Off-Shell Scalar Supermultiplet in the Unfolded Dynamics Approach

We show how manifestly supersymmetric action for Wess-Zumino model can be constructed within the unfolded dynamics approach. The off-shell unfolded system for N = 1, D = 4 scalar supermultiplet is found. The action is presented in the form of integral of a closed 4-form over any (4, 0) surface in superspace as well as a superspace integral of an integral form or a chiral integral form. The proposed method is argued to provide a most general tool for the analysis of manifestly supersymmetric functionals.


Introduction
Unfolded dynamics approach, originally developed for the description of higher-spin field dynamics [1], implies rewriting field equations in the form of some generalized covariant constancy conditions. In principle, any theory can be reformulated in such a way (e.g., in [2] this has been done for gravity and Yang-Mills theory). Unfolded formulation of a dynamical system allows one to control easily its gauge symmetries. The coordinate-free language of differential forms is particularly convenient for theories of gravity. Moreover, so-called universal unfolded equations [3], to which class belong all relevant examples, are insensitive to a particular spacetime where the fields live. The latter property allows one in particular to extend any unfolded supersymmetric system from physical space-time to superspace.
Another remarkable feature of the unfolded dynamics approach is that it provides a tool for the search for Lagrangians and conserved currents in terms of certain Q-cohomology associated with the system of unfolded equations [2]. The aim of this paper is to illustrate this scheme by systematic derivation of the manifestly supersymmetric superspace actions for the simplest supersymmetric model, namely 4d Wess-Zumino model. Our results provide an off-shell extension of the on-shell results of [4].
Naively, the proposed scheme may look obstructed by the fact that superforms do not support integration over superspace. This is avoided once, as we proceed in this paper, the action is either defined as an integral of a superform over an even submanifold arbitrarily embedded into the full superspace or as an integral form. In application to supersymmetric models, our method has much in common with the group manifold approach [5,6] which treats actions as invariant functionals on hypersurface embedded into the group manifold, as well as with the "ectoplasm" approach of [7,8,9] which gives component action by imposing d-closure condition on superLagrangian. The Q-cohomology used in the unfolded dynamics is related to de Rham cohomology by virtue of unfolded equations. As such, the Q-cohomology approach is more general and can be applied to any (not necessarily supersymmetric) model.
The paper is organized as follows. In Section 2 unfolded dynamics approach is overviewed and the cohomological method of computation of Lagrangians for a graded system is proposed. In Section 3 relevant aspects of superform integration are considered. In Section 4 we recall the formulation of Minkowski superspace in terms of flat superconnections. Section 5 contains an overview of the results of [4] for the on-shell massless scalar supermultiplet as well as its offshell generalization. In Section 6 we explore operator Q of the system in question and compute the cohomology of its highest grade part Q 3 . In Section 7 we derive and solve equations that determine supersymmetric invariant functionals of the model and find the particular solutions associated with action of Wess-Zumino model. Conventions and notations are collected in Appendix A. The full system of equations from Section 7 is stored in Appendix B. where d = dx m ∂ ∂x m is de Rham differential, W Ω (x) are degree p Ω differential forms and f Ω Υ 1 ...Υn W Υ 1 ...W Υn (2.2) are degree p Ω + 1 differential forms built from exterior products of forms W Υ (x) (wedge symbol is omitted in this paper).
Here Ω and Υ are indices carried by differential forms. The identity d 2 ≡ 0 implies the compatibility condition which has to be satisfied for all W Ω . It can be equivalently rewritten as (2.4) Unfolded equations are called universal [3,2] if compatibility condition (2.3) holds independently of the fact that any p-form with p > d is zero in d-dimensional space. In this case one can differentiate freely over W Ω (x), and equation (2.1) is invariant under gauge transformation where (p Ω − 1)-form gauge parameter ε Ω (x) is related to the p Ω > 0 form W Ω (x) (0-forms do not give rise to gauge parameters). For universal unfolded equations, condition (2.3) holds independently of the choice of a space-time manifold. Full information about local physical degrees of freedom of the unfolded system is contained in 0-forms at any given point of space-time. Since these data remain the same in any space, universal unfolded systems provide an equivalent description in a larger (super)space simply via addition of extra coordinates. Particular examples of this phenomenon have been presented in [10,11,12,4].
