Geometrical framework for picture changing operators in the pure spinor formalism

It is well known in NSR string theory, that vertex operators can be constructed in various ``pictures''. Recently this was discussed in the context of pure spinor formalism. NSR picture changing operators have an elegant super-geometrical interpretation. In this paper we provide a generalization of this super-geometrical construction, which is also applicable to the pure spinor formalism.


Introduction
Pure spinor formalism is very promising for studying strings in AdS, since it naturally includes the Ramond-Ramond fields [1,2]. However, the progress has been slowed down by the lack of explicit formula for vertex operators (see [3,4] for definitions and [5] for a simple example). Recently, a promising new method was suggested in [6]. The main idea is to construct the vertex in nonzero picture 3 . This means allowing delta-functions of the pure spinor ghosts. It was shown in [6], that the ansatz for 1/2-BPS vertex is simlified in the -8 picture. Then, the vertex operator in picture zero can be obtained by applying picture raising operators, as explained in [9]. However, this picture-raising procedure is not the usual one [10], because the pure spinor variables are constrained to live on a cone. In particular, it does not immediately fit into the geometrical framework of [11,12].
In this paper we will develop a generalization of the approach of [11,12], which does cover the construction of [6,9].
The pure spinor target space can be considered a generalization of the odd tangent bundle ΠT X over the super-space-time X (see Section 2.1). The generalization consists of imposing some constraints on the coordinates in the fiber. If X has coordinates x, θ, then ΠT X has coordinates x, θ, dx, dθ. We consider a submanifold C ⊂ ΠT X defined by some quadratic and linear equations -see Section 4 and [13]. The idea is to construct the action of some "odd loop group" ΠT G on C and then average over its orbits, using the natural measure on ΠT G. The integration removes the delta-functions of the pure spinor ghosts, and the result is the picture zero vertex operator.
Plan of this paper. We first review the geometrical construction of [11], [12] in Section 2, and discuss d-closed submanifolds in Section 3. Then we apply these concepts to pure spinor formalism in Section 4. In Section 5 and Section A we introduce a generalization of the geometrical picture changing procedure of [11,12] and reproduce the result of [9]. Finally, in Section 6 we discuss open questions.
2 Geometrical interpretation of picture changing 2.1 Reminder on odd tangent bundle ΠT X When studying a supermanifold X, it is often useful to consider, for any "test" supermanifold S, the space of maps: (see J. Bernstein's lectures in [14], which are also available online at "https://www.math.ias.edu/QFT/fall/"). This defines a contravariant functor S → F X [S] from supermanifolds to ordinary manifolds (although infinite-dimensional, spaces of maps). This is the "functor of points", F X [S] is called S-points of X.
If X is a supermanifold, then functor ΠT in the category of supermanifolds is defined as follows: Functions on ΠT X are called "pseudo-differential forms (PDFs) on X". In particular, for a Lie supergroup G, the odd tangent space ΠT G is also a Lie supergroup, which might be considered an odd analogue of the loop group of G. Let g = Lie(G) be the Lie algebra of G. The Lie algebra of ΠT G is usually called "cone of g" and denoted Cg:

Picture raising operators
Suppose that a Lie supergroup G acts on a supermanifold X. Then ΠT G acts on ΠT X. This means that pseudodifferential forms on X form a linear representation of ΠT G. Given a PDF ω ∈ Fun(ΠT X) we consider: We restrict ourselves to those ω for which this integral converges. We observe that: Eq. (5) implies that ω actually descends (as a PDF) on the space of orbits of G in X.
Eq. (6) implies that closed ω gives closed Γω. This construction is usually applied to the case when G is purely odd, i.e. take the simplest example G = R 0|1 and ΠT G = R 1|1 with the target supermanifold X parametrized by (Z M ). Suppose that ω contains enough delta-functions to absorb the integration along odd variables. In this case the convergence of the integral in Eq. (4) is guaranteed, and moreover ω → Γω is actually a local operation. To be explicit, suppose that the Lie algebra g = R 0|1 is generated by the odd vector fields ν: The action of the group R 0|1 × X → X is then given by: The corresponding action of ΠT G on ΠT X is: The integral of Eq. (4) is: After integrating on the odd parameter ǫ, we can arrive at the "usual" (in string theory literature) expression for the PCO: where ι ν = ν M ∂ ∂dZ M . This can be generalized to G = R 0|n and ΠT G = R n|n where g = R 0|n is generated by n odd vector fields ν a with a = 1, . . . , n. Then the integral becomes where g ǫ a ,dǫ a is the flow corresponding to the infinitesimal transformation

