Notes on the D = 11 pure spinor superparticle

The D = 11 pure spinor superparticle has been shown to describe linearized D = 11supergravityinamanifestlycovariantway. Anumberofauthorshaveproposedthat its correlation functions be used to compute amplitudes. The use of the scalar structure of the eleven-dimensional pure spinor top cohomology introduces a natural measure for computing such correlation functions. This prescription requires the construction of ghost number one and zero vertex operators. In these notes, we construct explicitly a ghost number one vertex operator but show the incompatibiliy of a ghost number zero vertex operator satisfying a standard descent equation for D = 11 supergravity.


Introduction
Since eleven-dimensional supergravity [1] is the low-energy limit of M-theory, worldline methods for computing D = 11 supergravity amplitudes may lead to new insights into M-theory. Worldline methods using the D = 11 superparticle in light-cone gauge were developed in [2], but a super-Poincaré covariant description of the D = 11 superparticle using pure spinors could be more powerful for making cancellations manifest and simplifying amplitude computations.
Pure spinors were introduced in D = 10 and D = 11 supersymmetric field theories in [3,4] and [5], and were introduced in the context of superstring theory in [6] as extra dynamical variables on the worldsheet. These extra variables allowed super-Poincaré covariant quantization using a simple BRST operator and simplified the computation of multiloop scattering amplitudes as compared to the other superstring formalisms. Pure spinors have also been used for worldline field theory computations in quantum field theories [7,8] where the ultraviolet behavior of the 4-point amplitude for ten-dimensional super Yang-Mills and Type II supergravity up to 5-loops was studied using power counting arguments.
The eleven-dimensional analogs of pure spinors that are discussed here for the superparticle were introduced in [9] and used by [10] to set up a framework for computing N -point correlation functions at tree and loop level using a worldline field theory framework. Some higher-loop computations using the non-minimal D = 11 pure spinor formalism of [11] have been performed in [12,13].
In this paper we provide evidence that contradicts some of the assumptions made in [10]. We construct the ghost number one vertex operator as a perturbation of the BRST operator. This will be BRST invariant only when the D = 11 supergravity equations of motion are imposed. This vertex operator takes the form

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where h α a , h α β , Ω αab come from small perturbations of the eleven-dimensional vielbeins and the structure equations of linearized D = 11 supergravity. They satisfy equations of motion and gauge freedoms arising from the D = 11 supergravity dynamical constraints. These determine their full θ-expansions as explained in [14] and these are required in correlation function prescriptions involving U (1) . The eleven-dimensional pure spinor prescription for computing tree-level N -point correlation functions given in [10] requires the existence of a ghost number zero vertex operator satisfying the standard descent relation where H = P 2 is the particle Hamiltonian. In this paper we will show that eq. (1.2) is incompatible with linearized D = 11 supergravity and discuss some possible ways to fix this problem without going into details. Our investigations were motivated by an attempt to extend the 11-dimensional pure-spinor superparticle to an 11-dimensional pure-spinor ambitwistor-string following [15,16], but the incompatibility of (1.2) appears to be an obstruction. The paper is organized as follows. In section 2 we review the D = 11 pure spinor superparticle. In section 3 we construct the ghost number one vertex operator by requiring that the pure spinor BRST operator be nilpotent at first order as an on-shell geometric deformation of the BRST charge. In section 4, we show the inconsistency between the descent equation (1.2) relating ghost number one and zero vertex operators and the structure equations of D = 11 supergravity. Finally, we give a self-contained review of the superspace formulation of D = 11 supergravity in appendix A.

D = 11 pure spinor superparticle
The eleven-dimensional pure spinor superparticle action in a flat background is given by [9,17] We will use lowercase letters from the beginning/middle of the Greek alphabet to denote SO(10, 1) tangent/curved-space spinor indices and we will let lowercase letters from the beginning/midle of the Latin alphabet denote SO(10, 1) tangent/curved-space vector indices. The superspace fermionic coordinate θ µ is an SO(10, 1) Majorana spinor and p µ is its respective canonical conjugate momentum, and P m is the momentum for X m . The variable λ α is a D = 11 pure spinor variable 1 satisfying λΓ a λ = 0, and w α is its conjugate momentum which is defined up to the gauge transformation δw α = (Γ a λ) α r m , for an arbitrary gauge parameter r m . The SO(10, 1) gamma matrices denoted by Γ a satisfy the Clifford algebra (Γ a ) αβ (Γ b ) βσ + (Γ b ) αβ (Γ a ) βσ = 2η ab δ σ α . In D = 11 dimensions there exist an antisymmetric spinor metric C αβ (and its inverse (C −1 ) αβ ) which allows us to lower (and raise) spinor indices.

