On the construction of integrated vertex in the pure spinor formalism in curved background

We have previously described a way of describing the relation between unintegrated and integrated vertex operators in AdS5xS5 which uses the interpretation of the BRST cohomology as a Lie algebra cohomology and integrability properties of the AdS background. Here we clarify some details of that description, and develop a similar approach for an arbitrary curved background with nondegenerate RR bispinor. For an arbitrary curved background, the sigma-model is not integrable. However, we argue that a similar construction still works using an infinite-dimensional Lie algebroid.


Introduction
The construction of the worldsheet sigma-model for the Type II superstring in the pure spinor formalism is a fundamental problem. It was more or less solved in [1]. However, we feel that some better understanding is possible. First of all, the sigma-model suggested in [1] is technically very special, and it is not clear why this is the most general solution. In particular, the formulation depends crucially on a special choice of fields. Indeed, the theory is not invariant under field redefinitions mixing matter fields with ghosts. It would be desirable to have an axiomatic formulation of the sigma-model. Something along these lines: • A sigma-model with two nilpotent symmetries, Q L and Q R , such that the current of Q L is holomorphic and the current of Q R is antiholomorphic, and there are symmetries U (1) L and U (1) R , such that Q L and Q R are appropriately charged under them.
However, we feel that this is not enough; the axiomatics sketched above is probably too weak, although likely enough to correctly describe small deformations of the flat space. The correct axiomatics should somehow encode the singularity of the pure spinor cone. Also, we believe that the worldsheet sigma-model should be formulated as a problem in cohomological perturbation theory. A small neighborhood of each point in space-time can be approximated by flat space: (1) with BRST symmetries: Then we say that a general background is obtained by the deformation of the action accompanied by some deformation 1 of Q L and Q R . The infinitesimal deformations at the linearized level are well known to correspond to the linearized SUGRA waves. They were classified in [3]. However, it was shown also in [3] that there is a potential obstacle to extending the deformations beyond the linearized level. The obstacle is a nonzero cohomology group, namely the ghost number three vertex operators. Without doubt, the obstacle actually vanishes (there is a nonzero cohomology group, but the actual class vanishes). This, however, is not well understood. As we explained in [3], one way to prove the vanishing of the obstacle is to consider the action of the b-ghost in cohomology. The formalism that would allow to do such calculation has not yet been fully developed. The definition of the b-ghost requires including the non-minimal fields which makes the lack of axiomatic formulation even more acute. And the b-ghost is nonpolynomial, opening the possibility that at some order the deformed action will also become nonpolynomial 2 . Again, such questions should be addressed together with the axiomatics. When we study the pure spinor formalism as a cohomological perturbation theory, one important technical aspect is the relation between integrated and unintegrated vertex operators. The deformation of the action is described by integrated vertices: It is very important, that such deformations are in one-to-one correspondence with the unintegrated vertices, which correspond to the cohomology of Q L + Q R . One of the goals of this paper is to better understand the correspondence between integrated and unintegrated vertices.
In [4,5,6] we have studied the relation between the pure spinor cohomology in AdS 5 ×S 5 and the Lie algebra cohomology, and argued that it is useful for understanding the relation between the integrated and unintegrated vertices. The pure spinor cohomology is the cohomology of the operator Q BRST acting on the space of functions F (g, λ L , λ R ): (Q BRST F )(g, λ L , λ R ) = λ α L L α + λα R Lα F (g, λ L λ R ) (5) 1 As we have shown in [2], the very leading effect will be actually the deformation of Q L and Q R leaving S undeformed; this corresponds to the linear dilaton. 2 We have no doubt that this does not happen, it is just that we don't know how to see this using the cohomological perturbation theory Here g ∈ G = P SU (2, 2|4) and L α , Lα are left shifts by some generators of psu(2, 2|4). We introduced an infinite-dimensional Lie superalgebra L tot , and shown that the cohomology of Q BRST is equivalent to some cohomology of L tot . Unintegrated vertices of the physical states correspond to the elements of the second cohomology group. Moreover, there is a Lax pair J + , J − taking values in L tot . Given a cohomology class represented by a cocycle ψ : Λ 2 L tot → Fun(G), the corresponding integrated vertex is ψ(J + , J − ). This construction uses special properties of AdS 5 × S 5 .
Here we will describe a similar construction for an arbitrary curved spacetime with the nondegenerate Ramond-Ramond field strength 3 . Instead of the superalgebra L tot we will use some super Lie algebroid. We will conjecture that the cohomology of this algebroid is equal to the BRST cohomology, i.e. unintegrated vertex operators. Moreover, there seems to be an analogue of a Lax pair, which allows to construct integrated vertices. However, this Lax pair takes values in the sections of a Lie algebroid (instead of a fixed Lie algebra), and presumably does not lead to integrability.
Better understanding of the integrated vertices could also help to explain the consistency of the higher orders of the deformation of the action (1). Superficially, this problem looks similar to the PBW theorem of quadraticlinear algebras which (coincidentally?) is also useful in the construction of the integrated vertex.
In eleven dimensional SUGRA, the analogous problem is the membrane worldsheet theory [7]. However, it appears more difficult than string worldsheet theory. But unintegrated vertices are more or less understood. Constructing integrated vertex operators is very close to understanding the worldsheet theory. Maybe some methods which we are developing here could be useful.

