Y-formalism and $b$ ghost in the Non-minimal Pure Spinor Formalism of
  Superstrings
arXiv:0704.1219v1  [hep-th]  10 Apr 2007
DPUR/TH/1
DFPD07/TH/06
April, 2007
Y-formalism and b ghost in the Non-minimal Pure Spinor
Formalism of Superstrings
Ichiro Oda 1
Department of Physics, Faculty of Science, University of the Ryukyus,
Nishihara, Okinawa 903-0213, Japan.
Mario Tonin 2
Dipartimento di Fisica, Universita degli Studi di Padova,
Instituto Nazionale di Fisica Nucleare, Sezione di Padova,
Via F. Marzolo 8, 35131 Padova, Italy
Abstract
We present the Y-formalism for the non-minimal pure spinor quantization of super-
strings. In the framework of this formalism we compute, at the quantum level, the explicit
form of the compound operators involved in the construction of the b ghost, their normal-
ordering contributions and the relevant relations among them. We use these results to
construct the quantum-mechanical b ghost in the non-minimal pure spinor formalism.
Moreover we show that this non-minimal b ghost is cohomologically equivalent to the
non-covariant b ghost.
1E-mail address: ioda@phys.u-ryukyu.ac.jp
2E-mail address: mario.tonin@pd.infn.it
http://arxiv.org/abs/0704.1219v1
1 Introduction
Several years ago, a new formalism for the covariant quantization of superstrings was proposed
by Berkovits [1]. Afterward, it has been recognized that this new formalism not only solves
the longstanding problem of covariant quantization of the Green-Schwarz (GS) superstring, but
also it is suitable to deal with problems that appear almost intractable in the Neveu-Schwarz-
Ramond (NSR) approach, such as those involving space-time fermions and/or backgrounds
with R-R fields.
In this approach, the GS superstring action (let us say in the left-moving sector) is replaced
with a free action for the bosonic coordinates Xa and their fermionic partners θα with their
conjugate momenta pα, plus an action for the bosonic ghosts λ
α and their conjugate momenta
ωα, where λ
α satisfy the ”pure spinor constraint” λΓaλ = 0. The ω − λ action looks like a free
action but is not really free owing to the pure spinor constraint, which is necessary to have
vanishing central charge and correct level of the Lorentz algebra. This formulation is nowadays
called ”pure spinor formulation of superstrings” and many studies [2]-[24] were devoted to it in
the recent years. 3
Another key ingredient in the pure spinor formulation is provided by the BRST charge
λαdα where dα ≈ 0 contains the constraints generating a fermionic κ symmetry in the
GS superstring and has the role of a spinorial derivative in superspace. The peculiar feature
associated with this BRST charge is that Q is nilpotent only when the bosonic spinor λα satisfies
the pure spinor condition. This peculiar feature is in fact expected since the constraint dα ≈ 0
in the GS approach involves both the first-class and the second-class constraints. Roughly
speaking, the pure spinor condition is needed to handle the second-class constraint of the GS
superstring, keeping the Lorentz covariance manifest.
Since the BRST charge Q is nilpotent, one can define the cohomology and examine its
physical content. Indeed, it has been shown that the BRST cohomology determines the physical
spectrum which is equivalent to that of the RNS formalism and that of the GS formalism in
the light-cone gauge [3]. Moreover, the BRST charge Q of the pure spinor formalism was found
to be transformed to that of the NSR superstring [4] as well as that of the GS superstring in
the light-cone gauge [18, 19].
Even if the pure spinor formalism provides a Lorentz-covariant superstring theory with
manifest space-time supersymmetry even at the quantum level, there are some hidden sources
of possible violation of Lorentz covariance.
One of such sources is related to the b field defined by T = {Q, b} with T being the stress-
energy tensor, which is necessary to compute higher loop amplitudes. Since the pure spinor
formulation is not derived from a diffeomorphism-invariant action and does not contain the
b− c ghosts of diffeomorphisms, the usual antighost b is not present in this approach. In [3] a
compound b field whose BRST variation gives the stress energy tensor, was obtained. However
this b field is not Lorentz-covariant.
The same b field follows from an attempt [10] to derive, at the classical level, the pure spinor
3Alternative formalisms to remove the constraint were proposed in [25, 26].
formulation from a (suitably gauge-fixed and twisted) N = 2 superembedding approach. In this
approach the b field is the twisted current of one of the two world-sheet (w.s.) supersymmetries
whereas the integrand of the BRST charge Q is the twisted current of the other supersymmetry,
suggesting an N = 2 topological origin of the pure spinor approach.
This b field turns out to be proportional to the quantity Yα =
where vα is a constant
pure spinor, such that bY = YαG
α where Gα is a covariant, spinor-like compound field, so that
bY is not only Lorentz non-covariant but also singular at vλ = 0.
A way to overcome the problem of the non-covariance and singular nature of bY was given
in [14] where a recipe to compute higher loop amplitudes was proposed, in terms of a picture-
raised b field constructed with the help of suitable covariant fields Gα, Hαβ, Kαβγ and Lαβγδ
and some picture-changing operators Z ′s and Y ′s. 4
Recently, a very interesting formalism called ”non-minimal pure spinor formalism” has been
put forward [27]. In this formalism, a non-minimal set of variables are added to that of the
(minimal) pure spinor formulation. These non-minimal variables form a BRST quartet and
have the role of changing the ghost-number anomaly from −8 to +3 without changing the
central charge and the physical mass spectrum. A remarkable thing is that, in this formalism,
one can define a Lorentz-covariant b ghost without the need of picture-changing operators.
With the help of a suitable regulator, a recipe has been given to compute scattering amplitudes
up to two-loop amplitudes. The OPE’s between the relevant operators that result in this
approach show that the (non-minimal) pure spinor formulation is indeed a hidden, critical,
N = 2 topological string theory. A significant improvement was obtained in [28]. Here a gauge
invariant, BRST trivial regularization of the b field is proposed, that allows for a consistent
prescription to compute amplitudes at any loop.
A further source of possible non-covariance arises at intermediate steps of calculations, since
the solution of the pure spinor constraint in terms of independent fields implies the breaking of
SO(10) to U(5). 5 To be more precise, the space of (Euclidean) pure spinors in ten dimensions
has the geometrical structure of a complex cone Q =
SO(10)
[21]. This space has been studied by
Nekrasov [29] and the obstructions to its global definition are analyzed. It was shown that the
obstructions are absent if the tip of the cone is removed. Then this complex cone is covered by
16 charts, U (α), (α) = 1, · · · , 16 and in each chart the local parametrization of the pure spinor,
which breaks SO(10) to U(5), is taken such that the parameter that describes the generatrix
of the cone is non-vanishing. This parametrization can be used to compute the relevant OPE’s
[1, 3] (U(5)-formalism).
In a previous work [30], we have proposed a new formalism named ”Y-formalism” for pur-
poses of handling this unavoidable non-covariance stemming from the pure spinor condition.
This Y-formalism is closely related to the U(5)-formalism, but has an advantage of treating all
operators in a unified way without going back to the U(5)-decomposition. It is based on writing
4The picture-lowering operators YC , which are needed to absorb the zero modes of the ghost λ
α, break the
Lorentz-covariance but this breaking is BRST trivial and then harmless.
5In the extended pure spinor formalism [26], the same non-covariance can be found in the ghost sector where
the ghosts are invariant under only U(5) group, but not S(10) group.
the fundamental OPE between ω and λ in a form that involves Yα =
. Strictly speaking, one
needs 16, orthogonal, constant pure spinors v(α) (and 16 Y (α)) for each chart, such that U (α)
(v(α)λ) 6= 0 in each chart. However, for our puposes it is sufficient to work in a given chart.
Actually, it turned out that the Y-formalism is quite useful to find the full expression of b
ghost [30]. More recently, the Y-formalism was also utilized to construct a four-dimensional
pure spinor superstring [31]. The Y -field also arises in the regularization prescription proposed
in [28].
The aim of the present paper is to extend the Y-formalism to the non-minimal case and
to discuss in the framework of this formalism the non-minimal, covariant b field in addition
to the fields Gα, Hαβ, Kαβγ and Lαβγδ, which are the building blocks of the b field. This will
be done not only at the classical but also at the quantum level, by taking into account the
subtleties of normal ordering. The consistent results which we will get in this article, could be
regarded as a good check of the consistency of the Y-formalism. Moreover we shall show that
the non-minimal, covariant b field is cohomologically equivalent to the non-covariant b field bY ,
improved by the term coming from the non-minimal sector.
In section 2, we will review the Y-formalism for the minimal pure spinor formalism. In
section 3, the operators Gα, Hαβ, Kαβγ and Lαβγδ, and their (anti-)commutation relations with
the BRST charge, will be examined from the quantum-mechanical viewpoint. In section 4,
we will construct the Y-formalism for the non-minimal pure spinor formalism. In section 5,
based on the Y-formalism at hand, we will construct the Lorentz-covariant quantum b ghost,
which satisfies the defining equation {Q, b} = T . We shall also show that it is cohomologically
equivalent to the non-covariant b ghost bY (improved by the term coming from the non-minimal
sector). Section 6 is devoted to conclusion and discussions. Some appendices are added.
Appendix A contains our notation, conventions and useful identities. In Appendix B, we will
review the normal-ordering prescriptions, the generalized Wick theorem and the rearrangement
theorem which we will use many times in this article. Finally in Appendix C we give some
details of the main calculations.
2 Review of the Y-formalism
In this section, we start with a brief review of the (minimal) pure spinor formalism of super-
strings [1], and then explain the Y-formalism [30]. For simplicity, we shall confine ourselves to
only the left-moving (holomorphic) sector of a closed superstring theory. The generalization to
the right-moving (anti-holomorphic) sector is straightforward.
The pure spinor approach is based on the BRST charge
dzλαdα, (2.1)
and the action
∂Xa∂̄Xa + pα∂̄θ
α − ωα∂̄λ
α), (2.2)
where λ is a pure spinor
λΓaλ = 0. (2.3)
This action is manifestly invariant under (global) super-Poincaré transformations. It is easily
shown that the action I is also invariant under the BRST transformation generated by the
BRST charge Q which is nilpotent owing to the pure spinor condition (2.3). Notice that in
order to use Q as BRST charge it is implicit that the pure spinor condition is required to vanish
in a strong sense.
Moreover, the action I is invariant under the ω-symmetry
δωα = Λa(Γ
aλ)α, (2.4)
where Λa are local gauge parameters. At the classical level the ghost current is
J0 = ωλ, (2.5)
and the Lorentz current for the ghost sector is given by
Nab0 =
ωΓabλ, (2.6)
which together with T0λ = ω∂λ are the only super-Poincaré covariant bilinear fields involving ω
and gauge invariant under the ω-symmetry. From the field equations it follows that p, θ, ω and
λ are holomorphic fields. At the quantum level, one obtains the following OPE’s 6 involving
the superspace coordinates ZM = (Xa, θα) and their super-Poincaré covariant momenta PM =
(Πa, pα):
< Xa(y)Xb(z) > = −ηab log(y − z),
< pα(y)θ
β(z) > =
y − z
δβα, (2.7)
so that
< dα(y)dβ(z) > = −
y − z
ΓaαβΠa(z),
< dα(y)Π
a(z) > =
y − z
(Γa∂θ)α(z), (2.8)
where
dα = pα −
(∂Xa +
θΓa∂θ)(Γaθ)α,
Πa = ∂Xa +
θΓa∂θ,
Π̄a = ∂̄Xa +
θΓa∂̄θ. (2.9)
6According to Appendix B, we should call them not the OPE’s but the contractions, but we have called
”OPE’s” since the terminology is usually used in the references of the pure spinor formulation.
As for the ghost sector, the situation is a bit more complicated owing to the pure spinor
condition (2.3). Namely, it would be inconsistent to assume a free field OPE between ω and
λ. The reason is that since the pure spinor condition must vanish identically, not all the
components of λ are independent: solving the condition, five of them are expressed nonlinearly
in terms of the others. Accordingly, five components of ω are pure gauge.
