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Published for SISSA by Springer

Received: April 14, 2013

Accepted: April 30, 2013

Published: May 10, 2013

Nilpotency of the b ghost in the non-minimal pure

spinor formalism

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: renannlj@ift.unesp.br

Abstract: The b ghost in the non-minimal pure spinor formalism is not a fundamental

field. It is based on a complicated chain of operators and proving its nilpotency is nontrivial.

Chandia proved this property in arXiv:1008.1778, but with an assumption on the non-

minimal variables that is not valid in general. In this work, the b ghost is demonstrated to

be nilpotent without this assumption.

Keywords: Superstrings and Heterotic Strings, Topological Strings

ArXiv ePrint: 1303.3966

c© SISSA 2013 doi:10.1007/JHEP05(2013)048

mailto:renannlj@ift.unesp.br
http://arxiv.org/abs/1303.3966
http://dx.doi.org/10.1007/JHEP05(2013)048


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Contents

1 Introduction 1

2 Review of the pure spinor formalism 2

2.1 Matter fields 2

2.2 Ghost fields 3

3 The b ghost 6

3.1 Definition and properties 7

3.2 Nilpotency 9

4 Conclusion 19

A Conventions and useful formulas 20

1 Introduction

The super Poincaré covariant quantization of the superstring was achieved in the year 2000,

with the development of the pure spinor formalism [1]. One of its oddest features is the

absence of a natural prescription for string loop amplitudes, that is manifest in the other

formalisms of bosonic and supersymmetric strings, due to the existence of the world-sheet

reparametrization invariance.

It is a well known fact that in gauge fixing the reparametrization symmetry, a (b, c)

system rises as the ghost-antighost pair. The c ghost is a conformal weight −1 field,

as it comes from the general coordinate transformation parameter, and the b ghost, the

conjugate of c, is a conformal weight +2 field.

Concerning amplitudes, the fundamental objects of study in quantum strings, the c

ghost appears at tree and 1-loop level. In these world-sheet topologies (respectively, the

sphere and the torus), the conformal Killing symmetries can be removed by fixing some

vertex positions. For the pure spinor formalism, Berkovits developed a prescription [1–3]

that successfully described superstring amplitudes, where a possible c ghost played no

role at all.

For the b ghost, however, the story is different. In a BRST-like description, b ghost

insertions lie in the heart of the BRST invariance of string loop amplitudes. The funda-

mental property is {Q, b} = T , where T is the energy-momentum tensor (since the BRST

charge has ghost number +1, the b ghost must have ghost number −1). Combined with

the Beltrami differentials, this property induces only a surface contribution in the moduli

space integration.

In the minimal pure spinor formalism, where the available ghost variables are the pure

spinor λα and its conjugate ωα, the b ghost is based upon a complicated chain of operators

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and can be implemented only in a picture raised manner [2], as there are no suitable ghost

number −1 fields.

With the addition of the ghost fields
(
λα, rα

)
and their conjugates (ωα, sα), the non-

minimal pure spinor formalism enables a much simpler construction of the b ghost [3].

More than that, the theory can be interpreted as a twisted N = 2 ĉ = 3 topological string,

where the BRST charge and the b ghost are the fermionic generators, while the ghost

number current and the energy-momentum tensor are the bosonic ones. This fact allowed

the covariant computation of multiloop superstring amplitudes without picture changing

operators, making the super Poincaré symmetry explicit in all the steps.

Since the b ghost is a composite field, its nilpotency, a crucial property in the topological

string interpretation, is not evident. In [4], the regularity of the b ghost OPE with itself

was derived, but in an incomplete manner,1 as will be explained here. Therefore, a rigorous

proof of such a fundamental property was still lacking.

This paper is organized as follows. Section 2 contains a review of the pure spinor

formalisms and section 3 presents the construction of the b ghost, its basic properties and

a rigorous derivation of the b ghost OPE with itself, proving its regularity. Appendix A

contains the conventions that are being used in this work and some ordering considerations.

2 Review of the pure spinor formalism

The pure spinor formalism will be reviewed here, establishing the fundamental fields that

will be used in the remaining sections.

2.1 Matter fields

The matter part of the action is

Sm =
1

2π

∫

d2z

(
1

α′
∂Xm∂Xm + pβ∂θ

β

)

, (2.1)

and the free field propagators are just

Xm (z, z̄)Xn (y, ȳ) ∼ −
α′

2
ηmn ln |z − y|2 (2.2a)

pα (z) θ
β (y) ∼

δβα
z − y

. (2.2b)

The action Sm is invariant under the supersymmetric charge, defined as

qα =

∮ [

pα +
1

α′
∂Xm (θγm)α +

1

12α′
(θγm∂θ) (θγm)α

]

. (2.3)

Note that

α′ {qα, qβ} = 2γmαβ

∮

∂Xm. (2.4)

1The flaw in the proof of [4] was pointed out by N. Berkovits, in a private communication.

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The construction of the supersymmetric invariants follows:

Πm = ∂Xm +
1

2
(θγm∂θ) , (2.5a)

dα = pα −
1

α′
∂Xm (θγm)α −

1

4α′
(θγm∂θ) (θγm)α . (2.5b)

So far, this is nothing but the left-moving sector of the Green-Schwarz-Siegel action in

the conformal gauge. The Virasoro constraint is ΠmΠm + α′dα∂θ
α = 0 and the fermionic

constraints (related to kappa symmetry) are dα = 0.

The related OPE’s are given by:

Πm (z)Πn (y) ∼ −
α′

2

ηmn

(z − y)2
, (2.6a)

dα (z)Π
m (y) ∼

γmαβ∂θ
β

(z − y)
, (2.6b)

dα (z) dβ (y) ∼ −
2

α′

γmαβΠm

(z − y)
. (2.6c)

The matter energy-momentum tensor (Virasoro constraint) is

Tmatter = −
1

α′
∂Xm∂Xm − pα∂θ

α, (2.7)

from which it follows that

Tmatter (z)Tmatter (y) ∼ −
11

(z − y)4
+ 2

Tmatter

(z − y)2
+
∂Tmatter

(z − y)
. (2.8)

Therefore, the free matter action yields a negative central charge, that will be cancelled

with the contribution coming from the ghost sector.

2.2 Ghost fields

Introducing a pure spinor λα variable and its conjugate, ωα, one is able to define

JBRST ≡ λαdα, (2.9)

and construct a BRST like charge,

Q =

∮

JBRST, (2.10)

where

{Q,Q} = −
2

α′

∮

(λγmλ)Πm = 0, (2.11)

and

λγmλ = 0 (2.12)

is the D = 10 pure spinor constraint, which implies that only 11 components of λα are

independent. Observe that an explicitly Lorentz invariant action for the ghost sector,

Sλ =
1

2π

∫

d2z
(
ωα∂̄λ

α
)
, (2.13)

must be gauge invariant under δǫωα = ǫm (γmλ)α, due to (2.12).

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The simplest gauge invariant objects are

Tλ = −ω∂λ, Nmn = −1
2ωγ

mnλ, Jλ = −ωλ,

respectively, the energy-momentum tensor, the Lorentz current and the ghost number

current.