The following terminology is used. The fields that can neither be expressed via derivatives of some other fields nor gauged away are called dynamical. The rest of the fields are referred to as auxiliary. (Let us note that the decomposition of fields into dynamical and auxiliary is not necessarily unambiguous). Differential conditions imposed by unfolded equations on dynamical fields are called dynamical equations. Other equations are either consequences of dynamical equations or constraints which express auxiliary fields via derivatives of the dynamical ones.
An example of unfolded equation can be constructed as follows. Let g be a Lie algebra with a basis {T a }. Consider a g-valued 1-form Ω 0 = Ω a 0 T a . For G = Ω 0 Ω 0 , equation (2.1) reads as The compatibility condition (2.3) gives usual Jacobi identity for the algebra g. Eq. (2.6) means that the connection Ω 0 is flat which is the standard way to describe g-invariant vacuum. Eq. (2.5) gives usual gauge transformations of the connection Ω 0 where ε 0 (x) is a 0-form valued in g. Given flat connection Ω 0 is invariant under the transformations with parameters obeying This equation is formally consistent by virtue of (2.3). Solutions of equations (2.8) describe the leftover global symmetry g of any solution of (2.6). Let us linearize unfolded equations (2.1) around fixed connection Ω 0 satisfying (2.6), where C are differential forms treated as small perturbations and hence contributing linearly to the equations. Let C i p be a subset of forms of a fixed degree p, enumerated by index i. In the linear approximation, the part of G which is bilinear in Ω 0 and C i p contributes, i.e. G = Ω a 0 (T a ) i j C j p . In this case, Eq. (2.3) implies that the matrices (T a ) i j form a representation of the algebra g in the space V where p-forms C i p are valued. Corresponding equation (2.1) is the covariant constancy condition where D Ω 0 ≡ d + Ω 0 is the covariant derivative in the g-module V . C i p transform properly under g gauge transformations. Indeed, eq. (2.5) gives for (2.9) where ε i p are gauge parameters related to C i p (for p > 0) and ε 0 are global g-symmetry parameters obeying (2.8).

σ − -cohomology
Classification of dynamical fields, gauge symmetries and dynamical equations of the unfolded systems can be performed in terms of so-called σ − -cohomology [13,3,2,4]. Let a linear unfolded system be of the form where C(x) are some differential form fields and operators σ i act algebraically (i.e. do not differentiate x n ).
In the σ − -cohomology technics, the decomposition of the fields into dynamical and auxiliary is controlled by the Z-grading G with respect to which auxiliary fields have higher grade than dynamical ones. The grading operator G has to be diagonalizable on the space of fields and to be bounded from below. d has grade zero. Usually, G counts a number of tensor indices of the fields.
σ − -cohomology technics applies if σ i contain operators of negative grades. Then σ − is the operator of the lowest grade and Eq. (2.11) takes the form with Σ denoting all operators that act algebraically and have G-grade higher than σ − . Since σ − has the lowest G-grade, from compatibility condition (2.3) (2.14) Using that the gauge transformation (2.5) for equation (2.12) is it can be shown [13,3,2] that, for p-forms C p from the space V , the cohomology H p−1 (σ − , V ), H p (σ − , V ) and H p+1 (σ − , V ) are, respectively, the spaces of differential gauge symmetries, dynamical fields and dynamical equations. The situation with several operators of negative grade is more complicated. As shown in [4], in this case usual σ − -analysis should be extended to the spectral sequence analysis of all such operators. The full field-theoretical pattern of the system is determined by the cohomology H σ

Unfolded actions and charges
Invariants of a general unfolded system such as actions and conserved charges are encoded by cohomology of the operator Q (2.4) [2].