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The BRST operator associated to this theory is defined to be where d α = p α − 1 2 (Γ a θ) α P a are the fermionic constraints of the D = 11 Brink-Schwarzlike superparticle. The nilpotency of this operator follows immediately from the pureness of λ α , and thus physical states can be defined as elements of its cohomology. As shown in [9], this BRST cohomology turns out to describe linearized D = 11 supergravity in its Batalin-Vilkovisky formulation. The D = 11 supergravity physical fields are found in the ghost number three sector of the cohomology. To see this, one can write the most general ghost number three superfield The physical state conditions will constrain the functional form of C αβδ to be where the fields C abc (x), h ab (x), χ α b satisfy the linearized D = 11 supergravity equations of motion and gauge invariances where Λ a , Λ bc , Λ α are arbitrary gauge parameters. The other BV fields of linearized D = 11 supergravity are placed into different ghost sectors up to ghost number 7.
Following D = 10 dimensions [7,8], one can attempt to define a pure spinor measure from the eleven-dimensional scalar top cohomology, namely λ 7 θ 9 = 1 in order to give a consistent prescription for computing N -point correlation functions. This measure is easily shown to be BRST-invariant and supersymmetric and has already been successfully used to get the kinetic terms of the D = 11 supergravity action from a second-quantized point of view [9]. Using this one can then propose that the N-point amplitude should be given by a correlation function of the form [10] A 11D In this expression U (3) is the ghost number three vertex operator described above, U (1) is a ghost number one vertex operator and V (0) is a vertex operator of ghost number zero. Although it is possible to write an alternative prescription involving the ghost number four vertex operator containing the antifields of the D = 11 supergravity physical fields, the existence of V (0) clearly plays a crucial role for the computation of the N-point correlation functions beyond N = 3 in this framework.
Having established the importance of the ghost number one and zero vertex operators, we now discuss their construction.

JHEP08(2019)178 3 Ghost number one vertex operator
The ghost number one vertex operator will be constructed from a small perturbation of the pure spinor BRST operator whose nilpotency will follow from the D = 11 linearized supergravity equations of motion and the pure spinor constraint. We give a detailed review of the superspace formulation of D = 11 supergravity in appendix A. 2 Let us write the eleven-dimensional vielbeins in their linearized form where These give dually to first order On the other hand, using eq. (A.2) one can show that at linear order where [·, ·} is the graded (anti)commutator. Using the D = 11 supergravity constraints (A.12), one then finds that Thus if one defines the BRST operator to be then its nilpotency property immediately follows from the e.o.m (A.23) After converting (3.6) into a worldline vector with ghost number 1 by replacing operators by corresponding worldline fields, one concludes that where and . . . means higher order terms. Thus {Q, Q} = 0 yields directly as desired.

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The e.o.m satisfied by the superfields in (3.9) can be easily found by plugging (3.3) into (3.4). From the relation {D α , D β }, one gets Moreover, the linearized supercurvature components can be written in terms of the super spin-conection using eq. (A.4) As a consistency check, one can verify that

Ghost number zero vertex operator
In order for a consistent standard equation to be satisfied, a ghost number zero vertex operator should exist and satisfy the relation where U (1) is the ghost number one vertex operator discussed above. To solve eq. (4.1), let us write first the most general ghost number zero vertex operator which is gauge invariant under the pure spinor constraint

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One can now compute the e.o.m that the superfields in (4.2) should satisfy such that (4.1) holds. After some algebraic manipulations one finds that The first equation can be automatically solved if one identifies G ab = h ab , Ψ α a = ψ α a as can be seen from (3.14). Replacing this into (4.4) one gets After taking a look at eq. (3.13), one concludes that this equation becomes an identity if one identifies W abc = Ω abc , −2(Γ a ) αγ P γβ = T αa β . However this solution for P αβ is inconsistent as will be shown now. If this identification were true, it would imply that If one now tries to recover T δa β by multiplying eq. (4.10) by −2(Γ a ) δα , one finds that which is clearly an inconsistency because of the eleven-dimensional structure of maximal supergravity (see eq. (A.21)). Further evidence that D = 11 supergravity is inconsistent with eqs. (4.3)-(4.8) can be found when trying to solve eq. (4.6). To see this, let us identify T ab α with one the D = 11 supergravity fields. Using dimensional analysis arguments one concludes that the most general expression for T ab α should have the form where a 1 , a 2 are numerical constants to be determined. Using eq. (A.36), one can relate the last two terms on the right hand side of (4.12) to the first one, since (Γ [a|c| ) α δ T b]c δ = T ab α and (Γ abcd ) α δ T cd δ = 2T ab α . This implies that T ab α = b 1 T ab α where b 1 is a constant normalization factor. After plugging this and eq. (4.10) into (4.6) one demonstrates that

