Nambu bracket and M-theory
Nambu proposed an extension of dynamical system through the introduction of a new bracket (Nambu bracket) in 1973. This article is a short review of the developments after his paper. Some emphasis are put on a viewpoint that the Nambu bracket naturally describes extended objects which appear in M-theory and the fluid dynamics. The latter part of the paper is devoted to a review of the studies on the Nambu bracket (Lie 3-algebra) in Bagger-Lambert-Gustavsson theory of multiple M2-branes. This paper is a contribution to the proceedings of Nambu memorial symposium (Osaka City University, September 29, 2015).
Nambu’s contributions to Physics are profound and diverse. While creating great ideas such as spontaneous symmetry breaking which becomes standard in the contemporary Physics, he sometimes presented ideas which were mysterious in the beginning but became gradually recognized after years. Nambu bracket  may be one of latter examples. The importance might not be so obvious even for himself. According to the paper, he kept the idea for more than twenty years before the publication. If we take it as was written, it started in early 50s when he moved from Osaka City University to Princeton. The reason why he needed so long period to decide the publication is understandable from his paper. Just after the definition of the bracket, he pointed out serious obstacles for his generalized dynamical system. During the long period that he kept his idea, he developed various new ideas which are useful and stimulating even from the current viewpoints.
As described in , there are two major challenges in the subject. One is how to quantize the Nambu bracket and the other is multi-variable extensions. This turned out to be difficult or impossible (there appeared the no-go theorems). We have to relax “natural” requirements of the Nambu bracket which are the direct generalization of the Poisson bracket. The ways to relax the conditions are not unique and depend on the problem we are considering. It explains the existence of many proposals to define (quantum) Nambu bracket.
The purpose of this article is to give a brief review of the Nambu bracket and to illuminate some applications in M-theory. In section 2, we explain the basic material in the original paper  where many ideas were already written. We also briefly quote some of the important results since then. It turned out that Nambu bracket fits with M-theory well and there appeared varieties of applications. We put some emphasis on the matrix model description of M-theory. In section 3, we review a proposal by Takhtajan  that the Nambu bracket naturally describes the extended object. For the 3-bracket case, it corresponds to strings. In this respect, it fits non-canonical string such as the self-dual string on M5-brane and the vortex in the incompressible fluid. We explain the quantization of Takhtajan’s action which might be relevant to describe these non-canonical strings. Finally in section 4, we review the developments of the Nambu bracket and associated Fillipov Lie 3-algebras to describe the multiple M2-branes by Bagger, Lambert and Gustavsson (BLG model) [3, 4, 5, 6]. Special emphasis is put on our works where we introduced varieties of Lie 3-algebras with Lorentzian signature in BLG formalism to describe different types of extended objects appearing in M-theory and string theory.
2 Nambu bracket
2.1 An introduction of Nambu bracket
In 1973 , Nambu proposed a generalization of Poisson bracket defined on a canonical pair
by the introduction of new dynamical system based on a canonical triple :
This bracket was later referred to as the Nambu bracket. Instead of the canonical Hamiltonian equation,
the time evolution is defined by the new bracket with two Hamiltonians ,
due to the antisymmetry of the bracket.
In this sense, it defines a dynamical system which has a generalized Liouville property (conservation of phase volume). This was one of the reasons why Nambu introduced such bracket.
