Equivalence of the 11D pure spinor and Brink-Schwarz-like superparticle cohomologies

The $D=11$ pure spinor formulation of the superparticle allows a simple realization of covariant quantization, unlike the $D=11$ Brink-Schwarz-like superparticle. We explicitly show the equivalence between the cohomologies of these two models in the context of two different group decompositions: $SO(10,1) \rightarrow SO(1,1)\times SO(9)$ and $SO(10,1) \rightarrow SO(3,1)\times SO(7)$. We also carry out a light-cone analysis of the pure spinor cohomology, and show that it correctly reproduces the $SO(9)$ equations of motion for $D=11$ linearized supergravity.


Introduction
It is well known that the D = 10 Brink-Schwarz formulation of the superparticle possesses first-and second-class constraints which cannot be separated out in a manifestly covariant way. If the physical spectrum is our main concern, we can always go to the lightcone gauge and follow Dirac's prescription to show that the physical spectrum consists of an SO (8) vector and spinor, which satisfy the D = 10 linearized Super Yang-Mills equations of motion [1]. However, light-cone gauge breaks the manifest covariance of the theory.
It is interesting and useful to look for covariant descriptions which manifestly preserve as many symmetries as possible. One candidate that addresses this point is the pure spinor version of the D = 10 Brink-Schwarz superparticle, known as the D=10 pure spinor superparticle [2,3]. This description preserves supersymmetry and Lorentz symmetry in a manifestly covariant way. The spectrum is defined as the cohomology of the BRST operator defined by Q = λ µ d µ , where λ µ is a D = 10 pure spinor and the d µ are the fermionic constraints of the D = 10 Brink-Schwarz superparticle. There are two ways to see that the pure spinor formulation indeed describes D = 10 linearized Super Yang-Mills. The first one is by looking at the Q-cohomology of the D = 10 pure spinor superparticle and realizing that the elements in this cohomology describe the BV version of D = 10 (abelian) Super Yang-Mills [2]. The second one is by showing that the cohomologies corresponding to the D = 10 Brink-Schwarz superparticle and the D = 10 pure spinor superparticle are identical [4].
As explained in [2][3][4][5], the D = 10 SYM physical fields are found in the ghost-number 1 vertex operator V = λ µ A µ , after imposing on it the pure spinor physical state condition. The light-cone analysis of this cohomology reproduces the SO(8) superfield A a satisfying the SYM equations of motion in D = 8 superspace [6].
In D = 11 the story is similar. The D = 11 Brink-Schwarz-like superparticle [7] possesses first-class and second-class constraints which do not allow a manifestly covariant quantization of the theory. However, it is possible to quantize the theory in the light-cone gauge and it can be shown that the spectrum is described by an SO(9) traceless symmetric tensor, an SO(9) Γ-traceless vector-spinor and an SO(9) 3-form which describe D = 11 linearized Supergravity. As before, this theory is no longer manifestly Lorentz covariant.
As in the D = 10 case, Berkovits formulated the so-called D = 11 pure spinor superparticle [5]. The physical states of this pure spinor version are defined as elements in the cohomology of the BRST operator Q = Λ α D α , where Λ α is a D = 11 pure spinor and D α are the fermionic constraints of the D = 11 Brink-Schwarz-like superparticle. The elements of this Q-cohomology describe the BV version of D = 11 linearized supergravity [5]. Unlike the D = 10 case there is not explicit proof of the equivalence between the cohomologies of the D = 11 Brink-Schwarz-like superparticle and the D = 11 pure spinor superparticle 1 . In this work we will demonstrate the equivalence of these two cohomologies by using two different group decompositions 2 .
As explained in [5], the D = 11 supergravity physical fields are found in the ghost number 3 vertex operator V = Λ α Λ β Λ δ C αβδ , after imposing the pure spinor physical state condition. The light-cone analysis of this cohomology will be described by the SO(9) superfields g jk ,ψ j A , C jkl , which satisfy a set of equations of motion in D = 9 superspace that match the linearized supergravity light-cone equations of motion [7].
