NEIP-00-008

hep-th/0003180

String Theory and Noncommutative Field Theories at One Loop

Adel Bilal, Chong-Sun Chu and Rodolfo Russo

Institute of Physics, University of Neuchâtel, CH-2000 Neuchâtel, Switzerland

Adel.B

Chong-Sun.C

Rodolfo.R

Abstract

By exploiting the boundary state formalism we obtain the string correlator between two internal points on the one loop open string world-sheet in the presence of a constant background -field. From this derivation it is clear that there is an ambiguity when one tries to restrict the Green function to the boundary of the surface. We fix this ambiguity by showing that there is a unique form for the correlator between two points on the boundary which reproduces the one loop field theory results of different noncommutative field theories. In particular, we present the derivation of one loop diagrams for and scalar interactions and for Yang–Mills theory. From the -point function we are able to derive the one loop -function for noncommutative gauge theory.

## 1 Introduction

Recently, noncommutative field theory has shown up as an effective description of string theory in a certain background [1, 2, 3, 4, 5, 6]. The non-standard commutation relation among space coordinates takes the form

(1.1) |

where is an antisymmetric real constant matrix of dimension length squared. In the dual language, the algebra of functions is described by the Moyal product

(1.2) |

which is associative and noncommutative. The noncommutative nature of string theory in the presence of a non-vanishing -field was emphasized in [3, 7]. There it was shown that when quantizing an open string with the boundary condition

(1.3) |

where is the gauge invariant Born-Infeld field strength, the noncommutativity is an unavoidable feature of open string dynamics. In fact one finds that the coordinates of the open string endpoints have to satisfy the new relation (1.1) and also the commutation relations for the modes of the string expansion are modified.

Recent interest in noncommutative quantum field theory was boosted by the paper of Seiberg and Witten [6] where it is systematically shown how tree-level open string theory, in the presence of a non-zero -field and of D-branes, leads to a noncommutative quantum field theory. One of the main observations in [6] was that, in the presence of a -field, the open string world-sheet Green’s function on the boundary of the disk is modified [8]. In particular, Seiberg and Witten showed that in the limit

(1.4) |

(1.5) |

the tree level amplitudes just consist of a phase factor, which corresponds to the vertices of a noncommutative field theory. The limit is necessary if one wants to kill the string propagators to get an irreducible field theory vertex from an -point string amplitude. In fact, the basic building block in open string theory is the 3-string vertex: thus in order to get higher point interactions (e.g. , ), one may sew together a couple of -string vertices and contract some of the propagators in between. This is accomplished by (1.5). Due to some existing confusions in the literature, we feel that it is worthwhile to stress that the “contraction of propagator” (1.5) should be imposed only when necessary and should not always be understood. Indeed, for our purposes of obtaining more general field theory amplitudes, it would be wrong to always insist on this limit, as this would kill all the propagators, including the loop propagators. However the other scaling limit (1.4) is necessary to obtain a noncommutative field theory and we will refer to it as the noncommutative limit. In this paper, we will show that by taking the limit (1.4) (and (1.5) only when necessary), one can reproduce from string theory different noncommutative field theories, at the tree and one loop levels.

The noncommutative scalar [9, 10, 11, 12] and gauge theories[13, 14, 15, 16, 17, 18] have been much studied in their own right. Intriguing phenomena occur, in particular there are important distinctions between planar and nonplanar Feynman diagrams even in theories of a single scalar. Moreover nonplanar diagrams are automatically regulated by an effective UV-cutoff , where is some combination of external momenta. This implies a non-analyticity in , and an IR singularity is generated from integrating out the high momentum modes. This UV/IR-mixing has attracted quite some attention. Within quantum field theory it appears as a puzzling feature, but if we think in terms of string theory there is no natural distinction between UV and IR since high energy open string loop excitations are mapped via a modular transformation to low energy closed string ones.

