Supergravity Actions with Integral Forms

Integral forms provide a natural and powerful tool for the construction of supergravity actions. They are generalizations of usual differential forms and are needed for a consistent theory of integration on supermanifolds. The group geometrical approach to supergravity and its variational principle are reformulated and clarified in this language. Central in our analysis is the Poincare' dual of a bosonic manifold embedded into a supermanifold. Finally, using integral forms we provide a proof of Gates' so-called"Ectoplasmic Integration Theorem", relating superfield actions to component actions.

tion. This is based on a formula which relates the superfield action to the component action via a density projection operator acting on a closed superform. This procedure incorporates the integration over the fermionic coordinates and the contributions due to the gravitons. We show here that the origin of that formula can be understood by interpreting the superfield action as an integral form. The relation between the density projection operator and the component action is achieved by partial integration using picture changing operators.
Three decades ago a group-based geometric approach to supergravity was put forward, known as group manifold approach [5,6], intermediate between the superfield and the component approaches. This framework provides a systematic algorithm to construct supergravities in any dimension. The starting point is a supergroup, and the fields of the theory are identified with the vielbein one-forms of (a manifold diffeomorphic to) the supergroup manifold. by many examples in ref. [5]. One of the advantages of this approach is that it yields the self-duality constraints of the D = 6 and D = 10 supergravities mentioned above as part of the equations of motion, besides allowing to construct the corresponding actions [7,8].
We show here how the variational principle of the group manifold approach can be reformulated and clarified by using integral forms and the Poincaré dual of the submanifold. In particular we derive the condition for the embedding independence of the submanifold. This coincides with the condition for local supersymmetry invariance of the spacetime action, and reduces to the vanishing of the contraction of dL along tangent vectors orthogonal to the submanifold.
The paper has the following organisation. In sec. 2, the integration on supermanifolds is briefly discussed and presented both from a mathematical point of view, and from a more intuitive/physical point of view. The integration on curved supermanifolds is also discussed.
In sec. 3, we describe, also for the case of supermanifolds, a simple and explicit form of the Poincaré dual as a singular localization form. The integration on a submanifold and the independence of the embedding is discussed. The construction of the actions in the groupgeometric approach is presented and the variational principle is explained. Finally, in sec. 4 we consider the relation between the integral of superforms in the ectoplasmic integration formalism and integral forms. The "ethereal conjecture" of Gates et al. [1,2,3,4] is proved using integral forms. Appendix A contains some additional material ancillary to the main text.

