BRST and supermanifolds

In this paper the subject of the BRST symmetry representation by means of superﬁelds is resumed and extended to N = 1 supersymmetric gauge theories. Then a new extension to diffeomorphisms is presented. Finally some speculations of a possible global representation of BRST symmetric theories by means of non-trivial supermanifolds are outlined.


Introduction
The BRST symmetry, [1,2], has been a fundamental breakthrough in quantum field theory. Not only did it provide an elegant and rigorous framework for the renormalization program of quantum gauge theories, but it also opened the way to an amazing number of applications in related fields. Suffice it to mention topological field theories and sigma models, string and superstring theories and string field theories. Whenever there is a gauge symmetry in a classical theory, at the quantum level a BRST symmetry appears that governs the quantum behavior of the theory. It is not exaggerated to say that its basic property is nilpotence: a twice repeated BRST transformation vanishes. This in turn, at least historically, has two origins: the first is the group theoretical nature of the BRST transform, the second the anticommuting property of the ghost and antighost fields, i.e. their wrong spin-statistic connection. The latter property is a con-sequence of the Faddeev-Popov quantization of gauge theories. If we perform two (classical) gauge transformations on a row and then repeat the operation in reverse order, the results of the two operations are related by a precise group theoretical rule. When, in the process of quantization, the classical gauge parameters are replaced by ghost fields, this rule becomes the nilpotence of the BRST transform.
In this paper in memory of Raymond Stora I would like to elaborate on these basic properties. The geometrical nature of the BRST transform was immediately sensed, [3,4]. However, the geometry in question is not that of a classical gauge theory, that is the geometry of a principal and associated fiber bundles, but rather that of a fiber bundle whose structure group is the infinite dimensional Lie group of gauge transformations. This was explained in [5]. But such a geometrical approach does not (or not yet) fit in the formalism of quantum field theory. A useful and simple synthesis of geometry and quantum field theory language seems to be provided by the supermanifold approach, [6,7]. One adds to spacetime additional anticommuting coordinates which faithfully mimic the above mentioned infinite dimensional geometry For instance, for a gauge field theory with Lie algebra valued potential form A, curvature F and ghost field c, one introduces an additional anticommuting coordinate ϑ, a superconnection A = φ + ϑψ, where φ and ψ are superfields, and a super-exterior derivative d = d + dϑ ∂ ∂ϑ . Then the condition that the supercurvature F =dA + AA equals F , horizontality condition, completely determines the BRST 'geometry'. It is enough to identify A with the lowest component of φ and c with the lowest component of ψ.
Such a superfield formalism has been subsequently enriched and applied to various models, see [8,9] and references therein. But altogether one can say that the superfield formalism for BRST has been regarded so far as an elegant decoration, whereas, for instance, superspace has been an important tool in the development of supersymmetric theories. In time supermanifolds have grown in importance and, especially in superstring theories, they have become an essential tool for the integration over the (super)moduli space, [10]. Therefore it is not beside the point to wonder if also for BRST supermanifolds can play an analogous role. In this paper I resume this subject. I would like to show that the superfield formalism is very flexible and applies also to symmetries for which the formalism has not been implemented so far: to this end I will develop the formalism for supersymmetric gauge theories in superspace formulation and, then, for diffeomorphisms. Secondly I would like to discuss the prospects of BRST supermanifolds.

The superfield formalism in supersymmetric gauge theories. A proposal
In this section I will formulate the 'BRST supergeometry' of an N = 1 supersymmetric gauge theory formulated in the superspace. To start with let me summarize the superspace presentation of this theory.

