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 $Q = \Lambda^{\alpha}D_{\alpha} + R^{\alpha}\bar{W}_{\alpha}$ 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 $\bar{\Sigma}^{i}$ constructed out of the fermionic spinor $D_{\alpha}$ and pure spinor variables. This simplified version will be shown to satisfy $\{Q, b\} = T$ and $\{b , b\} =$ BRST - trivial.

1 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], 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 vector Σ 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.
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: S = dτ (Ẋ m P m +θ µ p µ − 1 2 P m P m +λ µ w µ +w µλ µ +ṙ µ s µ ) (2.2) 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 (λγ mn r) (λλ)Γ mΓn (2.5) This simplified D = 10 b-ghost was shown to satisfy the property {Q, b} = T in [19], and as shown in Appendix I, the nilpotence property {b, b} = 0 easily follows from {Γ m ,Γ n } = 0 and [Γ m ,λλ] = 0.
3 D = 11 non-minimal pure spinor superparticle 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] 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 for the D=11 b-ghost: Therefore our candidate to play the analog role toΓ m is: is the only term containing D α 's. Using the identities (B.1), (B.2) in Appendix B, one finds thatΣ j satisfies the constraint: Furthermore, it will be shown in Appendix D 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 (B.10), (B.13), As shown in Appendix F, 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 F): After plugging (3.9) into (3.12), all of the terms explicitly depending on N ab are cancelled and we get (see appendix G): 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 H that {Σ j ,Σ k } is non-zero and is proportional to R α . This implies that 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 I 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 = 10 gamma matrix identities
In D = 10 we have chiral and antichiral spinors which will be denoted by χ α and χ α respectively. The product of two spinors can be decomposed into two forms depending on the chiralities of the spinors used: The 1-form and 5-form are symmetric, and the 3-form is antisymmetric. Furthermore, it is true that Two particularly useful identities are: From A.4 we can deduce the following: The Lorentz algebra satisfied by the ghost currents N mn = 1 2 (λγ mn w) is: [N pq , N rs ] = η qs N pr − η qr N ps − η ps N qr + η pr N qs (A.9) B 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 (B.4) it can be shown that ab,cd f abc = (ΛΓ abΛ )(ΛΓ cd R)f abc (B.10) where f abc , f abce are antisymmetric in all of their indices. Other useful identities: Some useful commutation relations

C Nilpotence of D=10 b-ghost
The nilpotency property satisfied by this object is not obvious to see so that we will check it in detail. The first step is to show that {Γ m ,Γ n } = 0. This can bee seen from the equation (2.5) and the use of the U (5) decomposition of the pure spinor variables [14]. If we choose the only non-zero component ofλ µ to beλ −−−−− = 0 then r −++++ , r +−+++ , r ++−++ , r +++−+ , r ++++− vanish as follows from the constraintλγ m r = 0. This implies that the only components of d µ and N mn appearing in (2.5)  Furthermore, from the constraint (γ mλ ) µΓm = 0 it is followed that the only non-zero components ofΓ m areΓ n with n = {(1 + 2i), (3 + 4i), (5 + 6i), (7 + 8i), (9 + 10i)}. This implies that the term (λγ mn r) in (2.5) is non-zero only for the cases m, n = {(1 − 2i), (3 − 4i), (5 − 6i), (7 − 8i), (9 − 10i)}. Now, because w µ can only appear with two or four plus signs in N pq and λ µ appears in (λγ mn r) at least with two plus signs the only relevant situation is when w µ has two plus signs and λ µ has three plus signs, however when this occurs the only components of r α which contribute are those with one minus sign making the whole expression vanish. Therefore the commutator [Γ m , (λγ pq r)]Γ pΓq = 0. This implies immediately that {b, b} = 0.
This result can also be shown from the following covariant computation: and the Lorentz algebra satisfied by N pq given in (A.9). The first term proportional to P m is zero because the pure spinor constraint and the bosonic nature ofλ α . From the identity (A.5) we can show that (λγ npq r)(λγ pqλ ) = 0. Therefore the terms proportional to this expression vanish.
