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Published for SISSA by Springer

Received: May 5, 2016

Accepted: June 14, 2016

Published: June 21, 2016

Untwisting the pure spinor formalism to the RNS and

twistor string in a flat and AdS5 × S5 background

Nathan Berkovits

ICTP South American Institute for Fundamental Research,

Instituto de F́ısica Teórica, UNESP — Universidade Estadual Paulista,

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

E-mail: nberkovi@ift.unesp.br

Abstract: The pure spinor formalism for the superstring can be formulated as a twisted

N=2 worldsheet theory with fermionic generators jBRST and composite b ghost. After

untwisting the formalism to an N=1 worldsheet theory with fermionic stress tensor jBRST+

b, the worldsheet variables combine into N=1 worldsheet superfields Xm and Θα together

with a superfield constraint relating DXm and DΘα. The constraint implies that the

worldsheet superpartner of θα is a bosonic twistor variable, and different solutions of the

constraint give rise to the pure spinor or extended RNS formalisms, as well as a new

twistor-string formalism with manifest N=1 worldsheet supersymmetry.

These N=1 worldsheet methods generalize in curved Ramond-Ramond backgrounds,

and a manifestly N=1 worldsheet supersymmetric action is proposed for the superstring in

an AdS5 × S5 background in terms of the twistor superfields. This AdS5 × S5 worldsheet

action is a remarkably simple fermionic coset model with manifest PSU(2, 2|4) symmetry

and might be useful for computing AdS5 × S5 superstring scattering amplitudes.

Keywords: Superstrings and Heterotic Strings, AdS-CFT Correspondence, Topological

Strings

ArXiv ePrint: 1604.04617

Dedicated to the memory of Mario Tonin.

Open Access, c© The Authors.

Article funded by SCOAP3.
doi:10.1007/JHEP06(2016)127

mailto:nberkovi@ift.unesp.br
http://arxiv.org/abs/1604.04617
http://dx.doi.org/10.1007/JHEP06(2016)127


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Contents

1 Introduction 1

2 Untwisting the pure spinor formalism 4

2.1 N=1 generators and superfields 4

2.2 Worldsheet supersymmetric action 6

2.3 U(1) generator 8

2.4 Massless vertex operators 9

2.5 Tree-level scattering amplitudes 10

2.6 Extended RNS formalism 11

2.7 Worldsheet supersymmetric action in curved background 13

3 Twistor string formalism 15

3.1 Twistor superfields 15

3.2 N=1 superconformal and U(1) generator 17

3.3 Worldsheet action in a flat background 17

3.4 Twistor string in AdS5 × S5 background 20

3.5 U(1) generator and d=12 pure spinors 22

1 Introduction

The pure spinor formalism for the superstring [1] has the advantage over the Ramond-

Neveu-Schwarz (RNS) formalism in that is manifestly spacetime supersymmetric. This

simplifies the computation of multiloop superstring amplitudes [2] since there is no sum over

spin structures, and allows the description of Ramond-Ramond superstring backgrounds

such as AdS5 × S5 [3]. However, the pure spinor formalism has the disadvantage that it is

not manifestly worldsheet supersymmetric. This complicates the construction of the b ghost

and integrated vertex operators, and introduces subtleties associated with regulators [4]

and contact terms [5] needed to preserve BRST invariance.

Although the pure spinor formalism is not manifestly worldsheet supersymmetric, it

has a twisted N=2 worldsheet supersymmetry in which the two fermionic N=2 generators

are the BRST current and the b ghost [6]. In this paper, the N=2 worldsheet supersym-

metry will be untwisted and the pure spinor formalism will be described in a manifestly

N=1 worldsheet supersymmetric and d=10 spacetime supersymmetric manner in terms of

the N=1 worldsheet superfields

Xm = xm + κψm, Θα = θα + κΛα, Φα = Ωα + κhα, (1.1)

where κ is the anticommuting coordinate, (xm, θα) are the usual d=10 superspace variables,

(ψm,Λα) are their worldsheet superpartners, and (Ωα, hα) are the conjugate momenta

to (Λα, θα).

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The N=1 worldsheet superfields of (1.1) are constrained to satisfy

λγmΦ = 0, (λγm)α

(
DXm −

1

2
DΘγmΘ

)
= 0, (1.2)

where λα is a fixed d=10 pure spinor satisfying λγmλ = 0. Although the constraints of (1.2)

break manifest Lorentz covariance, one can solve these constraints using three different

methods to produce three different Lorentz-covariant descriptions of the superstring.

The first method is to solve for ψm and Λα in terms of (xm, θα, λα) where λα is a d=10

pure spinor satisfying λγmλ = 0. This method produces the pure spinor formalism which is

manifestly spacetime supersymmetric but not manifestly worldsheet supersymmetric, and

where the N=1 fermionic generator is the sum of the pure spinor BRST current and b ghost.

The second method is to solve for θα and Λα in terms of (xm, ψm, θ′α, λα) where θ′α is

constrained to satisfy θ′γmλ = 0 and λα is constrained to satisfy λγmλ = 0. This method

is manifestly worldsheet supersymmetric where λα is the worldsheet superpartner of θ′α,

but is not manifestly spacetime supersymmetric. One can argue that (θ′α, λα) decouples

from physical vertex operators and scattering amplitudes, so this method produces an

“extended” version of the RNS formalism where Xm = xm + κψm plays the role of the

usual RNS matter superfield.

Finally, the third method is to solve for xm and ψm in terms of (Λα, θα) and its conju-

gate momenta (Ωα, hα). This method preserves both manifest worldsheet supersymmetry

and spacetime supersymmetry, and produces a twistor description of the superstring in

which (Λα,Ωα) are d=10 twistor variables which replace the xm spacetime variable.

There are several similarities of this worldsheet supersymmetric twistor description

with earlier twistor descriptions of the superstring in [7–14], however, these earlier twistor

descriptions were mostly for the heterotic superstring whereas this twistor description is

only for the Type II superstring. It would be very interesting to study the relation of these

twistor descriptions to each other, as well as to the more recent twistor superstrings which

describe either N=4 d=4 super-Yang-Mills [15, 16] or d=10 supergravity [17, 18].

In a flat background, the N=(1,1) worldsheet supersymmetric action for the Type II

twistor superstring is

S =

∫
d2zd2κ

[
−ΦαDΘα + Φ̂α̂DΘ̂α̂ − 1

8
(ΘγmDΘ)(Θ̂γmDΘ̂) +

1

8
(Θ̂γmDΘ̂)(ΘγmDΘ)

]
(1.3)

where D = ∂
∂κ+κ∂z and D = ∂

∂κ+κ∂z, (Θα,Φα, Θ̂
α̂, Φ̂α̂) are N=(1,1) worldsheet superfields

and α, α̂ = 1 to 16 are d=10 spinor indices of the same/opposite chirality for the Type

IIB/IIA superstring. This action is manifestly invariant under both N=(1,1) worldsheet

supersymmetry and d=10 N=2 spacetime supersymmetry which transforms the worldsheet

superfields as

δΘα = εα, δΘ̂α̂ = ε̂α̂, (1.4)

δΦα =
1

4
(εγmΘ + ε̂γmΘ̂)(γmDΘ), δΦ̂α̂ =

1

4
(εγmΘ + ε̂γmΘ̂)(γmDΘ̂).

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Surprisingly, when expressed in terms of these twistor superfields, the Type IIB super-

string action in an AdS5 × S5 background takes the extremely simple form

S = r2

∫
d2zd2κ

[
DΘJ

RDΘ̃R
J +DΘJ

SΘ̃S
KDΘK

R Θ̃R
J

]
(1.5)

where r is the AdS radius, R = 1 to 4 are SO(4, 2) spinor indices for AdS5, J = 1 to 4

are SO(6) spinor indices for S5, the 16 components of the superfield ΘJ
R are obtained by

decomposing Θα+iΘ̂α̂ under SO(4, 2)×SO(6), and the 16 components of the superfield Θ̃R
J

are obtained by decomposing Θα− iΘ̂α̂. This AdS5×S5 twistor-string action is manifestly

invariant under both N=(1,1) worldsheet supersymmetry and under PSU(2, 2|4) where the

32 spacetime supersymmetries transform the worldsheet superfields as

δΘJ
R = εJR + ΘJ

S ε̃
S
KΘK

R , δΘ̃R
J = ε̃RJ − ε̃RKΘK

S Θ̃S
J − Θ̃R

KΘK
S ε̃

S
J . (1.6)

Hopefully, the simple form of (1.5) will be useful for constructing vertex operators and

computing superstring scattering amplitudes in an AdS5×S5 background. But before con-

structing vertex operators and computing scattering amplitudes using this twistor-string

action in an AdS×S5 background, it will be necessary to better understand the vertex oper-

ators and scattering amplitudes using the twistor-string action in a flat background of (1.3).

In section 2.1, the N=2 worldsheet supersymmetry of the pure spinor formalism is un-

twisted to an N=1 worldsheet supersymmetry with the fermionic generator G = jBRST + b,

and the constrained N=1 worldsheet superfields [Xm,Θα,Φα] satisfying (1.2) are defined.

