Normal form of nilpotent vector field near the tip of the pure spinor cone

Pure spinor formalism implies that supergravity equations in space-time are equivalent to the requirement that the worldsheet sigma-model satisfies certain properties. Here we point out that one of these properties has a particularly transparent geometrical interpretation. Namely, there exists an odd nilpotent vector field on some singular supermanifold, naturally associated to space-time. All supergravity fields are encoded in this vector field, as coefficients in its normal form. The nilpotence implies, modulo some zero modes, that they satisfy the SUGRA equations of motion.

1 Introduction In the low energy limit of superstring theory, spacetime fields satisfy supergravity (SUGRA) equations of motion, which are super-analogues of the Einstein equations. It is one of the main principles of string theory, that these target space equations of motion are equivalent to the BRST invariance of the string worldsheet theory. When they are satisfied, the space of fields is an infinite-dimensional Q-manifold (a manifold with an odd nilpotent vector field [1]). But in the case of pure spinor string, the sigma-model also defines a finite-dimensional Q-manifold. Indeed, the action of the BRST operator on matter fields and pure spinor ghosts does not contain worldsheet derivatives. (The worldsheet derivatives will appear when we consider the action on the conjugate momenta to matter fields and pure spinor ghosts, but they can be considered separately.) This means that, if we think of the pure spinor ghosts as part of target space, the BRST operator defines on the target space an odd nilpotent vector field, which we denote Q. In other words, the target space of the pure spinor sigma-model (a finite-dimenisional supermanifold) is a Q-manifold. Moreover, in generic space-time (for example in AdS 5 × S 5 , but not in flat space-time) the energy-momentum tensor and the b-ghost can also be interpreted as symmetric tensors on the target space (see [2]). How to classify a generic odd nilpotent vector field Q? A vector field can usually be "simplified" by a clever choice of coordinates. This is called "normal form". If a vector field is non-vanishing, one can choose coordinates so that the it is ∂ ∂θ where θ is one of fermionic coordinates. If Q vanishes at some point, then the normal form would be (in the notations of [1]) η a ∂ ∂x a . But in out case, the target space is not a smooth supermanifold, because pure spinor ghosts live on a cone. The vector Q vanishes precisely at the singular locus, and the problem of classification of normal forms is a nontrivial cohomological computation. This is what we will do in this paper. We will find that the space of equivalence classes of odd nilpotent vector fields in a vicinity of the singular locus is equivalent to the space of the classical SUGRA solutions. This is true modulo some "zero modes" -a finite-dimensional subspaces of soultions (see [6]) which we ignore in this paper.
Some details of our computations can be found in the HTML version of this paper.

Definition of M
The particular singularity which we are interested in can be described as follows. Consider the space M with bosonic coordinates x m (m running from 1 to 10) and λ α L , λα R (α andα both running from 1 to 16), and fermionic θ α L and θα R , with the constraint: where Γ m are ten-dimensional gamma-matrices. These constraints are called "pure spinor constraints". We understand Eqs. (1) as specifying the singular locus in M , from the point of view of differential geometry. All we need from these equations is to know how M deviates from being smooth. The singular locus is the tip of the cone (1): Pure spinor constraints (1) are invariant under the action of the group The diagonal is called "ghost number symmetry". Infinitesimal ghost number symmetry is generated by λ α Consider an odd vector field Q satisfying the following properties: • Q has ghost number 1, i.e.: • Q 2 = 0 • Q is "smooth" in the sense that it can be obtained as a restriction to the cone (1) of a smooth (but not nilpotent) vector field in the space parametrized by unconstrained x, θ, λ We want to classify such vector fields modulo coordinate transformations. Coordinate transformations are supermaps (x, λ, θ) → (x,λ,θ) such thatλ satisfy the same constraints (1). Such a vector field is one of the geometrical structures associated to the pure spinor superstring worldsheet theory [3], [4]. In particular, flat background (empty ten-dimensional spacetime) corresponds to Q = Q flat : String worldsheet theory also has, besides Q, some other structures which are less geometrically transparent (various couplings in the string worldsheet sigma-model). All these structures should satisfy certain consistency conditions.
• Question: is it true, that just a nilpotent vector field Q already includes, as various coefficients in its normal form, all the supergravity fields, and the supergravity equations of motion are automatically satisfied (i.e. follow from Q 2 = 0)?
This may be false in two ways. First, it could be that some supergravity fields do not enter as coefficients in the normal form of Q (i.e. they would only appear as some couplings in the sigmamodel, but would not enter in Q). Second, it could be that just Q 2 = 0 would not be enough to impose SUGRA equations of motion (i.e. one would have to also require the Q-invariance of the worldsheet sigma-model action).