This problem is nicely resolved by introducing the Y-formalism. Let us first define the
non-covariant object
, (2.10)
such that
α = 1, (2.11)
where vα is a constant pure spinor Y Γ
aY = 0. Then it is useful to define the projector
(Γaλ)α(Y Γa)
β, (2.12)
which, since TrK = 5, projects on a 5 dimensional subspace of the 16 dimensional spinor space
in ten dimensions. The orthogonal projector is (1 − K)α
β. Now the pure spinor condition
implies
β = 0. (2.13)
Since K projects on a 5 dimensional subspace, Eq. (2.13) is a simple way to understand why
a pure spinor has eleven independent components.
Then we postulate the following OPE between ω and λ:
< ωα(y)λ
β(z) >=
y − z
(δβα −Kα
β(z)). (2.14)
It follows from Eq. (2.14) that the OPE between ω and the pure spinor condition vanishes
identically. Moreover, the BRST charge Q is then strictly nilpotent even acting on ω. It is
useful to notice that, with the help of the projector K, one can obtain a non-covariant but
gauge-invariant antighost ω̃ defined as
ω̃α = (1−K)α
ωβ. (2.15)
In the framework of this formalism one can compute [30] the OPE’s among the ghost
current, Lorentz current and stress energy tensor and one can obtain the quantum version of
these operators. Indeed, it has been shown in [30] that all the non-covariant, Y-dependent
contributions in the r.h.s. of the OPE’s among these operators disappear if the stress energy
tensor, the Lorentz current for the ghost sector, and the ghost current at the quantum level,
are improved by Y -dependent correction terms, those are
T = −
∂Xa∂Xa − pα∂θ
α + Tλ
ΠaΠa − dα∂θ
α + ωα∂λ
∂(Y ∂λ), (2.16)
Nab =
ωΓabλ−
∂Y Γabλ− 2Y Γab∂λ
, (2.17)
J = ωλ+
Y ∂λ. (2.18)
Then the OPE’s among T , Nab and J read
< T (y)T (z) >=
(y − z)2
T (z) +
y − z
∂T (z), (2.19)
< T (y)J(z) >=
(y − z)3
(y − z)2
J(z) +
y − z
∂J(z), (2.20)
< T (y)Nab(z) >=
(y − z)2
Nab(z) +
y − z
∂Nab(z), (2.21)
< J(y)J(z) >= −
(y − z)2
, (2.22)
< J(y)Nab(z) >= 0, (2.23)
< Nab(y)N cd(z) >= −
(y − z)2
ηd[aηb]c −
y − z
(ηa[cNd]b − ηb[cNd]a), (2.24)
which are in full agreement with [1, 3]. Note that although the correction terms in the currents
depend on the non-covariant Y-field explicitly, these can be rewritten as BRST-exact terms.
Now a remark is in order. It appears at first sight that, due to the correction terms, the
operators J , Nab and T are singular at vλ = 0 but the opposite is in fact true: it is clear from
Eqs. (2.19)-(2.24) that the Y -dependent correction terms have just the rôle of cancelling the
singularites which are present in the operators T0, N
0 and J0, owing to the singular nature of
the OPE (2.14) between ω and λ. 7
It will be convenient to rewrite (2.17), (2.18) and Tλ as
Nab =
[ΩΓabλ− 2Y Γab∂λ], (2.25)
J = Ωλ + 2Y ∂λ, (2.26)
7As anticipated in the notation , we will append a suffix ”0” when we refer to compound fields at the classical
level, that is, given in terms of T0, N
0 and J0, and we will reserve the notation without suffix ”0” in denoting
the corresponding quantities at the quantum level, given in terms of T , Nab and J .
Tλ = Ω∂λ + 3∂Y ∂λ +
Y ∂2λ, (2.27)
where we have introduced the quantity
Ωα = ωα −
∂Yα. (2.28)
The Y-formalism explained thus far is also useful to deal with the b field which plays an
important role in computing higher loop amplitudes. Its main property is
{Q, b(z)} = T (z), (2.29)
where T is the stress energy tensor. Since in the pure spinor formulation the reparametrization
ghosts do not exist, b must be a composite field. Moreover, since the b ghost has ghost number
−1 and the covariant fields, which include ωα and are gauge invariant under the ω-symmetry,
always have ghost number zero or positive, one must use Yα (which also has ghost number −1)
to construct the b ghost. Therefore b is not super-Poincaré invariant. The b ghost has been
constructed for the first time in [3] in the U(5)-formalism in such a way that it satisfies Eq.
(2.29). In the Y-formalism at hand, at the classical level it takes the form
b0Y =
ΠaY Γad+ ω(1−K)∂θ = YαG
0 , (2.30)
where
d)α −
0 (Γab∂θ)
. (2.31)
The last equality in (2.30) follows from the identity (A.3). The expression of bY at the quantum
level will be derived in section 5.
The non-covariance of bY is not dangerous since, as we shall show in section 5, the Lorentz
variation of bY (or of its improvement at the non-minimal level) is BRST-exact. However, this
operator cannot be accepted as insertion to compute higher loop amplitudes. Indeed, contrary
to the operators T , Nab and J , it has a true singularity at vλ = 0 of the form (vλ)−1. The point
is that there exists an operator ξ = Y θ, singular with a pole at vλ → 0, such that {Q, ξ} = 1
and the cohomology would become trivial if this operator is allowed in the Hilbert space, since
for any closed operator V , V = {Q, ξV }. Then, for consistency, operators singular at vλ → 0
must be excluded from the Hilbert space.
3 Fundamental operators and normal-ordering effects
When we attempt to construct a b ghost covariantly, either a picture-raised b ghost [14, 30] or
a covariant b ghost in the framework of the non-minimal approach [27], we encounter several
fundamental operators, Gα, Hαβ, Kαβγ and Lαβγδ [14, 30], which are a generalization of the
constraints introduced by Siegel some time ago in [32]. Thus, in this section, we will consider
those operators in order. We will pay a special attention to a consistent treatment of the
normal-ordering effects.
Let us notice that in addition to Gα, the totally antisymmetrized operatorsH [αβ], K [αβγ] and
L[αβγδ] are the more fundamental objects and are of particular interest since they are involved
in the construction of the b field in the non-minimal formulation. At the classical level, Gα is
defined in (2.31) and H [αβ], K [αβγ] and L[αβγδ] are given by
abc(dΓ
abcd+ 24Nab0 Π
[αβγ]
0 = −
abc (Γ
ad)γ]N bc0 ,
[αβγδ]
0 = −
(Γabc)
[αβ(Γade)γδ]N bc0 N0de. (3.1)
They satisfy the following recursive relations:
{Q,Gα0} = λ
0 ] = λ
[αβγ]
0 } = λ
[αβγδ]
0 ] = λ
βγδρ]
0 = 0, (3.2)
which one can verify easily. The full fields H
0 , K
0 and L
0 , which are involved in the
construction of the picture-raised b ghost, can be obtained by adding new terms symmetric
with respect to at least a couple of adjacent indices, and they satisfy the recursive relations
0 ] = λ
0 + · · · ,
0 } = λ
0 + · · · ,
0 ] = λ
0 + · · · ,
0 = 0 + · · · , (3.3)
where the dots denote ”Γ1-traceless terms”, i.e. terms that vanish if saturated with a Γ
αiαi+1
between two adjacent indices. The fields H
0 , K
0 and L
0 are defined modulo Γ1-traceless
terms.
In this section we wish to discuss these operators and their recursive relations at the quantum
level. A remark is in order. At the quantum level, in dealing with holomorphic operators
composed of fields with singular OPE’s, a normal-ordering prescription is needed for their
definition. As a rule, for the normal ordering of two operators A and B we shall adopt in
this paper the generalized normal-ordering prescription, denoted by (AB) in [33] since it is
convenient in carrying out explicit calculations. As explained in Appendix B, this prescription
consists in subtracting the singular poles, evaluated at the point of the second entry and it is
given by the contour integration
(AB)(z) =
w − z
A(w)B(z). (3.4)
Often, for simplicity, in dealing with this prescription the outermost parenthesis is suppressed
and the normal ordering is taken from the right so that, in general, A1A2A3...An means
(A1(A2(A3(· · ·An) · · ·))).
A different prescription denoted as : AB :, that we shall call ”improved”, consists in sub-
tracting the full contraction < A(y)B(z) >, included a possible finite term, as computed from
the canonical OPE’s (2.7) and (2.14). In many cases the two prescriptions coincide but when
they are different, it happens, as we shall see, that the final results look more natural if expressed
in the improved prescription.
3.1 Gα
Gα is obtained from (2.31) by replacing Nab0 and J0 with N
ab and J as defined in Eqs. (2.17)
and (2.18) and adding a normal-ordering term parametrized by a constant c1
Πa(Γad)
Nab(Γ
ab∂θ)α −
J∂θα + c1∂
≡ Gα1 +G
4 . (3.5)
The constant c1 will be determined from the requirement that G
α should be a primary field
of conformal weight 2. Then we have to compute the OPE < T (y)Gα(z) >. The three terms
Gα1 ≡
Πa(Γad)
α, Gα2 ≡ −
Nab(Γ
ab∂θ)α and Gα3 ≡ −
J∂θα are all products of two operators of
conformal weight 1 so that their OPE’s with the stress energy tensor can be easily calculated.
One finds that only Gα2 is a primary field. G
1 has a triple pole with residuum −5∂θ
α and Gα3
has a triple pole with residuum −2∂θα. Moreover, the normal-ordering term Gα4 ≡ c1∂
2θα also
has a triple pole with residuum 2c1∂θ
α. Therefore, putting them together, one has
< T (y)Gα(z) >=
−5− 2 + 2c1
(y − z)3
∂θα(z) +
(y − z)2
Gα(z) +
y − z
∂Gα(z). (3.6)
Hence, the requirement that Gα must be a primary field of conformal weight 2 is satisfied when
we select the constant c1 to be
In spite of the appearance, it turns out that this figure is in agreement with the result of [14]
where the value −1
is indicated as the coefficient in front of the normal-ordering term ∂2θα in
Gα. The difference is an artifact of the different normal-ordering prescriptions, the generalized
normal-ordering prescription in (3.5) and the improved one. Whereas the two prescriptions
coincide for Gα2 and G
3 , there appears a difference in G
1 . Indeed, since
Πa(x)dα(z) =
(x− z)2
[(Γaθ)α(z)− (Γ
aθ)α(x)]
(Γa∂θ)α(x)+ : Π
a(z)dα(z) : + · · · , (3.7)
we obtain
(Πa(Γad)
α) = −
: Πadα : . (3.8)
Substituting this result into Eq. (3.5), setting c1 =
, we have
Gα =:
Πa(Γad)
α : −
Nab(Γ
ab∂θ)α −
J∂θα −
∂2θα, (3.9)
which precisely coincides with the expression given in [14].
