A Note on Weyl invariance in gravity and the Wess-Zumino functional

It is shown that the explicit calculation of the Wess-Zumino functional pertaining to the breaking term of the Weyl symmetry for the Einstein-Hilbert action allows to restore the Weyl symmetry by introducing the extra dilaton field as Goldstone field. Adding the Wess-Zumino counter-term to the Einstein-Hilbert action reproduces the usual Weyl invariant action used in standard literature. Further consideration might confer to the Einstein-Hilbert action a new status.


Introduction and motivation
It is fair to say that the conformal Weyl symmetry which induces a local rescaling of a metric g µν (x) → Ω 2 (x)g µν (x) is still a fascinating local symmetry. As such, it can be considered on the same footing as a gauge symmetry, a standpoint we shall adopt in the paper.
Quite recently G. 't Hooft in [1] put an emphasis on the use of what he called the local conformal symmetry, and the rôle of a dilaton field with only renormalizable interactions, in particular when gravity couples to matter. Some years before, in [2] Weyl symmetry is shown to be related to the origin of mass. As 't Hooft says, local conformal symmetry is not well understood yet. Let us add in particular, its relationship with scale (or dilation) symmetry x µ → λx µ which confers the canonical dimension to a field. This canonical dimension can be reinterpreted as a weight when fields are geometrically considered as densities.
Both [2,1] used the conformal Weyl invariant local functional in gravitation in n-dimensional spacetime where ||∇σ|| 2 = g µν ∂ µ σ∂ ν σ is the Riemannian scalar product. The Weyl invariance is achieved when beside the Weyl rescaling of the metric g, the scalar field σ transforms according to σ → Ωσ. The latter is a Weyl compensating field (or dilaton) in order to restore Weyl invariance of the theory while maintaining the locality principle for the functional in the fields g and σ.
Former approaches showed that non-Weyl invariant theories (as the Einstein-Hilbert action for gravity is) can be turned into Weyl invariant ones by using a Weyl compensating scalar field. According to the existing literature on the subject, it would appear that this functional action was considered at the beginning of the 1970's by B. Zumino [3, in particular, reference 13 therein] and S. Deser [4], independently. This was around the same period of the occurrence of the so-called Wess-Zumino term [5]. Surprisingly enough, no relationship with the latter was put into evidence at that time and subsequently [6, formula (14)], according to the best of our knowledge.
Relying onto the standpoint of locality principle in QFT and considering the metric g as an external gravitational field, we shall adopt the attitude to consider the breaking of the Weyl rescalings (conformal transformations) of the Einstein-Hilbert action as an "anomaly". If it turns out to be the case, by making use of the BRST techniques, namely, it is a BRST 1-cocyle from which the Wess-Zumino term can be constructed. Recall that the latter integrates the anomalous term and can be used as counter-term to restore the symmetry. The price to pay usually is to add in the theory a new Goldstone field which carries a non linear transformation law.
This legitimately raises the natural question whether the introduction of the above dilaton field σ as compensating field pertaining to the Weyl conformal symmetry, and accordingly, the modification of the Einstein-Hilbert action into the action (1), stems from the usual generic construction of a Wess-Zumino term (or action or functional). This issue will be addressed in the present paper whose purpose is to provide a positive answer to this somewhat conceptual issue.
The construction of the Wess-Zumino functional as in [5], [7], [8, formula (4.33)], [9] and [10, see p. 164] is well-known for usual gauge theories. That is, when the gauge anomaly is of polynomial type, i.e. constructed through the so-called descent equations stemming from an invariant polynomial given by a characteristic class of the underlying principle fiber bundle. What about the construction of a Wess-Zumino functional in the case where the anomaly is not "polynomial", namely, there is no Q coming from the Chern-Simons transgression formula? For instance, diffeomorphism anomalies, Weyl anomaly fall into this category. The generic construction given in [11,12] takes fully into account both the cases. Let us call that construction the "Stora construction" 1 . We shall apply it in the context of local conformal symmetry in gravity. In order to achieve this goal, the general BRST treatment of anomalies will be used.
The paper is organised as follows. Section 2 will serve to fix the notation. In section 3, we shall show that the breaking term inferred by the Einstein-Hilbert action falls into the usual algebraic BRST cohomology. In section 4 the corresponding Wess-Zumino term will be computed explicitly. The paper is closed by some concluding remarks and open questions. Two appendices are devoted to the construction of the Wess-Zumino action according to Raymond Stora.