Suppose that system (2.1) is off-shell, i.e. it does not contain any dynamical equations, describing only a set of constraints. In the language of σ − -cohomology, this means that unfolded equations for (p − 1)-forms W Ω have H p (σ − ) = 0. Following [2], the action S of this system is defined as an integral over a manifold M d Taking into account that δL = ∂L/∂W Ω δW Ω and using (2.5), one easily obtains Assuming that M d has no boundary (or that fields decrease fast enough at infinity), the action remains invariant under gauge transformations (2.5).
If the Lagrangian L is Q-exact, i.e. L = G Ω ∂F ∂W Ω , by virtue of (2.1) and hence Q-exact Lagrangians lead to trivial local actions. Thereby nontrivial invariant actions of the off-shell system (2.1) are in one-to-one correspondence with its Q-cohomology.
If system (2.1) is on-shell (i.e. contains some dynamical equations) and a p-form L is a representative of the nonzero Q-cohomology class, the same formula (2.16) describes a conserved charge as an integral over a p-cycle Σ q =Σ L . (2.20) Let M d be embedded into some ambient space, M d ⊂ Md,d > d. Extending (2.1) to Md, by virtue of (2.17), which is equivalent to d-closure of L, action (2.16) is independent of the local form of this embedding.
The case where the algebra of functions of fields from which a Lagrangian is built admits a grading G bounded from below, is of particular interest. Let Q and L admit decompositions into finite sums of G-homogeneous parts It can be shown that the space of nontrivial Q-closed Lagrangians is isomorphic to some subspace of H(Q n ), where Q n is the part of Q of maximal G-grade. Indeed, from Q 2 = 0 at different G-grades it follows that Let the highest grade components be denoted as Q := Q n and L := L k . Since nontrivial Lagrangians are represented by Q-cohomology, if L = Qf it can be removed by the redefinition where L ′ is the highest grade part of L ′ . If L ′ = Qg, the subtraction L ′′ = L ′ − Qg reduces the maximal grade further. The process stops in a finite number of steps because G-grading is bounded below. Eventually, either the Lagrangian vanishes or its highest grade part L belongs to H (Q). Thus, any nontrivial Lagrangian is represented by some L ∈ H (Q). This does not mean however that any L ∈ H (Q) is associated with some nontrivial Lagrangian L ∈ H(Q). Two related phenomena may happen.
One is that L cannot be supplemented with the terms of the lowest degrees to form a Qclosed Lagrangian L. Indeed, Q n−1 L in Eq. (2.26) is Q-closed by virtue of (2.22), (2.23) and (2.25). If it is not Q-exact however, Eq. (2.26) admits no solutions. In other words, that Q n−1 L is in nontrivial Q-cohomology provides an obstruction for reconstruction of L in terms of L.
Another phenomenon is that if the same element of the Q-cohomology, that provides an obstruction for the extension to full Q-cohomology, is interpreted as a highest grade part of some other Lagrangian with L ′ = Q n−1 L, then such a highest grade part can be removed by adding a Q-exact term -QL.
More generally, a similar phenomenon may occur at any step of the analysis of Eqs. (2.25)-(2.27). In particular, the corresponding Q-trivial highest grade components have the form In the latter case, though being non-Q-exact, L can be removed by adding Q-exact terms − i QF i .
Note that somewhat similar situation took place in [14], where the deformation of Minkowski higher-spin vertices to AdS space was studied. There nontrivial vertices belong to cohomology of the nilpotent operator Q = Q f l + λ 2 Q sub , where −λ 2 is the cosmological constant. The grading G counts the number of derivatives in vertices, G Q f l = 1, G Q sub = −1. The AdS deformation (if exists) of a nontrivial vertex F in Minkowski space (where λ = 0 and Q = Q f l ) may in principle turn out to be trivial in AdS.
As a result, the space of nontrivial Q-closed Lagrangians is isomorphic to subspace of H (Q), which is formed by some highest grade terms L that cannot be represented in the form (2.29) with F i obeying (2.30)-(2.32).
The construction of invariant functionals presented so far works nicely for usual manifolds but is less obvious in the case of superspace which is of most interest in this paper. Since, naively, the differential superforms are not integrable over supermanifolds (see e.g., [15]), we have to specify the notion of an unfolded action in superspace.

