J
H
E
P
0
8
(
2
0
1
4
)
1
0
2

Published for SISSA by Springer

Received: July 2, 2014

Accepted: July 30, 2014

Published: August 18, 2014

Light-cone analysis of the pure spinor formalism for

the superstring

Nathan Berkovits and Renann Lipinski Jusinskas

ICTP South American Institute for Fundamental Research, Instituto de F́ısica Teórica, UNESP

- Univ. Estadual Paulista, Rua Dr. Bento T. Ferraz 271, 01140-070, São Paulo, SP, Brasil

E-mail: nberkovi@ift.unesp.br, renannlj@ift.unesp.br

Abstract: Physical states of the superstring can be described in light-cone gauge by

acting with transverse bosonic αj−n and fermionic q̄ȧ−n operators on an SO(8)-covariant

superfield where j, ȧ = 1 to 8. In the pure spinor formalism, these states are described

in an SO(9,1)-covariant manner by the cohomology of the BRST charge Q = 1
2πi

∮
λαdα.

In this paper, a similarity transformation is found which simplifies the form of Q and

maps the light-cone description of the superstring vertices into DDF-like operators in the

cohomology of Q.

Keywords: Superstrings and Heterotic Strings, BRST Symmetry

ArXiv ePrint: 1406.2290

Open Access, c© The Authors.

Article funded by SCOAP3.
doi:10.1007/JHEP08(2014)102

mailto:nberkovi@ift.unesp.br
mailto:renannlj@ift.unesp.br
http://arxiv.org/abs/1406.2290
http://dx.doi.org/10.1007/JHEP08(2014)102


J
H
E
P
0
8
(
2
0
1
4
)
1
0
2

Contents

1 Introduction 1

2 Superparticle 3

2.1 Review of the pure spinor superparticle 3

2.2 Similarity transformation 4

2.3 Relation with light-cone vertex operators 5

3 Superstring 6

3.1 Review of the pure spinor superstring 7

3.2 Similarity transformation 9

3.3 Relation with light-cone vertex operators 10

4 Conclusion 12

A SO(9,1) to SO(8) decomposition 13

1 Introduction

Although the covariant description of string theory is convenient for amplitude computa-

tions and for describing curved backgrounds, the light-cone description is convenient for

computing the physical spectrum and for proving unitarity. For the manifestly spacetime

supersymmetric string, the light-cone description was worked out over 30 years ago [1] but

the covariant description using the pure spinor formalism [2] is still being developed.

In this paper, the relation between this covariant and light-cone superstring descrip-

tions will be analyzed. As in other string theories, physical states in the pure spinor

formalism are covariantly described by the cohomology of a nilpotent BRST operator.

However, because the pure spinor worldsheet ghost is constrained, evaluation of the BRST

cohomology is not straightforward. By partially solving the pure spinor constraint, it was

proven in [3] that the BRST cohomology reproduces the correct light-cone spectrum. How-

ever, the proof was complicated and involved an infinite set of ghosts-for-ghosts. In this

paper, the proof will be simplified considerably and an explicit similarity transformation

will be given for mapping 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

QB = − 1

2πi

∮ {
1

2
c∂Xm∂Xm + ibc∂c

}
, (1.1)

– 1 –



J
H
E
P
0
8
(
2
0
1
4
)
1
0
2

can be mapped by a similarity transformation R to the operator

Q̂B = eRQBe
−R = −k+

∑
n6=0

c−nα
−
n + c0

kmkm
2

+
∑
n6=0

αi−nα
i
n − 1

 , (1.2)

where α−n , αin and cn are modes of the X−, Xi and c variables, and km is the momentum,

with
√

2k+ =
(
k0 + kd−1

)
assumed to be non-vanishing. Because of the quartet argu-

ment, the cohomology of Q̂B is independent of the (cn, bn, α
+
n , α

−
n ) modes, so the physical

spectrum is the usual light-cone spectrum constructed by acting with the transverse αi−n
modes on the tachyonic ground state. Furthermore, the similarity transformation R maps

light-cone vertex operators VLC which only depend on the transverse variables into the

physical DDF vertex operators VDDF = e−RVLCe
R which are in the cohomology of QB [5].

In the pure spinor formalism, the covariant BRST operator is

Q =
1

2πi

∮
(λαdα) , (1.3)

where α = 1 to 16 is an SO(9,1) spinor index, λα is a bosonic spinor ghost satisfying the

pure spinor constraint

λαγmαβλ
β = 0, (1.4)

and dα is the Green-Schwarz-Siegel fermionic constraint which has 8 first-class and 8 second-

class components. A similarity transformation R will be found which maps Q into

Q̂ = eRQe−R =
1

2πi

∮ [
λa

(
pa +

√
2

2
T̂ θa

)
+ λ̄ȧ

(
p̄ȧ +

√
2

2
∂X+θ̄ȧ

)]
, (1.5)

where θα = (θa, θ̄ȧ) and pα = (pa, p̄ȧ), and a (ȧ) represent the chiral (antichiral) SO(8)

spinor indices. T̂ will include part of the energy-momentum tensor and will impose the

usual Virasoro-like conditions.

