Symplectic realization of two interacting spin-two fields in three dimensions

We constructed a symplectic realization of the dynamic structure of two interacting spin-two fields in three dimensions. A significant simplification refers to the treatment of constraints: instead of performing a Hamiltonian analysis $\grave{a}\, la$ Dirac, we worked out a method that only uses properties of the pre-symplectic two-form matrix and its corresponding zero-modes to investigate the nature of constraints and the gauge structure of the theory. For instance, we demonstrate that the contraction of the zero-modes with the potential gradient, yields explicit expressions for the whole set of constraints on the dynamics of the theory, including the symmetrization condition and an explicit relationship between the coupling and cosmological constants. This way, we further identify the necessary conditions for the existence of a unique non-linear candidate for a partially massless theory, using only the expression for the interaction parameters of the model. In the case of gauge structure, the transformation laws for the entire set of dynamical variables are more straightforwardly derived from the structure of the remaining zero-modes; in this sense, the zero-modes must be viewed as the generators of the corresponding gauge transformations. Thereafter, we use an appropriate gauge-fixing procedure, the time gauge, to compute both the quantization brackets and the functional measure on the path integral associated with our model. Finally, we confirm that three-dimensional bi-gravity has two physical degrees of freedom per space point. With the above, we provide a new perspective for a better understanding of the dynamical structure of theories of interacting spin-two fields, which does not require the constraints to be catalogued as first- and second-class ones as in the case of Dirac's standard method.


I. INTRODUCTION
Although Einstein's General Theory of Relativity [1] is accepted as being the only physical theory that describes the geometrical structure of the space-time and the gravitational dynamics of massive bodies. The greatest puzzles in modern cosmology such as Dark Matter [2] and Dark Energy [3], which attempt to explain the primordial and late time accelerating expansion of the current universe, have been strong motivators for a plethora of alternative gravity theories beyond original Einstein's General Relativity (GR), both at the ultraviolet and infrared scale [4][5][6]. Besides that, the issue concerning the compatibility between renormalizability and the unitarity of gravity at the quantum regime [7,11], has also led many theorists to search for extensions of GR and their three-dimensional cousins; because they are, to our knowledge, another framework for addressing some conceptual features of the four-dimensional GR and some fundamental issues of quantum gravity [8][9][10]. Nonetheless, modifications and/or extensions to the theory of GR are highly restricted according to Lovelock's theorem [12,13]. Along these lines, given the assumption that GR is essentially the correct theory of gravity leading to the propagation of two physical degrees of freedom (DoF) corresponding to a single spin-two field (graviton) whose rest masses may be exactly zero [14][15][16], a natural extension to Einstein's gravity would be the addition of new DoF to the massless graviton in a consistent manner.
Massive gravity has been studied extensively over the past years as a straightforward extension of GR as it contains five DoF which correspond to a single massive spin-two field [17]. Furthermore, the study of massive gravity models has been mainly motivated by offering an alternative to the ΛCDM cosmological standard model [19][20][21] and by solving the problem of finding a consistent quantum theory of gravity [22]. This interest was triggered in particular by the discovery of a non-linear theory of massive gravity by de Rham, Gabadaze, and Tolley (dRGT) [23][24][25] which not only propagates the appropriate five DoF for a massive spin-two field, but is also free of ghost instabilities [26,27], i.e., unphysical DoF appearing generically at the non-linear level and having unbounded negative energy [28]. Motivated by the advances pioneered by dRGT, Hassan and Rosen (H-R) generalize the dRGT model as a theory of two interacting spin-two fields, dubbed as bi-gravity, which describes two gravitons interacting with each other, through non-derivative interaction terms [29]. It was shown in Refs. [30,31] that like the dRGT model the H-R theory also has a correct number of ghost-free DoF; namely it contains seven DoF corresponding to one massive (five DoF) and one massless (two DoF) graviton, in contrast to the five DoF for the massive graviton of dRGT theory. A remarkable feature of the theory of two interacting spin-two fields, is that it fills a gap in the list of consistent constraint, including the symmetrization constraints, and a unique condition for the partially massless sector of the theory. Having determined the constraints on the dynamics, we will show that the remaining zero-modes are the generators of the local gauge transformations. Thereafter, by choosing an appropriate gauge-fixing procedure, we will obtain the fundamental brackets structure for the dynamic variables and the functional measure for determining the quantum transition amplitude. Finally, we will also confirm that Zwei-Dreibein gravity has two physical degrees of freedom per space point.