The full set of OPE’s of the ghost sector is:

Tλ (z)Tλ (y) ∼
11

(z − y)4
+ 2

Tλ

(z − y)2
+

∂Tλ
(z − y)

,

Jλ (z)Tλ (y) ∼ −
8

(z − y)3
+

Jλ

(z − y)2
,

Nmn (z)Tλ (y) ∼
Nmn

(z − y)2
, Tλ (z)λ

α (y) ∼
∂λα

(z − y)
, Nmn (z)λα (y) ∼

1

2

(γmnλ)α

(z − y)
,

Nmn (z) Jλ (y) ∼ regular, Jλ (z)λ
α (y) ∼

λα

(z − y)
, Jλ (z) Jλ (y) ∼ −

4

(z − y)2
,

Nmn (z)Npq (y) ∼ 6
ηm[pηq]n

(z − y)2
+ 2

ηm[qNp]n + ηn[pN q]m

(z − y)
.

The non-minimal version of the pure spinor formalism includes a new set of fields,
(
λα, rα

)
. The former is also a pure spinor, that is

λγmλ = 0, (2.14)

whereas the latter is a fermionic spinor constrained through

λγmr = 0. (2.15)

Both constraints imply that there are only 11 independent components in each spinor.

Denoting their conjugates as (ωα, sα), the action for the non-minimal sector is

Sλ =
1

2π

∫

d2z
(
ω̄α∂̄λ̄α + sα∂̄rα

)
, (2.16)

which is gauge invariant by the following transformations

δǫ,φω̄
α = ǫm

(
γmλ̄

)α
+ φm (γmr)

α ,

δφs
α = φm

(
γmλ̄

)α
. (2.17)

There are several gauge invariant quantities that can be built out of ωα and sα.

N
mn

=
1

2

(
λ̄γmnω̄ − rγmns

)
,

Jλ̄ = −λ̄ω̄, Tλ̄ = −ω̄∂λ̄− s∂r, Φ = rω̄,

S = λs, Smn =
1

2
λ̄γmns, Jr = rs.

(2.18)

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Here, N
mn

is the Lorentz generator, Tλ is the energy-momentum tensor, and Jλ̄ and Jr
are the ghost number currents. Note that they are not all independent,2 since

Smn

(
rγmnλ

λλ

)

+ S

(
rλ

λλ

)

− 4Jr = 0, (2.19)

N
mn
(
rγmnλ

λλ

)

− Jλ

(
rλ

λλ

)

+ 3Jr

(
rλ

λλ

)

+ 4Φ = 0. (2.20)

The OPE’s between them can be summarized as follows:

Tλ (z)Tλ (y) ∼ 2
Tλ

(z − y)2
+

∂Tλ
(z − y)

, N
mn

(z)Tλ (y) ∼
N

mn

(z − y)2
,

Smn (z)Tλ (y) ∼
Smn

(z − y)2
,

Jλ̄ (z)Tλ (y) ∼ −
11

(z − y)3
+

Jλ

(z − y)2
, Φ (z)Tλ (y) ∼

Φ

(z − y)2
,

S (z)Tλ (y) ∼
S

(z − y)2
,

Jr (z)Tλ (y) ∼
11

(z − y)3
+

Jr

(z − y)2
, Tλ (z)λα (y) ∼

∂λα
(z − y)

,

Tλ (z) rα (y) ∼
∂rα

(z − y)
,

Φ (z)S (y) ∼ −
8

(z − y)2
−
Jλ + Jr

(z − y)
, Φ (z)Smn (y) ∼

N
mn

(z − y)
,

Φ (z)λα (y) ∼ −
rα

(z − y)
,

Φ (z) Φ (y) ∼ regular, N
mn

(z) Jλ̄ (y) ∼ regular,

N
mn

(z) Φ (y) ∼ regular,

N
mn

(z)N
pq
(y) ∼ 2

ηm[qN
p]n

+ ηn[pN
q]m

(z − y)
, Jλ̄ (z) Jr (y) ∼ −

3

(z − y)2
,

Jλ̄ (z) Jλ̄ (y) ∼ −
5

(z − y)2
,

N
mn

(z) Jr (y) ∼ regular, N
mn

(z)S (y) ∼ regular,

Jr (z) Jr (y) ∼
11

(z − y)2
,

N
mn

(z)λα (y) ∼ −
1

2

(
λγmn

)

α

(z − y)
, N

mn
(z) rα (y) ∼ −

1

2

(rγmn)α
(z − y)

,

Jλ̄ (z)λα (y) ∼
λα

(z − y)
,

2Some numerical coefficients of the corresponding relations in [3] are incorrect, as can be promptly

verified by the definitions in (2.18) and the gamma matrices identity (A.8).

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Jr (z) rα (y) ∼
rα (y)

(z − y)
, Jλ̄ (z) rα (y) ∼ regular,

Jr (z)λα (y) ∼ regular.

Note that there are no contributions to the central charge or to the level of the

Lorentz algebra.3

The non-minimal variables enter the formalism in a very simple way, as the BRST

charge is defined to be

Q ≡

∮

(λαdα +Φ)
︸ ︷︷ ︸

JBRST (z)

. (2.21)

The same notation was used for the BRST charge in the minimal formalism, but from

now on, only (2.21) will be referred to as Q. The cohomology of (2.21) is independent

of
(
λ, ω, r, s

)
, as can be seen from the quartet argument, and there is a state ξ that

trivializes it:

ξ =
λ · θ

λ · λ− r · θ
, {Q, ξ} = 1. (2.22)

Since rα and θα are grassmannian variables, ξ can be expanded as a finite power series in

terms of r · θ. Besides, rα has only 11 independent components, in such a way that

ξ =
λ · θ

λ · λ

11∑

n=0

(
r · θ

λ · λ

)n

. (2.23)

Therefore, one way of avoiding the appearance of ξ is limiting the amount of inverse

powers of λλ. However, this is a fundamental ingredient in the construction of the b

ghost, constituting the main obstruction for loop amplitude calculations in the pure spinor

formalism [3, 5].

3 The b ghost

The b ghost is a central field in string perturbation theory, being related to the BRST

invariance of string loop amplitudes. Its basic property is

{Q, b} = T. (3.1)

As there is not such a fundamental object in the pure spinor formalism, it must be build

out of the available fields of the theory.

As introduced in [3], the construction of the non-minimal b ghost is based on a chain

of operators satisfying some special relations, that will be reviewed below.

3The quadratic pole in Φ (z)S (y) is also absent in [3].

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3.1 Definition and properties

The full quantum version of the b ghost can be cast as

b = b−1 + b0 + b1 + b2 + b3, (3.2)

where

b−1 ≡ −sα∂λα, (3.3a)

b0 ≡

(

λα
(
λλ
) , Gα

)

+O, (3.3b)

b1 ≡ −2!

(

λαrβ
(
λλ
)2 , H

αβ

)

, (3.3c)

b2 ≡ −3!

(

λαrβrγ
(
λλ
)3 ,K

αβγ

)

, (3.3d)

b3 ≡ 4!