In section 2.2, the N=1 worldsheet supersymmetric action with manifest d=10 supersym-

metry is constructed for the superstring in a flat background in terms of these constrained

superfields. In section 2.3, the U(1) generator J = −λαwα is used to define physical states

whose integrated vertex operator is required to be N=1 superconformally invariant and have

zero or negative U(1) charge. In section 2.4, physical vertex operators are constructed for

massless states and the Siegel gauge-fixing condition b0 = 0 is clarified. In section 2.5, a

new tree amplitude prescription is given for the pure spinor formalism based on the un-

twisted approach which matches the RNS tree amplitude prescription in the F1 picture.

In section 2.6, an alternative solution to the superfield constraints of (1.2) is shown to pro-

duce an extended version of the RNS formalism where the [Θα,Φα] superfields decouple

from the Xm superfield. And in section 2.7, the N=1 worldsheet supersymmetric approach

to the pure spinor formalism is generalized to curved heterotic and Type II supergravity

backgrounds.

In section 3.1, a third solution to the superfield constraints of (1.2) for the Type II su-

perstring is described which replaces the usual spacetime variable xm with twistor variables

and solves for Xm in terms of [Θα,Φα]. In section 3.2, N=1 worldsheet superconformal

generators are constructed for this Type II twistor-string formalism and a U(1) generator

corresponding to the projective weight of d=10 twistors is used to define physical states.

In section 3.3, the N=(1,1) worldsheet supersymmetric twistor-string action in a flat back-

ground of (1.3) is constructed and shown to be equivalent to the usual pure spinor Type

II superstring action up to a BRST-trivial term. In section 3.4, this twistor-string action

is generalized in an AdS5 × S5 background to the remarkably simple action of (1.5) which

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has manifest PSU(2, 2|4) symmetry and reduces in the large radius limit to the action in

a flat background of (1.3). Finally, in section 3.5, the AdS5 × S5 twisor-string action is

written in SO(10, 2) notation and a U(1) generator involving d=12 pure spinors is used to

define physical states.

2 Untwisting the pure spinor formalism

2.1 N=1 generators and superfields

In a flat background, the left-moving variables of the pure spinor formalism for the super-

string are described in conformal gauge by the free worldsheet action

S =

∫
d2z

(
1

2
∂xm∂xm + pα∂θ

α + wα∂λ
α

)
, (2.1)

where (xm, θα) are the usual N=1 d=10 superspace variables for m = 0 to 9 and α = 1

to 16, pα is the conjugate momenta to θα, λα is a d=10 pure spinor variable satisfying

λγmλ = 0, and wα is the conjugate momentum to λα which is defined up to the gauge

transformation δwα = fm(γmλ)α.

As discussed in [6], this pure spinor formalism can be interpreted as a topologically

twisted N=2 worldsheet superconformal field theory with fermionic left-moving generators

G+ = jBRST = λαdα, (2.2)

G− = b = −wα∂θα +
1

2(λλ)

[
πm(λγmd) + (wγmλ)(λγm∂θ)

]
where

∮
G+ is the BRST charge used to define physical states, G− is the composite b

ghost used for computing loop amplitudes, λα is a fixed pure spinor on a patch defined by

(λαλα) 6= 0, and πm and dα are the spacetime supersymmetric operators

πm = ∂xm − 1

2
∂θγmθ, dα = pα −

1

2
∂xm(γmθ)α −

1

8
(θγm∂θ)(γmθ)α. (2.3)

Using the OPE’s from the free worldsheet action of (2.1), one can verify that G+ and G−

are nilpotent operators satisfying the relation{∮
G+, G−

}
= Ttwisted = −1

2
∂xm∂xm − pα∂θα − wα∂λα (2.4)

for any choice of λα. Although G− can be Lorentz-covariantized by treating λα as a non-

minimal worldsheet variable [6], this non-minimal version of the pure spinor formalism will

not be discussed here and λα will be assumed to be fixed on each patch. Furthermore, we

will be ignoring all normal-ordering terms and central charges thoughout this paper such

as the term proportional to λα∂
2θα in G−. Hopefully, the non-minimal formalism and

normal-ordering contributions will be treated in a later paper.

To untwist the N=2 generators of (2.2), define the N=1 generator

G = G+ +G− = λαdα − wα∂θα +
1

2(λλ)

[
πm(λγmd) + (wγmλ)(λγm∂θ)

]
(2.5)

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which satisfies the OPE of an N=1 superconformal stress tensor

G(y)G(z)→ 2(y − z)−1T (z) (2.6)

where

T = −1

2
∂xm∂xm − pα∂θα −

1

2
(wα∂λ

α − λα∂wα) (2.7)

is the untwisted stress tensor with (λα, wα) of +1
2 conformal weight, and the central charge

contribution in (2.6) is being ignored.

Under the N=1 worldsheet supersymmetry generated by G of (2.5), the bosonic world-

sheet superpartner Gθα of θα is

Λα = λα +
1

2(λλ)
πm(γmλ)α (2.8)

and the fermionic worldsheet superpartner Gxm of xm is

ψm =
1

2
Λγmθ − 1

2(λλ)
(λγmd). (2.9)

So one can define N=1 worldsheet superfields

Xm = xm + κψm, Θα = θα + κΛα (2.10)

where κ is an anticommuting parameter, which transform covariantly under N=1 world-

sheet supersymmetry transformations and transform under the d=10 spacetime supersym-

metry transformations as δΘα = εα, δXm = −1
2εγ

mΘ.

Furthermore, the conjugate momenta variables pα and wα can be combined into the

worldsheet superfield

Φα = wα −
1

2(λλ)
(wγmλ)(γmλ) + κ

[
dα −

1

2(λλ)
(dγmλ)(γmλ)α −

1

2(λλ)2
λα(λγmd)πm

]
(2.11)

where πm and dα are defined in (2.3). Note that Φα has conformal weight + 1
2 and trans-

forms covariantly under N=1 worldsheet supersymmetry, and is spacetime supersymmet-

ric. The N=1 superfields (Xm,Θα,Φα) are not independent and satisfy the worldsheet and

spacetime supersymmetric constraints

(γmλ)α
(
DXm −

1

2
DΘγmΘ

)
= 0, (λγm)αΦα = 0 (2.12)

where D = ∂
∂κ + κ ∂

∂z .

As will be shown later, different solutions of the constraints of (2.12) will describe

either the pure spinor formalism, an extended version of the RNS formalism, or a new

twistor formalism of the superstring.

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2.2 Worldsheet supersymmetric action

To construct the N=(1,0) worldsheet supersymmetric action for the heterotic superstring,

generalize the superfields of (2.10) and (2.11) to the off-shell N=(1,0) superfields

Xm = xm + κψm, Θα = θα + κΛα, Φα = Ωα + κhα, (2.13)

where (xm, ψm, θα,Λα,Ωα, hα) are treated as independent components. For the heterotic

superstring in a flat background, the N=(1,0) worldsheet action in terms of these super-

fields is

S =

∫
d2zdκ

[
Φα∂Θα +

1

2
Πm
κ Πzm +Bhet

κz + (λγmL)Πκm +Mm(λγmΦ)

]
(2.14)

where

Πm
κ = DXm− 1

2
DΘγmΘ, Πm

z = ∂Xm− 1

2
∂ΘγmΘ, Π

m
z = ∂Xm− 1

2
∂ΘγmΘ, (2.15)

Bhet
κz =

1

4
[(DΘγmΘ)∂Xm−DXm(∂ΘγmΘ)], Bhet

zz =
1

4
[(∂ΘγmΘ)∂Xm−∂Xm(∂ΘγmΘ)],

(2.16)

Bhet
zz is the usual heterotic Green-Schwarz two-form, Bhet

κz is obtained from Bhet
zz by replac-

ing ∂z with D, Lα and Mm are Lagrange multiplier superfields enforcing the constraints

of (2.12), and the right-moving fermions of the heterotic superstring [19] which generate

the SO(32) or E8 × E8 gauge groups will be ignored throughout this paper.

Performing the Grassmann integral over κ and imposing the constraints of (2.12), the

action of (2.14) is equal to

S =

∫
d2z

[
DΦα∂Θα+Φα∂DΘα+

1

2

(
Πm
z −

1

2
DΘγmDΘ

)
Πzm−

1

2
Πκm(∂Πm

κ +DΘγm∂Θ)

+Bhet
zz +

1

2
(DΘγm∂Θ)Πκ +

1

4
(DΘγmDΘ)Πz

]
(2.17)

=

∫
d2z

[
DΦα∂Θα + Φα∂DΘα +

1

2
Πm
z Πzm −Πm

κ DΘγm∂Θ +Bhet
zz

]
=

∫
d2z

[
hα∂θ

α + Ωα∂Λα −
(
ψm − 1

2
Λγmθ

)
(Λγm)α∂θ

α +
1

2
πmπm +Bhet

zz

]
(2.18)

where Ωα and hα are constrained to satisfy λγmΩ = λγmh = 0. Finally, one can define

Ωα = wα −
1

2(λλ)
(wγmλ)(γmλ), dα = hα − (Λγm)α

(
ψm − 1

2
Λγmθ

)
, (2.19)

to obtain the heterotic pure spinor action of (2.1)

S =

∫
d2z

[
dα∂θ

α + wα∂λ
α +

1

2
πmπm +Bhet

zz

]
(2.20)

=

∫
d2z

[
1

2
∂xm∂xm + pα∂θ

α + wα∂λ
α

]
where the relation of pα and dα is defined in (2.3).