The cohomology of Q̂ will be argued to consist of states which are independent

of (θa, pa) and which are constructed from the SO(8)-covariant light-cone superfields

fa(θ̄)eik·X of [4] by hitting with the transverse raising operators αj−n (bosonic) and q̄ȧ−n
(fermionic) as

VLC =
∏
n,j,ȧ

(
αj−n

)Nn,j (
q̄ȧ−n

)Nn,ȧ
λafa(θ̄)e

ik·X (1.6)

where kmkm = −2
∑
n (Nn,i +Nn,ȧ). Furthermore, it will be shown that the similarity

transformation R maps the light-cone vertex operators of (1.6) into DDF-like vertex oper-

ators VDDF = e−RVLCe
R, which are in the cohomology of the pure spinor BRST operator

and which will be described in a separate paper by one of the authors [6]. DDF-like ver-

tex operators in the pure spinor formalism were first constructed by Mukhopadhyay [7]

using a Wess-Zumino-like gauge choice which breaks manifest supersymmetry, whereas the

recent construction of Jusinskas [6] uses a supersymmetric gauge choice which simplifies

the analysis and enables an explicit SO(8) superfield description of the whole physical

spectrum.

– 2 –



J
H
E
P
0
8
(
2
0
1
4
)
1
0
2

In section 2, the superparticle will be discussed and a similarity transformation R will

be constructed which 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 transformation maps the light-cone vertex operators

of (1.6) into the DDF-like vertex operators of [6] in the cohomology of the pure spinor BRST

operator.

2 Superparticle

In this section, the pure spinor formulation of the superparticle will first be reviewed and

a similarity transformation will then be presented which makes the massless constraint

explicit in the BRST operator and maps the super-Yang-Mills vertex operator into the

light-cone SO(8) superfield of [4].

2.1 Review of the pure spinor superparticle

The pure spinor superparticle was extensively discussed in [8] and is described by the first

order action

S =

∫
dτ

{
ẊmP

m − 1

2
PmPm + λ̇αωα − iθ̇αpα

}
, (2.1)

containing the pure spinor ghost λα and the anti-ghost ωα. Note that the dot above the

fields represent derivatives with respect to τ .

The BRST charge is defined as

Q = λαdα, (2.2)

where

dα = pα −
i

2
Pm (γmθ)α (2.3)

is the supersymmetric derivative, and the supersymmetry generators are

qα = pα +
i

2
Pm (γmθ)α . (2.4)

Canonical quantization of (2.1) gives

[Xm, Pn] = iηmn, (2.5a)

{θα, pβ} = iδαβ . (2.5b)

Note that {qα, dβ} = [qα, Pm] = 0 and {dα, dβ} = Pmγ
m
αβ.

To compare this covariant description with the light-cone description, chiral and an-

tichiral SO(9,1) spinors will be decomposed into their SO(8) components as θα → (θa, θ̄ȧ)

and dα → (da, d̄ȧ) where a, ȧ = 1, . . . , 8, and the SO(9,1) vectors will be decomposed as

Xm → (Xj , X+, X−) for j = 1, . . . , 8. The precise conventions of this SO(8) decomposition

are discussed in appendix A, where the SO(9,1) gamma-matrices are expressed in terms of

the SO(8) Pauli matrices σiaȧ. In terms of these Pauli matrices, the D = 10 pure spinor

constraint λαγmαβλ
β = 0 takes the form

λaλa = 0, λ̄ȧλ̄ȧ = 0, λaσiaȧλ̄
ȧ = 0. (2.6)

– 3 –



J
H
E
P
0
8
(
2
0
1
4
)
1
0
2

2.2 Similarity transformation

Since the cohomology of (2.2) is described by the N = 1 D = 10 super Yang-Mills super-

field, the BRST operator must impose the Siegel constraint Pm (γ+γmd)a = 0, which is

the generator of 8 independent kappa symmetries [9] in the Green-Schwarz formalism. To

see that, first note that we can make the constraint explicit in Q by performing a similarity

transformation generated by

R = i
P iN̄i

P+
, (2.7)

where

N̄ i = − 1√
2

(
λaσiaȧω̄

ȧ
)
. (2.8)

Observe that

Q
′ ≡ eRQe−R

= λaGa + λ̄ȧd̄ȧ, (2.9)

where

Ga ≡
Pm

2P+
(γ+γmd)a = da −

Pi
(
σid̄
)
a

P+
√

2
(2.10)

and satisfies {
Ga, d̄ȧ

}
= 0, (2.11a)

{Ga, Gb} = −ηab√
2

(
P 2

P+

)
, (2.11b)

with P 2 = −2P+P− + PiP
i. It will be important to note that equation (2.11a) implies

that nilpotency of Q
′
p does not rely any more on the pure spinor constraint λaλ̄ȧσiaȧ = 0.

The next step in simplifying the BRST operator is to perform the further similarity

transformation generated by

R̂ ≡ iθa (pa −Ga) (2.12)

=
i√
2

(
θaσiaȧp̄

ȧ
) Pi
P+

.