It is thus quite natural to suspect that the Seiberg-Witten limit of string theory with a non-vanishing -field can be extended beyond tree-level. In fact, by now the low energy behavior () of string amplitudes is well understood also at one loop-level and has provided a reliable and flexible tool for analyzing various aspects of very different field theories. For instance, string amplitudes or string–inspired techniques were used to evaluate one–loop QCD scattering amplitudes [19, 20] (see also References therein) and renormalization constants [21, 22, 23]; graviton scattering amplitudes were computed and their relation to gauge amplitudes explored [24]; progress was made towards the extension of the method to more than one loop [25, 26, 27, 28, 29], and to off–shell amplitudes [30]. String techniques also served to stimulate the development of new techniques in field theory, that preserve some of the nice features of the string formalism [31]. Basically, the flexibility of these techniques has its root in the fact that string theory has a two-dimensional structure, describing the world-sheet dynamics beyond the usual space-time structure. If one can find a corner of string moduli space which at low energies yields the field theory under study, it is possible to use the string description to perform the calculations and thus exploit all the conformal theory features one has already studied for other problems. Thus, it is not really a surprise that field theory computations, which are largely independent of each other, appear to be related if viewed from the string theory point of view.

As already discussed above, Seiberg and Witten have pointed out a regime of string theory which at low energies is described by a non-commutative theory and thus in the spirit of the above papers it is natural to exploit string computations to derive field theory results. Specifically, we will show how the non-commutative parameter arises in the field theory limit of one-loop amplitudes of the simple open bosonic string. We would like to insist that string theory not only conceptually leads to the noncommutative quantum field theories, but it also represents a simplifying technique for the computation of perturbative amplitudes. As we will explicitly see, once we have computed the objects entering in the string master formula also in the presence of a non-vanishing -field, the one-loop diagrams of different noncommutative quantum field theories can be obtained by following the calculations performed in the commutative case; in particular, we refer to [23, 26, 28, 29]. A nice feature discussed in detail in those references is the existence of a one-to-one correspondence between Feynman diagrams and different corners of the integration region over the string parameters. It is worth to stress that this mapping is preserved in the non-commutative case and is identical to the one found for .

In order to incorporate a non-vanishing -field in one-loop string computations, the first non-trivial task is to obtain the conformal field theory propagator with on a world-sheet with topology of the annulus, rather than simply a disk. This will be done in section 2. Starting from the boundary state formalism, we discuss the possible ambiguity for the open string Green function that exists in the literature and provide an unambiguous computation to fix its form. Once this Green function is known, one can apply the techniques developed in [23, 26, 28, 29] in order to extract quantum field theory Feynman diagrams from string loop amplitudes. By means of this formalism, we compute various one-loop amplitudes in noncommutative scalar and field theories. In all cases we show that string theory exactly reproduces the previously known results obtained from quantizing the noncommutative field theory action. We then study the -gluon amplitude in noncommutative gauge theories and determine the leading and subleading singularity in . By exploiting the fact that in the field theory limit, string theory gives results in the background field method [23], we easily obtain the -function for the noncommutative gauge theory.

Note added: After completing this work and while the present paper was typed, two related papers [44, 45] appeared where also one-loop noncommutative field theory amplitudes are obtained from one-loop string theory amplitudes. However, the Green function of [44] differs from ours since the above mentioned ambiguity was differently resolved. In section 2 we will argue that to obtain the correct gluon two-point function imposes our choice for resolving the ambiguity.

##
2 One-loop open string Green function in the

presence of an -field

In this section we focus on the one-loop Green function of bosonic string theory and, in particular, we want to generalize the usual calculation to the case where a constant field is present. In fact, once the explicit form of the Green function is known, it is possible to write in a compact form a generic string amplitude with an arbitrary number of legs. The situation is thus very different from the one in field theory, where each diagram represents an independent calculation and one has always to start from the very first building blocks, i.e. the Feynman rules. As we said, this simplification is possible because string calculations rely on the world-sheet structure, which is described by a two-dimensional theory, more than on the space-time structure. In fact, the -loop bosonic string amplitude among -tachyons with inflowing momenta can be written as

(2.1) |

Here and are the normalization factors depending on the world-sheet topology and the string vertices respectively; their explicit form in terms of the dimensionless string coupling constant will be given later. The other building blocks of (2.1) have a clear geometrical interpretation: is the correlator of two world–sheet bosons located at on the boundary labeled , and at on the boundary ; is the measure of integration over the moduli space for an open Riemann surface with loops and punctures; are projective transformations which define local coordinate systems around each puncture . Here we do not give the explicit expressions of these quantities in general (see for instance [32]), but we want to stress that their definition depends only on the geometrical properties of the string world-sheet and in general on the two-dimensional conformal theory living on it. From this point of view it is natural that different computations are much more related to each other than in the usual field theory approach.

