Light-Cone Analysis of the Pure Spinor Formalism for the Superstring
aa r X i v : . [ h e p - t h ] J un Light-Cone Analysis of the Pure Spinor Formalism for theSuperstring
Nathan Berkovits ∗ andRenann Lipinski Jusinskas † ICTP South American Institute for Fundamental ResearchInstituto de Física Teórica, UNESP - Univ. Estadual PaulistaRua Dr. Bento T. Ferraz 271, 01140-070, São Paulo, SP, Brasil.
Abstract
Physical states of the superstring can be described in light-cone gauge by acting with transversebosonic α j − n and fermionic ¯ q ˙ a − n operators on an SO (8) -covariant superfield where j, ˙ a = 1 to .In the pure spinor formalism, these states are described in an SO (9 , -covariant manner by thecohomology of the BRST charge Q = πi ¸ λ α d α . In this paper, a similarity transformation is foundwhich simplifies the form of Q and maps the light-cone description of the superstring vertices intoDDF-like operators in the cohomology of Q . Contents ∗ [email protected] † [email protected] Conclusion 13A SO (9 , to SO (8) decomposition 14 Although the covariant description of string theory is convenient for amplitude computations and for de-scribing curved backgrounds, the light-cone description is convenient for computing the physical spectrumand for proving unitarity. For the manifestly spacetime supersymmetric string, the light-cone descriptionwas worked out over 30 years ago [1] but the covariant description using the pure spinor formalism [2] isstill being developed.In this paper, the relation between this covariant and light-cone superstring descriptions will beanalyzed. As in other string theories, physical states in the pure spinor formalism are covariantly describedby the cohomology of a nilpotent BRST operator. However, because the pure spinor worldsheet ghostis constrained, evaluation of the BRST cohomology is not straightforward. By partially solving thepure spinor constraint, it was proven in [3] that the BRST cohomology reproduces the correct light-conespectrum. However, the proof was complicated and involved an infinite set of ghosts-for-ghosts. In thispaper, the proof will be simplified considerably and an explicit similarity transformation will be given formapping light-cone superstring vertex operators constructed from the SO (8) -covariant superfields of [4]into DDF-like vertex operators in the cohomology of the pure spinor BRST operator.In bosonic string theory, the covariant BRST operator Q B = − πi ˛ (cid:26) c∂X m ∂X m + ibc∂c (cid:27) , (1)can be mapped by a similarity transformation R to the operator ˆ Q B = e R Q B e − R = − k + X n =0 c − n α − n + c k m k m X n =0 α i − n α in − , (2)where α − n , α in and c n are modes of the X − , X i and c variables, and k m is the momentum, with √ k + = (cid:0) k + k d − (cid:1) assumed to be non-vanishing. Because of the quartet argument, the cohomology of ˆ Q B isindependent of the ( c n , b n , α + n , α − n ) modes, so the physical spectrum is the usual light-cone spectrumconstructed by acting with the transverse α i − n modes on the tachyonic ground state. Furthermore, thesimilarity transformation R maps light-cone vertex operators V LC which only depend on the transversevariables into the physical DDF vertex operators V DDF = e − R V LC e R which are in the cohomology of Q B [5]. In the pure spinor formalism, the covariant BRST operator is Q = 12 πi ˛ ( λ α d α ) , (3)2here α = 1 to is an SO (9 , spinor index, λ α is a bosonic spinor ghost satisfying the pure spinorconstraint λ α γ mαβ λ β = 0 , (4)and d α is the Green-Schwarz-Siegel fermionic constraint which has first-class and second-class com-ponents. A similarity transformation R will be found which maps Q into ˆ Q = e R Qe − R = 12 πi ˛ " λ a p a + √
22 ˆ