The outline of this paper is as follows. In Section II, we will introduce the action principle corresponding to Zwei-Dreibein gravity. Later introducing the action principle, we will perform the 2 + 1 decomposition of the action in Section III, in order to identify the dynamical variables that make up the pre-symplectic matrix. We then show that the symplectic framework applied to Zwei-Dreibein gravity easily allows us to obtain the complete set of physical constraints and the partially massless sector of the theory. In Section IV, we show that the gauge transformations can be computed using the zero-modes of the pre-symplectic two-form matrix. Then we recovered the Diffeomorphisms and Poincaré symmetry by mapping the gauge parameters appropriately. In the penultimate Section V, we determine the fundamental quantization brackets, the functional measure on the path integral and the physical degrees of freedom associated with our model, by introducing a gauge-fixing procedure. Finally, Section VI, is devoted to our concluding remarks.

II. THE ACTION PRINCIPLE
We consider the action for three-dimensional bi-gravity in the first-order formalism [44][45][46]. The space-time M is a three-dimensional oriented smooth manifold and the action is simply given by the sum of two three-dimensional Einstein-Cartan actions with independent deibreins and connections, plus an interaction term: with Λ 1 = −1/l 2 and Λ 2 = −1/ l 2 the cosmological constants and, k 1 and k 2 the coupling. The fundamental fields of the action (1) are: a pair of dreibein one-forms e I = e I µ dx µ and l I = l I µ dx µ , and a pair of dualised spinconnection one-forms A I = ǫ IJK A µ JK dx µ and w I = ǫ IJK w µ JK dx µ valued on the adjoint representation of the Lie group SO(2, 2), so that, it admits an invariant totally anti-symmetric tensor ǫ IJK . Furthermore, R I and F I are the curvatures of the connections A I and w I severally, which explicitly read R I [A] = dA I + (1/2)ǫ IJK A J ∧ A K and F I [w] = dw I + (1/2)ǫ IJK w J ∧ w K . In particular, we defined two types of covariant derivative, acting on internal indices, by mean of the spin-connections A and w, respectively where ∂ is a fiducial derivative operator. In what follows, we denote the components of three-dimensional spacetime x as x µ , µ = 0, 1, 2 and those of space x as x a , a = 1, 2. Whereas, the Latin capital letters I correspond to Lorentz indices, I = 1, 2, 3. An arbitrary variation of the action functional (1) gives a bulk term, which defines the classical equations of motion, and a boundary term in the following form: where E Φ represents equations of motion for all dynamical fields collectively denoted by Φ, δΦ are variations of those fields, and Θ is the boundary term. Then the field equations have the form: E µI e = ε µαβ D α e I β = 0, while the boundary term look like which is understandable as the divergence of a phase space pre-symplectic potential, from which the corresponding symplectic structure is recognized: If we choose Σ to be a time slice x 0 = const, then the symplectic structure takes the form The corresponding non-vanishing fundamental Poisson brackets between pairs of basic fields read The above Poisson brackets should be employed in a Hamiltonian analysisà la Dirac, in order to find out the full structure of the constraints in terms of dreibeins and connections. However, in the following lines, we will discuss a new strategy for investigating the constraint structure and the associated gauge symmetry in Zwei-Dreibein gravity.

