ELS EV IER

19 May 1994

Physics Letters B 327 (1994) 234--240

PHYSICS LETTERS B

Non-polynomial string field theory and reparametrization of strings

F. Anton, A. Abdurrahman ~ Department of Theoretical Phystcs, Universtty of Oxford, 1 Keble Road, Oxford OX1 3NP, UK

J. Bordes 2 Departament de F{sica Te6rica, Umversitat de Valencia, Dr. Mohner 50, E-46100 Burjassot, Spain

Received 5 February 1994 Editor: L. Alvarez-Gaumd

Abstract

Reparametrization invariance of string amplitudes is closely related to the non-polynomial closed string field theory. We show that it is possible to formulate the terms of the non-polynomial action just by using the SL(2, C) properties of the so-called "restricted vertices". In this context we work out in detail the four point amplitude in the different channels in terms of the reparametrization parameters.

Recently, there has been a revived interest in the construction of a field theory for closed strings (CSB'r). While for open strings the problem is well settled [ 1 ], a CSFT has proved to be a difficult task to carry out: the requirement of gauge invariance and the correct covering of the relevant moduli spaces rules out a cubic action analogous to Witten's action for open strings [2,3].

In the past three years, strong indications that this theory should be non-polynomial have appeared in the literature. In fact a fully consistent covariant theory has been discussed by Zwiebach in [ 13]. In this theory, the action has an infinite number of terms describing the interaction of each time an increasing number of strings. These terms are not obtained from the usual cubic interaction by applying the Feynman rules; they

t Also at Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, UK. 2 Also at IFIC, Centru Mixto Universitat de Valencia-CSIC

0370-2693/94/$07.00 © 1994 Elsevier Science B.V. All fights reserved SSD103 70-269 3 ( 94 ) 00371 -D

are dictated by the requirement to cover the region of the modular space which is outside the usual channels. This "missing region" was first noted by Kaku [4]. Any practical formulation, however, which is based on functionals or operators of these non-polynomial the- ories is a formidable task since it involves the explicit calculation of the vertex operators for an arbitrary num- ber of strings. Furthermore, although these regions have been described, it is not clear whether some cor- relation between the integration regions of the modular space occurs at the different levels.

On the other hand, two years ago [5] it was sug- gested that closed strings can be treated in a similar manner to open strings as far as the interaction is con- cerned [7]. This approach of calculating closed string amplitudes, presented in [5], opened the possibility of writing them in a dual form that encodes the contribu- tion of the different channels as well as the abovemen- tioned missing region. In this approach, one splits the closed string in two halves, in exactly the same manner

F. Anton et al./Phystcs Letters B 327 (1994) 234-240 235

as for open strings, and the interaction is given by the overlap of two such pieces. The key point is the repar- ametrization invariance of the string amplitudes which can mimic the change of length of the overlapping parts in Zwiebach's theory and thus reproduce the kind of interaction proposed in the non-polynomial (classical) c s v r .

In this note we argue that the reparametrization approach mentioned above is closely related to the field theory and that in fact, it can be used as an effective tool to write explicitly the different terms appearing in the nonpolynomial CSFT. We will study the four point amplitude giving explicitly the form of the modular region of the different channels as well as the missing region. Due to the complications of the full quantum mechanical theory, we will only deal with the classical action in this note.

The paper is organised as follows: first we review those aspects of the non-polynomial CSFT relevant to our work. We also review the reparametrization approach of calculating amplitudes. Then we develop the link between both approaches by working out explicitly the four point amplitude. The different chan- nels as well as the missing region are identified in terms of the string reparametrization parameters. Finally we give our conclusions.

When trying to formulate a CSFT by introducing the naive generalization of Witten's 3-vertex to closed strings, one finds that the theory is inconsistent for several, closely related reasons. The first thing to notice is that, because we want only states which are invariant under rigid rotations of the string, the vertices cannot depend on the origin of the o-coordinate. This is achieved by inserting the operator P that projects out the states that do not satisfy this physical condition, into the vertices. Notice that, if the closed string vertex is built from the product of two open string vertices, the twist and cyclic properties of the former render the closed string vertex fully symmetric under the permu- tation of any pair of strings.