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For the Type II superstring, one generalizes the N=(1,0) superfields of (2.13) to

N=(1,1) off-shell superfields

Xm = xm + κψm + κψ̂m + κκfm, (2.21)

Θα = θα + κΛα + κρα + κκsα, Θ̂α̂ = θ̂α̂ + κΛ̂α̂ + κρ̂α̂ + κκŝα̂,

Φα = Ωα + κhα + κrα + κκξα, Φ̂α̂ = Ω̂α̂ + κĥα̂ + κr̂α̂ + κκξ̂α̂,

where (z, z, κ, κ) are the parameters of N=(1,1) worldsheet superspace and α and α̂ denote

spinors of the same/opposite chirality for the Type IIB/IIA superstring. In terms of these

N=(1,1) worldsheet superfields, the Type II worldsheet supersymmetric action in a flat

background is

S =

∫
d2zdκdκ

[
− ΦαDΘα + Φ̂α̂DΘ̂α̂ +

1

2
Πm
κ Πκm +BII

κκ (2.22)

+ (λγmL)Πm
k +Mm(λγmΦ) + (λ̂γmL̂)Π

m
κ + M̂m(λ̂γmΦ̂)

]
,

where [Lα,M
m] and [L̂α̂, M̂

m] are Lagrange multipliers for the left and right-moving con-

straints

Πm
κ (γmλ)α = 0, λγmΦ = 0, Π

m
κ (γmλ̂)α̂ = 0, λ̂γmΦ̂ = 0, (2.23)

λα and λ̂α̂ are two fixed pure spinors satisfying λαλ
α 6= 0 and λ̂α̂λ̂

α̂ 6= 0, D = ∂
∂κ + κ ∂

∂z

and D = ∂
∂κ + κ ∂

∂z ,

Πm
κ = DXm − 1

2
DΘγmΘ− 1

2
DΘ̂γmΘ̂, Πm

z = ∂Xm − 1

2
∂ΘγmΘ− 1

2
∂Θ̂γmΘ̂, (2.24)

Π
m
κ = DXm − 1

2
DΘγmΘ− 1

2
DΘ̂γmΘ̂, Π

m
z = ∂Xm − 1

2
∂ΘγmΘ− 1

2
∂Θ̂γmΘ̂, (2.25)

BII
κκ =

1

4

[
(DΘγmΘ−DΘ̂γmΘ̂)DXm −DXm(DΘγmΘ−DΘ̂γmΘ̂) (2.26)

− 1

2
(DΘγmΘ)(DΘ̂γmΘ̂) +

1

2
(DΘ̂γmΘ̂)(DΘγmΘ)

]
,

BII
zz =

1

4

[
(∂ΘγmΘ− ∂Θ̂γmΘ̂)∂Xm − ∂Xm(∂ΘγmΘ− ∂Θ̂γmΘ̂) (2.27)

− 1

2
(∂ΘγmΘ)(∂Θ̂γmΘ̂) +

1

2
(∂Θ̂γmΘ̂)(∂ΘγmΘ)

]
,

and BII
κκ is the usual Type II Green-Schwarz two-form Bzz field with ∂

∂z and ∂
∂z replaced

by D and D.

After shifting Φα and Φ̂α̂, integrating over κ and κ, and solving for auxiliary fields,

the action of (2.22) reduces to

S =

∫
d2z

[
hα∂θ

α + Ωα∂Λα+ĥα̂∂θ̂
α̂+Ω̂α̂∂Λ̂α̂−

(
ψm− 1

2
Λγmθ

)
(Λγm)α∂θ

α (2.28)

−
(
ψ̂m − 1

2
Λ̂γmθ̂

)
(Λ̂γm)α∂θ̂

α +
1

2
πmπm +BII

zz

]

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where [Ωα, Ω̂α̂, hα, ĥα̂] are constrained to satisfy λγmΩ = λ̂γmΩ̂ = λγmh = λ̂γmĥ = 0.

Defining

Ωα = wα −
1

2(λλ)
(wγmλ)(γmλ), dα = hα − (Λγm)α

(
ψm − 1

2
Λγmθ

)
, (2.29)

Ω̂α̂ = ŵα̂ −
1

2(λ̂λ̂)
(ŵγmλ̂)(γmλ̂), d̂α = ĥα̂ − (Λ̂γm)α̂

(
ψ̂m − 1

2
Λ̂γmθ̂

)
,

one obtains the Type II pure spinor action

S =

∫
d2z

[
1

2
πmπm +BII

zz + dα∂θ
α + wα∂λ

α + d̂α̂∂θ̂
α̂ + ŵα̂∂λ̂

α̂

]
(2.30)

=

∫
d2z

[
1

2
∂xm∂xm + pα∂θ

α + wα∂λ
α + p̂α̂∂θ̂

α̂ + ŵα̂∂λ̂
α̂

]
.

Although the manifestly worldsheet supersymmetric actions of (2.14) and (2.22) are

not manifestly Lorentz-covariant because of the presence of λα in the constraints of (2.12),

one can solve these constraints to obtain the manifestly Lorentz-covariant action of the

pure spinor formalism which is, however, not manifestly worldsheet supersymmetric. As

will be shown later, there are alternative ways to solve the constraints of (2.12) which

either lead to the extended RNS formalism or to the twistor string formalism. However,

before discussing the relation of (2.14) and (2.22) to the extended RNS and twistor-string

formalisms, it will be shown how to construct vertex operators and compute tree-level

scattering amplitudes using this N=1 worldsheet supersymmetric description of the pure

spinor formalism.

2.3 U(1) generator

As expected for an N=1 superconformal field theory, physical vertex operators V should

be N=1 superconformal primary fields of conformal weight + 1
2 so that the integrated

vertex operator
∫
GV =

∫
dzdκV is N=1 superconformally invariant. But after imposing

the constraints of (2.12) and fixing the N=1 superconformal invariance, the superfields

[Xm,Θα,Φα] contain 30 + 30 worldsheet variables. So one needs to impose additional

requirements if one wants to reproduce the usual superstring spectrum depending in light-

cone gauge on only 8 + 8 worldsheet variables.

To obtain the additional requirements, consider the U(1) generator

J = −λαwα (2.31)

which has the OPE’s

J(y)G(z)→ (y − z)−1(G+ −G−) (2.32)

where G = G+ + G− and G± are defined in (2.2) and carry ±1 U(1) charge with respect

to
∮
J . Since the integrated vertex operator

∫
GV is N=1 superconformally invariant, it

would be N=2 superconformally invariant if it had no poles with J since this would imply

that
∫
GV has no poles with either G+ or G−. Although this condition on the vertex

operator would be too restrictive, an appropriate condition is that
∫
GV must have only

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terms of zero or negative U(1) charge with respect to
∮
J . Defining

∫
(GV )n to be the term

in
∫
GV with U(1) charge n, this condition combined with N=1 superconformal invariance

implies that [
∮
G+,

∫
(GV )0] = 0.

It will later be shown that charge conservation implies that the terms with negative

U(1) charge in
∫
GV do not contribute to tree amplitudes. So at least for tree amplitudes,

the integrated vertex operator can be identified with
∫

(GV )0 which is annihilated by
∮
G+.

Furthermore, it will be required that the integrated vertex operator of zero U(1) charge,∫
(GV )0, is independent of the fixed pure spinor λα and is therefore globally defined on the

pure spinor space. So in addition to requring that
∫
GV is N=1 superconformally invariant,

it will also be required that
∫

(GV )n = 0 for n positive and that
∫

(GV )0 is globally defined

on the pure spinor space, i.e.
∫

(GV )0 is independent of λα and is invariant under the gauge

transformation δwα = fm(γmλ)α. By fixing the way that the vertex operator depends on

11 components of λα and θα and their conjugate momenta, these additional requirements

will reduce the degrees of freedom in physical vertex operators from 30 + 30 worldsheet

variables to 8 + 8 worldsheet variables.

2.4 Massless vertex operators

N=1 superconformal invariance implies that the open superstring unintegrated massless

vertex operator of conformal weight + 1
2 has the form

V = DΘαAα(X,Θ) + Πm
κ Am(X,Θ) + ΦαW

α(X,Θ) (2.33)

where (Aα, Am,W
α) are spacetime superfields with momentum km satisfying kmkm = 0.

By acting on V with the worldsheet superspace derivative D, the integrated vertex operator

is easily computed to be

GV = ∂ΘαAα + Πm
z Am +DΦαW

α + ΦαDΘβ∇βWα + ΦαΠm
κ ∂mW

α (2.34)

+DΘαDΘβ

(
−1

2
Amγ

m
αβ +∇βAα

)
+DΘαΠm

κ (∂mAα −∇αAm) + Πm
κ Πn

κ∂mAn.

The constraints of (2.12) imply that the κ = 0 component of the superfields Φα and Πm
κ

carry −1 U(1) charge, and the condition that (GV )2 = 0 implies the d=10 super-Yang-Mills

equation of motion γαβm1...m5∇αAβ = 0. So

(GV )0 = ∂θαAα + πmAm + hαW
α + λβΩα∇βWα (2.35)

+ λαπn
(γnλ)β

λλ
(−Amγmαβ +∇βAα +∇αAβ) + λαΠm

κ (∂mAα −∇αAm).

