Abstract
Higher dimensional ChernSimons theories, even though constructed along the same topological pattern as in 2+1 dimensions, have been shown recently to have generically a nonvanishing number of degrees of freedom. In this paper, we carry out the complete Dirac Hamiltonian analysis (separation of first and second class constraints and calculation of the Dirac bracket) for a group . We also study the algebra of surface charges that arise in the presence of boundaries and show that it is isomorphic to the WZW discussed in the literature. Some applications are then considered. It is shown, in particular, that ChernSimons gravity in dimensions greater than or equal to five has a propagating torsion.
hepth/9605159 
PARLPTHE9614 
ULBTH96/06 
IMAFFRCA9601 
The dynamical structure of higher
[5mm] dimensional ChernSimons theory
[10mm] Máximo Bañados, Luis J. Garay and Marc Henneaux
[5mm] Centro de Estudios Científicos de Santiago, Casilla 16443, Santiago, Chile
Centro de Física Miguel A. Catalán, Instituto de Matemáticas y Física Fundamental, CSIC, Serrano 121, E–28006 Madrid, Spain
Laboratoire de Physique Théorique et Hautes Energies,
Universités Paris VI et Paris VII, Bte 126, 4, Place Jussieu,
75252 Paris Cedex 05, France ^{1}^{1}1Permanent address: Université Libre de Bruxelles, Campus Plaine, C.P. 231, B1050, Bruxelles, Belgium
[5mm] 29 May 1996
1 Introduction
In a previous paper [1], we have shown that pure ChernSimons theories in spacetime dimensions greater than or equal to five possess local degrees of freedom in contrast to the familiar three dimensional case. The only exception is the ChernSimons theory based on the onedimensional group , which is devoid of local degrees of freedom for any spacetime dimension. However, whenever the gauge group is of dimension greater than 1, the ChernSimons action generically contains propagating degrees of freedom.
One way to understand this somewhat unexpected result is to observe that the equations of motion no longer imply that the curvature vanishes for dimensions greater than or equal to five. If there is only one gauge field, i.e., if the gauge group is , one can always bring its curvature to some fixed canonical form by using the diffeomorphism invariance (Darboux theorem). Thus, even though , one may assume that it has a fixed form and, therefore, the space of solutions of the equations of motion modulo gauge transformations is reduced to a single point. If, however, the gauge group is larger, there are more curvatures. One may bring one of them to a fixed canonical form as for , but once this is done, there is not enough invariance left to fix the other curvatures in a similar way. Thus, the space of solutions is now bigger and, consequently, there exist local degrees of freedom.
The number of local degrees of freedom was explicitly counted in Ref. [1] by using the Hamiltonian formalism. The phase space of the theory was constructed and all the constraints were exhibited. We also derived the number of second and first class constraints. This provided the necessary information to count the number of local degrees of freedom according to the formula,
(1.1) 
where is the dimension of phase space, is the number of first class constraints and is the number of second class constraints (see, e.g., Ref. [2]).
Although the analysis of Ref. [1] enables one conclude rigorously to the existence of local degrees of freedom, the second class constraints were not explicitly separated from the first class ones; only their number was given and the Dirac bracket associated to the elimination of the second class constraints was not computed either. As it is known, these steps are quite important and must be carried out before trying to quantize theory.
An interesting feature of higherdimensional ChernSimons theories, also displayed in Ref. [1], is that these are theories invariant under spacetime diffeomorphisms for which the generator of timelike diffeomorphisms is not independent from the other constraints. That is, the timelike diffeomorphisms can be expressed in terms of the spacelike diffeomorphisms and of the internal gauge transformations. Now, it is well known that the “superhamiltonian” constraint is usually the one that resists an exact treatment in the quantum theory. This appears quite strikingly in the loop representation approach to quantum gravity [3]. In the case of higherdimensional ChernSimons theories, however, the “hard constraints” are absent, even though there are local degrees of freedom. Thus, any state that is invariant under both the internal gauge symmetries and the spacelike diffeomorphisms is automatically a solution of all the quantum constraint equations. Solving the “kinematical” constraints associated with spacelike diffeomorphism invariance and internal gauge invariance, is, however, not entirely straightforward in higherdimensional ChernSimons theories. This is because the components of the connection have non trivial Dirac brackets and thus do not define simultaneously diagonalizable operators. It makes the issue of computing these brackets even more pressing.