Unlike R of (2.7), R̂ does not commute with the supersymmetry generators and transforms

the various operators as Ô ≡ eR̂Oe−R̂:

ˆ̄dȧ = p̄ȧ −
i√
2
P+θ̄ȧ, (2.13a)

Ĝa = pa +
iθa

2
√

2

(
P 2

P+

)
, (2.13b)

ˆ̄qȧ = p̄ȧ +
i√
2
P+θ̄ȧ, (2.13c)

q̂a = pa +
1√
2

(
σi ˆ̄q
)
a

Pi
P+
− iθa

2
√

2

(
P 2

P+

)
. (2.13d)

– 4 –



J
H
E
P
0
8
(
2
0
1
4
)
1
0
2

As expected, ˆ̄dȧ and Ĝa are supersymmetric with respect to the transformed supersym-

metry generators q̂a and ˆ̄qȧ. After performing this second similarity transformation, the

BRST charge Q̂ ≡ eR̂Q′e−R̂ takes the simple form

Q̂ = λa
(
pa +

i

2
√

2

P 2

P+
θa

)
+ λ̄ȧ

(
p̄ȧ −

i√
2
P+θ̄ȧ

)
, (2.14)

where the mass-shell constraint PmPm now appears explicitly.

2.3 Relation with light-cone vertex operators

Because of the simple form of Q̂, it is easy to compute its cohomology and show equivalence

to the light-cone vertex operators. Consider a state with momentum km (assuming k+ 6= 0)

which is represented by the ghost-number 1 vertex operator

Û = λaÂa + λ̄ȧ ˆ̄Aȧ. (2.15)

The λ̄ȧλ̄ḃ component of Q̂Û = 0 implies

ˆ̄Dȧ
ˆ̄Aḃ + ˆ̄Dḃ

ˆ̄Aȧ = ηȧḃΩ (2.16)

for some superfield Ω, where ˆ̄Dȧ = ∂̄ȧ − k+√
2
θ̄ȧ. The above equation implies that ˆ̄Aȧ =

− 1√
2k+

ˆ̄DȧΩ, which can be set to zero by the gauge transformation δÂα = − 1√
2k+

D̂αΩ.

In the gauge ˆ̄Aȧ = 0, the λaλ̄ȧ component of Q̂Û = 0 (together with the constraint

λaλ̄ȧσiaȧ = 0) implies that
ˆ̄DȧÂa = Âiσ

i
aȧ, (2.17)

for some superfield Âi. Equation (2.17) is precisely the constraint on the SO(8)-covariant

superfield described in [4], and the most general solution is

Âa = Φ (θ) fa(θ̄)e
ik·X , (2.18)

where Φ(θ) is a generic scalar function of θa and, as shown in [4], fa(θ̄) is a light-cone

super-Yang-Mills superfield depending on an SO(8) vector aj and an SO(8) chiral spinor

χa, denoting the transverse polarizations of the gluon and gluino. The explicit formula for

fa(θ̄) is

fa(θ̄) = ai
(
σlθ̄
)
a

{
ηil +

(
1

3!

)
θ̃il +

(
1

5!

)
θ̃ij θ̃jl +

(
1

7!

)
θ̃ij θ̃jkθ̃kl

}
+

(√
2

k+

)
χa (2.19)

+
(
χσiθ̄

) (
σlθ̄
)
a

{(
1

2!

)
ηil +

(
1

4!

)
θ̃il +

(
1

6!

)
θ̃ij θ̃jl +

(
1

8!

)
θ̃ij θ̃jkθ̃kl

}
where θ̃ij ≡

(
k+√
2

)
θ̄ȧθ̄ċσ

ij
ȧċ.

Finally, the λaλb component of Q̂Û = 0 implies that

fa(θ̄)D̂bΦ(θ) + fb(θ̄)D̂aΦ(θ) = δabΣ (2.20)

– 5 –



J
H
E
P
0
8
(
2
0
1
4
)
1
0
2

for some superfield Σ where

D̂b ≡
∂

∂θb
+

√
2

4

(
kmkm
k+

)
θb. (2.21)

Equation (2.20) can only be satisfied if D̂aΦ = 0. Since D̂aD̂a = 2
√

2
(
kmkm
k+

)
, D̂aΦ = 0

implies both that k2 = 0 and that ∂
∂θaΦ = 0 (i.e. Φ is constant). After rescaling the

polarizations by the constant Φ, one finally obtains that the states in the ghost-number

one cohomology of Q̂ are described by

Û = λafa(θ̄)e
ik·X , (2.22)

where fa(θ̄) is the SO(8)-covariant light-cone superfield of (2.19) and kmkm = 0.

States in the cohomology of the original pure spinor BRST operator are directly ob-

tained from Û by defining

U ≡ e−(R+R̂)Ûe(R+R̂) = λafa(θ̂)e
ik·X , (2.23)

where θ̂ȧ ≡ θ̄ȧ+ ki√
2k+

(
σiθ
)
ȧ
. Note that when the state has vanishing transverse momentum,

ki = 0, the similarity transformations R and R̂ of (2.7) and (2.12) vanish and

U |ki=0 = Û = λafa(θ̄)e
−ik+X− . (2.24)

It is easy to verify that U is the usual vertex operator λαAα(X, θ) in the gauge (γ+A)ȧ = 0.

Now we will proceed to the more intricate case of the superstring.