Moreover, as pointed out by Kaku in [4], the pres- ence of the operator P in the vertices is related to the fact that a cubic theory cannot give the correct four point amplitude.

A necessary condition for obtaining the correct region of modular space is that the transition between the different channels is smooth [8]. The geometry of 4 closed strings at the instant of interaction is depicted

in Fig. 1 (Notice the scattering of 4 closed strings in the s and the t channels). As the figure indicates, the s- channel interaction can take place when the product of strings 3 and 4 is displaced by an angle @ relative to the product of strings 1 and 2.

When they collide it is impossible to instantly deform the rotated s-channel graph to the t-channel graph where the product of strings 1 and 4 is the one that is displaced from the product of strings 2 and 3. In order to make the transition smooth, one is naturally led to introduce a higher interaction. An explicit calculation has been presented by Zwiebach [9] in which he shows that indeed a finite chunk of the modular space for the four point amplitude is missing.

The correct way in which this should be done has been discussed in [ 10,11].

In the following we will reproduce some of the argu- ments presented there to see how one introduces an order Ninteraction to cover correctly the modular space of the N-point scattering amplitude. The final result is a completely non-polynomial theory which is gauge invariant.

It is natural to expect that a field theory which covers correctly the relevant modular space be gauge invari- ant. In fact by requiring covariance, plus the correctness of all the tree level amplitudes one is led to a completely non-polynomial theory in which gauge invariance is a consequence of the conditions that the several vertices of the theory have to satisfy in order to cover correctly the modular space. These conditions are given in the form of recursion relations.

The general requirement of covariance suggests that all the external strings should have the same o-length, which we fix by convention to be 2~r. This will imply that the Feynman rules of the theory produce a sym- metrical decomposition of the modular space.

s-cha~nel t -channel

Fig. 1. Contact interaction for 4-closed string scattering m the s and t channels.

236 F Anton et al . /Phystcs Letters B 327 (1994) 234-240

The interaction vertices correspond to string over- lapping patterns on the two-dimensional sphere, with the following properties [ 10,11] :

(i) Only three edges join at each vertex. (ii) Since the length of a closed string is 2~" the

perimeter of any face also equals 2rr, i.e., the sum of the lengths l, of the edges of any face F satisfies

E l, =2¢r. (1) tc~F

From Euler's theorem one deduces that a polyhedron with N faces can be described by 2 N - 6 edges and, since there is a one-to-one correspondence between the N-punctured Riemann sphere and the patterns of N- string contact interactions [ 10], it follows that these edges play the role of the modular parameters of the N- point amplitude. Notice that 2 N - 6 is exactly the num- ber of modular parameters of the N-punctured sphere. As pointed out by Zwiebach [ 10], if we integrate over all possible values of the overlap patterns satisfying conditions (i) and (ii) we should get the correct N- point scattering amplitude. However, we are interested in formulating the field theory; therefore the N-point amplitude should be obtained via Feynman rules, and we must in some way restrict further the polyhedra so that the contribution from the N-vertex is just the region not covered by the lower order vertices. The correct constraint for cases N = 4 and 5 was found by Zwiebach in [ 10] and was generalized to any N by Kugo, Kuni- tomo and Suehiro [ 11 ], their result is:

(iii) Any closed path C surrounding two or more faces on a polyhedron has a length larger or equal to 2 ~

t, >_.2~. (2) t ~ C

The Feynman diagrams based on the string vertices satisfying (i), (ii) and (iii) cover the whole modular space correctly. These polybedra are called restricted.