As an example which is described by the new bracket, Nambu considered the rotational motion of a rigid body which is described by angular momentum . In this case, we have two conserved quantities, the energy and the total momentum:
where are the moment of inertial along each axis. We introduce the Nambu bracket by
Some computation shows that the equation (2.1) gives Euler’s equation for the rigid body:
2.2 Generalizations of Nambu bracket
2.2.1 Mathematical definition
Nambu bracket is defined in more abstractly through the following requirements which generalize those for the Poisson bracket. It is defined on the ring of functions with variables . The Nambu bracket in a generalized sense is defined by a map
which satisfies the following three conditions :
Alternation law (skew symmetry):
Derivative law (Leibniz rule):
Generalized Jacobi law (fundamental identity):
These rules are essential to define the time evolution of Nambu equation with Hamiltonians:
or a canonical transformation of variables defined by generating functions (for ):
They are natural in the sense to ensure the basic properties of the dynamics. Firstly, the alternation law I) ensures the Hamiltonians are constants of the motion222It implies that the Nambu dynamical system has higher conserved quantities . In this sense, it has some connection with the integrable models. See, for example , for a study in this direction. :
The derivative law II) implies Leibniz rule for the time derivative:
Finally the fundamental identity III) (in the following we abbreviate it as FI) implies the distribution law of the time derivative in the bracket:
2.2.2 Some properties of the generalized Nambu bracket
Here is a few comments on the generalized Nambu bracket and Liouville theorem:
The Jacobian 
satisfies all conditions I)–III) for . The time evolution defined by this bracket keeps the -dimensional phase volume , thus the dynamics satisfies the Liouville theorem.
In , possible solutions to the conditions I) II) III) are examined. The bracket which satisfies I) and II) may be written in the form:
where is anti-symmetric for the indices. The fundamental identity is written as the bilinear identities among Nambu tensor . It was proved that Nambu bracket should be decomposable
to satisfy the constraint . In particular, a natural multi-variable extension such as does not satisfy FI.
In order to keep the phase volume, it is possible to generalize (2.2.1) to
with Hamiltonians . These generalized Hamiltonians, however, are not preserved by the equation of motion. In terms of the canonical variables, the equation of motion is written as
The quantity is antisymmetric . The first equation is the most general form to preserve phase volume.
For case, the canonical equation is rewritten as
It was noted  that there are some arbitrariness in the choice of to give the same equation. Namely a different set of Hamiltonian gives the same equation of motion as long as it satisfies canonical transformation with as the canonical pair in the Poisson sense,
One may check the statement for the infinitesimal variations. Let us use and . The variation of the equation (24) is absorbed in the variation of as, which may be interpreted as the infinitesimal gauge transformation. It is obvious that it leads to the same equation of motion.
The other type of the hierarchy structure exists for general . Starting from arbitrary bracket which satisfies I)-III), one may define the bracket by using arbitrary ,
One may show easily that the new bracket satisfies the three conditions. By continuing the same procedure, one may obtain Nambu bracket from Nambu bracket for .
As an example, let us take the Nambu bracket for the rigid rotor. The original Nambu bracket was defined as
If we take , the Poisson bracket gives
which is the standard Poisson bracket for the angular momentum.
2.3 Difficulties in Nambu bracket
In , it was already mentioned some serious difficulties in the formulation. They are not the technical problems and there is no way to overcome them. All we can do is to relax some of the conditions I), II), III) as long as they do not produce serious troubles in the applications which we consider.
In Poisson bracket, it is straightforward to extend the formalism to canonical pairs, () as
It satisfies the consistency condition of the Poisson bracket (Jacobi identity),
for any . The existence of such identity is necessary for the compatibility of the time evolution (2.1).
In the Nambu bracket, the analog of (30) is played by the fundamental identity (FI). A difficulty of the Nambu bracket is that the FI is too strict that there is almost no room for the generalization. As already mentioned, a naive multi-variable extension of (2.1)
for variables (, ) does not satisfy FI. In , Nambu examined the canonical transformation defined by the bracket (31) and the generating function in (2.2.1) should be decomposed as from the consistency conditions. It implies that the variable set should transform within themselves. While the fundamental identity was not proposed explicitly but this analysis has already shown the difficulty in the multi-variable extension.
In the Poisson bracket, the quantization procedure is to replace the bracket into the commutator
The commutator satisfies a noncommutative version of the three consistency conditions.
For the Nambu bracket, the most straightforward generalization of the commutator is,
While it satisfies I), the conditions II) and III) are not kept.