Our results
In this paper we will derive the normal form of Q as a deformation of Q flat : Our analysis will be restricted to the terms linear in ǫ (i.e. Q 1 ). It turns out that Q 1 is parameterized by some tensor fields satisfying certain hyperbolic partial differential equations. These fields are in one-to-one correspondence with the fields of the Type II SUGRA, and our hyperbolic equations are the equations of motion of the linearized Type II SUGRA. It is useful to compare to the pure spinor description of the super-Yang-Mills equations. The super-Yang-Mills equations are equivalent to having an odd nilpotent operator: where t a are generators of the gauge group, and A a α (x, θ) is vector potential. Zero solution corresponds to A α = 0. In this sense, the SYM solutions can be considered as deformations of the differential operator: where the leading symbol (i.e. the derivatives) remains undeformed. Here we consider, instead, the deformations of the leading symbol.

Relation to partial G-structures
The variables λ L and λ R parametrize the normal direction to the singularity locus Z ⊂ M : The first infinitesimal neighborhood is a bundle over Z with the fiber C L × C R -the product of two cones. Filling the cones, we obtain a vector bundle over Z with the fiber V = C 32 . The vector field Q is power series in λ L , λ R , with zero at the tip of C L × C R . The derivative of Q at the zero locus defines a linear map: This map is not an isomorphism, since the image of Q * only covers a (0|32)-dimensional subbundle of T Z. We can interpret M as (C L × C R ) × G Z where Z is a partial frame bundle of Z and G is given by Eq. (3). It was shown in [5] that Q defines a connection in a partial G-structure on Z with some constraints on torsion, modulo some equivalence relation. The relation to previous work on SUGRA constraints [7], [8], [9], [10], [11], [12], [13], [14], [15] can be established along these lines.

Divergence of a nilpotent vector field
Let us fix some volume form vol, an integral form on M . Then we can consider, for any vector field V , its divergence div V . This is by definition the Lie derivative of vol along V : In particular, for a nilpotent odd vector field Q, we can consider the cohomology class: This cohomology class does not depend on the choice of the volume form vol. The cohomology class [div Q] plays two importan roles in our approach. First, they allow to reduce the study of Q to the first infinitesimal neighborhood of the singularity locus given by Eq.
(2). Second, it allows to prove that there is no obstacle to extending the linearized deformations (the term ǫQ 1 in Eq. (7)) to higher orders in ǫ. We will now explain this.

Requiring div Q = 0
We required that Q has ghost number one, see Eq. (5). But the "ghost number" λ α is coordinate-dependent. It is not invariant under a change of coordinates: In fact, we can relax Eq. (5) by replacing it with the following two requirements: Indeed Eq. (15) implies that Q is a vector field of ghost number one plus vector fields of ghost numbers two and higher. Although we have not computed H >1 Q flat , , the results of [6] suggest that div : is an isomorphism, and H >2 Q flat , is zero modulo finite-dimensional spaces. Then, Eq. (16) implies that the terms of the ghost number higher than one in Q can be removed by the coordinate redefinition. In other words, the normal form of Q can be always choosen to be of the ghost nuber one. It is enough to study the first infinitesimal neighborhood of the singularity locus.

Vanishing of obstacles to nonlinear solution
It is necessary to extend this analysis to full nonlinear SUGRA equations, i.e. higher order terms in Eq. (7). The potential obstacle to extending linearized solutions to the solution of the nonlinear equation Q 2 = 0 lies in H 2 Q flat , . We will not compute H 2 Q flat , in this paper, but the results of [6] suggest that H 2 Q flat , is actually nonzero. (We would expect it to be roughly isomorphic to H 1 Q flat , which we compute here.) But we also know that the actual obstacle is zero, because of the consistency of the nonlinear supergravity equations of [3]. This means that {Q 1 , Q 1 } must be a coboundary. In our language, this can be proven in the following way. Let us choose vol so that the divergence of Q flat is zero. The divergence of Q 1 , and therefore of {Q 1 , Q 1 }, is Q 0 -exact (this statement does not depend on the choice of vol). This is because div Q 1 has ghost number 1. The cohomology at ghost number 1 is finite-dimensional, and in fact those Q 1 with nonzero div Q 1 mod Q 0 (. . .) are non-physical (see [6] and references there). At the same time, the divergence of the elements of H 2 Q flat , is nonzero. Therefore the obstacle actually vanishes.

Notations
To avoid the discussion of reality conditions, we consider complex vector fields. The notation: means the space of all linear combinations of vectors v 1 , v 2 , . . . with complex coefficients. We introduce the abbreviated notations: Setup for cohomological perturbation theory 3.1 Definition of θ α L and θα R We define odd coordinates θ so that:

Flat Q and expansion around it
Flat spacetime corresonds to Let us consider Q as a small deformation of Q flat : to the first order in ǫ. Such deformations form a linear space. They correspond to odd vector fields Q 1 satisfying: modulo the equivalence relation, corresponding to the action of diffeomorphisms: where R is a ghost number zero vector field on M . Therefore, the classification of nilpotent vector fields of the form (20) is equivalent to the computation of the cohomology of the operator [Q flat , ] on the space of vector fields.
In the rest of this paper we will compute the cohomology of [Q flat , ] on the space of vector fields.