III. THE NATURE OF CONSTRAINTS IN THE SYMPLECTIC FRAMEWORK
To carry out symplectic analysis, we assume that the manifold M is globally hyperbolic such that it may be foliated as Σ × ℜ, with Σ being a Cauchy's surface without boundary (∂Σ = 0) and ℜ an evolution parameter. By performing the 2 + 1 decomposition of our fields, the action (1) can be written as (we recall that all spatial boundary terms will be neglected because Σ has no boundary) where we have defined R ab I = ∂ a A b I + (1/2)ǫ IJK A aJ A bK , D a e b I = ∂ a e b I + ǫ IJK A aJ e bK , F I ab = ∂ a w b I + (1/2)ǫ IJK w aJ w bK and ∇ a l b I = ∂ a l b I + ǫ IJK w aJ l bK . To start our analysis, and without loss of generality, we express the action functional (12) in the symplectic form as: which is first-order in time derivativeξ i , where hereafter ξ i stands for the collection of all the dynamical fields of the theory. They are usually referred as symplectic variables in the symplectic framework [53]. The first term in Eq. (13) defines the so-called canonical one-form a = a i (ξ)ξ i , whereas the second term represents the symplectic potential which could also be identified with the canonical Hamiltonian density H. For Zwei-Dreibein gravity, the set of initial symplectic variables and the components of the canonical one-form can be read easily from the first-order Lagrangian density in Eq. (12) as follows On the other hand, the corresponding symplectic potential reads In the symplectic picture, the equations of motion deduced from the above action principle (13) can be compactly written in a first-order form as At this point, it is important to notice that the dynamics of the theory is then characterized by the called pre-symplectic two-form matrix which is defined as a generalized curl of the canonical one-form, Clearly, F ij is an anti-symmetrical matrix that can be either singular or non-singular. According to the symplectic approach, if the matrix F ij is non-singular, then its inverse can be computed. As a consequence, the set of equations in Eq. (17) can immediately be solved for the time evolution of the fields ξ i , as follows, In our case, introducing the symplectic variables (14) and the canonical one-form (15) into the pre-symplectic matrix definition (18), we find that the corresponding pre-symplectic matrix turns out to be After constructing the pre-symplectic matrix, we can see clearly that it is manifestly degenerate in the sense that there are more degrees of freedom in the equations of motion (17) than physical degrees of freedom in the theory. As a consequence of this, there exist constraints that must remove the unphysical degrees of freedom. In the symplectic framework, the constraints emerge as algebraic relations necessary to maintain the consistency of the equations of motion. Since the matrix (20) is a singular one, it is straightforward to determine that it has the following the zero-modes: where v A0 , v w0 , v e0 , v l0 are totally arbitrary functions. In terms of the symplectic formalism, the zero-modes satisfy the equation v i 1,2,3,4 F ij = 0, and therefore, the multiplication of each of the zero-modes by the gradient of the symplectic potential (16) leads to the following conditions: and since v A0 , v w0 , v e0 , v l0 are arbitrary functions, we obtain the following constraints on the dynamics of the theory At this point, we can demand stability of these constraints, which guarantees its time-independence, so that its time derivative vanishesΦ I 1,2,3,4 = 0. This means that the constraints should remain on the constraints surface during their evolution. We note now that Φ I 1,2,3,4 depend only on the set of symplectic variables ξ i , so the consistency condition can be written as; Then, the consistency of the constraints Ω n (33), together with the equations of motion (17) can be rewritten in the following way such that Accordingly, the explicit form of the new matrix F Here we have abbreviated We now can easily verify that F (1) mj is also a singular matrix that has the following linearly independent zero-modes: Using the symplectic potential (16), we find that the matrix Z (1) m has the form Multiplying the zero-modes in Eqs. (37)- (40) to the two sides of Eq. (34), we get the following constraint relations (the integration symbols are omitted for simplicity): Restricting the above relations to the constraints surface, that is v m | Ωn=0 , we can identify the following set of integrability conditions: whose solution is given by The Eq. (50), known as the symmetrization condition, play a crucial role in the relation of the metric and first-order formulations bi-gravity theories [17,45,46]. It is straightforward to see that the above equations Eqs. (50)-(51) can be split into four equations; In this descomposition we can see that the functions (53) and (55) have fixed fields e I 0 , l I 0 , A I 0 and w I 0 . Whereas the functions (52) and (54) gives us two new constraints. In order to be consistent, the new constraints (52) and (54) must be preserved under time evolution, that isΨ = 0 andΥ = 0. Combining Eq. (34) with the consistency conditions on Ψ and Υ, we will obtain a group of new linear equations dx F (2) sj (x, y)ξ(x) j − Z (2) s (x, y) = 0, with Here Ξ l ∈ (Ψ, Υ) and s = m + l.