Next, we want to derive the quantum counterpart of the first (classical) recursive relations in
(3.2) and, for that, we need to compute {Q,Gα}. In doing this calculation, one must be careful
to deal with the order of the factors in the terms coming from the (anti)commutator among
Q and Gα and use repeatedly the rearrangement theorem, reviewed in Appendix B, in order
to recover the operator λαT . The details of this calculation are presented in Appendix C. As
expected from the covariance of {Q,Gα}, the terms involving Y , coming from the rearrangement
procedure, cancel exactly those coming from the Y -dependent correction terms of the operators
Nab and J (see (2.17) and (2.18)) present in the definition of Gα. The final result is
{Q,Gα} = λαT −
∂2λα. (3.10)
The normal-ordering term −1
∂2λα in (3.10) might appear to be strange at first sight, but it is
indeed quite reasonable. The point is that it is not λαT but λαT − 1
∂2λα that is a primary
field of conformal weight 2 when we take account of the normal-ordering effects. In fact, since
α(y)T (z) >≡
Rα1 (z)
y − z
∂λα(z)
y − z
, (3.11)
< T (y)(λαT )(z) > has a triple pole with residuum +∂2λ, and 1
∂2λ has the same triple pole, it
follows that
Bα1 = λ
∂2λα, (3.12)
is a BRST-closed primary operator of conformal weight 2. From now on, it is convenient to
define
α = Gα +
, (3.13)
so that (3.10) becomes
{Q, Ĝα} = λαT. (3.14)
Now we would like to study the operator λαGβ, that is expected to arise in the quantum
counterpart of the second recursive relations in (3.2). As before, λαGβ is not primary since
< λα(y)Gβ(z) > is different from zero. Indeed,
< λα(y)Gβ(z) >≡
2 (z)
y − z
, (3.15)
where
2 = −∂θ
αλβ +
Γαβa (∂θΓ
aλ). (3.16)
Note that since ∂λα∂θβ is also primary, there is an ambiguity in defining a primary operator,
say B
2 , associated to λ
αGβ. Given (3.16), for the symmetric one, one has
Γαβa [(λΓ
aG) +
∂(λΓa∂θ) + c+(∂λΓ
a∂θ)], (3.17)
while, for the antisymmetric one, one has
2 = λ
[αGβ] +
∂(λ[α∂θβ]) + c−∂λ
[α∂θβ]. (3.18)
Let us remark that λαGβ is not BRST-closed. Indeed {Q, λαGβ} = λαλβT− 1
λα∂2λβ. Whereas
(λ[α(λβ]T )) = (T (λ[αλβ])) = 0, one has ((λαΓaαβ(λ
βT )) = (T (λΓaλ)) − 2(λΓa∂2λ). Therefore
the requirement that B
2 and B
2 are BRST-closed implies c+ = −
and c− = −
so that
Γαβa [λΓ
λΓa∂2θ + ∂(λΓa∂θ)], (3.19)
2 = λ
[αGβ] +
λ[α∂2θβ] = λ[αĜβ]. (3.20)
3.2 Hαβ
A minimal choice for Hαβ is
Hαβ = H(αβ) +H [αβ], (3.21)
where
(αβ) =
Γαβa (N
abΠb −
JΠa + c2∂Π
a), (3.22)
H [αβ] =
dΓabcd+ 6NabΠc). (3.23)
First, we shall evaluate < T (y)Hαβ(z) > in order to fix the normal-ordering term. We can
easily show that H [αβ] and the first term in H(αβ) are primary fields whereas − 1
Γαβa JΠ
a and
Γαβa ∂Π
a have a triple pole with residua −4 1
Γαβa Π
a and 2c2
Γαβa Π
a, respectively. Thus, we
obtain
< T (y)Hαβ(z) >=
−4 + 2c2
(y − z)3
Γαβa Π
a(z) +
(y − z)2
Hαβ(z) +
y − z
∂Hαβ(z), (3.24)
thereby taking c2 = 2 makes H
αβ a primary field of conformal weight 2. This value agrees with
the value in the Berkovits’ paper [14]. Next, we wish to evaluate [Q,Hαβ]:
[Q,H(αβ)] =
(λΓabd)Πb +N
ab(λΓb∂θ) +
(λd)Πa
J(λΓa∂θ) + c2∂(λΓ
, (3.25)
[Q,H [αβ]] =
((Γdλ)ρΠd)(Γ
abcd)ρ −
(Γabcd)ρ(Γdλ)ρΠd
+ 3(λΓabd)Πc + 6Nab(λΓc∂θ) + 2c3(∂λΓ
abc∂θ)
. (3.26)
Then, after some algebra and taking into account the normal-ordering terms by the rearrange-
ment formula, we get for the symmetric part of Hαβ
[Q,H(αβ)] =
Γαβa [λΓ
λΓa∂2θ + ∂(λΓa∂θ)], (3.27)
and for the more interesting antisymmetric part H [αβ]
[Q,H [αβ]] = λ[αGβ] +
λ[α∂2θβ] = λ[αĜβ], (3.28)
in agreement with (3.19) and (3.20). Notice that the Y -dependent contributions coming from
rearrangement theorem cancel exactly those coming from the definitions (2.17) and (2.18) of
Nab and J (For details see Appendix C).
Since the term +∂(λΓa∂θ) in (3.27) is the BRST variation of ∂Πa, (3.27) can be rewritten
[Q, Ĥ(αβ)] =
Γαβa [λΓ
λΓa∂2θ], (3.29)
where we have defined as Ĥ(αβ) = H(αβ) − 1
Γαβa ∂Π
Now let us consider the composite operator λαHβγ. Since Hαβ has conformal weight 2 but
its contraction with λα does not vanish, one can expect that λαHβγ is not primary. Actually,
using the fact
< λα(y)Hβγ(z) >≡
3 (z)
y − z
, (3.30)
with R
3 being given by
3 = −
Γβγa [(Γ
abλ)αΠb − λ
αΠa]−
abc(Γ
abλ)αΠc, (3.31)
it turns out that a primary field of conformal weight 2 related to λαHβγ is
3 ≡ λ
3 . (3.32)
Again there is an arbitrariness in choosing the primary field related to λαHβγ since ∂λαΠa is
primary.
As in previous cases we are especially interested in the antisymmetric part B
[αβγ]
3 of B
Since, in D = 10, a field which is totally antisymmetric in its three, spinor-like indices contains
only the SO(10) irreducible representation (irrep.) 560 and R
3 in Eq. (3.31) does not contain
such an irrep., it follows that
[αβγ]
3 = 0, (3.33)
so that B
[αβγ]
3 simply becomes
[αβγ]
3 = λ
[αHβγ]. (3.34)
From Eqs. (3.28), (3.15) and (3.16), it is then easy to show that λ[αHβγ] is BRST-closed.
Indeed, one finds
[Q, λ[αHβγ]] = λ[α(λβĜγ])
= Ĝ[γλαλβ] + λ[α∂(λβ∂θγ]) + ∂(λ[α∂θγ)λβ]
= 0. (3.35)
3.3 Kαβγ
A covariant expression of Kαβγ is
Kαβγ = −
Γαβa (Γbd)
γNab −
abc(Γ
ad)γN bc
(Γbd)
αNab +
(Γad)αJ + c3(Γ
a∂d)α
abc(Γ
ad)αN bc
6 , (3.36)
whereas the totally antisymmetric part is given by
[αβγ] = −
abc (Γ
d)γ]N bc. (3.37)
The term including a constant c3 describes the normal-ordering contribution. As before, we
will calculate < T (y)Kαβγ(z) > in order to fix the normal-ordering term. One finds that all
the terms K
i are primary with conformal weight 2, except K
(Γad)αJ and
5 ≡ c3
Γβγa (Γ
a∂d)α which have triple poles in their OPE’s with T . In fact,
< T (y)K
4 (z) > =
(y − z)3
(Γad)α(z) +
(y − z)2
4 (z) +
y − z
4 (z),
< T (y)K
5 (z) > =
(y − z)3
(Γad)α(z) +
(y − z)2
5 (z) +
y − z
5 (z).
(3.38)
Therefore, one obtains
< T (y)Kαβγ(z) >=
12 + 2c3
(y − z)3
(Γad)α(z) +
(y − z)2
Kαβγ(z) +
y − z
∂Kαβγ(z), (3.39)
so that the condition of a primary operator of conformal weight 2 requires us to take c3 = −6,
which is a new result.
As for {Q,Kαβγ}, we will limit ourselves to considering only the antisymmetric part K [αβγ]
of Kαβγ
{Q,K [αβγ]} =
((ΓaΓdλ)γ]Πd)N
(Γad)γ](λΓbcd)
. (3.40)
As before, the Y -dependent contributions coming from rearrangement theorem are exactly
cancelled by those coming from the definition (2.17) of Nab, as expected from the covariance of
the l.h.s of (3.40). Then from the rearrangement theorem and with a few algebra one gets
{Q,K [αβγ]} = λ[αHβγ]. (3.41)
Given that the Y -dependent terms are absent, (3.41) can also been argued as follows: coho-
mology arguments based on Eq. (3.35) and the classical equivalence between {Q,K [αβγ]} and
λ[αHβγ] imply {Q,K [αβγ]} = λ[αHβγ]+Λ
[αβγ]
3 , where Λ
[αβγ]
3 is a primary field of conformal weight
2 satisfying [Q,Λ
[αβγ]
3 ] = 0. Then, notice that Λ
[αβγ]
3 has ghost number +1 and involves ∂λ
and Πa or ∂Πa and λα. However, using these fields, it is impossible to construct a 560 irrep. of
SO(10), so Λ
[αβγ]
3 vanishes identically.
As before, let us construct a primary field of conformal weight 2 from λαKβγδ. We define
< λα(y)Kβγδ(z) >≡
4 (z)
y − z
, (3.42)
where R
4 takes the form
(Γabλ)αΓβγa (Γbd)
(Γabλ)αΓ
abc(Γ
[(Γabλ)α(Γbd)
β + 3λα(Γad)β]Γγδa +
(Γabλ)α(Γcd)βΓ
abc. (3.43)
Provided that we define B
4 ≡ λ
βγδ +
4 , (3.44)
it is easy to get
< T (y)B
4 (z) >=
(y − z)2
4 (z) +
y − z
4 (z), (3.45)
which means that B
4 is a primary field of conformal weight 2 as expected. As before, there
is an arbitrariness in the choice of the primary field related to λαKβγδ since the field ∂λαdβ is
primary.
If one considers the completely antisymmetric component B
[αβγδ]
4 , one can notice that, in
D = 10, a field antisymmetric in its four, spinor-like indices contains only the irreps. 770 and
1050 which are absent in the expression (3.43) of R
[αβγδ]
4 so that one obtains
[αβγδ]
4 = 0. (3.46)
Consequently, we have
[αβγδ]
4 = λ
[αKβγδ]. (3.47)
Furthermore, Eq. (3.41) together with (3.33) gives us the equation
{Q, λ[αKβγδ]} = 0. (3.48)
3.4 Lαβγδ
In this final subsection, we wish to consider Lαβγδ. In our previous paper [30], at the classical
level, the form of Lαβγδ was fixed to be
0 = −
λα(ω̃Γa)β[λγ(ω̃Γa)
(ΓbΓaλ)
γ(ω̃Γb)δ], (3.49)
where ω̃ is defined in (2.15). One subtle point associated with this expression is that L
cannot be entirely expressed in terms ofNab0 and J0. However, we have found that the dangerous
terms involving ω̃Γa1a2a3a4λ cancel exactly in constructing the picture raised b ghost.
On the other hand, when we consider the totally antisymmetrized part of L
0 , these
dangerous terms never appear. In order to show that, let us notice that, given Eq. (3.49), one
can write:
0 + L
0 = −
(ω̃Λαβc λ)(ω̃Λ
γδcλ)−
λα(ω̃Γa)βλγ(ω̃Γa)
δ, (3.50)
where we have defined
ω̃Λαβc λ = (ω̃Γc)
αλβ −
(ω̃Γb)
α(ΓbΓcλ)
β. (3.51)
Then, taking the totally antisymmetrized part of Eq. (3.50) one gets
[αβγδ]
0 = −
(ω̃Λ[αβc λ)(ω̃Λ
γδ]cλ). (3.52)
Using (3.51) and (A.3), we can rewrite ω̃Λ[αβ]c λ as
ω̃Λ[αβ]c λ =
abc(ωΓ
0 . (3.53)
Hence, we have shown that L
[αβγδ]
0 is in fact expressed by N
In order to have a covariant expression for L[αβγδ], at the quantum level, the classical Lorentz
generator N bc0 must be replaced with N
bc as given in (2.17) so that L[αβγδ] is
[αβγδ] = −
(Γabc)
[αβ(Γade)γδ]N bcNde. (3.54)
From the OPE’s < T (y)Nab(z) > and < Nab(y)N cd(z) >, one can easily verify that L[αβγδ] is a
covariant, primary field of conformal weight 2.