In this paper, we complete the canonical analysis of higher dimensional ChernSimons theory. It turns out that the separation of first and second class constraints is technically intricate for an arbitrary gauge group . However, if one considers a ChernSimons action for the gauge group , the calculations become much simpler. The only requirement on the group is that it possesses a nondegenerate bilinear invariant form. In order to avoid uninteresting (and conceptually trivial) complications, we shall therefore complete the Dirac bracket analysis only in that case, as well as in the case, which has its own peculiarities.
Three dimensional ChernSimons theory is well known to induce a rich dynamics at the boundary [4]. One may therefore wonder whether this is also the case in higher dimensional spacetimes. The answer to this questions is affirmative. This problem has been already considered in the literature [5, 6]. Specifically, in Ref. [6], the connection between a ‘conformal’ field theory in four dimensions and a Kähler five dimensional ChernSimons theory was established. In this work, we explicitly exhibit the symmetry algebra arising at the boundary for the full ChernSimons theory with no extra assumption other than the boundary conditions. For definiteness, we consider ChernSimons theory in five dimensions and show that, if the gauge group is taken to be , as above, then the resulting symmetry algebra is just the WZW algebra (based on ) discussed in Ref. [6], with the curvature of the factor appearing as a Kähler form. [For a recent work dealing with the WZW algebra see Ref. [7].]
This paper is organized as follows. After a brief survey of our conventions, we review in Sec. 2 the results of Ref. [1]. We recall, in particular, the importance of the socalled generic condition that was introduced there. Then, we illustrate the generic condition in the physically interesting context of LovelockChernSimons gravity (Sec. 3), as well as for some seven dimensional ChernSimons theories (Sec. 4). In Sec. 5, we complete the Dirac analysis for the theory, which has no degrees of freedom. We show that in this theory one needs to break general covariance in order to separate the first and second class constraints. This is quite analogous to what happens for the superparticle [8, 9, 10] and the analogies are pointed out. We turn next (Sec. 6) to the separation of first and second class constraints in the more general theory with a gauge group and we work out the Dirac bracket between the basic dynamical variables. In contrast to the case, the analysis can be performed without breaking manifest covariance, by taking advantage of the peculiar group structure. Finally we discuss the global charges arising when the spatial manifold has a boundary, and show that they fulfill, in five dimensions, the WZW algebra found in Ref. [6]. We summarize and conclude in Sec. 8.
2 Local dynamics
2.1 Conventions and definitions
The ChernSimons action in higher odd dimensions is a direct generalization of the three dimensional case. Let us consider a Lie algebra of dimension . Let be the curvature 2form^{2}^{2}2We denote by the spacetime curvature 2form (greek indices run over spacetime while latin indices run over the spacelike hypersurfaces). The symbol will denote the spacelike curvature 2form, . associated to the gauge field 1form , where are the structure constants of the gauge group, and let be a rank , symmetric tensor invariant under the adjoint action of the gauge group. The ChernSimons Lagrangian in dimensions is defined through the formula
(2.1) 
The ChernSimons action is invariant, up to a boundary term, under standard gauge transformations
(2.2) 
It is also invariant under diffeomorphisms on the spacetime manifold , , because is a ()form. The diffeomorphisms on can be represented equivalently by
(2.3) 
This transformation differs from the Lie derivative only by a gauge transformation and it is often called improved diffeomorphism [11].
If the only symmetries of the ChernSimons action are the diffeomorphisms (2.3) and the gauge transformations (2.2), then we shall say that there is no accidental gauge symmetry. How this translates into an algebraic condition on the invariant tensor will be described precisely in Sec. 2.4. As we shall also indicate, the absence of accidental gauge symmetries is “generic”. Generic, however, does not mean universal and there exist examples with further gauge symmetries. A typical one is obtained by taking all the mixed components of equal to zero, so that the action is just the direct sum of copies of the action for a single Abelian field. This theory is then clearly invariant under diffeomorphisms acting independently on each copy. But there is no reason to take vanishing mixed components for . In fact, for a nonAbelian theory, the form of the invariant tensor is severely restricted. If the mixed components of differ from zero (and cannot be brought to zero by a change of basis), then the action is not invariant under diffeomorphisms acting independently on each gauge field component , because the invariance of the cross terms requires the diffeomorphism parameters for each copy to be equal, thus gluing all of them together in a single symmetry.