3 Superstring

In this section, we will repeat the analysis done for the superparticle. After reviewing the

pure spinor description of the superstring, we will show that the pure spinor BRST charge

Q = 1
2πi

∮
(λαdα) can be written after a similarity transformation as

Q̂ =
1

2πi

∮ [
λa

(
pa +

√
2

2
T̂ θa

)
+ λ̄ȧ

(
p̄ȧ +

√
2

2
∂X+θ̄ȧ

)]
, (3.1)

where

T̂ ≡ −1

2

(
∂Xm∂Xm

∂X+

)
+ i

(
p̄ȧ∂θ̄ȧ
∂X+

)
+

1

2
∂

(
Ĵ

∂X+

)

−

(
i
√

2

4

)
ˆ̄dȧ∂

ˆ̄dȧ

(∂X+)2
−
(

1

2

)
(∂ ln (∂X+))

2

∂X+
(3.2)

and

Ĵ ≡ −ω̄ȧλ̄ȧ, (3.3)

ˆ̄dȧ ≡ p̄ȧ +

√
2

2
∂X+θ̄ȧ. (3.4)

– 6 –



J
H
E
P
0
8
(
2
0
1
4
)
1
0
2

The structure of Q̂ for the superstring closely resembles (2.14) for the superparticle,

and one can verify that Q̂ is nilpotent using the OPE’s

T̂ (z) λ̄ȧ
ˆ̄dȧ (y) ∼ regular, (3.5)

T̂ (z) T̂ (y) ∼ regular. (3.6)

The cohomology of Q̂ will be shown to reproduce the usual light-cone superstring spectrum

where the similarity transformation maps the DDF-like vertex operators of [6] into the 8

transverse bosonic and fermionic operators, αj−n and q̄ȧ−n, which create massive superstring

states from the massless ground state in light-cone gauge.

3.1 Review of the pure spinor superstring

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

Smatter =
1

2π

∫
d2z

(
1

2
∂Xm∂̄Xm + ipβ ∂̄θ

β

)
, (3.7)

and the free field OPE’s

Xm (z, z̄)Xn (y, ȳ) ∼ −ηmn ln |z − y|2 , (3.8a)

pα (z) θβ (y) ∼ iδβα
z − y

. (3.8b)

The supersymmetry charge is

qα =
1

2π

∮ {
−pα +

1

2
∂Xm (γmθ)α +

i

24
(θγm∂θ) (γmθ)α

}
, (3.9)

satisfying {qα, qβ} = Pmγ
m
αβ, where Pm ≡ 1

2π

∮
∂Xm. The usual supersymmetric invariants

are

Πm = ∂Xm +
i

2
(θγm∂θ) , (3.10)

dα = pα +
1

2
∂Xm (γmθ)α +

i

8
(θγm∂θ) (γmθ)α , (3.11)

and the OPE’s among them are easily computed to be

Πm (z) Πn (y) ∼ − ηmn

(z − y)2
, (3.12a)

dα (z) Πm (y) ∼ −
γmαβ∂θ

β

(z − y)
, (3.12b)

dα (z) dβ (y) ∼ i
γmαβΠm

(z − y)
. (3.12c)

The main feature of the formalism is its simple BRST charge, given by

Q =
1

2πi

∮
(λαdα) , (3.13)

– 7 –



J
H
E
P
0
8
(
2
0
1
4
)
1
0
2

where λα is the bosonic ghost. Nilpotency of Q is achieved when λα is constrained by

λγmλ = 0, the pure spinor condition. The conjugate of λα will be represented by ωα and

they can be described by the Lorentz covariant action

Sλ =
1

2π

∫
d2z

(
ωα∂̄λ

α
)
, (3.14)

which has the gauge invariance δωα = φm (γmλ)α due to the pure spinor constraint. The

gauge invariant quantities are the Lorentz current, Nmn = −1
2 (ωγmnλ), the ghost number

current, J = −ωλ, and the energy-momentum tensor of the ghost sector, Tλ = −ω∂λ. The

pure spinor constraint also implies the classical constraints on the currents:

Nmn (γnλ)α +
1

2
J (γmλ)α = 0. (3.15)

Nmn (γmn∂λ)α + J∂λα = −4λαTλ. (3.16)

The physical open string spectrum is described by the ghost number one cohomology

of Q. For example, the massless states are described by the unintegrated vertex

U = λαAα (X, θ) , (3.17)

where Aα is a superfield composed of the zero-modes of (Xm, θα). Note that

{Q,U} = λαλβDαAβ, (3.18)

where

Dα ≡ i∂α −
1

2
(γmθ)α ∂m, (3.19)

with ∂α = ∂
∂θα , ∂m = ∂

∂Xm . Since λα is a pure spinor, λαλβ ∝ γαβmnpqr (λγmnpqrλ) and

{Q,U} = 0 implies the linearized super Yang-Mills equation of motion [11]:

DγmnpqrA = 0. (3.20)

The integrated version of (3.17) is given by

V =
1

2πi

∮
{ΠmAm + i∂θαAα + idαW

α +NmnFmn} , (3.21)

where Am and Wα are the super Yang-Mills fields, constrained by

Am ≡
1

8i

(
Dαγ

αβ
m Aβ

)
, (3.22a)

(γmW )α ≡ (DαAm + ∂mAα) , (3.22b)

and Fmn = 1
2 (∂mAn − ∂nAm).

– 8 –



J
H
E
P
0
8
(
2
0
1
4
)
1
0
2

3.2 Similarity transformation

To show that the cohomology of (3.13) describes the light-cone superstring spectrum, it will

be convenient to follow the same procedure as in the previous section for the superparticle.