At the classical level one has the identities
[αβγδ]
0 ] = λ
βγδρ]
0 = 0, (3.55)
where the last identity follows by noting that L
[αβγδ]
0 is proportional to λ
[α(ωΓa)
β(ωΓb)
γ(Γabλ)δ].
Since L[αβγδ] and λ[ǫLαβγδ] are covariant tensors and a possible quantum failure of these identities
would involve Yα, thereby inducing violation of Lorentz covariance, one should expect that these
identities hold at the quantum level as well. It is worthwhile to verify this result directly as a
nice check of the consistency of the Y-formalism.
The quantum counterpart of the former equation in Eq. (3.55) reads
[Q,L[αβγδ]] = λ[αKβγδ]. (3.56)
In this case there are no contributions from the rearrangement theorem and using (3.37) and
(3.54) one finds that both sides of Eq. (3.56) are equal to 1
(Γabc)
[αβ(Γade)γδ](dΓbcλ)Nde, thus
showing that (3.56) is true. It is a little more cumbersome to verify the quantum analog of the
latter equation in Eq. (3.55), which is given by
λ[ǫLαβγδ] = 0. (3.57)
To do that it is convenient to introduce the following notation that extends that in Eq. (3.51):
if Ψα and Φ
β are two spinors that (by the conventions which we adopt) belong to the 1̄6 and
the 16 of SO(10), respectively, we define
[αβ]Φ = (ΨΓc)
[αΦβ] −
(ΨΓb)
[α(ΓbΓcΦ)
β]. (3.58)
Then, from Eqs. (2.17) and (3.54), L[αβγδ] can be rewritten as
[αβγδ] = −
, (3.59)
where
ab = ΩΛ[αβ]c λ− 2Y Λ
c ∂λ, (3.60)
and Ω is defined in (2.28).
Using Eqs. (3.59) and (3.60), the l.h.s. of Eq. (3.57) splits in three parts:
αβγδ] = −
[λ[ǫL
αβγδ]
1 + λ
αβγδ]
2 + λ
αβγδ]
3 ], (3.61)
where we have defined
αβγδ]
1 = λ
[ǫ(ΩΛαβc λ)(ΩΛ
λ), (3.62)
αβγδ]
2 = −2
λ[ǫ(ΩΛαβc λ)(Y Λ
γδ]c∂λ) + λ[ǫ(Y Λαβc ∂λ)(ΩΛ
γδ]cλ)
, (3.63)
αβγδ]
3 = 4λ
[ǫ(Y Λαβc ∂λ)(Y Λ
γδ]c∂λ). (3.64)
To compute the l.h.s. of Eq. (3.57), one must shift the fields Ω to the left using the rearrange-
ment formula. Then
αβγδ]
1 = Ωσ(Ωτ (λ
[ǫ(Λαβc λ)
σ(Λγδ]cλ)τ )) + ΩσA
[ǫαβγδ]σ
1 + A
[ǫαβγδ]
0 , (3.65)
αβγδ]
2 = ΩσB
[ǫαβγδ]σ
[ǫαβγδ]
0 , (3.66)
where A1, A0, B1 and B0 are Ω-independent, Y -dependent fields.
The term quadratic in Ω in the r.h.s. of Eq. (3.65) vanishes since it contains the factor
λ[ǫλβ(Γbcλ)
δ]. An explicit calculation shows that the terms linear in Ω in (3.65) and (3.66)
cancel each other:
[ǫαβγδ]σ
1 + ΩσB
[ǫαβγδ]σ
1 = 0, (3.67)
and that the sum of the terms of zero-order in Ω in (3.65), (3.66) and (3.64) vanishes
[ǫαβγδ]
[ǫαβγδ]
0 + λ
αβγδ]
3 = 0, (3.68)
so that (3.57) is proved.
The details of this calculation are given in Appendix C.
4 Y-formalism for the non-minimal pure spinor formal-
In this section, we would like to construct the Y-formalism for the non −minimal pure spinor
formalism which has been recently proposed by Berkovits [27]. Before doing that, we will first
review the non-minimal pure spinor formalism briefly. The main idea is to add to the fields
involved in the minimal formalism a BRST quartet of fields λ̄α, ω̄
α, rα and s
α in such a way
that their BRST variations are δλ̄α = rα, δrα = 0, δs
α = ω̄α and δω̄α = 0. Here, λ̄α is a
bosonic field, rα is a fermionic field, and ω̄
α and sα are the conjugate momenta of λ̄α and rα,
respectively. These fields are required to satisfy the pure spinor conditions
λ̄Γaλ̄ = 0,
λ̄Γar = 0. (4.1)
The action is then obtained by adding to the conventional pure spinor action I in Eq. (2.2), Ī
given by the BRST variation of the ”gauge fermion” F = −
(s∂̄λ̄) so that
Inm ≡ I + Ī =
∂Xa∂̄Xa + pα∂̄θ
α − ωα∂̄λ
α + sα∂̄rα − ω̄
α∂̄λ̄α). (4.2)
In addition to the ω-symmetry Eq. (2.4), due to the conditions Eq. (4.1), this action is invariant
under new gauge symmetries involving ω̄ and s,
δω̄α = Λ(1)a (Γ
aλ̄)α − Λ(2)a (Γ
ar)α,
δsα = Λ(2)a (Γ
aλ̄)α, (4.3)
where Λ(1)a and Λ
a are local gauge parameters. Let us note that the conditions Eq. (4.1) and
these symmetries reduce the independent components of each field in the quartet to eleven com-
ponents. It is easy to show that the action Inm is invariant under the new BRST transformation
with BRST charge
Qnm =
dz(λαdα + ω̄
rα). (4.4)
Of course the quartet does not contribute to the central charge and has trivial cohomology with
respect to the (new) BRST charge.
As a final remark, it is worthwhile to recall that this new formalism can be interpreted
[27] as a critical topological string with ĉ = 3 and (twisted) N = 2 supersymmetry. Then it
is possible to apply topological methods to the computation of multiloop amplitudes where a
suitable regularization factor replaces picture-changing operators to soak up zero modes. The
covariant b field and the regulator proposed in [27] allow to compute loop amplitudes up to
g = 2. A more powerful regularization of b that allows to compute loop amplitudes at any g loop
has been presented in [28]. This regularization is gauge invariant but Lorentz non-covariant
since it involves the Y -field. However, this non-covariance is harmless since the regularized b
field differs from the covariant one by BRST-exact terms.
Now we are ready to present the Y-formalism for the non-minimal pure spinor quantization.
As in Eqs. (2.10) and (2.12), we first introduce the non-covariant object
Ȳ α =
, (4.5)
and the projector
(Γaλ̄)α(Ȳ Γa)β, (4.6)
where v̄α is a constant pure spinor so that we have
Ȳ ΓaȲ = 0. (4.7)
Let us note that the conditions (4.1) lead to relations λ̄αK̄
β = rαK̄
β = 0, which imply that
λ̄α and rα have respectively eleven independent components.
Next we postulate the following OPE’s among ω̄α, λ̄α, s
α and rα:
< ω̄α(y)λ̄β(z) >=
y − z
(δαβ − K̄
β(z)), (4.8)
< sα(y)rβ(z) >=
y − z
(δαβ − K̄
β(z)), (4.9)
< ω̄α(y)rβ(z) >=
y − z
[K̄α β(z)(Ȳ r)(z)−
(Γar)α(z)(Ȳ Γa)β(z)], (4.10)
< sα(y)λ̄β(z) >= 0. (4.11)
Then, with these OPE’s it is easy to check that the OPE’s between the conjugate momenta ω̄α
and sα, and the conditions (4.1) vanish identically:
< ω̄α(y)(λ̄Γaλ̄)(z) > = 0,
< ω̄α(y)(λ̄Γar)(z) > = 0,
α(y)(λ̄Γaλ̄)(z) > = 0,
< sα(y)(λ̄Γar)(z) > = 0. (4.12)
Notice that (4.10) follows for consistency by acting with the BRST charge Qnm on (4.8) (or
(4.9)).
Following [27], the only holomorphic currents involving ω̄ and s and gauge invariant under
(4.3) are:
• i) the bosonic currents
N̄ab =
(ω̄Γabλ̄− sΓabr),
J̄λ̄ = ω̄λ̄− sr,
Tλ̄ = ω̄∂λ̄− s∂r, (4.13)
those are, the Lorentz current, the ghost current and the stress energy tensor of the
non-minimal fields, respectively.
• ii) the fermionic currents
Sab =
sΓabλ̄,
S = sλ̄,
S(b) = s∂λ̄. (4.14)
• iii) the doublet
J0 = rs,
Φ0 = ω̄r. (4.15)
Using the fundamental OPE’s (4.8)-(4.11), one can compute the OPE’s among these operators.
The OPE’s of N̄ab, Tλ̄ and J̄λ̄ with λ̄ and r and the ones among themselves are canonical,
namely
< N̄ab(y)λ̄α(z) > =
y − z
(Γabλ̄)α(z), < N̄ab(y)rα(z) >=
y − z
(Γabr)α(z),
< J̄λ̄(y)λ̄α(z) > =
y − z
λ̄α(z), < J̄λ̄(y)rα(z) >=
y − z
rα(z),
< Tλ̄(y)λ̄α(z) > =
y − z
∂λ̄α(z), < Tλ̄(y)rα(z) >=
y − z
∂rα(z), (4.16)
< N̄ab(y)N̄cd(z) > = −
y − z
(ηc[bN̄a]d − ηd[bN̄a]c)(z),
< N̄ab(y)J̄λ̄(z) > = 0, < N̄ab(y)Tλ̄(z) >=
(y − z)2
N̄ab(z),
< J̄λ̄(y)J̄λ̄(z) > = 0, < J̄λ̄(y)Tλ̄(z) >=
(y − z)2
J̄λ̄(z),
< Tλ̄(y)Tλ̄(z) > =
(y − z)2
Tλ̄(z) +
y − z
∂Tλ̄(z). (4.17)
Notice that in contrast with the operators T , Nab and J in (2.16)-(2.18), the operators N̄ab,
Tλ̄ and J̄λ̄ do not involve Ȳ -correction terms since the Ȳ -dependent terms which arise in their
OPE’s are absent or cancel in the combinations (4.13). It is instructive to see explicitly how
this cancellation arises. Let us write N̄
(ω̄λ̄)
ω̄Γabλ̄ and N̄
sΓabr and consider for
instance the OPE between N̄ab = N̄
(ω̄λ̄)
ab − N̄
ab and rα. From Eq. (4.9), one obtains
ab (y)rα(z) >=
y − z
(Γabr)α +
y − z
(Γf Ȳ )α(λ̄Γ
fΓabr). (4.18)
Then, the second term in the r.h.s. of (4.18) is exactly cancelled by the contribution of the
OPE < N̄
(ω̄λ̄)
ab (y)rα(z) > in terms of Eq. (4.10). As a second example, consider the OPE
< N̄ab(y)N̄cd(z) >. The double poles coming from < N̄
(ω̄λ̄)
ab (y)N̄
(ω̄λ̄)
ab (z) > are cancelled by those
coming from < N̄
ab (y)N̄
ab (z) >. As for the simple poles, one has
(ω̄λ̄)
ab (y)N̄
(ω̄λ̄)
cd (z) > + < N̄
ab (y)N̄
cd (z) >
y − z
(ηc[bN̄a]d − ηd[bN̄a]c) +
[(sΓabΓf Ȳ )(rΓ
fΓcdλ̄) + (sΓcdΓf Ȳ )(rΓ
fΓabλ̄)], (4.19)
but the terms, which are independent of N̄ab in the r.h.s. of (4.19), are just cancelled by the
contributions stemming from −(< N̄
(ω̄λ̄)
ab (y)N̄
cd (z) > + < N̄
ab (y)N̄
(ω̄λ̄)
cd (z) >). For all the
remaining OPE’s in both (4.16) and (4.17), the spurious, Ȳ -dependent terms are absent or
cancelled in a similar way. Moreover, the OPE’s among Sab, S and S(b) are regular and those of
N̄ab, J̄λ̄ and Tλ̄ with S
ab, S and S(b) are canonical so that S
ab, S and S(b) are covariant primary
fields with weight 1 and ghost number 2 with respect to the ghost current J̄λ̄. Thus, as for N̄ab,
J̄λ̄ and Tλ̄, they do not have to include Ȳ -dependent corrections.