Using the definitions of (2.19), one can verify that (GV )0 is independent of λα if

∇αAβ +∇βAα = γmαβAm, ∇αAm − ∂mAα = γmαβW
β , (2.36)

which are the usual onshell superfield constraints for d=10 super-Yang-Mills. And after

imposing these super-Yang-Mills constraints, (GV )0 reproduces the pure spinor integrated

vertex operator

U = (GV )0 = ∂θαAα + πmAm + dαW
α +

1

4
(wγmnλ)Fmn (2.37)

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where ∇αW β = 1
4(γmn)βαFmn and dα ≡ hα − (λγm)αΠm

κ differs from the definition of dα
in (2.19) by a term with −2 U(1) charge which does not contribute to (2.37). Note that

(GV )−2 is nonzero and satisfies G−
∫

(GV )0 = −G+
∫

(GV )−2. This explains why the usual

pure spinor integrated vertex operator U of (2.37) is not annihilated by the b ghost [20]

but satisfies b−1

∫
U = Q

∫
Λ where Λ = −(GV )−2.1

2.5 Tree-level scattering amplitudes

In the pure spinor formalism, the usual tree-level N -point open string scattering amplitude

prescription is to take 3 vertex operators Vr of ghost-number one and conformal weight

zero, and N − 3 integrated vertex operator Ur of ghost-number zero and conformal weight

one. One then defines the tree amplitude A to be the correlation function

A = 〈V1(z1)V2(z2)V3(z3)

∫
dz4U4 . . .

∫
dzNUN 〉 (2.38)

where the (z1, z2, z3) are arbitrary points and the zero mode normalization is defined by

〈(λγmθ)(λγnθ)(λγpθ)(θγmnpθ)〉 = 1. Although this prescription only requires 5 of the 16

θ zero modes to be present in the integrand, it is spacetime supersymmetric since one can

show that any term in the integrand with more than 5 θ zero modes and +3 ghost-number

is not in the cohomology of Q =
∫
λαdα [1].

But before twisting, the vertex operators Vr of +1 ghost-number have conformal weight

+1
2 . So this prescription is only conformally invariant after twisting the pure spinor for-

malism and is inconsistent in the untwisted pure spinor formalism. Fortunately, there is

an alternative prescription one can define for tree-level amplitudes in the pure spinor for-

malism which only involves ghost-number zero vertex operators U and can be defined both

before and after twisting.

In this alternative prescription, one takes N integrated vertex operators Ur of ghost-

number zero and conformal weight one and defines the tree amplitude as

A = 〈(z1 − z2)(z2 − z3)(z3 − z1)U1(z1)U2(z2)U3(z3)

∫
dz4U4 . . .

∫
dzNUN 〉 (2.39)

where (z1, z2, z3) are arbitrary points and the zero mode normalization is defined by 〈1〉 = 1.

In this prescription, none of the 16 θ zero modes need to be present in the integrand. But it

is again spacetime supersymmetric since one can show that the only term in the cohomology

of Q =
∫
λαdα with zero ghost-number is the identity operator. So any term with θ zero

modes and zero ghost number will decouple since it is not in the cohomology of Q =
∫
λαdα.

For example, consider the N -point Yang-Mills tree amplitude where U is defined

in (2.37). For N external gluons, requiring an equal number of θα and pα zero modes

implies that the only term in (2.37) which contributes is

U = ∂xmAm(x) +
1

2
MmnFmn(x) (2.40)

1An interesting question is which vertex operators satisfy (GV )−2 = 0 and therefore are annihilated by

the b ghost and preserve N=2 worldsheet supersymmetry. By analyzing (2.34), one finds that (GV )−2 = 0

if and only if λαW
α = 0. Since ∇βWα = 1

4
(γmn)αβFmn, λαW

α = 0 implies that Fmn(γmnλ)α = 0, so λα is

a killing spinor in these backgrounds. I would like to thank Andrei Mikhailov for discussions on this point.

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where Mmn = 1
2(pγmnθ) + 1

2(wγmnλ). Using the fact that Mmn is a Lorentz current of

level 1 with the same OPE’s as the RNS Lorentz current ψmψn, one can easily verify that

the prescription of (2.39) reproduces the correct tree amplitudes.

A similar zero mode prescription was used by Lee and Siegel in [21], and is closely

related to the F1 picture for scattering Neveu-Schwarz states in the RNS formalism. To

compute N -point open string RNS tree amplitudes in this F1 picture, one chooses all N

Neveu-Schwarz vertex operators in the zero picture and uses the same zero mode regular-

ization 〈c(z1)c(z2)c(z3)〉 = (z1 − z2)(z2 − z3)(z3 − z1) as in the bosonic string. Although it

is unclear how to generalize this prescription to loop amplitudes in the RNS formalism, it

is easy to show that computations in the F1 picture reproduce the same tree-level ampli-

tude prescription as in the conventional F2 picture [22] where two Neveu-Schwarz vertex

operators are chosen in the −1 picture and one uses the zero mode regularization

〈c(z1)e−φ(z1)c(z2)e−φ(z2)c(z3)〉 = (z1 − z3)(z2 − z3). (2.41)

To prove the equivalence of RNS computations in the F1 and F2 pictures, multiply the

BRST-invariant state (c∂c∂2ce−2φ) appearing in (2.41) with two picture raising operators

to obtain a BRST-invariant state of ghost-number three and zero picture which includes

the term c∂c∂2c.

So for computing tree-level scattering amplitudes in the untwisted pure spinor for-

malism, the prescription of (2.39) can be used. Because of U(1) charge conservation with

respect to J = −λαwα and the absence of terms with positive U(1) charge in the worldsheet

action and vertex operators, terms with negative U(1) charge cannot contribute to tree am-

plitudes using this precription. Furthermore, it will be verified in the next section that the

tree amplitude prescription of (2.39) for Neveu-Schwarz states in the extended RNS for-

malism gives the same tree amplitudes as in the usual RNS formalism. Although it will

not be verified here, it is natural to conjecture that the prescription of (2.39) with the zero

mode normalization 〈1〉 = 1 can also be used to compute twistor-string tree amplitudes.

2.6 Extended RNS formalism

The N=1 worldsheet superfields in the pure spinor formalism are [Xm,Θα,Φα] satisfying

the constraints of (2.12) that (γmλ)α(DXm− 1
2DΘγmΘ) = 0 and (λγmΦ) = 0. If one shifts

Θα by defining

Θ′α ≡ Θα +Km(γmλ)α where Km = − 1

λDΘ′
DXm +

1

2

λΘ′

λDΘ′
D

(
DXm

λDΘ′

)
, (2.42)

(γmλ)α(DXm− 1
2DΘγmΘ) = 0 implies that (λγm)α(Θ′γmDΘ′) = 0. So the N=1 superfields

[Θ′α,Φα] satisfy the constraints

(λγm)α(Θ′γmDΘ′) = 0, λγmΦ = 0, (2.43)

and leave unconstrained the N=1 superfield Xm. Interpreting Xm as the usual N=1 world-

sheet superfield of the RNS formalism, the formalism including both Xm and (Θ′α,Φα) will

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be called the “extended RNS formalism”. In components, it is convenient to expand the

new superfield Θ′α as

Θ′α = θ′α + κλα + (γmλ)α(fm + κgm) (2.44)

where θ′α and λα are constrained to satisfy

θ′γmλ = 0, λγmλ = 0, (2.45)

and (λγm)α(Θ′γmDΘ′) = 0 implies that fm and gm are quadratic and higher-order in θ′α.

In terms of the N=(1,0) superfields [Xm,Θ′α,Φα], the heterotic worldsheet action in

the extended RNS formalism will be defined to be the sum of the usual RNS action with

an action for the [Θ′α,Φα] superfields as

S =

∫
d2zdκ

[
1

2
DXm∂Xm + Φα∂Θ′α + (λγmL)(Θ′γmDΘ′) +Mm(λγmΦ)

]
. (2.46)

And in terms of the N=(1,1) superfields [Xm,Θ′α, Θ̂′α̂,Φα, Φ̂α̂], the Type II worldsheet

action will be defined as

S =

∫
d2zdκdκ

[
1

2
DXmDXm − ΦαDΘ′α + Φ̂α̂DΘ̂′α̂ (2.47)

+ (λγmL)(Θ′γmDΘ′) +Mm(λγmΦ) + (λ̂γmL̂)(Θ̂′γmDΘ̂′) + M̂m(λ̂γmΦ̂)

]
.