Here we want to exploit the great flexibility of this technique in order to derive the one-loop Feynman diagrams of noncommutative field theories; from the string point of view noncommutativity is easily implemented: one changes the commutation relations of the open string modes [3] or, equivalently, the boundary conditions of the open string coordinates . At the tree and one-loop level this modification basically only shows up in the Green function. This means that the definition of the field theory limit of the string master formula is not modified by the presence of ; in particular, the mapping between the corners of integration over the moduli and Feynman diagrams can be read from the calculations of the usual commutative case [23]. Since in our approach all the differences between commutative and noncommutative field theory are resumed in the string Green function, we want to derive here this key ingredient from first principle.

### 2.1 Boundary state formalism

In [8], among other things,
the tree level Green function in the presence of a constant
-field was derived by solving the defining differential equation
with the following boundary condition
^{1}^{1}1Notice that there is a factor of
different from (1.3) due to a Wick rotation on the world-sheet.

(2.2) |

where () is the derivative parallel (normal) to the world-sheet border. There is however an ambiguity in this approach: one can always add to a given solution a constant piece (i.e. independent of the punctures ) with arbitrary dependence on and on the annulus width and obtain another Green function which gives different results in the limit. As we will see, these terms play a crucial role in the field theory limit of noncommutative amplitudes, so it is important to understand the actual form of the Green function appearing in the string master formula (2.1). In order to clarify this point, we derive the Green function in the boundary state formalism using a simple trick that reduces the actual calculation to the one encountered in the usual case .

Tree level

Let us consider the correlation function of two closed string tachyons on a disk, from which one may extract the tree-level Green function by simply looking at the term with explicit dependence on both punctures

(2.3) |

This same amplitude can be calculated in the boundary state formalism. In this approach one starts from a world-sheet with the topology of the sphere and thus the string coordinates depend on two independent sets of oscillators and . Then one introduces in the amplitudes a coherent state (see [35] and References therein) which basically identifies the left and the right sector of the closed strings with the appropriate boundary conditions and thus inserts a boundary on the string world-sheet

(2.4) |

where T is the radial ordering. The in the boundary state is put there to emphasize that in order to sew to a boundary state to a given Riemann surface, it has to contain a closed string propagator [33]. At tree-level its effect in the amplitudes is just to shift the positions of the external legs, so it does not modify the form of the Green functions. Notice that in (2.3) and (2.4) we have used two different ways to label the world-sheet coordinates. This is because the two approaches naturally give rise to different parameterizations of the string world-sheet. In (2.3) the poles of open string exchanges between the two vertex operators are manifest and the fundamental region is the upper half of the complex plane. On the other hand, the boundary state calculation is written in the “closed string channel” and the world-sheet is mapped inside the disk of unit radius.

The advantage of the boundary state formulation is that it is very simple to introduce a constant -field; in fact, one only needs to slightly modify the identification brought by the boundary state [34, 36] (here we use the convention of [36] in the particular case where there are no Dirichlet directions)

(2.5) |

From this identification it is easy to see that the part of the tachyon vertex depending on the non-zero modes satisfies the following relation

(2.6) |

where is the oscillator part of the string coordinate without the zero modes. In fact, since we only consider noncompact and Neumann directions in , both and vanish on the boundary; thus the zero mode contribution can be calculated separately and modifies only the -independent part of . Using this identification repeatedly, one can reduce (2.4) to the the usual computation of the expectation value of four open string-like vertex operators and, in general, a -point function of closed string on a disk is equivalent to a -point amplitude among open strings. The only difference is that the former part of the vertices is evaluated in the unusual image point and its Lorentz index is contracted with the external momentum through a non-trivial matrix depending on . After these considerations, it is easy to find the expression for the Green function which depends on two different points and

(2.7) |

Here the antisymmetric nature of the field has been used to rewrite the final result in the standard form where the index always precedes . It is clear that is symmetric under the simultaneous exchange and .

Note that the above result is written in the -coordinates, the parameterization chosen by the boundary state calculation. In order to do the comparison with the one of [8], it is necessary to perform a conformal transformation and rewrite the Green functions in terms of the -coordinates. A small subtlety in this mapping comes from the fact that the Green function in (2.7) is not a scalar under conformal transformations. The simplest thing to do is to transform the scalar combination () to the parameterization and read off from there. One finds indeed the same result (i.e. Eq. (2.15) of [8]).