Solutions to canonical quantization condition
While it does not satisfy the conditions, it may be possible to use it relaxing some conditions. In , Nambu tried to find a set of operators which satisfies an analog of canonical quantization condition:
while neglecting the constraints (2,3) for the moment. Here when and otherwise. Assuming the set () are the basis of some Lie algebra . By writing
for the first three generators and . Eq.(34) implies that
The right hand side is c-number and should commute with arbitrary generators in . So it may be implemented by Casimir operator for the Lie algebra. From this observation, assuming is semisimple, one may classify the possible algebras. The result is:
If the algebra is not semi-simple, there are futher choices after contractions:
Here is the euclidean algebra generated by (momentum and angular momentum operators). The others are similar algebra with different dimensions and signature.
Use of nonassociative algebras
Nambu also considered a possibility to use nonassociative algebra to define the quantization. In this case, the associator
does not in general vanish. If we require that the associator be skew symmetric with respect to all elements, the algebra is restricted to Cayley number. It nevertheless does not satisfy the derivative property.
He then modified the bracket to keep the derivative property:
for the Cayley number. This time, we do not have total skewness but only the partial one: . The time evolution generated by
generates the automorphism.333It looks like Nahm equation with holonomy if and are properly chosen. It may provide another link with M-theory. See for example, .
He also examined to use a commutative and nonassociative algebra (Jordan algebra). In this case, the derivative operator is written in the form:
Jordan algebra, in general, is written in terms of noncommutative and associative algebra by modification of the multiplication . If we use this realization, the derivative operator is rewritten as . So the equation of motion is reduced to the conventional Hamiltonian flow where Hamiltonian is written in the form .
2.4 Some attempts to quantize Nambu bracket
A natural approach to quantize the Nambu bracket is through the deformation quantization. It is a generalization of Moyal bracket,
The quantum Nambu bracket thus defined failed to satisfy FI . There are a few alternative approaches for the deformation quantization (see for example, [8, 7]). Later, Dito et. al.  proposed a deformation quantization based on Zariski quantization which satisfies FI. It is very different from conventional quantization method but some efforts have been made to use it for the M-theory .
Curtight and Zachos tried to formulate the quantum Nambu bracket in the line of (33). Instead of the modification of the bracket (33), they proposed an alternative to the fundamental identity . This reference contains a nice review on the Nambu bracket.
In the connection with the matrix model approach to M-theory , the Nambu dynamics is natural to realize the generalized uncertainty relation . Awata, Li, Minic and Yoneya  defined a quantization of Nambu bracket through the matrices as
which satisfies the fundamental identity. Very recently, Yoneya suggested a similar bracket  to describe the covariant M-theory matrix model.
In the context of M-theory, the degree of freedom is predicted to behave as for five-branes from AdS/CFT correspondence. In this sense, it may be natural that the quantum degree of freedom is described by a tensor with three indices (cubic matrix). Such direction was pursued by Kawamura in [16, 17]. The triple matrix for the cubic matrix was defined as
and quantum Nambu bracket is defined by anti-symmetrization. While FI is not satisfied with this bracket, a consistent dynamical system can be constructed if the Hamiltonians are restricted to the normal form,
Due to this restriction, the time evolution becomes essentially diagonal. We note that the choice of the product of the cubic matrix is not unique. For example, in , a different choice, was used. It is more natural to associate the cubic matrix with the triangle which covers the membrane: the index is assigned to the edges of a triangle and the triple product is interpreted as gluing edges of three triangles to produce three open edges. It is a natural framework to implement discretized quantum gravity  but the analog of FI is difficult to be realized.
3 Nambu bracket and the extended objects
3.1 Takhtajan’s action
In , Takhtajan introduced an action principle which describes the Nambu dynamics as the motion of the extended objects. Let new variables () describe a string-like object in (three spacial dimensions). We assume that the Hamiltonians are the functions of at the same world-sheet point.
Variation of the action gives
It implies the equation of motion for the string-like object,
The left hand side of the equation is Nambu’s equation and the right hand side is the arbitrariness due to the reprametrization invariance with respect to . When we need to consider more general Nambu action of the form (22), one may simply replace it by
In this case, the variable describes an -brane.
