Solving the open bosonic string in perturbation theory

Nuclear Physics B341 (1990) 5 13—610 North-Holland

SOLVING THE OPEN BOSONIC STRING IN PERTURBATION THEORY Stuart SAMUEL* Department of Physics, Indiana University, Bloomington, IN 47405, USA Received 19 October 1989

The integrand and integration region for the N-point amplitude in the open oriented bosonic string are obtained to all orders in perturbation theory. The result is derived from the Witten covariant string field theory by using on-shell and off-shell conformal methods and Riemann surface mathematics. Although only the off-shell g-loop tachyon amplitudes are computed explicitly, the methods generalize to other external states. We derive the g-loop ghost-Jacobi identity in which the ghost correlation function cancels the jacobian factor in changing from second-quantized to first-quantized variables. Moduli space is discussed from several viewpoints and it is shown that string field theory provides an algorithm for its determination.

1. Introduction 1.1. BACKGROUND REMARKS

Superstring theory has the possibility of uniting the four fundamental interactions, including gravity, into a finite quantum theory [1—7].The search continues for a string theory which correctly reproduces the known phenomenology. There are many challenges to overcome: compactification, zero cosmological constant, the hierarchy problem, the correct fermion spectrum and the proper breaking of gauge groups to SU(3) x SUL(2) x U~(1)and subsequently to SU(3) x UEM(1). If, some day, a new standard model based on a string is found, it is important to have the necessary computational apparatus ready to compute the corrections to the old standard model, the Weinberg—Salam—Glashow model [8]. This paper is written with the above in mind. The open bosonic string is solved to all orders in perturbation theory. At present, we are able to explicitly solve only this particular theory and only about the standard vacuum state which is known to be unstable due to the tachyon. Nevertheless, this is a step forward and perhaps further developments will permit the solution of the closed string, the superstring and strings in other vacuums. In the conclusion, we explain why these other theories are not yet accessible with our methods. *

Present address: Physics Department, The City College of New York, NY 10031, USA.

0550-3213/90/$03.50 © Elsevier Science Publishers By. (North-Holland)

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The general N-point g-loop amplitude involves two ingredients: the integrand and the integration region. To obtain both, high-powered mathematics is required. As a consequence, it is not always clear whether a given result is an abstract embellishment of advanced mathematics or a practical solution. For this reason, we have imposed upon ourselves a computability criterion: we require all results to be accessible to computer evaluation. With each mathematical formula, we require an algorithm to numerically calculate it. A program might require enormous hours of supercomputer time or the code might be laborious to write. We do not address the issue of efficiency. For the integration region, three approaches are provided. For certain configurations the problem is solved analytically (see sect. 7). For the other cases, a set of equations is presented. Unfortunately, as demonstrated in sect. 8.7, a numerical evaluation in a finite amount of compute time is not possible. Nevertheless, sect. 8 provides interesting results which, with further developments and techniques, might produce an efficient algorithm. In sect. 9, string field theory is used to determine the integration region although this approach is awkward to program in a computer. Let us explain why one might expect string theories, as opposed to particle theories, to be solvable in perturbation theory. To obtain the gth-loop result in particle physics one must enumerate all Feynman graphs involving g loops and determine the integrand for each graph. An amplitude is obtained by summing, over all graphs, the contribution from each graph. The contribution is obtained by integrating the integrand over the g-loop momenta. Since as g increases, the number of graphs increases and since the integrand for each graph is different, many integrands need to be determined. It is hard to specify the generic integrand without examining in detail the corresponding graph. In string field theory, the procedure is similar. There is one difference however. When the appropriate integration variables are used, the functional dependence of the integrand is the same for all graphs of a given loop order. For g fixed, only one integrand need be found. If the integration region is determined, then the gth-loop result is obtained. This is what we accomplish in the present work*. We use both first-quantized and second-quantized methods. Computation in the former is most easily carried out using conformal field theory [9—26].The latter is approached via covariant string field theory [27] plus developments [28—42]during the past few years which allow concrete computations [33—55]. At tree-level, the three-point couplings have been computed in refs. [34—41], on-shell agreement of the four-tachyon amplitude was achieved by Giddings [33], the off-shell Veneziano formula [56] was obtained in refs. [47, 48], and finally, ref. [49]derived the off-shell Koba—Nielsen formula [57—59].At one-loop, the two-point [51] and higher-point [52] functions have been calculated and produce some *

No attempt is made to sum the perturbation series.