The paper is organized as follows: In section 2 we review the D = 11 Brink-Schwarz-like superparticle. In section 3 we present the D = 11 pure spinor superparticle and show the equivalence between the cohomologies of this theory and the previous one by decomposing D = 11 objects into their SO(1, 1) × SO(9) and SO(3, 1) × SO(7) components. In section 4 we study the light-cone pure spinor cohomology and show that it is described by the usual SO(9) irreducible representations that describe D = 11 supergravity and satisfy linearized equations of motion in D = 9 superspace. 1 There is a brief discussion of this point in [8], which suggests following the same ideas developed in the D = 10 case. We will elaborate on the ideas mentioned there, and give another way to parametrize D = 11 pure spinors.
2 In [9, 10] I. Bandos relates these two models by using the Lorentz harmonics approach. We will address the problem in a different way, by focusing on the D = 11 light-cone Brink-Schwarz-like superparticle.
The action (2.1) is invariant under reparametrizations, SUSY transformations and κtransformations which are defined by the following equations: The conjugate momentum to Θ α is Therefore, this system possesses constraints, and considering that {Θ α , P β } P.B = iδ α β , we get the constraint algebra where {·, ·} denotes a Poisson bracket. One can show that K α = P m Γ αβ m D β are the firstclass constraints that generate the κ-symmetry. From (2.4), we realize that we have 16 firstclass constraints and 16 second-class constraints, and there is no simple way to covariantly separate them out. However, the physical spectrum can be easily found by using the semi light-cone gauge, which is defined by: In these light-cone coordinates one can use the κ-transformation to choose a gauge where (Γ + Θ) α = 0 3 . With this choice we can rewrite the action as follows where S A is an SO(9) Majorana spinor, which can be written in terms of SO(9) component of Θ α . The conjugate momentum to S A is: So, the constraints for this gauge-fixed system are: Hence, the constraint matrix is C AB = δ AB , and its corresponding inverse is (C −1 ) AB = δ AB . This allows us to compute the following Dirac Bracket: As is well known, the representation of the algebra (2.11) defines the space of physical states. These states will be denoted |IJ , |BI and |LM N , where we represent SO(9) vector indices by I, J, K, L, . . ., and spinor indices by A, B, C, D, . . .. These states correspond to an SO(9) traceless symmetric tensor, an SO(9) Γ-traceless vectorspinor and an SO(9) 3form, which, together, form the field content of D = 11 SUGRA. The action of the operators S A on the physical states is defined by We can check that these definitions indeed reproduce the desired algebra. Let us check the statement explicitly for the graviton |IJ : Analogously, Thus, the anticommutator is Now, let us consider the symmetry properties of the SO(9) Γ-matrices. The 1-form and 4-form are symmetric in their spinor indices, and the 2-form and 3-form are antisymmetric in their spinor indices. Therefore, as expected. One can similarly show that this algebra is satisfied for the action of S A on the other two fields. Therefore, we have shown that the D = 11 superparticle spectrum describes the physical degrees of freedom of D = 11 supergravity.

D=11 pure spinor superparticle
As for the D = 10 case [4], we will obtain the D = 11 pure spinor superparticle from the gauge-fixed Brink-Schwarz-like superparticle (2.7) by introducing a new set of variables (Θ α , P α ) and a new symmetry coming from the following first-class constraints: Let us check that these ones are indeed first-class constraints: Thus, the modified Brink-Schwarz-like action will be: where we have added the usual kinetic term for the variables (Θ α , P α ) and the last term takes into account the new constraint through the fermionic Lagrange multiplier f α . The standard BRST method gives us the following gauge-fixed action: and the BRST operatorQ once we choose the gauge e = − 1 2 and f α = 0. The ghosts c,Λ α come from gauge-fixing the reparametrization symmetry and the new fermionic symmetry, respectively. Now we will show that the cohomology of the BRST operatorQ is equivalent to the cohomology of a BRST operator Q = Λ α D α , where Λ α is a pure spinor. We will show this claim in two steps. First, we show that theQ-cohomology is equivalent to Q -cohomology, where Q = Λ αD α and Λ Γ + Λ = 0. Finally, we will prove that the Q -cohomology is equivalent to the Q-cohomology.
