Simplified D = 11 pure spinor b ghost

A b-ghost was constructed for the D = 11 non-minimal pure spinor superparticle by requiring that {Q, b} = T where Q=ΛαDα+RαW¯α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ Q={\Lambda}^{\alpha }{D}_{\alpha }+{R}^{\alpha }{\overline{W}}_{\alpha } $$\end{document} is the usual non-minimal pure spinor BRST operator. As was done for the D = 10 b-ghost, we will show that the D = 11 b-ghost can be simplified by introducing an SO(10, 1) fermionic vector Σ¯i\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\overline{\Sigma}}^i $$\end{document} constructed out of the fermionic spinor Dα and pure spinor variables. This simplified version will be shown to satisfy {Q, b} = T and {b, b} = BRST - trivial.


Introduction
The D = 11 pure spinor superparticle is a useful tool to describe D = 11 linearized supergravity in a manifestly covariant way [1]. This formalism describes physical states as elements of the cohomology of a BRST operator defined by Q min = Λ α D α , where Λ α is a D = 11 pure spinor 1 satisfying the constraint ΛΓ a Λ = 0, a is an SO(10, 1) vector index, and D α are the first-class constraints of the D = 11 Brink-Schwarz-like superparticle [3]. The spectrum found by using this formalism coincides with that obtained via the BV quantization of D = 11 linearized supergravity and includes the graviton, gravitino, and 3-form at ghost-number 3, as well as their ghosts and antifields at other ghost number [1,4], JHEP07(2017)115 each one of them satisfying certain equations of motion and gauge invariances as dictated by the BV prescription.
Motivated by the non-minimal version of the pure spinor superstring [5], Cederwall formulated the D = 11 non-minimal pure spinor superparticle by introducing a new set of variablesΛ α , R β and their respective momentaW α , S β , whereΛ α is a D = 11 bosonic spinor and R β is a D = 11 fermionic spinor satisfying the constraintsΛΓ aΛ = 0 and ΛΓ a R = 0 [6,7]. In order for the new variables to not affect the physical spectrum, the BRST operator should be modified to Q = Λ α D α +R αW α , as in the quartet argument of [8].
In the non-minimal pure spinor formalism of superstring, one can formulate a consistent prescription to compute scattering amplitudes by constructing a non-fundamental b ghost satisfying {Q, b} = T . Therefore, it is important to know if a similar b ghost can be constructed in the D = 11 superparticle case. The D = 11 b-ghost was first constructed in [9] in terms of quantities which are not manifestly invariant under the gauge symmetries of w α generated by ΛΓ a Λ = 0. This b-ghost was later shown in [10] to be Q-equivalent to one written in terms of the gaugeinvariant quantities N ab and J, and we will focus on this manifestly gauge-invariant version of the b-ghost.
The complicated form of the b-ghost in [10] makes it difficult to treat, so for instance its nilpotency property {b, b} has not yet been analyzed. A similar complication exists in D = 10 dimensions, however, it was shown in [11] that the D = 10 b-ghost could be simplified by defining new fermionic vector variables. In this paper, a similar simplification involving fermionic vector variables will be found for the D = 11 b-ghost which will simplify the computations of {Q, b} = T and {b, b}.
The paper is organized as follows: in section 2 we review the D = 10 non-minimal pure spinor superparticle, constructing the corresponding pure spinor b-ghost and its simplification. In section 3 we review the D = 11 pure spinor superparticle, constructing the manifestly gauge-invariant b-ghost and explaining how to translate the simplification of the D = 10 b-ghost to the D = 11 b-ghost by defining the SO(10, 1) composite fermionic vectorΣ j . Finally we construct the simplified D = 11 b-ghost and show that it satisfies the relations {Q, b} = T and {b, b} = BRST-trivial. Some comments are given at the end of the paper concerning the relation between the b-ghost found in [10] and this simplified b-ghost.