Plan of the paper
• In Section 2 we give a streamlined review of [4,5,6], also simplifying some of the proofs in those references • Section 3 develops a different point of view on the formalism of [1]; our approach emphasizes the similarity between the constraints of the Type IIB SUGRA and the constraints of the supersymmetric Yang-Mills theory • In Section 4 we study the worldsheet currents. We construct an object resembling the Lax pair of the AdS theory, but using an algebroid instead of a Lie algebra. We conjecture that the cohomology of this algebroid corresponds to integrated vertex operators 2 Brief review of the case of AdS 5 × S 5

Definition of L tot and PBW
The infinite-dimensional superalgebra L tot is obtained by "gluing together" two copies of the Yang-Mills algebra which we call L L and L R , in the following way. The L L is generated by letters ∇ L α , and L R is generated by ∇ R α , satisfying the super-Yang-Mills constraints: (The existence of such A L and A R are the constraints.) All we need to do is to explain how ∇ L α anticommutes with ∇ R α . For that we add a copy of the finite-dimensional algebra g0 = so(1, 4) ⊕ so(5) with the generators denoted t 0 [mn] . We impose the commutation relations: where the coefficients f •• • are the structure constants of g = psu(2, 2|4). One can consider the Lie algebra generated by the letters with the above relations, or the associative algebra generated by them. The associative algebra is the same as the universal enveloping U L tot . The algebra U L tot is an example of a quadratic-linear algebra. It is a filtered algebra; F p U L tot consists of those elements which can be represented by the product of p letters. For example A L m ∈ F 2 U L tot . One can also define a homogeneous quadratic algebra qU L tot as an associative algebra generated by the letters ∇ L α , ∇ R α , t 0 [mn] with the relations The algebra qU L tot is graded; gr p U L tot consists of those elements which consist of p letters. Theorem 1 (PBW): Proof uses the fact that qU L tot is a Koszul quadratic algebra. We will give a proof following Section 3.6.8 of [8]. We will need some standard language, which we will now review. Let V be the vector space generated by the letters . Consider the subspace R ⊂ V ⊗ V generated by the following elements (the relations of qU L tot ): Notice that qU L tot can be defined as the factorspace of the tensor algebra (=free algebra) T V modulo the ideal generated by R. The dual coalgebra U L ¡ tot is defined as the following subspace of T V : The coalgebra structure is induced from the standard coalgebra structure of the tensor product: Explanation of notations: For any coalgebra C, the coproduct ∆ acts from C to C ⊗ C. In our case, it so happens that C is itself defined as a tensor product. In this case it is common to use the notation C|C instead of C ⊗ C, just to avoid confusion. The spaces C|C| · · · |C form the so-called cobar complex, because there is a natural differential: The nilpotence of this differential is equivalent to the co-associativity of ∆. This complex is denoted Ω(C). As a vector space Ω(C) is: It is naturally an algebra, just a tensor (=free) algebra over C: Also notice that d respects the multiplication: This means that Ω(C) is a differential algebra. Let us consider the cohomology of d.
Proof: This is obvious from the definitions. So far the definition of U L ¡ tot only used the homogeneous relations of qU L tot , it is really (qU L tot ) ¡ rather than U L ¡ tot . We have to somehow take into acount the nonzero right hand sides of (7), (8), (9), (10). This is done by supplying U L ¡ tot with a differential d 1 , which is defined as follows 4 : The verification of the nilpotence of d 1 is equivalent to the verification of the Jacobi identity of L tot in filtration ≤ 3. There are the following cases to verify: Eq. (32) is derived as follows. After first time applying d 1 we get: Since [Γ mn , Γ m 1 ...m 5 ] is a five-form, the second application of d 1 results in zero. Eq. (31) is derived as follows: where we have taken into account that f βα k = Γ βα k and therefore the contraction with Γ αβ m 1 ...m 5 is zero.
Proof: This is also obvious from the definitions. Notice that until now we have not done anything nontrivial, just developed a language. But now we are ready to proceed with the proof of the PBW theorem (11). Before the proof, we probably have to explain why the statement is nontrivial. Let us consider, for example, the following element of L tot : This expression can be represented by the following element of V |V |V ⊂ Ω(U L ¡ tot ): The question is, how do we know that this element is nonzero? Maybe one can prove that it is zero, using the relations of L tot ? We know however that it is nonzero as an element of L L . (We are not going to prove it now; in fact this particular expression corresponds to the field strength superfield.) The L L is a homogeneous quadratic algebra. We want to prove that X it is also nonzero as an element of L tot , an inhomogeneous (quadratic-linear) algebra. The danger is that maybe there is some element Y 0 , for example in R|V |V ⊂ Ω(U L ¡ tot ), such that dY 0 = 0 and d 1 Y 0 = X. This would imply that (d + d 1 )Y 0 = X and therefore X is actually zero as an element of L tot . Or, perhaps there are Y 0 ∈ V |V |V |R and Y 1 ∈ V |V |R such that d 1 Y 1 = X and dY 1 = −d 1 Y 0 and dY 0 = 0; then again (d + d 1 )(Y 0 + Y 1 ) = X and therefore X is zero. We deal with such fears in the following manner. Suppose X = (d + d 1 )Y and Y 0 be the highest bar-order term of Y (the term with the highest number of |). Then dY 0 = 0. Because qL tot is Koszul 5 , this implies that Y 0 = dZ 0 . We therefore have (d + d 1 )(Y − (d + d 1 )Z 0 ) = X, and the bar-order of Y − (d + d 1 )Z 0 is one less than the bar-order of Y . We repeat this until the bar-order of Y is equal to the bar-order of X. Now (d+d 1 )Y = X implies that the leading order term in X is zero in qU L tot . This contradicts the assumption and completes the proof of the PBW theorem. Theorem 2: as a linear space (42) 5 The Koszul property implies that the cohomology group corresponding to d-closed Y 0 modulo d-exact Y 0 vanishes, see Section 3.6.8 of [8] for details. The algebra of functions of ten-dimensional spinors satisfying (λΓ m λ) = 0 is Koszul by the results of [9]. The SYM algebras L L and L R are both Koszul as quadratic duals to the Koszul algebra of pure spinors. The algebra qU L tot is the commutative product of U L L , U L R and Λg0, and therefore is Koszul by the Corollary 1.2 in Chapter 3 of [10] (where the commutative product is denoted ⊗ q=1 ).
Proof The Lie superalgebra L tot can be considered a subspace of U L tot , consisting of those elements which can be represented as nested commutators. Then (42) follows from the PBW theorem.
Comment: A physical interpretation of qU L tot could be the flat space limit of U L tot .