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interesting physics [51]. Particularly useful in obtaining many of these results is the adaptation of conformal field theory to the off-shell case [42].

1.2. STRING FIELD THEORY

Because we rely in part on the Witten covariant open string field theory [271it is useful to review some of its features. It is based on a Chern—Simons form [27] Action

=

f~p*QlJf+~g’f111*1I1*~11,

(1.1)

where Q is the first-quantized BRST charge [60, 61], ~I’is the string field, f and * are the string integral and star product [27] and g’ is a coupling constant. Eq. (1.1) is invariant under gauge transformations ~I’ QA + g~(1I~*A fl*i[t) Precise mathematical representations of all the string objects in this paragraph can be found in refs. [34—411. To commence perturbation theory, the gauge must be fixed [27—31]. A convenient choice is the Siegel—Feynman gauge, b0~I’ 0 [621. The propagator becomes =

—

=

a’b0( L0) -‘

=

a’bJ dT exp(

—

TL0).

(1.2)

The operator exp( TLO) produces a rectangular strip of world sheet of width ir and length T. The trilinear term in eq. (1.1) represents the following interaction. Let a- be the parameter characterizing a position on a string and let it range from 0 to ‘n-. Divide the string into two halves using aas the separation point. The interaction term “glues” the second half of the ith string onto the first half of the (i + 1)th string, for i 1, 2 and 3, with (i + 1) 1 for i 3, as illustrated in fig. la. A kink or curvature singularity is present at the common joining point where a- ir/2. The perturbation series generates Feynman graphs, each one of which is associated with a certain Riemann surface or string configuration [27, 281. External states are represented as semi-infinite strips of width Vertices like the one in fig. la join these strips and internal propagator rectangles as in fig. lb. Altogether the result is an open two-dimensional surface of strips joined via the Witten vertex. Examples of such surfaces at tree level and the one-loop level are displayed in fig. 1 of ref. [33] and fig. 2 of ref. [521.Figures 2c—4c show the two-loop vacuum bubble diagrams. There is a direct way of generating the Riemann surface from the Feynman graph. Replace the edges in a graph by dotted lines bordered by two solid lines. At a vertex join the solid boundary lines together and fusethe together all three dotted 2 and solid lines correspond lines. Let the dotted lines correspond to a- IT/ —

=

=

=

=

~-.

=

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st~ng2 ~

strIng 1

(a)

(b) string 3

sIrIng2

2

I

stringl

1

(c)

(dl

4 (e)

If

(f)

______________

(g)

(h)

Fig. I. The covariant string field interaction and propagator. Using o~as the parametrization variable, (a) displays how the three strings overlap in the interaction f ~P~’J.’~’I’. If short propagator strips are added to (a), (b) is obtained. In a Feynman graph, the interaction involves at three-point vertex as in (d). Replacing lines by dotted lines bordered by solid lines turns (d) into (c). To go between (b) and (c), identify the dotted lines (respectively, solid lines) in (c) as the centers (respectively, edges) of the strips in (b). The same procedure can be applied to a propagator: (e)—(h). From a propagator line (e), a solid—dotted line configuration (f) can be obtained; (f) can be thought of as the top view of’ (g). Unfolding (g) produces the propagator strip in (h).

to a- 0 and a- ~ so that the edges are replaced by strips of width ir. A Witten-type world-sheet string configuration is obtained. Figures lb—d and le—h, respectively, illustrate what happens locally to a vertex and a propagator. As an example, fig. 2c is the surface produced from the Feynman graph in fig. 2a. It can be unfolded to give the surface in fig. 2d. Figures 3 and 4 display the process for the other two 2-loop vacuum bubble diagrams. =

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(a)

TA~B

C~D

(dl

(e)

Fig. 2. A two-loop vacuum diagram with B = 3 and H = 0. Figure (a) is the Feynman graph. Replacing the propagator lines by a dotted line bordered by two solid lines turns (a) into (b). This can be thought of as the top view of (c). It, when unfolded, produces the string configuration in (d). Cutting (d) along A and B and along C and D and flattening to produce (e) shows how a plane domain becomes a string configuration; if, in (e), the A, B, C and D edges are glued together, (d) is reconstructed. The configuration has three boundaries and no handles.