Takhtajan’s action is relevant to the study of self-dual string on M5-brane [20, 21] and the fluid motion in 3 dimensions. The connection with the fluid motion is discussed in the next subsection. In the context of M-theory, the fundamental degree of freedom is described by M2-brane (and the dual M5-brane) whereas the effective description by supergravity is described by anti-symmetric 3-form field and its dual 6-form. In the low energy, the effective description of the membrane is given by Nambu-Goto type action and the coupling to three-form ,
where is the membrane tension and is the world volume of the membrane. Suppose we are considering an extreme situation where is constant and large enough such that one may neglect the Nambu-Goto part, we are left with the coupling of the membrane world-volume to the constant 3-form field. In the simplest case where , the latter term coincides with the Takhtajan action when the world-volume has the boundary since
It is known that the the boundary of M2-brane is located on M5-brane. On M5-brane, the two-form gauge field should be self dual, namely . In this sense, Takhtajan string describes the self-dual string on M5.
3.2 Connections with incompressible fluid dynamics
As Nambu himself pursued for a long time, (due to a review in ), the Nambu dynamics is a natural framework to describe the incompressible fluid motion. The incompressibility implies that the volume element does not change in the time evolution. It implies that the coordinates has to satisfy in the Lagrangian formulation where is the location of fluid which was at at . It implies that the time evolution should be written in the form,
In this subsection, we collect some descriptions of fluid motion by the Nambu-bracket.
3.2.1 Vortex string dynamics
Takhtajan’s action for the Nambu dynamics can be directly related with the vortex motion where there is no dissipation. In the following, we use the description in [23, 24]. We consider the Euler equation,
for the velocity . In such a system, the fluid motion is governed by the center of vorticity, described by strings localized at . As long as there is no dissipation, the delta-function shape vorticity retains its form and motion of the vortex string determines the flow. Here we assume there are vortex filaments and . The vorticity is described by
From this expression, one obtains the velocity field by Biot-Savart law,
Plug it into the Euler equation for the vorticity,
one finds that the Euler equation is solved if satisfies the equation,
It implies that , namely the velocity of the string is identical to the flow velocity up to reparametrization. The fact that the above equation takes the same form as (48) implies that the action can be written in the Takhtajan form:
The second term may be rewritten as
One may regard it as a generalization of Takhtajan action with the Hamiltonians replaced by with replaced by multiple indices .
3.2.2 Fluid dynamics in shallow water
where is the velocity for horizontal directions, is the fluid depth, is the vorticity, and is the energy density. It was realized that the equations can be written in the form of Nambu dynamics where and , where is an arbitrary function. The bracket is defined as the functional deferentiation by which is more involved. See for example, eq.(1.15) in .
3.3 Quantization of Takhtajan’s action
We note that in the action (47), the time derivative is contained in the first term. The momentum variable is therefore given as, Since it is expressed in terms of the coordinate variables, we have a constrained system with three constraints:
The Poisson brackets among the constraints are given by
This matrix has rank two. It implies that a combination of the constraints is the first class. By inspection, one finds that
has vanishing bracket and becomes first class. It satisfies a classical version of the Virasoro algebra,
The appearance of the Virasoro algebra is natural since we have the reparametrization invariance. One may turn the first class constraints into the second class by adding the gauge fixing condition. There are some choices. The simplest one is to use ”static gauge”,
The Dirac bracket associated with it gives
The other possibility is to use invariant gauge,
The Dirac bracket for this gauge choice gives
In either case, the Nambu dynamics is described in the form of Dirac bracket as
This procedure seems to produce a simple 2D conformal field theory. For example, the commutator (69) is the same as the commutator of ghosts. A subtlety is how to regularize the volume preserving diffeomorphism generator which are nonlinear functions of coordinates . It is also nontrivial how to recover the rotational symmetry . These issues have not been fixed in our understanding.