We will work out the N---4 case which is the first non-trivial one, where we have 2 modular paramemrs. Condition (1) implies that the tetrahedron is such that the edges that lie opposite to each other have equal length. If a,j represents the length of the edge separating strings i and j, these lengths satisfy al= = a3a, a~a = a a

and a14---a23. T h e y a re constrained by a ~ = + a t x +

a14 = 2q'r. This tetrahedron is represented in Fig. 2. Actually, there are two different ways of labeling the

0-,12

f ~24

g23

t~34

/ a l a

ah2 q- a13 q- a14 ---- 2a"

Fig, 2. Tetrahedron graph corresponding to the four point vertex.

faces on this tetrahedron, depending on the order in which faces are met when one travels along the perim- eter of a given face.

In order to distinguish between them, we shall use a different set of parameters (a,~) restricted by similar relations. The two sets of parameters define triangular regions in parameter space which should be glued across their edges because when al2 = a]2 = 0 the two tetrahedra are identical when al3 =a~3 and a l a = a i 4 ,

and so on. We therefore see that the modular space of the 4-punctured sphere is topologically a sphere with three cusps.

We are now interested in the characteristics of the restricted region. On the tetrahedron of Fig. 2 there are three different paths surrounding two faces separated by an edge. If we consider the path that surrounds the two faces on the right half circle of the figure, Eq. (2) gives 2(a13+ala)~>21r, so that ~>~a12. Hence all edges are restricted to be less than or equal to zr. The

? same applies to a, j , so that the restricted tetrahedra are determined by the conditions

{au, a ~ } < ~ ,

a n d Y'~j = 2,3,4a D ---- E~=2,3.4a], = 2~'. These restricted polybedra provide the necessary

terms in the action in order to build a CSFT. As Kugo and Suehiro showed [ 14], the restricted polyhedra sat- isfy the recursion relations (we use the Fock space language)

F Anton et a l /Physws Letters B 327 (1994) 234-240 237

N

(v~,l E O, = - E (~,,,l(,,N21v12), l = 1 NI +N2 - - 2 = N

(3)

where IVy2) is the sewing ket corresponding to the BPZ inner product [ 15] and N1, N2 and N are greater than or equal to three. Notice the distinction between the restricted vertices (vN[ (those that correspond to the restricted polyhedra) and the non-restricted ones (Vt~[ which have all the edges of the same (fixed) length and satisfy only conditions (i) and (ii). Apart from the P insertion, the VN vertices have the ghost operator bo implicitly inserted, which is needed in order to recover the correct physical amplitudes [10,11].

The terms on the right hand side of Eq. (3) corre- spond to Feynman diagrams in the limit when the prop- agator collapses. Therefore (3) is just the statement that the region of modular space covered by the N-th order polyhedra lies on opposite sides of the common boundary with those regions covered by lower order polyhedra. Eq. (3) is then the condition that the restricted vertices should satisfy in order that the mod- ular space is covered correctly,

The (classical) action constructed with the restricted polyhedra is [4,13]

g~-2 ~N S=½(~lQboPldP)+ ~ ~ , (4) N ~ 3

where

• "= ,2 N ( ,,,, I q~)l . . . I q') , , .

Due to (3), this non-polynomial action is invariant under gauge transformations [4,13 ] and constitute the final answer as far as the classical theory goes. As has been mentioned before, the quantum theory requires the introduction of another infinite set of the "higher genus" vertices [ 13 ].

New we turn our attention to the calculation of ampli- tudes based on the reparametrization invariance of the string amplitudes. It has been shown in [ 5 ], using a functional approach that, to any order in perturbation theory one can write directly the string amplitudes in a dual form. By that we mean that in a compact expres- sion the contribution of the different channels is included. Thus for closed strings, the contribution of the missing region (restricted polyhedra) to the ampli-

tude will somehow be encoded in this dual expression. Our next task is to identify this region and, at the same time, provide an explicit formulation of the different terms in the non-polynomial theory (4).

Before going we recall that, in the reparametrization approach, the string field is interpreted as a matrix whose elements are a function of the translational mode and the rows and columns are labelled by the two halves of the string #1. Interaction between two strings is then interpreted as the overlapping of two halves and, in matrix language, as the products and trace of the matri- ces representing the string fields. To obtain the com- plete amplitude (for simplicity we restrict ourselves to tree level), one inserts the dualizing operator which mimics the change in the length of the overlapping pieces.