Integration in superspace
Most of differential geometry admits straightforward generalization to supermanifolds. However extension of integration of differential forms over a supermanifold is not quite straightforward. One way to see this is to observe that superform transformation law does not match Berezin integral. Indeed, consider a supermanifold M p|q with local coordinates z M = (x m , θ µ ). To be coordinate-independent, the integration measure has to transform according to Berezin formulâ Superforms resulting from naive extension to odd coordinates obviously do not satisfy this condition and hence are not integrable. However, even forms can still be integrated over even cycles in superspace. Indeed, consider an even n-dimensional surface S n on M p|q : z M (t a ) = (x m (t a ) , θ µ (t a )), a = 1, ..., n, parametrized by some even parameters t a (surface coordinates). Let the integral of a n-superform ω = ω M 1 ...Mn dz M 1 ...dz Mn over S n be defined aŝ It is neither dependent on the choice of coordinates t i nor of z M . Obviously if ω is exact in superspace, the integration over S n amounts to that over its boundary ∂S n . By Cartan formula L V = {d , i V } for the Lie derivative of a vector field V N (z), the integral of a closed form ω (dω = 0) is independent of local variations of S n . This allows us to define unfolded superfield action as the integral of a closed p-superform L over an even p-dimensional surface in superspace. More generally, integration in superspace can be defined in terms of integral forms, as was originally proposed in [16] (see also [15]). To this end an integral of superfunction f (x, θ) in superspace can be rewritten aŝ where ς m , m = 1, ..., p and s µ , µ = 1, ..., q are treated as additional anticommuting and commuting integration variables, respectively. To contribute, F (x, θ, ς, s) should have the form To make the link with differential forms, one formally substitutes dx m and dθ µ for ς m and s µ in (3.4). Resulting objects are called integral forms. Note that δ p (dx) is just the usual volume form dx 1 . . . dx p while δ q (dθ) is the actual (even) δ-function, which is essentially non-polynomial. On the other hand, integration of usual superforms polynomial in dθ does not make sense, leading As mentioned in Introduction, our approach has much in common with the "ectoplasm" method [7,8,9] of construction of manifestly supersymmetric actions represented by integrals of superforms over space-time "hypersurface" in the full superspace M p|q with coordinates with the covariant derivative D M and torsion tensor T M N P . Then the integral over space-time hypersurface is independent of coordinates in M p|q and, by virtue of (3.6), of a particular choice of the integration hypersurface, provided that the latter is even and J falls down fast enough at spatial infinity. It is invariant under the transformation The more general case of a curved superspace can be considered analogously in terms of supervielbeins [8,9]. Similarity of the ectoplasm and unfolded approaches is obvious. Indeed, in the former dclosure of Lagrangian guarantees its manifest SUSY, while in the latter the same is achieved via Q-closure condition. As shown in Subsection 2.1, Q is an algebraic counterpart of de Rham differential. Although for particular supersymmetric models both methods lead to similar results, the unfolded approach is more general being applicable to any (not necessarily supersymmetric) theory and (generalized) space-time.

Supersymmetric vacuum
Following [4], to obtain unfolded description of the flat superspace we start with the N = 1 SUSY algebra (All other (anti)commutators are zero.) Gauge fields of supergravity are 1-forms of vierbein e a = e a m dx m , spin-connection ω a,b = ω a,b m dx m and gravitino φ α = φ α m dx m (see e.g. [17]). They are components of a 1-form connection Ω 0 valued in the SUSY algebra Supersymmetric flat background is represented by a connection Ω 0 obeying (2.6), which leads to the following component equations where D L ≡ d + ω is the Lorentz covariant derivative. As explained in Subsection 2.1, to promote these unfolded equations (which are obviously universal) to superspace it suffices to add fermionic coordinates x m → z M = x m , θ µ ,θμ extending properly indices of the differential forms where D is the Lorentz covariant derivative in superspace. Global SUSY transformations are described by those gauge transformations (2.8) of system (4.11)-(4.14), that leave invariant background connection Cartesian coordinate system is associated with the following solution of (4.11)-(4.14) In these coordinates, the explicit solution to system (4.15)-(4.18) is with ξ a , ξ a,b = −ξ b,a , ξ α ,ξα being free constants, which are parameters of global symmetries. Note that Lorentz covariant derivative in superspace can be rewritten in the form In Cartesian coordinates (4.25)

Unfolded free massless scalar supermultiplet
In this section we present an off-shell extension of the unfolded equations of motion for N = 1, D = 4 free massless scalar supermultiplet, obtained in [4].
