Integral invariants in flat superspace

We are solving for the case of flat superspace some homological problems that were formulated by Berkovits and Howe. (Our considerations can be applied also to the case of supertorus.) These problems arise in the attempt to construct integrals invariant with respect to supersymmetry. They appear also in other situations, in particular, in the pure spinor formalism in supergravity.


Introduction
In present paper we are solving for the case of flat superspace some homological problems that were formulated in the paper [1]. (Our considerations can be applied also to the case of supertorus.) These problems arise in the attempt to construct integrals invariant with respect to supersymmetry and in other situations.
Let us consider a flat superspace with coordinates z M ∼ (x m , θ α ). The supersymmetry Lie algebra susy is generated by transformations [e α , e β ] + = γ m αβ P m , with differential d = γ m αβ t α t β ∂ ∂C m Here e α are generators acting on the space of (x m , θ α ), P m is the translation operator, and γ m αβ are Dirac Gamma matrices.
To construct integrals invariant with respect to supersymmetry one should find closed differential forms expressed in terms of physical fields. This is the homological problem we are trying to solve (see [1] for more details).
It follows from d 2 = 0 that t 2 0 = 0 (1) This means, in particular, that t 0 can be considered as a differential; corresponding cohomology groups will be denoted by H p,q t . The differential t 0 can be identified with the differential γ m αβ t α t β ∂ ∂C m appearing in the calculation of cohomology group of the supersymmetry algebra susy; hence the cohomology groups H p,q t coincide with graded components of cohomology groups of the Lie superalgebra susy.
Cohomology groups of the Lie superalgebra susy were calculated in [3], this allows us to compute the groups H p,q t . However, this is not the end of the story: the operator d 1 induces a differential on H p,q t ; corresponding homology groups are denoted by H p,q s . (See [1].) We would like to calculate these groups. We give complete answers for zero momentum (independent of coordinate variables x m ) part of H p,q s . (It will be denoted by H p,q s .) Ordinary supersymmetry superspace, which we used in the above construction, can be replaces by a superspace that supports extended supersymmetry. We analyze the case of N = 2 supersymmetry in ten-dimensional space. Notice that in this case the group H 0,q s can be interpreted as the cohomology of the differential t L ∂ ∂θ L + t R ∂ ∂θ R where the spinors t L , t R obey the relaxed pure spinor condition t L γ m t L + t R γ m t R = 0. N. Berkovits informed us that the problem of calculation of this cohomology arises in pure spinor formalism of ten-dimensional supergravity; Section 3.2 answers his question. The same cohomology appears also in [4] We would like to thank N. Berkovits and A. Mikhailov for thought provoking discussions.
2. Groups H p,q t and H p,q s : general results In [3] we have calculated the cohomology groups H p (susy) as graded modules in polynomial algebra C[t] generated by t α . These groups can can be considered also as Autrepresentations where Aut stands for the group of automorphisms of susy; we have found the action of Aut on them.
The group H p,q t is a graded component of H p (susy). More precisely, H p,q t is a graded component of H p (susy) multiplied by the space of functions of θ α and x m . (The differential t 0 has the same form as the differential in the definition of Lie algebra cohomology, but it acts on the space of functions depending on variables θ α and x m in addition to the ghost variables t α and c m .) To calculate the group H p,q s we consider the differential As a first step in this calculation we will simplify this differential omitting the second term. The cohomology corresponding to this differential d 0 s will be denoted by H p,q s .(In other words, we consider the differential on functions that do not depend on x. This corresponds to zero momentum.) First of all we notice that the differential d 0 is acyclic (only constants give non-trivial cohomology classes). This follows immediately from the commutation relation hd 0 s + d 0 s h = N where h = θ α ∂ ∂t α and N counts the number of θ's and t's (denoted by θ α and t α respectively): N = α (θ α + t α ). It follows that a cocycle The cohomology H p (susy) is not a free module, but it has a free resolution [3]. In other words there exists a graded module Γ with differential δ acting from Γ i into Γ i−1 ; this differential is acyclic in all degrees except zero, the only non-trivial homology is isomorphic to H p (susy).
Let us consider now a differential D = d 0 s +δ acting in Γ p ⊗Λ[θ]. (we omit the dependence of p in these notations). We represent a cocycle v of this differential in the form Knowing v m we can find all v k inductively using the formula From the other side we can solve the equations for v k starting with v 0 . Then we can identify the cohomology of D with the cohomology of H p (susy) ⊗ Λ[θ] with respect to the differential d 0 s , i.e. with the direct sum of groups H p,q s for all q. It follows that this direct sum is isomorphic to A p = m µ p m To find the part of A p corresponding to the group H p,q s we should use (4) and the tdegrees of elements of µ p m (this information can be found in [3]). It is easy to check the following statement: an element of µ p m having t-degree r corresponds to an element of H p,q s where In other words the t-degree of this element is r − k. This follows from the remark that the operator h decreases the t-degree by 1 . It increases θ-degree by 1 , hence the θ-degree of this element is k. Let us denote by µ p k,r the part of µ p k consisting of elements of t-degree r. Then it follows from the above considerations that More precisely, µ p k,q+k is isomorphic (H p,q s ) k (to the subgroup of H p,q s consisting of elements of θ-degree k).
More rigorous treatment of the above calculations should be based on application of spectral sequences of bicomplex. Namely, the cohomology of D = d 0 s + δ can be calculated by means of spectral sequence starting with H(H(d 0 s ), δ) or by means of spectral sequence starting with H(H(δ), d 0 s ). Both spectral sequences collapse, hence the cohomology of D is equal to is isomorphic to µ p m and the differential δ acts trivially on this group.) Let us discuss the calculation of H p,q s . First of all we notice that the differential d s acting on generating supersymmetry Lie algebra. This means that H p,q s can be considered as a representation of this Lie algebra (moreover, it is a representation of super Poincare Lie algebra).
The differential d s can be represented in the form We can apply spectral sequence of bicomplex to calculate its cohomology. However, we will use more pedestrian approach considering the second summand in d s as perturbation, In other words, we are writing d s as d 0 s + ǫd 1 s and looking for a solution of the equation The operator d 1 s descends to a differential on the cohomology of the operator d 0 s . In the first approximation the cohomology of d s is equal to the cohomology of d 1 s in the cohomology of d 0 s . This is clear from the above formulas ( the cohomology class of a 0 in the cohomology of d 0 s is a cocycle of d 1 s ). It follows that in the first approximation the homology H p,q s is equal to Here ∇f stands for the gradient ∂f ∂x m , and R is considered as a map of Aut-modules where Aut stands for the automorphism group of susy, the letter V denotes the vector representation of Aut. Using (6) we can interpret R as intertwiner In many cases this intertwiner is unique (up to a constant factor); we can use the LiE program to establish this fact.
In the language of spectral sequences the first approximation is the E 2 term. There is a natural differential on E 2 ; its cohomology E 3 can be regarded as the next approximation. The calculation of the differential on E 2 can be based on the remark that the super Poincare group acts on homology.
3. Groups H p,q t and H p,q s in ten-dimensional space 3.1. N=1 superspace in 10D. Groups H p,q t were calculated in [3] . The direct sum q H p,q t can be considered as C[t] -module; the resolutions of these modules also were calculated in this paper. We denote by µ p k the k-th term of the resolution and by µ p k,r the part of µ p k consisting of elements of t-degree r. We use a shorthand notation [i 1 , . . . , i [n/2] ] for an irreducible representation V of SO(n) where i 1 , . . . , i [n/2] stand for coordinates of the highest weight of V .
Using (6) we obtain the groups H p,q s :  all other groups vanish. The results above are verified by SO character-valued Euler characteristics where k corresponds to the degree of θ. We use a shorthand notation [V ] for the character χ V (g) of the representation V . The formula (7) reflects the idea that the character-valued Euler characteristics should be the same for the chain complex C p,q,k := H p,q t ⊗ Λ k (θ) and its cohomology groups (H p,q s ) k . Since their associated differential is d 0 s = t α ∂ ∂θ α , we can do alternated summation by varying the degree of θ, while the degrees of p and q + k are invariant.