BRST complex 2.2.1 The structure of the dual coalgebra
Besides the PBW theorem, the Koszulity also implies that the complex (U L tot ) ¡ ⊗ U L tot is acyclic. Notice that (U L tot ) ¡ has the following structure. As a linear space: Any element ω ∈ (U L tot ) ¡ can be presented as a linear sum of the expressions of the "decomposable" elements of the form: where F α 1 ···αpβ 1 ···βq [m 1 n 1 ]···[mrnr] is symmetric in α 1 , . . . , α p and inβ 1 , . . . ,β q and antisymmetric in [m 1 n 1 ], . . . , [m r n r ] and satisfies: and . . . in (44) stand for the terms which are obtained from the first term by permutations of the tensor product, which are needed so that the resulting expression belong to the exterior product of p + q + r copies of the linear superspace generated by ∇ L , ∇ R and t 0 . For example, when p = 2 and r = 1, we get: The action of d 1 on this ω is: We have seen that the complex: provides a free resolution of C as a U L tot -module. Therefore, for any representation V of U L tot , we can compute the cohomology group: as the cohomology of the complex We will now interpret this complex in terms of ghosts. An element of (U L tot ) ¡ is the sum of expressions of the form (44). Notice that the dual space . In this language the BRST operator becomes Notice that the ghosts corresponding to g0 are non-abelian while the ghosts c α L and cα R are pure spinors. We will therefore call (51) the "mixed complex": it is the pure spinor BRST complex coupled with the Serre-Hochschild complex of the finite-dimensional Lie algebra g0. In the Serre-Hochschild complex of L tot Similarly, the decoupling of the c 0 ghosts in the Serre-Hochschild complex of L tot leads to the relative cohomology: This establishes the relation between the BRST cohomology and the relative Lie algebra cohomology [4].

Cohomology of the ideal
Consider the ideal I ⊂ L tot such that: By the Shapiro's theorem: This helps to identify various supergravity field strengths [4].

Integrated vertex 2.3.1 Generalized Lax operator
Consider a classical string solution in AdS 5 × S 5 , i.e. a field configuration in the worldsheet sigma-model solving the classical equations of motion. It was shown in [5,6] that one can construct the Lax pair: (56) where J ± and λ, w are worldsheet fields and t 0 , ∇, A, W generators of L tot , satisfying the zero curvature equations: and having simple BRST transformation laws: We will denote J ± the connections in L ± : Let us denote: J = J + dτ + + J − dτ − -an L tot -valued one-form on the worldsheet. For the purpose of calculations, it is convenient to assume that dτ + and dτ − anticommute with the worldsheet fields θ: We also introduce arbitrarily many anticommuting parameters 1 , 2 , . . ., which anticommute among themselves, with θ, and with dτ ± . With these notations, we have: Schematically: Also: Given an n-cochain ψ ∈ C n (L tot , g0; (U g) ), let us consider an inhomogeneous form ev(ψ) on the worldsheet which can schematically be defined by the following formula: This schematic notation is deciphered as follows. Notice that ψ is a function of the type: We first evaluate it on 1 (J − λ) ⊗ 2 (J − λ) ⊗ . . ., which gives us a function of the type U g → C. We then evaluate it on a "group element"g ∈ U g. The "group elements" are defined as expressions of the form g = e ξ where ξ ∈ g.
Being infinite series, they strictly speaking do not belong to U g. This rises the question of convergence, which we will ignore. Then we observe: The derivation of this formula, besides (65) and (66), also uses the fact that ψ is a relative cocycle, and therefore: In our case ψ is a 2-cocycle. Therefore: This means that the ghost number two part of ev 1 , 2 (ψ) is an unintegrated vertex, and the ghost number zero part of ev 1 , 2 (ψ) is an integrated vertex. Therefore our construction provides one way of thinking about the relation between unintegrated and integrated vertices.