T θ a ! + ¯ λ ˙ a ¯ p ˙ a + √ ∂X + ¯ θ ˙ a ! , (5)where θ α = ( θ a , ¯ θ ˙ a ) and p α = ( p a , ¯ p ˙ a ) , and a ( ˙ a ) represent the chiral (antichiral) SO (8) spinor indices. ˆ T will include part of the energy-momentum tensor and will impose the usual Virasoro-like conditions.The cohomology of ˆ Q will be argued to consist of states which are independent of ( θ a , p a ) and whichare constructed from the SO (8) -covariant light-cone superfields f a (¯ θ ) e ik · X of [4] by hitting with thetransverse raising operators α j − n (bosonic) and ¯ q ˙ a − n (fermionic) as V LC = Y n,j, ˙ a (cid:16) α j − n (cid:17) N n,j (cid:16) ¯ q ˙ a − n (cid:17) N n, ˙ a λ a f a (¯ θ ) e ik · X (6)where k m k m = − P n ( N n,i + N n, ˙ a ) . Furthermore, it will be shown that the similarity transformation R maps the light-cone vertex operators of (6) into DDF-like vertex operators V DDF = e − R V LC e R , which arein the cohomology of the pure spinor BRST operator and which will be described in a separate paper byone of the authors [6]. DDF-like vertex operators in the pure spinor formalism were first constructed byMukhopadhyay [7] using a Wess-Zumino-like gauge choice which breaks manifest supersymmetry, whereasthe recent construction of Jusinskas [6] uses a supersymmetric gauge choice which simplifies the analysisand enables an explicit SO (8) superfield description of the whole physical spectrum.In section 2, the superparticle will be discussed and a similarity transformation R will be constructedwhich maps the superparticle BRST operator into a simple quadratic form and maps the light-cone SO (8) -covariant superfield of [4] into the super-Yang-Mills vertex operator in the pure spinor formalism.In section 3, this construction will be generalized to the superstring such that the similarity transformationmaps the light-cone vertex operators of (6) into the DDF-like vertex operators of [6] in the cohomologyof the pure spinor BRST operator. In this section, the pure spinor formulation of the superparticle will first be reviewed and a similaritytransformation will then be presented which makes the massless constraint explicit in the BRST operatorand maps the super-Yang-Mills vertex operator into the light-cone SO (8) superfield of [4].3 .1 Review of the pure spinor superparticle The pure spinor superparticle was extensively discussed in [8] and is described by the first order action S = ˆ dτ (cid:26) ˙ X m P m − P m P m + ˙ λ α ω α − i ˙ θ α p α (cid:27) , (7)containing the pure spinor ghost λ α and the anti-ghost ω α . Note that the dot above the fields representderivatives with respect to τ .The BRST charge is defined as Q = λ α d α , (8)where d α = p α − i P m ( γ m θ ) α (9)is the supersymmetric derivative, and the supersymmetry generators are q α = p α + i P m ( γ m θ ) α . (10)Canonical quantization of (7) gives [ X m , P n ] = iη mn , (11a) { θ α , p β } = iδ αβ . (11b)Note that { q α , d β } = [ q α , P m ] = 0 and { d α , d β } = P m γ mαβ .To compare this covariant description with the light-cone description, chiral and antichiral SO (9 , spinors will be decomposed into their SO (8) components as θ α → ( θ a , ¯ θ ˙ a ) and d α → ( d a , ¯ d ˙ a ) where a, ˙ a = 1 , . . . , , and the SO (9 , vectors will be decomposed as X m → ( X j , X + , X − ) for j = 1 , . . . , .The precise conventions of this SO (8) decomposition are discussed in appendix A, where the SO (9 , gamma-matrices are expressed in terms of the SO (8) Pauli matrices σ ia ˙ a . In terms of these Pauli matrices,the D = 10 pure spinor constraint λ α γ mαβ λ β = 0 takes the form λ a λ a = 0 , ¯ λ ˙ a ¯ λ ˙ a = 0 , λ a σ ia ˙ a ¯ λ ˙ a = 0 . (12) Since the cohomology of (8) is described by the N = 1 D = 10 super Yang-Mills superfield, the BRSToperator must impose the Siegel