The ChernSimons equations of motion are easily found to be
(2.4) 
and they reduce to only in the threedimensional case (provided of course that is invertible).
2.2 The Hamiltonian action
In order to perform the Hamiltonian analysis, we assume that the spacetime manifold has the topology , where is a dimensional manifold. In this section, we will concentrate on the local properties of the theory, that is, we will not analyse the special features that arise if has a boundary or a nontrivial topology. The presence of boundaries will be considered in Sec. 7. We decompose the spacetime gauge field 1form as where the coordinate runs over and the are coordinates on . Although there is no spacetime metric to give any meaning to expressions such as timelike or spacelike, we will call time the coordinate and we will say that is a spacelike section as shorthand expressions.
It is easy to see that the ChernSimons action depends linearly on the time derivative of ,
(2.5) 
where is given by
(2.6) 
The explicit form of the function appearing in Eq. (2.5) is not needed here but only its “exterior” derivative in the space of spatial connections, which reads
(2.7)  
The equations of motion obtained by varying the action (2.5) with respect to are given by
(2.8) 
while the variation of the action with respect to yields the constraint
(2.9) 
Of course, Eqs. (2.8) and (2.9) are completely equivalent to Eq. (2.4). Despite the fact that Eqs. (2.8) are first order, they are not Hamiltonian. The reason is that the matrix is not invertible, as we will see. Indeed, on the surface defined by the constraint (2.9), the matrix has, at least, null eigenvectors. The noninvertibility of is a signal of a gauge symmetry. We shall see below that this symmetry is nothing but the diffeomorphism invariance.
To proceed with the Hamiltonian formulation of the action we shall use the Dirac method [2]. Since the action (2.5) is linear in the time derivatives of , the canonically conjugate momenta are subject to the primary constraints,
(2.10) 
These constraints transform in the coadjoint representation of the Lie algebra because the inhomogeneous terms in the transformation laws of and cancel out.
In principle one should also define a canonical momentum for . This would generate another constraint, , whose consistency condition yields the constraint . The constraint is first class and generates arbitrary displacements of . One can drop and keep as an arbitrary Lagrange multiplier for the constraint .
It turns out to be more convenient to replace the constraints by the equivalent set
(2.11) 
This redefinition is permissible because the surface defined by , is equivalent to the surface defined by , . The motivation to replace by is that the new constraints generate the gauge transformations (2.2) and therefore are first class. Indeed one can easily check that,
(2.12) 
The Hamiltonian action takes the form,
(2.13) 
where the Poisson brackets among the constraints are given by
(2.14)  
(2.15)  
(2.16) 
Here, are the structure constants of the Lie algebra under consideration.
It follows from the constraint algebra that there are no further constraints. The consistency condition
(2.17) 
is automatically fulfilled because is first class while the other consistency equation
(2.18) 
will just restrict some of the Lagrange multipliers .
2.3 First class constraints
Equations (2.15) and (2.16) reflect that the constraints are the generators of the gauge transformations and that the constraints transform in the coadjoint representation. This means, in particular, that the ’s are first class, as mentioned above.
The nature of the constraints is determined by the eigenvalues of the matrix . It turns out that the matrix is not invertible on the constraint surface and, therefore, not all the ’s are second class. Indeed, using some simple combinatorial identities, one can prove that and satisfy the identity
(2.19) 
This equation shows that, on the constraint surface , the matrix has, at least, null eigenvectors . The existence of these null eigenvectors of tells us that among the ’s, there are first class constraints. These constraints are given by
(2.20) 
and they generate the spatial diffeomorphisms (2.3), namely, . Thus, they satisfy the spatial diffeomorphism algebra, up to gauge transformations. The presence of these constraints is not surprising because the ChernSimons action is invariant under diffeomorphisms for any choice of the invariant tensor . What is perhaps more surprising in view of what occurs for ordinary ChernSimons theory in three dimensions, is that the constraints (2.20) are generically independent from the constraints generating local internal gauge transformations (, see below).