The superstring version of the similarity transformation of (2.7) is

R = − 1

2πi

∮ {
N̄iΠ

i

Π+

}
, (3.23)

and transforms λα and dα as

eRλae
−R = λa,

eRλ̄ȧe
−R = λ̄ȧ −

(
σiλ
)
ȧ

Πi√
2Π+

−

(√
2

4

) (
σiλ
)
ȧ
∂N̄ i

(Π+)2
,

eRdae
−R = da −

(
σi∂θ̄

)
a
N̄i

Π+
−
√

2
∂θaN̄iΠ

i

(Π+)2
,

eRd̄ȧe
−R = d̄ȧ −

(
σi∂θ

)
ȧ
N̄i

Π+
.

Using the above relations together with the properties N̄i

(
σiλ̄
)
a

=
√

2λaĴ and N̄iN̄
i = 0,

which follow from the SO(8) decomposition of (3.15), we obtain

Q
′ ≡ 1

2πi

∮
eR (λαdα) e−R (3.24)

=
1

2πi

∮ {
λa
(
Ga −

√
2
∂θa
Π+

Ĵ

)
+ λ̄ȧd̄ȧ −

(√
2

4

) (
λaσiaȧd̄

ȧ
)
∂N̄ i

(Π+)2

}
, (3.25)

where

Ga ≡ da −
(
σid̄
)
a√

2

(
Πi

Π+

)
. (3.26)

Note that although normal-ordering contributions are being ignored in the explicit compu-

tations, the only terms that can receive quantum corrections are

∂2θa
Π+

and ∂

(
∂θa
Π+

)
(3.27)

and their coefficients can be determined by requiring nilpotency of Q
′
.

The term proportional to ∂N̄ i in (3.25) did not appear in the superparticle BRST

operator of (2.9), however, it can fortunately be removed by performing a second similarity

transformation generated by

R′ = −
√

2

8π

∮ {
N̄i

(
∂θσid̄

)
(Π+)2

}
. (3.28)

After this transformation, the BRST operator Q′′ = eR
′
Q′e−R

′
is

Q
′′

=
1

2πi

∮ {
λa

(
Ga + ∂θbHab −

(√
2

2

)
∂θaĴ

Π+

)
+ λ̄ȧd̄ȧ

}
, (3.29)

– 9 –



J
H
E
P
0
8
(
2
0
1
4
)
1
0
2

where

Hab = −Hba =

(
i

4

) (
σid̄
)
a

(
σid̄
)
b

(Π+)2
, (3.30)

and the possible normal-ordering contributions have the same form of those in (3.27) and

can be determined in an analogous manner. Similar to the superparticle BRST charge Q
′

of (2.9), Q
′′

of (3.29) is manifestly supersymmetric and is nilpotent without requiring the

pure spinor constraint
(
λσiλ̄

)
= 0 because(

Ga + ∂θbHab −

(√
2

2

)
∂θaĴ

Π+

)
(z)
(
λ̄ȧd̄ȧ

)
(y) ∼ regular. (3.31)

To reduce Q′′ to the form of Q̂ in (3.1), one needs to perform a further similarity

transformation which is a generalization of R̂ presented in (2.12) for the superparticle.

Expanding in powers of θa, one finds that

R̂ =
1

2πi

∮ {
i√
2

∂Xi

∂X+

(
θσip̄

)
+

√
2

8

(
θσip̄

) (
∂θσiθ̄

)
∂X+

−
√

2

8

(
θσi∂θ̄

) (
θσi ˆ̄d

)
∂X+

− 1

8

(
∂θσi ˆ̄d

)(
θσi ˆ̄d

)
(∂X+)2

+
i

2
√

2

(
θ∂θ

∂X+

)
Ĵ + . . .

}
(3.32)

where . . . denotes terms which are at least cubic order in θa, θij = θaθcσijac and

ˆ̄dȧ ≡ p̄ȧ +

√
2

2
∂X+θ̄ȧ.

The first term in (3.32) is the same as in the superparticle R̂ of (2.12) while the second

is required to transform the supersymmetry generator ˆ̄qȧ ≡ eR̂q̄ȧe−R̂ to the simple form

ˆ̄qȧ = − 1

2π

∮ {
p̄ȧ −

√
2

2
∂X+θ̄ȧ

}
. (3.33)

The terms in the second line of (3.32) commute with ˆ̄qȧ and are necessary so that Q̂ ≡
eR̂Q

′′
e−R̂ has at most linear dependence on θa. Using the explicit terms in (3.32), it was

verified up to linear order in θa that

Q̂ =
1

2πi

∮ [
λa

(
pa +

√
2

2
T̂ θa

)
+ λ̄ȧ

(
p̄ȧ +

√
2

2
∂X+θ̄ȧ

)]
, (3.34)

where T̂ is defined in (3.2).

3.3 Relation with light-cone vertex operators

To compute the cohomology of Q̂ of (3.34), note that the zero mode structure of Q̂ is the

same as in the superparticle Q̂ of (2.14), so the superstring ground state describing the

massless states is

Û = λafa(θ̄)e
ik·X (3.35)

of (2.22) where fa(θ̄) is the SO(8)-covariant light-cone superfield of (2.19) and kmkm = 0.