To relate the pure spinor heterotic action of (2.14) to the extended RNS heterotic

action of (2.46), substitute into (2.18) the component form of (2.42) which is

θα = θ′α −
[

1

λλ
ψm − (λθ′)

2(λλ)2

(
∂xm − ψm (λ∂θ′)

(λλ)

)
+ fm

]
(γmλ)α, (2.48)

where fm is quadratic and higher-order in θ′α. In terms of θ′α, the action of (2.18) is

S =

∫
d2z

[
hα∂θ

′α+Ωα∂λ
α−ψm(λγmγnλ)∂

(
1

λλ
ψn− (λθ′)

2(λλ)2
∂xn

)
+

1

2
∂xm∂xm +O(θ′)

]
=

∫
d2z

[
(hα−λαrm∂xm)∂θ′α+(Ωα−2ψmr

mλα)∂λα−ψm∂ψm+
1

2
∂xm∂xm+O(θ′)

]
(2.49)

where O(θ′) denotes terms linear or higher-order in θ′α (counting its conjugate momentum

hα as an inverse power of θ′α) and

rm = ψn
λγmγnλ

2(λλ)2
. (2.50)

Defining Φα = Ω̃α + κh̃α in the extended RNS action of (2.46) where

Ω̃α ≡ Ωα − 2ψmr
mλα, h̃α ≡ hα − rm∂xmλα, (2.51)

one can easily verify that (2.46) reproduces (2.49) if one ignores terms proportional to O(θ′).

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To understand why these O(θ′) terms can be ignored, note that physical vertex opera-

tors will be required to carry zero U(1) charge with respect to J = −λαΩ̃α and be globally

defined on the pure spinor space, i.e. physical vertex operators must be independent of λα
and be invariant under the gauge transformations

δΩ̃α = ξm(γmλ)α + ρm(γmθ′)α, δh̃α = (γmλ)αρm, (2.52)

generated by the constraints of (2.45) where ξm and ρm are arbitrary parameters. It can

be verified that all quantities with zero U(1) charge which are gauge-invariant under (2.52)

must contain non-negative powers of θ′α (where h̃α counts as an inverse power of θ′α), e.g.

the Lorentz current Mmn = 1
2(Ω̃γmnλ+h̃γmnθ′). And since the tree amplitude prescription

vanishes unless there are an equal number of θ′α’s and h̃α’s in the correlation function, one

can ignore any O(θ′) terms in the worldsheet action which are linear or higher-order in θ′α.

Finally, it will be argued that the computation of tree-level scattering amplitudes of

physical states using the prescription of (2.39) in the extended RNS formalism is equivalent

to the computation of Neveu-Schwarz states using the usual RNS prescription in the F1

picture. To prove this equivalence, one needs to show that the extra fields (λα, θ′α, Ω̃α, h̃α)

in the extended RNS formalism do not contribute to tree-level scattering amplitudes using

the zero mode normalization where 〈1〉 = 1.

Any physical vertex operator in the extended RNS formalism must be gauge-invariant

under (2.52) and be a worldsheet primary field with zero U(1) charge. Examples of such

gauge-invariant operators with zero U(1) charge which involve h̃α and Ω̃α are Mmn =
1
2(Ω̃γmnλ+h̃γmnθ′) and its derivatives. But since Mmn has level zero, i.e. the OPE of Mmn

with Mpq has no double pole proportional to the identity operator, it is not possible for a

correlation function involving Mmn and its derivatives to be proportional to the identity

operator. It seems reasonable to conjecture that all gauge-invariant opeartors depending

on h̃α or Ω̃α are of this type and cannot produce the identity operator in their OPE’s.

Therefore, any terms in the vertex operator which depend on h̃α or Ω̃α will decouple from

the tree amplitudes. Furthermore, since the tree amplitude vanishes unless there are an

equal number of h̃α’s and θ′α’s in the correlation function, any terms in the vertex operator

which depend on θ′α will also decouple. So the only terms in the vertex operator which

can contribute to tree amplitudes are terms that only depend on the superfield Xm. But

worldsheet N=1 superconformal primary fields which only depend on Xm are the usual

Neveu-Schwarz states in the RNS formalism. So tree amplitudes of physical states in the

extended RNS formalism are equivalent to the tree amplitudes of Neveu-Schwarz states in

the usual RNS formalism.

2.7 Worldsheet supersymmetric action in curved background

By adding integrated vertex operators to the worldsheet action in a flat target-space

background, one can generalize the N=1 worldsheet supersymmetric actions to a curved

background. For the heterotic superstring in the pure spinor description, the N=(1,0)

worldsheet supersymmetric action of (2.14) generalizes in an N=1 d=10 supergravity

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background to

S =

∫
d2zdκ

[
1

2
ηabE

a
MDZ

MEbN∂Z
N +

1

2
Bhet
MNDZ

M∂ZN + ΦαE
α
M∂Z

M (2.53)

+ (λγaL)EaMDZ
M +Ma(λγ

aΦ)

]
where M = (m,µ) are curved-space indices for m = 0 to 9 and µ = 1 to 16, A = (a, α)

are tangent-space indices for a = 0 to 9 and α = 1 to 16, ZM = (Xm,Θµ), EAM is the

super-vierbein, and Bhet
MN is the graded antisymmetric tensor superfield.

And for the Type II superstring in the pure spinor description, the N=(1,1) worldsheet

supersymmetric action of (2.22) generalizes in an N=2 d=10 supergravity background to

S =

∫
d2zdκdκ

[
1

2
ηabE

a
MDZ

MEbNDZ
N +

1

2
BII
MNDZ

MDZN (2.54)

− ΦαE
α
MDZ

M + Φ̂α̂E
α̂
MDZ

M − Fαα̂ΦαΦ̂α̂

+ (λγaL)EaMDZ
M +Ma(λγ

aΦ) + (λ̂γaL̂)EaMDZ
M + M̂a(λ̂γ

aΦ̂)

]
,

where M = (m,µ, µ̂) are curved-space indices, A = (a, α, α̂) are tangent-space indices,

ZM = (Xm,Θµ, Θ̂µ̂), EAM is the super-vierbein, BII
MN is the graded antisymmetric tensor

superfield, and Fαα̂ is the superfield whose lowest components are the Type II Ramond-

Ramond bispinor field strengths.

After imposing the constraints from varying the Lagrange multipliers, one can expand

the actions of (2.53) and (2.54) in components in terms of the pure spinor worldsheet vari-

ables (ZM , dα, λ
α, wα, λα). Although it will not be verified here, it is expected that when

the supergravity fields are onshell, all terms in the action will have either zero or negative

U(1) charge with respect to (2.31), and the terms with zero U(1) charge will be independent

of λα and reproduce the pure spinor worldsheet action in a curved background of [23].

Using the extended RNS description, the heterotic superstring action of (2.46) can be

generalized in a Neveu-Schwarz background and the resulting action is

S =

∫
d2zdκ

[
1

2
(gmn(X) + bmn(X))DXm∂Xm + Φα(∇zΘ′)α (2.55)

+ (λγaL)(Θ′γa∇κΘ′) +Ma(λγ
aΦ)

]
where (∇zΘ′)α = ∂Θ′α + ∂Xmωmβ

α(X)Θ′β , (∇κΘ′)α = DΘ′α + DXmωmβ
α(X)Θ′β , and

ωnβ
α is the spin connection. Similarly, the Type II worldsheet action of (2.47) generalizes

in a Neveu-Schwarz/Neveu-Schwarz background to

S =

∫
d2zdκdκ

[
1

2
(gmn(X) + bmn(X))DXmDXn − Φα(∇κΘ′)α + Φ̂α̂(∇κΘ̂′)α̂ (2.56)

+ (λγaL)(Θ′γa∇κΘ′) +Ma(λγ
aΦ) + (λ̂γaL̂)(Θ̂′γa∇κΘ̂′) + M̂a(λ̂γ

mΦ̂)

]
,

where (∇κΘ′)α = DΘ′α + DXmωmβ
α(X)Θ′β , (∇κΘ̂′)α̂ = DΘ̂′α̂ + DXmω̂mβ̂

α̂(X)Θ̂′β̂ , and

ωnβ
α and ω̂nβ̂

α̂ are the left and right-moving spin connections.

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3 Twistor string formalism

3.1 Twistor superfields

By choosing different solutions of the superfield constraints of (2.12),

(γmλ)α
(
DXm −

1

2
DΘγmΘ

)
= 0, (λγm)αΦα = 0, (3.1)

one obtains different worldsheet supersymmetric descriptions of the superstring. Expanding

the superfields in component fields as

Xm = xm + κψm, Θα = θα + κΛα, Φα = Ωα + κhα, (3.2)

the pure spinor description solves for ψm and hα in terms of dα through the equations (2.9)

and (2.19). And in the extended RNS description, one solves for θα in terms of ψm and

a constrained θ′α satisfying (λγmθ′) = 0 by shifting Θα to Θ′α as in (2.42). In both of

these descriptions, the bosonic component fields Λα and Ωα are solved in terms of xm and

(λα, wα) where λα is a pure spinor and

Λα = λα +
1

2(λλ)

(
∂xm− 1

2
∂θγmθ

)
(γmλ)α, Ωα = wα −

1

2(λλ)
(λγmw)(γmλ)α. (3.3)

In this section, a new twistor-like solution for the constraints of (3.1) will be presented

in which the superfield Xm = xm +κψm is solved in terms of the other superfields. In this

description, the superfield Φα is shifted to Φ′α = Φα − 1
2X

m(γmDΘ)α whose components

Φ′α ≡ Ω′α + κh′α no longer satisfy λγmΩ′ = λγmh′ = 0. It will be convenient to expand the

bosonic component fields Λα and Ω′α which appear in Θα and Φ′α as

Λα = λα +
1

2(λλ)
(γmλ)α(λγmν), Ω′α = µα + wα −

1

2(λλ)
(γmλ)α(wγmλ), (3.4)

where λα and µα are constrained to satisfy

λγmnµ = µγmµ = λγmλ = 0. (3.5)

The variables wα and να in (3.4) are the conjugate momenta to λα and µα and are

defined up to the gauge transformations

δwα = fm(γmλ)α, δνα = cmn(γmnλ)α + hm(γmµ)α, (3.6)

for arbitrary parameters cmn, hm and fm. Note that (λα, µα) satisfying (3.5) contain 16

independent components and can be interpreted as d=10 twistor variables. As discussed

in [24, 25], twistors in d spacetime dimensions are pure spinors in d+ 2 dimensions which

transform covariantly under SO(d, 2) conformal transformations. The spinors (λα, µα) sat-

isfying (3.5) can therefore be interpreted as the 16 independent components of a d=12 pure

spinor UA satisfying the pure spinor condition UAγMN
AB UB = 0 where A = 1 to 32, M = 0

to 11, and γM are the d = 12 gamma-matrices. So (3.4) decomposes the 32 components of

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Λα and Ω′α into the 16 independent components of (λα, µα) describing a d=12 pure spinor,

and the 16 gauge-invariant components of its conjugate momenta (wα, ν
α).