One could also expect the presence of another first class constraint, namely, the generator of timelike diffeomorphisms. However, this symmetry is not independent from the other ones and hence its generator is a combination of the first class constraints and . This can be viewed as follows. The action of a timelike diffeomorphism parameterized by acting on is [see Eq. (2.3)],
(2.21) 
Now, the equations of motion (2.8) are . Let us assume (this assumption will be clarified below) that the only null eigenvectors of are those given above, then there must exist some such that . Inserting this result in Eq. (2.21), we obtain
(2.22) 
which is a spatial diffeomorphism with parameter . Hence, timelike diffeomorphisms are equivalent to spacelike diffeomorphisms onshell. For that reason, there is no constraint associated to normal deformations of the surface. If the theory has ‘accidental symmetries’, i.e. the matrix has more zero eigenvalues, the above analysis shows that the timelike diffeomorphisms can still be written in terms of spatial diffeomorphisms plus the extra ‘accidental’ symmetries.
2.4 Generic theories
We now examine whether the first class constraints and are independent and constitute a complete set. At this point, we must distinguish between the and cases. Indeed, it turns out that for the Lie algebra , the theory cannot be obtained as a limiting case from the theory.
Consider first the general case with . The eigenvalues of , which determine the nature of the constraints , depend on the properties of the invariant tensor and, for a definite choice of , they also depend on the phase space location of the system since the constraint surface of the ChernSimons theory is stratified into phase space regions where the matrix has different ranks. For example, is always a solution of the equations of motion and, for that solution, is identically zero. There exist, however, other solutions of the equations of motion for which . The rank of classifies the phase space into regions with different number of local degrees of freedom. The key ingredient controlling the maximum possible rank of is the algebraic structure of the invariant tensor .
We will say that an invariant tensor is generic if and only if it satisfies the following condition: There exist solutions of the constraints such that

the matrix (with as row index and as column index) has maximum rank , so that the only solution of is and therefore the null eigenvectors are linearly independent;

the matrix has the maximum rank compatible with , namely ; in other words, it has no other null eigenvectors besides .
We will also say that the solutions of the constraints such that and hold are generic. The reason for this name comes from the following observation. For a given generic tensor , a solution fulfilling both conditions and will still fulfill them upon small perturbations, since maximum rank conditions correspond to inequalities and therefore, they define open regions. Conversely, a solution not fulfilling conditions or , i.e. located on the surface where lower ranks are achieved (defined by equations expressing that some non trivial determinants vanish), will fail to remain on that surface upon generic perturbations consistent with the constraints. Thus, non generic solutions belong to subsets of the constraint surface of smaller dimension.
The physical meaning of the above algebraic conditions is straightforward. They simply express that the gauge transformations (2.2) and the spatial diffeomorphisms (2.3) are independent and that the are the only first class constraints among the ’s.
As we stressed above, the theory cannot be obtained as a limiting case from the theory. The definition of what is meant by “generic” in the case must therefore be amended as follows. For , the invariant tensor has a single component , which we assume, of course, to be different from zero. The constraint is then
(2.23) 
which implies that the matrix cannot be invertible. The solutions of Eq. (2.23) are the set of matrices with zero determinant. Thus, the equation does not imply and the constraint (2.23) prevents us from finding solutions of the theory satisfying the generic condition of the case. This means, in particular, that the spatial diffeomorphisms are not all independent. Solutions of Eq. (2.23) such that has the maximum rank compatible with the constraint (2.23) will be called generic. This rank is clearly . The complete Hamiltonian analysis for the theory is performed in Sec. 5.
2.5 Degrees of freedom count
When the condition is satisfied, the count of local degrees of freedom goes as follows. We have, canonical variables (), first class constraints associated with the gauge invariance, first class constraints associated with the spatial diffeomorphism invariance, and second class constraints (the remaining ). Hence, we have
(2.24)  
local degrees of freedom (for , ). It should be stressed here that this formula gives the number of local degrees of freedom associated to the open region of phase space defined by generic solutions.