Integration on Supermanifolds
In this section we give a short introduction to the theory of integration on supermanifolds (see for example the review by Witten [9]). The translation of the picture changing operators into supergeometry has been explored in [10,11]. More recently, the application to target space supersymmetry and Chern-Simons theories have been discussed in [12]. Thom classes for supermanifolds have been constructed in [13]. The picture changing operators have been introduced in string theory in [14], from world sheet point of view, and in [15], from target space point of view.
We start, as usual, from the case of the real superspace R n|m with n bosonic (x i , i = 1, . . . , n) and m fermionic (θ α , α = 1, . . . , m) coordinates. We take a function f (x, θ) in R n|m with values in the real algebra generated by 1 and by the anticommuting variables, and we expand f as a polynomial in the variables θ : If the real function f m (x) is integrable in some sense in R n , the Berezin integral of f (x, θ) is defined as: formal symbol that has nothing to do neither with "exterior products", nor with "top forms" mainly because if θ is a fermionic quantity, dθ is bosonic (dθ ∧ dθ = 0).
An important property of [d n x d m θ] is elucidated by the following simple example: consider in R 1|1 the function f (x, θ) = g(x)θ (with g(x) integrable function in R ). We have: A brief review of the formal properties of integral forms [10,12] is given in Appendix A.
Here we elaborate on their definition and on the computation of integrals.
The usual integration theory of differential forms for bosonic manifolds can be conveniently rephrased to shed light on its relations with Berezin integration.
We start again with a simple example: consider in R the integrable 1-form ω = g(x)dx (with g(x) integrable function in R ). We have: Observing that dx is an anticommuting quantity, and denoting it by ψ, we could think of ω as a function on R 1|1 : This function can be integratedà la Berezin reproducing the usual definition: Note that (as above) the symbol [dxdψ] is written so as to emphasize that we are integrating on the two variables x and ψ, hence the dx inside [dxdψ] is not identified with ψ.
This can be generalized as follows. Denoting by M , a bosonic differentiable manifold with dimension n, we define the exterior bundle Ω • (M ) = n p=0 p (M ) as the direct sum of p (M ) (sometimes denoted also by Ω p (M )). A section ω of Ω • (M ) can be written locally as where the coefficients ω [i 1 ...ip] (x) are functions on M and the indices [i 1 , . . . , i p ] are antisymmetrized. The integral of ω is defined as: At first sight this might seem a bit strange, but we are actually saying that in the definition of the integral only the "part of top degree" of ω is involved. This opens the way to the relations between the integration theory of forms and the Berezin integral, that can be exploited by If the manifold is equipped with a metric g (that for the moment we assume globally defined), we can expand a generic form ω on the basis of forms ψ a = e a i dx i (a = 1, . . . , n) such that g = ψ a ⊗ ψ b η ab where η ab is the flat metric on the tangent space T (M ) and we have that where e = det(e a i ), and g = det(g ij ). Again, we use the Berezin integral to select the top degree component of the form. Notice that the last integral can be computed if suitable convergence conditions are satisfied according to Riemann or Lebesgue integration theory.
In the following we will need also distributions, and therefore we consider expressions that for covariance under diffeomorphisms) and into a test functionω(x) (for example, belonging to the space of fast decreasing functions). In this case: where in the last term we evaluate the expression at x i = 0. In this case, the compactness of the space or other convergence conditions do not matter, since the measure is concentrated in the point x i = 0. The points x i where the integral is localised can be moved by suitable diffeomorphisms.