Let us start by defining the operator Q 0 = Λ α 0D α . Notice that when Λ α 0 is equal tô Λ α or Λ α , Q 0 becomes the first term ofQ or Q , respectively. Now, let V be a state such that Q 0 V = (Λ 0 Γ + Λ 0 )W , for some W. Because of the property that Λ α satisfies, V is annihilated by Q . Also, using (3.3), we find that We can then show that the stateV = V − 2iP + cW is annihilated byQ: where we have assumed that b annihilates physical states. Now, let us show that if a state V is BRST-trivial (in the Q -cohomology), we can find a stateV = V − 2iP + cW which is also BRST-trivial (in theQ-cohomology). Let V be a state which satisfies where we used the fact that b annihilates Ω as well as the resultQ 0 Y = W − i 2P + P m P m , which follows from the definition of V . Hence, we obtain Therefore, we have proven that for each state V in the Q -cohomology, we can find a statê V in theQ-cohomology. If we reverse the arguments given above we can show that any state in theQ-cohomology corresponds to a state in the Q -cohomology.
The last step is to show that the Q -cohomology is equivalent to the Q-cohomology. We will do this by using two different approaches.
The SO(10, 1) spinors Λ α and D α can be expressed in terms of their SO(8) components in the following way: where a,ȧ = 1, . . . , 8. The constraint Λ Γ + Λ = 0 can be written in terms of these SO(8) components as follows λ ȧ λ ȧ +λ aλ a = 0 (3.11) The particular representation for SO(10, 1) Γ-matrices used in this section is studied in detail in Appendix A. Now, we find it useful to break SO(9) into SU (2) × SU (4). The branching rule for the spinor representation is 16 → (2, 4) + (2,4). Explicit expressions for the SU (2) × SU (4) components corresponding to S a ,Sȧ, dȧ,d a , λ a ,λ ȧ are given below: where the SO (9) spinor S A has been expressed in terms of its SO(8) components: andÂ,Ā = 1, . . . , 4. It should be clear in (3.12) that fields in the same representation of SU (4) (4 or4) form SU (2) doublets. So, for instance, dẪ dÂ transforms under (2, 4), λ Ā λ Ā transforms under (2,4), etc. Notice that the representations 4 and4 are defined by the null spinor (Γ + Λ ) A by using the fact that one can always choose an SU (4) subgroup under which this spinor is invariant. Therefore we define the antifundamental representation (4) in such a way that (Γ J ) (ΥĀ)A (Γ + Λ ) A = 0, where J = 1, . . . , 9, Υ is an SU (2) vector index and A is an SO(9) spinor index. After making the following shifts: the operator Q will change by the similarity transformation: This result can be expanded by using the BCH formula: where X = Q = Λ αD α and Z = K(SĀdÂ −SĀdÂ). The first term is just Q , which can be cast as To find the second term in (3.17), it is necessary to compute the SU (4) (anti)commutation relations, which can be obtained from the SO(8) relations: Using these, together with (3.12), leads us to the following SU (4) relations: Hence, we get From this expression it is easy to see that: and so the third term and all of the other ones in (3.17) (which were represented by . . .) vanish.