This formula does not apply to because the spatial diffeomorphisms are not independent in that case. One finds instead that there are no local degrees of freedom (see Sec. 5). For a similar reason, this formula does not apply to () where diffeomorphism invariance is completely contained within the ordinary YangMills gauge invariance.
2.6 The generic condition in the Lagrangian equations of motion
It is instructive to study the implications of the generic condition in the context of the Lagrangian equations of motion. This provides also an equivalent method of counting the number of local degrees of freedom. The Lagrangian equations of motion written in a covariant way are given in Eq. (2.4). Upon a decomposition, these equations acquire the expressions given in Eqs. (2.8) and (2.9).
If the generic condition is fulfilled, then Eq. (2.8) can be rewritten in the useful and simple form,
(2.25) 
where the are arbitrary functions of spacetime. This form of the equations of motion clearly shows that the time evolution is generated by a gauge transformation (with parameter ) plus a diffeomorphism (with parameter ). [Eq. (2.25) follows directly from Eq. (2.8), and the fact that, in the generic case, has only null eigenvectors given in Eq. (2.19).] Therefore, the Lagrangian equations of motion can be replaced by the constraint (2.9) plus Eq. (2.25). Due to the simplicity of Eq. (2.25) we can study its space of solutions, modulo gauge transformations.
Equation (2.25) is invariant under standard gauge transformations,
(2.26) 
where is the commutator in the Lie algebra. Eq. (2.25) is also invariant under spatial diffeomorphisms
(2.27) 
where now the symbol denotes the Lie bracket of two spatial vectors. Of course, the constraint equation (2.9) is also invariant under both gauge transformations and diffeomorphisms.
We shall study the solutions of the equations of motion in the gauge
(2.28) 
In this gauge, Eq. (2.25) simply says that , hence, the configurations are time independent. It is important to note that the above gauge choice does not exhaust all the gauge symmetry. The conditions (2.28) are preserved by gauge transformations and diffeomorphism that do not depend on time, i.e., by transformations whose parameters satisfy and .
Thus, in the time gauge, we are left only with the constraint equation (2.9) with the extra condition that the fields are time independent. Equation (2.9) is invariant under the residual gauge group consisting in timeindependent gauge transformations and diffeomorphisms. In summary, we have arbitrary functions of the spatial coordinates, . These functions are restricted by equations, the constraints (2.9). Also, there is a dimensional residual gauge group which can be used to set functions equal to zero.
Therefore, the number of arbitrary functions in the solutions of the equations of motion is
(2.29) 
These are the Lagrangian “integration functions” for the equations of motion, which are twice the number of local degrees of freedom, in agreement with Eq. (2.24).
3 LovelockChernSimons gravity
The goal of this section is twofold. On the one hand, we exhibit the LovelockChernSimons theory as a concrete example with a non abelian internal group for which the generic condition defined in Sec. 2.4 is fulfilled. On the other hand, the analysis of the dynamics of the LovelockChernSimons action reveals a rather unexpected result. It turns out that the torsion tensor is dynamical in this theory, hence the Palatini and second order formalisms are not equivalent in higher dimensional ChernSimons gravity, contrary to what happens in three dimensions.
The LovelockChernSimons theory (in higher dimensions) is not defined by the HilbertEinstein action but, rather, it contains higher powers of the curvature tensor. However, the equations of motion are first order in the tetrad and spin connection, and, if the torsion is set equal to zero, they are second order in the metric. The gravitational ChernSimons action is a particular case of the socalled Lovelock action [12]. For this reason, we call this theory the LovelockChernSimons (LCS) theory. It is a natural extension to higher dimensions of the formulation of 2+1 gravity given by Achúcarro and Townsend [13] and Witten [14]. As we shall see, however, the dynamical content is quite different. The construction of the LovelockChernSimons theory has been carried out in Ref. [15]. Here we briefly review its main features.
Let be a connection for the group [we recall that ] and its curvature 2form (here, the capital indices run over ). The connection can be split in the form,
(3.1) 
where parameterizes the cosmological constant and and transform, respectively, as a connection and as a vector under the action of the Lorentz subgroup . Hence, will be called the spin connection and the vielbein. Similarly, the curvature has the form
(3.2) 
where and is the torsion tensor.
The ChernSimons Lagrangian is defined by making use of the LeviCivita invariant tensor,
(3.3) 
Since is an invariant tensor of , we say that is a ChernSimons Lagrangian for the (adS) group . When written in terms of the vielbein and spin connection, the Lagrangian defined in Eq. (3.3) is a particular case of the Lovelock Lagrangian considered in Ref. [12]. Black hole solutions for this action have been found in Ref. [16].
The equations of motion for this theory are
(3.4) 
which are explicitly invariant under . Splitting the curvature as in Eq. (3.2), these equations separate into the two sets of equations,
(3.5)  
(3.6) 
which are the equations of motion following from varying the action with respect to the vielbein and spin connection, respectively. After the split has been made, the equations are explicitly invariant only under but, in view of Eq. (3.4), they are in fact invariant under the larger group . Note that, as mentioned above, is always a solution of Eq. (3.6) and, in dimensions, it is the only solution. We now prove that, for , this is not the most general solution and dynamical modes exist.
For simplicity, we consider the five dimensional case. As it has been shown in Ref. [17], the Lovelock Lagrangian written in the second order formalism () has the same number of degrees of freedom as the Hilbert Lagrangian. Thus, in five dimensions, the theory with zero torsion carries local (physical) degrees of freedom. On the other hand, since the Lagrangian defines a ChernSimons theory we can count the number of degrees of freedom by using the formula (2.24). However, before we can apply that formula we need to prove that the LovelockChernSimons theory is generic in the sense defined in Sec. 2.4.
The constraint for this theory is
(3.7) 
where are the spatial projections of the 2form curvature. To prove that this theory is generic, it is enough to find one solution for which the matrix has maximum rank. The 2form curvature given by
(3.8)  
with all other components equal to zero satisfies the constraint (3.7) and has maximum rank. The proof of this statement is straightforward. One looks at the equation
(3.9) 
where is a one form. Due to the fact that and are nondegenerate 2forms, one easily obtains that this equation possesses only four independent solutions, that is, has the maximum rank.
Note that given in Eq. (3.8) does not have any zero column in the indices therefore this solution clearly has a nonzero torsion [see Eq. (3.2)]. Note also that the above curvature can be derived from the connection
(3.10)  
with all other components equal to zero. Thus, given in Eq. (3.8) represents an allowed physical configuration.
The existence of the above solution ensures that this theory is generic and therefore we can apply the formula (2.24) to count the number of degrees of freedom. In five dimensions, the LovelockChernSimons theory is a ChernSimons theory for the Lie algebra , of dimension 15. Hence, formula (2.24) gives local degrees of freedom. Thus, this theory indeed has more degrees of freedom than the metric theory. This reflects the fact that setting the torsion equal to zero eliminates degrees of freedom. In other words, Lovelock theory, at least in the case considered here, has a dynamical torsion.
4 The seven dimensional case
In Ref. [1], examples fulfilling the generic condition were explicitly given only in five dimensions. We exhibit in this section seven dimensional examples for which the generic condition is satisfied. In this case, there exists a ‘simple’ choice for the invariant tensor: We can take the rankfour invariant symmetric tensor given by
(4.1) 
where is an invariant metric on the Lie algebra. We prove in this section that if is invertible, then the associated ChernSimons theory is generic. This means, in particular, that the seven dimensional theory with the choice (4.1) — or with any other choice of invariant tensor sufficiently close to it — is generic for any simple Lie algebra.
The constraint in this case reduces to the simpler form
(4.2) 
where the internal indices are raised and lowered with . Here , and we regard the constraint as a 6form. Similarly, is a 4form given by,
(4.3) 
We want to find solutions to Eq. (4.2) so that has maximum rank. Thus, we are interested in the number of solutions of the null eigenvalue problem
(4.4) 
where is a 1form vector, and satisfies Eq. (4.2). We already know that this equation has six independent solutions which are of the form . We will now show that there exists a solution to the constraint (4.2) for which does not have any other null eigenvectors.
The constraint (4.2) is solved by the following expression for :
(4.5) 
where
(4.6) 
and , are two vectors satisfying , . Here we have assumed that only for simplicity. The analysis can be carried out for any invertible .
