###### Abstract

We present the Y-formalism for the non-minimal pure spinor quantization of superstrings. 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 ghost, their normal-ordering contributions and the relevant relations among them. We use these results to construct the quantum-mechanical ghost in the non-minimal pure spinor formalism. Moreover we show that this non-minimal ghost is cohomologically equivalent to the non-covariant ghost.

DPUR/TH/1

DFPD07/TH/06

April, 2007

Y-formalism and ghost in the Non-minimal Pure Spinor Formalism of Superstrings

Ichiro Oda
^{1}^{1}1
E-mail address:

Department of Physics, Faculty of Science, University of the Ryukyus,

Nishihara, Okinawa 903-0213, Japan.

and

Mario Tonin
^{2}^{2}2
E-mail address:

Dipartimento di Fisica, Universita degli Studi di Padova,

Instituto Nazionale di Fisica Nucleare, Sezione di Padova,

Via F. Marzolo 8, 35131 Padova, Italy

## 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 and their fermionic partners
with their conjugate momenta ,
plus an action for the bosonic ghosts and their conjugate
momenta , where satisfy the ”pure spinor
constraint” .
The 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}^{3}3Alternative formalisms to remove the constraint
were proposed in [25, 26].

Another key ingredient in the pure spinor formulation is provided by the BRST charge where contains the constraints generating a fermionic 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 is nilpotent only when the bosonic spinor satisfies the pure spinor condition. This peculiar feature is in fact expected since the constraint 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 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 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 field defined by with 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 ghosts of diffeomorphisms, the usual antighost is not present in this approach. In [3] a compound field whose BRST variation gives the stress energy tensor, was obtained. However this field is not Lorentz-covariant.

The same field follows from an attempt [10] to derive, at the classical level, the pure spinor formulation from a (suitably gauge-fixed and twisted) superembedding approach. In this approach the field is the twisted current of one of the two world-sheet (w.s.) supersymmetries whereas the integrand of the BRST charge is the twisted current of the other supersymmetry, suggesting an topological origin of the pure spinor approach.

This field turns out to be proportional to the quantity where is a constant pure spinor, such that where is a covariant, spinor-like compound field, so that is not only Lorentz non-covariant but also singular at .

A way to overcome the problem of the non-covariance and singular nature of
was given
in [14] where a recipe to compute higher loop amplitudes was
proposed, in terms of a picture-raised field constructed with the
help of suitable covariant fields , ,
and and
some picture-changing operators and . ^{4}^{4}4The picture-lowering
operators , which are needed to absorb the zero modes of the
ghost , break the Lorentz-covariance but this breaking is
BRST trivial and then harmless.

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 to without changing the central charge and the physical mass spectrum. A remarkable thing is that, in this formalism, one can define a Lorentz-covariant 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, topological string theory. A significant improvement was obtained in [28]. Here a gauge invariant, BRST trivial regularization of the 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 to .
^{5}^{5}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 group, but not group. To be more
precise, the space of (Euclidean) pure spinors in ten dimensions has
the geometrical structure of a complex cone
[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, and in each chart the local parametrization of the pure spinor,
which breaks to , 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 purposes of handling this unavoidable non-covariance stemming from the pure spinor condition. This Y-formalism is closely related to the -formalism, but has an advantage of treating all operators in a unified way without going back to the -decomposition. It is based on writing the fundamental OPE between and in a form that involves . Strictly speaking, one needs 16, orthogonal, constant pure spinors (and 16 ) for each chart, such that 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 ghost [30]. More recently, the Y-formalism was also utilized to construct a four-dimensional pure spinor superstring [31]. The -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 field in addition to the fields , , and , which are the building blocks of the 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 field is cohomologically equivalent to the non-covariant field , 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 , , and , 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 ghost, which satisfies the defining equation . We shall also show that it is cohomologically equivalent to the non-covariant ghost (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 superstrings [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

(2.1) |

and the action

(2.2) |

where is a pure spinor

(2.3) |

This action is manifestly invariant under (global) super-Poincaré transformations. It is easily shown that the action is also invariant under the BRST transformation generated by the BRST charge which is nilpotent owing to the pure spinor condition (2.3). Notice that in order to use as BRST charge it is implicit that the pure spinor condition is required to vanish in a strong sense.