3.2 Group decomposition SO(9) → U (1) × SO (7) We will express SO(10, 1) spinors in terms of their SO(3, 1) × SO (7) components: where i = 1, . . . , 7. The notation ± and the representation of the SO(10, 1) gamma matrices used here are explained in detail in Appendix B. Using this notation, we can express the (anti)commutation relations studied above in the SO(3, 1) × SO (7) language: and any other anticommutator vanishes. Under a certain subgroup U (1) × SO(7) ⊂ SO (9), the null spinor (Γ + Λ ) A will be invariant up to rescaling. This subgroup is chosen in such a way that where we have dropped out the minus sign associated to the first U (1) charge, and j = 1, . . . , 7. The BRST operator Q can be expressed in terms of SO(3, 1) × SO (7) variables: After performing the following shifts: the BRST operator will changed by where Λ Γ + Λ = 0, the resulting BRST operator can be written as From this last expression, we can conclude that the space of physical states will not depend on the canonical variables S −+0 , S −−i , Λ ++0 , Λ +−i , or their respective conjugate momenta S −−0 , S −+i , W −−0 , W −+i . Therefore the BRST operator takes the simple form where Λ α is a D = 11 pure spinor. Therefore, we have proved that the modified Brink-Schwarz like superparticle action (3.5) is equivalent to the theory described by the manifestly Lorentz covariant action and the BRST operator Q = Λ α D α , where ΛΓ m Λ = 0. This theory is the D = 11 pure spinor superparticle.

Light-cone analysis of the pure spinor cohomology
In this section it will be shown that the pure spinor physical condition implies lightcone equations of motion for D = 11 linearized supergravity in D = 9 superspace, which coincide with those found in [7]. To see this, let us write Q in SO (9)

= Λ
where G A is defined by the relation whereî is an SO(8) vector index. This object can be written in the compact form It will be useful to keep in mind the following SO(9) relations which can be deduced from (2.3), (2.4): where D A ,D A are given by or in a more compact form where Γ I AB = (−i(γ 9 γî) AB , iδ AB ), Γ IĀ B = (i(γ 9 γî) AB , iδ AB ). Using the equations (4.5), (4.6) one can show that Notice that the nilpotency ofQ no longer requires the validity of the SO(9) pure spinor constraint Λ A Γ I ABΛ B = 0 as can be seen from (4.11). A further similarity transformation induced by the operatorR will transform the operatorsD A , G A intô Hence the pure spinor BRST operator will take the form The supersymmetry invariance of this operator follows from the supersymmetry invariance ofĜ A andD A under the operatorŝ which are theR-transformed versions of the supersymmetry generators

Light-cone equations of motion
The physical fields are contained in the ghost number 3 superfield V = Λ α Λ β Λ σ C αβσ [5]. This superfield can be written in SO(9) notation as where the signs ± come from the splitting SO(10, 1) → SO(1, 1) × SO (9). The use of the gauge transformation δV =QΩ, with Ω being an arbitrary ghost number 2 superfield, allows us to cancel out the last three terms in (4.21): , Ω (+B)(+C) . Therefore we are left with where we have dropped the SO(1, 1) index for convenience. TheQ-closedness condition for V implies the following equations for C BCD : where χ DA , ξ CD , C JCD , C JKCD , C JKLCD are SO(9) p-form-bispinors. Each of these possesses a certain symmetry determined by (4.23), (4.24). To find the physical spectrum and the corresponding equations of motion, we should solve these equations subject to the constraints: A way to solve this constrained system of equations is the following: Let us choose the only non-zero component of the spinor Λ A to be Λ +0 . This choice will implyΛ −i = Λ +0 = 0, where i is the usual SO(7) vector index. With these constraints, the onlyD A that act non-trivially on C (+0)(+0)(+0) areD −i andD +0 . Therefore, we will have 2 8 states in C (+0)(+0)(+0) : 128 bosonic and 128 fermionic states. The other componens of C ABC can be shown to be related to C (+0)(+0)(+0) by SO(9) rotations (see Appendix C) given by the operator which satisfies the algebra The 128 fermionic states can be adequately represented by the