Using Ψ and Υ to obtain F sj , we see that the sub-matrix (δΞ/δξ) li of F (2) sj to be whereas F mj is defined in Eq. (36). Again, one can easily verify that the matrix F (2) sj in Eq. (57) is also singular, and so, it has the following linearly independent zero-modes: If we multiply the first four zero-mode (59)-(62) by Z s , we will find the same constraint obtained previously, whereas from the zero-mode v + (k 1 e αI + k 2 l αI ) (k 1 e βJ + k 2 l βJ ) (e µK − l µK ) + (k 1 e αI + k 2 l αI ) (Λ 1 e βJ e µK − Λ 2 l βJ l µK )] , These conditions must hold only on the constraints surface, i.e., when Ω n , Ξ l = 0. Imposing the above condition, we obtain the following scalar relation between coupling and cosmological constants, plus a parameters-free relationship fixing the field A I 0 in terms of w I 0 , Since the expression in Eq.(67) mixes the canonical variables A I a , w I a , e I a and l I a with the Lagrange multipliers A I 0 , w I 0 , e I 0 and l I 0 , this condition becomes in an scalar equation, establishing the most general relationship between the coupling parameters of the theory, rather than acts as a new constraint on the dynamics of the theory. Hence, there are no further constraints in Zwei-Dreibein gravity and therefore our method to obtain new constraints via the consistency condition has finished. Now, it is very important that, according to our result in Eq. (67), for the conformal case in which the dreibeins are proportional to each other; we obtain from Eq. (67) the following condition for the parameter values: where we have used ε αβµ e I α e J β e I µ = eǫ IJK with e = det|e I α |. At this stage, we already note that the Eq. (70) (first determined in Ref. [33] in the metric formalism via field equations) immediately suggest the following solution: An interesting feature of this solution is that it satisfies the Higuchi bound [33]. Moreover, by substituting (69) and (71) into Eq. (1), the bi-gravity action simply reduces to two copies of Einstein-Cartan gravity for e I α with a cosmological constant, explicitly, Remarkably, only for these parameter values (71), we have that the original theory defined in Eq. (1) reduces to a non-linear partially massless bi-gravity theory, where the mass value of a very special massive spin-two particle, referred to as partially massless is m 2 = −Λ 1 [33][34][35][36][37]. In this case, the partially-massless spin-two field has 1 degree of freedom more than those of massless field describing 3D pure gravity and 1 less that those of massive spin-two field in three dimensions.
On the other hand, having at our disposal the explicit form of the constraints (30), (31), (32), (32), (52) and (54), and according to the symplectic procedure, we shall now incorporate them into the kinetic part of the Lagrangian density (13) through the corresponding Lagrangian multipliers to define a new one. As a result, the new Lagrangian density is: whereλ 1I ,λ 2I ,λ 3I ,λ 4I ,λ 5 andλ 6 stand for multipliers associated to the constraints that enforce the stability of the full set of constraints in time. Furthermore, one can note that the symplectic potential, which can also be identified with the total Hamiltonian density H, turns out to be which drop from the Lagrangian density after being evaluated on the constraint surface, along with H| Ωn,Ξ l = 0. This fact shows the general covariance of the theory, and therefore, the dynamics will be governed by the constraints. Thus, the Lagrangian density (73) contains all the necessary information to describe the dynamics of the three-dimensional bi-gravity theory. Now the new set of symplectic variables is identified easily as: ξ new i = (A aI , w aI , e aI , l aI , λ 1I , λ 2I , λ 3I , λ 4I , λ 5 , λ 6 ), .
This permits us to identify the components of the canonical one-form Making use of the definition of the pre-symplectic two-form matrix in Eq. (18), and after a bit of calculation, we can show that the explicit expression for the corresponding square matrix F New Here we have abbreviated Notwithstanding the above positive results, it is worth noting that the corresponding pre-symplectic matrix F new ij remains singular, however, we have shown that no further constraints are generated by the zero-modes. Thus, the above observation implies that there might be gauge degrees of freedom in the theory that must be fixed through gauge conditions in order to obviate the singularity. In this way, the quantization-bracket structure can be determined and the procedure can be achieved in terms of the physical degrees of freedom.