(

λαrβrγrλ
(
λλ
)4 , Lαβγλ

)

, (3.3e)

and

O ≡ −∂

(

λαλβ
(
λλ
)2

)

λα∂θβ, (3.4a)

Gα =
1

2
γαβm (Πm, dβ)−

1

4
Nmn (γ

mn∂θ)α −
1

4
Jλ∂θ

α + 4∂2θα, (3.4b)

Hαβ =
1

4 · 96
γαβmnp

(
α′

2
dγmnpd+ 24NmnΠp

)

, (3.4c)

Kαβγ = −
1

96

(
α′

2

)

Nmnγ
[αβ
mnp (γ

pd)γ] , (3.4d)

Lαβγλ = −
3

(96)2

(
α′

2

)

(Nmn, N rs) ηpqγ[αβmnpγ
γ]λ
qrs . (3.4e)

Note that the subscript n in bn is the r charge qr of the operators, defined as

∫

dz {Jr (z)O (y)} = qr (O)O (y) . (3.5)

The building blocks of bn satisfy:

{
Q,−sα∂λα

}
= Tλ, {Q,Gα} = (λα, Tλ + Tmatter) ,

[

Q,Hαβ
]

=
(

λ[α, Gβ]
)

,
{

Q,Kαβγ
}

=
(

λ[α, Hβγ]
)

,
[

Q,Lαβγλ
]

=
(

λ[α,Kβγλ]
)

,
(

λ[α, Lβγλσ]
)

= 0.

(3.6)

Some observations should be made concerning the above operators:

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• the ordering here, implemented through

(A,B) (y) ≡
1

2πi

∮
dz

z − y
A (z)B (y) , (3.7)

plays a major role, allowing a correct manipulation of the quantum corrections to the b

ghost. Obviously, a different ordering prescription must not conflict with {Q, b} = T .

• the operator O defined above is required because

((
λαrβ − λβrα

)
λα

(
λλ
)2 , Gβ

)

−

(

λαrβ
(
λλ
)2 ,
(

λα, Gβ
)

−
(

λβ , Gα
)
)

6= 0, (3.8)

and (

λα
(
λλ
) , (λα, Tλ)

)

− Tλ 6= 0. (3.9)

One can see that {Q,O} precisely matches the above inequalities. In [6], besides (3.7),

an alternative prescription was used, that conveniently absorbs the operator O.

• the quantum contribution to Gα is proportional to ∂2θα. The coefficient can be

fixed by comparing the cubic pole in the OPE of the energy-momentum tensor

with both sides of the equation {Q,Gα} = (λα, T ), or directly through the usual

U (5) decomposition.4

The last observation is directly related to the fact that the b ghost is a conformal

weight 2 primary field [6],

T (z) b (y) ∼ 2
b

(z − y)2
+

∂b

(z − y)
. (3.10)

This result is reproduced in appendix A. Note also that b is a Lorentz scalar and manifestly

supersymmetric.

Another interesting property of the b ghost is the pole structure of its OPE with the

BRST current:

JBRST (z) b (y) ∼
3

(z − y)3
+

J

(z − y)2
+

T

(z − y)
, (3.11)

where

J = Jλ + Jr − 2
λ∂λ

λλ
+ 2

r∂θ

λλ
− 2

(rλ)
(
λ∂θ

)

(
λλ
)2 .

= Jλ − Jλ −

{

Q,

(

S + 2
λ∂θ

λλ

)}

. (3.12)

4That explains why the coefficient used here differs from the one used in [2, 3], where Gα was required

to be primary. Equation (A.24) shows that this is not the case, since (λα, T ) is not a primary field.

[6] contains a detailed discussion on this subject. There, Gα and Ĝα denote the primary and the non-

primary constructions, respectively.

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With a BRST transformation, the U(1) current can be brought into a more natural form,

without changing the ghost numbers of the BRST charge and the b ghost.

To verify the interpretation of J as the ghost number current, it is worth noting that,

T (z) J (y) ∼ −
3

(z − y)3
+

J

(z − y)2
+

∂J

(z − y)
, (3.13)

J (z) JBRST (y) ∼
JBRST

(z − y)
, (3.14)

J (z) b (y) ∼ −
b

(z − y)
. (3.15)

Together, b, T , JBRST and J may describe a twisted N = 2 ĉ = 3 critical topological

string [3]. The untwisted version would satisfy

T ′ (z)T ′ (y) ∼
(9/2)

(z − y)4
+ 2

T ′

(z − y)2
+

∂T ′

(z − y)
, T ′ (z) J (y) ∼

J

(z − y)2
+

∂J

(z − y)
,

T ′ (z)G+ (y) ∼
3

2

G+

(z − y)2
+

∂G+

(z − y)
, T ′ (z)G− (y) ∼

3

2

G−

(z − y)2
+

∂G−

(z − y)
,

J (z)G+ (y) ∼
G+ (y)

(z − y)
, J (z)G− (y) ∼ −

G− (y)

(z − y)
,

J (z) J (y) ∼
3

(z − y)2
,

G+ (z)G− (y) ∼
3

(z−y)3
+

J

(z−y)2
+
T ′ + 1

2∂J

(z−y)
,

G+ (z)G+ (y) ∼ regular, G− (z)G− (y) ∼ regular,

where G+ = JBRST , G
− = b and T ′ = T − 1

2∂J . The twist here means T ′ → T ′ − 1
2∂J ,

which modifies the conformal weights of the ghosts λ and r from 1
2 to 0 and turns the

central charge off.

By examining this set of OPE’s, one notes that b must be nilpotent in order for the

non-minimal pure spinor formalism to be viewed as a topological string. This property will

now be rigorously demonstrated.

3.2 Nilpotency

The OPE of the b ghost with itself can be cast as

b (z) b (y) ∼
O0

(z − y)4
+

O1

(z − y)3
+

O2

(z − y)2
+

O3

(z − y)
, (3.16)

for there are no (covariant, supersymmetric) negative conformal weight fields in the theory.

Due to its anticommuting character, b (z) b (y) = −b (y) b (z), implying that

b (z) b (y) ∼
O1

(z − y)3
+

1

2

∂O1

(z − y)2
+

O3

(z − y)
. (3.17)

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Furthermore, since {Q, b} = T and b is a primary field of conformal weight 2,

{Q, b (z)} b (y)− b (z) {Q, b (y)} = T (z) b (y)− b (z)T (y)

∼ regular, (3.18)

or, equivalently,

{Q, b (z) b (y)} ∼
{Q,O1}

(z − y)3
+

1

2

∂ {Q,O1}

(z − y)2
+

{Q,O3}

(z − y)
. (3.19)

Comparing both expressions, one concludes that O1 and O3 are BRST closed.

Taking now into account the specific form of the b ghost for the non-minimal pure

spinor formalism, given in (3.2), it is a simple task to verify that the cubic poles are all

proportional to the constraints (2.14) and (2.15). The possible terms will be listed below:

• b−1 may give rise to cubic poles only in the OPE with b3, due to ordering effects.

The different terms are proportional to

(
λγmnpr

) (
∂λγpqrr

) (
λγmnλ

) (
λγqrλ

)
,

(
λγmnpr

) (
∂λγpqrr

) (
λγmnγqrλ

)
,

(
λγmnp∂λ

)
(rγpqrr)

(
λγmnλ

) (
λγqrλ

)
,

(
λγmnp∂λ

)
(rγpqrr)

(
λγmnγqrλ

)
.

(3.20)

• b0 has cubic poles with itself, b1, b2 and b3:

– in b0 (z) b0 (y), it comes from the multiple contractions of Πm (γmd)
α with itself

and from its single contraction with ∂2θβ, both proportional to Πm
(
λγmλ

)
.

– for b0 (z) b1 (y), it will arise in the contractions of (dγmnpd)
(
λλ
)−2

with all the

terms in b0, being proportional to
(
λγmnpr

) (
λγmnpd

)
.