(a)

~ (b)

(c)

Fig. 3. A two-loop vacuum diagram with B = 3 and H = 0: (a) is the Feynman graph. Performing the construction in fig. 1 generates the surface in (c) via (b). Figs. 2 and 3 are the only two vacuum bubble diagrams with B = 3 and H = 0.

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(a)

(b)

(c)

Fig. 4. The two-loop vacuum diagram with B = 1 and H = 1. (a) is the unique two-loop Feynman vacuum graph which produces via the procedure in fig. 1 a surface with one boundary and one handle. The surface is shown in (c). It was one of the prototypes used in ref. [321to check the covering of moduli space.

1

~

2 (a)

/ 1

~

2

‘~

K

(b) Fig. 5. The rigid vertex. The cyclic ordering at a vertex is important. With if as in fig. Ia, the first half of string i (mod 3) overlaps with the second half of string i + 1 (mod 3) in (a). In (b), which has the opposite cyclic ordering, the second half of string i (mod 3) overlaps with the first half of string i + 1 (mod 3).

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Although the Witten vertex is cyclic, it is not permutation invariant. Relabelling string states by 1 2 —~3 1 leaves the vertex invariant but 1 2 does not. This is similar to the three gauge-boson interaction in a non-abelian gauge theory. “Rigid” vertices are used. The cyclic ordering dictates whether the second half of string 1 overlaps with the first half of string 2 or string 3 (see fig. 5). Consequently, Feynman graphs 3a and 4a are different and in fact produce Riemann surfaces of different topology: the former is a sphere with three disks removed; the latter is a torus with one disk removed. An external state is produced by placing a vertex operator at the asymptotic position, i.e., at the end of the semi-infinite strip. For tachyons, the operator is c exp(k X) where k is the momentum of the state and X~(z)and c(z) are first quantized string and ghost variables, respectively. —‘

—*

~

1.3. A PEDESTRIAN DESCRIPTION OF THE COMPUTATIONAL METHOD

Our computations and results are somewhat complicated. In this section we explain in words how we have proceeded. The calculation reduces to computing the bc-ghost and X’~correlation functions on the Witten-type Riemann surface. Even though free fields are involved, the calculation is non-trivial because of the complicated nature of the surface. To overcome this, we map the string configuration onto some canonical domain for which the correlation functions are known. Let p(z) denote the map. Then as z varies in the canonical domain, p(z) varies over the Riemann surface. In carrying out this transformation, conformal factors are generated. For a tachyon operator on the rth leg, c exp(kr X) —s exp(hN~~)c exp(kr X), where h a’kT k’~— 1 is the conformal weight and N1~is a zero—zero Neumann function .

=

for the surface. A formula valid on any Witten-type surface for N1~has been derived in refs. [49,52]. For other states, other conformal factors are produced but they all can be calculated from the map p [42]. The conformal factors are absent for on-shell physical states. For example, for the tachyon, exp(hN~) 1 since h 0 when the mass-shell condition k~ 1/a’ is imposed. It is believed that the string field theory produces a single covering of moduli space [281*. We assume this result so that when one varies the lengths, T (in eq. (1.2)), of the propagator strings and sums over all Feynman graphs, moduli space is covered. Since each point in moduli space corresponds to a distinct marked Riemann surface we are summing over Riemann surfaces once. This is the prescription of the first-quantized Polyakov approach [63]. To concretely characterize a Riemann surface we rely on the uniformization theorem [64—66].It is discussed in detail in sect. 2.1. Roughly speaking, it says that =

=

*

=

Rigorous results have been obtained for the vacuum bubble diagram involving one boundary and an arbitrary number of handles [32], the N-point tree-level amplitudes [33,49] and the one-loop N-point functions [521.