General curved superspace
The pure spinor description of the Type IIB SUGRA emphasizes the local Lorentz symmetry of the supergravity theory. More specifically, the Type IIB superstring combines left and right sectors, and there are two copies of the local Lorentz group.
We will now describe some structure on the superspace, which we call "SUGRA data". We first describe it as an abstract geometrical structure, and then explain how it emerges in the sigma-model.

Weyl superspace
The formulation of the pure spinor sigma-model in [1] uses the so-called Weyl superspace. In this formalism, besides the local Lorentz symmetry, there are also two copies of the local dilatation symmetries: Let S L ⊕ S R denote the spinor representation ofĥ. We will also denoteĤ the Lie group corresponding toĥ. To summarize: We will now start describing the SUGRA data.
The first part of the SUGRA data is: • for every point x ∈ M , an orbit of the action ofĤ on some linear map notice that the map D itself does not enter into the SUGRA data, only its orbit (with the action ofĤ on it) Let M π − → M be the principal bundle over M whose fiber over a point x ∈ M is that orbit. In other words, a point of M is a pair (x, D). Let π denote the natural projection: More explicitly, any linear map D is of the form: Sometimes we will simply write E α and Eα instead of The second part of the SUGRA data is a map satisfying the following properties: • D commutes with the action ofĤ • D is "fixed modulo Vect M /M " in the following sense: for any point (in other words, only the vertical component of D is non-obvious; the projection to T M is tautological) • SUGRA constraints: where: -A L m and A R m are some sections of T M and -R LL αβ , R RṘ αβ and R LR αβ some sections of T M /M (i.e. vertical vector fields); they are essentially "curvatures" Notice that satisfying the SUGRA constraints does depend on the vertical component of D.

Moreover:
• there is an equivalence relation, which we will describe in Section 3.2.4

SUGRA constraints, oversimplified
Let M be the super-space-time. In supergravity, M comes equipped with the distribution S ⊂ T M . The SUGRA constraints are conditions on the Frobenius form of S, which go roughly speaking as follows. We choose some vector fields ∇ α , α ∈ {1, . . . , dimS} tangent to S and say that: where A m are some other vector fields. (The point of the constraint being that the RHS is proportional to Γ m αβ .) It is important to remember that when we write such conditions, we need to fix a basis of S, i.e. a set of ∇ α . If we choose some linear combination: then, generally speaking, ∇ α will not satisfy the constraint (84). If we want ∇ α to satisfy the constraint, we should require that X ∈ so(1, 9) ⊕ Ran antisymmetric matrix plus a scalar. This means that S actually comes with an additional structure, namely an orbit of the action of SO(1, 9) × R × on some linear map D : S → S where S is the spinor representation of so(1, 9) ⊕ R. As we said in Section 3.1, the map D itself does not enter into the SUGRA data, only its orbit (with the action of SO(1, 9) × R × on it). Given a point x ∈ M , and an orbit of SO(1, 9) × R × in S(x), we can choose a point D in this orbit, then choose any set of vector fields ∇ α such that ∇ α (x) = D α , and verify Eq. (84).
This means that it is useful instead of M to consider M , which is the SO(1, 9) × R × -bundle over M whose fiber over x ∈ M is that SO(1, 9) × R ×orbit in Hom C (S, S(x)) which we should have received as part of our SUGRA data. It is natural to think that the matter fields live in M rather than M , except that the fiber is a gauge degree of freedom. The fiber can be gauged away because, as we said, the map D ∈ Hom C (S, S(x)) itself does not enter into the SUGRA data, only its orbit. This is how the AdS 5 × S 5 sigmamodel is formulated [11]. In that case M is P SU (2, 2|4)/(SO(1, 4) × SO (5)) and M is P SU (2, 2|4). The sigma model has the SO(1, 4) × SO(5) gauge symmetry which gauges away the fiber. It is SO(1, 4) × SO(5) rather than SO(1, 9) × R × because in that particular case some of the gauge symmetry can be canonically fixed.
In the sigma-model we couple matter fields with the ghosts λ which belong to the pure spinor cone C ⊂ S. As D ∈ Hom C (S, S) can be thought of as linear functions from S to S, it make sense to apply it to λ ∈ S. The resulting vector field D(λ) describes the action of the BRST operator on the matter fields:

Sigma-model
The target space of the sigma-model is M , but as we explained there is a gauge symmetry which reduces M → M . The action, copied from [11], is: In a generic background, one can integrate out d,d and get a simpler-looking action. It is postulated that the field d should be the same as the density of the BRST charge. This is, essentially, a restriction on the choice of fields.
Notice that the form of the Lagrangian (87)