constraint P m ( γ + γ m d ) a = 0 , which is the generator of independentkappa symmetries [9] in the Green-Schwarz formalism. To see that, first note that we can make theconstraint explicit in Q by performing a similarity transformation generated by R = i P i ¯ N i P + , (13)4here ¯ N i = − √ (cid:0) λ a σ ia ˙ a ¯ ω ˙ a (cid:1) . (14)Observe that Q ′ ≡ e R Qe − R = λ a G a + ¯ λ ˙ a ¯ d ˙ a , (15)where G a ≡ P m P + ( γ + γ m d ) a = d a − P i (cid:0) σ i ¯ d (cid:1) a P + √ (16)and satisfies (cid:8) G a , ¯ d ˙ a (cid:9) = 0 , (17a) { G a , G b } = − η ab √ (cid:18) P P + (cid:19) , (17b)with P = − P + P − + P i P i . It will be important to note that equation (17a) implies that nilpotency of Q ′ p does not rely any more on the pure spinor constraint λ a ¯ λ ˙ a σ ia ˙ a = 0 .The next step in simplifying the BRST operator is to perform the further similarity transformationgenerated by ˆ R ≡ iθ a ( p a − G a ) (18) = i √ (cid:0) θ a σ ia ˙ a ¯ p ˙ a (cid:1) P i P + . Unlike R of (13), ˆ R does not commute with the supersymmetry generators and transforms the variousoperators as ˆ O ≡ e ˆ R O e − ˆ R : ˆ¯ d ˙ a = ¯ p ˙ a − i √ P + ¯ θ ˙ a , (19a) ˆ G a = p a + iθ a √ (cid:18) P P + (cid:19) , (19b) ˆ¯ q ˙ a = ¯ p ˙ a + i √ P + ¯ θ ˙ a , (19c) ˆ q a = p a + 1 √ (cid:0) σ i ˆ¯ q (cid:1) a P i P + − iθ a √ (cid:18) P P + (cid:19) . (19d)As expected, ˆ¯ d ˙ a and ˆ G a are supersymmetric with respect to the transformed supersymmetry generators ˆ q a and ˆ¯ q ˙ a . After performing this second similarity transformation, the BRST charge ˆ Q ≡ e ˆ R Q ′ e − ˆ R takesthe simple form ˆ Q = λ a (cid:18) p a + i √ P P + θ a (cid:19) + ¯ λ ˙ a (cid:18) ¯ p ˙ a − i √ P + ¯ θ ˙ a (cid:19) , (20)5here the mass-shell constraint P m P m now appears explicitly. Because of the simple form of ˆ Q , it is easy to compute its cohomology and show equivalence to the light-cone vertex operators. Consider a state with momentum k m (assuming k + = 0 ) which is represented bythe ghost-number 1 vertex operator ˆ U = λ a ˆ A a + ¯ λ ˙ a ˆ¯ A ˙ a . (21)The ¯ λ ˙ a ¯ λ ˙ b component of ˆ Q ˆ U = 0 implies ˆ¯ D ˙ a ˆ¯ A ˙ b + ˆ¯ D ˙ b ˆ¯ A ˙ a = η ˙ a ˙ b Ω (22)for some superfield Ω , where ˆ¯ D ˙ a = ¯ ∂ ˙ a − k + √ ¯ θ ˙ a . The above equation implies that ˆ¯ A ˙ a = − √ k + ˆ¯ D ˙ a Ω , whichcan be set to zero by the gauge transformation δ ˆ A α = − √ k + ˆ D α Ω .In the gauge ˆ¯ A ˙ a = 0 , the λ a ¯ λ ˙ a component of ˆ Q ˆ U = 0 (together with the constraint λ a ¯ λ ˙ a σ ia ˙ a = 0 )implies that ˆ¯ D ˙ a ˆ A a = ˆ A i σ ia ˙ a , (23)for some superfield ˆ A i . Equation (23) is precisely the constraint on the SO (8) -covariant superfielddescribed in [4], and the most general solution is ˆ A a = Φ ( θ ) f a (¯ θ ) e ik · X , (24)where Φ( θ ) is a generic scalar function of θ a and, as shown in [4], f a (¯ θ ) is a light-cone super-Yang-Millssuperfield depending on an SO (8) vector a j and an SO (8) chiral spinor χ a , denoting the transversepolarizations of the gluon and gluino. The explicit formula for f a (¯ θ ) is f a (¯ θ ) = a i (cid:0) σ l ¯ θ (cid:1) a (cid:26) η il + (cid:18) (cid:19) ˜ θ il + (cid:18) (cid:19) ˜ θ ij ˜ θ jl + (cid:18) (cid:19) ˜ θ ij ˜ θ jk ˜ θ kl (cid:27) + √ k + ! χ a (25) + (cid:0) χσ i ¯ θ (cid:1) (cid:0) σ l ¯ θ (cid:1) a (cid:26)(cid:18) (cid:19) η il + (cid:18) (cid:19) ˜ θ il + (cid:18) (cid:19) ˜ θ ij ˜ θ jl + (cid:18) (cid:19) ˜ θ ij ˜ θ jk ˜ θ kl (cid:27) where ˜ θ ij ≡ (cid:16) k + √ (cid:17) ¯ θ ˙ a ¯ θ ˙ c σ ij ˙ a ˙ c .Finally, the λ a λ b component of ˆ Q ˆ U = 0 implies that f a (¯ θ ) ˆ D b Φ( θ ) + f b (¯ θ ) ˆ D a Φ( θ ) = δ ab Σ (26)for some superfield Σ where ˆ D b ≡ ∂∂θ b + √ (cid:18) k m k m k + (cid:19) θ b . (27)Equation (26) can only be satisfied if ˆ D a Φ = 0 . Since ˆ D a ˆ D