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

TL;DR

This paper develops the Y-formalism for non-minimal pure spinor superstring quantization, explicitly constructs the quantum $b$ ghost, and demonstrates its cohomological equivalence to the non-covariant version.

Contribution

It introduces the Y-formalism into the non-minimal pure spinor framework and explicitly constructs the quantum $b$ ghost, advancing the understanding of superstring quantization.

Findings

Explicit quantum $b$ ghost construction in non-minimal formalism

Demonstration of cohomological equivalence to non-covariant $b$ ghost

Clarification of operator relations at the quantum level

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 $b$ 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.

Equations

$$Q=\oint dz{\lambda }^{\alpha }{d}_{\alpha },$$

$$I=\int d^{2}z(\frac{1}{2}\partial X^a\bar{\partial }{X}_a+p_{\alpha }\bar{\partial }{\theta }^{\alpha }-{\omega }_{\alpha }\bar{\partial }{\lambda }^{\alpha }),$$

$$\lambda {\Gamma }^a\lambda =0.$$

$$\delta {\omega }_{\alpha }={\Lambda }_a({\Gamma }^a\lambda {)}_{\alpha },$$

$$J_0=\omega \lambda ,$$

$$N_0^{ab}=\frac{1}{2}\omega {\Gamma }^{ab}\lambda ,$$

$$<X^a(y)X^b(z)>$$

$$<p_{\alpha }(y){\theta }^{\beta }(z)>$$

$$<d_{\alpha }(y)d_{\beta }(z)>$$

$$<d_{\alpha }(y){\Pi }^a(z)>$$

$$d_{\alpha }$$

$${\Pi }^a$$

$${\bar{\Pi }}^a$$

$$Y_{\alpha }=\frac{v_{\alpha }}{v\lambda },$$

$$Y_{\alpha }{\lambda }^{\alpha }=1,$$

$$K_{\alpha }{}^{\beta }=\frac{1}{2}({\Gamma }^a\lambda {)}_{\alpha }(Y{\Gamma }_a{)}^{\beta },$$

$${\lambda }^{\alpha }{K}_{\alpha }{}^{\beta }=0.$$

$$<{\omega }_{\alpha }(y){\lambda }^{\beta }(z)>=\frac{1}{y-z}({\delta }_{\alpha }^{\beta }-K_{\alpha }{}^{\beta }(z)).$$

$${\tilde{\omega }}_{\alpha }=(1-K{)}_{\alpha }{}^{\beta }{\omega }_{\beta }.$$

$$T$$

$$\displaystyle N^{ab}=\frac{1}{2}\Big{[}\omega\Gamma^{ab}\lambda-\frac{3}{2}\partial Y\Gamma^{ab}\lambda-2Y\Gamma^{ab}\partial\lambda\Big{]},$$

$$J=\omega \lambda +\frac{7}{2}Y\partial \lambda .$$

$$<T(y)T(z)>=\frac{2}{(y-z{)}^2}T(z)+\frac{1}{y-z}\partial T(z),$$

$$<T(y)J(z)>=\frac{8}{(y-z{)}^3}+\frac{1}{(y-z{)}^2}J(z)+\frac{1}{y-z}\partial J(z),$$

$$<T(y)N^{ab}(z)>=\frac{1}{(y-z{)}^2}{N}^{ab}(z)+\frac{1}{y-z}\partial N^{ab}(z),$$

$$<J(y)J(z)>=-\frac{4}{(y-z{)}^2},$$

$$<J(y)N^{ab}(z)>=0,$$

$$<N^{ab}(y)N^{cd}(z)>=-\frac{3}{(y-z{)}^2}{\eta }^{d[a}{\eta }^{b]c}-\frac{1}{y-z}({\eta }^{a[c}{N}^{d]b}-{\eta }^{b[c}{N}^{d]a}),$$

$$N^{ab}=\frac{1}{2}[\Omega {\Gamma }^{ab}\lambda -2Y{\Gamma }^{ab}\partial \lambda ],$$