From pure spinor Q to SUGRA constraints
We just said that the target space of the sigma-model is M . This, however, is not the full truth, because there are also ghosts. With ghosts, the target space is a cone in the associated vector bundle of the principal bundle M corresponding to the spinor representation ofĤ: where C L is the pure spinor cone in S L and C R the pure spinor cone in S R . The BRST operator of the sigma-model is a nilpotent odd vector field: Generally speaking, consider a coset space X/G, where the action of G on X is free and transitive. Then vector fields on X/G can be described as follows. Let us start by considering the subalgebra A ⊂ Vect(X) which consists of those vector fields which are invariant under the action of G, i.e. g * v = v for any g ∈ G (a.k.a "Atiyah algebroid"). One can check that the vertical vector fields (those which are tangent to the orbits of G) are an ideal I ⊂ A. The factoralgebra is isomorphic to the algebra of vector fields on X/G: Let us see how this description works in the particular case: Let us fix some lift Consider Lift * Q -a vector field on the image of Lift. Notice that (Lift * Q) 2 = 0, but don't forget that Lift * Q is not a vector field on the whole M × (C L × C R ), but only on a submanifold -the image of Lift. However, we can define anĤ-invariant vector field on the whole M × (C L × C R ) using the fact that M ×(C L ×C R ) is foliated by the translations of the image of Lift by elements ofĤ: This means that we can extend Lift * Q to the whole M × (C L × C R ) in an h-invariant matter, simply by translating. In other words, let Q ↑ be the nilpotent vector field on M × (C L × C R ) such that: A different choice of Lift will result in another Q ↑ , but the difference in Q ↑ will be in adding a vertical vector field, i.e. and element of I. This is precisely (90). Notice that the vertical component of Q ↑ isĤ-invariant. Moreover, C L × C R is an orbit of the action ofĤ on S L ×S R . In other words, any pair of pure spinors (λ L , λ R ) can be obtained from a fixed pair (λ (0) L , λ (0) R ) by the action of some element h ∈Ĥ. Therefore exists a vertical vector field ω such that the following vector field on M × (C L × C R ): acts trivially on C L × C R . In other words, Q is a vector field on M .
To clarify the construction, let us describe it in coordinates. A point of M ×(C L ×C R ) is described in coordinates as follows: (Z, (E L α ), (E R α ), λ L , λ R ). A point of M ×Ĥ (C L × C R ) is described in the same way but with the equivalence relation: Our Lift is essentially gauge fixing. It is described by specifying the functions: The way it works, for every point in M ×Ĥ(C L ×C R ), to calculate its lift we use the equivalence relations (96) to bring its coordinates (Z, (E L α ), (E R α ), λ L , λ R ) to the form satisfying (97). The resulting (Z, (E L0 α (Z)), (E R0 α (Z)), λ new L , λ new R ) specifies a point in M × (C L × C R ), which is the lift. The lift of the BRST field Lift * Q is of the form: We must stress that Lift * Q is only defined on the image of Lift. To extend this vector field to the whole M × (C L × C R ), we must relax the gauge fixing (97). We observe that any E L α and E R α can be presented in the form: Let us use (Z, g L , g R , λ L , λ R ) as coordinates on M × (C L × C R ). Then we have: -a vector field on M . Notice that Q L depends linearly on λ L . Therefore, Q defines sixteen vector fields D L α : These are the vector fields which were postulated in Section 3.1.
Ambiguity However, the definition of Q L , and therefore of D L α , contains an ambiguity. It is possible to add to Q L a vertical vector field: such that λ α L ω L α ∈ St(λ L ) ⊂ĥ L . This corresponds to the "shift gauge transformations" of [1]. We will now describe such ω α .

Shift gauge transformations
Let us modify D L α by adding to it a vector field in T M /M (i.e. tangent to the fiber) of the form (cf. Eq. (61) of [1]): The characteristic property of such ω L α is that λ α L ω L α ∈ St(λ L ) ⊂ĥ L ; in other words: Similarly, we can modify D R α by adding to it some ω R α defined in a similar way; we stress that ω L takes values inĥ L and ω R takes values inĥ R . Obviously, these "shift transformations" depend on two parameters: h αn L and hα n R . In terms of Section 3.1 this modifies D:

SUGRA fields
The action (87) involves various SUGRA fields, which are either sections of associated vector bundles over M , or connections on them. They enter the action through their pullback on the string worldsheet.
Sections For example, P αα is a section of M ×Ĥ (S L ⊗ S R ) π → M . Such sections can be interpreted asĤ-invariant maps 6 M → S L ⊗ S R . From this point of view we consider P αα as a function P αα (Z, (E L β ), (E R β )) such that: Connections Connections are needed to define the kinetic terms for the ghost fields. A connection on the associated vector bundle M ×Ĥ (S L ⊗S R ) π → M is constructed from a connection on the principal bundle M π → M . We will now remind how this works. For any vector field ξ ∈ Vect(M ), a connection in the principal bundle defines a lift ξ ∈ Vect( M ), which isĤ-invariant in the sense that for any χ ∈ĥ the corresponding vector field v(χ) commutes with ξ : [v(χ), ξ ] = 0 for any χ ∈ĥ where L is the Lie derivative. Eq. (108) implies that for any σ satisfying (109), L ξ σ also satisfies (109). This means that the lift ξ → ξ consistently defines the action of ξ on the sections of the associated vector bundle. Let us explain how a connection in the principal bundle M → M defines a kinetic term for the ghosts. Consider the ghost λ L ; in the flat space limit it is a left-moving field. In the general curved space, the kinetic term for λ L should involve the derivative ∂ − λ L . A point of the target space is (Z, (E L α ), (E R α ), λ L , λ R ). The worldsheet is foliated by the characteristics. Let us consider the right-moving characteristic τ + = const. It is parametrized by the τ − : Let us choose a representative so that dZ(τ − ) dτ − ) is a horizonthal vector, in the sense defined by the principal bundle connection in M . Then the kinetic term is: where w L + is the conjugate momentum to λ L .