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Since this equation is antisymmetric in (β, γ), it should be true for the all antisymmetric gamma matrix projections of it, namely C βγ , (Γ f gh ) βγ , (Γ f ghi ) βγ . In particular, the 3-form projection requires (Γ f gh ) βγ (Γ ab ) α γ T ab β = 0 (4.14) However, the use of eq. (A.36) allows one to show that which is non-zero and thus inconsistent with (4.6). Thus it is not possible to obtain a ghost number zero vertex operator from the D = 11 supergravity fields that satisfy the standard descent equation (4.1).

Discussion
In this paper we have constructed a ghost number one vertex operator involving more terms in its definition compared to that presented in [10]. In principle, there is no physical reason to ignore them in the 3-point function computations, so it would be interesting to see how they affect the results found in [10]. Moreover, an explicit relation between U (1) and U (3) would be important to understand the structures underlying the eleven-dimensional pure spinor framework, for example to prove permutation invariance of the correlator. A first step in this direction was provided in [18] where it was shown that and Ξ a is a ghost number two operator. It should be emphasized that this superfield Φ b has been successfully used to study D = 11 supergravity from a secondquantized perspective [11,12,18]. In particular, note that Φ b is contained in U (1) after contraction with the momentum P b . On the other hand, we have shown that it is not possible to write a ghost number zero vertex operator made out of the D = 11 supergravity superfields satisfying a standard descent equation. One possible resolution is to extend the present framework to its nonminimal version by introducing the standard non-minimal pure spinor variables. In this setting, a ghost number zero vertex operator can be defined using the relation where b was found in [12] and simplified in [19] where it was shown that b is nilpotent up to BRST-exact terms. Since b satisfies {Q, b} = H,Ṽ (0) satisfies the standard descent equation Unlike the ten-dimensional case [8],Ṽ (0) cannot be split into a function depending only on minimal variables plus a BRST-exact term as follows from our result in section 4. This implies thatṼ (0) will be a complicated function depending on minimal and non-minimal variables. It would be interesting to determine whether the non-minimal sector decouples from the theory perhaps with the use of some appropriate measure in the correlation functions.

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Another approach is to impose additional constraints on the eleven-dimensional pure spinor in such a way that more terms in the ghost number zero vertex operator are allowed. For example, one could impose the full Cartan purity condition λΓ ab λ = 0 as was considered in [5,20], and it would be interesting to see if there is some relation of this constraint with the present work.

A Review of superspace formulation of D = 11 supergravity
In this appendix we will review the original superspace formulation of D = 11 supergravity given in [21]. This turns out to be useful to fix conventions and get consistently the equations of motion for the dynamical superfields, which in turn play a crucial role when constructing the ghost number one and zero vertex operators of sections 3 and 4. Let us start by fixing notation. Latin capital letters from the beginning/middle of the alphabet will be used to represent tangent/coordinate superspace indices. The vielbein and spinconnection will be defined to be 1-forms on superspace as follows and the 2-form supercurvature as the covariant derivative of the 1-form super spinconnection

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As usual, we will constrain the super spin-connection components to satisfy and all the other components to vanish. This choice automatically implies that which in component notation read where [·, ·} is a graded antisymmetrization. Furthermore, a 4-form superfield can be also introduced which will be required to satisfy dH = 0. This condition gives rise to a new identity, which in component notation takes the form In order to put the theory on-shell we will impose the standard conventional and dynamical constraints, namely In this way, the only dynamical superfields of D = 11 supergravity are H abcd , T aα β , T ab α .

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The first equation is just a consistency check. The second equation (A.14) tells us that which implies that (Γ a ) α δ T αb δ = 0 and This implies that T αa β can be written in terms of H abcd . Using symmetry arguments one finds that The use of eq. (A.14) tells us that where the identity (Γ b ) βα ∇ α H bcde = − 3 7 (Γ [c ) β δ (Γ f g ) δγ ∇ γ H de]f g was used.
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