The equation (A.2) allows putting this expression into the form Now we can use the GAMMA package [21] to do gamma matrix manipulations. The expansion of this expression, the use of the pure spinor constraint and the bosonic nature ofλ µ give us the following result Using the reasons mentioned above we can writeλγ tuvmrλ =λγ tu γ vmrλ =λγ tuv γ mrλ . Therefore after using the identity (A.7) and the constraintλγ m r = 0 we obtain the result desired Using this we can calculate {b, b} directly: We should figure out which are the D α 's appearing in the expressions forΣ i and the b-ghost. For this we will decompose the eleven dimensional Lorentz group in the following way: SO(10, 1) → SO(3, 1) × SO(7) and we will break the Lorentz invariance by choosing a special direction forΛ α : ΛΓ aΛ = 0 → We choose the only non-zero component ofΛ to be:Λ ++0 = 0 (D.1) and 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 (I.1) in terms of these components: Now, we can 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. Furthermore, if we multiplyΣ j 0 by Λ −+j Λ −−0 we get: where we used the pure spinor constraint (D.3). In a similar way we obtain: Therefore 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 ((D.10), (D.11), (D.12), (D.13), (D.14)).
Let us define the quantity: Now we will assume that there exist a matrix (M −1 ) β α such that: and let us make the following ansatz for (M −1 ) β α : Next we will check that this proposal for (M −1 ) β α is right: where the identity (B.3) was used in the penultimate line. Therefore we have the relation: It can be shown thatΣ 0 i can be written in terms of H α : Therefore by using the constraint (ΛΓ abΛ )Σ b 0 = 0, we find 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 (B.8), (B.9) allow us simplify this object: The third term of this expression requires more careful manipulations, so we will do them in detail Furthermore, if we use (B.4) this result can be cast as Plugging this result into (F.1), we find After applying the transformation (3.11) and using the identities (B.2), (B.3), (B.4) one can show that this expression is invariant as mentioned above.
Therefore we can replace the inverse relation (E.4) in (3.10). Let us do this for each term in (F.4): Replacing these expressions in (F.4) and putting all together in (3.10) we obtain G 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 Now let us focus on the contributions proportional toΛR: The last term can be written as As a result, we get Now let us focus on the term proportional to (ΛΓ ci R): Now we use the identity (B.9): Therefore, Now let us simplify the remaining terms in (G.1): Now we apply the identity (B.8) to each term: Therefore J i 3 takes the form And we also have The sum of these quantities gives us the following result where we have used that (ΛΓ mn R)(ΛΓ mn Λ)(ΛΓ ab R)(ΛΓ ab Λ) = 0. After using the identity (B.4) this expression simplifies to Now we will simplify the expressions containing N mn explicitly: For convenience let us focus first on the last three terms: The same manipulations for the first term of S i give us Therefore we get When summing S 1 1 + S i 2 we obtain Thus we have a full cancellation J i + S i = 0.

H Calculation of {Σ i ,Σ j }
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 usē Σ i 0 to denote the part depending on D α . Thereforē It is easy to see that {Σ i 0 ,Σ j 0 } = 0: using the identity (B.3).
The next step is to compute the anticommutator {Σ i 0 ,Σ j 1 }. To this end let us writeΣ j 1 explicitly: and denote each term byΣ in a more convenient way: after using the identity (B.9). Therefore and by doing the same forΣ we obtain Hence we get Analogously we obtain and thus the sum of (H.8) and (H.9) is Now we will simplify the expression corresponding toΣ j 1 : Lets us call Y j 1 to the first term of this expression and expand it as follows Plugging this result into the equation (H.11) This expression is invariant under the gauge symmetry generated by the pure spinor constraint, as it should. Now let us make the following definitions: Consequently to compute {Σ i 1 ,Σ j 1 } we should calculate the anticommutator between each pair of these W j {1,2,3,4,5} variables. Explicitly this computation works as follows because of the identities (B.28) and (ΛR)(ΛR) = 0.
because of the identity (B.2).