Moreover, the action is invariant under the -symmetry

(2.4) |

where are local gauge parameters. At the classical level the ghost current is

(2.5) |

and the Lorentz current for the ghost sector is given by

(2.6) |

which together with are the
only super-Poincaré covariant bilinear fields involving and
gauge invariant under the -symmetry.
From the field equations it follows that , , and
are holomorphic fields. At the quantum level, one obtains
the following OPE’s ^{6}^{6}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.
involving the superspace coordinates
and their super-Poincaré covariant momenta :

(2.7) |

so that

(2.8) |

where

(2.9) |

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 and . The reason is that since the pure spinor condition must vanish identically, not all the components of are independent: solving the condition, five of them are expressed nonlinearly in terms of the others. Accordingly, five components of 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

(2.11) |

where is a constant pure spinor . Then it is useful to define the projector

(2.12) |

which, since , projects on a 5 dimensional subspace of the 16 dimensional spinor space in ten dimensions. The orthogonal projector is . Now the pure spinor condition implies

(2.13) |

Since 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 and :

(2.14) |

It follows from Eq. (2.14) that the OPE between and the pure spinor condition vanishes identically. Moreover, the BRST charge is then strictly nilpotent even acting on . It is useful to notice that, with the help of the projector , one can obtain a non-covariant but gauge-invariant antighost defined as

(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 -dependent correction terms, those are

(2.16) | |||||

(2.17) |

(2.18) |

Then the OPE’s among , and read

(2.19) |

(2.20) |

(2.21) |

(2.22) |

(2.23) |

(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 , and are singular at
but the opposite is in fact true: it is clear from
Eqs. (2.19)-(2.24)
that the -dependent correction terms have just the rôle of cancelling
the singularites which are present in the operators , and
, owing to the singular nature of the OPE (2.14)
between and .
^{7}^{7}7As anticipated in the notation , we will append a suffix
when we refer to compound fields at the classical level, that is, given
in terms of , and , and we will reserve the notation
without suffix in denoting the corresponding quantities at the
quantum level, given in terms of , and .

It will be convenient to rewrite (2.17), (2.18) and as

(2.25) |

(2.26) |

(2.27) |

where we have introduced the quantity

(2.28) |

The Y-formalism explained thus far is also useful to deal with the field which plays an important role in computing higher loop amplitudes. Its main property is

(2.29) |

where is the stress energy tensor. Since in the pure spinor formulation the reparametrization ghosts do not exist, must be a composite field. Moreover, since the ghost has ghost number and the covariant fields, which include and are gauge invariant under the -symmetry, always have ghost number zero or positive, one must use (which also has ghost number ) to construct the ghost. Therefore is not super-Poincaré invariant. The 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

(2.30) |

where

(2.31) |

The last equality in (2.30) follows from the identity (A.3). The expression of at the quantum level will be derived in section 5.

The non-covariance of is not dangerous since, as we shall show in section 5, the Lorentz variation of (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 , and , it has a true singularity at of the form . The point is that there exists an operator , singular with a pole at , such that and the cohomology would become trivial if this operator is allowed in the Hilbert space, since for any closed operator , . Then, for consistency, operators singular at must be excluded from the Hilbert space.

## 3 Fundamental operators and normal-ordering effects

When we attempt to construct a ghost covariantly, either a picture-raised ghost [14, 30] or a covariant ghost in the framework of the non-minimal approach [27], we encounter several fundamental operators, , , and [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 , the totally antisymmetrized operators , and are the more fundamental objects and are of particular interest since they are involved in the construction of the field in the non-minimal formulation. At the classical level, is defined in (2.31) and , and are given by

(3.1) |

They satisfy the following recursive relations:

(3.2) |

which one can verify easily. The full fields , and , which are involved in the construction of the picture-raised 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

(3.3) |

where the dots denote ”-traceless terms”, i.e. terms that vanish if saturated with a between two adjacent indices. The fields , and are defined modulo -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 and we shall adopt in this paper the generalized normal-ordering prescription, denoted by 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

(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, means .