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any Riemann surface beyond i-loop can be represented as a region of the upper-half complex plane. This region is bounded by an even number of arcs of circles of various radii with centers on the real axis. The arcs are identified pairwise using fractional linear transformations with real coefficients, i.e., SL(2, l1) elements of the form (az + b)/(c + dz). The region in the upper-half plane exterior to the circle arcs is the fundamental domain, F, for our map p. Via the uniformization theorem a concrete realization of F is achieved. Quite important is the map, p, which maps the fundamental domain onto the string configuration. It is given in sect. 4. The object (dp/dz)2 is uniquely determined by its zeroes and its poles and by a normalization condition. With it in hand, we are able to compute the correlation functions and the amplitude integrand. We do this by using conformal field theory [9—26,42] and by deriving a ghost-Jacobi identity. The overall normalization is fixed in appendix E using world-sheet factorization by taking the limit in which the circle arcs shrink in size. The remaining ingredient in the integration region. It is given in sects. 7, 8 and 9. To obtain moduli space, we use string field theory itself and the assumption that it gives a proper covering. In short, by using string field theory, off-shell and on-shell conformal field theory, advanced Riemann surface theory [64—81] plus a lot of algebra, the perturbative solution to the open string theory is obtained.

1.4. NOTATION AND CONTENTS

Sect. 2 discusses how to represent Riemann surfaces via the uniformization theorem. Other topics concerning surfaces are treated. Sect. 3 is an exposition on various functions and quantities associated with a given Riemann surface. These include the holomorphic differentials, q-differentials, the period matrix, the @functions and the prime form. The Riemann vanishing theorem is also presented. The i-loop case is used to illustrate the concepts. The map p is given in sect. 4 in terms of e-functions, the prime form and the cr-function. Sect. 3 provides formulae for all of the above in terms of the g-holomorphic differentials defined on the double of the Riemann surface. For open surfaces, the holomorphic differentials can be computed from the Poincaré series of Burnside [81]. The the correlation functions for X’~and b—c are presented in sect. 5. In going from the T-type integration variables in eq. (1.2) to Koba—Nielsen-like variables and variables characterizing the Riemann surface, a jacobian is generated. Sect. 6, with the assistance of appendices A—D, demonstrates how the jacobian cancels most of the b—c correlation function, leaving one with an integrand consistent with what one would expect from conformal field theory or from the analog model [9, 10,59,82]. Moduli space is discussed in sects. 7, 8, 9 and appendices F—I.

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We follow the notation and conventions of previous work [22,42,49,52, 83]. Two vertical lines stand for absolute value. A bar, over a variable indicates complex conjugation; over a set it indicates closure. The variable, z, is associated with the fundamental domain, F, whereas p is the string configuration variable. The constants Z~and Y are, respectively, the asymptotic positions and the interaction points in F. Much of the notation associated with Riemann surfaces is established in sects. 2 and 3. Although g usually denotes the 1oop level and coincides with the genus of the double of the surface, it also stands for the on-shell three-tachyon coupling. Group elements of SL(2, l~)are generically indicated by h, with the exception of the identity element, e. Holomorphic differentials are denoted by o~. In integrals they implicitly contain a dz factor; in algebraic formulas the dz factor is absent. Here is work related to or useful to the present endeavor: for light-cone string field theory see refs. [4, 84—90]; for conformal field theory in string theory there are refs. [4,25, 26,28,29,42,44,49,52,91—94]; for the early work on the use of Schottky groups in loop computations see refs. [95—100];for results related to multiloop methods and correlation functions see refs. [4, 77—79, 101—136]; review articles can be found in refs. [1—4,23, 24, 39—41, 136—143]. —,

2. Representation of open Riemann surfaces The surface associated with a string field theory diagram is somewhat complicated. It involves a curvature singularity at a- ~-/2 at each interaction point. As a consequence, computations on such a surface are difficult. As discussed in the introduction, it is preferable to map conformally to some more standard and manageable representation of the surface. We must therefore have a good way to represent Riemann surfaces. Fortunately, mathematicians have studied this subject extensively. The purpose of this section is to review these results. Use of them is made throughout the rest of the paper. For more on this topic, see refs. [64—79]. =

2.1. THE UNIFORMIZATION THEOREM

The uniformization theorem [64—66] states that, with the exception of the surfaces listed below, an arbitrary Riemann surface, ~/, is conformally equivalent to F \ I-I ~J’=—F\11,

(2.1)

where [-Iis the extended upper-half complex plane (not containing the real axis but containing the point at infinity) and F is a discontinuous subgroup of SO(2, l~) without fixed points in I-I. Below, we explain in more detail the content of eq. (2.1).

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The exceptional Riemann surfaces are [64—66]: (a) The Riemann sphere the extended complex plane C U {oo}. (b) The once-punctured Riemann sphere the complex plane C. (c) The disk El {z E C, IzI <11. (d) The twice-punctured Riemann sphere the complex plane minus a point = C\{0). (e) The punctured disk Ill \ (0). (f) The torus family of surfaces F \ C, where F is the free group generated by the two translations z —*z + 1 and z —~z+T, ImT>0. These enter in the perturbative expansion of string theory as follows. The once-punctured sphere appears in closed-string tadpole computations, the disk occurs in tree-level open-string scattering processes, the twice-punctured Riemann sphere enters in the closed-string propagator, the punctured disk is associated with the decay of one closed-string into open strings and the torus arises in the one-loop vacuum diagram in closed strings. The punctured disk also appears as a limiting case of the one-loop open-string scattering processes. Such processes involve, the annulus family, AT=F\ I-I, where F is generated by z eTz. The limit T ~ of AT is C \ (01 or C \ (01 depending on how the limit is taken. The perturbative contributions beyond one loop in string theory are associated with the non-exceptional surface. Hence, the representation we need in the present work is F \ H. Let us explain the terminology used in the statement of the uniformization theorem. SL(2, Fl) is the group under composition of möbius transformation with real coefficients [64—68].It consists of elements, h(z) of the form =

=

=

=

=

=

=

=

=

=

—~

—‘

h(z) where ad matrix

—

bc

=

=

(az+b)/(cz+d),

(2.2)

1, a, b, c and d being real. Sometimes h is represented as a h=(a

~),

(2.3)

and composition is identical to matrix multiplication. An h e SL(2, Fl), h ~e, has either one or two fixed points. When the fixed points are distinct, h can be written as h(z)—z~

2z—z~ z—z

,

(2.4)

2, know as the multiplier, is real and A> 1, and z~and z are the fixed where A points: h(z÷)= z~and h(z) = z. The point z~ is attractive because, if z ~ z, the nth reiterate of h converges to z~:~ The attractive fixed

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point of the inverse of h, h’, is z_: limp h~’~(z) = z, for z * z~.A subgroup F of SL(2, Fl) is said to have no fixed point in H if the fixed points of any h a- F, h * e, are on the real axis. Each element of F maps F-I into itself. Via F, an equivalence relation on points in I-I is established: z and z’ are equivalent if and only if there exists an h a- F such that z’ = h(z). The symbol F \ I-I means H modulo this equivalence relation. Let D be a set of points in H such that no two points in D are equivalent under F and any point in H is equivalent to some point in D. D is known as a fundamental set. The set F \ H is D. A fundamental domain (or fundamental region), F, is an open connected subset of H such that no two points in F are equivalent under F and any point in I-I is equivalent to some point in the closure, F, of F in H. Certain points in F are identified under F and “gluing” these together produces the Riemann surface [66—68]. Let h~be a series of distinct elements of F. Let z,, = h~(z)for any z a- H. If {z~} has a convergent subsequence, let z~be a limit. The point z~is called a limit point of F. The group F is discontinuous if all its limit points lie on the real axis [64,68, 69]. It is useful to make manifest the relation between the representations of h in eqs. (2.2) and (2.4) -.