The story is completely different for the currents Jr and Φ. Indeed, using the OPE’s (4.8)-
(4.11), one finds
< (rs)(y)N̄ab(z) >=
(y − z)2
Ȳ Γabλ̄. (4.20)
And since one has
< (Ȳ ∂λ̄)(y)N̄ab(z) >=
(y − z)2
Ȳ Γabλ̄, (4.21)
the Ȳ -dependent term in < Jr(y)N̄
ab(z) > disappears if one assumes, as definition of Jr at
quantum level,
Jr = rs− 3Ȳ ∂λ̄. (4.22)
With this definition, the OPE’s of Jr with N̄ab, J̄λ̄, Tλ̄, S
ab, S and S(b) read
< Jr(y)Jr(z) > =
(y − z)2
< Jr(y)N̄
ab(z) > = 0,
< J̄λ̄(y)Jr(z) > =
(y − z)2
< Jr(y)Tλ̄(z) > =
(y − z)3
(y − z)2
< Jr(y)S
ab(z) > =
y − z
< Jr(y)S(z) > =
y − z
< Jr(y)S(b)(z) > =
y − z
S(b). (4.23)
In particular, note that the coefficient 8 of the double pole in the contraction < J̄λ̄(y)Jr(z) >
emerges from the arithmetic 8 = 11 − 3 where 11 comes from the first term and −3 from the
second term in (4.22).
In a similar manner, for Φ one has
< (ω̄r)(y)N̄ab(z) > = −
(y − z)2
[Ȳ Γabr − (Ȳ r)(Ȳ Γabλ̄)],
< (ω̄r)(y)Sab(z) > = −
(y − z)2
Ȳ Γabλ̄. (4.24)
Therefore, at quantum level Φ must be defined as
Φ = ω̄r + 3[Ȳ ∂r − (Ȳ r)(Ȳ ∂λ̄)] = ω̄r + 3∂(Ȳ r), (4.25)
in order to avoid spurious Ȳ -dependent terms. With this new definition, one can also derive
< Φ(y)N̄ab(z) > = 0,
< Φ(y)J̄λ̄(z) > = 0,
< Φ(y)Tλ̄(z) > =
(y − z)2
< Φ(y)Sab(z) > =
y − z
< Φ(y)S(z) > =
(y − z)2
y − z
J̄λ̄,
< Φ(y)S(b)(z) > =
(y − z)3
(y − z)2
y − z
< Φ(y)Jr(z) > =
y − z
Φ. (4.26)
The operator Φ is part of the BRST current and S(b) is a contribution of the b ghost as will be
seen in the next section.
From the definitions (4.13) and (4.14), one finds that the operators N̄ab, J̄λ̄ and Tλ̄ are the
BRST variations of the operators Sab, S, and S(b), respectively. Moreover, contrary to what
happens for the operators in (2.16)-(2.18), the correction term of Jr in (4.22) is not BRST-exact
but its BRST variation is just the correction term for −Φ in (4.25), so that Φ is just the BRST
variation of −Jr. These properties are fully consistent with the OPE’s we have computed thus
far. 8
As a final remark, let us note that in all the derivations of this section (but the second
equality of (4.25)) we have never used the fact that v̄ in (4.5) is constant and therefore all the
equations in this section remain true even if one replaces Ȳ α with Ỹ α ≡ λ
8Apart from a difference in the OPE < ΦS > where we find a double pole with residuum 8, not present in
[27] (perhaps a misprint in [27]), our results agree with those computed in [27] by using the U(5)-formalism.
5 A quantum b ghost in the non-minimal pure spinor
formalism
In Ref. [27], Berkovits has obtained an expression for a covariant b ghost in the framework of
non −minimal formalism. His idea was triggered by the observation that in this formalism the
non-covariant Yα field can be replaced by a covariant field λ̃α (which will be defined soon) and
then one can look for a new, covariant b ghost satisfying the defining equation
{Qnm, bnm(z)} = T (z) + Tλ̄(z) ≡ Tnm(z), (5.1)
by starting with bnm = λ̃αG
α + sα∂λ̄α + · · ·. The result, given in [27], is
bnm = s
α∂λ̄α + λ̃αG
α − 2λ̃β r̃αH
+ 6λ̃γ r̃β r̃αK
[αβγ] − 24λ̃δ r̃γ r̃β r̃αL
[αβγδ], (5.2)
where we have defined
λ̃α =
(λ̄λ)
r̃α =
(λ̄λ)
. (5.3)
Note that λ̃α and r̃α are primary fields of conformal weight 0 with respect to Tnm.
In this section, we will construct a covariant, quantum-mechanical b ghost in the non −minimal
pure spinor formalism on the basis of our Y-formalism, taking care of normal-ordering effects.
Furthermore, we shall show that this covariant b ghost is cohomologically equivalent to the
non-covariant b̃Y ghost, improved by the non-minimal term s
α∂λ̄α which takes the form at the
classical level
b̃0Y = YαG
0 + s
α∂λ̄α. (5.4)
It is now convenient to consider the following operators:
ρ[αβ] ≡
(r̃αλ̃β − r̃βλ̃α) ≡ r̃[αλ̃β],
ρ[αβγ] ≡ −r̃[αr̃βλ̃γ],
ρ[αβγδ] ≡ −r̃[αr̃β r̃γ λ̃δ],
ρ[αβγδǫ] ≡ r̃[αr̃β r̃γ r̃δλ̃ǫ], (5.5)
that satisfy the recursive relations
Qnm, λ̃α
= λβρ[αβ],
{Qnm, ρ[αβ]} = λ
γρ[αβγ],
Qnm, ρ[αβγ]
= λδρ[αβγδ],
{Qnm, ρ[αβγδ]} = λ
ǫρ[αβγδǫ]. (5.6)
Next, let us also recall the results which were obtained in section 3 and hold at the quantum
level:
{Q, Ĝα} = λαT,
Q,H [αβ]
= λ[αĜβ],
{Q,K [αβγ]} = λ[αHβγ],
Q,L[αβγδ]
= λ[αKβγδ],
λ[αLβγδρ] = 0, (5.7)
where Ĝα is defined in (3.13).
It is also useful to compute the contractions:
β(y)ρ[βα](z) > =
y − z
< H [βγ](y)ρ[γβα](z) > =
y − z
< K [βγδ](y)ρ[δγβα](z) > =
y − z
< L[βγδǫ](y)ρ[ǫδγβα](z) > =
(y − z)2
y − z
. (5.8)
After a simple calculation, it turns out that R1α is given by
R1α = −2ρ[αβ][λ
β(λ̃∂θ)−
(Γaλ̃)β(λΓa∂θ)], (5.9)
but the second term in the square bracket vanishes when contracted with ρ[αβ] due to the con-
ditions (4.1). As for R2α, R3α, R4α and R̃4α, they all contain (at least) a factor
abc(λ̃Γ
abλ) ≡
λ̃Λ[αβ]c λ and therefore vanish when contracted with ρ[βγ···] by taking into account (5.5), (3.58)
and (4.1). 9 To summarize, we have the following results:
R1α = −2ρ[αβ]λ
β(λ̃∂θ),
R2α = R3α = R4α = R̃4α = 0. (5.10)
As already noted, the non-minimal b field is expected to be of the form:
bnm = S(b) + λ̃G+ · · · . (5.11)
The anticommutator of Qnm with S(b) = s∂λ̄ is
{Qnm, S(b)} = Tλ̄. (5.12)
9In R̃4α, there is also a term proportional to ρ[αβγδ](Γabc)
αβ(Γdec)γδ(λ̃ΓabΓdeλ) that vanishes for the same
reason.
Now let us compute the anticommutator {Qnm, (λ̃αĜ
{Qnm, (λ̃αĜ
α)} = λ̃α(λ
αT ) + (λβρ[αβ])Ĝ
α. (5.13)
Using the rearrangement theorem and some algebra, (5.13) can be rewritten as
{Qnm, (λ̃αĜ
α)} = T + ρ[αβ](λ
βĜα) + {Qnm, ∂λ̃∂θ − (λ̃∂λ)(λ̃∂θ)}. (5.14)
Here it is of interest to remark that the term ∂λ̃∂θ − (λ̃∂λ)(λ̃∂θ) that arises in the r.h.s.
of (5.14) is just the difference between the generalized normal ordering (· · ·) in [33] and the
improved one : · · · : of λ̃αĜ
α, that is
(λ̃αĜ
α) =: λ̃αĜ
α : +∂λ̃∂θ − (λ̃∂λ)(λ̃∂θ), (5.15)
so that (5.14) becomes
{Qnm, : λ̃αĜ
α :} = T + ρ[αβ](λ
βĜα). (5.16)
With the help of the second recursive equations in (5.6) and (5.7) the last term in the r.h.s. of
Eq. (5.16) reads
ρ[αβ](λ
[βĜα]) = ρ[αβ]([Qnm, H
[βα]])
= {Qnm, ρ[αβ]H
[αβ]} −
(ρ[αβγ]λ
[α)Hβγ]. (5.17)
In this case, the rearrangement theorem does not give extra contributions since
(ρ[αβγ]λ
[α)Hβγ] − ρ[αβγ](λ
βγ]) = R2γ∂λ
γ + ∂ρ[αβγ]R
[αβγ]
3 , (5.18)
and the r.h.s. vanishes from (5.10) and (3.33) . Therefore, Eq. (5.17) can be rewritten as
ρ[αβ](λ
α]) = {Qnm, ρ[αβ]H
[αβ]} −
ρ[αβγ](λ
βγ]). (5.19)
For the last term in the r.h.s. of this equation, one can repeat the same procedure using the
third recursive equations in (5.6) and (5.7). Again the contributions from the rearrangement
theorem are absent since they involve the operators R
[αβγδ]
4 and R3α that vanish according to
(3.46) and (5.10). As a result, one obtains
ρ[αβγ](λ
[αHβγ]) = {Qnm, ρ[αβγ]K
[αβγ]}+
(ρ[αβγδ]λ
[α)Kβγδ]. (5.20)
As a last step, one can express the last term in (5.20) in terms of {Qnm, ρ[αβγδ]L
[αβγδ]} by
using the fourth recursive equations in (5.6) and (5.7). Again the contributions from the
rearrangement theorem are absent as before, so we have
(ρ[αβγδ]λ
[α)Kβγδ] = −{Qnm, (ρ[αβγδ]L
[αβγδ])}, (5.21)
where we have disregarded the term ρ[αβγδǫ]λ
[ǫLαβγδ] that vanishes according to (3.57).
Finally, using (5.12) and (5.16)-(5.21) we arrive at the result
{Qnm, bnm} = Tnm, (5.22)
where
bnm = s
α∂λ̄α+ : λ̃αĜ
α : −2(λ̃β r̃α)H
+ 6(λ̃γ r̃β r̃α)K
[αβγ] − 24(λ̃δ r̃γ r̃β r̃α)L
[αβγδ]. (5.23)
In conclusion, we have confirmed Eq. (5.2) provided that one interprets the compound field
α as the operator : λ̃αĜ
α : which is normal-ordered according to the improved prescription
(For the other terms in (5.23) the generalized and the improved normal-ordering prescriptions
coincide). Incidentally, we have also checked that this bnm possesses conformal weight 2
It might appear from (5.23) and the definition of λ̃ and r̃ that bnm is singular at λ̄λ → 0
with poles up to fourth order. However, as explained in [28], this singularity is not dangerous.
Indeed in this case, the analogous of the operator ξ = Y θ that would trivialize the cohomology,
ξnm =
λ̃λ+ r̃θ
= λ̄θ
(−rθ)n−1
(λλ̄)n
, (5.24)
since {Qnm, ξnm} = 1. However, ξnm diverges like (λλ̄)
−11 and to have a nontrivial cohomology
it is sufficient to exclude from the Hilbert space operators that diverge like ξnm or stronger.