– in the OPE b0 (z) b2 (y), the multiple contractions of Nmn (γpd)α will give cubic

poles like:
(
λγmnpr

)
Nmn

(
λγpr

)
,

(
λγmnpr

)
J
(
λγmnλ

) (
λγpr

)
,

∂
[(
λγmnλ

)
λα
] (
λγmnpr

)
(γpr)α .

(3.21)

– finally, in b0 (z) b3 (y), the cubic poles are of the form:

(
λ∂θ

) (
λγmnλ

) (
λγqrλ

) (
λγmnpr

)
(rγpqrr) ,

(
λγmn∂θ

) (
λγqrλ

) (
λγmnpr

)
(rγpqrr) .

(3.22)

• b1 has cubic poles with itself, b2 and b3:

– in b1 (z) b1 (y), they are of the form

∂
(
λγmnpr

) (
λγmnpr

)
,

(
λγmnpr

) (
λγmnqr

)
Np

q,

∂
(
λγmnpr

) (
λγpqrr

) (
λγmnλ

) (
λγqrλ

)
.

(3.23)

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– for b1 (z) b2 (y), the only possible cubic poles are proportional to

(
λγmnpr

) (
λγqrsr

) (
λγmnλ

)
(rγqrsγp∂θ) ,

(
λγmnpr

) (
λγmnqr

)
(rγqγ

p∂θ) .
(3.24)

– the cubic poles arising in b1 (z) b3 (y) come from the multiple contractions of

NmnΠp
(
λλ
)−2

with b3, and are given by

(
λγmnpr

) (
λγmnqr

)
(rγqrsr)

(
λγrsλ

)
Πp,

(
λγmnpr

) (
λγmnλ

)
Πp
(
λγqrsr

) (
λγqrλ

) (
rγstur

) (
λγtuλ

)
,

(
λγmnpr

) (
λγmnλ

)
Πp
(
λγqrsr

) (
λγqrγtuλ

) (
rγstur

)
.

(3.25)

• b2 has cubic poles with itself and with b3:

– in b2 (z) b2 (y), they are of the form

(
λγmnpr

) (
λγqrsr

) (
rγpstr

)
Πtη

mqηnr,
(
λγmnpr

) (
λγmnλ

) (
λγqrsr

) (
λγqrλ

) (
rγpstr

)
Πt.

(3.26)

– for b2 (z) b3 (y), dα appearing in b2 is inert and there are only contractions in-

volving the ghost Lorentz currents:

(
λγmnpr

)
(rγpd)

(
λγmnqr

)
(rγqrsr)

(
λγrsλ

)
,

(
λγmnpr

) (
λγmnλ

)
(rγpd)

(
λγqrsr

) (
λγqrλ

) (
rγstur

) (
λγtuλ

)
.

(3.27)

• the cubic poles of b3 (z) b3 (y) involve all possible contractions of the the Lorentz

generators and will give similar results to the ones above, only with more r’s.

Due to the pure spinor constraints,

(
λγmn

)

α

(
λγmnpr

)
=
(
λγmn

)

α

(
λγmnp∂λ

)
=
(
λγmnpr

) (
λγmnp

)α
= 0, (3.28)

and every expression listed contains at least one of these types of contractions. Conse-

quently, O1 = 0 and

b (z) b (y) ∼
O3

(z − y)
. (3.29)

It is clear from (3.2), that O3 can only be composed with supersymmetric invariants:

matter fields (Πm, dα, ∂θ
α); ghost currents from the minimal sector (Nmn, J); ghost fields

(λα,λα,rα); and, in principle, their partial derivatives.

In [4], the vanishing of O3 has been argued as follows. The author assumed that all

partial derivatives of rα that may appear in the OPE (3.29) can be removed due to the

pure spinor constraint, since

λγm∂r = −∂λγmr. (3.30)

Based on that assumption, all the rα dependence of O3 could be made explicitly through

O3 = Ω+ rαΩ
α + rαrβΩ

αβ + . . . (3.31)

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where the Ω’s are supersymmetric, ghost number −2, conformal weight 3, BRST closed

operators. Since the BRST charge can be split into two pieces according to the r-charge

Q = Q0 +Q1, (3.32a)

Q0 =

∮

(λαdα) , (3.32b)

Q1 =

∮

(ωαrα) , (3.32c)

requiring [Q,O3] = 0, implies [Q0,Ω] = 0. Then, it has been shown that there are no Ω

with the above requisites satisfying [Q0,Ω] = 0, so it vanishes identically. Then, Ω = 0

implies [Q0,Ω
α] = 0. Again, this can be demonstrated to vanish. Pursuing this argument,

the nilpotency of the b ghost was obtained in [4].

However, the absence of ∂nrα in O3 is incorrect, as will be illustrated soon, which

means that the cohomology argument of [4], summarized above, must be extended, as will

now be done.

The computation of (3.29) is organized according to the r-charge of the operators,

that is

O3 = (bb)0 + (bb)1 + (bb)2 + (bb)3 + (bb)4 + (bb)5 + (bb)6 . (3.33)

To make the expressions more clear, the ordering notation will be dropped and α′ will be

set to 2.

The first term, (bb)0, is given by

(bb)0 ≡

∫

dz {b0 (z) b0 (y) + b−1 (z) b1 (y) + b1 (z) b−1 (y)} (3.34)

= α01
Nmn

(
λγmn∂θ

) (
λ∂θ

)

(
λλ
)2 + α02

(
λγmnp∂λ

)
NmnΠp

(
λλ
)2

+α03
Πm

(
λ∂θ

) (
λγmd

)

(
λλ
)2 + α04

(
λγmnp∂λ

)
(dγmnpd)

(
λλ
)2

+α05
Πm

(
λγm∂

2λ
)

(
λλ
)2 + α06

(
λ∂θ

) (
λ∂2θ

)

(
λλ
)2 + α07

(
λ∂θ

) (
∂λ∂θ

)

(
λλ
)2 , (3.35)

where α0n are just numerical coefficients. By a direct computation, it is relatively simple to

show the vanishing of (bb)0. It is enough to compute [Q, (bb)0] and use the BRST argument

mentioned above. Note that [Q,O3] = 0 implies the vanishing of

[Q0, (bb)0] = α01
Nmn

(
λγmn∂λ

) (
λ∂θ

)

(
λλ
)2 − α01

1
2 (dγ

mnλ)
(
λγmn∂θ

) (
λ∂θ

)

(
λλ
)2

−α01
Nmn

(
λγmn∂θ

) (
λ∂λ

)

(
λλ
)2 − α02

(
λγmnp∂λ

)
1
2 (dγ

mnλ)Πp

(
λλ
)2

+α02

(
λγmnp∂λ

)
Nmn (λγp∂θ)

(
λλ
)2 − α03

Πm
(
λ∂θ

) (
λγmγn∂θ

)
Πn

(
λλ
)2

– 12 –



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+α03
Πm

(
λ∂λ

) (
λγmd

)

(
λλ
)2 + α03

(λγm∂θ)
(
λ∂θ

) (
λγmd

)

(
λλ
)2

+α04
2
(
λγmnp∂λ

)
(dγmnpγqλ)Πq

(
λλ
)2 + α06

(
λ∂λ

) (
λ∂2θ

)

(
λλ
)2 − α06

(
λ∂θ

) (
λ∂2λ

)

(
λλ
)2

+α05
(λγm∂θ)

(
λγm∂

2λ
)

(
λλ
)2 + α07

(
λ∂λ

) (
∂λ∂θ

)

(
λλ
)2 − α07

(
λ∂θ

) (
∂λ∂λ

)

(
λλ
)2 .

