Light-cone Gauge NSR Strings in Noncritical Dimensions II —Ramond Sector—
Nobuyuki Ishibashi***e-mail: firstname.lastname@example.org and Koichi Murakami†††e-mail: email@example.com
Institute of Physics, University of Tsukuba,
Tsukuba, Ibaraki 305-8571, Japan
Okayama Institute for Quantum Physics,
Kyoyama 1-9-1, Kita-ku, Okayama 700-0015, Japan
Light-cone gauge superstring theory in noncritical dimensions corresponds to a worldsheet theory with nonstandard longitudinal part in the conformal gauge. The longitudinal part of the worldsheet theory is a superconformal field theory called CFT. We show that the CFT combined with the super-reparametrization ghost system can be described by free variables. It is possible to express the correlation functions in terms of these free variables. Bosonizing the free variables, we construct the spin fields and BRST invariant vertex operators for the Ramond sector in the conformal gauge formulation. By using these vertex operators, we can rewrite the tree amplitudes of the noncritical light-cone gauge string field theory, with external lines in the (R,R) sector as well as those in the (NS,NS) sector, in a BRST invariant way.
The light-cone gauge string field theory [1, 2, 3, 4, 5, 6] takes a simple form and it can therefore be a very useful tool to study string theory. Being a gauge fixed theory, it can be formulated in noncritical spacetime dimensions. In the conformal gauge, such noncritical string theories correspond to worldsheet theories with nonstandard longitudinal part. In our previous works [7, 8], we have studied the longitudinal part of the worldsheet theory which is an interacting CFT called CFT. It has the right value of the Virasoro central charge, so that one can construct a nilpotent BRST charge combined with the transverse part and the reparametrization ghosts.
In the conformal gauge formulation, the amplitudes can be calculated in a BRST invariant manner. In Ref.  we have shown that the tree level amplitudes in the light-cone gauge coincide with the BRST invariant ones in the conformal gauge, in the case of the bosonic noncritical strings. For superstrings, the equivalence of the amplitudes in the two gauges has been shown for the cases where all the external lines are in the (NS,NS) sector111In our previous works and in this work, we discuss closed strings. .
We would like to extend this analysis into the case in which external lines in the Ramond sector are involved. In the conformal gauge formulation, the vertex operators corresponding to the external lines in the Ramond sector should involve the spin fields in the CFT. Since the CFT is an interacting theory, it is not straightforward to construct spin fields. In this paper, we formulate a free field description of the CFT consisting of the CFT and the reparametrization ghosts. Namely, we construct free variables which can be expressed in terms of and the ghosts. We provide a formula to express the correlation functions of this interacting CFT in terms of these free variables. Bosonizing the free variables, we define the spin fields and thereby construct the vertex operators in the Ramond sector.
In the conformal gauge, the amplitudes can be expressed by the vertex operators thus constructed. It turns out that the closed superstring theory in noncritical dimensions generically does not include spacetime fermions. We show that the tree amplitudes with not only external lines in the (NS,NS) sector but also those in the (R,R) sector can be written by using the vertex operators.
This paper is organized as follows. In section 2, we consider the system of the bosonic CFT combined with the reparametrization ghosts and construct free variables. We show how the correlation functions on the complex plane can be expressed by those of the free variables. In section 3, we supersymmetrize the analyses in section 2 and formulate the free field description of the supersymmetric CFT. In section 4, we first study how the BRST invariant vertex operators in the Neveu-Schwarz sector can be described in terms of free variables obtained in section 3. Then we construct those in the Ramond sector, using the free variables. In section 5, we show that the tree amplitudes involving external lines in the (R,R) and the (NS,NS) sectors of the noncritical strings can be expressed in a BRST invariant way using the BRST invariant vertex operators. Section 6 is devoted to conclusions and discussions. In appendix A, we explain some details of the action for the strings in the (R,R) and the (NS,NS) sectors of the light-cone gauge string field theory in noncritical dimensions. In appendix B, we present a proof of a relation which we use in section 5.