– 10 –



J
H
E
P
0
8
(
2
0
1
4
)
1
0
2

To construct massive states in the cohomology, first note that the integrated vertex

operators

αjn ≡
1

2πi

∮
∂Xj exp

(
in

k+
X+

)
, (3.36a)

q̄ȧn ≡ −
1

2π

∮ (
p̄ȧ −

√
2

2
∂X+θ̄ȧ

)
exp

(
in

k+
X+

)
, (3.36b)

are in the cohomology of Q̂ for any value of n. The relation to the usual Laurent modes

becomes clear when X+ (z) = −ik+ ln (z), i.e. in light-cone gauge where X+ is the world-

sheet time coordinate. In this gauge, exp
(
in
k+
X+
)

= zn and we recover the usual Laurent

expansion.

One interpretation of the integrated vertex operators of (3.36) is as massless integrated

vertex operators for the 8 physical polarizations of the gluon and gluino with momenta pj =

p+ = 0 and p− = n
k+

. However, as will be discussed in [6], another interpretation of (3.36)

is as DDF-like operators which act on the ground state vertex operator of (3.35) with ki = 0

to create excited state vertex operators that describe the massive superstring states. If X+

in (3.36) 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 αj−n and q̄ȧ−n around the ground state vertex

operator Û = λafa(θ̄)e
ik·X will produce the excited state vertex operators

αj−nÛ =
1

(n− 1)!

(
∂nXj + . . .

)
λafa(θ̄)e

i
(
kjXj−k+X−−

(
k−+ n

k+

)
X+

)
, (3.37a)

q̄ȧ−nÛ =
1

(n− 1)!

[
∂n−1

(
p̄ȧ +

ik+

n
√

2
∂θ̄ȧ
)

+ . . .

]
λafa(θ̄)e

i
(
kjXj−k+X−−

(
k−+ n

k+

)
X+

)
,

(3.37b)

where . . . denotes terms proportional to derivatives of X+. One can similarly act with any

number of αj−n and q̄ȧ−n operators on the ground state vertex operator to construct the

general excited state vertex operator∏
n>0

∏
ȧ

∏
j

(
αj−n

)Nn,j (
q̄ȧ−n

)Nn,ȧ
Û . (3.38)

Since αj−n and q̄ȧ−n commute with Q̂, it is clear that the vertex operators of (3.38) are

BRST-closed. And it is easy to see they are not BRST-exact since the worldsheet variables

∂Xj and
(
p̄ȧ −

√
2
2 ∂X

+θ̄ȧ

)
only appear in Q̂ through T̂ . Furthermore, one expects that

there are no other states in the cohomology of Q̂ as the terms (λapa) and
(
λ̄ȧ ˆ̄dȧ

)
in (3.34)

imply that the cohomology is independent of (θa, pa), and depends on
(
θ̄ȧ, p̄ȧ

)
only through

combinations that anticommute with ˆ̄dȧ. Also, as in bosonic string theory, the dependence

on (X+, X−) is completely fixed by T̂ . So the cohomology of Q̂ is expected to be described

by the states of (3.38) which are in one-to-one correspondence with the usual light-cone

Green-Schwarz states of the superstring spectrum.

Since the original pure spinor BRST operator Q is related to Q̂ by Q =

e−R−R
′
e−R̂Q̂eR̂eR+R

′
, where the similarity transformations R, R

′
and R̂ are defined

– 11 –



J
H
E
P
0
8
(
2
0
1
4
)
1
0
2

in (3.23), (3.28) and (3.32), covariant BRST-invariant vertex operators in the pure spinor

formalism can be related to the vertex operators of (3.38) by acting with these same simi-

larity transformations.

To see how this works, it will be useful to interpret αjn and q̄ȧn of (3.36) as integrated

vertex operators for a massless gluon and gluino with momenta pj = p+ = 0 and p− = n
k+

,

and to combine them into a super-Yang-Mills vertex operator by contracting them with

the gluon and gluino polarization aj and χ̄ȧ as

V̂−n ≡ ajαj−n − iχ̄ȧq̄ȧ−n. (3.39)

After performing the similarity transformation with R̂ of (3.32), one finds (up to terms

quadratic in θa) that

V
′
−n ≡ e−R̂V̂−neR̂ =

1

2πi

∮ {
ΠiA

i +

(
i∂θ̄ȧ +

n√
2k+

d̄ȧ

)
Aȧ
}

(3.40)

where DaAȧ = iσjaȧAj , with Da = ∂a − n
k+
√
2
θa. Aȧ is an SO(8)-superfield that depends

only on θa and X+, in an exact parallel to fa
(
θ̄
)
e−ik

+X− which depends only on θ̄ȧ and

X− and satisfies the constraint (2.17). See [6] for further details on Aȧ and how it emerges

in the pure spinor cohomology.

Finally, the gauge fixed version of the integrated massless vertex of (3.21) is obtained

by acting with the similarity transformation R+R′ of (3.23) and (3.28) which transforms

V ′−n into

V−n ≡ e−R−R
′
V
′
−ne

R+R
′

(3.41)

=
1

2πi

∮ {(
Πi − i

n

k+
N̄i

)
Ai +

(
i∂θ̄ȧ +

n√
2k+

d̄ȧ

)
Aȧ

}
.