$$J=\Omega \lambda +2Y\partial \lambda ,$$

$$T_{\lambda }=\Omega \partial \lambda +3\partial Y\partial \lambda +\frac{3}{2}Y{\partial }^2\lambda ,$$

$${\Omega }_{\alpha }={\omega }_{\alpha }-\frac{3}{2}\partial Y_{\alpha }.$$

Full text

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 $X^{a}$ and their fermionic partners $\theta^{\alpha}$ with their conjugate momenta $p_{\alpha}$, plus an action for the bosonic ghosts $\lambda^{\alpha}$ and their conjugate momenta $\omega_{\alpha}$, where $\lambda^{\alpha}$ satisfy the ”pure spinor constraint” $\lambda\Gamma^{a}\lambda=0$. The $\omega-\lambda$ 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. 333Alternative formalisms to remove the constraint were proposed in [25, 26].

Another key ingredient in the pure spinor formulation is provided by the BRST charge $Q=\oint\lambda^{\alpha}d_{\alpha}$ where $d_{\alpha}\approx 0$ contains the constraints generating a fermionic $\kappa$ 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 $\lambda^{\alpha}$ satisfies the pure spinor condition. This peculiar feature is in fact expected since the constraint $d_{\alpha}\approx 0$ 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 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 $Y_{\alpha}={{v_{\alpha}}\over{v\lambda}}$ where $v_{\alpha}$ is a constant pure spinor, such that $b_{Y}=Y_{\alpha}G^{\alpha}$ where $G^{\alpha}$ is a covariant, spinor-like compound field, so that $b_{Y}$ is not only Lorentz non-covariant but also singular at $v\lambda=0$.

A way to overcome the problem of the non-covariance and singular nature of $b_{Y}$ 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 $G^{\alpha}$, $H^{\alpha\beta}$, $K^{\alpha\beta\gamma}$ and $L^{\alpha\beta\gamma\delta}$ and some picture-changing operators $Z^{\prime}s$ and $Y^{\prime}s$. 444The picture-lowering operators $Y_{C}$, which are needed to absorb the zero modes of the ghost $\lambda^{\alpha}$, 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 $-8$ 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 $U(5)$. 555In 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. To be more precise, the space of (Euclidean) pure spinors in ten dimensions has the geometrical structure of a complex cone $Q=\frac{SO(10)}{U(5)}$ [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, $U^{(\alpha)},(\alpha)=1,\cdots,16$ 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 purposes 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 the fundamental OPE between $\omega$ and $\lambda$ in a form that involves $Y_{\alpha}=\frac{v_{\alpha}}{v\lambda}$. Strictly speaking, one needs 16, orthogonal, constant pure spinors $v^{(\alpha)}$ (and 16 $Y^{(\alpha)}$) for each chart, such that $U^{(\alpha)}$ $(v^{(\alpha)}\lambda)\neq 0$ 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 $G^{\alpha}$, $H^{\alpha\beta}$, $K^{\alpha\beta\gamma}$ and $L^{\alpha\beta\gamma\delta}$, 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 $b_{Y}$, 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 $G^{\alpha}$, $H^{\alpha\beta}$, $K^{\alpha\beta\gamma}$ and $L^{\alpha\beta\gamma\delta}$, 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 $b_{Y}$ (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

[TABLE]

and the action

[TABLE]

where $\lambda$ is a pure spinor

[TABLE]

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 $\omega$-symmetry

[TABLE]

where $\Lambda^{a}$ are local gauge parameters. At the classical level the ghost current is

[TABLE]

and the Lorentz current for the ghost sector is given by

[TABLE]

which together with $T_{0\lambda}=\omega\partial\lambda$ are the only super-Poincaré covariant bilinear fields involving $\omega$ and gauge invariant under the $\omega$-symmetry. From the field equations it follows that $p$, $\theta$, $\omega$ and $\lambda$ are holomorphic fields. At the quantum level, one obtains the following OPE’s 666According 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 $Z^{M}=(X^{a},\theta^{\alpha})$ and their super-Poincaré covariant momenta $P_{M}=(\Pi_{a},p_{\alpha})$:

[TABLE]

so that

[TABLE]

where

[TABLE]

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

This problem is nicely resolved by introducing the Y-formalism. Let us first define the non-covariant object

[TABLE]

such that

[TABLE]

where $v_{\alpha}$ is a constant pure spinor $Y\Gamma^{a}Y=0$. Then it is useful to define the projector

[TABLE]

which, since $TrK=5$, projects on a 5 dimensional subspace of the 16 dimensional spinor space in ten dimensions. The orthogonal projector is $(1-K)_{\alpha}\ ^{\beta}$. Now the pure spinor condition implies

[TABLE]

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 $\omega$ and $\lambda$:

[TABLE]

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

[TABLE]

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

[TABLE]

[TABLE]

[TABLE]

Then the OPE’s among $T$, $N^{ab}$ and $J$ read

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

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 $J$, $N^{ab}$ and $T$ are singular at $v\lambda=0$ 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 $T_{0}$, $N_{0}^{ab}$ and $J_{0}$, owing to the singular nature of the OPE (2.14) between $\omega$ and $\lambda$. 777As 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 $T_{0}$, $N_{0}^{ab}$ and $J_{0}$, and we will reserve the notation without suffix $"0"$ in denoting the corresponding quantities at the quantum level, given in terms of $T$, $N^{ab}$ and $J$.

It will be convenient to rewrite (2.17), (2.18) and $T_{\lambda}$ as

[TABLE]

[TABLE]

[TABLE]

where we have introduced the quantity

[TABLE]

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

[TABLE]

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 $\omega_{\alpha}$ and are gauge invariant under the $\omega$-symmetry, always have ghost number zero or positive, one must use $Y_{\alpha}$ (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

[TABLE]

where

[TABLE]

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

The non-covariance of $b_{Y}$ is not dangerous since, as we shall show in section 5, the Lorentz variation of $b_{Y}$ (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 $T$, $N^{ab}$ and $J$, it has a true singularity at $v\lambda=0$ of the form $(v\lambda)^{-1}$. The point is that there exists an operator $\xi=Y\theta$, singular with a pole at $v\lambda\rightarrow 0$, such that $\{Q,\xi\}=1$ and the cohomology would become trivial if this operator is allowed in the Hilbert space, since for any closed operator $V$, $V=\{Q,\xi V\}$. Then, for consistency, operators singular at $v\lambda\rightarrow 0$ 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, $G^{\alpha}$, $H^{\alpha\beta}$, $K^{\alpha\beta\gamma}$ and $L^{\alpha\beta\gamma\delta}$ [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 $G^{\alpha}$, the totally antisymmetrized operators $H^{[\alpha\beta]}$, $K^{[\alpha\beta\gamma]}$ and $L^{[\alpha\beta\gamma\delta]}$ 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, $G^{\alpha}$ is defined in (2.31) and $H^{[\alpha\beta]}$, $K^{[\alpha\beta\gamma]}$ and $L^{[\alpha\beta\gamma\delta]}$ are given by

[TABLE]

They satisfy the following recursive relations:

[TABLE]

which one can verify easily. The full fields $H_{0}^{\alpha\beta}$, $K_{0}^{\alpha\beta\gamma}$ and $L_{0}^{\alpha\beta\gamma\delta}$, 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

[TABLE]

where the dots denote ”$\Gamma_{1}$-traceless terms”, i.e. terms that vanish if saturated with a $\Gamma_{a}^{\alpha_{i}\alpha_{i+1}}$ between two adjacent indices. The fields $H_{0}^{\alpha\beta}$, $K_{0}^{\alpha\beta\gamma}$ and $L_{0}^{\alpha\beta\gamma\delta}$ are defined modulo $\Gamma_{1}$-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

[TABLE]

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, $A_{1}A_{2}A_{3}...A_{n}$ means $(A_{1}(A_{2}(A_{3}(\cdots A_{n})\cdots)))$.

A different prescription denoted as $:AB:$, that we shall call ”improved”, consists in subtracting the full contraction $<A(y)B(z)>$