D = 10 non-minimal pure spinor superparticle
The D = 10 (minimal ) pure spinor superparticle action is given by [12]: where m, µ are SO(9, 1) vector/spinor indices, θ µ is an SO(9, 1) Majorana-Weyl spinor, p µ is its corresponding conjugate momentum and P m is the momentum. The variable λ µ is a D = 10 pure spinor satisfying the constraint λγ m λ = 0 where m is an SO(9, 1) vector index, and w µ is its corresponding conjugate momentum. Because of the pure spinor constraint this SO(9, 1) antichiral spinor is defined up to the gauge transformation δw µ = (γ m λ) µ f m ,
In the non-minimal version of the pure spinor superparticle [5,16], one introduces a new pure anti-Weyl spinorλ µ , and a fermionic field r µ satisfying the constraintλγ m r = 0, together with their respective conjugate momentaw µ , s µ . In order to not affect the cohomology corresponding to Q min , the non-minimal BRST operator is defined as Q non−min = λ µ d µ +w µ r µ . Thus the D = 10 non-minimal pure spinor superparticle is described by the action: and the BRST operator Q = λ µ d µ +w µ r µ . By construction, the physical spectrum also describes BV D = 10 (abelian) Super Yang-Mills.

D = 10 b-ghost
As discussed in [16,17] a consistent scattering amplitude prescription can be defined using a composite b-ghost satisfying {Q, b} = T , where Q is the non-minimal BRST operator and T = − 1 2 P a P a is the stress-energy tensor. This superparticle b-ghost is obtained by dropping the worldsheet non-zero modes in the superstring b ghost and is where N mn = 1 2 λγ mn w. The complicated nature of this expression makes it difficult to prove nilpotence [18], however it was shown in [11] that the b-ghost can be simplified by introducing an SO(9, 1) composite fermionic vectorΓ m satisfying the constraint (γ mλ ) µΓm = 0. In the expression (2.3), the terms involving d µ always appear in the combination The D = 11 non-minimal pure spinor superparticle action is given by [1] We use letters of the beginning of the Greek alphabet (α, β, . . .) to denote SO(10, 1) spinor indices and henceforth we will use Latin letters (a, b, . . . , l, m, . . .) to denote SO(10, 1) vector indices, unless otherwise stated. In (3.1) Θ α is an SO(10, 1) Majorana spinor and P α is its corresponding conjugate momentum, and P a is the momentum for X a . The variables Λ α , Λ α are D = 11 pure spinors and W α ,W α are their respective conjugate momenta, R α is an SO(10, 1) fermionic spinor satisfyingΛΓ a R = 0 and S α is its corresponding conjugate momentum. The SO(10, 1) gamma matrices denoted by Γ a satisfy the Clifford algebra 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 (e.g. (Γ a ) αβ = C ασ C βδ (Γ a ) σδ , (Γ a ) α β = C ασ (Γ a ) σβ , etc). The physical states described by this theory are defined as elements of the cohomology of the BRST operator Q = Λ α D α + R αW α where D α = P α − P a (Γ a Θ) α and describe D = 11 linearized supergravity.

D = 11 b-ghost and its simplification
As in the D = 10 case, a composite D = 11 b-ghost can be constructed satisfying the properties {Q, b} = T where T = −P a P a , and was found in [9,10,20] to be: and means antisymmetrization between each pair of indices. The D = 11 ghost current is defined by N ij = ΛΓ ij W . To simplify this complicated expression for the D=11 b-ghost, we shall mimic the procedure explained above for the D=10 b-ghost and look for a similar object toΓ m . A hint comes from looking at the quantity multiplying the momentum P i in the expression JHEP07(2017)115 ab,cd 2(ΛΓ abc ki Λ)N dk Therefore our candidate to play the analog role toΓ m is: is the only term containing D α 's. Using the identities (A.1), (A.2) in appendix A, one finds thatΣ j satisfies the constraint: Furthermore, it will be shown in appendix B that the D α 's appearing inΣ i 0 are the same as those appearing in the b-ghost. Therefore a plausible assumption for the simplification of the b-ghost would be b = P iΣ i + O(Σ 2 ). As will now be shown, the simplified form of