A couple of remarks are in order. Notice that the tree-level Green function in the -coordinates cannot satisfy a boundary condition similar to the one in (2.2). This peculiarity has already been stressed in [8] where it was pointed out that the condition (2.2) may be in contradiction with the equation of motion because Gauss theorem requires that the sum of all boundary integrals is equal to . This is indeed the case in the coordinate and the boundary condition is modified as

(2.8) |

As already anticipated in the amplitudes among closed string vertices on a disk, one also gets a contribution depending on a single point: this comes from the contraction of the former left and right moving part of the same vertex, since the oscillators of these two parts are now identified by the presence of a boundary (2.5)

(2.9) |

One loop

The same approach easily generalizes to one-loop, where now two boundary states must be inserted. Since the noncommutativity we are interested in at present is related to the global , we choose the boundary states to each enforce the same identification on the oscillators sets of closed strings,

(2.10) |

Again, one can use (2.6) in order to eliminate the dependence of the two vertices on the right moving oscillators, so that the scalar product in this sector transforms into a trace over the remaining left moving ’s. The final evaluation of the trace is most easily performed by using coherent states and canonical forms [32]; let us here report and comment on the three basic pieces of the result. First the measure. The two propagators in (2.10) combine and give rise to the usual factor present in one-loop amplitudes. It is clear from this point of view that the contribution to (2.10) not appearing in the exponent depends trivially on only through the normalization of the boundary state [36]. This part is absorbed in the definition of the measure (the relation between and is given below) which thus becomes

(2.11) |

Notice here that there is no in since we did not have to perform any Gaussian integral in the closed string channel. Next, by looking at the exponent of the result (2.10), one can extract the one-loop Green function

Finally as in the tree-level case, the full amplitude contains also contractions between the left and the right part of a single vertex which are encoded in the following

We note that (2.1) is exactly the Green function obtained in [8] (Eq. (3.6) there), which satisfies a certain particular form of boundary condition. We remark that the result (2.1) are written in the “closed string channel” and the natural modular parameter is related to the length of the surface viewed as a cylinder. In this case a fundamental region for the string world-sheet is the annulus with inner radius and outer radius 1. The result of [8] is obtained by setting and rescaling the coordinate .

In order to extract from the string amplitude the contribution of the “open string channel”, where the world-sheet degenerates into a circle, one has to perform a modular transformation on both the Green function and the measure. In particular, at one-loop level the relation between and -coordinates is [37]

(2.14) |

Notice that this identification fixes the cut of the log function in
the complex plane. In fact we want that the segment
of the negative real axis is mapped by (2.14) on the inner border of
the -parameterization; thus we take
^{2}^{2}2Of course, insisting on makes
the logarithm a single valued function at the expense of continuity.

(2.15) |

However, before explicitly performing the modular transformation (2.14) on the various building blocks of the string amplitude, we want to make two important remarks about the Green function (2.1).

First as one can see from (2.1), the string amplitude does not contain simply , but its combination with the derivatives of the local coordinates around each punctures [32], and this combination has conformal weight zero. Usually the dependence drops out on-shell, since the factor coming from the exponent cancels against the one present in the definition of the the measure [26]

(2.16) |

However, in order to exploit the off-shell continuation of the string results which is possible in the field theory limit [23, 26], it is more useful not to perform this simplification. Thus we use in the amplitudes the measure instead of the one of (2.11), and a shifted Green function

(2.17) |

Here, as in [23], we have related the derivative of the local coordinate to the one-loop Abelian differential since this is the only well defined object on the annulus having conformal dimension 1. In particular we have .