Weyl rescalings
Let M be a n-dimensional smooth manifold and consider Met(M ) the infinite dimensional space of (pseudo-)Riemannian metrics on M .
A Weyl rescaling is a mapping on Met(M ) defined by g(x) → ϕ(x)g(x) where ϕ ∈ C ∞ (M, R * + ) is a smooth positive function defined on M . It is also named a conformal change of the metric g. Such transformations yield an abelian transformation group acting on the space Met(M ).
We shall parametrize for convenience ϕ(x) = Ω 2 (x) = e 2φ(x) so that the Weyl rescaling of a metric reads for respectively, a finite local conformal change of the metric g, or infinitesimally linearized version. Notice or under an infinitesimal Weyl transformation as where the Laplacian acting on scalar functions is given by (see e.g. [15, § 2.7]) If one considers the quotient space Met(M )/C ∞ (M, R * + ), this is the infinite dimensional space of the conformal classes [g] of the metrics. Morally, a Weyl (scale) invariant physical theory is supposed to depend only on the conformal classes of metrics. Considering the local Weyl rescalings as a gauge group, one can pass to the quotient space Met(M ) . However, as Raymond Stora used to insist on, owing to the locality principle, the (gauge) Weyl invariant action functional must depend on a representative of the conformal class, with the inherent problem of the ambiguity on the choice of this representative, say g. This amounts one to working on the space Γ loc (Met(M )) of local functionals in the metric. However, in gauge theory, observables are functions on orbit space, or equivalently, gauge invariant functions on field space (which is here Met(M )) [16].
For the local conformal symmetry the corresponding (classical) Ward identity acting on local functionals reads Let us now consider the Einstein-Hilbert action which is such a local functional of g where vol(g) is the volume form on M and R(g) is the scalar curvature, both associated to the metric g. The next step will be the study of the behavior of the Einstein-Hilbert action under local conformal rescalings.

Another Weyl "anomaly"
As is well known, the Einstein-Hilbert gravitation theory yields a spontaneous breaking of the conformal Weyl symmetry. We shall study this breaking of conformal symmetry in the framework of the BRST differential algebra [17,18]. Thus, turning the Weyl parameter φ into the Faddeev-Popov ghost field, still denoted by φ with φ 2 = 0 (abelian Lie algebra), the corresponding Slavnov operation s acting on the field generators -see e.g. [19,20]-is defined by In particular, one has This Slavnov operation s will act on the functional space Γ loc (Met(M )). A direct computation shows that the variation of the Einstein-Hilbert action under Weyl transformations is infinitesimally given by This means that the Ward identity (4) is already broken at the classical level by the Einstein-Hilbert action. By inspection, the functional A(φ, g) is local in g and linear in the ghost argument φ, and it turns out to fulfill the celebrated Wess-Zumino consistency condition [5,18] in its BRST formulation, namely, as it can be checked explicitly. In the course of the computation, an integration by parts must be performed in order to get an integrand of the type ||∇φ|| 2 which vanishes by Faddeev-Popov argument. Hence, since (10) expresses a 1-cocycle condition, one can analogously treat A(φ, g) as an "anomaly" even if it is not a quantum breaking of the Weyl symmetry. Here, it is rather a geometrical breaking. However, one might speculate that the Einstein-Hilbert action could be considered as a local "vacuum functional" depending on an external gravitational field g. The latter ought to come from a field theory coupled to a gravitational field, similarly to the approach followed in [20,21] for the bosonic string in the 2D case, where by a gauge fixing condition the gravitational field becomes external. Let us also mention that the Einstein-Hilbert action may be seen as resulting from the spectral action principle initiated in [22] together with the Standard Model Lagrangian (at least at the tree level).
The main issue now is to restore the local Weyl conformal symmetry by proceeding along the line used in gauge theory to reabsorb the anomaly.

The Wess-Zumino functional
The reader is recalled that the Wess-Zumino functional is defined to integrate the anomaly, see e.g. [10,12] for a BRST treatment of this issue, that is (according to our situation at hand) In fact, the anomaly is trivialized at the cost of introducing an extra field with values in the Lie algebra of the gauge group. Two appendices give a detailed account on Raymond Stora's ideas on the construction of the Wess-Zumino action as an important item in gauge theory. 2 In our case, the anomaly does not seem to be a "polynomial anomaly" (namely of Adler-Bardeen type). A priori, there is no Chern-Simons transgression formula. However, as said in section 1, there is still a way to construct the Wess-Zumino functional [12].
To this end, take a 1-parameter subgroup for t ∈ [0, 1], γ t = e −tτ in the Weyl gauge group from the identity element γ 0 = 1 (constant function) to the positive function γ 1 = e −τ =: γ. Accordingly, its action on a given metric g defines a path in the conformal class [g] of the metric g (namely, in the Weyl gauge orbit of the latter) Consider the pull-back to the interpolating family γ t of the Maurer-Cartan form on the gauge group associated to the Weyl group which is found to be no longer depending on the t-parameter. Following [10,11] one extends the Slavnov operation s to the added scalar field τ by requiring It is like a gauge fixing on the conformal component of the metric field. Rather, it might be viewed as a change of variables within the field space which g and τ belong to [23]. Hence, the field τ carries 1 as Weyl conformal weight and will play the rôle of dilaton as we shall see.
Raymond Stora [12] defines the Wess-Zumino functional, Γ WZ , by integrating over the interpolating family the anomalous term according to upon using (13). One has to perform the integration over t of the functional The overall factor 2 in the r.h.s. comes from the linearity of A in its first argument. Upon using (3) and (12) and after some algebra, one gets the expression Performing the integration over t, the corresponding Wess-Zumino functional reads Upon defining σ = e τ (= 1/γ) (the inverse element to γ) with s σ = φσ, 3 as a compensating field of conformal weight one, the Wess-Zumino action can be rewritten as and next by using once more the Weyl transformation (3) for g → σ −2 g, one is led to which is nothing but the difference Since the rôle of the counter-term Γ WZ is to cancel the anomaly, one thus has by construction which is consistent actually with the constraint (14). Finally, this yields the Weyl invariant local functional Remembering that σ = e τ , one can check that this formula, up to an integration by parts and up to an overall factor − 1 2 , is nothing but the Weyl invariant action (1) given in [3,4] and used in [2,1]. At this stage some comments are in order.