Computation of {Q,Σ j }
To show that the b-ghost of (3.8) satisfies {Q, b} = T , it will be convenient to first compute {Q,Σ i } where, using the identities (A.10), (A.13),

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As shown in appendix D, this expression in invariant under the same gauge transformations under whichΣ i 0 is invariant: where (Λ ′ ) α = 1 2η (ΛΓ mnΛ )(ΛΓ mn ) α is a pure spinor, and f ij is an antisymmetric gauge parameter. Therefore we can write all D α 's in this object in terms ofΣ i 0 , and the result is (see appendix D): After plugging (3.9) into (3.12), all of the terms explicitly depending on N ab are cancelled and we get (see appendix E): Using (3.13) it is now straightforward to compute {Q, b}: To make the computations transparent, each term in (3.14) involving {Q,Σ i } will be simplified separately: Putting together all the terms in (3.14): Recalling that T = −P 2 is the stress-energy tensor, we have checked that {Q, b} = T .

{b, b} = BRST-trivial
In the D=10 case, the identity {Γ m ,Γ n } = 0 was crucial for showing that {b, b} = 0. However, in the D=11 case, it is shown in appendix F that {Σ j ,Σ k } is non-zero and is proportional to R α . This implies that for some G α (Λ,Λ, R, W, D).

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Note that [Q, {b, b}] = 0 since [b, T ] = 0 where T = −P a P a . Since Q = Λ α D α + R αW α , the quartet argument implies that the cohomology of Q is independent of R α , which allows us to conclude that {b, b} = BRST-trivial. It would be interesting to investigate if this BRST-triviality of {b, b} is enough for the scattering amplitude prescription using the bghost to be consistent.

Remarks
We have succeeded in finding a considerably simpler form in (3.8) for the D=11 b-ghost than that of equation (3.2) which was presented in [10]. Although this simplified version is not strictly nilpotent, it satisfies the relation {b, b} = BRST-trivial which may be good enough for consistency.
It is natural to ask if the simplified D = 11 b-ghost (3.8) is the same as the b-ghost presented in (3.2). These two expressions are compared in appendix G and we find that they coincide up to normal-ordering terms coming from the position of N mn in each expression. Note that the product of N mn 's appears as an anticommutator in (3.2) whereas it appears as an simple ordinary product in (3.8). However, because we have ignored normal-ordering questions in our analysis, we will not attempt to address this issue.

A D = 11 pure spinor identities
We list some pure spinor identities in eleven dimensions: where f ac , g bd are antisymmetric in (a, c), (b, d) respectively. In addition, using (A.4) it can be shown that

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On the other hand, from the pure spinor constraint ΛΓ a Λ = 0 we have: where j = 1, . . . , 7 and we have assumed that Λ −−0 = 0. This allows us to expand the quadratic term in D α in the b-ghost in terms of these components: Now, we writeΣ i 0 in the convenient form: After using the particular direction chosen above,Σ i 0 presents the following SO(3, 1)×SO(7) components: where k, j = 1, . . . , 7. Therefore, after using the pure spinor constraint, we see that the expression for b 1 contains the same combinations of D α 's as those contained in the expression forΣ i 0 ((B.9), (B.10), (B.11)).