IV. GAUGE TRANSFORMATIONS AND ITS GENERATORS
We thus proceed towards the discussion of the gauge symmetry in the symplectic framework. It is worth noting that, the degeneracy of the pre-symplectic matrix (77) and the fact that its remaining zero-modes are orthogonal to the gradient of the potential that means that the remaining zero-modes generate degenerate gauge directions inside the symplectic potential (74). As a consequence, such zero-modes must be identified as the generators of the corresponding gauge symmetry 'δ G ', that is, the components of the zero-modes give the transformation properties related to the underlying (gauge) symmetry that leaves the action invariant [54,56]. The local infinitesimal transformations on the symplectic variables generated by v i A can be expressed as: where v i A are the independent zero-modes of the singular symplectic matrix F New ij and η A are the gauge parameters. For the singular pre-symplectic matrix (77), these zero-modes turn out to be: these zero-modes turn out to be orthogonal to the gradient of the symplectic potential and at the same time generate local displacements on the isopotential surface. In this way, one can infer from Eq. (78), that the infinitesimal gauge transformations that leave the original Lagrangian invariant take the form where η I , ς I , σ I and ̺ I are the arbitrary gauge transformation parameter. It is worth remarking that (83), (84), (85) and (86) correspond to the fundamental gauge symmetry of the theory, though the diffeomorphisms symmetry 'δ Diff ' have not been found yet. However, it is well-known that an appropriate choice of the gauge parameters does generate the diffeomorphism on-shell. Let us redefine the gauge parameters: with ζ β an arbitrary three-vector. Consequently, from the fundamental gauge symmetries (83)-(86), we obtain δ Diff e I α = L ζ e I α + ε αβµ ζ β E µI e , δ Diff l I α = L ζ l I α + ε αβµ ζ β E µI l .
which are precisely diffeomorphisms on-shell. Besides, diffeomorphism invariant theories have the Poincaré gauge transformations 'δ PGT ', i.e. local Lorentz rotations and translations, as off-shell symmetries by construction [60,61]. Thus, to recover the Poincaré symmetry we need to map the arbitrary gauge parameters of the fundamental gauge symmetry δ G into those of the Poincaré symmetry. This is achieved by a mapping of the gauge parameters [62][63][64], e.g.: such that θ µ and ω I are related to local coordinate translations and local Lorentz rotations, respectively, which together constitute the 6 independent gauge parameters of Poincaré symmetries in 3D. By using this map, the gauge symmetries reproduce the Poincaré symmetries modulo terms proportional to the equations of motion: We thus conclude that the Poincaré symmetry δ P GT , as well as the diffeomorphisms δ Dif f , are not independent symmetries: they are contained in the fundamental gauge symmetry δ G as on-shell, that is, only when the equations of motion are imposed. Besides, the generators of such gauge transformations are represented in terms of the zeromodes, thereby making evident that the zero-modes of the pre-symplectic two-form encode all the information about the gauge structure of this theory.

V. FIXING GAUGE
As was already mentioned in Sec. III, in theories with gauge symmetry, the pre-symplectic matrix obtained at the end of the procedure is still singular. Nevertheless, to obtain a non-singular symplectic matrix and to determine the quantization-bracket structure between the dynamical fields, we must impose a gauge-fixing procedure, that is, new gauge constraints. In this case, we now partially fix the gauge by imposing the one most natural manner is to choose the time-gauge, namely, A I 0 = 0 (λ 3I = const) and e I 0 (λ 1I = const). In this manner, we also introduce new Lagrange multipliers relative to the gauge conditions, namely,λ 7I andλ 8I . Thus, the final Lagrangian density after gauge fixing can be written as Clearly, from the Lagrangian density (97), the final set of symplectic variables is taken as ξ i Final = (A aI , w aI , e aI , l aI , λ 1I , λ 2I , λ 3I , λ 4I , λ 5 , λ 6 , λ 7I , λ 8I ), (98) From the above set of the symplectic variables, we finally obtain the pre-symplectic matrix defined in Eq. (18), given by a block matrix of the form Here the explicit form of each sub matrix A, B and D in Eq. (100) turns out to be Here, we can easily see that A is invertible.
