M5-brane as a Nambu-Poisson geometry of a Multi D1-brane theory

We introduce a Nambu-Poisson bracket in the geometrical description of the D=11 M5-brane. This procedure allows us, under some assumptions, to eliminate the local degrees of freedom of the antisymmetric field in the M5-brane Hamiltonian and to express it as a D=11 p-brane theory invariant under symplectomorphisms. A regularization of the M5-brane in terms of a multi D1-brane theory invariant under the $SU(N)\times SU(N)$ group in the limit when $N\to\infty$ is proposed. Also, a regularization for the D=10 D4-brane in terms of a multi D0-brane theory is suggested.


I. INTRODUCTION
Although the geometrical and algebraic structure of the M5-brane has been considered by several authors, it is far from been completely understood. There are natural questions concerning the structure of the spectrum of its Hamiltonian that remain without answer.
In that sense an important aspect to analyze is the role of the volume preserving diffeomorphisms as a residual gauge symmetry of the canonical action once the light cone gauge has been fixed. In the case of the D = 11 Supermembrane, the residual gauge symmetry is the area preserving diffeomorphisms group, which coincides with the symplectomorpisms preserving a canonical two-form ω. In two dimensions ω may be expressed in terms of the totally antisymmetric tensor density and the volume element which is naturally introduced by the light cone gauge fixing procedure. In this case, then, both groups coincide, however, for higher dimensional p-branes or D-Branes, the symplectomorphisms are only a subgroup of the volume preserving diffeomorphisms. The understanding of this residual gauge symmetry is directly related to the problem of finding an interesting regularization of the M5-brane in terms of multi D0 or D1-branes. That regularization was crucial in understanding the spectrum of the D = 11 Supermembrane. In that case, the regularization was performed in terms of a quantum mechanical system which may be interpreted as a SU(N), N → ∞, Yang-Mills theory on zero spatial dimensions [1]or as a multi D0-brane theory [2]. It was then understood that the existence of a continuous spectrum from 0 to ∞, in the supersymmetric case, is directly related to the existence of string-like spikes which may be attached to any membrane without changing its energy. The supersymmetry was relevant since it annihilates an effective basin shaped bosonic potential arising from the zero point energy of the harmonic oscillators associated to the membrane potential. It is already known [3], [4] the existence of 4,3,2 and 1-branes spikes, which may be attached to the M5-brane without changing its classical energy. This is also a general property of all p-Branes [5]. It is not known however its relation to the spectrum of the M5-brane quantum Hamiltonian. Even more, it is not known if there exists a suitable regularization for the M5-brane. In this letter we start to analyze some of these questions.
We will show how the volume preserving generators and the M5-brane Hamiltonian fits into a description where the world volume is endowed with a Nambu-Poisson structure. This algebraic structure was already proposed for p-branes in [6]. The Nambu-Poisson structure over the world volume is directly related to the canonical Poisson structure over the fields describing the M5-brane. We will then consider the reduction of the volume preserving diffeomorphisms, by taking a suitable partial gauge fixing, to symplectomorphisms preserving a symplectic two-form constructed from the antisymmetric field of the 5-brane. We will show there is no local dynamics associated to the antisymmetric field. This one only contributes to global degrees of freedom. We finally use this partial gauge fixing to construct a regularization for the M5-brane in terms of a multi 1-brane theory with gauge symmetry SU(N) × SU(N), N → ∞. The analogous construction for the D = 10 4-brane yields a regularization in terms of a multi 0-brane theory, but without any residual gauge symmetry.

II. THE ALGEBRAIC STRUCTURE OF THE M5-BRANE HAMILTONIAN
We start recalling the M5-brane Hamiltonian for the bosonic sector in the light cone gauge that was obtained in [3], where and Θ 5i , Θ j , Λ αβ are the Lagrange multipliers associated to the remaining constraints where (5) and (6) are the first class constraints that generate the gauge symmetry associated to the antisymmetric field and (7) is the volume preserving constraint. √ W is a scalar density introduced by the LCG fixing procedure, it may be interpreted as the square root of the determinant of an intrinsic metric over the spatial world volume. P µν and Π M are the conjugate momenta to B µν and X M , respectively. In our notation caps Latin letters are transverse light cone gauge indices M, N = 1, . . . , 9, Greek ones are spatial world volume indices ranging from 1 to 5, and small Latin letters denote spatial world volume indices from 1 to 4.