For the heterotic superstring, Φ′α does not have enough degrees of freedom to solve for

Xm, but for the Type II superstring, one also has the right-moving superfields

Θ̂α̂ = θ̂α̂ + κΛ̂α̂, Φ̂′α̂ = Ω̂′α̂ + κĥ′α̂, (3.7)

with component expansions

Λ̂α̂ = λ̂α +
1

2(λ̂λ̂)
(γmλ̂)α̂(λ̂γmν̂), Ω̂′α̂ = µ̂α̂ + ŵα̂ −

1

2(λ̂λ̂)
(γmλ̂)α̂(ŵγmλ̂). (3.8)

The shifted superfields Φ′α and Φ̂′α̂ are defined by

Φ′α = Φα −
1

2
Xm(γmDΘ)α, Φ̂′α̂ = Φ̂α̂ −

1

2
Xm(γmDΘ̂)α̂, (3.9)

and no longer satisfy the constraints λγmΦ′ = 0 and λ̂γmΦ̂′ = 0. Since the Φ′α and Φ̂′α̂
superfields are related to the Xm superfield by

λγmΦ′ = −1

2
(λγmγnDΘ)Xn, λ̂γmΦ̂′ = −1

2
(λ̂γmγnDΘ̂)Xn, (3.10)

the bosonic spinor variables (µα, λ
α) and (µ̂α̂, λ̂

α̂) in (3.4) and (3.8) are related to the

spacetime vector variable xm by the usual twistor relation

µα = −1

2
xm(γmλ)α, µ̂α̂ = −1

2
xm(γmλ̂)α̂. (3.11)

For the Type IIA superstring, the twistor relation of (3.11) can be inverted to solve

for xm in terms of µα and µ̂α as

xm = − 1

(λλ̂)
(λ̂γmµ+ λγmµ̂) (3.12)

where it is assumed that λαλ̂α 6= 0. Similarly, the Type IIA superfield Xm can be expressed

in terms of Φ′α and Φ̂′α as

Xm = − 1

(DΘDΘ̂)
(DΘ̂γmΦ′ +DΘγmΦ̂′). (3.13)

Although there is no analogous solution for the uncompactified Type IIB superstring, one

can use the standard T-duality relation of Type IIB with Type IIA to solve for xm if at

least one direction of the Type IIB superstring is compactified on a circle. For example, if

x9 is compactified on a circle, define

µα = −1

2
x̃m(γmλ)α, µ̂α = −1

2
x̃m(γ9γmγ9λ̂)α, (3.14)

where x̃m is the T-dual to xm defined by

x̃m = xmL + xmR for m = 0 to 8, x̃9 = x9
L − x9

R, (3.15)

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and xmL and xmR are the left and right-moving parts of xm defined by xmL (z) =
∫ z
dy∂xm(y)

and xmR (z) =
∫ z
dy∂xm(y). Using (3.14), one can invert to solve for x̃m in terms of µα

and µ̂α as

x̃m = − 1

(λγ9λ̂)
(λ̂γ9γmµ+ λγmγ9µ̂) (3.16)

where it is assumed that (λγ9λ̂) 6= 0.

3.2 N=1 superconformal and U(1) generator

The left-moving N=1 superconformal stress tensor for the twistor-string is T = 1
2DΦ′αDΘα−

1
2Φ′α∂Θα, which in components is

G = h′αΛα − Ω′α∂θ
α, T = −1

2
Ω′α∂Λα +

1

2
∂Ω′αΛα − h′α∂θα. (3.17)

As in the other worldsheet supersymmetric descriptions of the superstring, physical states

will be required to be N=1 superconformal primary fields whose integrated vertex operators

have zero or negative charge with respect to a U(1) generator J . In the twistor-string

description, the U(1) generator will be defined as

J = −λαwα + µαν
α (3.18)

which splits G into

G+ = h′αλ
α − µα∂θα, G− =

1

2(λλ)
(h′γmλ)(λγmν)− wα∂θα +

1

2(λλ)
(wγmλ)(λγm∂θ)

(3.19)

where (λα, µα, wα, ν
α) are defined in (3.4). Note that the U(1) generator J of (3.18) counts

the projective weight of the d=10 twistor variables where (λα, µα) carry projective weight

+1 and (wα, ν
α) carry projective weight −1. So the integrated vertex operator GV will be

required to carry zero or negative projective weight, and the term (GV )0 of zero projective

weight will be required to be globally defined on pure spinor space, i.e. independent of λα
and invariant under the gauge transformations of (3.6).

3.3 Worldsheet action in a flat background

Under spacetime supersymmetry, (3.10) and δXm = −1
2(εγmΘ + ε̂γmΘ̂) implies that the

Φ′α and Φ̂′α̂ superfields transform as

δΘα = εα, δΘ̂α̂ = ε̂α̂, (3.20)

δΦ′α =
1

4
(εγmΘ + ε̂γmΘ̂)(γmDΘ), δΦ̂′α̂ =

1

4
(εγmΘ + ε̂γmΘ̂)(γmDΘ̂).

And under spacetime translations, δXm = cm implies that the Φ′α and Φ̂′α̂ superfields

transform as

δΘα = δΘ̂α̂ = 0, δΦ′α = −1

2
cm(γmDΘ), δΦ̂′α̂ = −1

2
cm(γmDΘ̂). (3.21)

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The N=(1,1) worldsheet supersymmetric action for the Type II twistor-string in a flat

background should be invariant under these super-Poincaré transformations and will be

defined in terms of the (Θα, Θ̂α̂,Φ′α, Φ̂
′
α̂) superfields as

S =

∫
d2zd2κ

[
−Φ′αDΘ + Φ̂′α̂DΘ̂− 1

8
(ΘγmDΘ)(Θ̂γmDΘ̂) +

1

8
(Θ̂γmDΘ̂)(ΘγmDΘ)

]
.

(3.22)

After integrating out auxiliary variables and shifting wα and ŵα̂, (3.22) reduces to

S =

∫
d2z

[
h′α∂θ

α + ĥ′α̂∂θ̂
α̂ + wα∂λ

α + ŵα̂∂λ̂
α̂ + µα∂ν

α + µ̂α̂∂ν̂
α̂ (3.23)

− 1

2

(
νγmλ− 1

2
θγm∂θ

)(
ν̂γmλ̂−

1

2
θ̂γm∂θ̂

)]
,

with the spacetime supersymmetry generators

qα =

∫
dzdκΦ′α +

1

4

∫
dzdκ(γmΘ)α(Θ̂γmDΘ̂) (3.24)

=

∫
dzh′α −

1

2

∫
dz

(
λ̂γmν̂ − 1

2
θ̂γm∂θ̂

)
(γmθ)α,

q̂α̂ =
1

4

∫
dzdκ(γmΘ̂)α̂(ΘγmDΘ) +

∫
dzdκΦ̂′α̂

= −1

2

∫
dz

(
λγmν − 1

2
θγm∂θ

)
(γmθ̂)α̂ +

∫
dzĥ′α̂,

Pm =
1

2

∫
dzdκΘγmDΘ +

1

2

∫
dzdκΘ̂γmDΘ̂

=

∫
dz

(
λγmν −

1

2
θγm∂θ

)
+

∫
dz

(
λ̂γmν̂ −

1

2
θ̂γm∂θ̂

)
.