Let us consider in more detail the case of zero-dimensional cohomology in ten-dimensional space. These calculations can be compared with computations in [2]. Higher dimensional cohomology can be analyzed in similar way.
It is easy to see that H 0 (susy) = q H 0,q t is the space of polynomial functions on pure spinors (in other words, this is a quotient of C[t] with respect to the ideal generated by γ m αβ t α t β ). According to [3]  where µ k are considered as representations of the group Aut = SO(10).
Here the degree is calculated with respect to t, the differential δ = δ k preserves the degree (this means, for example, that δ 5 is a multiplication by a quadratic (with respect to t) polynomial).
There are two ways to calculate these elements explicitly (up to a constant factor). First of all the SO(10)-invariance of the resolution permits us to guess the expression for δ k . It is easy to see that the maps specify an SO(10)-invariant complex. From another side using LiE code one can check that SO(10)-invariance specifies this choice of differentials uniquely (up to a constant factor). Knowing δ k we can calculate the elements belonging to the cohomology classes in H p,q s using (4). We obtain the elements s , [(tγ m θ)(tγ n θ)(θγ mnp θ)] ∈ H 0,2 s , [(tγ m θ)(tγ n θ)(tγ p θ)(θγ mnp θ)] ∈ H 0,3 s . Another way is based on the remark that we know the degrees and transformation properties of these elements. Again using LiE code we can find all possible answers. 3.2. N = 1 superspace in 11D. In out previous paper [3], we already obtained the cohomology groups of H p,q t in 11D. Then based on the Euler characteristics of µ i , namely the Eq.(109) in our paper [3], we could find µ k by matching the grading in the LHS and RHS of the equation (assuming there is no highest weight vector representation with the same degree appearing in two µ k 's). The results for the case when p = 0 are verified with Movshev's result [4].
• p = 0, 3.3. N = 2 superspace in 10D. In this superspace we have coordinates z M ∼ (x m , θ L , θ R ) (vector and two spinors). We take a basis in the space of 1-forms given by formulas The differential that is used to calculate the cohomology H t (= the cohomology of susy 2 , i.e. of the Lie algebra of N=2 supersymmetry) has the form It acts on the space of functions of commuting variables t α L , t α R and anticommuting variables C m . It is useful to make a change of variables t + = t L + it R , t − = t L − it R ,, then the differential takes the form The automorphism group of N = 2, D = 10 susy is SO(10) × GL(1). The variables t + , t − are SO(10) spinors having the weight +1 and −1 with respect to GL(1), the variables C m constitute a vector of weight 0.
The cohomology H t has three gradings: with respect to the number of ghosts (of Cvariables, and denoted by C), to the number of t + -variables (denoted by t + ) and to the number of t − -variables (denoted by t − ). Hence we could denote its components by H Sometimes instead of t + and t − gradings, it is convenient to use t-grading t = t + + t − and GL(1)-grading gl = t + − t − .
One can calculate it using the methods of [3]. Let us formulate the answers for the case when the number of ghosts is 0 (zero-dimensional cohomology).
Again using Macaulay2 one can find the dimensions and gradings of the components of the resolution of the zero-th cohomology considered as a module over the polynomial ring C[t + , t − ]. Using the LiE code we find the action of SO(10) on these components. We represent the resolution as an exact sequence .... . This group has an additional grading (the number of θ's); using the same considerations as in the proof of 6 we obtain the component with respect to this grading is given by the formula Comparing to the case of N = 1 supersymmetry mentioned in Section 2, we see that To prove 8 we use the fact that h decreases the t-grading by 1 and does not change the GL(1) grading gl = t + −t − +θ + −θ − (we are assuming that θ + , θ − have the GL(1)-gradings +1 and −1).
We where N is defined by Eq. 9, and Generally, a similar formula for higher cohomology group in term of resolutions of modules H p t (the p−th cohomology with respect to the differential t) has the form (H p,t,gl s ) k = µ p k,t+k,gl where µ p k,t,gl stands for k-th term in the resolution of H p,t,gl t .