Comments and outlook
On the one hand, since s(σ −2 g) = 0, it turns out to be obvious that s S EH (σ −2 g) = 0. But, on the other hand, the Weyl invariance has been restored by mimicking a construction coming from Lagrangian gauge theory, by adding the Wess-Zumino counter-term to the well-known Einstein-Hilbert action and thus introducing the so-called compensating field σ in order to be consistent with the locality principle. 4 In this respect, and to parallel some Raymond Stora's viewpoints (see section 2) this rises the following question: Does the Weyl invariant combination σ −2 g gives a substitute to parametrize the conformal class [g] (of the metric g) compatible with the locality principle? And to "mirror" [1]: Does the construction of the Wess-Zumino term provides a canonical way to isolate the dilaton component of the metric?
Moreover, according to [3, p.464] an action which is invariant under both Einstein and Weyl symmetries is invariant in the Minkowski flat limit under the 15-dimensional conformal group. One is led to make contact also with [19,20] , and one may also remark that if it would be possible to set σ = (det g) 1/2n by gauge fixing or as an equation of motion, then s σ = φσ, since s (det g) = 2nφ(det g). Accordingly, σ −2 g = (det g) −1/n g and det(σ −2 g) = 1 so that σ −2 g =ĝ, the associated unimodular matrix to the metric g, will also serve as a representative of the conformal class [g] of the metric g. It ought to be useful to investigate more in that direction.
Going back to the construction of the Wess-Zumino action, in particular the rôle of the interpolating family, formula (17) can be simply recast as Not only the integration over the family can be explicitly performed for computing the Wess-Zumino action, but it highlights the Einstein-Hilbert action since one may write along a path in the gauge orbit given by the conformal class [g] of the metric g. Following Stora's tricks [12], at the algebraic level (here, the Lie algebra is abelian) the Maurer-Cartan equation d t ω t = 0, together with d t g t = −d M ω t (d M is the de Rham differential on spacetime M ), yields a differential algebra which is similar to the BRS one given in (6). The Wess-Zumino consistency condition leads to d t A(ω t , g t ) = 0. Since d 2 t = 0, equation (21) indicates that the Wess-Zumino term turns out to be independent of the interpolating family up to smooth deformations and depends only on the bounds.
Thus S EH (g t ) interpolates along a family of conformally related metrics to the metric g. That is within a fiber of Met(M ) with respect to the conformal symmetry. This might indicate that the Einstein-Hilbert action plays a special role in the construction. If one was able to pass to the quotient space Met(M )/C ∞ (M, R * + ), as configuration space, the "genuine" physical theory ought to depend only on the conformal classes [g] of the metrics g. More investigation deserves to be performed in this matter.