The Lorentz generators Nmn appear in three terms. It is straightforward to check that

they are not related by a Fierz decomposition of the spinors, implying that α01 = α02 = 0.

Now, there is only one term that contributes with one dα and two ∂θα, so α03 = 0, which,

on the other hand, imply that α04 = 0, since the term with one dα and one Πm cannot

be cancelled anymore. The vanishing of α05, α06 and α07 is evident, since they do not

possibly cancel each other. There is no linear combination of the above operators that can

be annihilated by Q0, therefore (bb)0 = 0.

The second term, (bb)1, is

(bb)1 ≡

∫

dz {b0 (z) b1 (y) + b1 (z) b0 (y) + b−1 (z) b2 (y) + b2 (z) b−1 (y)} (3.36)

= α11

(
λγmnpr

)
NmnΠp

(
λ∂θ

)

(
λλ
)3 + α12

(
λγmnpr

)
Nmn

(
∂λγpd

)

(
λλ
)3

+α13

(
λγmnp∂λ

)
Nmn (rγpd)

(
λλ
)3 + α14

(
λγmnpr

)
(dγmnpd)

(
λ∂θ

)

(
λλ
)3

+α15

(
λγmnpr

)
Πm

(
∂λγnp∂θ

)

(
λλ
)3 + α16

(
λγm∂

2λ
)
(rγmd)

(
λλ
)3 . (3.37)

Since [Q1, (bb)0] = 0, [Q0, (bb)1] must also vanish:

[Q0, (bb)1] = α11

1
2

(
λγmnpr

)
(dγmnλ)Πp

(
λ∂θ

)

(
λλ
)3 − α11

(
λγmnpr

)
Nmn (λγp∂θ)

(
λ∂θ

)

(
λλ
)3

−α11

(
λγmnpr

)
NmnΠp

(
λ∂λ

)

(
λλ
)3 + α12

1
2

(
λγmnpr

)
(dγmnλ)

(
∂λγpd

)

(
λλ
)3

+α12

(
λγmnpr

)
Nmn

(
∂λγpγqλ

)
Πq

(
λλ
)3 − α13

1
2

(
λγmnp∂λ

)
(dγmnλ) (rγpd)

(
λλ
)3

+α13

(
λγmnp∂λ

)
Nmn (rγpγqλ)Πq
(
λλ
)3 − α14

2
(
λγmnpr

)
(dγmnpγqλ)Πq

(
λ∂θ

)

(
λλ
)3

−α14

(
λγmnpr

)
(dγmnpd)

(
λ∂λ

)

(
λλ
)3 − α15

(
λγmnpr

)
(λγm∂θ)

(
∂λγnp∂θ

)

(
λλ
)3

−α15

(
λγmnpr

)
Πm

(
∂λγnp∂λ

)

(
λλ
)3 + α16

(
λγm∂

2λ
)
(rγmγnλ)Πn
(
λλ
)3 .

There is only one term that contains one Lorentz generator Nmn and two ∂θα, so α11 =

0. Now, there are two other terms that contain Nmn, but they are unrelated to any

– 13 –



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Fierz decomposition, implying that α12 = α13 = 0. The remaining terms are obviously

independent: α14 = 0, since it is the only one with (dγmnpd); α15 = 0, as no other term

contains two ∂θα; and α16 = 0, for there is nothing else to cancel it. As (bb)0, (bb)1 is not

BRST closed for any set of coefficients α1n and (bb)1 = 0 is the single possibility left.

Going on,

(bb)2 ≡

∫

dz {b0 (z) b2 (y) + b2 (z) b0 (y) + b1 (z) b1 (y) + b−1 (z) b3 (y) + b3 (z) b−1 (y)}

(3.38)

can be written as

(bb)2 = α21

(
λγmnpr

)
(rγpd)Nmn

(
λ∂θ

)

(
λλ
)4 + α22

(
λγm∂r

)
(rγmd)

(
λ∂θ

)

(
λλ
)4

+α23

(
λγmnpr

) (
∂λγpqrr

)
NmnNqr

(
λλ
)4 + α24

(
λγm∂r

) (
λγnd

)
(rγmn∂θ)

(
λλ
)4

+α25

(
λγmnpr

) (
∂λγqr

)
NmnNpq

(
λλ
)4 + α26

(
λγmnpr

) (
rγp∂2λ

)
Nmn

(
λλ
)4

+α27

(
λγm∂r

) (
rγm∂2λ

)

(
λλ
)4 + α28

(
rγm∂

2λ
) (
λγm∂r

)

(
λλ
)4 . (3.39)

The last line of the expression is Q0-closed. In computing [Q0, (bb)2],

[Q0, (bb)2] = α21

1
2

(
λγmnpr

)
(rγpd) (dγmnλ)

(
λ∂θ

)

(
λλ
)4 − α21

(
λγmnpr

)
(rγpd)Nmn

(
λ∂λ

)

(
λλ
)4

−α21

(
λγmnpr

)
(rγpγqλ)ΠqN

mn
(
λ∂θ

)

(
λλ
)4 − α22

(
λγm∂r

)
(rγmγnλ)Πn

(
λ∂θ

)

(
λλ
)4

−α22

(
λγm∂r

)
(rγmd)

(
λ∂λ

)

(
λλ
)4 − α23

1
2

(
λγmnpr

) (
∂λγpqrr

)
(dγmnλ)Nqr

(
λλ
)4

−α23

1
2

(
λγmnpr

) (
∂λγpqrr

)
Nmn (dγqrλ)

(
λλ
)4 − α24

(
λγm∂r

) (
λγnd

)
(rγmn∂λ)

(
λλ
)4

+α24

(
λγm∂r

) (
λγnγpλ

)
Πp (rγmn∂θ)

(
λλ
)4 − α25

1
2

(
λγmnpr

) (
∂λγqr

)
Nmn (dγpqλ)

(
λλ
)4

−α25

1
2

(
λγmnpr

) (
∂λγqr

)
(dγmnλ)Npq

(
λλ
)4 − α26

1
2

(
λγmnpr

) (
rγp∂2λ

)
(dγmnλ)

(
λλ
)4 ,

the terms that contain matter fields or the Lorentz current do not vanish for any set α2n of

coefficients: α21 = 0, for it is the single term that contains Nmn and Πm; α22 = α24 = 0,

since they are the only ones that contribute with one Πm and one ∂θα, but independently;

α23 = α25 = 0, because they are the remaining (and also independent) terms containing

the Lorentz generator; and α26 = 0, for it is not BRST closed.