The matrix evaluated for this solution is equal to
where
(4.8)  
An immediate set of null eigenvectors of comes from the observation that and are, respectively, null eigenvectors of and . These eigenvectors are easily seen to correspond to the diffeomorphisms eigenvectors .
To prove that this theory has maximum rank, it is now enough to prove that the matrices and do not have any further null eigenvectors. This is most easily shown by going to the particular basis in which,
(4.9) 
Then, the vectors and have the form
(4.10) 
and , and have the block form
(4.11)  
where is the identity in the subspace orthogonal to and . Accordingly, and have rank showing that each of them has only one null eigenvector. These eigenvectors are, respectively, , and .
Thus, we have proved that the seven dimensional ChernSimons theory defined by the invariant tensor given in Eq. (4.1) provides another example of generic theories.
5 The Abelian theory
The theory was first studied in Ref. [18] were the absence of degrees of freedom for this theory was pointed out. Here we shall analyse this theory along the lines introduced in Sec. 2. Our main goal is to display the differences between the and theories.
As it was pointed out at the end of Sec. 2.4, the theory needs a special treatment. We defined in that section the generic solutions of the constraint as those for which the matrix has the maximum possible rank .
Let us split the spatial coordinates into where and , so that the gauge field takes the form
(5.1) 
and the curvature can be written as
(5.2) 
The generic condition for the theory can be implemented by requiring that the matrix appearing in Eq. (5.2) is invertible, i.e.
(5.3) 
so that has the maximum rank . By a change of coordinates, one can always make a generic to fulfill the condition (5.3).
5.1 Dirac brackets and first class algebra
Once the maximum rank condition over is imposed, the constraints split naturally into first and second class. To see this, we first note that (the projection of along the coordinates ) can be written as
(5.4) 
where is the Pfaffian of
(5.5) 
and is the LeviCivita tensor in the 2dimensional manifold labeled by the coordinates . Since the determinant of is different from zero, the matrix is invertible. Let be its inverse:
(5.6) 
which satisfies .
The invertibility of implies, from Eq. (2.14), that the two constraints are second class. Consequently, their associated Lagrange multipliers can be solved from Eq. (2.18),
(5.7) 
As we have seen, one may take as first class constraints the combinations . In the case, there are only independent constraints among the ’s, since the matrix is of rank . If we recall that the matrix is invertible, we can take the independent first class constraints to be . The system of constraints () provides a system equivalent to the system (), in which the constraints are manifestly split into second and first class. Upon elimination of the second class constraints, we obtain the corresponding Dirac bracket
(5.8) 
The smeared generators and , where satisfy the Dirac bracket algebra,
(5.9)  
(5.10)  
(5.11) 
where is the Lie bracket of the two vectors and . The above algebra is self explanatory; generates gauge transformations and generates diffeomorphisms in the directions. Equation (5.10), on the other hand, tells us that transforms as a scalar under diffeomorphisms.
This completes the problem of relating the first class algebra with the symmetries of the ChernSimons action. A different question is whether the above constraints encode all the symmetries of the action. Two evident missing pieces are diffeomorphisms along the directions and timelike diffeomorphisms. It turns out that these symmetries are generated by the above constraints. The proof of this statement is straightforward. The key point is the fact that the equations of motion trivialize the timelike (as in the case ) and diffeomorphisms. That is, when acting on the space of solutions of the equations of motion, these symmetries reduce to the identity. For this reason, there are no independent first class constraints associated to them. In a canonical language, the generators of timelike and diffeomorphisms are linear combinations of the generators of gauge transformations and diffeomorphisms. This is reminiscent of the 2+1 theory where the whole diffeomorphism invariance can be expressed in terms of the local gauge transformations.
There is thus a gradation that can be summarized as follows: in three dimensions, the diffeomorphisms can be expressed in terms of the internal gauge transformations for any choice of the gauge group. In higher dimensions, some of the diffeomorphisms become independent gauge symmetries (in the generic case). These are of the spatial diffeomorphisms if the gauge group is onedimensional, and all the spatial diffeomorphisms otherwise. The timelike diffeomorphisms are not independent gauge symmetries in any (generic) case; they can always be expressed in terms of the other symmetries.