a = 2 √ (cid:16) k m k m k + (cid:17) , ˆ D a Φ = 0 implies both that6 = 0 and that ∂∂θ a Φ = 0 ( i.e. Φ is constant). After rescaling the polarizations by the constant Φ , onefinally obtains that the states in the ghost-number one cohomology of ˆ Q are described by ˆ U = λ a f a (¯ θ ) e ik · X , (28)where f a (¯ θ ) is the SO (8) -covariant light-cone superfield of (25) and k m k m = 0 .States in the cohomology of the original pure spinor BRST operator are directly obtained from ˆ U bydefining U ≡ e − ( R + ˆ R ) ˆ U e ( R + ˆ R ) = λ a f a (ˆ θ ) e ik · X , (29)where ˆ θ ˙ a ≡ ¯ θ ˙ a + k i √ k + (cid:0) σ i θ (cid:1) ˙ a . Note that when the state has vanishing transverse momentum, k i = 0 , thesimilarity transformations R and ˆ R of (13) and (18) vanish and U | k i =0 = ˆ U = λ a f a (¯ θ ) e − ik + X − . (30)It is easy to verify that U is the usual vertex operator λ α A α ( X, θ ) in the gauge ( γ + A ) ˙ a = 0 .Now we will proceed to the more intricate case of the superstring. In this section, we will repeat the analysis done for the superparticle. After reviewing the pure spinordescription of the superstring, we will show that the pure spinor BRST charge Q = πi ¸ ( λ α d α ) can bewritten after a similarity transformation as ˆ Q = 12 πi ˛ " λ a p a + √
22 ˆ
T θ a ! + ¯ λ ˙ a ¯ p ˙ a + √ ∂X + ¯ θ ˙ a ! , (31)where ˆ T ≡ − (cid:18) ∂X m ∂X m ∂X + (cid:19) + i (cid:18) ¯ p ˙ a ∂ ¯ θ ˙ a ∂X + (cid:19) + 12 ∂ ˆ J∂X + ! − i √ ! ˆ¯ d ˙ a ∂ ˆ¯ d ˙ a ( ∂X + ) − (cid:18) (cid:19) ( ∂ ln ( ∂X + )) ∂X + (32)and ˆ J ≡ − ¯ ω ˙ a ¯ λ ˙ a , (33) ˆ¯ d ˙ a ≡ ¯ p ˙ a + √ ∂X + ¯ θ ˙ a . (34)The structure of ˆ Q for the superstring closely resembles (20) for the superparticle, and one can verify7hat ˆ Q is nilpotent using the OPE’s ˆ T ( z ) ¯ λ ˙ a ˆ¯ d ˙ a ( y ) ∼ regular , (35) ˆ T ( z ) ˆ T ( y ) ∼ regular . (36)The cohomology of ˆ Q will be shown to reproduce the usual light-cone superstring spectrum where thesimilarity transformation maps the DDF-like vertex operators of [6] into the transverse bosonic andfermionic operators, α j − n and ¯ q ˙ a − n , which create massive superstring states from the massless ground statein light-cone gauge. The matter (holomorphic) sector of the pure spinor formalism is constructed from the Green-Schwarz-Siegel variables of [10] and is described by the free action S matter = 12 π ˆ d z (cid:18) ∂X m ¯ ∂X m + ip β ¯ ∂θ β (cid:19) , (37)and the free field OPE’s X m ( z, ¯ z ) X n ( y, ¯ y ) ∼ − η mn ln | z − y | , (38a) p α ( z ) θ β ( y ) ∼ iδ βα z − y . (38b)The supersymmetry charge is q α = 12 π ˛ (cid:26) − p α + 12 ∂X m ( γ m θ ) α + i
24 ( θγ m ∂θ ) ( γ m θ ) α (cid:27) , (39)satisfying { q α , q β } = P m γ mαβ , where P m ≡ π ¸ ∂X m . The usual supersymmetric invariants are Π m = ∂X m + i θγ m ∂θ ) , (40) d α = p α + 12 ∂X m ( γ m θ ) α + i θγ m ∂θ ) ( γ m θ ) α , (41)and the OPE’s among them are easily computed to be Π m ( z ) Π n ( y ) ∼ − η mn ( z − y ) , (42a) d α ( z ) Π m ( y ) ∼ − γ mαβ ∂θ β ( z − y ) , (42b) d α ( z ) d β ( y ) ∼ i γ mαβ Π m ( z − y ) . (42c)8he main feature of the formalism is its simple BRST charge, given by Q = 12 πi ˛ ( λ α d α ) , (43)where λ α is the bosonic ghost. Nilpotency of Q is achieved when λ α is constrained by λγ m λ = 0 , thepure spinor condition. The conjugate of λ α will be represented by ω α and they can be described by theLorentz covariant action S λ = 12 π ˆ d z (cid:0) ω α ¯ ∂λ α (cid:1) , (44)which has the gauge invariance δω α = φ m ( γ m λ ) α due to the pure spinor constraint. The gauge invariantquantities are the Lorentz current, N mn = − ( ωγ mn λ ) , the ghost number current, J = − ωλ , and theenergy-momentum tensor of the ghost sector, T λ = − ω∂λ . The pure spinor constraint also implies theclassical constraints on the currents: N mn ( γ n λ ) α + 12 J ( γ m λ ) α = 0 . (45) N mn ( γ mn ∂λ ) α + J∂λ α = − λ α T λ . (46)The physical open string spectrum is described by the ghost number one cohomology of Q . Forexample, the massless states are described by the unintegrated vertex U = λ α A α ( X, θ ) , (47)where A α is a superfield composed of the zero-modes of ( X m , θ α ) . Note that { Q, U } = λ α λ β D α A β , (48)where D α ≡ i∂ α −
12 ( γ m θ ) α ∂ m , (49)with ∂ α = ∂∂θ α , ∂ m = ∂∂X m . Since λ α is a pure spinor, λ α λ β ∝ γ αβmnpqr ( λγ mnpqr λ ) and { Q, U } = 0 impliesthe linearized super Yang-Mills equation of motion [11]: Dγ mnpqr A = 0 . (50)The integrated version of (47) is given by V = 12 πi ˛ { Π m A m + i∂θ α A α + id α W α + N mn F mn } , (51)where A m and W α are the super Yang-Mills fields, constrained by A m ≡ i (cid:0) D α γ αβm A β (cid:1) , (52a) ( γ m W ) α ≡ ( D α A m + ∂ m A α ) , (52b)9nd F mn = ( ∂ m A n − ∂ n A m ) . To show that the cohomology of (43) describes the light-cone superstring spectrum, it will be convenientto follow the same procedure as in the previous section for the superparticle. The superstring version ofthe similarity transformation of (13) is R = − πi ˛ (cid:26) ¯ N i Π i Π + (cid:27) , (53)and transforms λ α and d α as e R λ a e − R = λ a ,e R ¯ λ ˙ a e − R = ¯ λ ˙ a − (cid:0) σ i λ (cid:1) ˙ a Π i √ + − √ ! (cid:0) σ i λ (cid:1) ˙ a ∂ ¯ N i (Π + ) ,e R d a e − R = d a − (cid:0) σ i ∂ ¯ θ (cid:1) a ¯ N i Π + − √ ∂θ a ¯ N i Π i (Π + ) ,e R ¯ d ˙ a e − R = ¯ d ˙ a − (cid:0) σ i ∂θ (cid:1) ˙ a ¯ N i Π + . Using the above relations together with the properties ¯ N i (cid:0) σ i ¯ λ (cid:1) a = √ λ a ˆ J and ¯ N i ¯ N i = 0 , which followfrom the SO (8) decomposition of (45), we obtain Q ′ ≡ πi ˛ e R ( λ α d α ) e − R (54) = 12 πi ˛ ( λ a (cid:18) G a − √ ∂θ a Π + ˆ J (cid:19) + ¯ λ ˙ a ¯ d ˙ a − √ ! (cid:0) λ a σ ia ˙ a ¯ d ˙ a (cid:1) ∂ ¯ N i (Π + ) ) , (55)where G a ≡ d a − (cid:0) σ i ¯ d (cid:1) a √ (cid:18) Π i Π + (cid:19) . (56)Note that although normal-ordering contributions are being ignored in the explicit computations, theonly terms that can receive quantum corrections are ∂ θ a Π + and ∂ (cid:18) ∂θ a Π + (cid:19) (57)and their coefficients can be determined by requiring nilpotency of Q ′ .The term proportional to ∂ ¯ N i in (55) did not appear in the superparticle BRST operator of (15),however, it can fortunately be removed by performing a second similarity transformation generated by R ′ = − √ π ˛ ( ¯ N i (cid:0) ∂θσ i ¯ d (cid:1) (Π + ) ) . (58)10fter this transformation, the BRST operator Q ′′ = e R ′ Q ′ e − R ′ is Q ′′ = 12 πi ˛ ( λ a G a + ∂θ b H ab − √ ! ∂θ a ˆ J Π + ! + ¯ λ ˙ a ¯ d ˙ a ) , (59)where H ab = − H ba = (cid:18) i (cid:19) (cid:0) σ i ¯ d (cid:1) a (cid:0) σ i ¯ d (cid:1) b (Π + ) , (60)and the possible normal-ordering contributions have the same form of those in (57) and can be determinedin an analogous manner. Similar to the superparticle BRST charge Q ′ of (15), Q ′′ of (59) is manifestlysupersymmetric and is nilpotent without requiring the pure spinor constraint (cid:0) λσ i ¯ λ (cid:1) = 0 because G a + ∂θ b H ab − √ ! ∂θ a ˆ J Π + ! ( z ) (cid:0) ¯ λ ˙ a ¯ d ˙ a (cid:1) ( y ) ∼ regular . (61)To reduce Q ′′ to the form of ˆ Q in (31), one needs to perform a further similarity transformation whichis a generalization of ˆ R presented in (18) for the superparticle. Expanding in powers of θ a , one finds that ˆ R = 12 πi ˛ ( i √ ∂X i ∂X + (cid:0) θσ i ¯ p (cid:1) + √ (cid:0) θσ i ¯ p (cid:1) (cid:0) ∂θσ i ¯ θ (cid:1) ∂X + − √ (cid:0) θσ i ∂ ¯ θ (cid:1) (cid:16) θσ i ˆ¯ d (cid:17) ∂X + − (cid:16) ∂θσ i ˆ¯ d (cid:17) (cid:16) θσ i ˆ¯ d (cid:17) ( ∂X + ) + i √ (cid:18) θ∂θ∂X + (cid:19) ˆ J + . . . (62)where ... denotes terms which are at least cubic order in θ a , θ ij = θ a θ c σ ijac and ˆ¯ d ˙ a ≡ ¯ p ˙ a + √ ∂X + ¯ θ ˙ a . The first term in (62) is the same as in the superparticle ˆ R of (18) while the second is required totransform the supersymmetry generator ˆ¯ q ˙ a ≡ e ˆ R ¯ q ˙ a e − ˆ R to the simple form ˆ¯ q ˙ a = − π ˛ ( ¯ p ˙ a − √ ∂X + ¯ θ ˙ a ) . (63)The terms in the second line of (62) commute with ˆ¯ q ˙ a and are necessary so that ˆ Q ≡ e ˆ R Q ′′ e − ˆ R has atmost linear dependence on θ a . Using the explicit terms in (62), it was verified up to linear order in θ a that ˆ Q = 12 πi ˛ " λ a p a + √
22 ˆ
T θ a ! + ¯ λ ˙ a ¯ p ˙ a + √ ∂X + ¯ θ ˙ a ! , (64)where ˆ T is defined in (32). 11 .3 Relation with light-cone vertex operators To compute the cohomology of ˆ Q of (64), note that the zero mode structure of ˆ Q is the same as in thesuperparticle ˆ Q of (20), so the superstring ground state describing the massless states is ˆ U = λ a f a (¯ θ ) e ik · X (65)of (28) where f a (¯ θ ) is the SO (8) -covariant light-cone superfield of (25) and k m k m = 0 .To construct massive states in the cohomology, first note that the integrated vertex operators α jn ≡ πi ˛ ∂X j exp (cid:18) ink + X + (cid:19) , (66a) ¯ q ˙ an ≡ − π ˛ ¯ p ˙ a − √ ∂X + ¯ θ ˙ a ! exp (cid:18) ink + X + (cid:19) , (66b)are in the cohomology of ˆ Q for any value of n . The relation to the usual Laurent modes becomes clearwhen X + ( z ) = − ik + ln ( z ) , i.e. in light-cone gauge where X + is the worldsheet time coordinate. In thisgauge, exp (cid:0) ink + X + (cid:1) = z n and we recover the usual Laurent expansion.One interpretation of the integrated vertex operators of (66) is as massless integrated vertex operatorsfor the physical polarizations of the gluon and gluino with momenta p j = p + = 0 and p − = nk + .However, as will be discussed in [6], another interpretation of (66) is as DDF-like operators which acton the ground state vertex operator of (65) with k i = 0 to create excited state vertex operators thatdescribe the massive superstring states. If X + in (66) is treated as a holomorphic variable with the OPE X + ( z ) X − ( y ) ∼ ln ( z − y ) and n is a positive integer, the contour integral of α j − n and ¯ q ˙ a − n around theground state vertex operator ˆ U = λ a f a (¯ θ ) e ik · X will produce the excited state vertex operators α j − n ˆ U = 1( n − (cid:0) ∂ n X j + . . . (cid:1) λ a f a (¯ θ ) e i ( k j X j − k + X − − ( k − + nk + ) X + ) , (67a) ¯ q ˙ a − n ˆ U = 1( n − (cid:20) ∂ n − (cid:18) ¯ p ˙ a + ik + n √ ∂ ¯ θ ˙ a (cid:19) + . . . (cid:21) λ a f a (¯ θ ) e i ( k j X j − k + X − − ( k − + nk + ) X + ) , (67b)where . . . denotes terms proportional to derivatives of X + . One can similarly act with any number of α j − n and ¯ q ˙ a − n operators on the ground state vertex operator to construct the general excited state vertexoperator Y n> Y ˙ a Y j (cid:16) α j − n (cid:17) N n,j (cid:16) ¯ q ˙ a − n (cid:17) N n, ˙ a ˆ U . (68)Since α j − n and ¯ q ˙ a − n commute with ˆ Q , it is clear that the vertex operators of (68) are BRST-closed.And it is easy to see they are not BRST-exact since the worldsheet variables ∂X j and (cid:16) ¯ p ˙ a − √ ∂X + ¯ θ ˙ a (cid:17) only appear in ˆ Q through ˆ T . Furthermore, one expects that there are no other states in the cohomologyof ˆ Q as the terms ( λ a p a ) and (cid:16) ¯ λ ˙ a ˆ¯ d ˙ a (cid:17) in (64) imply that the cohomology is independent of ( θ a , p a ) , anddepends on (cid:0) ¯ θ ˙ a , ¯ p ˙ a (cid:1) only through combinations that anticommute with ˆ¯ d ˙ a . Also, as in bosonic stringtheory, the dependence on ( X + , X − ) is completely fixed by ˆ T . So the cohomology of ˆ Q is expected12o be described by the states of (68) which are in one-to-one correspondence with the usual