Appendix A The Stora construction of the Wess-Zumino action
In [8,9], [7,Section III] or [10,25,11] the construction of the Wess-Zumino functional was mainly performed in the case where the gauge anomaly was of polynomial type, i.e. constructed through the so-called descent equations stemming from an invariant polynomial coming from a characteristic class of the underlying principle fiber bundle. What about the construction of a Wess-Zumino functional in the case where the anomaly does not come from a polynomial, namely there is no local functional coming from the Chern-Simons transgression formula. For instance, diffeomorphism anomalies, Weyl anomaly or presently, the "anomalous" term obtained from the Einstein-Hilbert action as a spontaneous breaking of the conformal Weyl invariance, they all fall into this latter type.
In this appendix, we would like to report on a more general construction which deals such a situation, even more, with any generic situation, for any Lie algebra Lie G and any representation space.
Historically and according to the best of our knowledge, a preliminary construction was explicitly given in [13, see appendix F] or [12]. It could be viewed as a most formalized version of [9, see pages 485 and 486] and exploits results in [11, section II and appendix]. Latter, Raymond Stora guessed an homotopy after a thorough discussion at CERN-TH in the mid of the 1990's with one of us (SL) and evoked by Raymond Stora himself in a seminar given at CPT (Marseilles) in November 1995.
Let us give a short account on this construction mainly recorded in [26] in order to bring this nice construction out of the shadows as a part of the wide legacy left by Raymond Stora.
To start with, let us denote the gauge group by G = {g : U → G} the set of local maps with values in a compact, simple symmetry group G. It carries a group law inherited from that of G. As field representation spaces for G, one can distinguish • when elements of G are considered as fields, one has the so-called gauge group G = {γ ∈ G, γ g = g −1 γg}, which is compatible with the group law in G.
• the gauge transformation of gauge potentials with the usual action of G given by the right action a γ = γ −1 aγ + γ −1 dγ. where d is the de Rham differential on M . For instance, in a Yang-Mills theory, the Slavnov operation s is explicitly given by

Consider a local consistency anomaly as an element in
where [ , ] is the graded Lie algebra bracket. Recall that c ∈ (Lie G) * ⊗ Lie G and is the generator of cochains on Lie G [11] or [28, section 6.10]). A(c, a) with sA(c, a) = 0, one can construct a functional on the field space of gauge fields a and on field u ∈ G The Wess-Zumino trick [5,8] is to use a seesaw mechanism between left and right actions by extending the Slavnov operation s on the gauge group G by su = −cu, where u is considered as a field, in order to guarantee s(a u ) = 0. This is, in the BRST language, the infinitesimal version of the gauge invariance of a composite field a u := Ad(u −1 )a + u −1 du under the gauge transformations of both elementary fields a and u (the latter are considered as belonging to representation spaces of G -a right group action) according to

Theorem. [5] Given an anomaly
where γ is a gauge group element, γ ∈ G. Let us stress that the composite field a u must not be considered as a gauge transformation of a because a u is no longer a connection owing to the fact that u carries a non-linear transformation law u γ = γ −1 u which is different from the required transformation law γ ′ γ = γ −1 γ ′ γ for a genuine gauge group element considered as a field in the 'adjoint' representation, (as recalled above (A.1)).
In fact, a u may be interpreted as a change of variable in the field space of the (a, u)'s; see discussion in [23,29, see in particular section 2 in both of those references] on what we called the dressing field method, a construction which goes back to Dirac [30] and which, in turn, enters in the construction of the Wess-Zumino functional.
The Stora construction. Given a family {γ t } in G, 0 ≤ t ≤ 1, with γ 0 = e and γ 1 = u, its action on a gives a family {a t := a γt }, interpolating from a 0 = a to a 1 = a u . In the field space, consider the interpolating family {u t , a t } defined by (a t ) ut = a u for all t. This constraint implies u t = γ −1 t u (in the case of a transitive action of G), that is u t is a family of dressing fields with u 0 = u and u 1 = e.
Requiring the invariance of (a t ) ut under the gauge transformations given in (A.6) infers This shows the gauge invariance of (a t ) ut under the following gauge transformations on the family It is worthwhile to notice that γ −1 t γγ t is a family which stays within the gauge group G as a field space for the 'adjoint' representation. Let us turn to the infinitesimal version of the latter in a BRST language. To sum up, one has the family in field space (the latter is the adjoint action of the family γ t on (Lie G) * ) and it can be checked that Along some ideas given in [10], the Stora's trick is to introduced a homotopy for s on the family through an even derivation k t defined as where d t is an antiderivation along the 1-parameter family induced by γ t .
Since the differential algebra (A.7) is similar to the BRST algebra (A.3), one has the consistency condition (A.1) along the whole family, namely, sA(c t , a t ) = 0, or s∆(c t , a t ) + d∆ ′ (c t , a t ) = 0. Therefore, by (A.8), one gets where on the r.h.s. the integrand used in the Wess-Zumino-Stora formula (A.4) occurs. Integration in t yields The smooth deformation τ → γ t,τ defines, for each t, a curve in G passing through γ t at τ = 0 with velocity ∂ ∂τ (γ t,τ ) |τ =0 . This gives rise to a t-dependent family of tangent vectors (as it will be explicitly seen later on) defines along the family {γ t } by where T γt G is the tangent space to G at the point γ t . We are interested in computing the variation of the smooth map namely, one has indeed to compute one of the derivatives either ∂ ∂τ The derivative on the r.h.s. allows to work easier with differential forms on M × G. In this respect, interesting developments might be found in [24, p.192ff.]. By smoothness, one has