– 14 –



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(bb)3 can be cast as:

(bb)3 ≡

∫

dz {b0 (z) b3 (y) + b3 (z) b0 (y) + b1 (z) b2 (y) + b2 (z) b1 (y)} (3.40)

= α31

(
λγmnpr

)
(rγpqrr)NmnNqr

(
λ∂θ

)

(
λλ
)5 + α32

(rγmnpr)
(
λγp∂r

)
Nmn

(
λ∂θ

)

(
λλ
)5

+α33

(
λγm∂r

)
(rγm∂r)

(
λ∂θ

)

(
λλ
)5 + α34

(
λγm∂r

) (
λγn∂r

)
(rγmn∂θ)

(
λλ
)5

+α35

(
λγm∂r

) (
λγn∂r

)
(rγmnλ)

(
λ∂θ

)

(
λλ
)6 . (3.41)

It is straightforward to see that the first two terms are not BRST closed. One of the

contributions of the first one contains two Lorentz generators, that cannot be cancelled,

so α31 = 0. The same happens for the second one, which has a contribution in [Q0, (bb)3]

with one Lorentz generator, not balanced by any other, thus α32 = 0. The result of the

computation of [Q1, (bb)2] + [Q0, (bb)3] with the remaining terms is

[Q1, (bb)2] + [Q0, (bb)3] = α27
4
(
λγm∂r

) (
rγm∂2λ

)
(rλ)

(
λλ
)5 − α27

(rγm∂r)
(
rγm∂2λ

)

(
λλ
)4

−α27

(
λγm∂r

) (
rγm∂2r

)

(
λλ
)4 + α28

4
(
λγm∂

2λ
)
(rγm∂r) (rλ)

(
λλ
)5

−α28

(
rγm∂

2λ
)
(rγm∂r)

(
λλ
)4 − α34

(
λγm∂r

) (
λγn∂r

)
(rγmn∂λ)

(
λλ
)5

−α28

(
λγm∂

2r
)
(rγm∂r)

(
λλ
)4 − α33

(
λγm∂r

)
(rγm∂r)

(
λ∂λ

)

(
λλ
)5

−α35

(
λγm∂r

) (
λγn∂r

)
(rγmnλ)

(
λ∂λ

)

(
λλ
)6 .

Obviously, there is no nontrivial solution to {α27, α28, α33, α34, α35} that may lead to the

vanishing of this equation, thus (bb)2 = (bb)3 = 0. Note that

(
λγm∂r

)
(rγm∂r)

(
λ∂θ

)

(
λλ
)5 (3.42)

does not allow the removal of partial derivatives acting on r, which contradicts the assump-

tion of [4].

So far, the pure spinor constraints only have been used to reduce the number of in-

dependent terms in the OPE computation. It turns out that for (bb)4, (bb)5 and (bb)6, all

possible terms being generated vanish due to the constraints.

For

(bb)4 ≡

∫

dz {b1 (z) b3 (y) + b3 (z) b1 (y) + b2 (z) b2 (y)} , (3.43)

the simple poles are given by:

– 15 –



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• terms with two N ’s and one Π, like

(
λγmnpr

) (
λγqrsr

) (
rγpqtr

)
NmnN r

tΠ
s

(
λλ
)6 . (3.44)

Since (rγmnpr) = (rγmγnγpr) and
(
λγmnpr

)
(rγp)

α = (rγmnpr)
(
λγp
)α

,

(
λγmnpr

) (
λγqrsr

) (
rγpqtr

)
= (rγmnpr) (rγqrsr)

(
λγpγtγqλ

)
, (3.45)

which vanishes because
(
λγmnpλ

)
= 0.

• terms with one N , one Π and one partial derivative (Taylor expansion of a quadratic

pole), as
(
∂λγmnpr

) (
λγqrsr

)
(rγpqrr)NmnΠs

(
λλ
)6 , (3.46)

which vanishes, since

(
λγqrsr

)
(rγpqrr) =

(
λγqrγsr

)
(rγqrγpr)

= 4
(
λγmr

)
(rγsγmγ

pr)

−2
(
λγsr

)
(rγpr)− 8 (rγsr)

(
λγpr

)

= 0. (3.47)

• terms with one N and two d’s, like

(
λγmnpr

) (
λγmqrr

)
Nn

r (rγ
pd) (rγqd)

(
λλ
)6 . (3.48)

Since λγmnpr is equal to λγmγnγpr, this term is proportional to
(
λγm

)α (
λγm

)β
,

and, according to equation (A.15), it vanishes.

• terms with two d’s and one partial derivative, such as

(
λγmnpr

) (
∂λγmnqr

)
(rγpd) (rγqd)

(
λλ
)6 . (3.49)

Decomposing
(
∂λγmnpr

)
as
(
∂λγmnγpr

)
+ ηnp

(
λγm∂r

)
− ηmp

(
λγn∂r

)
, it is possible

to rewrite the expression as follows,

(
λγmnpr

) (
∂λγmnqr

)
=
(
λγmnγpr

) (
∂λγmnγqr

)
+ 2ηnq

(
λγmnpr

) (
λγm∂r

)

=
(
λγm∂λ

)
(rγpγmγ

qr)− 2
(
λγpr

) (
∂λγqr

)

−8
(
λγqr

) (
∂λγpr

)
+ 2ηnq

(
λγmγnpr

) (
λγm∂r

)

= 0, (3.50)

showing that this term also vanishes.

– 16 –



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• and terms with one Π and two partial derivatives (Taylor expansion of a cubic

pole), like
(
∂λγmnp∂r

) (
λγmnqr

)
(rγpqrr)Πr

(
λλ
)6 . (3.51)

Decomposing
(
∂λγmnp∂r

)
as

(
∂λγmnγp∂r

)
− ηnp

(
∂λγm∂r

)
+ ηmp

(
∂λγn∂r

)
,

the expression
(
∂λγmnp∂r

) (
λγmnqr

) (
rγpqrr

)
(3.52)

can be split into two pieces. One of them is similar to the ones presented before and

also vanishes. The other one is proportional to

(
λγm∂r

) (
λγn∂r

)
(rγmnpr) = (rγm∂r)

(
λγn∂r

) (
λγmnpr

)

= − (rγm∂r)
(
λγn∂r

) (
λγnγmpr

)
, (3.53)

and vanishes, since
(
λγm

)α (
λγm

)β
= 0.

For

(bb)5 ≡

∫

dz {b3 (z) b2 (y) + b2 (z) b3 (y)} , (3.54)

all contributions to the simple pole will have dα:

• there are terms with two N ’s, as

(
λγmnpr

)
(rγpd)

(
λγqrsr

) (
rγstur

)
NmqηnrNtu

(
λλ
)7 . (3.55)

Note that

(
λγmnpr

) (
λγqrsr

)
ηnr =

(
λγnγmpr

) (
λγrγqsr

)
ηnr

=
(
λγm

)α (
λγm

)β
(. . .)αβ

= 0, (3.56)

gives a vanishing contribution.

• terms with one N and one partial derivative, as

(
λγmnp∂r

)
(rγpd)

(
λγmnqr

)
(rγqrsr)N

rs

(
λλ
)7 . (3.57)

It is easy to extract the pure spinor constraint out of this expression:

(
λγmnp∂r

) (
λγmnqr

)
=
(
λγmnγp∂r

) (
λγmnγqr

)

− 2
(
λγm∂r

) (
λγmγnqr

)
ηnp

= 4
(
λγmλ

)
(∂rγpγmγ

qr)− 10
(
λγp∂r

) (
λγqr

)

−2
(
λγm∂r

) (
λγm0γnqr

)
ηnp.

= 0. (3.58)

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• and terms with two partial derivatives, coming from the cubic poles, like
(
∂λγmnp∂r

)
(rγpd)

(
λγmnqr

)
(rγqrsr)

(
λγrsλ

)

(
λλ
)8 . (3.59)

Note that (rγqrsr)
(
λγrsλ

)
has the same structure of (3.28) and also vanishes.