light-coneGreen-Schwarz states of the superstring spectrum.Since the original pure spinor BRST operator Q is related to ˆ Q by Q = e − R − R ′ e − ˆ R ˆ Qe ˆ R e R + R ′ , wherethe similarity transformations R , R ′ and ˆ R are defined in (53), (58) and (62), covariant BRST-invariantvertex operators in the pure spinor formalism can be related to the vertex operators of (68) by actingwith these same similarity transformations.To see how this works, it will be useful to interpret α jn and ¯ q ˙ an of (66) as integrated vertex operatorsfor a massless gluon and gluino with momenta p j = p + = 0 and p − = nk + , and to combine them into asuper-Yang-Mills vertex operator by contracting them with the gluon and gluino polarization a j and ¯ χ ˙ a as ˆ V − n ≡ a j α j − n − i ¯ χ ˙ a ¯ q ˙ a − n . (69)After performing the similarity transformation with ˆ R of (62), one finds (up to terms quadratic in θ a )that V ′ − n ≡ e − ˆ R ˆ V − n e ˆ R = 12 πi ˛ (cid:26) Π i A i + (cid:18) i∂ ¯ θ ˙ a + n √ k + ¯ d ˙ a (cid:19) A ˙ a (cid:27) (70)where D a A ˙ a = iσ ja ˙ a A j , with D a = ∂ a − nk + √ θ a . A ˙ a is an SO (8) -superfield that depends only on θ a and X + , in an exact parallel to f a (cid:0) ¯ θ (cid:1) e − ik + X − which depends only on ¯ θ ˙ a and X − and satisfies the constraint(23). See [6] for further details on A ˙ a and how it emerges in the pure spinor cohomology.Finally, the gauge fixed version of the integrated massless vertex of (51) is obtained by acting withthe similarity transformation R + R ′ of (53) and (58) which transforms V ′− n into V − n ≡ e − R − R ′ V ′ − n e R + R ′ (71) = 12 πi ˛ (cid:26)(cid:16) Π i − i nk + ¯ N i (cid:17) A i + (cid:18) i∂ ¯ θ ˙ a + n √ k + ¯ d ˙ a (cid:19) A ˙ a (cid:27) . It is straightforward to see that R ′ commutes with V ′ − n , and that R is responsible for reintroducing theghost Lorentz current ¯ N i in the vertex.A detailed discussion of the properties of the DDF-like operators (71) is presented in [6]. As shownthere, the superstring spectrum is obtained by acting with the above operators on the SO (8) -covariantground state ˆ U | k i =0 of (65), allowing a systematic description of all massive pure spinor vertex operatorsin terms of SO (8) superfields. In this paper, the pure spinor BRST operator Q = πi ¸ ( λ α d α ) was mapped by the similarity transform-ations of R , R ′ and ˆ R of (53), (58) and (62) into the nilpotent operator ˆ Q = 12 πi ˛ " λ a p a + √
22 ˆ
T θ a ! + ¯ λ ˙ a ˆ¯ d ˙ a , (72)13here ˆ T is defined in (32). The cohomology of ˆ Q is the usual light-cone Green-Schwarz superstringspectrum and is described by the vertex operators Y n> Y ˙ a Y j (cid:20) ∂ n − (cid:18) ¯ p ˙ a + ik + n √ ∂ ¯ θ ˙ a (cid:19) + . . . (cid:21) N n, ˙ a (cid:0) ∂ n X j + . . . (cid:1) N n,j λ a f a (¯ θ ) e i ( k j X j − k + X − − ˜ k − X + ) , (73)where ˜ k − = 1 k + " k i k i X n n ( N n, ˙ a + N n,j ) , (74) f a (¯ θ ) is the SO (8) -covariant superfield (25) of reference [4], and ... involves derivatives of X + . Finally,the similarity transformations of R , R ′ and ˆ R were argued to map the vertex operators of (73) into purespinor BRST-invariant vertex operators constructed using the DDF-operators described in [6]. Acknowledgements
