# Operator Formulation of Green-Schwarz Superstring

in the Semi-Light-Cone Conformal Gauge

###### Abstract

In this article we present a comprehensive account of the operator formulation of the Green-Schwarz superstring in the semi-light-cone (SLC) gauge, where the worldsheet conformal invariance is preserved. Starting from the basic action, we systematically study the symmetry structure of the theory in the SLC gauge both in the Lagrangian and the phase space formulations. After quantizing the theory in the latter formulation we construct the quantum Virasoro and the super-Poincaré generators and clarify the closure properties of these symmetry algebras. Then by making full use of this knowledge we will be able to construct the BRST-invariant vertex operators which describe the emission and the absorption of the massless quanta and show that they form the appropriate representation of the quantum symmetry algebras. Furthermore, we will construct an exact quantum similarity transformation which connects the SLC gauge and the familiar light-cone (LC) gauge. As an application BRST-invariant DDF operators in the SLC gauge are obtained starting from the corresponding physical oscillators in the LC gauge.

[4cm]UT-Komaba 10-5

TU-872

August, 2010

## 1 Introduction

With the advent of the idea of the D-branes[1] and the subsequent discovery of the AdS/CFT correspondence,[2, 3, 4] the choice of the worldsheet formalism of the superstring theory underwent a notable change. The Lorentz-covariant Ramond-Neveu-Schwarz (RNS) formalism,[5, 6] which had been dominating over the alternative Green-Schwarz (GS) formalism,[7, 8] was to be used less frequently. The reason is that in the RNS formalism the spacetime spinors are described by the composite spin fields which are not easy to handle and hence this formalism is not so suitable for the description of the Ramond-Ramond (RR) bispinor fields characteristically produced by the D-branes. It was thus inevitaible that the first formulation of the superstring in the background with the RR flux[9] was made in the GS formalism, where the target space spinor fields are among the basic variables. Despite its non-covariance at the quantum level, the GS formalism regained its raisons d’être.

It was not long before another scheme containing fundamental spinor variables, called pure spinor (PS) formalism, was invented.[10] The great advantage of this formalism is that, although rigorously speaking the super-Poincaré covariance is broken by the underlying quantization procedure, the rules for the computations of the amplitudes can be made completely covariant. Moreover, the rules for the multiloop amplitudes look quite similar to those of the bosonic or topological string.[11] Consequently they are formally much simpler than those of the RNS formalism and with judicious regularization procedure a number of powerful results have been obtained.[12, 13] Also since the target space spinor variables are built in, as in the GS formalism, it is suitable for the description of a superstring in curved spacetimes with RR flux relevant to AdS/CFT. In fact the action in the background was written down already in the original work[10] that introduced this formalism.

Although it has many advantages as sketched above, the PS formalism is not entirely without shortcomings. One feature is that the structure of the worldsheet conformal symmetry is not explicitly seen. The BRST operator does not contain the energy-momentum tensor and this presents a difficulty in constructing the string field theory based on this formalism. Correspondingly, the construction of the“ -ghost” is quite complex. Another problem is that due to the presence of the quadratic pure spinor constraints, the hermiticity property of the pure spinor variables is peculiar.[14] This leads to the difficulty in constructing the D-brane boundary states together with their conjugates. In the context of the study of the AdS/CFT correspondence, PS formalism so far has not been particularly useful in actually solving the quantum dynamics in the relevant curved background. For instance, the spectrum of the superstring in the plane-wave background, which was obtained exactly in the light-cone gauge GS formalism,[15] has not been reproduced in this formalism. PS formalism needs to be further developed for such purposes.

Let us now go back to the GS formalism and assess some of its features. In the past, the study and the use of the GS formalism have been made overwhelmingly in the light-cone (LC) gauge, where half of the light-cone components of the fermions are set to zero and the bosonic light-cone variable is identified with the worldsheet time. These conditions make the Lorentz invariance non-manifest and moreover break the conformal invariance. With the lack of these symmetries the computations of the amplitudes become less organized and cumbersome. This is certainly a big disadvantage of this formalism. On the other hand, GS formalism in the LC gauge deals directly with the physical degrees of freedome and is powerful in studying the physical property of the system, such as the spectrum. In this regard, we have already mentioned the celebrated exact solution of the spectrum for a string in the plane-wave background using the LC gauge,[15] which played a crucial role in the understanding of the AdS/CFT correspondence in this background.[16]