We denote now by M a supermanifold with coordinates (x i , θ α ) (with i = 1, . . . , n and α = 1, . . . , m) and we consider the "exterior" bundle Ω • (M) as the direct sum of bundles of fixed degree forms. The local coordinates in the total space of this bundle are ( where (x i , dθ α ) are bosonic and dx j , θ β fermionic. In contrast to the pure bosonic case, a top form does not exist because the 1− forms of the type dθ α commute among themselves dθ α ∧ dθ β = dθ β ∧ dθ α . Then we can consider forms of any degree (wedge products are omitted in the following) where the coefficients ω of simply expanding formally a generic form ω(x, θ, dx, dθ) in dθ, we can consider analytic functions of the bosonic variables dθ and in addition we will admit also distributions acting on the space of test functions of dθ. In this way, the exterior bundle in the dθ directions is a conventional bosonic manifold with coordinates dθ α and the superforms become distributionvalued on that space. In particular, we introduce the distributions δ(dθ α ) that have most (but not all!) of the usual properties of the Dirac delta function δ(x). As explained in Appendix A, one must have: Therefore, the product of all Dirac's delta functions δ m (dθ) ≡ m α=1 δ(dθ α ) serves as a "top form".
An integral form ω (p|q) belonging to Ω (p|q) (M) is characterised by two indices (p|q): the first index is the usual form degree and the second one is the picture number which counts the number of delta's. For a top form, that number must be equal to the fermionic dimension of the space. Consequently, an integral form reads: The dθ α appearing in the product and those appearing in the delta functions are reorganised respecting the rule dθ α δ(dθ β ) = 0 if α = β. We see that if the number of delta's is equal to the fermionic dimension of the space no dθ can appear; if moreover the number of the dx is equal to the bosonic dimension the form (of type ω (n|m) ) is a top form.
Notice that ω (p|q) as written above is not the most generic form, since we could have added the derivatives of delta functions (and they indeed turn out to be unavoidable and will play an important role). They act by reducing the form degree according to the rule dθ α δ (dθ α ) = −δ(dθ α ), where δ (x) is the first derivative of the delta function with respect to its variable. (We denote also by δ (p) (x) the p-derivative).
We also define as a superform a 0-picture integral form Ω (p|0) Integral top forms (with maximal form degree in the bosonic variables and maximal number of delta forms) are the only objects we can hope to integrate on supermanifolds.
In general, if ω is a form in Ω • (M), its integral on the supermanifold is defined as follows: (in analogy with the Berezin integral for bosonic forms): where the last integral over M is the usual Riemann-Lebesgue integral over the coordinates x i (if it is exists) and the Berezin integral over the coordinates θ α . The expressions denote those components of (2.10) with no symmetric indices.
Note that under the rescaling invariant quantity, in fact it is locally a "product measure", and we know that . This can be extended to general coordinate transformations, and the outcome is It is clear now that we cannot integrate a generic ω (x, θ, dx, dθ) . Suppose that the Riemann-Lebesgue integrability conditions are satisfied with respect to the x variables; the integrals over dx and θ (being Berezin integrals) pose no further problem but, if ω (x, θ, dx, dθ) has a polynomial dependence in the (bosonic) variables dθ, the integral diverges unless ω (x, θ, dx, dθ) depends on the dθ only through the product of all the "distributions" δ (dθ α ). 2 This solves the problem of the divergences for all the dθ α variables because Summing up we can integrate only integral forms ω, the integral selecting only forms contained in ω with top degree in bosonic variables and top picture number, namely the so-called integral top forms.
In order to shorten notations, when the "variables of integration" are evident, we will omit in the integrals all the "integration measures symbols" such as In the case of curved supermanifolds, by expressing the 1-forms dx i and dθ α in terms of the supervielbeins E A M ≡ (e i M , e α M ) (where A runs over the flat indices i and α, and M runs over the curved indices), we have . As usual this definition is invariant under (orientation preserving super) diffeomorphisms.

Poincaré duals and Variational Principles on Submanifolds
As discussed in the introduction, we consider a submanifold S of a bigger space M -that could be also a supermanifold -and we give a recipe to construct an action I on that submanifold.

Poincaré Duals
We start with a submanifold S of dimension s of a differentiable manifold M of dimension n. We take an embedding i : and a compact support form L ∈ Ω s (M). The Poincaré dual of S is a closed form η S ∈ Ω n−s (M) such that ∀L: where i * is the pull-back of forms. We are not interested here in a rigorous mathematical treatment (see [16]) and we take a heuristic approach well-adapted for the generalization to the supermanifold case. In the symbol I[L, S], we have recalled the dependence upon the embedding of S into M.
If we suppose that the submanifold S is described locally by the vanishing of n − s coordinates t 1 , . . . , t n−s , its Poincaré dual can also be described as a singular closed localization form (the correct mathematics is the de Rham current theory [17]): This distribution-valued form is clearly closed (from the properties of the delta distributions d δ(t) = δ (t)dt and from dt i ∧ dt i = 0). This form belongs to Ω n−s (M) and is constructed in such a way that it projects on the submanifold t 1 = · · · = t n−s = 0 and orthogonally to dt 1 ∧ . . . ∧ dt n−s . Thus, by multiplying a given form L ∈ Ω s (M) by η S , the former is restricted to those components which are not proportional to the differentials dt i .
Observing that the Dirac δ-function of an odd variable (dt is odd if t is even) coincides with the variable itself (as can be seen using Berezin integration), we rewrite η S as a form that will turn out to be very useful for generalization (omitting wedge symbols): which heuristically corresponds to the localisation to t 1 = · · · = t n−s = 0 and dt 1 = · · · = dt n−s = 0. Note that if a submanifold S is described by the vanishing of n − s functions f 1 (t) = · · · = f n−s (t) = 0 the corresponding Poincaré dual η S is: This form, when written completely in terms of the t i coordinates, contains also the derivatives of the δ's because of the expansion of δ(f ) and δ(df ) in terms of t i .
If we change (in the same homology class) the submanifold S to S the corresponding Poincaré duals η S and η S are known to differ by an exact form: This can be easily proved by recalling that the Poincaré duals are closed dη S = 0 and any variation (denoted by ∆) of η S is exact: Given the explicit expression of η S , it is easy to check eq. (3.4) by expanding both members (assuming that ∆ follows the Leibniz rule) and using the distributional laws of δ's.
Using this property we can show that, then the action does not depend on the embedding of the submanifold. Indeed varying the embedding amounts to vary the Poincaré dual, so that the variation of the integral reads The same arguments apply in the case of supermanifolds. Consider a submanifold S of dimension s|q of a supermanifold M of dimension n|m. We take an embedding i : and an integral form L ∈ Ω s|q (M) (integrable in the sense of superintegration when pulled back on S). The Poincaré dual of S is a d-closed form η S ∈ Ω n−s|m−q (M) such that: Again we can write: where the f 's are the functions defining (at least locally) the submanifold S. Here some of them are even functions and some of them are odd functions, accordingly the Poincaré dual is a closed integral form that, written in the coordinates (x, θ), contains delta forms and their derivatives.
Again it is easy to check that any variation of η S is d-exact: Note that the two formulae (3.4) and (3.6) for the variation of η S can be combined in a formula that holds true in both cases: Indeed, one has δ (df ) = 1 or δ(f ) = f when f is respectively bosonic or fermionic.
Before considering some examples, we have to spend a few words on the general form of the Poincaré dual in the case of supermanifolds: where we have added l-derivatives ∂ l on the Dirac delta functions (for the moment we have not Let us consider for example R (0|1) as a submanifold of R (0|2) , which has two coordinates θ 1 We can compute the integral in two ways: the first is by using θ 1 = − b a θ 2 and by re-expressing L in terms of the coordinate θ 2 only. Thus and the integral gives: The second way is as follows. The Poincaré dual of The first delta function can be rewritten as a θ 1 + b θ 2 because of the anticommutativity of θ's. Multiplying η S by L we obtain: a which coincides with the computation above. The integral depends upon the embedding parameters (a, b) (and is not defined for a = 0). Repeating the same computation with a closed form (for example θ 1 δ(dθ 1 )), it is easy to see that the integral equals 1 and does not depend on the embedding parameters as expected.