It is straightforward to see that R
′

commutes with V
′
−n, and that R is responsible for

reintroducing the ghost Lorentz current N̄i in the vertex.

A detailed discussion of the properties of the DDF-like operators (3.41) is presented

in [6]. As shown there, the superstring spectrum is obtained by acting with the above

operators on the SO(8)-covariant ground state Û |ki=0 of (3.35), allowing a systematic

description of all massive pure spinor vertex operators in terms of SO(8) superfields.

4 Conclusion

In this paper, the pure spinor BRST operator Q = 1
2πi

∮
(λαdα) was mapped by the similar-

ity transformations of R, R
′

and R̂ of (3.23), (3.28) and (3.32) into the nilpotent operator

Q̂ =
1

2πi

∮ [
λa

(
pa +

√
2

2
T̂ θa

)
+ λ̄ȧ ˆ̄dȧ

]
, (4.1)

where T̂ is defined in (3.2). The cohomology of Q̂ is the usual light-cone Green-Schwarz
superstring spectrum and is described by the vertex operators∏

n>0

∏
ȧ

∏
j

[
∂n−1

(
p̄ȧ +

ik+

n
√

2
∂θ̄ȧ
)

+ . . .

]Nn,ȧ (
∂nXj + . . .

)Nn,j
λafa(θ̄)ei(k

jXj−k+X−−k̃−X+),

(4.2)

– 12 –



J
H
E
P
0
8
(
2
0
1
4
)
1
0
2

where

k̃− =
1

k+

[
kiki

2
+
∑
n

n (Nn,ȧ +Nn,j)

]
, (4.3)

fa(θ̄) is the SO(8)-covariant superfield (2.19) of reference [4], and . . . involves derivatives of

X+. Finally, the similarity transformations of R, R′ and R̂ were argued to map the vertex

operators of (4.2) into pure spinor BRST-invariant vertex operators constructed using the

DDF-operators described in [6].

Acknowledgments

NB would like to thank CNPq grant 300256/94-9 and FAPESP grants 2009/50639-2 and

2011/11973-4 for partial financial support, and RLJ would like to thank FAPESP grant

2009/17516-4 for financial support.

A SO(9,1) to SO(8) decomposition

Given an SO(9,1) chiral spinor λα (antichiral λ̄α), one can write down its SO(8) components

through the use of the projectors PαI and (PαI )−1 ≡ P Iα, where I generically indicates the

SO(8) indices, defined in such a way that

λα = Pαa λ
a + Pαȧ λ

ȧ, λa = P aαλ
α, λȧ = P ȧαλ

α,

λ̄α = P aα λ̄a + P ȧα λ̄ȧ, λ̄a = Pαa λ̄α, λ̄ȧ = Pαȧ λ̄α.

Being invertible, they satisfy

Pαa P
b
α = δba, Pαȧ P

ḃ
α = δḃȧ,

Pαa P
ȧ
α = 0, δαβ = Pαa P

a
β + Pαȧ P

ȧ
β ,

where a, ȧ = 1, . . . , 8 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 can define a spinorial metric, ηab (ηȧḃ), and its inverse, ηab (ηȧḃ), such that ηacη
cb =

δba (ηȧċη
ċḃ = δḃȧ) and are responsible for lowering and raising spinorial indices, respectively,

acting as charge conjugation. For example,
(
σi
)ȧa

= ηabηȧḃ
(
σi
)
bḃ

.

Using the projectors, one can build a representation for the gamma matrices γm in

terms of the 8-dimensional equivalent of the Pauli matrices,

(
γi
)αβ ≡ (σi)ȧa (Pαȧ P βa + Pαa P

β
ȧ

)
,
(
γi
)
αβ
≡
(
σi
)
aȧ

(
P aαP

ȧ
β + P ȧαP

a
β

)
,(

γ−
)αβ ≡ √2ηabPαa P

β
b ,

(
γ−
)
αβ
≡ −
√

2ηȧḃP
ȧ
αP

ḃ
β,(

γ+
)αβ ≡ √2ηȧḃPαȧ P

β

ḃ
,

(
γ+
)
αβ
≡ −
√

2ηabP
a
αP

b
β,

(A.1)

– 13 –



J
H
E
P
0
8
(
2
0
1
4
)
1
0
2

where (
σi
)
aȧ

(
σj
)ȧb

+
(
σj
)
aȧ

(
σi
)ȧb

= 2ηijδba, (A.2a)(
σi
)ȧa (

σj
)
aḃ

+
(
σj
)ȧa (

σi
)
aḃ

= 2ηijδȧ
ḃ
, (A.2b)(

σi
)
aȧ

(σi)cċ +
(
σi
)
cȧ

(σi)aċ = 2ηacηȧċ, (A.2c)(
σij
)a
b

(
σij
)c
d

= 8ηacηbd − 8δadδ
c
b , (A.2d)(

σij
)a
b

(
σij
)ȧ
ḃ

= 4σȧai σ
i
bḃ
− 4δab δ

ȧ
ḃ
, (A.2e)

and ηij is the flat SO(8) inverse metric, with i, j = 1, . . . , 8. As usual, ηikη
kj = δji . Note

that {
γi, γj

}
= 2ηij ,{

γ+, γ−
}

= −2,{
γ±, γi

}
=
{
γ+, γ+

}
=
{
γ−, γ−

}
= 0.