Spectral sequence
The grading operator:

First page
The first page of this spectral sequence is the cohomology of: on the space of vector fields on M . For a set of coordinates x, y, . . . we denote Fun(x, y, . . .) the space of functions of x, y, . . . and Vect(x, y, . . .) the space of vector fields (i.e. differentiations of Fun(x, y, . . .)). Let us introduce the following complexes: R decomposes as follows: (We do not need to take care about the completions of the tensor products, since all our functions are polynomials in θ and λ.) The cohomology of C fun L and C fun R is well known, see e.g. the review part of [16]: Parts of the cohomology of C vect L and C vect R which are relevant to this work will be computed in Section 4.
4 Cohomology of Q (0) in the space of vector fields 4

.1 Notations
Let X denote the singular supermanifold parametrized by bosonic λ α and fermionic θ α satisfying the pure spinor constraint: (The space M introduced in Section 1.1 is the direct product of two copies of X, and the space parametrized by x m .) Let O(X) denote the algebra of polynomial functions on X, and V ect(X) = Der(O(X)) the space of polynomial vector fields. Consider the odd nilpotent vector field Q (0) : The commutation [Q (0) , −] is a nilpotent operator on V ect(X). We will now compute the cohomology of this operator.
Any vector field V ∈ V ec(X) can be written as The condition (λγ m ) α ξ α = 0 is needed because λ α is constrained to satisfy Eq. (28).
Consider the subsheaf U ⊂ T X consisting of vectors of the form u α ∂ ∂θ α (in other words, ξ α = 0). Its space of sections is: We observe that Γ(U ) ⊂ Vect(X) is invariant under the action of [Q (0) , ]. Therefore, we can think of both Γ(U ) and Γ(T X/U ) as complexes with the differential [Q (0) , ].

Summary of results for H 1 (V ect(X))
Using the notations of Section 2: In the rest of this Section we will explain the computation.

Exact sequences
Consider the short exact sequence of complexes: The corresponding long exact sequence in cohomology of [Q (0) , ] is:

Summary of result
We use the following segment of the long exact sequence: The cohomology groups participating in this segment have the following description: H 0 (Γ(T X/U )) even = C D, M mn of Eq. (39) This implies: We will now explain the computation.

Computation
Γ(T X/U ) The space Γ(T X/U ) is generated as an O(X)-module, by the following vector fields: However Γ(T X/U ) is not a free O(X) module, because there is a relation: δ : H 0 (Γ(T X/U )) even −→ H 1 (Γ(U )) odd It is zero because both D and M mn can be extended to elements of Vect(X) commuting with Q (0) : H 1 (Γ(T X/U )) odd and δ : H 1 (Γ(T X/U )) odd → H 2 (Γ(U )) even For any tensor A lmn , consider vector fields of the form: Such vector fields generate Z 1 (Γ(T X/U )) odd . But some of them are Q (0) -exact: Therefore the vector fields of the form Eq. (43) with A lmn of the form: are zero in H 1 (Γ(T X/U )) odd . This implies that H 1 (Γ(T X/U )) odd is generated by the vector fields of the form: A vector field of Eq. (43) is zero in cohomology iff: Vector fields of the form (46) correspond to: Notice that the section of Γ(T X/U ) defined by Eq. (46) can be extended to a [Q (0) , ]-closed section of T X: This means that δ : H 1 (Γ(T X/U )) odd → H 2 (Γ(U )) even is zero. Eq. (44) has the following refinement: 4.5 Computation of H 1 (Γ(T X)) even

Computation
We use the following segment of the long exact sequence: H 0 (Γ(T X/U )) odd is generated by: H 1 (Γ(U )) even is generated by: The linear map Y q α → (Γ m Γ q ) β α Y m β is a bijection. More precisely: Therefore H 0 (Γ(T X/U )) odd cancels with H 1 (Γ(U )) even .