5.2 Absence of local degrees of freedom
Having determined the first and second class constraints we can now proceed to count the number of local degrees of freedom of this theory. We have canonical variables and canonical momenta . To this number, , we subtract the number of second class constraints, namely 2, and twice the number of first class constraints, , so that
(5.12) 
Thus, there are no local degrees of freedom. This is a pure topological field theory and the only degrees of freedom that may be present are the global ones. We must stress again, however, that this is a peculiarity of the theory, which is in that sense a poor representative of the general case.
5.3 Reduced action
It is instructive to write down the reduced action once the second class constraints have been solved. Since the constraints are linear in the momenta, this is easily achieved. Upon inserting the solution inside the action, one obtains a reduced action for the relevant dynamical fields and Lagrange multipliers,
(5.13) 
which must be varied with respect to all its arguments.
The symplectic structure is not canonical due to the presence of the factor in the first kinetic term. The symplectic form
(5.14) 
is, by construction, invertible and its inverse provides the Dirac bracket
(5.15) 
which is equivalent to that defined in Eq. (5.8), in agreement with the general theory [2].
5.4 Comparison with superparticle
The canonical analysis of the ChernSimons theory in higher dimensions presents many similarities with that of the superparticle (see, e.g., Ref. [9] and references therein). In both cases, although the original action is manifestly covariant to begin with, one cannot reach a complete canonical formulation without breaking explicitly this manifest covariance. This is because one cannot isolate covariantly the second class constraints [8]. Furthermore, although one can write down a complete set of first class constraints that transform covariantly (here, the constraints ), these first class constraints are redundant, implying in a BRST treatment the presence of ghosts of ghosts. Again, one cannot isolate covariantly a complete, irreducible, set of first class constraints.
There is another interesting similarity: if one chooses to work with the covariant, redundant, first class constraints , one finds that the reducibility identities are
(5.16) 
(on the constraint surface ) with
(5.17) 
These reducibility identities, in turn, are not independent since (weakly), and this reducibility of the reducibility is itself not irreducible, etc. One is thus led to an infinite tower of reducibility identities, requiring an infinite set of ghosts of ghosts in the BR ST formulation, exactly as in the superparticle case [9, 10].
6 The theory
The Hamiltonian analysis performed so far in the case is incomplete because the second class constraints have not yet been eliminated and in the case of manifolds with boundaries, it is known that the Hamiltonian has to be supplemented with some boundary terms [19], a problem not yet discussed. These two issues are cumbersome on the computational side — even though conceptually easy — if one works with an arbitrary Lie group and, actually, they cannot be treated in a general covariant way. It is surprising, therefore, that a drastic simplification takes place if one couples a factor to the group . To avoid unessential technical difficulties, we shall restrict our attention to that case, which illustrates all the conceptual features. In this section, we solve all second class constraints and compute the Dirac bracket. In the next section, we deal with the boundary terms necessary to make the Hamiltonian welldefined. For simplicity, we work explicitly in five dimensions but we shall indicate how the results obtained here can be extended to any odd dimensional spacetime.
6.1 The invariant tensor for
Consider the ChernSimons action in five dimensions for the Lie group . [In this section capital Latin indices run over . Small Latin indices run over , and denotes .] It is straightforward to see that the invariance condition on the tensor implies the following restrictions on its components. The components , and must separately be invariant under the adjoint action of , and is an arbitrary constant.
We shall now impose three extra conditions on the group and its invariant tensors. First we assume that is zero. Usually, this is not an additional requirement because, in general, there is no vector invariant vector under the adjoint action of . Second, we assume that admits an invariant nondegenerate quadratic form , as it is the case if is semisimple, and we take
(6.1) 
Finally, we impose
(6.2) 
This condition is justified on simplicity grounds since, as we shall see below, it allows for a simple separation between first and second class constraints. Note that the gauge fields associated with and respectively are not decoupled in the action, because .
6.2 Dirac Brackets
An immediate consequence of this choice for the invariant tensor is that the second class constraints can be explicitly isolated and solved, at least in a generic region of phase space.
The constraint equations (), in this case, are