(A.3)

Given a SO(9,1) vector Nm, the SO(8) decomposition used in this work is,

N± ≡ 1√
2

(
N0 ±N9

)
, (A.4)

and N i, with i = 1, . . . , 8, stands for the spatial components. Therefore, the scalar product

between Nm and Pm is given by NmPm = −N+P− −N−P+ +N iPi.

For a rank-2 antisymmetric tensor Nmn, the SO(8) components will be represented as{
N ij , N i = N−i, N̄ i = N+i, N = N+−} .

Open Access. This article is distributed under the terms of the Creative Commons

Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in

any medium, provided the original author(s) and source are credited.

References

[1] J.H. Schwarz, Superstring Theory, Phys. Rept. 89 (1982) 223 [INSPIRE].

[2] N. Berkovits, Super Poincaré covariant quantization of the superstring, JHEP 04 (2000) 018

[hep-th/0001035] [INSPIRE].

[3] N. Berkovits, Cohomology in the pure spinor formalism for the superstring, JHEP 09 (2000)

046 [hep-th/0006003] [INSPIRE].

[4] L. Brink, M.B. Green and J.H. Schwarz, Ten-dimensional Supersymmetric Yang-Mills

Theory With SO(8) - Covariant Light Cone Superfields, Nucl. Phys. B 223 (1983) 125

[INSPIRE].

[5] Y. Aisaka and Y. Kazama, Relating Green-Schwarz and extended pure spinor formalisms by

similarity transformation, JHEP 04 (2004) 070 [hep-th/0404141] [INSPIRE].

[6] R.L. Jusinskas, Spectrum generating algebra for the pure spinor superstring,

arXiv:1406.1902 [INSPIRE].

[7] P. Mukhopadhyay, DDF construction and D-brane boundary states in pure spinor formalism,

JHEP 05 (2006) 055 [hep-th/0512161] [INSPIRE].

– 14 –

http://creativecommons.org/licenses/by/4.0/
http://dx.doi.org/10.1016/0370-1573(82)90087-4
http://inspirehep.net/search?p=find+J+Phys.Rept.,89,223
http://dx.doi.org/10.1088/1126-6708/2000/04/018
http://arxiv.org/abs/hep-th/0001035
http://inspirehep.net/search?p=find+EPRINT+hep-th/0001035
http://dx.doi.org/10.1088/1126-6708/2000/09/046
http://dx.doi.org/10.1088/1126-6708/2000/09/046
http://arxiv.org/abs/hep-th/0006003
http://inspirehep.net/search?p=find+EPRINT+hep-th/0006003
http://dx.doi.org/10.1016/0550-3213(83)90096-2
http://inspirehep.net/search?p=find+J+Nucl.Phys.,B223,125
http://dx.doi.org/10.1088/1126-6708/2004/04/070
http://arxiv.org/abs/hep-th/0404141
http://inspirehep.net/search?p=find+EPRINT+hep-th/0404141
http://arxiv.org/abs/1406.1902
http://inspirehep.net/search?p=find+EPRINT+arXiv:1406.1902
http://dx.doi.org/10.1088/1126-6708/2006/05/055
http://arxiv.org/abs/hep-th/0512161
http://inspirehep.net/search?p=find+EPRINT+hep-th/0512161


J
H
E
P
0
8
(
2
0
1
4
)
1
0
2

[8] N. Berkovits, Covariant quantization of the superparticle using pure spinors, JHEP 09

(2001) 016 [hep-th/0105050] [INSPIRE].

[9] W. Siegel, Hidden Local Supersymmetry in the Supersymmetric Particle Action, Phys. Lett.

B 128 (1983) 397 [INSPIRE].

[10] W. Siegel, Classical Superstring Mechanics, Nucl. Phys. B 263 (1986) 93 [INSPIRE].

[11] P.S. Howe, Pure spinors lines in superspace and ten-dimensional supersymmetric theories,

Phys. Lett. B 258 (1991) 141 [Addendum ibid. B 259 (1991) 511] [INSPIRE].

– 15 –

http://dx.doi.org/10.1088/1126-6708/2001/09/016
http://dx.doi.org/10.1088/1126-6708/2001/09/016
http://arxiv.org/abs/hep-th/0105050
http://inspirehep.net/search?p=find+EPRINT+hep-th/0105050
http://dx.doi.org/10.1016/0370-2693(83)90924-3
http://dx.doi.org/10.1016/0370-2693(83)90924-3
http://inspirehep.net/search?p=find+J+Phys.Lett.,B128,397
http://dx.doi.org/10.1016/0550-3213(86)90029-5
http://inspirehep.net/search?p=find+J+Nucl.Phys.,B263,93
http://dx.doi.org/10.1016/0370-2693(91)91221-G
http://inspirehep.net/search?p=find+J+Phys.Lett.,B258,141

	Introduction
	Superparticle
	Review of the pure spinor superparticle
	Similarity transformation
	Relation with light-cone vertex operators

	Superstring
	Review of the pure spinor superstring
	Similarity transformation
	Relation with light-cone vertex operators

	Conclusion
	SO(9,1) to SO(8) decomposition