Therefore bnm is allowed as insertion to compute higher loop amplitudes. To do actual calcula-
tions at more than two loops [28], bnm must be regularized properly. In fact, in [28] a consistent
regularization has been proposed.
Now let us come back to the non-covariant b ghost b̃0Y in (5.4). As a first step, let us derive
a quantum counterpart of b̃0Y , which is denoted as b̃Y . From the first equation in (5.7), one has
{Qnm, YαĜ
α} = Yα(λ
αT ). (5.25)
Moreover, since Yα(λ
αT )− (Yαλ
α)T = 2∂Y ∂λ from the rearrangement theorem, one obtains
{Qnm, YαĜ
α − 2∂Y ∂θ} = T. (5.26)
As before, the term 2∂Y ∂θ is just the difference between (YαĜ
α) and : YαĜ
α : and therefore
the quantum non-covariant b ghost takes the form
b̃Y =: YαĜ
α : +(s∂λ̄), (5.27)
and it satisfies
{Qnm, b̃Y } = Tnm. (5.28)
Even if b̃Y is non-covariant, its Lorentz variation is BRST-exact. Actually, one has
δLb̃Y =
Qnm, 2(L
αYβYγ)H
, (5.29)
where Lβα are (global) Lorentz parameters.
From (5.22) and (5.28), it follows that b̃Y − bnm is BRST-closed and then it is plausible
that it is also exact. Indeed in [34], we have shown that, at the classical level, the covariant
non-minimal b ghost (5.2) and the non-covariant one (5.4) are cohomologically equivalent. In
this respect, we wish to verify the cohomological equivalence between bnm and b̃Y even at the
quantum level
bnm = b̃Y + [Qnm,W ], (5.30)
where
W = 2(λ̃βYα)H
[αβ] + 3!(λ̃γ r̃βYα)K
[αβγ] + 4!(λ̃δr̃γ r̃βYα)L
[αβγδ] +WR, (5.31)
with WR being a quantum contribution coming from the rearrangement theorem, which will be
determined later.
In order to verify (5.30), let us compute the (anti)-commutators of Qnm with the first three
terms in the r.h.s. of (5.31). We have
2[Qnm, (λ̃βYα)H
[αβ]] = −ραβH
[αβ] + (Y[γραβ]λ
γ)H [αβ] + (λ̃βYα)(λ
[αĜβ])
= −2(λ̃β r̃α)H
[αβ] − 3!(Y[γ r̃αλ̃β])(λ
γHαβ) + (λ̃αĜ
α)− (YαĜ
+ RH +RG, (5.32)
where RH and RG are the contributions coming from the rearrangement theorem of the last
two terms in the first row of this equation. Then
3!{Qnm, (Yαr̃βλ̃γ)K
[αβγ]} = 3!(r̃αr̃βλ̃γ)K
[αβγ] − 4!(Yαr̃β r̃γ λ̃δ)(λ
[δKαβγ])
+ 3!(Yαr̃βλ̃γ)(λ
βγ]) +RK , (5.33)
where RK arises from rearrangement theorem. Finally, we have
Qnm, (Yαr̃β r̃γ λ̃δL
[αβγδ])
= 4!r̃αr̃β r̃γ λ̃δL
[αβγδ] + 4!(Yαr̃β r̃γλ̃δ)(λ
[αKβγδ]) +RL, (5.34)
where RL comes from rearrangement formula. The quantum contributions RG, RH , RK and
RL are explicitly given by
RG = −[∂λ̃∂θ − (λ̃∂λ)(λ̃∂θ)− 2∂Y ∂θ] + 2[Qnm, (Yαλ̃β)W
R1 ]−
GΠa, (5.35)
RH = 3![Qnm, (Yαr̃βλ̃γ)W
[αβγ]
R2 ]−
AaHα(dΓa)
AaGΠa, (5.36)
RK = 4![Qnm, (Yαr̃β r̃γλ̃δ)W
[αβγδ]
R3 ] +
AaHα(dΓa)
AcKαβN
c , (5.37)
RL = −
AcKαβN
c +BL, (5.38)
where
((Y + λ̃)Γa)
[α∂λβ]Πa,
[αβγ]
((Y + 2λ̃)Γa)
β(Γad)γ],
[αβγδ]
((Y + 3λ̃)Γa)
[α∂λβNγδ]a, (5.39)
AaG = 3!Y[αr̃βλ̃γ]λ
γ((Y + 2λ̃)Γa)α∂λβ ,
AaHα = 4!Y[αr̃β r̃γλ̃δ]λ
δ((Y + 3λ̃)Γa)β∂λγ ,
AaKαβ = 5!Y[αr̃β r̃γ r̃δλ̃ǫ]λ
ǫ((Y + 4λ̃)Γa)δ∂λγ . (5.40)
The Y -dependent operators AaG, A
Hα and A
Kαβ cancel when (5.35)-(5.38) are summed up.
As for BL, it turns out that it is BRST-exact:
BL = 4!
Qnm, (Yαr̃β r̃γλ̃δ)W
[αβγδ]
Qnm, ∂((Yαr̃β r̃γ λ̃δ)W
[αβγδ]
, (5.41)
where
[αβγδ]
[(ΓcY )[α(Γb(Y + 3λ̃))β(Γbc∂λ)
γ∂λδ] − (ΓcY )[α(Γb(Y + 3λ̃))β(Γbcλ)
γ∂2λδ]
+ 3(λ̃ΓcY )(Γb(Y + 2λ̃))[α∂λβ(ΓcΓ
aλ)γ(ΓbΓa∂λ)
δ]], (5.42)
[αβγδ]
(ΓcY )[α(Γb(Y + 3λ̃))β[(Γbcλ)
γ∂λδ] −
aλ)γ(ΓbΓa∂λ)
δ]]. (5.43)
Some details on the derivations of these results will be given in Appendix C. From (5.15), one
finds that the term −∂λ̃∂θ+ (λ̃∂λ)(λ̃∂θ)− 2∂Y ∂θ transforms (λ̃αĜ
α)− (YαĜ
α) to : λ̃αĜ
α : − :
Collecting Eqs. (5.32)-(5.43), one recovers (5.30) where bnm and b̃Y are given in (5.23) and
(5.27), respectively and
W = 2(λ̃βYα)(H
[αβ] +W
R1 ) + 3!(λ̃γ r̃βYα)(K
[αβγ] +W
[αβγ]
+ 4!(λ̃δ r̃γ r̃βYα)(L
[αβγδ] +W
[αβγδ]
R3 +W
[αβγδ]
R4 ) + 4!∂[(λ̃δ r̃γ r̃βYα)W
[αβγδ]
R5 ]. (5.44)
6 Conclusion
In this article, using the Y-formalism [30], we have calculated the normal-ordering contributions
existing in various composite operators in the pure spinor formalism of superstrings. These
operators naturally appear when we try to construct a b ghost. Moreover, we have constructed
the Y-formalism for the non-minimal sector. Using these information, we have presented a
quantum-mechanical expression of the b ghost, bnm, in the non-minimal formulation and we
have shown, in this case, that the non-covariant b field bY and bnm, are equivalent in cohomology.
The consistent results we have obtained in this article could be regarded as a consistency
check of the Y-formalism in the both minimal and non-minimal pure spinor formulation of
superstrings.
In the case of the non-minimal formulation, due to its field content and structure, it is
natural to ask if it is possible to reach a fully covariant system of rules for the OPE’s in the
minimal and non-minimal ghost sectors, by replacing the non-covariant fields Yα and Ȳ
α with
the covariant ones λ̃α =
and Ỹ α = λ
, respectively. As for the replacement of Ȳ α with Ỹ α,
that is of v̄α with λα for the non-minimal sector, we do not see any problem, as noted at the
end of section 4 because v̄α and λα are both BRST invariant and all the OPE’s among the
currents of the non-minimal sector remain unchanged.
On the contrary, a naive, straightforward replacement of Yα with λ̃α looks problematic.
Indeed, even if the OPE’s among the Lorentz current Nab, the ghost current J , and the stress
energy tensor Tλ of the minimal ghost sector are unchanged, those among these operators and
that of the non-minimal sector become different from zero, since the correction terms in (2.16)-
(2.18) now acquire a dependence from λ̄. Therefore the OPE’s among the total Lorentz current,
ghost current and stress energy tensor of the (minimal and non-minimal) ghost sector do not
close correctly. Moreover, the BRST variation of (2.14) appears to be inconsistent. We cannot
exclude a possibility that these problems could be overcome by a smart modification of the
basic OPE’s, but it is far from obvious that a consistent modification could be found. Thus, in
this paper, we have refrained from exploring this possibility further and we hope to come back
to this question in future.
Acknowledgements
The work of the first author (I.O.) was partially supported by the Grant-in-Aid for Scientific
Research (C) No.14540277 from the Japan Ministry of Education, Science and Culture. The
work of the second author (M.T.) was supported by the European Community’s Human Po-
tential Programme under contract MRTN-CT-2004-005104 ”Constituents, Fundamental Forces
and Symmetries of the Universe”.
A Notation, Conventions and Useful identities
In this appendix, we collect our notation, conventions and some useful formulae employed in
this paper.
As usual, in ten space-time dimensions, Γa are the Dirac matrices γa times the charge
conjugation matrix C, that is, (Γa)βα = (γaC)αβ and (Γa)βα = (C
−1γa)αβ; they are 16 ×
16 symmetric matrices with respect to the spinor indices, and satisfiy the Clifford algebra
{Γa,Γb} = 2ηab. Our metric convention is ηab = (−,+, · · · ,+).
The square bracket and the brace respectively denote the antisymmetrization and the sym-
metrization of p indices, normalized with a numerical factor 1
so that, for instance A[µBν] =
(AµBν − AνBµ). As for the products of Γ
a, Γa1···ap = Γ[a1···ap]. These antisymmetrized
products of Γ have definite symmetry properties, which are given by (Γab)α β = −(Γ
(Γabc)αβ = −(Γ
abc)βα, (Γ
abcd)α β = (Γ
abcd)β
α, (Γabcde)αβ = (Γ
abcde)βα, etc.
The product of generic spinors fα and gβ can be expanded in terms of the complete set of
gamma matrices as
fαgβ =
Γaαβ(fΓag) +
16× 3!
Γabcαβ (fΓabcg) +
16× 5!
Γabcdeαβ (fΓabcdeg). (A.1)
Similarly, for spinors fα and g
β we have
δβα(fg) +
16× 2!
(Γab)α
β(fΓabg) +
16× 4!
(Γabcd)α
β(fΓabcdg). (A.2)
A useful identity, involving three spinor-like operators Aα, B
β and Cγ is
(BΓabA)(ΓabC)
(BA)Cα = (BβA
α)Cβ −
((ΓaB)αAβ)(ΓaC)β. (A.3)
B Normal ordering, the generalized Wick theorem and
rearrangement theorem
In this appendix, we explain the prescription of normal ordering, the generalized Wick theorem
and rearrangement theorem, which are used in this paper. The more detail of them can be seen
in the texbook of conformal field theory [33].
B.1 Normal ordering
In conformal field theory, we usually consider normal ordering for free fields where the OPE
contains only one singular term with a constant coefficient. Then, normal ordering is defined
as the subtraction of this singular term. This definition of normal ordering is found to be
equivalent to the conventional normal ordering in the mode expansion where the annihilation
operators are placed at the rightmost position. However, we sometimes meet the case for
which the fields are not free in this sense. One of the well-known examples happens when
we try to regularize the OPE between two stress enery tensors T (y)T (z). In this case, we
have two singular terms where one singular term contains the quartic pole whose coefficient is
proportional to the central charge while the other singular term contains the quadratic pole
whose coefficient is not a constant but (2×) stress energy tensor itself. The ususal normal
ordering prescription amounts to subtraction of the former, most singular term, but the latter
singular term is still remained. Let us note that in the present context, the OPE between ω and
λ is not free owing to the existence of the projection K reflecting the pure spinor constraint.