Using the standard identity for any matrix where A and D are square matrices, but B and C need not be square. We can see that Hence, making use of (105), and after some algebra, it is possible to show that the determinant of F Final here we have defined ∆ = e J c l c J . This result leads us to conclude that F Final ij is not singular, and therefore the inverse of this matrix exists: it is dubbed as the symplectic two-form matrix. Now, it is interesting to note that according to Toms approach [57], the functional measure on the path integral associated with our model, in the time gauge, is On the other hand, since the matrix (100) is of block form, its formal inversion is of the form In this setup, the form of the above matrix F Final After some algebraic manipulations, we have Then the above brackets all turn out to be either zero unlike those derived in Eq. (11). Now, we need a bracket for observables on Σ. Such a bracket should agree with the commutator in the classical limit. In this regard, for any two observable O 1 , O 2 defined on the phase space which owns itself a symplectic structure as ξ Final i , ξ Final j , we can define the following relation: This can be taken as the definition of the Poisson bracket. The canonical quantization can be fully made at tree level by the replacement of classical observables and Poisson brackets by the quantum operators commutators, respectively, according to: whereÔ is any operator associated with an observable (or constraint) and |ψ is any quantum state. Finally, the number of propagating degrees of freedom may be calculated in the phase space from the relation where N 1 is the number of field components in ξ i = A I 0 , A I a , w I 0 , w I a , e I 0 , e I a , l I 0 , l I a , N 2 is the number of fields eliminated A I 0 , w I 0 , e I 0 , l I 0 , and N 3 is the number constraints including gauge fixing conditions Φ I 1 , Φ I 2 , Φ I 3 , Φ I 4 , Ψ, Υ, A I 0 = 0, e I 0 = 0 . Hence, it is concluded that contrary to standard 3D pure gravity, bi-gravity in three dimensions has 1/2 (36 − 12 − 20) = 2 physical degrees of freedom; 2 degrees of freedom for a massive spin-two field and 0 for a massless spin-2 field in three dimensions, which is the desired result in agreement with [46].

VI. FINAL REMARKS
In this paper we have discussed the construction of a symplectic realization for a theory describing two interacting spin-two fields in three dimensions, called Zwei-Dreibein gravity. The construction was done within the symplectic framework developed originally by Faddeev and Jackiw [53,58,59]. A central point in our discussion is that the analysis be focused on the properties of the pre-symplectic two-form matrix and its corresponding zero-modes which are associated with the constrained dynamics of the theory. The remarkable feature of such a construction is that it does not need to classify the constraints into first-and second-class ones as in the case of the standard Hamiltonian procedures. For instance, using only the zero-modes of the corresponding pre-symplectic matrix, we have explained how to extract, systematically and consistently, the structure of all the physical constraints on the dynamics of the theory. This has led us to infer the necessary conditions under which a candidate for a partially massless theory at the non-linear level exists. Furthermore, upon using the remaining zero-modes, we explicitly found out the full gauge transformations for all the fundamental variables, while showing that the remaining zero-modes are indeed the generators of the local gauge symmetry under which all physical quantities are invariant. In particular, we successfully recovered the Diffeomorphisms and Poincaré symmetry by mapping the gauge parameters appropriately. This leads to a significant reduction of labor compared to the framework of Dirac for constrained systems.
Moreover, we have shown that the time-gauge condition on the Lagrangian density (73) renders a non-degenerate symplectic matrix F Final ij (100), whose determinant allows us to identify the functional measure for determining the quantum transition amplitude according to Toms [57], and whose inverse allows one to identify the quantizationbrackets. As a consequence, we have confirmed that Zwei-Dreibein gravity has two physical degrees of freedom per space point, as expected. Our work suggests that this symplectic method can be straightforwardly applied in massive-and bi-gravity theories written in first-order formalism in three-and four dimensions, trivializing the issue of identification of the physical constraints and gauge structure of such theories.

VII. ACKNOWLEDGMENTS
This work has been supported by the National Council of Science and Technology (CONACyT) of México under postdoctoral Grant No. 290692.