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C D α in terms ofΣ j 0 Let us define the quantity: Now we will assume that there exist a matrix (M −1 ) β α such that: and let us check that the following ansatz for (M −1 ) β α : is right. This can be seen easily as follows where the identity (A.3) was used. Therefore we have the relation: Furthermore, from the constraint (ΛΓ abΛ )Σ b 0 = 0, one immediately concludes that which is the inverse relation betweenΣ k and H α .
We will show that the D α 's appearing in (3.10) are invariant under the gauge transformations (3.11). Therefore they are the same D α 's as those contained in the definition ofΣ i . In this appendix and the next ones we have made use of the GAMMA package [21] because of the heavy manipulation of gamma matrix identities which computations demanded. Let us call I i to the terms containing D α 's explicitly in (3.10). The identities (A.8), (A.9) allow JHEP07(2017)115 us simplify this object: The third term of this expression requires more careful manipulations: Furthermore, the identity (A.4) allows us to cast this result as Plugging this result into (D.1), we find After applying the transformation (3.11) and using the identities (A.2), (A.3), (A.4) one can show that this expression is invariant as mentioned above. Therefore we can replace the inverse relation (C.4) in (3.10). After doing this for each term in (D.4), we get:

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Replacing these expressions in (D.4) and putting all together in (3.10) we obtain E Cancellation of all of the N ab contributions in the equation (3.12) We will show this cancellation in two steps. First we will simplify the expression depending explicitly onΣ i 0 and then simplify the expression depending explicitly on N ab . Finally we will see that these two expressions identically cancel out. We start with the following equation One can show that the term proportional toΛR can be cast as The use of the identity (A.9) allows us to write the term proportional to (ΛΓ ci R) in the form Finally, with a little algebra and the use of the identities (A.4), (A.8) one gets the following result

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Now we will simplify the expressions containing N mn explicitly: The first term in (E.5) can be written as follows The last three terms in (E.5) can be put into the form: When summing S 1 1 + S i 2 we obtain Thus we have a full cancellation J i + S i = 0.
The objectΣ i has a part depending on D α and other part depending on N mn , as it can be seen in (3.9). The part depending on N mn will be calledΣ i 1 and as before we useΣ i 0 to denote the part depending on D α . Thereforē

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It is easy to see that {Σ i 0 ,Σ j 0 } = 0. To compute the anticommutator {Σ i 0 ,Σ j 1 } we writeΣ j 1 in the more convenient way: The result is Analogously, one can show that {Σ i 1 ,Σ j 1 } depends linearly and quadratically on R α . This allows us to find the R α -dependence of {b, b} which turns out to be of the form: This can be used to check that {b, b} = QΩ where Ω is an arbitrary function of pure spinor variables and the constraints D α . To see this let us expand Ω in terms of R α : Thus the action of the BRST operator Q = Q 0 + R αW α on Ω gives us The comparison of this result with the equation (F.4) determines the functions Ω (k) for k = 1, . . . , 23: . . .

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Therefore if we make the following definitions: βδσρλ =Λ α f In this appendix we will show explicitly the terms contained in O(Σ 2 ) in the expression for the simplified D = 11 b-ghost. We will work with the expression The contributions proportional to D 2 are: where the identities (A.3), (A.4) were used.

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The terms proportional to η −3 , η −4 can be found by using the identities (A.2), (A.4) and (A.7). The result is In order to compare the two ways for the b-ghost, we should move all of the N ab 's at the end of the expressions showed above. After doing this one concludes that b simpl changes by the factor − 1 3η 3 (ΛΓ efΛ )(ΛΓ gc R)(ΛΓ mn R)(ΛΓ ef ghi Λ)(ΛΓ chimn D), and b simpl does not receive any contribution. Therefore, the simplified b-ghost has the following form: ab,cd 2(ΛΓ abc ki Λ)N dk The quadratic term in D α is easy to obtain using the identity (A.10) b (2) = 1 η 2 (ΛΓ abΛ )(ΛΓ cd R)(ΛΓ a D)(ΛΓ bcd D) (G.7) With a little algebra the terms proportional to η −3 , η −4 can be found. which should be compared with the analog expressions corresponding to b simpl , equations (G.3), (G.4). The differences between this equation (G.6) and (G.5) is the non-zero extra term proportional to (ΛΓ chimn D). This might be related to normal-ordering ambiguities.
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