Putting all together, the result is One of the useful things that can be extracted from this result is the fact that {Σ i ,Σ j } depends linearly and quadratically on R α . This allows us to find the R α -dependence of {b, b} which it 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 (H.36) determines the functions Ω (k) for k = 1, . . . , 23: . . .
Therefore if we make the following definitions: βδ =Λ α f

I Expanding the simplified D = 11 b-ghost
In this Appendix we will reproduce the terms contained in O(Σ 2 ) in the expression for the simplified D = 11 b-ghost. First we will reproduce the quadratic term in D α in the expression for the b-ghost (3.2) We will work with the expression It is useful to writeΣ i 0 in the convenient way: which is a direct consequence of the identity (B.1). Now we will expandΣ i as it was done in (3.9): Using this equation we can writeΣ j ,Σ c ,Σ k in the following way: Replacing these expressions in (I.2) we have Hence the contributions proportional to D 2 are: The next term to be computed is that proportional to η −3 . This term can be calculated in two steps. First we focus on the part proportional to (ΛΓ a D) which will be called K 1 and then on the part proportional to (ΛΓ bcd D) which will be called K 2 . Thus We can use the identity (I.4) to simplify this expression: Now it is useful to use the following identity which is followed from (B.4): With the additional use of (B.2) we obtain Now let us move on to compute the term proportional to (ΛΓ abc D), this term comes from the following contribution: where we have just used the identities (B.4) and (B.7). Therefore The last term to be calculated is that proportional to η −4 . The relevant terms are (after using (B.7)): So our simplified D = 11 b-ghost has the following expansion: ab,cd [2(ΛΓ abc ki Λ)N dk We can compare this result with the expansion of the b-ghost in (3 The quadratic term in D α is easy to obtain using the identity (B.10) Now we will find the term proportional to η −3 . Let us do this in two steps: First let us focus on the term proportional to ΛΓ a D (which will be called K 1 ) and then on the term proportional to ΛΓ abc D (which will be called K 2 ): ab,cd,ef ( where the identity (B.12) was used from the first to the second line. Now we make use of the identities (B.14) and (B.15): where we have made use of the identity (B.4). By using the antisymmetry in (b, c, e) we show that: Now let us focus on the term proportional to (ΛΓ bcd D). This term appears in the expression (3.2) in the form (after using (B.12)): Putting these results together we obtain which it should be compared with the analog expression corresponding to b simpl , equation (I. 16).
The last term to be computed is that proportional to η −4 in (3.2): Let us use some identities in order to write this expression in a simpler way. It is more convenient to do this in two steps: First we will focus on the first term (P 1 ) and then on the second term (P 2 ): The simplifications made here are result of repeated uses of the identities (B.10) and (B.11). Now let us focus on P 2 : This expression differs from (I.18) in two points. First, the position of N hi in the last term proportional to η −3 is not at the end of the expression as it is in (I.31). Second, in the terms proportional to η −4 we do not have the anticommutator of N ab 's in (I.18) as we do in (I.31), and once again the position of N hi is not at the end of the expressions in (I. 18) as it is in (I.31).
In order to have a clearer idea on what is happening, we will move all of the N ab 's at the end of the expressions mentioned above in (I. 18). Let us start with the term proportional to η −3 . We should put the ghost current N hi to the right hand side of (ΛΓ cmn D). For this purpose we compute the commutator between N hi and (ΛΓ cmn D) with the symmetry properties written in (I.15): simpl changes by the factor − 1 3η 3 (ΛΓ efΛ )(ΛΓ gc R)(ΛΓ mn R)(ΛΓ ef ghi Λ)(ΛΓ chimn D) when N mn is placed at the end of the full expression. Now we move on to the terms proportional to η −4 . We should move N hi to the right hand side of (ΛΓ clmnq Λ): simpl does not change when we move N mn to the end of the full expression. We can summarize the result in the following expression for the simplified b-ghost: ab,cd [2(ΛΓ abc ki Λ)N dk