The supermanifold formulation of SYM
From ch. XIII of [11], a supersymmetric gauge theory can be introduced as follows. One starts from a torsionful (but flat) superspace with supercoordinates z M = (z m , θ μ , θμ) and introduces a supervielbein basis where A = (a, α, α) are flat indices. The vielbein satisfy The vielbein are chosen to be In such a type of supergeometry one has de a = −2ie α σ a αα eα, de α = 0, The flat indices derivatives The superconnection is defined by where v r m is the ordinary non-Abelian potential and T r are the Hermitean generators of the gauge Lie algebra.
The gauge curvature is given by the superform On the flat basis this becomes where the torsion term is the first on the RHS. We have F ab θ=θ =0 = iT r v r ab . The dynamics is determined by the super-Bianchi identity, DF = dF − [φ, F ] = 0. They are solved by with further restrictions coming from: This allows us to write Similarly Moreover the W 's must satisfȳ
For pedagogical reasons I will discard for the moment θ and consider only ϑ and postpone to the next subsection the more complete treatment.
Notation. In this section the square bracket notation [ , ] denotes a graded commutator, with grading according to total Grassmannality of the two entries The total Grassmannality includes both the one related to supersymmetry and to the BRST symmetry. I will call supersuperfield (ss-field) a supersymmetric superfield that is a function also of the coordinate ϑ . In terms of ZM = (x m , θ μ , θμ, ϑ) = (z M , ϑ) we havẽ where f (z) and g(z) are ordinary supersymmetric superfields. We introduce also d = dZM ∂ ∂ZM = d + dϑ ∂ ∂ϑ . Next we can introduce the super-super-connection (ss-connection) We choosẽ φ A , ψ A , ϕ ϑ and ψ ϑ are ordinary superfields valued in the gauge Lie algebra with generators T r . The BRST interpretation is: The ss-curvature can be writteñ the horizontality condition is F = F . Thus the terms proportional to dϑ and dϑ 2 must vanish. From The second equation is identically satisfied once we satisfy the first This is a definition for ψ ϑ . The lowest component of ϕ ϑ is an anticommuting scalar valued in the gauge Lie algebra, and is to be identified with the ghost field c = c r (x) T r . From the term proportional to dϑ we get two conditions where D A denotes the super-gauge-covariant derivative. Inserting the first into the LHS of the second we get which is the covariant derivative of (15) and thus vanishes. Therefore the independent relations are the first of (14) and (16). They define the BRST transform of ϕ ϑ and φ A , respectively. The surviving term in (13) is The term de A is the superspace (super)torsion. The term linear in ϑ is the BRST transform of the supercurvature. But some of the components of F actually vanish, so we have to verify that their BRST transform also vanish. At this point we could easily write the Bianchi identity DF = 0 in a BRST-covariant form. Since this is a very cumbersome procedure we prefer to BRST-covariantize the constraints extracted from it in chapter XIII of [11]. For instancẽ Since F αβ = 0 due to (4), it must be that This can be verified using (16), thanks again to (4).
In the α, β case we must have instead We notice that Therefore the BRST transform is consistent with the constraint F αβ = 0. In fact one must notice that, for all indices, Thanks to this equation also the constraints (5) can be written in a BRST-covariant form (that is, for instance, with F replaced by F ). So far the constraints (4) are consistently satisfied. Next we have to satisfy the analog of (8). We can introduce the BRST covariant definitions for W α and Wα: The BRST covariant analogs of (8) can be written as follows: where we definẽ Using these definitions we find consistently. The other constraints can be rewritten in the same way.

The ϑ , θ superfield formalism
Now I will switch on the second 'ghost' coordinate θ and I will call supersuperfield (ss-field) a superfield that is a function also of the coordinate θ . In terms of ZM = (x m , θ μ , θμ, ϑ, θ ) = (z M , ϑ, θ ) we havẽ where f (z), g(z), ḡ(z) and h(z) are ordinary supersymmetric superfields. The BRST-anti-BRST interpretation is We introduce also the ss-exterior derivative d = dZM ∂ We choosẽ φ A , ψ A , . . . , θ are ordinary superfields valued in the gauge Lie algebra with generators T r . The ss-curvature can be writteñ and the horizontality condition is It gives rise to the following set of equations Eqs. (34), (35) yield the identification The remaining equations are identically satisfied. The lowest component of ϕ ϑ is an anticommuting scalar valued in the gauge Lie algebra, and is to be identified with the ghost field c = c r (x) T r . Its BRST transform is ψ ϑ . ϕ ϑ is the BRST transform parameter. The lowest component of ϕθ is to be identified with the dual ghost field c =c r (x) T r . Its anti-BRST transform is ψθ .
Eq. (36) gives the relation which is to be interpreted as the Curci-Ferrari relation, [12], and ψ ϑ +ψθ are the Lautrup-Nakanishi superfields. Using (40) the remaining relations turn out to be identically verified. Let us come next to the constraint (32). It implies the definitions and the identities while from (33) we get the definitions and the identities The superfield ψ A , ψ A , π A are easily recognized (anti)BRST transform. The equivalence of (43) and (46) can be proven by means of the CF condition. Next let us come to (31). In general, using (3), one can show that In proving this a particular attention must be paid to the (A, B) = (α, β ) case. The definition (3) includes in this case also a contribution from the supertorsion; but this contribution is exactly canceled by an analogous term coming from the first commutator (2). From (48) it this evident that the constraints (4) can be covariantly implemented in the BRST formalism. Also here instead of solving the ss-Bianchi identity, we prefer to covariantize the constraints extracted from it in chapter XIII of [11]. In the same way as (4) also (5) can be covariantly implemented in the BRST formalism. Moreover, using (6), (7), we can introduce the ss-field expressions for W α and an analogous one for Wα . The next issue is now to BRST-covariantize the constraints (8).
Let us use the compact notation α to denote both α and α and introduce the BRST supercovariant derivativẽ then it is lengthy but straightforward to prove that This allows us to write down the constraints (8) in a BRST covariant form.
Of course this is only the beginning of the story when quantizing an N = 1 supersymmetric Yang-Mills theory. Then one has to fix the gauge and show that the overall action can be written in an invariant way in the enlarged superspace. Finally one should not forget that concrete calculations are carried out in the Wess-Zumino gauge. But at least this section shows that the BRST formalism can be consistently embedded in a supermanifold that encompasses also the supersymmetric spinorial directions.