Coefficients of normal form satisfy wave equations
Modulo F 4 Vect we can choose the coordinates so that: where E, Ω, P are some functions of x. Indeed, using Section 3.4: • H 1 (C fun L ) odd ⊗ ∂ ∂x enters on Line (55), • Second part of H 1 (C vect L ) odd (see Eq. (38)) and H 1 (C fun L ) odd ⊗ H 0 (C vect R ) even on Line (56), • First part of H 1 (C vect L ) odd (see Eq. (38)) and H 1 (C fun L ) even ⊗ H 0 (C vect R ) odd on Line (57)

Equations for tetrad and spin connection
where e L and e R are infinitesimal. We assume summation over repeated indices. We can choose a freedom of so(10) ⊕ C redefinitions of both (λ L , θ L ) and (λ R , θ R ) to fix: At this point, the only remaining freedom in redefinition of λ and θ is overall rescaling of (λ L , λ R , θ L , θ R ). We fixed both (so(10) ⊕ C) L ⊕ (so(10) ⊕ C) R down to the diagonal C.  38)). This will cancel by P LL and P RR , see Section 5.2.1.
Let us denote: and similar definition for Ω R m,nk in terms of Ω L R R . This notation is useful, because for any vector V l : From {Q L , Q R } = 0, the coefficient of ((λ L θ L )) m ((λ R θ R )) n ∂ ∂x : ∂ m e Rn,k + Ω R m,nk = ∂ n e Lm,k + Ω L n,mk Equivalently: Eq. (67) is zero torsion of the "average" (i.e. left plus right) connection.

Einstein equations
Let us denote: Then Eq. (67) implies the existence of a m such that: Infinitesimal coordinate redefinitionx µ = x µ + εv µ , followed a compensating rotation of θ and λ in order to preserve Eq. (60), corresponds to: The overall rescaling corresponds to: Considering the scalar part, we conclude that a m satisfies the Maxwell equations: and g mn satisfies: ∂ p] +∂ m a n + δ mn ∂ p a p = −∂ n b m It follows from the symmetry m ↔ n that exists φ such that b m = a m − ∂ m φ. Therefore: The rescaling Eqs. and therefore a m can be gauged away: fixing the overall rescaling gauge symmetry of Eq. (70).

Equations following from {Q
Considering terms proportional to ((λ R θ R ))((λ L θ L θ L )) ∂ ∂θ L and similar terms with L ↔ R, we need to require that they cancel similar terms in Section 5.1.2.
This implies that modulo zero modes: The antisymmetric tensor field H lmn should be identified with the field strength of the NSNS B-field: Now consider the terms proportional to ((λ R θ R θ R ))((λ L θ L )) ∂ ∂θ L : It is cancelled by adding: leading to the extra term: There is a similar contribution with R ↔ L. For them to cancel each other, we need:

Fermionic fields
In Section 5 we restricted ourselves with Q L and Q R parameterized by even functions E L , E R , . . .. We will now add the terms parameterized by odd functions. According to Section 4.5 these terms are: The first terms in both Q ′ L and Q ′ R are of grade 1, and the rest of the terms are of grade 3.

Grade 3 terms are determined by the grade 1 terms
Let us first assume that the grade 1 terms are zero, i.e ξβ LRm (x) = 0 and ξ β RLm (x) = 0. Considering the coefficient of ((λ L θ L θ L ))((λ R θ R )) ∂ ∂x , we deduce that ψ αµ L satisfies: and a similar equation for ψα µ R . This implies (see Section A) that modulo finite dimensional subspaces (which we ignore): The coefficients ξα LRm and ξ α RLm come with gauge transformations: we conclude that ξ LR (and similarly ξ RL ) should satisfy the Maxwell equations: We will now consider the anticommutator of Q ′ R with Q L . It is convenient to start by completing is removed by further modifying (cp Eq. (53)): to: Then, when we commute with (λ l Γ m θ L ) ∂ ∂x m , this modification produces: Indeed, let us denote: (95) The total contribution is: We observe: In fact: To cancel Eq. (96) we must impose the Dirac equation on ξ α RLk , in the following sense. Require that exists η RLα such that: Then we cancel Eq. (96) by choosing To summarize: and a similar formula for Q ′ R .

Comparison to SUGRA
The only fermionic superfields of [3]

Supersymmetries and dilatation
The vector field Q flat of Eq. (6) This means that any infinitesimal symmetry can be brought to the form: where . . . stand for terms of the higher order in the grading defined by Eq. (23). The vanishing of the coefficients of ((λ R θ R )) ∂ ∂θ L and ((λ L θ L )) ∂ ∂θ R imply S α L (x) = S α L0 and Sα R (x) = Sα R0 (do not depend on x).

Acknowledgments
This work was supported in part by ICTP-SAIFR FAPESP grant 2016/01343-7, and in part by FAPESP grant 2019/21281-4. We want to thank Nathan Berkovits and Andrey Losev for useful discussions.

A Higher spin conformal Killing tensors
Consider tensor fields on the flat N -dimensional space R N with coordinates: f (x, y) = y m 1 · · · y mn f m 1 ...mn (x) Homogeneous polynomialsf (x, y) of x, y of the order N form a finite-dimensional representation of sl(2, R), with the generators defined as follows: Eq. (103) implies thatf (x, y) is a highest weigh vector: On the other hand,f being a polynomial of the order n in y implies: