice that the first correction in both cases, polynomial and non-polynomial, is a function of the first curvature of the world line. This result is general and it also holds when we consider the alternative procedure, already mentioned in the introduction, of constructing Diff invariant actions through the IH mechanism (1.6).

Particle in three dimensions
In the particular case of three dimensions we can construct a pseudo invariant Lagrangian from a H invariant two form L 01 ∧ L 02 constructed from the MC forms of broken Lorentz rotations, we have therefore we can consider an action of a particle in three dimensions The EOM of γ gives the orthonormality of Φ's and that of Φ giveṡ Using the second one the first equation becomeṡ Saturating this last equation with Φ a 0 we get e = −γ 00 and On the other hand, saturating Eq.(3.36) with Φ a b ′ , we obtaiṅ which is the correction of the IH condition (1.7). Saturating Eq.(3.37) withẋ a we can express e as where β 2 term comes from both e and β m (Φ a1 Φ a 2 ) terms in (3.34). 23 Here A| N G means A is evaluated at O(β 0 ). It is written as (3.44) 23 Different parametrizations of Φ's connected by local SO (2) give equivalent action since The O(β) term κ 2 is the torsion and κ 1 in O(β 2 ) term is the curvature, all defined in (3.27)- (3.28). Particle dynamics with the torsion term has been discussed in [39] and in non-covariant gauge -corresponding to another choice instead of (3.41),(3.41)in [40] [41]. Both Lagrangians differ by a total derivative.

String
The NG action for the string is constructed from the SO(1, 1) × SO(D − 2) invariant two form 1 2 ǫ ab L a ∧ L b = L 0 ∧ L 1 ≡ µ, which is the volume form of the world-sheet (2.9). The Hodge operator for one form is then the Hodge operations on L a and dξ i are where g ij is the inverse of the metric of the world-sheet g ij = e i a e i b η ab and e = det e = − 1 2 (ǫ ab ǫ ij e i a e j b ). Using (3.6) we can write the invariant scalar local density (3.8) for the string as The dimension of the string tension is [κ] = m 2 and the β's are dimensionless. The Lagrangian is described by x a , Φ a b and the Lagrange multipliers γ cd . The first term is the NG Lagrangian and the second term is a total divergence. If we ignore the surface term Eq.(3.47) becomes For small β's we may solve the EOM for Φ perturbatively and the resulting geometrical Lagrangian depends on higher order derivatives of x.
The latter equality holds identically due to the local SO(1, 1) invariance of L ′ under δΦ a b = α b c Φ a c , (α bc + α cb = 0). Plugging (3.50) and (3.51) into (3.49), Now, saturating Eq.(3.52) with Φ a c ′ and using the orthogonality relations we get which is the correction of the IH condition. Multiplying ∂ i x a on (3.52) where g ji is the induced metric, (3.57) Here A| N G means A is evaluated at O(β 0 ) order, where the IH condition ∂ i x a Φ a a ′ = 0 holds and Φ a a 's are tangential to the world-sheet. Up to SO(1, 1) gauge freedom we can choose, for example, The R| N G is the scalar curvature of the world-sheet and the K c ′ = K c ′ | N G is the extrinsic curvature written as here is the g ij covariant d'Alembertian and n a c ′ = Φ ac ′ | N G are normal unit vectors of the world-sheet. The O(β 2 ) term of the Lagrangian in (3.57) should give higher order curvatures of the world-sheet.

Conclusions
In this paper we have constructed terms of the effective action of a tensionfull pbrane using the non-linear realization method. We have considered the Goldstone bosons associated to the broken and unbroken translations, coordinates of the pbrane, and the Goldstone bosons associated to the broken Lorentz rotations. In order to avoid using an explicit parametrization of the Lorentz transformations we have introduced also Goldstone bosons associated to the unbroken rotations SO(1, p) × SO (D − (p+1)). The Goldstone bosons of unbroken generators could be eliminated using the corresponding gauge transformations of local rotations.
We have seen using the action with the lowest order of derivatives that the Goldstone bosons of the broken Lorentz rotations are non dynamical and can be expressed as functions of the coordinates and momenta of the p-brane. We have also seen how the unperturbed action leads in a natural way to the canonical action of the NG p-brane. In other words the action obtained through the non-linear realizations of space-time symmetries is an action of phase space type, this result generalizes analogous results for the particle [15] [16] to the p-brane.
We have constructed the lower order corrections up to velocities in x and Φ to the NG action. The correction terms in the spirit of the effective action are small. We do not use the IH constraint [12], instead we solve perturbatively the equations of motion for the Lorentz variables Φ which, upon substitution in the original Lagrangian, give higher order geometrical terms for the effective action. For the case of the particle and string, the actions obtained in this way are written in terms of natural geometrical objects.
Our method does not have the completeness of the IH method, which gives any possible correction terms for the NG Lagrangian of a p-brane. However we want to remark that it naturally selects some specific terms within the wider set of potential correction terms. It appears that there is a very natural, an non-trivial, geometric interpretation of the first terms obtained in our approach, which may suggest that there could be physical reasons behind the selection of the seeds that give raise to these terms. At this point we do not have a compelling argument for these choices, but we think that they have enough interest to be considered.

A MC equations of Poincare group. Geometrical Aspects
The MC forms of the Poincare algebra (1.2) satisfy the MC equations which mean that the target space is torsionless and has no curvature. We can consider the pullback to the world-volume Σ and L a * = dξ i e i a as p + 1bein of Σ and L ab * = dξ i ω i ab as the spin connection. Using (A.1) 24 Here T a is torsion two form of Σ, which vanishes for the NG p-brane since L b ′ = 0 using the EOM. R ab is the curvature two form, R ab = 1 2 dξ i dξ j R ij ab . It is related to the scalar curvature R as follows. In general p + 1 dimensions (ǫ 01.. In addition, in this case, because ω 0c ω c 1 ≡ 0 for the string. Then the curvature R ab is an exact form and, as a consequence, the scalar density eR is a surface term in 2-dimensions.

B Canonical formalism
In this Appendix we present the canonical formalism of the action (2.8) and show how the NLR action leads in a natural way to the canonical action of the NG p-brane. We will also show that the Goldstone bosons of the broken Lorentz generators are functions of the phase space variables of the p-brane generalizing the result of the particle in the introduction. The canonical momenta are 25 Note that the combination ee b 0 does not depend on the velocitiesẋ a . Therefore all the definitions of momenta (B.2)-(B.4) yield primary constraints, The Hamiltonian is a sum of primary constraints, where λ a , Λ a b and λ γ ab are arbitrary functions of ξ at this moment. In order to compute the Hamiltonian equations of motion we introduce the Poisson brackets, ( ξ 0 is temporal and ξ m , m = 1, ..., p, are spatial world-volume coordinates) from which we obtain Hamilton's equations for the configuration variables, 12) and the momenta, The relations (B.12) determine the velocitiesẋ a ,Φ b a andγ γ ab in terms of the multipliers λ a , Λ a b and λ γ ab respectively. Here and hereafter the velocitiesẋ a ,Φ b a andγ γ ab , when appear in the Hamiltonian analysis, are to be understood as the multipliers λ a , Λ a b and λ γ ab . The consistency condition of the primary constraintsφ = 0 reproduces the EL equations of γ ab , Φ a b and x a , where γ bc is symmetric by definition. The b = b ′ components of Eq.(B.18) are (D − (p + 1))(p + 1) + (D − (p + 1))(D − p) 2 secondary constraint, and (D − (p + 1)) conditions on the multipliers λ a =ẋ a for i = 0, The D equations (B.16) do not give secondary constraints, but only (D − (p + 1)) relations among the arbitrary functions. Let us find them. Using the notational convention mentioned after Eq.(B.13) we havė Then, using Eq.(B.22) and the constraints (B.17), we obtain, forφ a , showing that only D − (p + 1) components ofφ a , which can be conveniently taken aṡ φ a Φ a a ′ , give independent conditions, where ee b m , (m = 1, ..., p) are linear functions of λ a =ẋ a .
We should further examine the stability of the secondary constraints. Let us first consider the conditions associated with the constraints χ ab in (B.17), If we redefine the arbitrary functions Λ a b in terms of Ω a b by which are p(R − p − 1) linear relations among multipliers Ω b b ′ and λ a . This finishes the analysis of constraints. No tertiary constraints arise because the dynamical consequences of the secondary constraints boil down to the partial determination of the arbitrary multipliers. This is a consistent dynamical system.
Let us examine for further use this partial determination of the arbitrary multipliers. Combining (B.33) with (B.28), we obtain (p + 1)(D − (p + 1)) equations that can be solved for (p + 1)(D − (p + 1)) components Ω ba ′ = −Ω a ′ b in terms of λ a . To do it we use the fact that (ee a 0 , e m a ), (a = 0, 1, ..., p) is a complete set basis in terms of canonical variables, that is, for any A a we can make the decomposition where g mℓ is the inverse of the p × p matrix of g mℓ = e m a e ℓ b η ab , g 00 = e a 0 e b 0 η ab = det g ℓm g = g −e 2 , g = det g ij and g = det g mℓ . It follows the completeness relation Let us remind that only p + 1 components of λ a are independent, as expressed in (B. 24), and that the anti-symmetric parts of Ω ab and Ω a ′ b ′ remain undetermined.
In summary we have the primary constraints (with their number in parenthesis)  , The remaining arbitrary Hamiltonian multipliers are Ω ab (in number , andλ a ≡ λ a Φ a a (in number p + 1). As expected, they correspond respectively to the local SO(p + 1), SO(D − (p + 1)) and Diff p+1 gauge invariances.