It was shown in [5] that, at the tree level,

1 f dO, AN= ~ ,~NSL(2; C)

X Tr(O1A 1 O2A2 ...ONAN) (5)

gives the correct dual amplitudes. This expression is just a generalization to closed strings of a similar result for open strings [7]. (In (5), a trivial integration over the translational mode of the string has been omitted).

The dualizing operator O, affects the set of fields on its right and this explains why each string overlaps with more than one (we will work out the N = 4 case to clarify this point). The explicit expression of O is given by

O(q~, A) =ei~(t°-zT°)e aM, dO, =dA,d~,e 4;t ,

where M is given by

M = Mleft + / ~ n g h t ----- LI - L_ 1 +/~'1 - L_ 1 •

(L. (£.) are the Virasoro operators for the left (right) movers.) Its action on the strings is such that it keeps invariant the time variable ( r = 0) whereas it repara- metrizes the tr coordinate, the explicit expression was given in [5].

Notice that, for every i, the operator O, contributes with two parameters to the total number of modular parameters of the amplitude, if we factor out the infinite

~t The theory, however, turns out to be local since only a particular combination of the singled points play the role of the translational mode of the string [6].

238 F. Anton et al. /Phystcs Letters B 327 (1994) 234-240

volume of the group SL(2; C) we only keep 2 N - 6 modular parameters in agreement with the number of modular parameters of the restricted polyhedra of order N.

Now, from the complete amplitude one could restrict the region of integration so that the restricted polyhedra appearing in the field theory are obtained. For N~>4, the proper region of integration has to be found in such a way that the modular space of the field theory is covered correctly.

For definiteness, let us see how Eq. (5) can be used to represent the contribution of the restricted polyhedra v4 in the action, (4).

We can take care of the infinite volume of SL(2; C) by fixing six of the modular parameters to arbitrary values. In this way the restricted 4-vertex is

1'4 = / dAd~b Tr(Al "A2" O(A3 "A4)) , (6) t - D 4

where/)4 is the region of the modular space correspond- ing to the restricted polyhedra.

In the above expression, e ~ reparametrizes the string field (As 'A4) , while e '*(t°-&) rotates it. The same happens to the string field (AI "A2), however, using the symmetry of the trace it is easy to see that we only need to consider the latter.

In Fig. 3 we have given the interaction in this picture. Using the same notation as before for the overlapping

parts of the string, it is clear that e x~ changes the length of a~2 and that e '~(t°-£°) affects both a~3 and a~4. When A roams through the interval ( - 0% oo), a[2 cov- ers the interval [0, 2~r] monotonically. Let us denote the (finite) change in a 42, produced by e xu, 0(A):

e ~ : a~2 --~a~2 - O(A) .

The explicit relation between 0 and A can be found, however we only need the conditions O(A --> O) ~ 0 and O(A-~ +_~)-~ -kw. If we start from the "initial" (point interaction) configuration a 12 - a 13 = a~4 =0, then O(A) ~ [ - T r, ~r]. Geometrically this is given in Fig. 4 (we take qb/> 0 counterclockwise).

Thus we arrive at the following relation between the parameters a,~, used to describe the restricted tetrahe- dra, and (th, O):

a ] 2 = ~ r - 0 ( X ) , a ] 3 = ~ b , a ] 4 = ' / r + 0 ( A ) - ~ b

(4,>/o).

They satisfy E~=2,s,4a~j =2~-trivially. Furthermore, we still have to impose 0 ~ a ~2 ~< 2~r (same for a ~3 and ai4).

Before going on, let us see what happens if ~b ~ O. In this case we have an analogous picture with

al2 = w - 0(A) , a13 = -- ~ , a~4 = ~r+ 0(h) + ~b

(4,<0).

! a 14

2

Fig 3. Overlapping of strings for the four point amplitude.