It should now be mentioned that the LC gauge is not the only gauge in which the GS string can be quantized. Although there is no way to make the Lorentz symmetry manifest in the quantum GS formalism, there exists a more symmetric gauge in which the conformal invariance can be retained. This is the so-called semi-light-cone (SLC) gauge,[17, 18, 19] where only the fermionic gauge conditions are imposed to fix the local -symmetry. As for the worldsheet reparametrization symmetry, the usual conformal gauge condition is adopted so that the Virasoro symmetry still remains and can be treated by the BRST formalism.

One of the main issues of the SLC gauge formalism when it was introduced was whether the theory suffers from conformal and related anomalies. Although early studies[17, 18, 19, 20, 21, 22] claimed that there is no anomaly, a subsequent work[23] revealed the existence of the 10-dimensional Lorentz anomaly. More complete study was made in Ref. \citenBastianelli:1990xn, Porrati:1991ts, which confirmed the result of Ref. \citenKraemmer:1989af as well as pointed out that the conformal and the related anomalies can be cancelled by adding appropriate counter terms to the action and the transformation rules.

All such studies were made in the path-integral formalism. The first operator formulation in the SLC gauge was attempted in Ref. \citenChu:1990jt. In this work, to avoid dealing with the second class constraints, the Batalin-Fradkin formalism[27, 28] was adopted, which makes use of graded brackets in the extended phase space with new fields. The BRST and the super-Poincaré operators were constructed with quantum modifications but their structures were quite complicated. Much more recently, a simpler BRST formalism for the GS superstring in the SLC gauge was introduced in Ref. \citenBerkovits:2004tw for the purpose of showing the equivalence of the PS and GS formalisms and it was subsequently utilized in some related works.[30, 31, 32] Although the work of Ref \citenBerkovits:2004tw contained a number of important ideas, it was not intended for a systematic development of the operator formalism for the GS superstring in the SLC gauge.

A brief sketch of the development of the SLC gauge formulation given above reveals that despite its long history surprisingly little has been known about its fundamental structures: Among other things, quantum symmetry structure has not been fully clarified and no vertex operators have yet been constructed. The primary purpose of the present work is to fill this gap and lay the foundation of the GS superstring in this important and unique gauge.

In particular, we will systematically develop the operator formalism, which is best suited for studying the quantum symmetry structure of the theory, starting from the basic action of the GS superstring. We will construct the quantum Virasoro and super-Poincaré generators and clarify the structure of their algebras in full detail for the first time. This knowledge in turn is indispensable for the construction of the BRST-invariant vertex operators. We will demonstrate this by constructing the vertex operators for the massless states of the open superstring in completely explicit manner. The way all the quantum symmetry algebras are realized in the space of these vertex operators is quite intricate and non-trivial.

Another new result achieved in this work is the construction of an exact quantum similarity transformation that connects the SLC gauge and the LC gauge. Among many expected applications of this mapping, in this article we will use it to construct all the so-called DDF operators,[33] which generate the BRST-invariant states in the SLC gauge, from the simple physical oscillators in the LC gauge.

As we wished to clarify the fundamental structures of the theory fully in a self-contained manner, this article has become rather long. Therefore we will now give the outline of our work in some detail so that the reader can grasp the scope of the manuscript.

We begin, in section 2, by reviewing the action and its symmetries of the type II Green-Schwarz superstring in the flat dimensional spacetime.

Then in section 3 we describe the gauge-fixing procedure for the local symmetries. To keep the worldsheet conformal invariance, the reparametrization symmetry is fixed by imposing the conformal gauge condition. As this condition is not invariant under the original local -symmetry transformation, one must redefine the -transformation by adding a judicious compensating reparametrization transformation. Subsequently, we will fix this modified -symmetry by imposing the semi-light-cone (SLC) gauge condition. This procedure in turn breaks the global super-Poincaré invariance and we must add appropriate compensating -transformations to modify the super-Poincaré transformations in order to stay in the SLC gauge. Finally we check that after all this process the conformal symmetry is still preserved.

In section 4, we develop the phase space formulation and quantize the theory. We will compare it to the canonical quantization approach and emphasize several advantages of the phase space formulation. One practical point is that in this formulation the compensating transformations, which are often complicated, are automatically taken care of by the use of the Dirac brackets. Another feature is that the phase space formulation can be useful for quantizing a non-linear system for which the complete classical solutions are not available.

Having completed all the necessary ground work, we will begin the detailed study of the quantum operator formulation of the GS superstring in SLC gauge. In section 5, we will first clarify the structure of the quantum symmetry algebras. Besides being important in its own right, this will be indispensable for the construction of the vertex operators. The characteristic feature of the SLC gauge is that in contrast to the full light-cone gauge the conformal symmetry is retained. The corresponding Virasoro operators are constructed first at the classical level and then at the quantum level. At the quantum level, one needs to add a quantum correction[29] in order to cancel the conformal anomaly. The nilpotent BRST operator is obtained in the usual way with this modification. We then discover that this correction, the origin of which was rather mysterious previously,[29] naturally shows itself up as one computes the quantum supersymmetry algebra. The algebra closes only up to a BRST-exact term, where the BRST operator automatically contains the correct quantum modification. The rest of the super-Poincaré algebra turned out to possess similar features. With a judicious quantum correction added to a part of the Lorentz generators, the algebra precisely closes up to BRST-exact terms.

We then proceed, in section 6, to the construction of the vertex operators for the massless excitations. (For simplicity we will consider the type I super-Maxwell sector.) By definition they must be BRST-invariant and form a correct representation of the quantum super-Poincaré algebra established in the previous section, up to BRST-exact expressions. The computations were quite involved, requiring various non-trivial -matrix and spinorial identities, but we could obtain the desired vertex operators which satisfy all the requirements consistently.

In order to deepen our understanding of the GS superstring in the SLC gauge further, we will study in section 7 its connection to the much-studied formulation in the usual (full) light-cone gauge. We will be able to do this in the most direct way, namely by constructing an explicit quantum similarity transformation which connects the operators in the two formulations. The basic method used is the one in Ref. \citenAisaka:2004ga, which was developed to relate Green-Schwarz and an extended version of the pure spinor superstring. Although the presence of the extra term in the BRST operator demanded additional new ideas and observations, we have obtained the desired similarity transformation exactly. As an application, we have been able to construct the BRST-invariant DDF operators in the SLC gauge from the physical operators in the light-cone gauge by performing this similarity transformation.

Finally, section 8 will be devoted to discussions, where we examine some problems which are not solved in this work and indicate future directions. Several appendices are provided to describe our conventions and supply additional technical details.

## 2 Classical action and its symmetries

We begin by reviewing the classical action for the Green-Schwarz superstring and its symmetries before gauge-fixing,[7, 8] mainly to set up our notations and to make this article self-contained.

### 2.1 Action

The action invariant under the super-Poincaré transformations is most easily constructed using the supercoset method.[35] It consists of the kinetic part and the Wess-Zumino (WZ) part and can be expressed compactly in terms of the worldsheet differential forms in the following way:

(1) | ||||

(2) | ||||

(3) |

Here, is the string tension^{3}^{3}3As for the fundamental
length scale, we will use the string length , related to by .
, and the 1-forms are defined as

(4) | ||||

(5) | ||||

(6) |

are the string coordinates, are the two sets of 16-component real chiral
spinors^{4}^{4}4In this article we specifically deal with the type IIB case.
Type IIA case can be easily described by adjusting certain signs. and denote the worldsheet coordinates. The convention for the spinors and the -matrices^{5}^{5}5For convenience, we use different notations for the -matrices with lower and upper indices, namely,
and . See appendix A for more details.
are elaborated in appendix A.
We take the flat target space metric to be and the signature of the worldsheet metric as .
The wedge product and the Hodge dual “” with respect to the worldsheet metric are given for the basic coordinate 1-form as

(7) | ||||

(8) |

This gives . With these formulas, the Lagrangian density can be written in terms of components as

(9) |

It is sometimes useful to note that the WZ term can be written in a slightly different form:

(10) |

This is due to the identity .

### 2.2 Symmetries of the action

The action presented above enjoys four types of symmetries, namely, the worldsheet reparametrization invariance, the target space Lorentz invariance, the supersymmetry and the symmetry. Since the first two symmetries are obvious, we will review the latter two.

The supercoset construction guarantees that the action is invariant under the global supersymmetry transformations

(11) |

where the supersymmetry(SUSY) parameters are constant real chiral spinors^{6}^{6}6We reserve the commonly used letter for the anti-chiral
components of , to appear later..
In fact since
are SUSY invariant the kinetic term is manifestly invariant even at the Lagrangian level. However, the Lagrangian for the WZ part
is not invariant (because are not invariant) and transforms into a total derivative. As this is important in deriving the Noether current, let us quickly review how this comes about.

For this purpose, it is convenient to use the form of given in (10). It is easy to see that although is not an invariant, it transforms into a total derivative as

(12) |

where . Applying this to (10) and rearranging, we readily obtain

(13) |

The first term is manifestly a total derivative. To show that the second term is also a total derivative, we need to use the well-known Fierz identity. Define

(14) | ||||

(15) |

Contracting the expression with the Fierz identity

(16) |

we get . Using this relation, the total derivative , which equals , becomes . Hence, we get a non-trivial identity

(17) |

Applying this to (13), we readily obtain

(18) | ||||

(19) | ||||

(20) |

From this result, it is easy to get the conserved SUSY Noether currents as^{7}^{7}7
Up to an overall normalization and the interchage of and , this agrees with the expression given in Ref. \citenGreen:1983sg.

(21) | ||||

(22) |

Another important symmetry is the -symmetry. The action is invariant under the off-shell symmetry transformations given by

(23) | ||||

(24) |

Here , are local fermionic parameters, which are anti-chiral spinors in spacetime and vectors on the worldsheet, and the projection operators are given by

(25) |

For invariance, the parameters must satisfy the conditions

(26) |

where .

## 3 Gauge-fixing and compensating transformation

In this section, we perform the gauge-fixing of the local symmetries. We will do this in two steps: First, we pick the conformal gauge to fix the reparametrization symmetry. Since the original -transformation acts on the worldsheet metric as in (24), we must modify the -transformation rule by adding an appropriate compensating reparametrization in order to stay in the conformal gauge. We will then fix the -symmetry by adopting the so-called semi-light-cone (SLC) gauge. As this gauge is not invariant under the supersymmetry transformation nor the Lorentz transformation, we must include suitable compensating -transformations in these transformations in order to keep the SLC gauge. Below we describe these procedures in some detail.

### 3.1 Conformal gauge fixing

As we wish to keep the worldsheet conformal invariance, we will choose the conformal gauge, where the Weyl-invariant combination takes the flat form . Since the original -transformation (24) changes this value, to remain in the conformal gauge we must make a judicious compensating reparametrization transformation so that

(27) |

holds at the conformal gauge point. Under the reparametrization transformation, are scalars while and are covariant and contravariant tensors. So they transform like

(28) | ||||

(29) |

where . As for , it transforms as transforms like . Combining these results, one easily finds that at the conformal gauge point,

(30) |

Now let us denote the -transform of at the conformal gauge point by . This quantity is symmetric and traceless. Then (27) reduces to

(31) |

Applying on both sides, we immediately obtain (where ) and hence we can solve for as

(32) |

In the present case, is given by (see (24))

(33) |

where . The conditions (26) on the parameters become

(34) |

Putting (33) into (32), we get the appropriate
compensating reparametrization transformation^{8}^{8}8In the
historic paper,[8] where this was first discussed, only the case of a very special
transformation was considered. Namely the authors imposed
the extra conditions .
In this case, the expression for simplifies to
. In general, however, one should
not (and need not) impose such dynamical restrictions on the -parameters. .
With the function obtained in (32),
the modified -transformations
for and are given by

(35) | ||||

(36) |

where denotes the original transformation (23). These transformations leave the conformal-gauge-fixed action invariant.

### 3.2 Semi-light-cone gauge fixing

#### 3.2.1 SLC gauge condition and the Lagrangian

Next, let us fix the -symmetry by imposing the SLC gauge conditions given by

(37) |

where denotes the anti-chiral components. Here and hereafter, we will often make use of the decompositions of the spinors and the -matrices. Our conventions and the properties of the -matrices are summarized in appendix A. In this gauge various terms in the action simplify considerably. First, the only non-vanishing components of are because is non-vanishing only for . This immediately leads to the formulas

(38) |

The kinetic and the WZ parts of the Lagrangian become

(39) | ||||

(40) | ||||

(41) |

Although there are still cubic terms in , the equations of motion for all the fields reduce to “free field forms”, namely , and , because the interaction terms vanish on-shell.