Lorentz superspace
There is a way to canonically fix R × L × R × R . In this paper we will use the variant of the formalism which has R × L × R × R fixed. For us the gauge algebra is: This version of the formalism is called "Lorentz superspace". We will now review how the Lorentz superspace is derived, as much as we understand.
Consider the sigma-model (87) and let us integrate out d α anddβ. It turns out that it is always possible to choose the gauge so that the coupling to the ghosts is only through the traceless currents 7 : The u(1) combinations (w L + λ L ) and (w R − λ R ) appear only in the kinetic terms (w L + ∂ − λ L ) and (w R − ∂ + λ R ). This fixes the gauge fromĥ L ⊕ĥ R to h L ⊕ h R . In the language of the present paper this simply means that we can use a slightly simpler M . A point of this simplified M is a point x ∈ M and a point in the orbit of H in S(x) ⊂ T x M ; the simplification is in replacing the orbit ofĤ = Spin (1,9) (1,9) R . As in Section 3.2.3, we can still trade the BRST operator for an H-invariant vector field on M . This statement is somewhat nontrivial, because what if the BRST operator Q involves a rescaling of λ L and λ R ? Let us consider the action of Q R on λ L : In particular, the Q R variation of the kinetic term w L α+ ∂ − λ α L gives the term w L α+ Xα α β λ β L ∂ − λα R which has nothing to cancel unless if Xα is traceless, i.e. if Xα α α = 0. (In this case it is cancelled by the variation of the connection on which w L α+ ∂ − λ α L depends, implicitly in our language.) Now consider the action of Q L on λ L : Now it is even meaningless to ask if X α is traceless or not, because X α is only defined by (115) up to a shift transformation of Section 3.2.4. We therefore use these shift transformations to remove the trace of X α . Then we have to remember that when we work in the Lorentz superspace formalism, the shift transformations have their parameter restricted to:

Worldsheet currents, quadratic-linear algebroid
Let us consider the algebroid A over M freely generated by D satisfying Eq. We introduce a bidirectional filtration on A in the following sense. For n > 0, we will say that ξ ∈ A ≤n if ξ can be represented as a nested supercommutator of ≤ n generators D L α . For n < 0, we will say that ξ ∈ A ≥n if ξ can be represented as a nested supercommutator of ≤ |n| generators D R α . Notice that the expression containing nested supercommutators of both D L α and D R α can be reduced to expressions containing either all D L α or all D R α .

Basic consequences of the defining relations
We observe that the basic commutation relations of (83) imply the existence of W α L such that: Furthermore, notice the existence of F [mn] such that: Indeed: and

Worldsheet currents
Remember that the string worldsheet is spanned by the left-moving characteristics τ − = const. Consider an observer moving along a characteristic with the constant velocityτ + = 1. The velocity vector can be decomposed via the worldsheet currents: Here we used the abbreviation: where t 0 [mn] are generators of h L . Notice that the "currents" J

L[mn] 0+
, I L 0+ , J α + , Π m + , ψ α+ are local functions on the phase space. We will denote the space of such functions Loc(X ): X = phase space Loc(X ) = the space of local functions on X At the same time, a(t L0 ), a(D L α ), a(A L m ) and a(W α L ) are vector fields on M . Notice that a function f (Z) on M and a point (τ + , τ − ) on the worldsheet define a function on X , namely f (Z(τ + , τ − )). In this sense, we should think of ∂ + Z as an element of the space: This is not an algebroid over X , because genarally speaking there is no way to lift a vector field on M to a vector field on the phase space. But this is possible if the vector field generates a symmetry of the sigma-model. When two elements X ∈ V and Y ∈ V both correspond to some symmetry of the sigma-model, then it is possible to define the commutator [X, Y ]. Another way of turning V into an algebroid is to go off-shell, i.e. replace the X with the space of off-shell configurations X OS .