F. Anton et al. ~Physics Letters B 327 (1994) 234-240 239

=+Of ~,) n+O(k)-O

,~ n+O0.)

Fig 4. Action of the operator//.

As before, these parameters correspond to the different labeling of the faces of the tetrahedron.

From the preceding two sets of equations, we get varying intervals of the reparametrization parameters.

Alternatively, we could describe the different regions in the (A, 4,) plane. Using Eq. (4.2) in [5], one finds

A=½ log( c°s ¼0-sin ¼0 / kcos ~O+sin ¼0J"

Again, the variation of O(A) e [ - or, w] gives the val- ues of A in the real line. At this point it is worth noticing that a detailed analysis of the former relation will lead

to the identification of the different channels in the usual modular representation of the string amplitude #2.

Eq. (6) gives the correct scattering amplitude if we integrate over the whole triangle B A B ' in Fig. 5. Now let us see how this region is (partially) covered by the s-, t- and u-channel Feynman diagrams. At the' 'initial" configuration (~b, 0) = 0 and/'2 = 1 (recall 0 = 0 if and only if A = 0), so the trace in (6) is just

Tr(AI "A2 "A3 "A4) •

We expect to be able to write this trace as two 3-vertices

¢2 We acknowledge the referee for pointing out flus point.

O(1)

B Q c

U

-;~ p ~[~ x~ R

C t

A'

Q I

2~ p ,

B '

Fig. 5. Modular space for the four point amplitude in (0, 4,) space.

240 F Anton et al. ~Physics Letters B 327 (1994) 234-240

glued together across one of their legs. The operation of gluing is realized by the gluing operator or 2-vertex; it corresponds to the limit when the propagator between two 3-vertices collapses and the intermediate angle of rotation is ~b = 0. This is a special case of the General- ized Gluing and Resmoothing Theorem (GGRT) of [ 8 ]. Furthermore, as we let th vary from 0 to 7r, keeping A = 0, we are just varying the intermediate angle of rotation between the string fields (A1"A2) and (A3 "A4), so we can apply this theorem at every point on the segment R 'P ' o f Fig. 5. Hence this segment corresponds to the border when the propagator of the s-channel contribution collapses. In this way the GGRT provides the link between (6) and the Feynman rules of the field theory. To see that we are, indeed, talking about the s-channel let 0 = - I t (along t h = 0 say), at this point A = - o o and the surface in Fig. 3 gets "p inched" , separating (AI 'A2) from (A3"A4); this corresponds to the limit when the propagator is infi- nitely long in the s-channel diagram. The same argu- ment is valid when th ~< 0, so the complete s-channel contribution is the triangle P P ' A in Fig. 5. Similarly we can get the t- and the u-channel contributions in the same figure. The regions marked with x and x ' are not covered by these channels, hence giving the missing region D4 needed to construct the restricted tetrahedron. Once again we see the same symmetric decomposition of the modular space as in [ 10]. In the same way as for that case, this implies that (3) is satisfied. The lines B Q C , B P A and CRA are identified with their homolo- gous giving the sphere with three cusps.

To conclude, in this note we have argued that the interaction terms needed to be included in the non- polynomial action in CSFT can be written in terms of reparametrization of the strings. This gives a calcula- tional approach to the field theory. In the reparametri- zation approach, on the other hand, half-strings play an important role in a similar manner as they do in open strings. Overlapping of half strings reproduces the interaction. Then, the introduction of the dualizing operators which reparametrize the string, completes the

picture necessary to reproduce the interaction in the field theory. Using this approach, we have computed the four point amplitude and we have found the relation between the modular parameters and the reparametri- zation ones. The missing region as well as the different channels have been identified. The higher N-point amplitudes can be treated in a similar manner, however the identification of the missing region is more involved. The inclusion of higher genus poses no prob- lems since, as it was shown in [5] , one can write the amplitude just employing the same tools used at tree level.

The work of J.B. has been supported by CICYT- Spain under grant AEN 93.0234 and that of A.A has been supported by the Secretariat of Scientific Research, Tripoli, Libya.

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