Finally, for the last term in the b (z) b (y) OPE, where only the ghost fields appear,

(bb)6 ≡

∫

dz {b3 (z) b3 (y)} , (3.60)

• there are terms with three N ’s, like
(
λγmnpr

)
(rγpqrr)

(
λγmstr

)
(rγtuvr)NqrN

n
sN

uv

(
λλ
)8 . (3.61)

Since λγmnpr = λγmγnγpr,
(
λγmnpr

) (
λγmqrr

)
vanishes, as shown above.

• terms with two N ’s and one partial derivative, like

∂
(
λγmnpr

)
(rγpqrr)

(
λγmnsr

)
(rγstur)NqrN

tu

(
λλ
)8 , (3.62)

which has the same structure presented before, being proportional to the pure spinor

constraints.

• terms with one N and two partial derivatives, coming from triple poles, such as

∂2
(
λγmnpr

)
(rγpqrr)

(
λγmnsr

)
(rγqstr)N

t
r

(
λλ
)8 , (3.63)

which are similar to the above ones and vanish.

• and terms with three partial derivatives, like
(
∂λγmnp∂r

)
(rγpqr∂r)

(
λγmnsr

)
(rγqrsr)

(
λλ
)8 , (3.64)

that can be rewritten as
(
∂λγm∂r

)
(rγq∂r)

(
λγmnpr

)
(rγnpqr)

(
λλ
)8 (3.65)

and vanish, since

(
λγmnpr

)
(rγnpqr) =

(
λγnpγmr

)
(rγnpγqr)

=
(
λγnr

)
(rγmγnγqr)

−2
(
λγmr

)
(rγqr)− 8 (rγmr)

(
λγqr

)

= 0. (3.66)

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Summarizing, in the OPE computation several terms vanish identically due to the pure

spinor constraints (in particular, (bb)4, (bb)5 and (bb)6 do not present nontrivial contribu-

tions). The remaining terms are excluded through the BRST argument, since they were

shown to be not BRST closed. Therefore,

(bb)1 = (bb)2 = (bb)3 = (bb)4 = (bb)5 = (bb)6 = 0, (3.67)

and the pure spinor b ghost is, indeed, nilpotent:

b (z) b (y) ∼ regular. (3.68)

4 Conclusion

In this work, some properties of the b ghost in the non-minimal pure spinor formalism were

reviewed and confirmed. The main object of study was the nilpotency of the non-minimal

b ghost.

From general arguments, the b (z) b (y) OPE is reduced to

b (z) b (y) ∼
O1

(z − y)3
+

1

2

∂O1

(z − y)2
+

O3

(z − y)
,

where O1 and O3 are BRST closed.

As was already known from [4], the different terms in the cubic pole, O1, are all

proportional to the pure spinor constraints

λγmr = λγmλ = 0.

However, the demonstration that the simple pole (O3) vanishes, was incomplete, due to a

wrong assumption on the absence of rα derivatives.

A counter-example to that assumption is very simple,

(
λγm∂r

)
(rγm∂r)

(
λ∂θ

)

(
λλ
)5 .

Note that the fundamental ingredient here is (rγm∂r), an object that does not allow, in

general, the removal of the partial derivatives acting on rα.

Knowing this flaw, the proof that O3 = 0 was carried out in a straightforward manner.

First, a careful analysis was made, obtaining all terms that could be generated in the

OPE computation. For some of them, the cancellation is very simple to obtain and no

BRST argument is needed. However, for most of the terms (those that appear in ordering

rearrangements, for example), a direct check is very hard to perform. However, they

were shown to be not BRST closed. Since the possible poles appearing in b (z) b (y) must

commute with the BRST charge, O3 vanishes, and this constitutes a rigorous proof of the b

ghost nilpotency in the non-minimal pure spinor formalism, confirming the interpretation

of the theory as a ĉ = 3 N = 2 topological string.

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Acknowledgments

I would like to thank Nathan Berkovits, Ido Adam and Ilya Bakhmatov for useful dis-

cussions. Also, for reading the manuscript, Thales Agricola and Chrysostomos Kalousios.

This work was supported by FAPESP grant 2009/17516-4.

A Conventions and useful formulas

Conventions. Indices:

{

m,n, . . . = 0, . . . , 9 space-time vector indices,

α, β, . . . = 1, . . . , 16 space-time spinor indices,

The indices antisymmetrization is represented by the square brackets, meaning

[I1 . . . In] =
1

n!
(I1 . . . In + all antisymmetric permutations) . (A.1)

For example,

γ[mγn] =
1

2
(γmγn − γnγm) = γmn, (A.2)

or,

λ[αHβγ] =
1

3!

(

λαHβγ − λαHγβ + λβHγα − λβHαγ + λγHαβ − λγHβα
)

. (A.3)

Concerning OPE’s, the right-hand sides of the equations are always evaluated at the

coordinate of the second entry, that is,

A (z)B (y) ∼
C

(z − y)2
+

D

(z − y)
(A.4)

means C = C (y) and D = D (y).

Gamma matrices. The gamma matrices γmαβ and γαβm satisfy

{γm, γn}αβ = (γm)ασ γnσβ + (γn)ασ γmσβ = 2ηmnδαβ . (A.5)

The Fierz decompositions of bispinors are given by

χαψβ =
1

16
γαβm (χγmψ) +

1

3!16
γαβmnp (χγ

mnpψ) +
1

5!16

(
1

2

)

γαβmnpqr (χγ
mnpqrψ) , (A.6a)

χαψ
β =

1

16
δβα (χψ)−

1

2!16
(γmn)

β
α (χγ

mnψ) +
1

4!16
(γmnpq)

β
α (χγ

mnpqψ) , (A.6b)

where

γαβm = γβαm , γαβmnp = −γβαmnp, γ
αβ
mnpqr = γβαmnpqr. (A.7)

The main gamma matrix identity that is being used in this work is

(γmn)αβ (γmn)
γ
λ = 4γmβλγ

αγ
m − 2δαβ δ

γ
λ − 8δαλδ

γ
β , (A.8)

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which can be deduced from (A.6). The other relevant one is given by

ηmn

(
γmαβγ

n
γλ + γmαγγ

n
βλ + γmαλγ

n
γβ

)
= 0. (A.9)

There are several other identities that can be derived from (A.8):

(γmn)αβ γ
γλ
mnp = 2 (γm)αγ (γpm)λ

β
+ 6γαγp δλβ − (γ ↔ λ) , (A.10)

(γmn)
α
β γ

mnp
γλ = −2 (γm)βλ (γ

pm)αγ + 6γpβλδ
α
γ − (γ ↔ λ) , (A.11)

γαβmnp (γ
mnp)γλ = 12

[

γαλm (γm)βγ − γαγm (γm)βλ
]

, (A.12)

γαβmnpγ
mnp
γλ = 48

(

δαγ δ
β
λ − δαλδ

β
γ

)

. (A.13)

All of them are very helpful in extracting the pure spinor constraints out of product of

bispinors containing space-time vector indices contracted. For example:

(
λγmnpr

) (
λγmnλ

)
= 2

(
λγmλ

)
(rγpmλ) + 6 (rλ)

(
λγpλ

)

−2
(
λγmr

) (
λγpmλ

)
− 6

(
λλ
) (
λγpr

)

= 0. (A.14)

The last identity that is often used in the calculations is

γmγn1...nkγm = (−1)k (10− 2k) γn1...nk , (A.15)

which is particularly useful since it implies that (γmλ)α (γmλ)β = 0 for λ being a

pure spinor.