NB would like to thank CNPq grant 300256/94-9 and FAPESP grants 2009/50639-2 and 2011/11973-4for partial financial support, and RLJ would like to thank FAPESP grant 2009/17516-4 for financialsupport. A SO (9 , to SO (8) decomposition Given an SO (9 , chiral spinor λ α (antichiral ¯ λ α ), one can write down its SO (8) components throughthe use of the projectors P αI and ( P αI ) − ≡ P Iα , where I generically indicates the SO (8) indices, definedin such a way that λ α = P αa λ a + P α ˙ a λ ˙ a , λ a = P aα λ α , λ ˙ a = P ˙ aα λ α , ¯ λ α = P aα ¯ λ a + P ˙ aα ¯ λ ˙ a , ¯ λ a = P αa ¯ λ α , ¯ λ ˙ a = P α ˙ a ¯ λ α . Being invertible, they satisfy P αa P bα = δ ba , P α ˙ a P ˙ bα = δ ˙ b ˙ a ,P αa P ˙ aα = 0 , δ αβ = P αa P aβ + P α ˙ a P ˙ aβ , where a, ˙ a = 1 , . . . , are the SO (8) spinorial indices, representing different chiralities.Note that upper and lower indices in the SO (8) language do not distinguish chiralities, i.e. , one candefine a spinorial metric, η ab ( η ˙ a ˙ b ), and its inverse, η ab ( η ˙ a ˙ b ), such that η ac η cb = δ ba ( η ˙ a ˙ c η ˙ c ˙ b = δ ˙ b ˙ a ) andare responsible for lowering and raising spinorial indices, respectively, acting as charge conjugation. Forexample, (cid:0) σ i (cid:1) ˙ aa = η ab η ˙ a ˙ b (cid:0) σ i (cid:1) b ˙ b .Using the projectors, one can build a representation for the gamma matrices γ m in terms of the14 -dimensional equivalent of the Pauli matrices, (cid:0) γ i (cid:1) αβ ≡ (cid:0) σ i (cid:1) ˙ aa (cid:16) P α ˙ a P βa + P αa P β ˙ a (cid:17) , (cid:0) γ i (cid:1) αβ ≡ (cid:0) σ i (cid:1) a ˙ a (cid:16) P aα P ˙ aβ + P ˙ aα P aβ (cid:17) , ( γ − ) αβ ≡ √ η ab P αa P βb , ( γ − ) αβ ≡ −√ η ˙ a ˙ b P ˙ aα P ˙ bβ , ( γ + ) αβ ≡ √ η ˙ a ˙ b P α ˙ a P β ˙ b , ( γ + ) αβ ≡ −√ η ab P aα P bβ , (75)where (cid:0) σ i (cid:1) a ˙ a (cid:0) σ j (cid:1) ˙ ab + (cid:0) σ j (cid:1) a ˙ a (cid:0) σ i (cid:1) ˙ ab = 2 η ij δ ba , (76a) (cid:0) σ i (cid:1) ˙ aa (cid:0) σ j (cid:1) a ˙ b + (cid:0) σ j (cid:1) ˙ aa (cid:0) σ i (cid:1) a ˙ b = 2 η ij δ ˙ a ˙ b , (76b) (cid:0) σ i (cid:1) a ˙ a ( σ i ) c ˙ c + (cid:0) σ i (cid:1) c ˙ a ( σ i ) a ˙ c = 2 η ac η ˙ a ˙ c , (76c) (cid:0) σ ij (cid:1) ab (cid:0) σ ij (cid:1) cd = 8 η ac η bd − δ ad δ cb , (76d) (cid:0) σ ij (cid:1) ab (cid:0) σ ij (cid:1) ˙ a ˙ b = 4 σ ˙ aai σ ib ˙ b − δ ab δ ˙ a ˙ b , (76e)and η ij is the flat SO (8) inverse metric, with i, j = 1 , . . . , . As usual, η ik η kj = δ ji . Note that (cid:8) γ i , γ j (cid:9) = 2 η ij , { γ + , γ − } = − , (cid:8) γ ± , γ i (cid:9) = { γ + , γ + } = { γ − , γ − } = 0 . (77)Given a SO (9 , vector N m , the SO (8) decomposition used in this work is, N ± ≡ √ (cid:0) N ± N (cid:1) , (78)and N i , with i = 1 , . . . , , stands for the spatial components. Therefore, the scalar product between N m e P m is given by N m P m = − N + P − − N − P + + N i P i .For a rank- antisymmetric tensor N mn , the SO (8) components will be represented as (cid:8) N ij , N i = N − i , ¯ N i = N + i , N = N + − (cid:9) . References [1] J. H. Schwarz, “Superstring Theory,” Phys. Rept. , 223 (1982).[2] N. Berkovits, “Super Poincare covariant quantization of the superstring”, JHEP , 018 (2000)[hep-th/0001035].[3] N. Berkovits, “Cohomology in the pure spinor formalism for the superstring”, JHEP , 046 (2000)[hep-th/0006003].[4] L. Brink, M. B. Green and J. H. Schwarz, “Ten-dimensional Supersymmetric Yang-Mills TheoryWith SO(8) - Covariant Light Cone Superfields”, Nucl. Phys. B , 125 (1983).155] Y. Aisaka and Y. Kazama, “Relating Green-Schwarz and extended pure spinor formalisms by simil-arity transformation,” JHEP , 070 (2004) [hep-th/0404141].[6] R. L. Jusinskas, “Spectrum generating algebra for the pure spinor superstring,” arXiv:1406.1902[hep-th].[7] P. Mukhopadhyay, “DDF construction and D-brane boundary states in pure spinor formalism,” JHEP , 055 (2006) [hep-th/0512161].[8] N. Berkovits, “Covariant quantization of the superparticle using pure spinors,” JHEP , 016(2001) [hep-th/0105050].[9] W. Siegel, “Hidden Local Supersymmetry in the Supersymmetric Particle Action,” Phys. Lett. B , 397 (1983).[10] W. Siegel, “Classical Superstring Mechanics”, Nucl. Phys. B , 93 (1986).[11] P. S. Howe, “Pure spinors lines in superspace and ten-dimensional supersymmetric theories,” Phys.Lett. B , 141 (1991) [Addendum-ibid. B259