First we consider the problem in Minkowski space. It is described by a solution of system (4.7)-(4.10) with φ α =φα = 0, i.e.
Massless scalar field in Minkowski space is described by the unfolded equations [1,13] Compatibility of the system is provided by flatness condition (2.6) and the identity ( which follows from σ-transversality and the fact that spinor indices take two values. Extension to superspace is trivially achieved via addition of fermionic coordinates

7)
Dχ a(k) The resulting system is universal. As explained in Subsection 2.1, this implies that all its symmetries are preserved. Hence, system (5.7), (5.8) is supersymmetric.
To check that these equations indeed describe free massless scalar supermultiplet, one has to single out independent dynamical superfields and dynamical equations with the help of σ −cohomology technics. As shown in [4], this gives the following result. The only dynamical superfield is C(z). All other fields are auxiliary, being expressed via its derivatives. For instance, which are standard equations of motion of a massless scalar supermultiplet [19].
To construct the action, we should find an off-shell modification of system (5.7)-(5.8) which implies no dynamical equations. As a guiding example, first consider the off-shell formulation of system (5.2), (5.3). It results from relaxing the tracelessness condition for C a(k) and for χ a(k) α as well as the σ-transversality condition for the latter. Then Eqs. (5.2), (5.3) just represent a set of constraints which express higher rank tensors in terms of derivatives of the dynamical fields.
However, supersymmetric extension (5.5)-(5.6) of the resulting off-shell system via introducing connections for full SUSY algebra ceases to obey (2.3), i.e. becomes inconsistent. Indeed, the compatibility condition for Eq. (5.6) requires In the on-shell case, it holds by virtue of σ-transversality, which is relaxed in the off-shell case. Inconsistency of the system means, in addition, that its gauge transformations (2.5) do not obey SUSY algebra (4.1)-(4.5) any more, i.e. the system lost SUSY. To restore both off-shell consistency and SUSY, a set of auxiliary fields F a(k) should be introduced. Supersymmetric off-shell system of equations acquires the form This system is consistent, obeying (2.3). By virtue of Eq. (5.14), auxiliary fields F a(k) are higher derivatives of the ground auxiliary field F familiar for Wess-Zumino model. Not surprisingly, the fields F a(k) provide closure of SUSY algebra of the off-shell system. As explained in Subsection 2.1, consistency of system (5.12)-(5.14) implies its global SUSY invariance. Corresponding SUSY transformations in Minkowski space are In Cartesian coordinates with e a m = δ a m , φ α =φα = 0, D L = d system (5.12)-(5.14) implies C a = −∂ a C, (χ α ) a = −∂ a χ α . As a result, where ξ α andξα are global SUSY parameters. These are standard supertransformations of the chiral supermultiplet [19]. Extension of system (5.12)-(5.14) to superspace is again achieved via extension of all functions to superspace Resulting system imposes, however, differential equations with respect to odd coordinates, i.e. strictly speaking it is not fully off-shell in superspace. Indeed, using (4.24) it is easy to obtain from (5.21)-(5.23) thatDα C a(k) = 0, D α F a(k) = 0 . These are chirality condition for the fields C a(k) and antichirality condition for the fields F a(k) . As is well known [19], these conditions do not impose differential equations in Minkowski space, where the system remains off-shell.