B.1 First and Second class constraints
We classify the constraints into the second class constraints that reduce the dependent canonical variables and the first class constraints that generate the gauge transformations. The constraints (B.41) and (B.43)-(B.45) are the second class constraints that allow us to reduce the canonical pairs (γ, p γ ), The D 2 constraints (B.40) are rearranged as 26 The D constraints φ a in (B.39) are projected intõ The symmetric constraints K ab 's, in number D(D+1)

2
, are combined with the same number of χ ab to form D(D+1) 2 pairs of second class constraints, (B.56) The (p + 1)(D − (p + 1)) constraints J ab ′ in (B.52) are paired with p(D − (p + 1)) χ m a ′ in (B.46) and (D − (p + 1))φ b ′ in (B.55). They are actually second class pairs since they satisfy non singular set of Poisson brackets; The constraints appearing in the Hamiltonian (B.8) with arbitrary multipliers are the first class constraints. Let us start with the original Hamiltonian and obtain the primary first class constraints present in this Hamiltonian (B.58) as the combinations of primary constraints that still keep arbitrary multipliers attached to them. In doing so we use the conditions of the multipliers (B.47)-(B.50) obtained from the consistency conditions of the constraints, and (B.50). In addition to above conditions on the multipliers we repeatedly use the fact that product of two constraints vanish as strong equation in the Hamiltonian [29] .
First we note using the completeness (B.37) whereλ ⊥ andλ m are introduced bŷ The first term of (B.8) φ a λ a is written as Using the completeness (B.37) The last term becomes where the square of constraint term (φ a Φ a a ′ )(∂ m x b Φ b a ′ ) has been dropped from second to third equation 27 . In the last second line of (B.63) we omit the square of constraint φ a 2 term and have also been dropped from the second to the third equation. We arrive at where The second term of (B.8) is where we have used the anti-symmetry of Ω. Note that Ω ba and Ω b ′ a ′ are arbitrary while Ω a ′ b is given in (B.50). Let us write J b a ′ Ω a ′ b as a linear combination ofλ ⊥ and λ m . To do this we use (B.60) and and take into account that in (B.50) Ω a ′ b is expressed in terms ofλ ⊥ andλ m . After some work we arrive at In summary the Hamiltonian is written as with the first class constraints and with a redefinition of arbitrary functions, The (p+1)p 2 constraints J ab and the (D−(p+1))(D−p−2) 2 constraints J a ′ b ′ are respectively the SO(1, p) and SO(D−(p+1)) local generators for the Φ variables, satisfying the algebras, ( and satisfy a closed algebra as The last equation means that the commutators of two local Diff transformations generated by H ⊥ results in a local transverse Diff H m and associated local Lorentz transformations SO(1, p) and SO(D − (p + 1)). Finally H ⊥ becomes ("prime" means ξ 1 derivative) where (ee [d 1 e b] 0 ) = ǫ db and the second term commutes with the first term. One can easily check the closure of the algebra for the string case,

B.2 Reduced phase space and Gauge fixing
The second class constraints are used to reduce the phase space variables. In order to find them explicitly it is convenient to rewrite the D 2 variables Φ a b in terms of same numbers of coordinates φ a b ; (p + 1) 2 , ϕ a b ′ ; (p + 1)(D − (p + 1)), ϕ b ′ a ; (p + 1)(D − (p + 1)), φ a ′ b ′ ; (D − (p + 1)) 2 . (B.81) where B 1 and B 2 are symmetric matrices defined by It means that φ a b is an element of SO(p, 1) and φ a ′ b ′ is an element of SO(D −(p+1)). One can adapt the canonical variables to the redefinition (B.82) of the Φ configuration variables. Indeed, making a canonical transformation Φ → (φ, φ ′ , ϕ,φ) generated by W (Π, φ, φ ′ , ϕ,φ) = Π a b Φ a b (φ, φ ′ , ϕ,φ), (B.85) we obtain the new momenta as π φ = ∂W ∂φ , etc. The primary constraints Π = 0 in (B.40) become, in terms of the new momenta, as π φ = π φ ′ = π ϕ = πφ = 0. (B.86) The gauge SO(p + 1) and SO(D − (p + 1)) degrees of freedom can be eliminated by imposing gauge fixing constraints for the SO(p + 1) and SO(D − (p + 1)) matrices. The simplest choice for this gauge fixing is Note now that πφ = 0 andφ − ϕ = 0 are the second class constraints to eliminate the (p + 1)(D − (p + 1)) canonical pairs (φ a b ′ , π ã ϕ b ′ ) = (ϕ a b ′ , 0). (B.88) Using them Φ is expressed in terms of ϕ only as in [10], see also [30], The ϕ's are further determined by solving the second class constraints χ m a ′ = 0 and the (D − (p + 1)) constraintsφ a ′ = 0. In the gauge (B.87) they are They are combined with π ϕ = 0 in (B.86) to form (p + 1)(D − (p + 1)) second class constraints pairs to eliminate ϕ a b ′ in terms of p a and ∂ m x a as in (2.43). Notice that ϕ a b ′ are written as functions of the phase space variables of the p-brane. Now all second class constraints are used to reduce the phase space to that of (x a , p a ). There remain the first class constraints H ⊥ = H m = 0, and the first class Hamiltonian becomes H = d p ξ (λ ⊥ H 1 ⊥ +λ m H 1 m ), (B.92) The action of p-brane obtained from the non-linear realization in the reduced space leads to the canonical action of a Dirac-Nambu-Goto p-brane action in configuration space.