The elimination of second class constraints from the formulation in [7], [8] and [9] to produce a canonical Hamiltonian with only first class constraints, was achieved at the price of loosing the manifest 5 dimensional spatial covariance. In this way , the spatial world volume splits into M 5 = M 4 × M 1 . We will exploit this decomposition in our analysis of the Hamiltonian. The supersymmetric version of this theory was given in [4] and [10].
We will see now how a formulation in terms of Nambu-Poisson brackets arises naturally from the analysis of the volume preserving diffeomorphisms in more detail.
The algebra of the first class constraints Ω αβ on M 5 is where ξ αβ are the antisymmetric parameters of the infinitesimal transformation and The transformation of scalar fields X M under these infinitesimal diffeomorphisms is and that one for their corresponding conjugate momenta is since Π M are scalar densities.
We notice that the parameter ξ α = 1 ensuring that that is , the diffeomorphisms generated by Ω αβ are the volume preserving ones over M 5 .
All the above mentioned properties are in general valid for the volume preserving diffeomorphisms in any p-dimensional world volume. In the case p = 2 an explicit solution for ξ α is then the transformation rule for any scalar field Φ becomes where is a Poisson bracket with symplectic two-form ω αβ = √ W ǫ αβ . The structure constants of the area preserving diffeomorphisms, in the p = 2 case, were interpreted as the N → ∞ limit of SU(N) in [11], leading to a SU(N) regularization of the D = 11 Supermembrane and its further quantum mechanical analysis of the canonical Hamiltonian [1]. In p = 2, the area preserving diffeomorphisms are exactly the same as the symplectomorphisms preserving ω αβ . This property of the p = 2 case was an essential ingredient in the noncommutative formulation of D = 11 Supermembranes presented in [12] and of its quantum mechanical analysis in [13] [14].For p ≥ 2, the symplectomorphisms are only a subgroup of the volumepreserving diffeomorphisms.
We may now consider an explicit solution for p ≥ 2. There always exist consequently, the volume preserving transformation of any scalar field over the p-brane world volume becomes which leads to the introduction of the Nambu bracket [15] with p-entries.
It turns out that the commutator of the volume preserving diffeomorphisms may now be expressed in terms of the Nambu algebraic structure: where the parameters may be expressed as It is interesting that the algebraic properties leading to (22) and (23) arise only from the generalized Jacobi identity for n-Lie algebras defined by skew-symmetric n-brackets, n ≤ p acting on the ring of C ∞ functions, with no need of an explicit expression for the Nambu-Poisson bracket. The fundamental identity or generalized Jacobi identity [16]is For any bracket satisfying this identity (24) and the Leibnitz rule [17], there exists an n-vector field such that Any bracket satisfying (24) and the Leibnitz rule is called a Nambu-Poisson bracket. When n ≥ 3, ω is decomposable at its regular points, that is this means that any Nambu-Poisson bracket may be expressed at its regular points as a Nambu bracket [18] {f 1 , . . . , f n } N P = {f 1 , . . . , f n , g 1 . . . g p−n } for some g 1 . . . g p−n .
Moreover, an n-bracket with n ≥ 3 entries is Nambu-Poisson if and only if by fixing an argument one obtains an (n − 1) Nambu-Poisson bracket [19], related work may be found in [20]. This is the kind of bracket we will find in the Hamiltonian of the M5-brane.
We conclude then that it is possible to endow any p-brane world volume with a Nambu-Poisson structure. In [21] it was proposed such kind of algebraic structure for world volume p-branes. The Nambu-Poisson structure over the world volume coexists with the canonical structure of the field theory.