4 Nambu bracket in M-theory
In string theory, the Lie algebra is needed when one promotes the low energy effective theory of a single D-brane  to that of a stack of multiple D-branes . Similarly, in M theory, the Nambu bracket is needed to promote the theory of a single membrance  to multiple membranes [3, 4, 5]. On the other hand, the commutator is needed for the noncommuative D-brane in the -field background [33, 34, 35], and similarly the Nambu bracket is needed to formulate an M5-brane in the -field background [36, 37, 38]. 444 The theory of a single M5-brane [39, 40, 41] was promoted to that of multiple M5-branes in  when they are compactified on a finite circle, yet only the Lie bracket is used. In this section, we review these theories of M-branes and D-branes in which the Nambu bracket and its generalizations appear to characterize the effect of interactions among branes, or the interaction with a particular background.
4.1 As an extension of M(atrix) theories
The low-energy effective theories of D-branes are well known to be supersymmetric Yang-Mills theories , in which transverse coordinates of the target space are represented by matrices. It was learned in the study of M(atrix) theories that higher dimensional branes can be constructed out of lower dimensional ones through certain matrix configurations . For instance, solutions to the Nahm equation 
for the multiple D1-brane theory describe a bound state of D1-branes ending on a D3-brane . (The parameter is the spatial world-sheet coordinate of the D1-brane.) This was generalized to the Basu-Harvey equation 
to describe M2-branes ending on an M5-brane. Here is the spatial coordinate of the M2-branes parametrizing their extension orthogonal to the M5-brane, and ’s are the matrices representing transverse coordinates. The 4-bracket is defined as a sum over permutations of 4 indices:
As the matrix is fixed, effectively a three-bracket appears here. Note that a 3-bracket structure must appear as the M5-brane is 3-dimensional higher than an M2-brane. Although the 3-bracket defined this way does not enjoy enough nice algebraic properties to allow one to define a supersymmetric action for multiple M2-branes, this is one of the first hints that one should replace the Lie bracket by something like the Nambu bracket when one considers M theory. Another hint for the relevance of the 3-bracket to M theory was obtained through calculations of scattering amplitudes of membranes in the -field background .
As an alternative to the use of the matrix algebra to realize the Nambu bracket, one can also define Lie 3-algebra abstractly as an analogue of the Lie algebra. The Lie 3-algebra is defined as a linear space equipped with a totally anti-symmetrized bracket of 3 slots , which maps three elements to an element in the linear space. For a given basis of the linear space, the Lie 3-bracket
is given in terms of the structure constants . The Lie 3-bracket is required to satisfy the fundamental identity
for all elements of the algebra. Lie 3-algebra is essentially the algebra of the Nambu bracket without demanding algebraic rules of multiplication among the elements. Hence we will refer to the Lie 3-algebra bracket also as the Nambu bracket.
A symmetric bilinear map that maps two elements to a number is said to be an invariant metric if we have
for all elements .
Unlike Lie algebra, it is not clear how to realize Lie 3-algebras in terms of matrices. Let denote a Lie 3-algebra. Then the Lie 3-bracket defines a set of maps as derivatives acting on for every anti-symmetric pair of elements . Define to be the set of such maps; it is obviously a Lie algebra, of which is a representation. The fundamental identity (73) implies that the Lie bracket of is given by 555 This expression is not manifestly antisymmetric in the exchange of with , but the skew-symmetry is guaranteed by the fundamental identify (73). One can thus think of Lie 3-algebras as a special class of Lie algebras with additional internal structures.
Note that whenever there is a continuous symmetry, there is an associated Lie group and hence a Lie algebra. The appearance of and its Lie bracket is always implied by the Lie 3-algebra.
One can define gauge theories for a Lie 3-algebra by identifying the Lie algebra as the gauge symmetry. For a Lie 3-algebra with generators , the generators of the Lie algebra are . A matter field taking values in changes by
under a gauge transformation with the transformation parameters . Equivalently,
where is the Lie 3-algebra structure constant in the basis , and is defined by
The gauge potential takes its value in the Lie algebra :
The covariant derivative on the base space with coordinates is thus