Tautological Lax pair
Consider the sigma model of the classical string in AdS 5 ×S 5 . It is classically integrable. There is a Lax pair, which depends on the spectral parameter z.
At some particular value of z, the Lax pair becomes tautological, the zero curvature equations being just the Maurer-Cartan equation for the worldsheet currents. We will now briefly review how this goes. The current is J = −dgg −1 . For any representation of g with generators t a , it is straightforward to verify the Maurer-Cartan equation: We will need a slight variation of this construction. Letg be the Lie superalgebra obtained from g by changing the sign of the anticommutators (all the commutators are the same, but all the anticommutators have the opposite sign). The left regular representation ofg on the space of functions on the group manifold of G is defined as follows: This means that: We get 8 : Eq. (127) is almost the particular case of (124) corresponding to the left regular representation. The only difference is that the left regular representation, as we defined it, is the representation ofg and not g. But at the same time, notice that the odd-odd terms in L(J + ) are of the form: . . =t α is the corresponding generator ofg, let us call itt α . Notice thatt α anticommutes with J α + , while in (124) by definition J a commute with t a . In spite of this subtlety, the two definitions are actually equivalent. Given a Lax pair in the sense of (127), let us replace every term of the form J αt α with J αt α (−) F . Notice thatt α (−) F are the generators of some representation of g (which should also be called "left regular"), and also thatt α (−) F commutes with J α . Therefore we obtained the Lax pair in the sense of (124).
We can interpret the operator ∂ ∂τ ± + L(J ± ) in the following way. Consider the space X OS of all field configurations (off-shell) in the classical sigmamodel. Let Loc(X OS ) denote the space of all local functions on X OS . Let us consider the space: This is, obviously, the same as Loc(X OS ). Let us, however, define the action of the Lax operator on this space, as follows: ∂ ∂τ ± acts only on Loc(X OS ) and L(J ± ) acts only on Fun( M ). Our point here is that this action is correctly defined. For example, the action of ∂ ∂τ + + αL(J + ) with α = 1 would not be correctly defined on (128), because it would act differently on f ⊗ φ and 1 ⊗ f φ.

General case
Consider the velocity of the coordinate function Z M pulled back on the string worldsheet: We write Z M instead of simply Z M , to stress that the coordinates include also the fiber. In the AdS 5 ×S 5 language, Z M would parametrize P SU (2, 2|4) rather than AdS. The terms J Then ∂ ∂τ + , ∂ ∂τ − Z M = 0 leads to the tautological zero curvature equation: In this formula ∂ ∂τ + in the first line and ∂ ∂τ − in the second line only act on the currents J, Π, ψ and do not act on a(D), a(A), a(W). The commutator is the commutator of the vector fields, e.g.: Consequences of [Q L , ∂ + ] = 0 Let us consider the BRST variation: We have two vector fields on the phase space, Q L and ∂ ∂τ + . They commute: In particular, we can say something about Q L ψ α+ . Let us define the superfield C α βγ (Z) by the following formula: where mod A [0,2] stands for a linear combination of a(D L α ) , a(A L m ), a(t L [mn] ) and a(t R [mn] ). From (133) we read: 2] (135) and therefore: Similarly we have: The nilpotence of Q L implies: Happily, the only part of Q L Π m + which gives a nonvanishing contribution is proportional to ψ α+ ; let us extract its coefficient:

Identification of ψ α+
We will now show that ψ α+ can be identified as the matter part of the BRST charge density. Remember that w L + is the momentum conjugate to λ L -see Eq. (87). Let us define ψ α+ and d α+ as follows: It follows: Here " mod ( ) m+ Γ m αγ λ γ L " means "up to adding u m+ Γ m αγ λ γ L with some arbitrary u m+ ". Notice that λ α L d α+ is the left BRST current. This follows from the fundamental property of the formalism: the U (1) L charge of the left BRST current is +1. We have: It follows from Eq. (139) that: (144) Therefore: Proof: Comparing Eqs. (145) and (142) we get: (the same equation as (136)). It follows from the analysis of Eq. (147) in the flat space limit that any ζ α+ satisfying (147) is of the form: where φ = φ(Z) and B β γα = B β γα (Z) are some functions. Indeed, in the flat space limit, in the neighborhood of any point of M , if θ and λ scale as R −1/2 and x as R −1 and w ± as R −3/2 , then ζ α+ should be of the order R −3/2 ; this means that the coefficients of J + and Π + in ζ α+ are zero. The leading term in the flat space expansion of ζ α+ is then of the form φ β α d β+ , and its BRST variation is in the leading order (Q L φ β α )d β+ + φ β α Γ m βγ Π m + λ γ L . Therefore the vanishing of the leading term in Eq. (147) up to ( ) m+ Γ m α• λ • implies that φ β α is proportional to δ β α .
Notice that Eq. (147) is satisfied when φ = const and B β γα = 0. When φ is not constant, the vanishing of the coefficient of d + in (147) implies: and the vanishing of the coefficient of w + λ L λ L implies: This is equivalent to the following equation being satisfied for any pure spinor Substitution of (149) gives: This implies that either C β •• λ • L λ • L = 0 for any λ L , which is generally speaking not the case, or λ • L D L • φ = 0, which implies implies that φ = const. In the case of AdS 5 × S 5 we know ζ α+ is zero. Therefore φ = 0 and Eq. (146) follows.
This means that in terms of the sigma-model (87):

Identification of C β αγ
Let us consider the SUGRA superfields C βα γ and P αα defined in Eq. (87). (Notice that we use the same letter C as for C β αγ , but with a different set of indices; we hope this will not lead to confusion.) Eq. (154) implies that: In particular, this implies that in the Lorentz superspace formalism (112): (This is not stated in [1].) One difference of our approach with [1] is that we do not require that T α βγ = 0. In fact, it is difficult to define T α βγ in our language.
We will now confirm this by comparing the "shift" gauge transformations defined in Eq. (61) of [1]. They correspond to the following variation of ∇ L α : where h αn is a gauge parameter. In the Lorentz superspace formalism (112) the shift parameter satisfies: Let us determine the transformation of ψ α+ and C α βγ .
In other words: where we used the gamma-matrix identity: (167) and therefore: This implies: At the same time: where the term −4h k• Γ k •β W α L corresponds to the trace part of ω. Therefore: This implies that 9 : Therefore: Given (159), this is in agreement with (155).