III. THE NAMBU-POISSON STRUCTURE OF THE M5-BRANE HAMILTO-NIAN
In two dimensions, the area preserving diffeomorphisms are the same as the symplectomorphisms. This feature has very interesting consequences for the supermembrane since the intrinsic symplectic structure arising, for example, from a non-trivial central charge in the supersymmetric algebra gives then rise to a formulation of the theory, in terms of a noncommutative geometry [12]. In higher dimensions, the symplectomorphisms are a subgroup of the full volume preserving diffeomorphisms. It is then interesting to analyze how the geometry of the M5-brane allows the reduction from the volume-preserving diffeomorphisms to only symplectomorphisms . In the supermembrane case it is assumed that the spatial part of the world-volume is a Riemann surface. Those are complex manifolds which admit a symplectic structure, moreover, they are Kähler manifolds. In our formulation the spatial five dimensional world-volume M 5 has the structure M 4 × M 1 . This is required to eliminate second class constraints. We will assume that M 4 admits a symplectic structure denoted as ω 0 . It is convenient to identify the scalar density √ W with the one arising from the symplectic structure over M 4 .
We will now show how the interacting terms of the M5-brane Hamiltonian (1) may be expressed directly in terms of the Nambu-Poisson bracket in five dimensions. Let us analyze it term by term. We first notice that g, the determinant of the induced metric may be re-expressed in a straightforward manner as a bracket g = 1 5! ǫ ν 1 ,...,ν 5 ǫ µ 1 ,...,µ 5 g µ 1 ν 1 . . . g µ 5 ,ν 5 Let us consider now the third term, it depends on the antisymmetric field B µν . It is invariant under the action of the first class constraints (5) and (6). To eliminate part of these constraints, we proceed to make a partial gauge fixing on B µν , following [3] we take which, together with the constraint (6), allow us a canonical reduction of the Hamiltonian (1). We notice that the contribution of this partial gauge fixing to the functional measure is 1. We are then left with the constraint which generates the gauge symmetry on the two-form B B as a two-form over M 4 may be decomposed using the Hodge decomposition theorem in an exact form plus a co-exact form plus an harmonic form. With this at hand, an admissible gauge fixing consists to impose the exact form part to be zero. In this gauge, we may express P ij as where ω is a closed two form. We also obtain The kinetic term in the action associated to the antisymmetric field becomes then where the co-exact part of B ij is conjugate to the exact part of ω kl , while the harmonic parts are conjugate to each other.
Noticing that l 5i is divergenceless, it may be rewritten as This decomposition in terms of scalars is always valid for any four dimensional divergenceless smooth vectorial density. The co-exact part of B kl may then be expressed in a unique way, modulo the symmetry generated by the group of rigid transformations that leaves invariant (35), in terms of the triplet φ a . Now we decompose the tensor density l ij into where ω is a closed two form. Comparing ω with ω, we notice that part of ω has been absorbed into the first term of the right hand side of (36). This decomposition exists and it is unique.
It is now possible, as a consequence of the Darboux's theorem , to express ω kl in terms of the two-form ω 0 over M 4 with k, m = 1, 2, 3, 4. When ω is non degenerate, there exists an atlas {U} such that, on each open set U, Ψ k describe the diffeomorphisms which reduces ω jl to the two-form ω 0 .
When ω is degenerate the decomposition (37) is still valid. In that case pairs of Ψ k may be zero.
Although there are four fields Ψ k , they represent only three physical degrees of freedom, since on an open set where ω has constant rank , one may reduce locally ω to ω 0 by a change of variables and still have one gauge symmetry left, the diffeomorphisms which preserve ω 0 .
The infinitesimal group parameter of those diffeomorphisms can be expressed as Using (35), (36) and (37) the interacting term in the Hamiltonian involving the antisymmetric field and its conjugate momentum may be expressed in terms of the Nambu-Poisson brackets Moreover, in the case of a non degenerate ω, the second Nambu-Poisson bracket may be re-expressed, on any open set of a Darboux atlas, in terms of a Poisson bracket constructed with the symplectic two-form ω 0 ij by fixing the volume preserving diffeomorphisms: In this case, the Hamiltonian is still invariant under the symplectomorphisms which preserves ω 0 . The interacting terms of the canonical Lagrangian may then be expressed in terms of the Nambu-Poisson bracket constructed with the antisymmetric tensor ǫ µναβγ / √ W and the Poisson bracket constructed with the symplectic two-form ω 0 .