Variational Principle
The action I[L, S] is a functional of L and S, and therefore varying it means varying both L and S. The latter corresponds to varying η S . The variational principle leads to independent of S. This is somehow rather obvious, but it is interesting to notice that in many cases dL = 0 holds only on a subset of the equations of motion, and in some cases it holds completely off-shell.
As an example we consider 3d−euclidean gravity on a 3d−submanifold S (for example a 3d−topological sphere) embedded into R 4 . The Poincaré dual is given by The action is given by where ω is the spin connection, R ab = dω ab − ω a c ∧ ω cb , V a is the dreibein and f is the embedding function. The equations of motion are given by where the torsion is defined as T a = dV a + ω a b ∧ V b . Notice that the equation on the first line are valid for any f , and this implies that R ab = 0, T a = 0 on the entire space R 4 . The last equation is a consequence of the first two equations together with the Bianchi identity DR ab = 0 (where D is the Lorenz covariant derivative), but we observe that only one is sufficient to guarantee the vanishing of the last equation. Namely, for a torsionless connection ω, dL = 0 off-shell and ω can be expressed in terms of V a (second order formalism).
We consider now 4d-Einstein gravity and we would like to embed the Einstein-Hilbert We denote by inner components the coefficients along V a ∧ V b and outer the remaining ones.
The EH action is written as where η S is the Poincaré dual of S in M. Under the conditions discussed above the equations of motion on the big space M are The Recalling that dη S = 0, this term reduces, after integration by parts, to (−) s M dL ∧ ι η S . Thus varying only L under y-diff's leaves the action (3.18) invariant if dL = 0. We conclude that y-diff's applied to the fields µ are invariances of the action I if dL = 0.
Actually the condition for y-diff's on µ to be invariances of I is weaker: indeed it is sufficient to have ι dL = 0. This can be checked directly by varying L in the action under y-diff.s: Integrating by parts the second term and recalling that dη S = 0 proves that the action I is invariant under y-diff 's applied to the fields µ when ι dL = 0.
The last equation (3.22) can also be used to study the dependence of the action upon the embedding functions. We know that M L ∧ L η S = − M L L ∧ η S from eq. (3.19).
Thus any variation of the embedding (generated by L , with an arbitrary outside S) can be compensated by a y-diff's on L. On the other hand we have seen that y-diff's on L do not change the action when ι dL = 0 with in the y-directions, and therefore this is also the condition for I to be independent on the particular embedding of S.
Let us come back to our example, pure gravity in the group manifold approach, where the "big space" M is (a smooth deformation of) the Poincaré group manifold, and the "small space" S is the usual Minkowski spacetime. Usual x-diff's on the fields V a and ω ab leave the action invariant, while y-diff's, i.e. diffeomorphisms along the Lorentz directions of M, are invariances when applied to V a and ω ab if ι t dL = 0 (t = t ab ∂ y ab being the tangent vectors in the Lorentz directions, dual to the spin connection ω ab ). Let us check whether this condition holds. Replacing again the exterior derivative d with the covariant exterior derivative D, and using the Bianchi identity DR ab = 0 and definition of the torsion, we find the condition: Using now the Leibniz rule for the contraction, and ι t (V a ) = 0, leads to the condition that all outer components of R ab and T a must vanish. These conditions are part of the field equations previously derived. In particular they do not involve the "inner" field equations, i.e. the Einstein equations. On this "partial shell" the action is invariant under y-diff's ("Lorentz diffeomorphisms") applied to the fields.
The vanishing of outer components of the curvature is also called horizontality of the curvature.
When horizontality of R A in the y-directions holds, the dependence of fields µ A (x, y) on

Field transformation rules
Let us have a closer look at the variation of the fields µ under (infinitesimal) diffeomorphisms.
The transformation rule is given by the action of the Lie derivative on µ: When µ A is the vielbein of a (deformed) group manifold M (the index A running on the Lie algebra of G), the variation formula (3.25) takes the suggestive form: where C B CD are the G-structure constants, = A t A is a generic tangent vector expanded on the tangent basis t A dual to the cotangent (vielbein) basis µ B , and ∇ is the G-covariant exterior derivative. To prove this one just uses the definition of the group curvatures: that allow to re-express dµ A in terms of R A and bilinears of vielbeins.

Supersymmetry
In the group-geometric approach to supergravity theories, the "big" manifold M is a su- becomes, using the structure constants of the Lie superalgebra: defining respectively the supertorsion, the gravitino field strength and the Lorentz curvature.
All forms live on M = (deformed) super-Poincaré group manifold. The action is a 4-form integrated on a S (diffeomorphic to Minkowski spacetime) submanifold of M: The field equations, when projected on all the M directions, give the following conditions on the curvatures: where the spacetime (inner) components R ab cd , ρ ab satisfy the propagation equations The situation is drastically different when auxiliary fields are available to close the supersymmetry algebra off-shell. Then one finds that the fermionic contractions of dL vanish identically without requiring any condition. This can be checked for example in the so-called new minimal D = 4, N = 1 supergravity (or Sohnius-West model [18]), where the super-Poincaré algebra is enlarged and auxiliary fields (a 1-form and a 2-form) enter the game. In fact in this case the natural algebraic framework is that of free differential algebras [5], a generalization of Lie algebras, whose dual formulation in terms of Cartan-Maurer equations is generalized to contain also p-form fields.