Some properties of ΠT
Some computations are simplified (see Section 5) by considering the iterated application of ΠT . Consider µ : ΠT ΠT X → ΠT X induced by the diagonal map There is a canonical nilpotent odd vector field d ∈ Vec(ΠT X) generating shifts along R 0|1 . For any odd vector field Q ∈ (Vec(X))1 we define the odd flux map and an even vector field ι Q ∈ (Vec(ΠT X))0 with the flux: If we denote the coordinates of ΠT ΠT X as x, dx, Dx, Ddx, the projection µ is: 3 Submanifolds in ΠT X Consider a submanifold C ⊂ ΠT X which is closed under d. (This means that the ideal generated by PDFs vanishing on C ⊂ ΠT X is closed under d.) We explained in Section 2.2 that the action of the group ΠT G on ΠT X can be obtained from the action of G on X. This, however, would not work for us here, because the resulting action of ΠT G would not preserve C ⊂ ΠT X. But in fact, the validity of Eqs. (5) and (6) does not depend on how we constructed the action of ΠT G on ΠT X. For any action of ΠT G on ΠT X, commuting with the action of d, the transformation ω → Γω defined by Eq. (4) will satisfy Eqs. (5) and (6). We will construct some action preserving C ⊂ ΠT X, i.e. the vector fields representing Cg will be all tangent to C. Then, we will define Γ by Eq. (4).
4 Pure spinor target space as a subspace in ΠT X

Supersymmetry generators and invariant derivatives
It is always possible to find coordinates x, θ such that the supersymmetry generators have the form: To construct supersymmetry-invariant objects, it is useful to know the vector fields commuting with q L|R . Besides translations ∂ ∂x m , there are fermionic vector fields commuting with q L|R . They are: The only non-zero commutators are Then ΠT X is parametrized by the coordinates (x m , θ α L , θ α R , dx m , dθ α L , dθ α R ) where "dx", "dθ" are considered one letter. Then, to describe the pure spinor string in a flat background, we choose as a target the subspace of C ⊂ ΠT X defined by the following conditions The constraint defined by Eq. (19) is the pure spinor constraint, it is essentially postulated. The constraint of Eq. (20) is characterized by: where ∇ L|R are defined by Eq. (18). The main property of the ideal generated by Eqs. (19) and (20) is that it is d-closed. Usually one denotes: The BRST operator Q BRST is just d: Eq. (21) is promising for constructing the action of ΠT G on ΠT X as described in Section 3. We also observe: However, let us keep in mind that:

Solving the pure spinor constraint
The ten-dimensional gamma matrices can be decomposed to give the eight-dimensional ones and also chiral projectors where i, j, a, b,ȧ,ḃ = 1, . . . , 8. Then, we will use the Kronecker deltas δ ab , δȧ˙b, δ ab , δȧ˙b to raise and lower spinor indices. The Pauli matrices are such that σ i aḃ = σ i ba and (σ j )ȧ b = δȧ˙bδ ba σ j aḃ . These matrices satisfy In SO(8) components, the pure spinor constraints for λ α = (λ a ,λȧ) are: Both λ L and λ R satisfy these constraints. They can be solved as follows. Let us define: Eqs. (19) imply: We can solve these equations for λ a − in terms of the rest of variables: where λ a + is unconstrained and (λȧ L ,λȧ R ) are still subject to the conditions (λ L ) 2 = (λ R ) 2 = 0. In light-cone coordinates the conditions (20) are written as Consider the subspace of functions which only depend on x + and do not on x − nor x i . On this subspace: An action of CR 0|8 on C Let us consider the coordinates on the fiber of C: (λ a + ,λ a + ,λ a − ). Let us introduce the following vector fields on C: These i a are vertical vector fields (tangent to the fiber). They commute: [i a , i b ] = 0. 4 Next, we define: In deriving Eq. (34), the following observation is useful. If υ is an even vertical vector field on ΠT X, then: 4 Equivalently, we can start with unconstrained λ, and define: where the derivative is of the RHS of Eq. (30). This vector field is tangent to the cone. This is the same as to consider the vector field of Eq. (32) on C in coordinates (λ a + ,λ a + ,λ a − ).
Indeed, the RHS of Eq. (34) is fixed by: We observe that λ b − is a linear function of λ a + . This implies This means that d, i a , l b define an action of the differential Lie superalgebra CR 0|8 on C.