Weighing anchor
Consider a linear map κ: such that: Notice that the following operator: is the projection to ker(a) along A [0,3] . Let us unapply the anchor from the RHS of (121): Notice that: Indeed, let us for example look at the λ L λ L part: This implies that the calculation of the action of Q 2 L on L + does not lead out of A [0,3] . Therefore the calculation is the same as it would be under the anchor, and the result is zero.
Also notice that: However it is not true that (Q L + Q R ) L + = 0; we will therefore correct L + by adding to it some expression with zero anchor.

General theory
Let us consider deforming: where ∆L + does not contain the derivative ∂ + and is an anchorless element of A such that: We also require that ∆L + be h-invariant. Let us denote Y + the linear space of all expressions X + ∈ A satisfying the following properties: 1. X + is h-invariant 2. X + has conformal dimension (1, 0) 3. (Q L + Q R ) 2 X + = 0 By definition ∆L + belongs to Y + .

Lemma 3:
The cohomology of the operator Q L + Q R acting in Y + is zero.
Proof: Let us prove that the cohomology of Q L is zero. First of all let us prove this statement in flat space. In flat space the algebroid is homogeneous, it is defined by the same relations as qU L tot . Given an expression annihilated by Q L , let us consider the term with the lowest number of the letters ∇. It is Q L -closed. Since the cohomology of Q L in the expressions of the conformal dimension (1, 0) is trivial, this means that this lowest order term is exact. This completes the proof that the cohomology of Q L is zero in flat space. In a general curved space, let us use the near-flat-space expansion (see [2] for details). For an element φ ∈ A let us define its degree deg(φ) so that deg(θ) = deg(λ) = 1, deg(w) = 3, deg(x) = 2 and deg(D L ) = deg(D R ) = −1. The proof follows from the following observations: • the action in the associated graded space is the same as in flat space, • we have just proven that the cohomology in flat space is zero. Lemma 3 implies the existence of such a Y + ∈ Y + that: We therefore denote:

Explicit construction
(If the reader is not familiar with the construction of the Lax operator for AdS 5 × S 5 [12] we would recommend to first look at [13].) We will now show that the leading term of Y + is in degree four, i.e. L + is of the form: where P αβ γδ is the projector on the zero-form plus two-form. In other words, where a ⊥ is defined in (179). We have to verify that Y + satisfies (186). First of all, notice that ψ α+ in Eq. (188) is by its definition 10 the same as ψ α+ defined in Eq. (140). This implies that Q L L + falls into A [0,3] . But at the same time, the anchor of Q L L + is zero. This implies that Q L L + = 0.
Lemma 4: Proof Unfortunately we did not manage to prove it directly, but we have an indirect argument. Consider the action of Q R on L + . Let us look at the leading term (which is in A 4 ): The direct examination of the action shows that Q R w L α+ = 0. (If Q R w L α+ were nonzero, the variation of the kinetic term w + ∂ − λ L would result in the term with the structure λ R w L + ∂ − λ L which would have nothing to cancel with.) Therefore Q R L + falls into A [−1,3] . Since the anchor is automatically zero, it remains to prove that Q R L + actually falls into A [0,3] . Let us look at the component of Q R L + in grading −1. It is of the form: We know that Q L Q R L + = 0. This implies that Q L φα + = 0. We conclude that φα + has conformal dimension (1, 0), ghost number (0, 1) and is Q L -closed. But there are not such operators, therefore φα + = 0. Unfortunately we did not manage to prove it directly, but we have an indirect argument. We know that [L + , L − ] is a dimension (1, 1) operator with components in A [−4,4] , annihilated by both Q L and Q R . Let us consider the highest component: We know that X is of the conformal dimension (1, 1) and ghost number zero. It follows that Q R X = 0. But there are no operators with such properties (as can be seen from the flat space limit). Therefore the components of [L + , L − ] span A −4,3 . Then we can consider [L + , L − ] mod A ≤2 and so on.

Relation between integrated and unintegrated vertex
There must be some analogue of the Koszul duality for algebroids, which should imply that the cohomology of the BRST operator λ α L a(D L α )+λα R a(D R α ) is equivalent to the Lie algebroid cohomology of A. Let us define J + and J − from the Lax pair: Then, given a 2-cocycle ψ representing the Lie algebroid cohomology, we can construct the corresponding integrated vertex as in [6]: Moreover, the Koszul duality must also imply the consistency of the definition of the algebroid A (PBW).
We leave the details for future work.
Is A an overkill? Notice that J ± only requires a small part of the A; indeed, J + belongs to A [0,4] and J − belongs to A [−4,0] . This suggests that our definition of A is quite an overkill.