A different prescription denoted as , that we shall call ”improved”, consists in subtracting the full contraction , 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

is obtained from (2.31) by replacing and with and as defined in Eqs. (2.17) and (2.18) and adding a normal-ordering term parametrized by a constant

(3.5) | |||||

The constant will be determined from the requirement that should be a primary field of conformal weight . Then we have to compute the OPE . The three terms , and are all products of two operators of conformal weight so that their OPE’s with the stress energy tensor can be easily calculated. One finds that only is a primary field. has a triple pole with residuum and has a triple pole with residuum . Moreover, the normal-ordering term also has a triple pole with residuum . Therefore, putting them together, one has

(3.6) |

Hence, the requirement that must be a primary field of conformal weight is satisfied when we select the constant to be .

In spite of the appearance, it turns out that this figure is in agreement with the result of [14] where the value is indicated as the coefficient in front of the normal-ordering term in . 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 and , there appears a difference in . Indeed, since

(3.7) | |||||

we obtain

(3.8) |

Substituting this result into Eq. (3.5), setting , we have

(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 . 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 and and use repeatedly the rearrangement theorem, reviewed in Appendix B, in order to recover the operator . The details of this calculation are presented in Appendix C. As expected from the covariance of , the terms involving , coming from the rearrangement procedure, cancel exactly those coming from the -dependent correction terms of the operators and (see (2.17) and (2.18)) present in the definition of . The final result is

(3.10) |

The normal-ordering term in (3.10) might appear to be strange at first sight, but it is indeed quite reasonable. The point is that it is not but that is a primary field of conformal weight 2 when we take account of the normal-ordering effects. In fact, since

(3.11) |

has a triple pole with residuum , and has the same triple pole, it follows that

(3.12) |

is a BRST-closed primary operator of conformal weight . From now on, it is convenient to define

(3.13) |

so that (3.10) becomes

(3.14) |

Now we would like to study the operator , that is expected to arise in the quantum counterpart of the second recursive relations in (3.2). As before, is not primary since is different from zero. Indeed,

(3.15) |

where

(3.16) |

Note that since is also primary, there is an ambiguity in defining a primary operator, say , associated to . Given (3.16), for the symmetric one, one has

(3.17) |

while, for the antisymmetric one, one has

(3.18) |

Let us remark that is not BRST-closed. Indeed . Whereas , one has . Therefore the requirement that and are BRST-closed implies and so that

(3.19) |

(3.20) |

### 3.2

A minimal choice for is

(3.21) |

where

(3.22) |

(3.23) |

First, we shall evaluate in order to fix the normal-ordering term. We can easily show that and the first term in are primary fields whereas and have a triple pole with residua and , respectively. Thus, we obtain

(3.24) |

thereby taking makes a primary field of conformal weight . This value agrees with the value in the Berkovits’ paper [14]. Next, we wish to evaluate :

(3.25) | |||||

and

(3.26) | |||||

Then, after some algebra and taking into account the normal-ordering terms by the rearrangement formula, we get for the symmetric part of

(3.27) |

and for the more interesting antisymmetric part

(3.28) |

in agreement with (3.19) and (3.20). Notice that the -dependent contributions coming from rearrangement theorem cancel exactly those coming from the definitions (2.17) and (2.18) of and (For details see Appendix C).

Since the term in (3.27) is the BRST variation of , (3.27) can be rewritten as

(3.29) |

where we have defined as .

Now let us consider the composite operator . Since has conformal weight but its contraction with does not vanish, one can expect that is not primary. Actually, using the fact

(3.30) |

with being given by

(3.31) |

it turns out that a primary field of conformal weight related to is

(3.32) |

Again there is an arbitrariness in choosing the primary field related to since is primary.

As in previous cases we are especially interested in the antisymmetric part of . Since, in , a field which is totally antisymmetric in its three, spinor-like indices contains only the irreducible representation (irrep.) and in Eq. (3.31) does not contain such an irrep., it follows that

(3.33) |

so that simply becomes

(3.34) |

From Eqs. (3.28), (3.15) and (3.16), it is then easy to show that is BRST-closed. Indeed, one finds

(3.35) | |||||

### 3.3

A covariant expression of is

(3.36) | |||||

whereas the totally antisymmetric part is given by

(3.37) |

The term including a constant describes the normal-ordering contribution. As before, we will calculate in order to fix the normal-ordering term. One finds that all the terms are primary with conformal weight , except which have triple poles in their OPE’s with . In fact, and

Therefore, one obtains

(3.39) |

so that the condition of a primary operator of conformal weight requires us to take , which is a new result.

As for , we will limit ourselves to considering only the antisymmetric part of