Type IIB supergravity vertex operator at picture (-8)
In coordinates y + = x + − 2θ a + θ −a : We have lowered their +, − superscripts such that ∇ ±a := ∇ ∓ a and q ±a := q ∓ a . These coordinates simplify the θ-expansion of the dilaton superfield Φ(y + , θ ± ) since it satisfies ∇ − Φ = 0. We have where the constants C a 1 ...a k are bosonic (fermionic) when k is even (odd). When working with the vertex operator corresponding to type IIB supergravity is useful to know the following identity We start by considering the vertex operator in a (−8)-picture that corresponds to the type IIB supergravity scalar state and by applying the q −a supersymmetry generators, we can generate the full type IIB supergravity multiplet using (39)

Type IIB supergravity vertex operator at picture (0)
We will now use the vector fields ρ(l a ), ρ(i a ) constructed in Section 5 to transform the supergravity vertex from picture -8 to picture 0. As we explained in Section 2.2 and Section 3, this amounts to computing the integral: The explicit expressions for these vector fields are: After some computation we can verify that these vector fields satisfy Now we can integrate along the orbits of ΠT R 0|8 , as in Eq. (12): where in the last line we have used that the only λ ± -dependence of V −8 is through δ(λ a + ) which means that the integral on dǫ eliminates all deltas at once. To proceed with the computation first notice that there is no dependence on coordinate dx + , so we can drop the ∂ ∂dx + -part in ρ(l a ). We also change to coordinates y + = x + − 2θ a + θ −a to obtain We want to further transform this expression. First, we substitute Eq. (30) for ∂λ − ∂λ + . Then, we anticommute all ∇ + 's all the way to right and make use of: (Also, remember that ∇ − a = ∇ +a and ∇ + a = ∇ −a ). The computation uses some identities for the commutators of the SUSY-invariant derivatives, namely Eqs. (51), (52) and (53) in Appendix Section A. The result is the following expression for the type IIB supergravity vertex operator V 0 = ΓV −8 : This formula reproduces the result of [9] where the computation was performed using the PCOs in their standard form Γ a = [Q BRST , Θ(ω + a )]. Our computation is, in some sense, more streamlined. In particular, we do not have any poles in λ a + . (The computation of [9] has such poles at the middle steps; they appear each time a Θ(ω + a ) hit a δ(λ a + ).) Still, our computation does suffer from the poles with denominators (λ LλR ). It was proven in [9] that they cancel in V 0 . The proof uses certain identities for the (σ j ) aȧ matrices. And, thus the zero-picture vertex operator is actually of the form V 0 =λȧ Lλḃ R Aȧ˙b(x + , θ ± ).

Open Questions
1. There should be some proving that the poles with denominators (λ LλR ) cancel 2. We constructed some action of CR 0|8 on the pure spinor target space. How flexible is this construction? Is it in some sense natural? Given V −8 , is there a canonical way to construct V 0 ? (This question is very important for the computation of amplitudes. If there is no canonical map V −8 → V 0 , then using the -8 picture in computation of amplitudes is not apriori justified.) 3. It is not immediately clear how to apply our method in AdS. A naive analogue of our vector fields i a and l a from Section 5 is not well-defined in AdS (it would not be gauge invariant).
4. The use of delta-functions on the cone is potentially dangerous. We would want to understand, when generalized functions of the form of products of δ(λ) are well-defined.