Ectoplasmic Integration with Integral Forms
We would like to put in relation the so-called Ectoplasmic technique (Ethereal Integration Theorem) with integral forms. The main point is to prove, by using the integral forms, the so-called "ectoplasmic integration theorem". This theorem states that, given a function L of the superspace (also known as superspace action) on a curved supermanifold M whose geometry is described by the supervielbein E A M (see eq. (2.14)), its integral where E is the superdeterminant of E A M , is equal to the following integral where e is the determinant of the vielbein e a n of the bosonic submanifold S of M (more precisely, S is identified with the bosonic submanifold of M obtained by setting to zero the fermionic coordinates). The expression D m L | θ=0 denotes the action of a differential operator D m on the function L evaluated at θ = 0. D m is a symbol denoting a differential operator of order m in the super derivatives. The form of the differential operator is difficult to compute by usual Berezin integration since one has to evaluate the supervielbein E A M (at all orders of the θ-expansion), compute its superdeterminant and finally expand the product EL.
That procedure leads to the form D m L | θ=0 , where D m is a combination of super derivatives, ordinary derivatives and non-derivative terms and the coefficients depend upon curvature, torsion and higher derivative supergravity tensors. The relation between I M and I S is easy in the case of flat superspace since there is no superdeterminant to be computed and all supergravity tensors drop out.
In order to circumvent this problem, Gates et al. proposed a new method to evaluate D m L | θ=0 . First, one has to select a closed superform (that we will denote by L (n|0) ) with degree equal to the dimension of the bosonic submanifold. The form must be closed on the complete supermanifold, namely dL (n|0) = 0, where d is the differential on the full supermanifold. The closure of the superform (and also its non-exactness) and the existence of a constant tensor imply that a given component of L (n|0) can be written in terms of this tensor times an arbitrary function Ω(x, θ) on the supermanifold. All other components of L (n|0) are either vanishing or written as combination of derivatives of the arbitrary function Ω(x, θ).
The coefficients of those combinations are related again to supergravity tensors. The total result L (n|0) is a superform whose coefficients are given in terms of Ω(x, θ), a combination of derivatives and supergravity fields. The Ethereal conjecture is that the unknown function Ω(x, θ) coincides with the superspace action L evaluated at θ = 0.
The first step is to translate the definitions given by Gates et al. in term of integral forms.
Then, we show that the integrals of eq. (4.1) and eq. (4.2) can be viewed as integrals of integral forms that can be related via the Poincaré dual. Finally, by changing the Poincaré dual by a different embedding of the bosonic submanifold into the supermanifold, we are able to show that indeed the function Ω(x, θ) does coincide with the superspace action L.

From Ectoplasm to Integral Forms
The integral of L (n|0) (which we will denote in the following with ω (n|0) ) on the bosonic submanifold I S is defined as follows where S ≡ M n ⊂ M (n|m) ≡ M is the bosonic submanifold (obtained by setting to zero the anticommuting variables in the transition functions) andω (n|0) θ=0 is obtained from ω (n|0) by setting to zero both the dependence on θ and on 1-forms dθ Notice that this superform can be integrated on the bosonic submanifold being a genuine where we have denoted by Latin letters a 1 , . . . , a n the flat indices and e is the determinant of the vielbein e a i . The first crucial observation is that I S can be also rewritten, following the prescription described in sec. 2, as follows where, as usual, we denote by θ m the product of all fermionic coordinates θ α and by δ m (dθ) the wedge product of all Dirac delta functions δ(dθ α ). Then, the Poincaré dual in this case is η S = θ m δ m (dθ) which is the product of "picture changing operators" embedding the bosonic submanifold S into the supermanifold M in the simplest way θ 1 = θ 2 = · · · = 0.
The integration is performed over the entire supermanifold. A simple computation leads to the original result (4.3). This is clear since integrating over the dθ has the effect that all components of ω (n|0) in the dθ directions are set to zero, leading toω (n|0) . The Berezin integral over the coordinates θ is simplified since the presence of the product θ m forces us to pick up the first component ofω (n|0) , namelyω (n|0) | θ=0 leading to the integral.

Closure and Susy
The important point about (4.3) is the invariance under supersymmetry. The variation under supersymmetry ofω (n|0) is given by a local translation in superspace where the first equality is due to the variation of the field components in the expression of ω (n|0) (and therefore it does not matter whether it is computed at θ = 0), the second equality is just the expression of a susy transformation as a supertranslation in superspace. The last term can be rewritten as follows: where we have used the closure of the superform ω (n|0) = ω (n|0) where the superindices M 1 , . . . , M n+1 are graded-symmetrized. In this way, the r.h.s. of We also notice that if ω (n|0) belongs to the d-cohomology H * (Ω (n|0) ), so does the integral form, since θ m δ m (dθ) is in the d-cohomology H * (Ω (0|m) ) (which are the class of forms with zero form degree and highest picture number, see [19]). However, the converse is not true: dω (n|0) cannot be proportional to δ(dθ) since it must be a picture-zero form and f α must belong to Ω (n|0) while g α to Ω (n−1|0) . However, by consistency we have d f α θ α + g α dθ α = 0, which implies that df α = 0 and f α = −dg α . This yields f α θ α + g α dθ α = −d(g α θ α ) which can be reabsorbed into a redefinition of ω (n|0) , leading to a closed form.
Again we can check the susy invariance of I S in the form (4.6). Performing the susy transformations leads to where ∆ θ α = α and ( θ m−1 ) ≡ α 1 ...αm α 1 θ α 2 . . . θ αm . Due to the closure of ω (n|0) we are in the same situation as above: the partial derivative w.r.t. to θ can be re-expressed as an x-derivative and its integral is then zero. The second piece is zero because we integrate over θà la Berezin and, since ω (n|0) is computed at θ = 0 (being multiplied by θ m ), the integral vanishes.

Density Projection Operator
Now, we need to understand the integral obtained in (4.3) in terms of the superform ω (n|0) by using the closure of it. We adopt the description given by Gates and we follow the same derivation.
Let us now compute the expression in (4.3), namely we computeω (n|0) by passing to non-holonomic coordinates as follows where we denote by Σ the non-holonomic super indices. So, we have: The requirement that the superform must be closed, dω (n|0) = 0, expressed in terms of the non-holonomic basis, implies that where T Γ [Σ 1 Σ 2 ) are the components of the torsion computed in the non-holonomic basis. The form ω (n|0) is defined up to gauge transformations the notation |Γ| excludes the index Γ from the graded symmetrization.
The coefficients of the torsion satisfy the Bianchi identities The exponent m denotes the maximal number of spinorial derivative. This conclude this review part on the density projection operator and we are finally in position to present a proof of the theorem.

Proof of the Ectoplasmic Integration Theorem
At this point we need to study the other side of the Ectoplasmic Integration Theorem, namely we have to describe the integral I M in terms of a superform. For that we recall eq. (2.14): the integral of a top integral form ω (n|m) reads  (4.20). Thus, to prove the ectoplasmic integration formula one has to verify that they indeed coincide. In order to do that we observe that the superformω (n|0) belongs to the space Ω (n|0) which has vanishing picture number. Thus, in order to integrate it we need to change its picture by inserting Picture Changing Operators of the form where ψ A are the gravitino superfields (also denoted by E A in the present work) and the dots stand for terms with higher derivative of Dirac delta functions. The functions M A (x, θ), Therefore, we can construct a top integral form fromω (n|0) as which has the correct picture number and the correct form number to be integrated on M (n|m) .
The symbol Φ A i denotes a function of θ's such that dΦ A δ(ψ A ) = 0, giving a new arbitrary expression for the picture changing operator. Then it is easy to show that, by integrating by parts (using the superderivative appearing in the coefficients ofω (n|0) ), the integral form In other words, this corresponds to modifying the picture changing operators, but remaining in the same cohomology class. That implies that the two integrals are indeed equal since the delta functions appearing in Y soak up the gravitinos appearing in the density projection operator D m .
Let summarize the main steps of the proof. We start by showing that both the integral I M and I S can be written in terms of integral forms. The former is viewed as an integral of a density which is the coefficient of a top form of Ω (n|m) . The second integral I S is converted into an integral of an integral form by introducing a suitable picture changing operator θ m δ m (dθ).
However, the choice of the picture changing operator is arbitrary and therefore it can be changed into the new form (4.21), such that the gravitons ψ A appear as arguments of the delta functions. Finally, the computation of the integral I S projects out all components of the combination except a superfield Ω(x, θ) which can be chosen to be equal to the density of

Acknowledgments
We thank L. Andrianopoli, R. D'Auria and M. Trigiante for useful discussions concerning integral forms and their use in supergravity theories.