#### 3.2.2 Supersymmetry in the SLC gauge

Let us now begin the discussion of the modification of the symmetry transformations needed in the SLC gauge. First consider the SUSY transformations. As we need to distinguish between the chiral and anti-chiral parts, we will split the SUSY parameters as and write the SUSY transformations in the form

(42) |

Since only the anti-chiral transformations violate the gauge conditions, the compensating -transformations will involve only half of the -parameters. Indeed we will find that it is enough to keep and set to zero.

It is useful to note that in such a case we can ignore the reparametrization part of the modified transformations obtained in (35, 36) and simply use the original transformations. The reason is as follows: The expression given in (33) is made up of the expressions of the form

(43) |

This vanishes in the SLC gauge for and hence the reparametrization parameter vanishes.

Thus, the parameter for the compensating -transformation should be determined by the requirement

(44) |

where denotes the original transformation (42). Writing this out explicitly with set to zero, we obtain the equation

(45) |

Recalling that only half of are independent, as in (34), we can write this as

(46) | ||||

(47) |

where we introduced the worldsheet light-cone coordinates and the derivatives as and . Therefore, as long as do not vanish, we can solve for the parameters:

(48) | ||||

(49) |

Throughout we assume that the zero-mode part of , i.e. the momentum , is non-vanishing and define the operators such as by expanding around .

We can now write down the -SUSY transformations for the fields and , modified by the -transformations. The -transformation for consists solely of the -transformation and is given by . Substituting (48) and (49), this becomes

(50) | ||||

(51) |

As for the -transformation of , it is given by

(52) |

From the property of , the first term is non-vanihsing only for , while the second term exists only for . Using the results (50) and (51), we get

(53) |

We will end the discussion of the SUSY in the SLC gauge by giving the form of the supercharge densities in this gauge. Before gauge-fixing, they are given as the time-components of the currents (21) and (22) as

(54) | ||||

(55) |

The expressions in the conformal gauge are obtained by substituting , and . As we further impose the SLC gauge condition , the following simplifications occur. First, for any vector we have . Next, are simplified as and . Finally, one can check the useful identity . Becaue of this, all disappear and we obtain

(56) | ||||

(57) |

For later convenience, let us make the decomposition. Using , , we get

(58) | ||||

(59) | ||||

(60) | ||||

(61) |

where we defined^{9}^{9}9Once decomposed into components, the
spinor indices can be trivially raised and lowered. .
We will later quantize them in the phase space formulation and study the quantum
algebra of the supercharges.

#### 3.2.3 Lorentz symmetry in the SLC gauge

The next subject is the form of the Lorentz transformations in the SLC gauge. Before gauge-fixing, the infinitesimal Lorentz transformations for and are given by

(62) | ||||

(63) |

where are infinitesimal antisymmetric parameters and . The SLC gauge conditions are broken in general by the Lorentz transformations, which act on as . From the structure of the -matrices discussed in appendix A, it is not difficult to see that is non-vanishing only for (and of course ). Thus for such transformations, we must add compensating transformations in order to stay in the SLC gauge. It turns out that, just as in the case of the supersymmetry, one can find such transformations using only, with set to zero. Therefore, as explained around (43), the modification of the -transformation due to reparametrization can be ignored. Hence the condition to fix the compensating parameters takes the form

(64) |

where denotes the original Lorentz transformation with the parameter . Recalling and , we easily find the solution:

(65) | ||||

(66) |

Accordingly we must change the transformation for and by adding the -transformation with these field-dependent parameters.

Consider first . The direct Lorentz transformation vanishes since . The transformation for induced by the transformation is

(67) |

This can be slightly simplified by noting and ( since ). After the simplification, together with the similar result for , we obtain

(68) | ||||

(69) |

Next consider the transformation of . Since the -transformation for in the SLC gauge is , only the component is affected by this transformation since is the only non-vanishing component. For the case with the parameter , this induced piece is in fact the only contribution to , because the original Lorentz transformation vanishes. The explicit expression for is simplified by using the following relation

(70) |

where we used the Clifford algebra and the identity . In this way we obtain