lowest order term inf JD : wheref JD is Γ-traceless. The 128 bosonic states can be accommodated in the SO(9) traceless symmetric tensor g JK and the 3-form H LM N . Therefore we can write After replacing (4.29), (4.30) in (4.23) one obtains Next we use the SO(9) Fierz identities which can be found by using the Mathematica package GAMMA [12], to obtain where the constants a, b will be determined from supersymmetry. To do this we should know howD A acts on g JK and H KLM . An educated guess based on linearity and symmetry properties isD where the factor −2 √ 2P + was chosen for convenience. These equations of motion should satisfy the supersymmetry algebra (4.25). This requirement fixes the values of a, b to be a = 1 4 , b = 1 72 . Therefore the whole set of light-cone equations of motion iŝ The mass-shell condition can be obtained from (4.26) after using the tracelessness condition for C BCD , which is necessary to have a non-trivial vertex operator V . This condition gives rise to the equation: which has solution only ifĜ A C BCD = 0. This result, together with (4.26), implies that k m k m = 0, where k m is the momentum. Consequently, C BCD depends only onΘ, C BCD = C BCD (Θ). To obtain the pure spinor vertex operator in the Q-cohomology one just performs the similarity transformation generated by −(R +R). The result is

Remarks
The equivalence of cohomologies for the D = 11 Brink-Schwarz-like superparticle and the D = 11 pure spinor superparticle is strong evidence that the two models describe the same physical theory. Our method to demonstrate the equivalence uses ideas that were applied previously to the D = 10 case (e.g., the group decomposition SO(10, 1) → SO(1, 1) × SO(9)), and introduces a parametrization of D = 11 objects (the group decomposition SO(10, 1) → SO(3, 1) × SO (7)) which was useful for analyzing the light-cone pure spinor cohomology.
The equations of motion in D = 9 superspace found in this paper, by studying the light-cone pure spinor cohomology, match the light-cone equations of motion presented in [7]. We conclude that the D = 11 pure spinor superparticle is a good model to study D = 11 linearized supergravity in a manifestly covariant way.
which reflects the fact that we don't have Weyl (anti-Weyl) spinors in this case. However, we can have Majorana spinors. It is easy to see that C = Γ 0 satisfies the definition of the charge conjugation matrix 5 CΓ m = −(Γ m ) T C. For two Majorana spinors Θ and Ψ, we haveΘΓ m Ψ = Θ T CΓ m Ψ. This result can be viewed in terms of SO(9, 1) components: where m = 0, . . . , 9 and γm µν , and (γm) µν are the SO(9, 1) γ-matrices. It is useful to mention that the index structure of the charge conjugation matrix is C αβ . So, the Γ-matrices have index structure (Γ m ) α β and when are multiplied by the charge conjugation matrix (or its inverse) we obtain the corresponding matrices (Γ m ) αβ and (Γ m ) αβ .
To construct the above representation of the Γ matrices, we used a basis convenient for dealing with SO(8) objects. Hence, an arbitrary D = 11 spinor χ α is written in this basis as This was the convention used in (3.10). This is useful when SO(8) objects are needed, as in Section 3. However, when analyzing the light-cone structure of the pure spinor cohomology and vertex operators, we need to deal with SO(9) objects. So, we define the following change of basis matrix: where each entry represents an 8 × 8 matrix. Using this matrix we find the corresponding Γ matrices in this new basis: where I AB is the SO(9) identity matrix, A, B are SO(9) spinor indices, andî = 1, . . . , 8.
Each entry in the above matrices is 16 × 16.
The octonion mutiplication table can be written in the form e i e j = −δ ij + ijk e k (C. 5) which is equivalent to e i e j = δ ij − i ijk e k (C. 6) where ijk is a totally antisymmetric tensor with value +1 when (ijk) = (123), (145), (176), (246), (257), (347), (365). Now we can identify these octonions as the gamma matrices of the SO(7) Clifford algebra: This equation can be thought of as the 7-dimensional generalization of the 3-dimensional case τ i τ j = δ ij + ie ijk τ k , (C. 8) where τ i are the ordinary Pauli matrices.