2 Free variables: bosonic case
As a warm-up, we would like to present the free field description for bosonic CFT formulated in Ref.  and show how the correlation functions are expressed by using the free variables.
2.1 Bosonic Cft
In the conformal gauge, the longitudinal part of the worldsheet theory for the noncritical light-cone gauge string theory is described by a conformal field theory with the energy-momentum tensor
is the Schwarzian derivative.
Such a conformal field theory can be studied using the path integral formalism . In order to make the theory well-defined, we always consider the situations where the vertex operators of the form are inserted so that has an expectation value and it is invertible except for sporadic points on the worldsheet. Indeed, for a functional of , one can calculate the correlation function with the insertion on the complex plane as
Thus one can see that acquires an expectation value . The expectation value of is proportional to . has poles at and zeros at . coincides with the Mandelstam mapping of a tree light-cone diagram for strings and are the interaction points.
The variables can be shown to satisfy the OPE’s
where denotes the left-moving part of . Expanding the right hand side of the third equation in terms of with the assumption , one gets the form of the OPE given in Ref. . Using these OPE’s, one can show that the energy-momentum tensor (2.1) satisfies the Virasoro algebra with central charge . Thus, with the reparametrization ghosts and the transverse part, the worldsheet theory becomes a CFT with the total central charge .
2.2 Free fields
Let us consider a 2D CFT which consists of the CFT and the system of reparametrization ghosts . One can show that this theory can be described by free variables . Free variables , are defined as
The OPE’s between can be derived from the OPE’s of and one can see that they are free variables. It is straightforward to show that the energy-momentum tensor of the system
can be written as
in the form of the energy-momentum tensor for the free fields , . The fields are with conformal weight respectively. It is also easy to express in terms of the free variables.
2.3 Correlation functions
Since one can express all the fields in the theory in terms of the free variables and vice versa, it should be possible to describe the theory using these free variables. Let
denote the correlation function on the complex plane in the CFT we are considering. As we mentioned above, in the CFT, we are mainly interested in the the correlation functions with insertions of . In our setup, the correlation functions to be considered are of the form
where and are local operators made from and their derivatives. is inserted to soak up the ghost zero modes.
The correlation function (2.11) should be expressed by using the free variables. Let us define the correlation function for the free theory on the complex plane as
Naively, one might expect that the correlation function (2.11) should be expressed in terms of the free variables as
on the right hand side of which , and are considered to be expressed by the free variables using the relations (2.6). Eq.(2.13) would hold if the relations (2.6) were not singular anywhere on the complex plane. However, if the expectation value of has zeros and poles, the relations (2.6) are not well-defined at these points. Then we need to modify eq.(2.13) and insert operators at these points on the right hand side.
2.4 Operator insertions
The necessity of such insertions can be seen by considering the case where all the do not involve derivatives of in eq.(2.13). Because of eq.(2.3), in the correlation function can be replaced by its expectation value in such a case.
If is replaced by , the relations between the ghost variables are the ones which were studied in Refs. [10, 11, 12]. They showed that the correlation functions of can be expressed by those of with extra operator insertions at . For example, if , are regular at ,222Here we assume in eq.(2.13) the generic configuration in which . The special cases where coincides with one of these points are realized as a limit of the generic ones. the relation (2.6) implies that , are singular at , because . One can see
for . Such singularities are induced by the insertions , where is defined so that . Therefore the correlation functions of with no insertions at should correspond to those of with insertions of . Thus we can see that eq.(2.13) cannot be true as it is. It should at least be modified as
in order to be consistent with the singularities of the ghost variables.
In our case, is dynamical and eq.(2.15) is still inconsistent. If one inserts the energy-momentum tensor into the correlation functions in eq.(2.15), the left hand side should be regular at  but the right hand side is not because of . Instead of , we therefore need to insert an operator which is conformal invariant and induces the same singularities for as . We find that
has such properties. Indeed, replacing by its expectation value , one can see that is equivalent to
and behave as eq.(2.14) in the presence of . Moreover, the OPE with the energy-momentum tensor can be calculated as
On the assumption that one can replace by its expectation value333Here we have also assumed , which is generically true. implies that coincides with another interaction point . Such cases are considered as a limit of the generic cases, in which we should insert and at the same point.