Ordering considerations. This part intended to present some aspects of the ordering

prescription that is being used in this work.

Classical relations between currents are now corrected with ordering contributions.

For example,

Nmn
cl (γnλ)α =

1

2
Jcl (γ

mλ)α (A.16)

is valid for any pure spinor λ. Its quantum version is given by

(

Nmn, λβ
)

γpαβηnp −
1

2

(

Jλ, λ
β
)

γmαβ = 2 (γm∂λ)α , (A.17)

showing that some of the 45 Lorentz generators can be written in terms of the others (in

fact, only 10 are independent components).

Another important example is the equation

4λαTcl + Jcl∂λ
α +Nmn

cl (γmn∂λ)
α = 0, (A.18)

which establishes a connection between the energy-momentum tensor and the other cur-

rents. Implementing the ordering leads to

(λα, T ) + 4∂2λα = −
1

4
(Jλ, ∂λ

α)−
1

4
(Nmn, (γ

mn∂λ)α) . (A.19)

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This relation appears in the construction of the quantum b ghost, as well as

(
1

4

)

γβαmnp (N
mn, λγp∂θ) = 8∂λ[α∂θβ] +

(

λ[α, Nmn (γ
mn∂θ)β]

)

+
(

λ[α, Jλ∂θ
β]
)

, (A.20)

which is the ordered version of

γαβmnpN
mn
cl (λγp∂θ) + 4λ[αNmn

cl (γmn∂θ)
β] + 4λ[αJcl∂θ

β] = 0. (A.21)

A further application is the Sugawara construction of the energy-momentum tensor

for the minimal ghost sector,

Tλ = −
1

20
(Nmn, Nmn)−

1

8
(Jλ, Jλ) + ∂Jλ, (A.22)

which correctly reproduces the related OPE’s.

OPE computations are more systematic5 within the prescription (3.7). As an example,

it will be shown here that the b ghost for the non-minimal formalism is a primary field.

Concerning b−1, the ordering does not matter and it is straightforward to see that

T (z) b−1 (y) ∼ 2
b−1

(z − y)2
+

∂b−1

(z − y)
. (A.23)

For b0, however, there are some subtleties. Analyzing Gα first,

T (z)Gα (y) ∼ 2
Gα

(z − y)2
+

∂Gα

(z − y)
+

∂θα

(z − y)3
. (A.24)

Note that the cubic pole receives contributions from Jλ (the ghost current anomaly), ∂2θα

and
(

Πm, γαβm dβ

)

:

T (z) Jλ (y) ∼
8

(z − y)3
+

Jλ

(z − y)2
+

∂Jλ
(z − y)

, (A.25)

T (z) ∂2θα (y) ∼ 2
∂θα

(z − y)3
+ 2

∂2θα

(z − y)2
+

∂3θα

(z − y)
, (A.26)

T (z)
(

Πm, γαβm dβ

)

(y) ∼

(
Πm

(z − y)2
+

∂Πm

(z − y)
, γαβm dβ

)

+

(

Πm,
γαβm dβ

(z − y)2
+
γαβm ∂dβ
(z − y)

)

. (A.27)

According to the ordering prescription, the first term in the last OPE can be rewritten as

1

2πi

∮
dw

(w − y)

{
1

(z − w)2
Πm (w) +

1

(z − w)
∂Πm (w)

}

γαβm dβ (y) =

− 10
∂θα

(z − y)3
+

(

Πm, γαβm dβ

)

(z − y)2
+

(

∂Πm, γαβm dβ

)

(z − y)
. (A.28)

5See chapter 6 of [7], where the normal ordering is presented in details.

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Therefore,

T (z)
(

Πm, γαβm dβ

)

(y) ∼ −10
∂θα

(z − y)3
+ 2

(

Πm, γαβm dβ

)

(z − y)2
+
∂
(

Πm, γαβm dβ

)

(z − y)
. (A.29)

Adding up all the contributions, equation (A.24) is reproduced. For the whole b0,

T (z) b0 (y) ∼

(
1

(z − y)
∂

(
λα

λλ

)

, Gα

)

+

(
λα

λλ
, 2

Gα

(z − y)2
+

∂Gα

(z − y)
+

∂θα

(z − y)3

)

+ 2
O

(z − y)2
+

∂O

(z − y)
. (A.30)

Again, the first term on the right-hand side can be rewritten as

1

2πi

∮
dw

(w − y)

1

(z − w)
∂w

(
λα

λλ

)

(w)Gα (y)

=
1

(z − y)

(

∂

(
λα

λλ

)

, Gα

)

−
1

(z − y)3

(
λα∂θ

α

λλ

)

. (A.31)

Replacing this equation in (A.30), the cubic pole disappears, yielding a primary field.

For b1, b2 and b3, there are no contributions like the one in b0 (they are all propor-

tional to the pure spinor constraints), therefore the b ghost, given by equation (3.2), is a

primary field:

T (z) b (y) ∼ 2
b

(z − y)2
+

∂b

(z − y)
. (A.32)

References

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

[hep-th/0001035] [INSPIRE].

[2] N. Berkovits, Multiloop amplitudes and vanishing theorems using the pure spinor formalism

for the superstring, JHEP 09 (2004) 047 [hep-th/0406055] [INSPIRE].

[3] N. Berkovits, Pure spinor formalism as an N = 2 topological string, JHEP 10 (2005) 089

[hep-th/0509120] [INSPIRE].

[4] O. Chand́ıa, The b ghost of the pure spinor formalism is nilpotent,

Phys. Lett. B 695 (2011) 312 [arXiv:1008.1778] [INSPIRE].

[5] N. Berkovits and N. Nekrasov, Multiloop superstring amplitudes from non-minimal pure

spinor formalism, JHEP 12 (2006) 029 [hep-th/0609012] [INSPIRE].

[6] I. Oda and M. Tonin, Y-formalism and b ghost in the non-minimal pure spinor formalism of

superstrings, Nucl. Phys. B 779 (2007) 63 [arXiv:0704.1219] [INSPIRE].

[7] P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory, Springer, New York

U.S.A. (1997).

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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/10.1088/1126-6708/2004/09/047
http://arxiv.org/abs/hep-th/0406055
http://inspirehep.net/search?p=find+EPRINT+hep-th/0406055
http://dx.doi.org/10.1088/1126-6708/2005/10/089
http://arxiv.org/abs/hep-th/0509120
http://inspirehep.net/search?p=find+EPRINT+hep-th/0509120
http://dx.doi.org/10.1016/j.physletb.2010.10.058
http://arxiv.org/abs/1008.1778
http://inspirehep.net/search?p=find+EPRINT+arXiv:1008.1778
http://dx.doi.org/10.1088/1126-6708/2006/12/029
http://arxiv.org/abs/hep-th/0609012
http://inspirehep.net/search?p=find+EPRINT+hep-th/0609012
http://dx.doi.org/10.1016/j.nuclphysb.2007.04.032
http://arxiv.org/abs/0704.1219
http://inspirehep.net/search?p=find+EPRINT+arXiv:0704.1219

	Introduction
	Review of the pure spinor formalism
	Matter fields
	Ghost fields

	The b ghost
	Definition and properties
	Nilpotency

	Conclusion
	Conventions and useful formulas