The second remark is related to an ambiguity in the determination of from the boundary state approach. In fact, as we have already seen, the computation of the amplitude (2.10) always gives a combination of and . Thus, by exploiting momentum conservation, it is possible to shift terms of a particular form between and . In particular one can extract from the closed string interaction on the annulus a different definition for and which is still compatible with the final result of (2.10)

(2.18) | |||||

(2.19) |

The general form of this ambiguity is the addition to of a term with satisfying the Laplace equation, as well as in order to preserve the exchange symmetry discussed after (2.7). Moreover, in order to ensure that a shift of the form (2.18), (2.19) does not change the string amplitude for all mass levels, one also needs the following properties , . Thus and are related by . This freedom in the definition of the Green function also appears in the calculation of [8] where is derived by solving the Laplace equation on the world-sheet. In fact, as in the tree level case, also here it is not possible to strictly impose on the Green function the same boundary condition imposed on the string coordinates (1.3); and hence there is a certain degree of freedom in the choice of what constraint is satisfied by . Indeed, the shift in (2.18) corresponds simply to a redefinition of the boundary condition of the Green function which leaves unmodified the value of the integral that is fixed by Gauss’ theorem. We stress that and give the same results as those obtained with and when they are used in the contraction of closed string fields (where thus must be a point not located on the boundary of the surface). However, when one wants to restrict the Green function to the boundary in order to calculate open string amplitudes, the two Green functions may give different results.

For the case of scalar amplitudes, we note that the kind of shift in (2.18) does not modify the amplitude at all, as it is antisymmetric in and and the Green function is contracted only with external momenta. Therefore the new contributions due to the additional term sum up to zero using momentum conservation. Thus, the two Green functions and actually give the same result for tachyon amplitudes. However this is not the case for gluon amplitudes as the string master formula will involve derivatives of the punctures, see (3.2) below. Using this in the next subsection, we will find that the correct Green function is given by (2.18).

### 2.2 One-loop open string Green function with

We begin by evaluating the Green function of eq. (2.17), as obtained from the boundary state approach, when the arguments take values on any of the two boundaries. Then we will argue that we need to exploit the above-mentioned ambiguity and shift to as in (2.18) in order to correctly reproduce the gluon two-point amplitude in the noncommutative field theory limit.

First, we rewrite in the following form, splitting it into its symmetric and antisymmetric part (in ):

(2.20) |

where

(2.21) | |||||

It is easy to see that is invariant under , (equivalently , ) which maps the outer boundary of the annulus to the inner one and vice versa. Note also that is single valued on the annulus.

As we have already said, the field theory limit of string amplitudes is more easily performed in the Schoktty representation of the annulus, since there the open string contributions are manifest. To go to the coordinate, the conformal transformation (2.14) implies the following transformation

(2.22) |

Let us discuss the symmetric part of first. The non-planar open string Green function has one argument on each boundary, i.e. and corresponding to and , or vice versa. For the planar case, a priori, one has to distinguish the case where both arguments are on the outer boundary ( corresponding to ) from the case where both arguments are on the inner boundary ( corresponding to ). It is easy to see that for both planar cases ()

(2.23) |

where we have separated the term, which will eventually combine with the modular transformation of the measure, from the usual Green function

(2.24) |

For the nonplanar case we have and taking into account (2.15),

(2.25) |

so that

(2.26) |

with given by

(2.27) |

Next one has to transform the other piece to the coordinate, and then restrict the values of and to the appropriate boundaries. While this is straightforward, it is simpler to remark, that when first restricting and to the boundaries one easily sees that

(2.28) |

As a result, we obtain for the symmetric part

(2.29) |

where we have identified

(2.30) |

As mentioned before, the noncommutative field limit is defined by with fixed and hence is a fixed quantity in field theory.

Now we turn to the antisymmetric part . We will transform this to the coordinates and then evaluate it on the boundaries. Since

As already pointed out, has an ambiguity that cannot be fixed in the boundary state computation. This ambiguity does not affect the scalar amplitudes as it gives a vanishing contribution when substituted into the string master formula (2.1). However there is an important difference in the gluon amplitude. Indeed, the gluon master formula (3.2) contains terms that depend on the derivatives with respect to the punctures and there is only one form of the Green function that can give the correct field theory result. The ambiguity can be most easily fixed by looking at the gluon 2-point function. We recall from field theory that the planar gluon 2-point function is independent of . It is easy to see that if , and using this in the master formula would lead to a -dependence of the planar 2-gluon amplitude. Indeed on the boundary we have

(2.31) |

where () refers to the outer (inner) boundary. However, the field theory result can be reproduced if is shifted as follows

(2.32) |

which amounts to choose the introduced in the previous section as . Thus, the final result is

(2.33) |

where is the step function that is 1 or for positive or negative . Note that this open string Green function is to be taken only on the boundaries (real ). Note also that is symmetric under the exchange of the two particles:

(2.34) |

We see that the only modification in the planar case is
a step function and it gives rise
to the usual phase factor of [38]^{3}^{3}3
Note that putting the labels in
increasing order in clockwise or anticlockwise direction
is a matter of convention, changing to
and hence in individual Feynman diagrams.
However the total amplitude is always an
even function of . We will take the convention of clockwise
ordering in this paper.
.