Highest grades
Now we are in a position to look for Lagrangians, representing cohomology of the operator Q. These are built from background 1-forms Ω a,b , E a , E α ,Ēα and supermultiplet 0-form fields . First of all, we observe that Lagrangian should be Ω-independent since the terms resulting from the action of Ω a,c Ω c b ∂ ∂Ω a,b in (6.2) cannot be canceled against other terms. Indeed, consider for instance a function Ω a,b A ab , where A ab is built from 1-forms E a , E α ,Ēα and 0-forms of supermultiplet fields. Then the part of QΩ a,b A ab bilinear in Ω contains three terms of the form Ω a,c Ω c b A ab which do not cancel. Similarly one proceeds with terms of higher orders in Ω. More precisely, though nonlinear Ω-dependent terms can be present, all of them can be removed by adding Q-exact terms, thus representing the trivial class of Q-cohomology.
Clearly, for Ω-independent Lagrangian, the condition Q Ω L = 0 (6.12) implies that L is Lorentz invariant, i.e. all indices are contracted with Lorentz-invariant flat metric and σ-matrices. As a result Q-cohomology amounts to cohomology of the Ω-independent partQ of Q. It is convenient to introduce the following grading G of the background 1-forms According to Subsection 2.3, nontrivial Lagrangians can be represented by cohomology H (Q 3 ). The computation considerably simplifies in spinor notations. The dictionary between vector and spinor indices is provided by σ-matrices. For example, for a Lorentz vector A a In spinor notations An efficient tool to compute cohomology is provided by the homotopy lemma (see e.g. [20]).
and Q be a grade -1 nilpotent operator The homotopy lemma states, that if the homotopy operator Hence, only v, that are eigenvectors of H with zero eigenvalue, can belong to H (Q, V ).
To apply this technique let us introduce the operator The homotopy operator associated with (6.16) and (6.20) is As explained in Section 3, in supersymmetric models one can look for different types of unfolded superLagrangians, depending on whether they are polynomials or distributions with respect to commuting odd differentials dθ or, in terms of gauge fields, with respect to gravitino 1-forms E α andĒα. In the both cases H has zero G-grade, and is diagonalizable so that the homotopy lemma applies.
It is convenient to characterize expressions in question by the number of 1-forms E αα they contain. Being anticommutative, E αα can appear only in the following combinations: Note that for the last term in (6.21) the following relations hold using (A.6) and that E γγ E δγ is symmetric in γ and δ due to anticommutativity of E αα . Derivative of δ-function is defined as usual viaĒ˙γδ ′β Ēα = ǫ˙γβδ 2 Ēα . Analogous relations hold for the third term in (6.21). Now let us use the homotopy lemma to compute Q 3 -cohomology in the class of real superforms polynomial in E α andĒα. Depending on the number of E αα , there are five options: ..E αĒα ...Ēαλ α(m),α(n) + h.c., where λ α(m),α(n) are 0-forms symmetric over indices α andα. Here we have HΛ 0 = mnΛ 0 + h.c. = 0, (6.24) which is true when m = 0 or n = 0, i.e.

Four-form Lagrangian
We look for a Lagrangian as a Q-closed 4-superform where 0-forms ℓ i are built Lorentz-covariantly from the supermultiplet fields C a(k) ,C a(k) , χ a(k) α , χ a(k) α , F a(k) ,F a(k) . That this is the most general Lorentz-invariant 4-superform Ansatz follows from the results of Section 6.2. (For instance, the term ǫ abcd E a E b E α (σ c ) ααĒα ℓ d , that has the same G-grade as (6.39), is not added as not representing nonzero Q 3 -cohomology.) The equationQL = 0 can now be analyzed in different grade sectors, starting from the highest one. The full set of equations is presented in Appendix B.