From the physical point of view, we want to subtract all the singular terms in the OPE’s, so
we have to generalize the definition of normal ordering.
To this end, we introduce the generalized normal ordering which is usually denoted by
parentheses, that is, explicitly, the generalized normal ordering of operators A and B is written
as (AB)(z). A definition of the generalized normal ordering is given by the contour integration
(AB)(z) =
w − z
A(w)B(z). (B.1)
Then the OPE of A(z) and B(w) is described by
A(z)B(w) =< A(z)B(w) > +(A(z)B(w)), (B.2)
where < A(z)B(w) > denotes the contraction containing all the singular terms of the OPE
and (A(z)B(w)) stands for the complete sequence of regular terms whose explicit forms can be
extracted from the Taylor expansion of A(z) around w:
(A(z)B(w)) =
(z − w)k
(∂kA ·B)(w). (B.3)
Another definition of the generalized normal ordering is provided by the mode expansion.
If the OPE of A and B is written as
A(z)B(w) =
{AB}k(w)
(z − w)k
, (B.4)
where N is some positive integer, the definition of the generalized normal ordering reads
(AB)(z) = {AB}0(z). (B.5)
Incidentally, in this context, the contraction is expressed by
< A(z)B(w) >=
{AB}k(w)
(z − w)k
, (B.6)
In this paper, we adopt the definition of the contour integration (B.1). Moreover, for
simplicity, we do not write explicitly the outermost parenthesis representing the generalized
normal ordering whenever we can easily judge from the context whether some operators are
normal-odered or not.
B.2 The generalized Wick theorem
Relating to the generalization of the normal-ordering prescription, we also have to reformulate
the Wick theorem for interacting fields. In general, the Wick theorem relates the time-ordered
product to the normal-ordered product of free fields. However, such a relation cannot be
generalized to interacting fields in a straightforward manner. Hence, the generalized Wick
theorem is defined by generalizing a special form of the Wick theorem for the contraction of
free fields. More explicitly, the generalized Wick theorem is simply defined as
< A(z)(BC)(w) >=
[< A(z)B(x) > C(w) +B(x) < A(z)C(w) >]. (B.7)
From this definition, it is important to notice that the first regular term of the various OPE’s
always contributes. If we would like to calculate < (BC)(z)A(w) >, we first calculate <
A(z)(BC)(w) >, then interchange w and z, and finally expand the fields evaluated at z in the
Taylor series around w.
B.3 Rearrangement theorem
We often encounter the situation where many of operators are normal-ordered, e.g., (A(BC))(z).
With the generalized normal ordering, some complication occurs since there is no associativity
in such normal-ordered operators
(A(BC))(z) 6= ((AB)C)(z). (B.8)
To deal with normal ordering of such composite operators, we make use of the rearrangement
theorem. The useful formulae are given by
(AB) = (BA) + ([A,B]), (B.9)
(A(BC)) = (B(AC)) + (([A,B])C), (B.10)
((AB)C) = (A(BC)) + (A([C,B])) + (([C,A])B) + ([(AB), C]), (B.11)
where A, B, and C are all the Grassmann-even quantities. Note that if the Grassmann-odd
quantities are involved, we must change the sign and the commutator in a suitable manner. For
instance, for the Grassmann-even A and the Grassmann-odd B and C, the last rearrangement
theorem is modified as
((AB)C) = (A(BC))− (A({B,C}))− (([C,A])B) + ({(AB), C}). (B.12)
In making use of these rearrangement theorems, we are forced to evaluate the generalized
normal ordering of the (anti-)commutator ([A,B]). Then, we rely on the useful formula
([A,B])(z) =
(−1)k+1
∂k{AB}k(z). (B.13)
Note that field-dependent singular terms contribute to the normal-ordering (anti-)commutator
while the non-singular term {AB}0 does not. In this paper, we make heavy use of these formulae
in evaluating various normal-ordered products of operators.
C Some details about the calculations
C.1 BRST variation of Gα
To compute the BRST variation of Gα it is convenient to use the following notation
gα(B,A,C) = −
(BΓabA)(ΓabC)
(BA)Cα
= (BβA
α)Cβ −
((ΓaB)αAβ)(ΓaC)β, (C.1)
where Aα, Bβ , and C
γ are generic spinors and the last step is the identity (A.3). Then, given
(3.5), one has
{Q,Gα1} = −
λα(ΠaΠa) +
(λΓa∂θ)(Γ
ad)α. (C.2)
Moreover,
{Q,Gα2 +G
3} = −g
α(d, λ, ∂θ) + gα(Ω, λ, ∂λ)− 2gα(Y, ∂λ, ∂λ)− (Y ∂λ)∂λα. (C.3)
The last three terms come from the definitions (2.25) and (2.26) of Nab and J .
Using the rearrangement formula (cf. (B.12)), one has
gα(d, λ, ∂θ) = λα(d∂θ) + 8∂2λα +
(λΓa∂θ)(Γ
ad)α, (C.4)
gα(Ω, λ, ∂λ) = (Ωβλ
α)∂λβ −
((ΓaΩ)αλβ)(Γa∂λ)β . (C.5)
Using the rearrangement theorem, the first term in the r.h.s. of Eq. (C.5) becomes
α)∂λβ = λα(Ω∂λ)−
(Y Γa)α(∂λΓa∂λ) +
, (C.6)
whereas the second term can be rewritten as
((ΓaΩ)αλβ)(Γa∂λ)β = −
∂2λα +
(Y ∂2λ)λα + (Y ∂λ)∂λα
+ 3(∂Y ∂λ)λα + 2gα(Y, ∂λ, ∂λ) +
(Y Γa)α(∂λΓa∂λ), (C.7)
so that from (C.5)-(C.7), one obtains
gα(Ω, λ, ∂λ) = λα(Ω∂λ) + 4∂2λα +
(Y ∂2λ)λα
+ (Y ∂λ)∂λα + 3(∂Y ∂λ)λα + 2gα(Y, ∂λ, ∂λ). (C.8)
Adding Eqs. (C.2), (C.3) and {Q,Gα4} = c1∂
2λα with c1 =
,taking into account (C.4),
(C.8) and using the definition (2.16) of the stress energy tensor T , we finally obtain
{Q,Gα} = λαT −
∂2λα. (C.9)
C.2 BRST variation of Hαβ
Now let us consider the BRST variation of Hαβ. Eq. (3.25) can be rewritten as
[Q,H(αβ)] =
Γαβa h
, (C.10)
where
(λΓaΓbd)Π
b +Nab(λΓb∂θ)−
J(λΓa∂θ) + 2∂(λΓa∂θ). (C.11)
The first term in the r.h.s. of this equation can be rewritten as
(λΓaΓbd)Π
(λΓaΓbΠ
bd) + 5∂(λΓa∂θ). (C.12)
With the notation
Λαβ ≡
∂λ[αλβ],
Λ̃[αβ] ≡ −
(ΓcΛΓc)[αβ], (C.13)
the vector
V a = Nab(λΓb∂θ)−
J(λΓa∂θ), (C.14)
becomes
V a =
(ΩΓaΓbλ)(λΓb∂θ)− J(λΓ
a∂θ) + 4(Y ΛΓa∂θ) + 4(Y ΓaΛ̃∂θ) + 2(∂λΓa∂θ). (C.15)
The first term in the r.h.s. of (C.15) vanishes modulo a rearrangement contribution:
(ΩΓaΓbλ)(λΓb∂θ) = −4(Y Γ
aΛ̃∂θ)− 4(Y ΛΓa∂θ) + 4(∂λΓa∂θ), (C.16)
so that ha becomes
(λΓaΓbΠbd) + 5∂(λΓ
a∂θ)− J(λΓa∂θ)
− 2(∂λΓa∂θ) + 2∂(λΓa∂θ). (C.17)
On the other hand,
λΓaG =
(λΓaΓbΠ
bd) +
(λΓa∂2θ) + Ṽ a, (C.18)
where Ṽ a = −1
(λ̃ΓaN bcΓbc∂θ)−
(λΓaJ∂θ). Then, using (2.25) and (2.26)
Ṽ a = −
(λΓc(ΩΓ
aΓcλ)∂θ) +
(λΓcΓb(ΩΓ
bΓcλ)Γa∂θ)− (λΓa(Ωλ)∂θ)
− 4(Y ΓaΛ̃∂θ) − 4(Y ΛΓ
a∂θ) + 2(Y ΛΓa∂θ) + (∂λΓa∂θ). (C.19)
But the first two terms in the r.h.s. of (C.19) vanish modulo the Y-dependent term 4[(Y ΓaΛ̃∂θ)+
(Y ΛΓa∂θ)] − 6(Y ΛΓa∂θ) coming from rearrangement theorem, so that we have
Ṽ a = −(λΓa(Ωλ)∂θ) − 4(Y ΛΓa∂θ) + (∂λΓa∂θ)
= −J(λΓa∂θ) + 4∂(λΓa∂θ)− 4(λΓa∂2θ), (C.20)
and therefore
λΓaG =
(λΓaΓbΠ
bd) +
(λΓa∂2θ)− J(λΓa∂θ)− 4(λΓa∂2θ) + 4∂(λΓa∂θ). (C.21)
Then comparing (C.17) with (C.21), one gets Eq. (3.27).
Next let us consider the BRST variation of H [αβ]. Eq. (3.26) can be rewritten as
[Q,H [αβ]] =
(λΓabcΓdΠdd) + 6(λN
abΓc∂θ)
+ 4(λΓabc∂2θ) + (∂λΓabc∂θ)
, (C.22)
where the last two terms in the r.h.s. of this equation come from normal ordering.
On the other hand,
λΓabcĜ =
(λΓabcΓdΠdd) + 4(λΓ
θ) + 6(λN [abΓc]∂θ)
+ 3(λΓfN
f [aΓbc]∂θ) +
(λΓfΓgΓ
abcNfg∂θ)−
(λΓabcJ∂θ). (C.23)
Using (2.30), (2.31) and the notation introduced in (C.13) the quantity in the second row of
(C.23), that is,
[abc] = +3(λΓfN
f [aΓbc]∂θ) +
(λΓfΓgΓ
(λΓabcJ∂θ), (C.24)
can be rewritten as
[abc] = −
(λΓfΓ
[ab(ΩΓc]Γfλ)∂θ) +
(λΓfΓgΓ
abc(ΩΓfΓgλ)∂θ)− 12(Y ΓaΛ̃Γbc∂θ)
− 6(Y ΛΓabc∂θ) + (∂λΓabc∂θ). (C.25)
On the other hand, by reordering, the sum of the first two, Ω-dependent terms in (C.25) yields
12(Y ΓaΛ̃Γbc∂θ) + 6(Y ΛΓabc∂θ) so that h[abc] = ∂λΓabc∂θ and (C.23) becomes
λΓabcĜ =
(λΓabcΓdΠdd) + 4(λΓ
θ) + 6(λN [abΓc]∂θ) + (∂λΓabc∂θ). (C.26)
By comparing (C.22) with (C.26) one gets Eq. (3.28).