Before substituting (2.29) and (2.33) into the string master formula for the open string amplitude, we note that one has to scale them by a factor of first. The factor of 2 is simply because a different normalization for the Green function was adopted for the boundary state formalism and the open string amplitude (for instance, because of (2.1), the exponent of (2.3) is proportional to , while in (2.1) a factor of is present). As for the scaling , it is needed when one passes from the closed to the open string amplitudes. The reason is simple. Let’s consider the case of tachyon states whose vertex operator is when . As usual one may read the mass of the ground state described by the above vertex by simply looking at the Virasoro constraint . When the non-commutative parameter is turned on, the commutation relations for the modes become [3, 7],

(2.35) | |||

(2.36) | |||

(2.37) |

where and

(2.38) |

is the mode expansion for the open string coordinates. The change in (2.35) and (2.36) is gentle since it is simply an dependent rescaling [8, 39],

(2.39) |

In terms of these operators, the commutation relations (2.35), (2.36) take the standard form, with the -dependence concentrated on

(2.40) |

However since does not show up in , the computation of the mass parallels the calculation for and is found to be dependent. On the other hand, in the field theories we want to reproduce, the presence of the noncommutative parameter has no effect on the quadratic part of the Lagrangian, so the mass of the fields do not depend on . In order to reproduce this feature in the string amplitude, we rescale quantum numbers like the external momenta and polarizations by an appropriate factor . Notice, as a check, that this rescaling exactly absorbs the overall -dependent normalization of the measure found in the boundary state calculation (2.11). Equivalently one can introduce the hatted variables as in the previous subsection. With this, the mass of the tachyon takes the usual value . We remark that this step of rescaling the modes by is equivalent to using a vertex operator with the rescaled open string coordinate .

As we have mentioned before, we note that there is no factor in the closed string amplitude just like the case, but there is a power of in the open string amplitude. There are three sources that these factors can arise when passing from the closed string to the open string amplitude: from the modular transformation (2.2); from the measure of integration over the moduli; and from the partition function. All these factors are independent of and they combine to give the desired dependence of the open string amplitude.

Summarizing, the open string Green function with constant -field in the Schottky representation of the annulus is given by

(2.41) |

with

(2.42) |

in the planar case; and the nonplanar Green function is

(2.43) |

with

(2.44) |

The sign in (2.43) refers to the outer (inner) borders. Note that is still symmetric with respect to the exchange of particles

(2.45) |

Exactly these open string Green functions are obtained from the open string operator formalism [46].

##
3 String amplitudes in the presence of a constant

-field and their field theory limits

After this detailed discussion of the Green functions and the measure in the previous section, we now turn to the actual evaluation of the one-loop open string amplitudes. In section 3.1 we start our analysis by focusing on scalar interactions; this is the simplest example since the scalar amplitudes involve only the ground state of open bosonic string theory (that is the tachyon). In section 3.2, we study the Yang-Mills case.

As a general remark, we want to stress that string amplitudes yield the correct overall normalization of the various Feynman diagrams without having to calculate the combinatorial factors typical of field theory and this agreement holds also in the non-planar case [29]. It is natural then to expect that also the coefficient of noncommutative amplitudes are reproduced by the string master formula. We find that this is indeed the case without having to change the definition of and . We have [23, 26, 33, 34, 29]

(3.1) |

where is the open string coupling constant and is related to the normalization chosen for the Chan-Paton factor . Note, in particular, that the vertex normalization is independent of the particular string state chosen [23] and will be used to derive both scalar and “photon” interactions.

### 3.1 One-loop amplitudes in scalar theories

We are now in the position to derive from the master formula
(2.1) a compact expression that will generate the
noncommutative Feynman diagrams for scalar theories^{4}^{4}4Notice
that the overall normalization, in terms of is different
from the one of [29] because there the convention has been
used. Here we focus on interaction and thus is more natural to
fix .

(3.2) | |||||

The string projective invariance has been used to choose the fixed points of the single Schottky generator as and