Appendix A.
We do not wish here to give an exhaustive and rigorous treatment of integral forms. A systematic exposition of the matter can be found in the references quoted in Section 2 .
As we said in section 2, the problem is that we can build the space Ω k of k-superforms out of basic 1-superforms dθ i and dx i and their wedge products, however the products between the dθ i are necessarily commutative, since the θ i 's are odd variables. Therefore, together with a differential operator d, the spaces Ω k form a differential complex which is bounded from below, but not from above. In particular there is no notion of a top form to be integrated on the superspace R p|q .
The space of integral forms is obtained by adding to the usual space of superforms a new set of basic "forms" δ(dθ), together with the derivatives δ (p) (dθ), (derivatives of δ(dθ) must be introduced for studying the behaviour of the symbol δ(dθ) under sheaf morphisms i.e.
coordinate changes, see below) that satisfies certain formal properties.
These properties are motivated and can be deduced from the following heuristic approach.
In analogy with usual distributions acting on the space of smooth functions, we think of δ(dθ) as an operator acting on the space of superforms as the usual Dirac's delta "measure" (more appropriately one should refer to the theory of de Rham's currents [17]), but this matter will not be pursued further). We can write this as where f is a superform. This means that δ(dθ) kills all monomials in the superform f which contain the term dθ. The derivatives δ (n) (dθ) satisfy f (dθ), δ (n) (dθ) = − f (dθ), δ (n−1) (dθ) = (−1) n f (n) (0), like the derivatives of the usual Dirac δ measure.
The most noticeable relation is the unfamiliar minus sign appearing in δ(dθ)δ(dθ ) = −δ(dθ )δ(dθ) (indeed this is natural if we interpret the delta "forms" as de Rham's currents) but can be also easily deduced from the above heuristic approach. To prove this formula we recall the usual transformation property of the usual Dirac's delta function An interesting and important consequence of this procedure is the existence of negative degree forms, which are those that by multiplication reduce the degree of a forms (e.g. δ (dθ) has degree −1).
We introduce also the picture number by counting the number of delta functions (and their derivatives) and we denote by Ω r|s the space of r-forms with picture s. For example, in the case of R p|q , the integral form is an r-from with picture s. All indices K i are antisymmetrized, while the first r − l indices are symmetrized and the last s are antisymmetrized. By adding derivatives of delta forms δ (p) (dθ), even negative form-degree can be considered, e.g. a form of the type: δ (n 1 ) (dθ i 1 ) . . . δ (ns) (dθ is ) is a −(n 1 + · · · + n s )-form with picture s. Clearly Ω k|0 is just the space Ω k of superforms, for k ≥ 0.
Integral forms form a new complex as follows . . .  We now briefly discuss how these forms behave under change of coordinates, i.e. under sheaf morphisms. For generic morphisms it is necessary to work with infinite formal sums in Ω r|s as the following example clearly shows.
Suppose (θ 1 ) = θ 1 + θ 2 , (θ 2 ) = θ 2 be the odd part of a morphism. We want to compute (δ dθ 1 ) = δ dθ 1 + dθ 2 in terms of the above forms. We can formally expand in series about, for example, dθ 1 : Recall that any usual superform is a polynomial in the dθ, therefore only a finite number of terms really matter in the above sum, when we apply it to a superform. Indeed, applying the formulae above, we have for example, Notice that this is equivalent to the effect of replacing dθ 1 with −dθ 2 . We could have also interchanged the role of θ 1 and θ 2 and the result would be to replace dθ 2 with −dθ 1 . Both procedures correspond precisely to the action we expect when we apply the δ (dθ 1 + dθ 2 ) Dirac measure. We will not enter into more detailed treatment of other types of morphisms.