Note on Constrained Cohomology

The cohomology of the BRS operator corresponding to a group of rigid symmetries is studied in a space of local field functionals subjected to a condition of gauge invariance. We propose a procedure based on a filtration operator counting the degree in the infinitesimal parameters of the rigid symmetry transformations. An application to Witten's topological Yang-Mills theory is given.


Introduction
All the symmetries of a given quantum field theory, including possibly the gauge invariance, may be grouped together in a single extended BRS operator D [1]. Here, whereas the ghosts associated to the gauge symmetry are represented by local fields, those associated with rigid symmetries are constant 7 . Both types of ghosts have a Grassmann parity opposite to that of the corresponding generator. The proof of renormalizability is then a matter of computing the cohomology of D in the space of local field polynomials with dimensions restricted by power counting [3].
A typical example is provided by the supersymmetric Yang-Mills theories quantized in the Wess-Zumino gauge. There, such a procedure appears to be necessary [4,5] since the supersymmetry generators do not form a closed algebra because of the presence of gauge transformations [6].
Very often, it may be desirable to single out the individual symmetries from this unifying picture. This is needed, for instance, in the construction of gauge invariant operators belonging to some representation of a rigid symmetry. This is a problem of constrained cohomology, i.e. of the cohomology of D in a space constrained by the requirement of gauge invariance.
In the case of the super Yang-Mills theories, the rigid symmetry is the supersymmetry itself, and an important set of gauge invariant operators is given by the components of the supercurrent multiplet [7], i.e. the supermultiplet of currents which includes the R-axial current, the spinor current and the energy-momentum tensor. The constrained cohomology approach we want to describe here was introduced [5] in this context, in order to construct the supercurrent [8].
It turns out that the structure involved is of the type studied by Witten [9] in the framework of topological Yang-Mills theories.
The method and its relevance for quantum field theory are described in Section 2. In particular, we reproduce the algebraic structure, called "BRS extension", found by Henneaux [10] in a Hamiltonian approach.
In Section 3 we briefly discuss the example of Witten's topological Yang-Mills theory. The reader may look at [5] for the supersymmetric gauge theoretical example.

Covariance of Gauge Invariant Operators Under a Group of Rigid Symmetries
The quantization of gauge theories possessing a rigid symmetry is conveniently performed by introducing an extended nilpotent BRS operator D which, besides the BRS operator s associated with gauge invariance, also includes the infinitesimal transformations of the rigid symmetry [4,5]. Let us write the infinitesimal transformations of the fields ϕ i , in the classical theory, as 8 with, in the most general case 9 δ rigid may act nonlinearly on the fields, and the algebra may happen to close only modulo equations of motion and field-dependent gauge transformations 10 . The extended BRS operator is then defined by where the infinitesimal parameter ε A is a constant ghost: it is an anticommuting (resp. commuting) number if the generator δ A is a bosonic (resp. fermionic) operator. The term involving the structure constants c AB C ensures the nilpotency of D: D 2 = 0 (modulo eqs. of motion) , (2.5) which expresses in a compact way the whole algebra of infinitesimal generators. Terms quadratic in ε are present in (2.4) if the rigid algebra (2.2) closes modulo field dependent gauge transformations, as it is the case, for instance, in the supersymmetric gauge theories in the Wess-Zumino gauge [4,5].
8 Summation over repeated indices is always understood. 9 In the case of a superalgebra, the bracket [·, ·] is a graded commutator, i.e. an anticommutator if both its arguments are fermionic, and a commutator otherwise. We shall keep this notation throughout the paper. 10 In the special case where the action of δ A is linear: and the algebra closes off-shell, this corresponds to the matrix algebra The invariance of the quantum theory under the extended BRS transformations (2.4), i.e. under the gauge BRS symmetry s and the rigid symmetry δ A , is expressed by a Slavnov-Taylor identity -we assume that there is no anomaly - Here Γ(ϕ, ϕ * ) is the vertex functional, i.e. the generating functional of the 1-particleirreducible, amputated Green functions, and the ϕ * i are external fields coupled to the extended BRS transformations of the fields ϕ i , introduced at the classical level by adding to the invariant action the terms The terms quadratic in the external sources are necessary if the whole algebra, as expressed by the nilpotency of D, holds only on-shell [11].
The Slavnov-Taylor identity also defines the quantum form of the field transformations. The extended BRS transformations of a local quantum composite field operator O, whose vertex functions are generated by the insertion functional where the Γ-dependent linear functional operator B Γ is the linearized Slavnov-Taylor operator This operator is automatically nilpotent: if Γ obeys the Slavnov identity (2.6). The action of B Γ on the fields ϕ and ϕ * defines their extended BRS transformations at the quantum level.
The problem we want now to solve is: how to extract the gauge-BRS and the rigid transformation laws of the quantum fields, together with their algebra, from the nilpotent extended BRS transformations containing all the symmetries of the theory ?
The answer begins by introducing a filtration operator whose eigenvalues are the nonnegative integers, its application being restricted to the polynomials in ϕ, ϕ * and ε. The vertex functional and the quantum BRS operator (2.9) may be expanded according to these eigenvalues: The nilpotency of B Γ implies, at order 0, 1 and 2, the equations (2.14) B 0 is interpreted as the gauge-BRS operator. It is natural to interpret B 1 as the operator of rigid transformations. More precisely, we can write The first equation defines the generators of the rigid transformations we were looking for. The second equation defines "second order" transformations. The algebra of the gauge-BRS and rigid operators is easily deduced from (2.14): (2.16) One sees that the rigid generators X A do not fulfill in general a closed (super)Lie algebra 11 . But this will however be the case if we restrict their action to the space of gauge invariant operators. A gauge invariant local field operator [O] is defined as a cohomology class of invariant composite insertion O · Γ (c.f. (2.8)) : There is a natural definition of the action of the rigid generators X A on these cohomology classes, since these generators commute with the gauge-BRS operator (see (2.16)). It is then clear that or, equivalently, Remark: The nilpotency condition of B 1 on the gauge invariant operators may be called a constrained nilpotency condition. The computation of the cohomology of B 1 in the space of the gauge invariant operators is known as a problem of invariant cohomology [12], or "constrained cohomology". We shall apply the above formalism in the next section to a simple derivation of Witten's observables [9] in four-dimensional topological Yang-Mills theory.