D1-BRANES
We will discuss in this section a partial gauge fixing of the M5 brane which yields a formulation of the theory invariant under symplectomorphisms only, allowing a regularization with a similar approach to the one of the D = 11 supermembrane [1].In here, we will assume ω to be non-degenerate. After fixing ω to ω 0 we may resolve the volume-preserving constraint for φ a a = 1, 2, 3. We are then left still with one constraint, The left hand member generates the symplectomorphisms preserving ω 0 . The five dimensional volume preserving diffeomorphisms have been reduced by the gauge fixing procedure to only that generator. We are then left with a formulation in terms of X M and its conjugate momenta Π M , invariant under symplectomorphisms. The antisymmetric field B µν and its conjugate momenta P µν have been reduced to ω 0 , there is no local dynamics related to them. All the dynamics may be expressed in terms of (X M , Π M ). We may then perform the explicit 4 + 1 decomposition on the spatial sector of the world-volume. We obtain We may also rewrite the Nambu-Poisson bracket in terms of the Poisson bracket as where the brackets on indices denote cyclic permutation.
The contribution of the other terms in the Hamiltonian may be expressed in a similar way. The scalar fields φ a may be eliminated from the volume preserving constraints. This elimination is algebraically possible since they appear linearly in them. We will show the complete analysis for the D4-brane in 10 dimensions. The Hamiltonian may be obtained from the one for the M5-brane by considering the potential gauge fixing X 11 = σ 5 , together with the assumption that all other fields are independent of σ 5 . π 5 is eliminated from one of the constraints.
The resulting Hamiltonian acquires then, as a result of the Π 5 elimination, a quadratic term in P ij , i, j = 1, 2, 3, 4. The bosonic sector of the Hamiltonian may then be expressed The last term on the right hand side is, in this case, the determinant of the Riemannian metric constructed from the symplectic structure and the integrable almost complex struc- while the third term requires the solution of the volume preserving constraints. We obtain where θ is a scalar field which becomes determined from the divergenceless property of the left hand member is the derivative of θ in the direction of the gradient of the scalar field(48).
The third term of (47) may thus be written as All the interacting terms of the Hamiltonian (47) can then be expressed in terms of the Poisson bracket.
Finally, in order to obtain a regularization of the above Hamiltonian, we express X M and Π M in terms of a complete orthonormal basis over M 4 , {Y a }, in the Hilbert space of L 2 functions. Since is a scalar function over M 4 , we may reexpress it in terms of the basis, and obtain where f abc is given by and is totally antisymmetric on a, b, c We then have consequently, Furthermore, we may introduce Which is again valid since Y a Y b is also a scalar function over M 4 . We get which becomes totally symmetric in a, b, d.
All the interacting terms in the Hamiltonian may be rewritten using X M a , their conjugate momenta and the structure constants f abc and C abd . As an example, the second term in the Hamiltonian (1) is given by which now takes the form of the potential of a condensate D1-branes over a non trivial background. The geometry of the background depends on the symplectic structure of the M 4 manifold. We may then regularize the Hamiltonian by taking a truncation in the range of the index a. The truncated Hamiltonian for the M5 brane may then be interpreted as a SU(N) × SU(N) gauge theory. While the D4-brane Hamiltonian corresponds to a gauge fixed SU(N) × SU(N) theory. The potential contains powers of X M a up to 8, moreover, the quadratic term in the momenta is contracted with a nontrivial background metric. Nevertheless, the corresponding quantum mechanical problem seems reasonable to be analyzed.
This regularized version of the D4-brane in terms of D0-branes as well as the one for the M5-brane in terms of D1-branes are both formulated in terms of X M and Π M as in the case of the supermembrane. In the latter case and that one for the M5-brane , there is a gauge freedom generated by symplectomorphisms. The antisymmetric field on the M5 and D4-branes only contributes with global degrees of freedom, not with local ones.

V. CONCLUSIONS
We introduced a Nambu-Poisson structure in the description of the D = 11 M5-brane and in the D = 10 D4-brane. By partial gauge fixing the volume preserving diffeomorphisms, we showed that the symplectic sector of the M5-brane Hamiltonian may be expressed as a D = 11 p-brane theory invariant under symplectomorphisms preserving a symplectic twoform constructed from the antisymmetric field of the M5-brane. All the degrees of freedom of the antisymmetric field reduce to global ones. We then propose a regularization of the M5-brane in terms of a multi D1-brane theory invariant under SU(N) × SU(N) in the limit when N → ∞. We also propose a regularization for the D = 10 D4-brane in terms of a multi D0-brane theory.