C.3 BRST variation of K [αβγ]
Now let us check (3.41). Let us rewrite (3.40) as
{Q,K [αβγ]} = k
[αβγ]
1 + k
[αβγ]
2 , (C.27)
where
[αβγ]
2 = −
(Γad)
(Γad)βλγ] −
(Γbd)
βΓbaλγ]
abc(dΓ
abcd), (C.28)
[αβγ]
aΓdλ)[αNβγ]a
aΓdλ)[α[
(ΩΛβγ]a λ)− (Y Λ
a ∂λ)]. (C.29)
The first term in the r.h.s. of (C.29) can be elaborated as follows:
aΓdλ)[α(ΩΛβγ]a λ) =
Πd(ΩΓb)[αλβ(ΓbΓdλ)
γ] +∆[αβγ]
Πdλ[α(ΩΛ
d λ) + ∆
[αβγ] + ∆̂[αβγ], (C.30)
where ∆[αβγ] and ∆̂[αβγ] are the contributions of rearrangement theorem and are given by
∆[αβγ] =
abc (Γ
aΓd∂KΓ
bcλ)γ]
Πf (ΓaY )[α[∂λβ(ΓaΓfλ)
γ] + (ΓaΓf∂λ)
βλγ] −
(ΓfΓb∂λ)
β(ΓaΓ
bλ)γ]]
Πd(ΓdY )
[α∂λβλγ], (C.31)
∆̂[αβγ] =
Πd(ΓfY )[α(∂λΓfΛ
d λ). (C.32)
Therefore, k
[αβγ]
1 becomes
[αβγ]
Πdλ[αN
d + {Π
d(λ[αY Λ
d ∂λ)−
aΓdλ)[α(Y Λβγ]a ∂λ)
+ ∆̂[αβγ] +∆[αβγ]}. (C.33)
With a little algebra, it is easy to show that the terms in the curly bracket in the r.h.s. of
(C.33) vanish so that (C.33) becomes
[αβγ]
λ[αΠdN
d . (C.34)
Then, (C.27), together with (C.28), (C.34) and (3.23), reproduces Eq. (3.41).
C.4 The vanishing of λ[ǫLαβγδ]
Now let us consider λ[ǫLαβγδ] in order to verify that it vanishes. As discussed at the end of
section 3, it consists of three terms:
αβγδ]
1 = λ
[ǫ(ΩΛαβc λ)(ΩΛ
γδ]cλ)
= ΩσA
[ǫαβγδ]σ
1 + A
[ǫαβγδ]
0 , (C.35)
αβγδ]
2 = ΩσB
[ǫαβγδ]σ
[ǫαβγδ]
0 , (C.36)
αβγδ]
3 = 4λ
[ǫ(Y Λαβc ∂λ)(Y Λ
γδ]c∂λ) (C.37)
where ΩσB
[ǫαβγδ]σ
[ǫαβγδ]σ
1 = −4Ωσλ
[ǫ(Y Λ[αβc ∂λ)(Λ
γδ]cλ)σ, (C.38)
and ΩσA
1 , A0 and B0 are rearrangement contributions coming when Ωσ is shifted to the left.
The explicit calculation of ΩσA
1 gives
[ǫαβγδ]σ
1 = Ωσ(Λ
c[αβλ)σ(∂λΓfΛ
c λ)(Γ
fY )ǫ] + Ωσ(Λ
c[αβΓfY )σ(∂λΓfΛ
c λ)λ
ǫ]. (C.39)
The first term in the r.h.s. of this equation can be rewritten as 4Ωσλ
[ǫ(Y Λ[αβc ∂λ)(Λ
γδ]cλ)σ −
2Ωσ(Λ
c[αβλ)σ(ΓcY )
γ∂λδλǫ] and the second one as 2Ωσ(Λ
c[αβλ)σ(ΓcY )
γ∂λδλǫ] so that we have
[ǫαβγδ]σ
1 = 4Ωσλ
[ǫ(Y Λαβc ∂λ)(Λ
γδ]cλ)σ. (C.40)
Then, using (C.38) and (C.40), Eq. (3.67) can be derived.
As for A
[ǫαβγδ]
0 and B
[ǫαβγδ]
0 , the explicit calculation gives
[ǫαβγδ]
(ΓfY )[ǫ(Y Λαβc λ)(∂λΓfΛ
(ΓfY )[ǫ(Y Λαβc ∂λ)(∂λΓfΛ
∂((ΓfY )[ǫ(Y Λαβc λ)(∂λΓfΛ
γδ]cλ)), (C.41)
[ǫαβγδ]
0 = −2(Γ
fY )[ǫ(Y Λαβc ∂λ)(∂λΓfΛ
γδ]cλ) + 4λ[ǫ(Y Λαβc ∂λ)(Y Λ
γδ]c∂λ)
+ 2∂λ[ǫ(Y Λαβc λ)(Y Λ
γδ]c∂λ)− 2∂(λ[ǫ(Y Λαβc ∂λ)(Y Λ
γδ]cλ)), (C.42)
so that we obtain
[ǫαβγδ]
[ǫαβγδ]
(ΓfY )[ǫ(Y Λαβc λ)(∂λΓfΛ
γδ]c∂λ)−
(ΓfY )[ǫ(Y Λαβc ∂λ)(∂λΓfΛ
γδ]cλ)
+ 4λ[ǫ(Y Λαβc ∂λ)(Y Λ
γδ]c∂λ) + 2∂λ[ǫ(Y Λαβc λ)(Y Λ
γδ]c∂λ)
∂{(ΓfY )[ǫ(Y Λαβc λ)(∂λΓfΛ
λ))− 4λ[ǫ(Y Λαβc ∂λ)(Y Λ
λ)}. (C.43)
In order to verify (3.68), one needs three useful identites:
(ΓfY )[ǫ(Y Λαβc λ)(∂λΓfΛ
λ) = 4λ[ǫ(Y Λαβc ∂λ)(Y Λ
λ), (C.44)
(ΓfY )[ǫ(Y Λαβc λ)(∂λΓfΛ
γδ]c∂λ) = 5λ[ǫ(Y Λαβc ∂λ)(Y Λ
γδ]c∂λ)
+ 10∂λ[ǫ(Y Λαβc λ)(Y Λ
γδ]c∂λ), (C.45)
(ΓfY )[ǫ(Y Λαβc ∂λ)(∂λΓfΛ
γδ]cλ) = 3λ[ǫ(Y Λαβc ∂λ)(Y Λ
γδ]c∂λ) + 2∂λ[ǫ(Y Λαβc λ)(Y Λ
γδ]c∂λ).(C.46)
From the identity (C.44), the derivative term in the last row of the r.h.s. of (C.43) vanishes.
Then, removing the first two terms in the r.h.s. of (C.43) by means of the two identites (C.45)
and (C.46), one gets
[ǫαβγδ]
[ǫαβγδ]
0 = −4λ
[ǫ(Y Λαβc ∂λ)(Y Λ
γδ]c∂λ), (C.47)
which is nothing but Eq. (3.68). In this way, we have succeeded in proving Eq. (3.57).
C.5 Equivalence in cohomology of bY and bnm
As a last remark, let us report briefly about the derivation of the rearrangement terms RG, RH ,
RK , RL and BL, which appear at the end of section 5. In particular we shall show that BL is
BRST-exact.
From the recipe given in Appendix B.3 and using the OPE (3.15), one can compute RG =
(λ̃βYα)(λ
[αĜβ])− (λ̃βYαλ
[α)Ĝβ] with the result
RG = −[∂λ̃∂θ − (λ̃∂λ)(λ̃∂θ)− 2∂Y ∂θ] + R̂G, (C.48)
where
R̂G = Y[αλ̃β]((Y + λ̃)Γa)
α∂λβ(λΓa∂θ). (C.49)
With the replacement
λΓa∂θ = [Qnm,Π
a], (C.50)
and some simple algebra, R̂G can be rewritten as
R̂G = 2[Qnm, (Yαλ̃β)W
R1 ]−
AaGΠa, (C.51)
where W
R1 is defined in (5.39) and A
G, defined in (5.40), comes from
GΠa =
Qnm, Y[αλ̃β]((Y + λ̃)Γa)
Πa, (C.52)
by using a simple algebra.
In a similar way, RH is given by
RH = −
AaGΠa + R̂H , (C.53)
where R̂H contains the factor (Γ
aλ)γ which can be replaced by −{Qnm, (Γ
cd)γ} and then,
working as before, one arrives at
R̂H = 3!
Qnm, Y[αr̃βλ̃γ]W
AaHα(dΓa)
α, (C.54)
where W
[αβγ]
R2 and A
Hα are defined in (5.39) and (5.40), respectively. Moreover,
AaHα(dΓa)
α + R̂K , (C.55)
where
R̂K =
(Y[αr̃β r̃γλ̃δ]((Y + 3λ̃)Γ
c)α∂λβ [Qnm, N
Qnm, (Y[αr̃β r̃γλ̃δ])W
[αβγδ]
+ AcK[αβN
c , (C.56)
where again W
[αβγδ]
R3 and A
Kαβ are defined in (5.39) and (5.40), respectively.
Now let us move on to RL which, according to (5.34), is
RL = 5!
(Y[αr̃β r̃γ r̃δλ̃ǫ]λ
ǫ)(N cαβNγδc ). (C.57)
With rearrangement formula and using (3.58), RL becomes
RL = 5!
K[αβγδǫ]λ
ǫ, N cαβNγδc
K[αβγδǫ], N
(−2)(K[αβγδǫ]((Y + 4λ̃)Λ
c λ)∂λ
ǫ)N cαβ + R̂L, (C.58)
where we have defined K[αβγδǫ] = Y[αr̃β r̃γ r̃δλ̃ǫ]. (The expression of R̂L will be given below.) To
the first term in the last row of Eq. (C.58), adding and subtracting the term defined by
((Y[αr̃β r̃γ r̃δλ̃ǫ])((Y + 4λ̃)Σ
c λ)∂λ
ǫ)N cαβ, (C.59)
where we have also defined
Ỹ Σ[αβ]c λ = (Ỹ Γc)
[αλβ] +
(Ỹ Γ)
bΓcλ)
β], (C.60)
Ỹα = Yα + 4λ̃α, (C.61)
RL is then reduced to
RL = −
AcKαβN
c +R0 + R̂L. (C.62)
Here we have introduced the quantity
R̂L = 5!
(∂R1 +R2 +R3), (C.63)
where R1, R2 and R3 are defined by
Y[αr̃β r̃γ r̃δλ̃ǫ]λ
ǫ(ΓaY )α(ΓbỸ )β(ΓbΓ
cλ)γ [(ΓaΓc∂λ)
δ + 2δac∂λ
R2 = −
Y[αr̃β r̃γ r̃δλ̃ǫ]λ
ǫ(ΓcY )α(ΓbỸ )β∂[(Γbcλ)
γ∂λδ],
R3 = Y[αr̃β r̃γ r̃δλ̃ǫ]λ
ǫ(λ̃ΓfY )(Γc(2Y + 5λ̃))α(ΓcΓ
bλ)β(ΓfΓb∂λ)
γ∂λδ. (C.64)
It is of importance that R1, R2 and R3 are all BRST-exact:
Qnm, Y[αr̃β r̃γ λ̃δ](Γ
aY )α(Γb(Y + 3λ̃))β(ΓbΓ
cλ)γ[(ΓaΓc∂λ)
δ + 2δac∂λ
R2 = −
Qnm, Y[αr̃β r̃γλ̃δ](Γ
cY )α(Γb(Y + 3λ̃))β∂[(Γbcλ)
γ∂λδ]
Qnm, Y[αr̃β r̃γ λ̃δ](λ̃Γ
fY )(Γc(Y + 2λ̃))α(ΓcΓ
bλ)β(ΓfΓb∂λ)
. (C.65)
On the other hand, by rearrangement theorem, R0 can be rewritten as
Ωσ(Y[αr̃β r̃γ r̃δλ̃ǫ](Ỹ Σ
c λ)∂λ
ǫ(Λαβcλ)σ)
Y[αr̃β r̃γ r̃δλ̃ǫ]λ
ǫ(ΓcỸ )α∂λβ(ΓbY )γ(ΓbΓc∂λ)
δ. (C.66)
The first term in the r.h.s. of (C.66) vanishes and the second one is BRST-exact. Indeed, one
[Qnm, Y[αr̃β r̃γ λ̃δ](Γ
c(Y + 3λ̃))α∂λβ(ΓbY )γ(ΓbΓc∂λ)
δ]. (C.67)
Eq. (C.62) is just Eq. (5.38) with BL = R0 + R̂L. Then, from Eqs. (C.63), (C.65) and (C.67),
one can reproduce Eqs. (5.41)-(5.43).
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