Diffeomorphisms and the superfield formalism
A first proposal of a superfield formalism for diffeomorphisms was made by [7]. Here I present another approach to the same problem, closer in spirit to the standard (commutative) geometrical approach and to the discussion in the next section.
Diffeomorphisms, or general coordinate transformations, are defined by means of a local parameter ξ μ (x): x μ → x μ + ξ μ (x). In a quantized theory this is promoted to an anticommuting field. The BRST transformation are δ ξ g μν = ξ λ ∂ λ g μν + ∂ μ ξ λ g λν + ∂ ν ξ λ g μλ , for a scalar field, a vector field, the metric and ξ . These transformations are nilpotent. We will introduce another anticommuting field, ξ , and a δξ transformation, which transforms a scalar, vector, the metric and ξ in the same way as δ ξ , and, in addition, The overall transformation δ ξ + δξ is nilpotent: Introducing this additional field seems to be completely unmotivated, but in fact it is needed if we want an invertible supermetric.

The superfield formalism
One introduces the superspace X M = (x μ , ϑ, θ ), where ϑ , θ are anticommuting. A diffeomorphism is represented by a superspace transformation X M = (x μ , ϑ) →X M = (x μ − ϑξ μ − ϑξ μ , ϑ, θ ), where ξ , ξ and ϑθ, anticommute. The horizontality condition is formulated by selecting appropriate covariant expressions in ordinary spacetime and identifying them with the same expressions extended to the superspace. Below we work out explicitly the case of a scalar, a vector field and the metric.

The scalar
For instance, a scalar field ϕ is embedded in the superfield and gets transformed into Horizontality means It is easy to prove that That is the diffeomorphism transforms of ϕ are its superpartner in the superfield.

The vector
Let us extend this to a vector field. In order to apply the horizontality condition we have to identify the appropriate expression. This is a 1-superform: and Horizontality means that Expanding we find where ξξ · ∂ 2 = ξ μξ ν ∂ μ ∂ ν . The commutation prescriptions are: x μ , ϑ , θ , ξ μ commute with dx μ , dϑ; ξ μ anticommute with ϑθ. Working out (66) we obtain the following identifications: and One can prove that In particular Analogously
superstring theory in the RNS formulation, the relevant supermanifold is generally not holomorphically split (a refinement of the previous notion of split supermanifold).
But let us return to the previous case of a (non-supersymmetric) gauge theory, and let us suppose that the gauge has been fixed and a BRST invariant action is at hand. In order to compute a given amplitude one has further to integrate over the moduli space M, i.e. over the space of gauge orbits. In this case the relevant geometry is that of a family of superspaces, that is a fibered superspace over M where each fiber is a copy of M. The problem is well-known and not yet satisfactorily solved for 4d gauge theories, see for instance [14,15]. The difficulty is related to the nonexistence of a continuous section over the moduli space, i.e. of a continuous choice of a representative for each orbit. When fixing the gauge-slice locally, for instance the Landau gauge, the problem manifests itself with the appearances of Gribov copies at the horizon. These are global aspects of the problem and they certainly signal the lack of a global gauge-slice, but also the failure of BRST symmetry in foliating the space of gauge connections in a regular way. At this point it is illuminating to look at Ref. [16]. In this paper the authors analyze the observable of 2d topological gravity. The local correlators they compute come from integrating over the supermoduli space of the Riemann surfaces with punctures, and turn out not to be globally defined and simultaneously to violate the BRST Ward identities. It is by using these BRST anomalies (cocycles) and the Č eck-De Rham complex that it is possible to construct globally defined correlators which locally reduce to the previous ones. Although this problem is different (in some sense more complicated but in general far simpler) than the gauge theory one, it teaches us an important lesson. The appearance of Gribov copies at the horizon can be seen as a breakdown of the local BRST symmetry. This symmetry should be 'bent' or 'deformed' in some way in order to provide a regular foliation of the connection space and avoid copies. But this means that the linear vector space structure of the odd fibers of M breaks down, and, as a consequence, M cannot be anymore split. On the other hand, the example of [16] tells us that it may be possible to 'repair' these fractures of the gauge theory texture by a more complicated construction, hopefully a nontrivial supermanifold with non-split structure. If the above conjecture makes any sense it would mean that the superfield formulation (of any theory) can store very significant information about its quantum structure.