Using that Φ′α+ 1
2X

m(γmDΘ)α and Φ̂′α̂+ 1
2X

m(γmDΘ̂)α̂ are spacetime supersymmetric,

the action of (3.22) can be written in manifestly spacetime supersymmetric notation as

S =

∫
d2zd2κ

[
−
(

Φ′α +
1

2
Xm(γmDΘ)α

)
DΘα +

(
Φ̂′α̂ +

1

2
Xm(γmDΘ̂)α̂

)
DΘ̂α̂ +BII

κκ

]
(3.25)

where BII
κκ is defined in (2.26). Note that this action is related to the Type II worldsheet

action of (2.22) by dropping the term

1

2

∫
d2zdκdκ Πm

κ Πκm. (3.26)

Since Πm
κ Πκm has left and right-moving U(1) charge (−1,−1), the term of zero U(1) charge

in (3.26) can be expressed as the BRST-trivial term

− 1

2

∫
d2z

∮
G+

∮
Ĝ+(Πm

κ Πκm). (3.27)

So if all vertex operators are annihilated by
∮
G+ and

∮
Ĝ+, it seems reasonable to assume

that dropping the term of (3.27) will not affect the scattering amplitudes since one can pull

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the countour integrals of G+ and Ĝ+ off of the surface. However, since vertex operators

have not yet been constructed in the twistor-string formalism, this assumption has not yet

been verified by explicit computations.

To relate (3.22) with the usual component form of the Type IIA pure spinor worldsheet

action, substitute µα = −1
2x

m(γmλ)α and µ̂α = −1
2x

m(γmλ̂)α into (3.23) and vary να and

ν̂α to obtain the equations of motion

(λ̂γmν̂)(γmλ)α = πm(γmλ)α, (λγmν)(γmλ̂)α = πm(γmλ̂)α, (3.28)

which implies

(λ̂γmν̂) =
(λ̂γmγnλ)

2λλ̂
πn, (λγmν) =

(λγmγnλ̂)

2λλ̂
πn (3.29)

where the equations of motion ∂λα = ∂λ̂α = 0 have been used. Plugging the auxiliary

equations of (3.29) back into the action of (3.23) and ignoring terms which vanish when

∂λα = ∂λ̂α = 0 (and can therefore be cancelled by an appropriate shift of wα and ŵα), one

finds that

S =

∫
d2z

[
h′α∂θ

α + ĥ′α̂∂θ̂
α̂ + wα∂λ

α + ŵα̂∂λ̂
α̂ (3.30)

+
1

2
πmπm −

(λγmγnλ̂)

2λλ̂
πmπn −

1

8
(θγm∂θ)(θ̂γ

m∂θ̂)

]
=

∫
d2z

[
pα∂θ

α + p̂α̂∂θ̂
α̂ + wα∂λ

α + ŵα̂∂λ̂
α̂ +

1

2
∂xm∂xm

]
(3.31)

−
∫
d2z

(λγmγnλ̂)

2λλ̂
πmπn,

where

pα = h′α +
1

2
xm(γm∂θ)α +

1

2

(
∂xm +

1

4
θγm∂θ +

1

8
θ̂γm∂θ̂

)
(γmθ)α, (3.32)

p̂α = ĥ′α +
1

2
xm(γm∂θ̂)α +

1

2

(
∂xm +

1

4
θ̂γm∂θ̂ +

1

8
θγm∂θ

)
(γmθ̂)α,

and the first line of (3.31) is the Type IIA worldsheet action of (2.30). Furthermore, (3.32)

implies that G+ and Ĝ+ of (3.19) are mapped into the pure spinor BRST currents

G+ = λαh′α − µα∂θα → G+ = λαdα, (3.33)

Ĝ+ = λ̂αĥ
′α − µ̂α∂θ̂α → Ĝ+ = λ̂αd̂

α,

where

dα = pα−
1

2

(
∂xm+

1

4
θγm∂θ

)
(γmθ)α, d̂α = p̂α− 1

2

(
∂xm+

1

4
θ̂γm∂θ̂

)
(γmθ̂)

α (3.34)

are defined as in (2.3) and the equations of motion ∂θα = ∂θ̂α = 0 have been used.

Finally, note that

−
∫
d2z

(λγmγnλ̂)

2λλ̂
πmπn =

∫
d2z

∮
G+

∮
Ĝ+

(
1

2λλ̂
dαd̂

α

)
(3.35)

in the second line of (3.31) is BRST-trivial. So up to this BRST-trivial term related

to (3.27), the Type II twistor-string action of (3.22) is equal to the pure spinor action

of (2.30).

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3.4 Twistor string in AdS5 × S5 background

In principle, one can obtain the worldsheet action for the twistor string in an AdS5 ×
S5 background by deforming the Type IIB action in a flat background with the vertex

operator for the Ramond-Ramond five-form field strength and computing the back-reaction.

However, a simpler method is to find a PSU(2, 2|4)-invariant action which reduces in the

large radius limit to the Type IIB twistor string action in a flat background of (3.22).

To describe the AdS5 × S5 background, it is convenient to start with the standard

representation of AdS5 × S5 where g(Z) takes values in the supercoset PSU(2,2|4)
SO(4,1)×SO(5) and

define the supervierbein EAM by

g−1∂g = EAM∂Z
M (3.36)

where A = (a, α, α̂) denote the 10 bosonic and 32 fermionic generators of PSU(2,2|4)
SO(4,1)×SO(5)

and ZM = [xm, θµ, θ̂µ̂]. Using the notation ηαβ̂ = (γ01234)αβ̂ and ηαβ̂ = (γ01234)αβ̂ , the

Ramond-Ramond field strength is given by Fαβ̂ = ηαβ̂ and one can choose the gauge

where

BMNDZ
MDZN = ηαα̂

(
EαME

α̂
N + EαNE

α̂
M )DZMDZN (3.37)

= ηαα̂((g−1Dg)α(g−1Dg)α̂ + (g−1Dg)α̂(g−1Dg)α
)
.

Just as the term 1
2

∫
d2z

∫
dκdκΠm

κ Πκm of (3.26) was dropped from the twistor-string

action in a flat background, the twistor-string action in an AdS5 × S5 background will be

defined by dropping the analog of this term in the curved background action of (2.54)∫
d2z

∫
dκdκ

1

2
ηabE

a
ME

b
NDZ

MDZN =

∫
d2z

∫
dκdκ

1

2
ηab(g

−1Dg)a(g−1Dg)b. (3.38)

So the twistor-string action is

S = r2

∫
d2zdκdκ

[
1

2
ηαα̂((g−1Dg)α(g−1Dg)α̂ + (g−1Dg)α̂(g−1Dg)α) (3.39)

− Φ′α(g−1Dg)α + Φ̂′α̂(g−1Dg)α̂ − ηαα̂Φ′αΦ̂′α̂

]
where r is the AdS5 radius and Xm is determined in terms of [Θα, Θ̂α̂,Φ′α, Φ̂

′
α̂] through the

AdS5 × S5 generalization of (3.13).

Since the AdS5 × S5 superstring is a Type IIB superstring, defining Xm in terms of

[Θα, Θ̂α̂,Φ′α, Φ̂
′
α̂] will require T-dualizing one of the AdS5 × S5 directions as explained

in (3.14). A convenient choice which will hopefully be explored in a future paper is

to T-dualize one of the S5 directions, which breaks the manifest SU(4) R-symmetry to

SO(4)×U(1). This is the same manifest symmetry as the spin-chain, and T-dualizing in

this direction means that the spin-chain ground state Tr(Zn) where Z is the scalar with

+1 U(1) charge is described by a string with winding number n.

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Integrating out the auxiliary superfields Φ′α and Φ̂′α̂ in (3.39), the AdS5 × S5 twistor-

string action simplifies to

S = r2

∫
d2zdκdκ

1

2
ηαα̂

[
(g−1Dg)α(g−1Dg)α̂ − (g−1Dg)α̂(g−1Dg)α

]
. (3.40)

Surprisingly, this action is not only invariant under the local transformation δg = gΩ when

Ω ∈ SO(4, 1) × SO(5), it is also invariant under δg = gΩ when Ω ∈ SO(4, 2) × SO(6).

So the bosonic elements of the coset can be gauged away and the action of (3.40) can be

expressed as

S = r2

∫
d2zdκdκ

1

2
ηαα̂

[
(G−1DG)α(G−1DG)α̂ − (G−1DG)α̂(G−1DG)α

]
(3.41)

where G is a fermionic coset taking values in PSU(2,2|4)
SO(4,2)×SO(6) .2

Using SU(2, 2)× SU(4) notation, this action can be expressed as

S = r2

∫
d2zdκdκ(G−1DG)JR(G−1DG)RJ (3.42)

where J = 1 to 4 is a spinor representation of SO(4, 2) = SU(2, 2) and R = 1 to 4 is a spinor

representation of SO(6) = SU(4). Note that the Maurer-Cartan equations imply that∫
d2zdκdκ(G−1DG)JR(G−1DG)RJ = −

∫
d2zdκdκ(G−1DG)RJ (G−1DG)JR (3.43)

up to a surface term. Defining the coset representative of G as

G = eΘJRT
R
J eΘ̃RJ T

J
R (3.44)

where T JR and TRJ are the generators of the 32 fermionic isometries, the left-invariant

forms are

(G−1∂G)JR = ∂ΘJ
R, (G−1∂G)RJ = ∂Θ̃R

J + Θ̃R
K∂ΘK

S Θ̃S
J , (3.45)

and under the 32 global fermionic isometries generated by δG = (εJRT
R
J + ε̃RJ T

J
R)G, the

superfields of (3.44) transform as

δΘJ
R = εJR + ΘJ

S ε̃
S
KΘK

R , δΘ̃R
J = ε̃RJ − ε̃RKΘK

S Θ̃S
J − Θ̃R

KΘK
S ε̃

S
J . (3.46)

Plugging in the left-invariant forms of (3.45) into the action of (3.42), one obtains the

remarkably simple action

S = r2

∫
d2zdκdκ(DΘJ

RDΘ̃R
J +DΘJ

RΘ̃R
KDΘK

S Θ̃S
J ). (3.47)

To show that the action of (3.47) reduces in the large radius limit to the flat space

action of (3.22), rescale

ΘJ
R →

1√
r

ΘJ
R, Θ̃R

J →
1√
r

Θ̃R
J , (3.48)

2This action has vanishing beta function since the coset is a symmetric space with PSU(2, 2|4) in the

numerator and there is no WZW term [26].