There are two complex conjugated equations in the highest grade G = 9 These hold true for any ℓ 6 andl 6 , because the corresponding terms belong to H (Q 3 ). Indeed, e.g. Eq. (7.2) is proportional to ǫ abcd (σ ab )αβ (σ c )˙γ αĒαĒβĒγ . From relation (A.9) it follows that to be symmetric over three dotted spinor indices ǫ abcd (σ ab )αβ (σ c )γ α must be antisymmetric with respect to the three undotted indices, which is zero because the latter take just two values. In the grade G = 8 we obtain four equations. From the first twō we find that ℓ 6 andl 6 have to obeyqαℓ 6 = 0 andq αl6 = 0, respectively. Using (A.9), one easily finds that the last two equations are solved by ℓ 5α = √ 2 6q α ℓ 6 andl 5α = √ 2 6qαl 6 . Continuation of this analysis gives the following Q-closed superfield Lagrangian where the 0-forms W andW are arbitrary functions of C a(k) ,F a(k) andC a(k) , F a(k) , respectively, so thatqαW = 0 andq αW = 0. By virtue of (5.24), (5.28) this means that W is chiral and W is antichiral. Note that, as follows from (6.51), W of the form W =q a f a lead to trivial Lagrangians. By construction, Lagrangian (7.8) is manifestly supersymmetric and the corresponding action is independent of the local variation of the integration surface. A particular solution, that reproduces free Wess-Zumino action [19], results from (7.8) An important comment is that, though the unfolded on-shell system from Section 5 describes free dynamics of massless scalar supermultiplet, its off-shell modification represents just an infinite set of constraints. The form of these constraints is independent of whether the model is free or nonlinear. As a result, it can be used for description of a massive interacting theory. Nonlinear (starting from cubic) representatives of Q-cohomology determine Lagrangians with interactions. In particular, if W = W (C) depends only on C (respectivelyW =W C ), (7.8) describes the superpotential.

Lagrangian as integral form
As explained in Section 3, a superspace Lagrangian can also be formulated as an integral form. This can be written as where ℓ [a 1 ...am] is some Lorentz-covariant 0-form built from the supermultiplet fields.
To analyze whether or not such actions contain trivial parts one has to consider Q-images of the expressions containing derivatives of delta-functions. Indeed, for we obtain As a result, (7.12) describes trivial Lagrangians if ℓ =q α f α + h.c. for some f α . In particular, Lagrangians with ℓ =q αα f αα are trivial because, as follows from (6.47), in this case In tensor indices the Lagrangian reads as The Lagrangian which reproduces the free Wess-Bagger Lagrangian [19] is (7.17) as follows from the formula (5.24) which implies that C (z) is a chiral superfield.
However, superpotentials are not represented in the integral form (7.16). This is expected since they represent chiral functions to be integrated over chiral subspaces [19]. In our approach such terms also are most conveniently represented in the form intermediate between the 4-form Lagrangians and integral-form Lagrangians, i.e. as integrals over chiral superspace.

Chiral superspace
To introduce superpotentials we introduce the following chiral integral forms, where W is the Lorentz-covariant 0-form built from chiral functions C a(k) andF a(k) (so, Q − 2 Λ = 0). Such forms are integrable over chiral superspace C m+n|2 in a standard way described in Section 3. HereĒα without δ-functions describe the pullback of the respective 1-forms to the chiral superspace.
Let us explore equation HΛ i = 0 in spinor notations, where Λ i from (7.21) contains i factors of E αα . For the equation HΛ = 0 has nontrivial solutions (with arbitrary 0-forms W ) only at m = 0 . This is easy to understand by noting that, because δ 2 (E α )Ēα = Q 3 i 4 δ ′ β (E α ) E βα all such Λ, being Q 3 -closed due to δ-functions, are not Q 3 -exact only if they are independent ofĒα.