Application: Topological Yang-Mills Theory
Witten's topological Yang-Mills field theory [9] in D = 4 Euclidean space is described by the following set of fields: a gauge field A µ with its associated ghost c, and a fermionic vector field ψ µ with its associated ghost ϕ.
These fields take their value in the adjoint representation of the gauge group, chosen as an arbitrary compact Lie group with structure constants 12 f ab c .
The invariances of this theory, namely the gauge symmetry and a supersymmetric-like shift symmetry, are grouped together in an extended BRS operator D, whose action on the fields is given by The supersymmetry infinitesimal parameter ε is taken to be commuting and plays the role of a constant ghost. Usually, ε is absorbed in a redefinition of the fields ψ µ and ϕ, which accordingly acquire ghost numbers 1 and 2, respectively. We shall nevertheless keep ε explicit to better illustrate our procedure. We remark that the presence of the term quadratic in ε in the transformation law of the ghost field c is necessary for the nilpotency of D. In this parametrization all the fields introduced up to now have ghost number 0 except c and ε, of ghost number 1.
This theory is renormalizable, and even ultraviolet finite. The proof [15] is based on the triviality of the cohomology of the extended BRS operator D. However, in order to maintain the exposition simple, we shall keep ourselves at the level of the classical theory described by the action (3.3), the generalization to the quantum theory being straightforward.
The filtration operator (2.11) and the expansion (2.12) take here the form where we factorized the powers of ε. The transformation laws of the operators D n and their algebra -up to order 2 -read Following the lines drawn in the previous section for the general case, we consider the space H (0) of gauge invariant local operators, i.e. the cohomology of D 0 . Within this space, the analogous of (2.19) reads which must be intended according to cohomology classes.
As it is shown in the Appendix, it follows from the transformation laws (3.7) that the the constrained cohomology of D 1 , i.e. the cohomology of D 1 in H (0) , is given by the D 0 -cohomology classes represented by the functions F (P inv (c), Q inv (ϕ)) , (3.10) whose arguments are group invariant polynomials which depend only on the fields c and ϕ without their derivatives.
The result (3.10) shows that, to the contrary of the cohomology of D, the constrained cohomology is not empty. The filtration we introduced in the global ghosts ε allowed to see this easily.
According to the definition of [9], the observables O of the model should be independent of c, thus excluding for instance terms like Tr c 2n+1 . The consequence of the c-independence is that the elements of the constrained space in which calculating the cohomology of D 1 are invariant under D 2 . This property, together with the third of eqs. (3.8), implies the exact nilpotency of D 1 (and not in the sense of the cohomology classes): This means that the zero ghost cohomology of D in H (0) is identical to that of D 1 . From (3.10), requiring c-independence, we recover the Witten's observables of the topological Yang-Mills theory: which are invariant polynomials of the field ϕ only, with the exclusion of its derivatives. As a comment, we remark that the above result (3.12) has been obtained after a simple cohomology calculation in a restricted functional space (see the Appendix), while usually the Witten's observables are characterized as what is called the "equivariant", or "basic", cohomology [16], which requires a specific, and more complicated, analysis [17].

Conclusions
We have been able to extract the individual symmetries, namely the gauge invariance and the rigid symmetries, from the general BRS operator of the theory. We have shown in particular that, although the generators constructed in this way do not form a closed algebra, they do so if their action is restricted to the space of gauge invariant quantities defined by the cohomology of the gauge BRS operator. Only in this restricted space it makes sense to study the cohomology corresponding to the rigid invariance, and we stress that this holds for any gauge theory characterized by an additional rigid invariance satisfying the general algebra (2.2). As an example, we recovered the Witten's observables of the topological Yang-Mills theory, which are usually found in the more complicated context of the equivariant cohomology. Moreover, the formalism proposed in this paper is well adapted to the quantum theory, since the constraints such as gauge invariance, as well as the symmetry generators, are expressed in a functional way, i.e. as constraints and operations on the generating functionals of Green functions.
Moreover, it also follows from the latter property that the kernel of D 1 in H (0) is isomorphic to its kernel inH (0) : Indeed, the D 1 -invariance of an element of H (0) represented by∆ ∈H (0) is expressed on any of its representatives by the condition which means that D 1∆ = D 0 (· · ·), i.e. D 1∆ = 0 sinceH (0) does not contain D 0 -exact elements.
The results (A.3) and (A.4) show that the cohomologies of D 1 in H (0) and iñ H (0) are isomorphic. We are thus left to compute the latter cohomology.
In this basis the BRS operators read and obey the algebra (3.8). In particular D 1 is not nilpotent, when applied to the variables (A.5), although it does when applied to the elements ofH (0) . Extending a well known argument [3] to the present case of an operator which is not nilpotent, we define the operators