Therefore seems to have the right properties to be inserted in the free field expression. We will prove the fact that the OPE becomes regular without any assumptions, in the next subsection.
With similar reasonings, one can deduce the singular behaviors of at the points and as well, from which one can infer the ghost operators to be inserted at these points. For , we can see that should be inserted. Combined with the insertion and the operator to be introduced in eq.(2.23) at , this ghost operator reproduces the correct OPE with the energy-momentum tensor. For , the ghosts should behave as
and one can see that , which is of weight , should be inserted. We can define a conformal invariant combination,
to implement such an insertion.
We should also take care of the singular behavior of in the CFT. From the results of Ref. , one can see that possesses logarithmic singularities at and , where is the interaction point at which the th string interacts. Therefore it is necessary to insert
The latter should be made into a conformal invariant combination
The conformal invariance of and can be proved in a similar way to that of in eq.(2.20) on the assumption that can be replaced by its expectation value . In the next subsection, we will prove that this assumption is not necessary as in the case.
2.5 Correlation functions in terms of the free variables
We have shown what kind of operator insertions are necessary. We would like to show that they are actually enough and the correlation functions of the system can be expressed in terms of the free variables only with the insertions obtained above. To be precise, we will prove
where is a numerical constant.
Let us first consider the simplest case and check if
The left hand side was evaluated to be up to a constant multiplicative factor in Ref. , where is defined as
Here is a Neumann coefficient given by
It is easy to calculate the free field correlation function on the right hand side, and we find that this also becomes up to a constant multiplicative factor, using the identity,
Thus eq.(2.26) holds.
Let us then consider the next simplest case where all the do not include derivatives of . In this case, in the correlation function can be replaced by its expectation value . Therefore the problem is reduced to the case where are made of ghost fields. Since the operator insertions on the right hand side was fixed so that the ghost variables have the same singularity structure as the quantity on the left hand side, it is easy to see
On the other hand, we have
Using eq.(2.26) we can show that these two are proportional to each other.
Now let us turn to the cases where derivatives of are included in . Once eq.(LABEL:eq:general) is proved for made of the derivatives of and the ghosts, one can get the free field expression of the correlation functions with insertions by differentiating eq.(LABEL:eq:general) with respect to . As an example, let us consider the correlation function with one insertion of
It can be expressed in terms of the one with no insertions. One can show
Here the factors are included so that the limit becomes smooth. On the right hand side, we have inserted sources and with , to generate the insertions of . With these sources, has the expectation value , where . has interaction points. In the limit , two of them, which we denote by and , tend to and , and the other ’s go to the interaction points of , which are denoted with the same subscripts .
Now let us rewrite the expression on the right hand side using the free variables. By making use of eq.(2.26) with replaced by , we have
where and are respectively and with being replaced by . Since and in the limit which we should eventually take, we get some operator insertions at and as a result. These insertions should correspond to . The behaviors of , and in the limit are given by eqs.(D.1), (D.2) and (D.4) of Ref.  respectively, where is a Neumann coefficient corresponding to . In the free field correlation function on the right hand side of eq.(2.34), ’s appear only in the form of the vertex operator . Therefore one can replace by its expectation value and vice versa. Using these facts, we can show that in the limit ,
Substituting eqs.(2.35) and (2.36) into eq.(2.34), we obtain444Here (and in eqs.(LABEL:eq:general)(2.41)) which appears in the definitions of are taken to be the interaction point which correspond to the Mandelstam mapping .
It is straightforward to prove eq.(LABEL:eq:general) for more general insertions. The key relation is eq.(2.35), which is valid in the free field correlation functions in which ’s appear only in the form of the vertex operator . For example, let us consider the correlation functions with two insertions of
Using eq.(2.5), the right hand side is expressed as