However, for Hence, Λ 4 (7.26) can represent Q 3 -cohomology at any m. Thus Q 3 -cohomology in the class of chiral integral forms is represented by To be integrated over C 4|2 , the only appropriate expression from (7.27)-(7.31) is Note that G (δ 2 (E α )) = −4, because the δ-function has the degree of homogeneity −2 and G (E α ) = 2, as a result G (L 0 ) = 0. To single out trivial Lagrangians, we write down Eqs. (2.29)-(2.32) for which represents the only expression from those in (7.27)-(7.31), whose Q-image can contribute to (7.32). This gives We immediately conclude that F −2 with G = −2 here can only have the form (G δ ′ β (E α ) = −6 as follows from the definition E γ δ ′ β (E α ) = ǫ γβ δ 2 (E α ) and G (δ (E α )) = −4.) Eq. (7.34) is satisfied as F −1 ∈ H (Q 3 ). Term with F −1 in (7.35) is zero because (7.33) contains δ-function and f ββ is built from C a(k) andF a(k) , so F −2 = 0. Finally, Eq. (7.36) has a solution We conclude that (7.32) describes nontrivial Lagrangians only if W =q αα f αα for some f αα . Also, analogously to Subsection 7.2, we have to consider expressions with derivatives of δfunction whose Q-images can lead to trivial Lagrangians in (7.32). It is easy to see that the only appropriate elements from Ker (Q 3 ) are However the former has G (K 1 ) = 2 while the latter has G (K 1 ) = 0. Since G (L 0 ) = 0 for (7.32), the system (2.29)-(2.32) admits no nonzero solutions in these cases. Next, besides Q 3 L 0 = 0, L 0 (7.32) obeys The first equation holds because W in (7.32) is built from chiral C a(k) andF a(k) . The second one holds due to the δ-function. The third one is true because L 0 contains the maximal number of E αα . So L 0 (7.32) is Q-closed and hence represents the general form of an unfolded chiral Lagrangian. (Note that (7.27)-(7.30) with nonzero W are not Q-closed, because they contain less than four E αα and hence are not annihilated by Q 1 = 1 2 E ααq αα .) In tensor notations, general chiral Lagrangian is where Lorentz-invariant 0-form W is built from C a(k) andF a(k) , and W =q a f a . For the Lagrangian to be real, (7.40) should be supplemented by the complex conjugated expression to be integrated over antichiral superspace with the 0-formW built fromC a(k) and F a(k) . Combination of (7.40) (plus conjugated expression) and (7.16) gives the unfolded superspace action of the interacting theory.
To reproduce superpotential of the Wess-Zumino model [19] we choose W = kC + m 2 CC + g 3 CCC with arbitrary constants k, m, g. Then superpotential takes the form The kinetic term − 1 16 CF in (7.42) results from integration of (7.17) overθα taking into account that from (5.25)-(5.27) it follows thatF = 2DDC.
To obtain conventional field-theoretic Lagrangian one has to express tensors of the rank k 1 in terms of derivatives of zero-rank tensors with the help of the unfolded equations and to substitute resulting expressions, for instance, into (7.9). Then, choosing Minkowski space as an integration surface, we see that integral of (7.9) reproduces the component action of the free scalar supermultiplet. Alternatively, one can use Lagrangians (7.17) or (7.42), arriving at the standard Wess-Bagger superfield action.

Conclusion
In this paper, unfolded off-shell formulation of the free massless scalar supermultiplet is presented and the system of equations, that determines all superfield Lagrangians of the model, is derived and analyzed. The particular solutions leading to superfield actions of the Wess-Zumino model in the form of integrals of a 4-superform, integral form and chiral integral form are obtained. Explicit relations between these forms of superspace actions are established. It is shown in particular how usual superspace action for the Wess-Zumino model can be rewritten as an integral of a 4-superform.
In some sense, the construction of chiral action is intermediate between the one of Section 7.1 and that of Section 7.2. In fact, this is a particular example of a very general phenomenon that the full action may have a form of integral over (super)manifolds of different dimensions. As long as even dimension is kept fixed, integration over supercoordinates will result in one or another space-time action. We expect that in more complicated theories like higher-spin theories and their further multiparticle extensions, invariant functionals resulting from integrals over space-times with different even dimensions may all contribute to the final result.
Being based on unfolded dynamics, the proposed method is most general, providing maximal flexibility in the construction of supersymmetric actions. Applied to on-shell unfolded system it provides a systematic tool for the analysis of on-shell counterterms in supersymmetric systems. It would be interesting to explore its applications to more complicated models including extended SUSY and, in the first place, to theories whose manifestly supersymmetric formulation is yet lacking, like N = 1, D = 10 or N = 4, D = 4 super Yang-Mills theories, as well as to harmonic superspace formulation [21] of the models of extended SUSY.