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and write (3.47) as

S =

∫
d2zdκdκ

[
−Φ′αDΘα + Φ̂′α̂DΘ̂α̂ − 1

r
ηαα̂Φ′αΦ̂′α̂ +DΘJ

RΘ̃R
KDΘK

S Θ̃S
J

]
, (3.49)

where in terms of the Θα and Θ̂α̂ superfields of (3.39) written in d=10 spinor notation with

α, α̂ = 1 to 16, ΘJ
R only involves the linear combination Θα + iΘ̂α̂ and Θ̃R

J only involves

the linear combination Θα − iΘ̂α̂. After expressing the quartic term in (3.49) in terms of

Θα and Θ̂α̂,

S =

∫
d2zdκdκ

[
− Φ′αDΘα + Φ̂′α̂DΘ̂α̂ − 1

r
ηαα̂Φ′αΦ̂′α̂ (3.50)

+ e(DΘγaΘ +DΘ̂γaΘ̂)(DΘγaΘ +DΘ̂γaΘ̂)

+ fabcdef (DΘγabcΘ +DΘ̂γabcΘ̂)(DΘγdefΘ +DΘ̂γdef Θ̂)

]
where e and fabcdef are constants which are invariant under SO(4, 1)× SO(5) transforma-

tions and come from expressing DΘJ
RΘ̃R

KDΘK
S Θ̃S

J in d=10 notation. When r → ∞, the

ηαα̂Φ′αΦ̂′α̂ term drops out of (3.50) and, after appropriately shifting Φ′α and Φ̂′α̂ to cancel

terms proportional to DΘα and DΘ̂α̂, the quartic terms in (3.50) can be reduced to

e
[
(DΘγaΘ)(DΘ̂γaΘ̂)− (DΘ̂γaΘ̂)(DΘγaΘ)

]
. (3.51)

Finally, the coefficient e can be scaled to −1
8 by scaling [Φ′α,Θ

α, Φ̂′α̂, Θ̂
α̂] appropriately so

that the AdS5 × S5 action reduces to the flat space action of (3.22).

3.5 U(1) generator and d=12 pure spinors

Expanding in components, the action of (3.47) is

S = r2

∫
d2z
[
(G−1∂G)JR(G−1∂G)RJ (3.52)

+ ΛJR(∇Λ̃)RJ + Λ̂JR(∇ ˜̂
Λ)RJ + ΛJRΛ̃RKΛ̂KS

˜̂
ΛSJ − Λ̃RJ ΛJS

˜̂
ΛSKΛ̂KR

]
,

where the bosonic components are defined by

ΛJR = (G−1DG)JR|κ=κ=0, Λ̃RJ = (G−1DG)RJ |κ=κ=0, (3.53)

Λ̂JR = (G−1DG)JR|κ=κ=0,
˜̂
ΛRJ = (G−1DG)RJ |κ=κ=0,

(∇Λ̃)RJ = ∂Λ̃RJ + (G−1∂G)RS Λ̃SJ − Λ̃RK(G−1∂G)KJ ,

(∇ ˜̂
Λ)RJ = ∂

˜̂
ΛRJ + (G−1∂G)RS

˜̂
ΛSJ −

˜̂
ΛRK(G−1∂G)KJ .

In order to construct the U(1) generator needed to define physical states, it is useful to

combine the SO(4, 2) × SO(6) spinors (G−1∂G)JR and (G−1∂G)RJ into an SO(10, 2) spinor

– 22 –



J
H
E
P
0
6
(
2
0
1
6
)
1
2
7

(G−1∂G)A for A = 1 to 32 and write (3.52) in SO(10, 2) notation as

S = r2

∫
d2zd2κ εAB(G−1DG)A(G−1DG)B (3.54)

= r2

∫
d2z
[
εAB(G−1∂G)A(G−1∂G)B + εABΛA(∇Λ)B + εABΛ̂A(∇Λ̂)B (3.55)

+RMNPQ(ΛγMNΛ)(Λ̂γPQΛ̂)
]
,

where A = 1 to 32 are d=12 spinor indices and M = 0 to 11 are d=12 vector indices, εAB
is the Lorentz-covariant antisymmetric tensor used to raise and lower d=12 spinor indices,

γM are the d=12 gamma matrices, γMN
AB = γMN

BA are products of gamma matrices, and

RMNPQ = −ηMP ηNQ + ηNP ηMQ when 0 ≤M,N,P,Q ≤ 5, (3.56)

RMNPQ = ηMP ηNQ − ηNP ηMQ when 6 ≤M,N,P,Q ≤ 11.

As in a flat background, a U(1) generator J can be defined which splits G = G+ +G−

where G± carries ±1 U(1) charge. This U(1) generator is constructed by first splitting

the bosonic components ΛA and Λ̂B of (3.55) into left and right-moving d = (10, 2) pure

spinors UA and ÛA satisfying the constraints

UA(γMN )ABU
B = 0, ÛA(γMN )ABÛ

B = 0, (3.57)

together with their conjugate momenta VA and V̂A. One then defines the left and right-

moving U(1) generators as

J = −UAVA, J = −ÛAV̂A, (3.58)

so that UA and ÛA carry +1 charge and VA and V̂A carry −1 charge.

Just as a d = 10 pure spinor parameterizes SO(10)
U(5) ×C and has 11 independent complex

components, a d = (10, 2) pure spinor parameterizes SO(10,2)
U(5,1) × C and has 16 independent

complex components. So J splits the 32 components of ΛA into the 16 components of UA

and 16 components of VA. To relate ΛA with UA and VA, introduce a fixed d = (10, 2)

pure spinor UA on the patch of pure spinor space where UAUA 6= 0, and define

ΛA = UA + εABVB +
1

(UU)

[
1

4
(γMNU)A(UγMNV ) + 2UA(UV )

]
, (3.59)

where the coefficients in (3.59) have been chosen such that UγMNΛ = UγMNU . Note

that (3.59) is invariant under the gauge transformation

δVA = ΩMN (γMNU)A (3.60)

for any ΩMN , which allows 16 components of VA to be gauged to zero.

When written in terms of (UA, VA) and (ÛA, V̂A), the worldsheet action of (3.55) has

no terms with positive U(1) charge and the term with zero U(1) charge is

S = r2

∫
d2z
[
εAB(G−1∂G)A(G−1∂G)B + VA(∇U)A + V̂A(∇Û)A (3.61)

+RMNPQ(V γMNU)(V̂ γPQÛ)
]

– 23 –



J
H
E
P
0
6
(
2
0
1
6
)
1
2
7

where RMNPQ is defined in (3.56). As expected, the term of zero U(1) charge in (3.55) is

independent of the fixed pure spinors U and Û and gauge-invariant under (3.60).

Under the SO(4, 2)× SO(6) subgroup of SO(10, 2), UA decomposes into (ŨRJ , U
J
R) and

the d = (10, 2) pure spinor constraint of (3.57) decomposes as

ŨRJ U
J
S =

1

4
δRS Ũ

T
J U

J
T , UJRŨ

R
K =

1

4
δJKU

L
RŨ

R
L , (3.62)

εJKLM ŨRJ Ũ
S
K = εRSTUULT U

M
U , εJKLMU

J
RU

K
S = εRSTU Ũ

T
L Ũ

U
M .

Note that the second line of (3.62) is not invariant under the U(1) ‘bonus’ symmetry which

rotates

UJS → eiφUJS , ŨSJ → e−iφŨSJ (3.63)

and enlarges the R-symmetry group from SU(4) to U(4). So although the action of (3.52)

is invariant under this U(1) bonus symmetry, J of (3.58) is not invariant which implies that

the action of (3.61) with zero U(1) charge is also not invariant under this bonus symmetry.

Acknowledgments

I would like to thank CNPq grant 300256/94-9 and FAPESP grants 2011/11973-4 and

2014/18634-9 for partial financial support, Nikita Nekrasov and Edward Witten for sug-

gesting to look for an N=1 worldsheet supersymmetric description of the pure spinor for-

malism, and Andrei Mikhailov, Warren Siegel, Cumrun Vafa and Pedro Vieira for useful

discussions.

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.

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	Introduction
	Untwisting the pure spinor formalism
	N=1 generators and superfields
	Worldsheet supersymmetric action
	U(1) generator
	Massless vertex operators
	Tree-level scattering amplitudes
	Extended RNS formalism
	Worldsheet supersymmetric action in curved background

	Twistor string formalism
	Twistor superfields
	N=1 superconformal and U(1) generator
	Worldsheet action in a flat background
	Twistor string in AdS(5) x S**5  background
	U(1) generator and d=12 pure spinors