a -

—

d ±V(tr h)2 2c

____

—

4 when c*0,

when c=0,

(2.5)

d—a where the ±sign on the left-hand side does not necessarily correspond to the ± sign on the right-hand side. The trace of h is trh =a +d=A +A1, trh

+

~I(trh)2 2

4

(2.6)

and h=

1 _____

ZZ÷

z_A1 —z~A z z(A —A’) —A+A’ z_A—z~A~ ±

,

(2.7)

which, using eq. (2.3), gives a, b, c and d in terms of A, z÷and z. By possibly multiplying a, b, c and d by 1, which leaves h unchanged, the trace of h can always be chosen positive. In this section we make this choice. When this is done, —

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the + (respectively, —) sign on the right-hand side of eq. (2.5) corresponds to the + (respectively, —) sign on the left-hand side.

2.2. GENERATING THE COVERING GROUP

The group F is known as the covering group and is isomorphic to the fundamental group, ir1C/), of ~/ [64, 66]. The universal covering surface of the non-exceptional Riemann surfaces is H. Two Riemann surfaces, F \ H and F’ \ H are conformally equivalent1if where and only if their groups are related by an SL(2, IF) conjugation: F’ = h o F h denotes composition and h a- SL(2, Fl), i.e., for every f’ a- F’, there is an fa- F such °

that f’=hofoh1 [65,66]. It turns out that at loop level g, F is a free group generated by g elements, h 5,h2,. .,hg [70,145]. An arbitrary ha-F is a “word” constructed out of h1, h2,.. hg and their inverses .

.,

h=hoh~”~11...

oh7l,

(2.8)

where m ~ 0 (m = 0 means h = e) is the number of “letters” in the word and i,,, for p = ito m, is 1,..., g —1 org so that h. is h1,h2,.. .,hg1 or hg. Finally n~ is + 1 or 1, the latter indicating the inverse element. The1o inverse of a letter is h not permitted as a neighboring letter in a word, e.g., h1 o h~ 2 is excluded; such an h would be written simply as h2. The same letter can, however, appear several times in a row, e.g., h1 o h1 o h2, also written as h~ h2, is permitted. In the rest of this article the group operation o is omitted and implicit so that h~h2means h1 h~oh2 —

°

2.3. THE FUNDAMENTAL DOMAIN

The fundamental domain, F, for F N H is not unique and there are many ways to construct F. We review two constructions in this section. The first method makes use of the non-euclidean Poincaré metric, ds = IdZI/Im z [64,68—70]. The corresponding geodesics are either arcs of circles whose centers are on the real axis or segments of vertical lines. The distance d(P1, P2) between two points, P5 and P2, is obtained by finding the geodesic between them and integrating ds along the geodesic arc: d(P1,P2)= If~2dsI.Let P~and P~’denote the points on the real-axis on the geodesic closest to P1 and P2, respectively. Then

d(P1, P2)

=

log

(Pr’ —P2)(P, .....p*) —P1)(P2 p*) ~

(2.9)

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Distances are preserved under SL(2, IF) transformations d(h(P1),h(P2))=d(P1,P2),

forhESL(2,IF).

(2.10)

Let z0 be a fixed point in H; we take z0 = i in this paper. For any h a- F, define [65, 68—70] 5(z Ch = ~z~d(z,h 0)) = d(z,zo)},

(ZId(z,h~(zo))
‘h=

1(z (z(d(z,h 0)) > d(z,z~)}. (2.11) Any z a- H is in either C,,, ‘h or 0,,. C,, is a semi-circle centered on the real-axis separating H into two open non-euclidean half-planes: an inside region I,, and an outside region 0,,. We call C,, the h-circle. When h is of the form in eq. (2.4), z is in the closure of I,, in H U (the real axis) = I-il U IF = H. Likewise z~is “inside” 0,, (see fig. 6a). The following hold =

h(Ch)

= Ch-I,

h(Ih)

=

0h1,

h(Oh)

=

I,,_ (2.12)

Z0a-O,,,

so that h maps its circle C,, onto the circle, C,,-, of h~.It also sends its inside (respectively, outside) region to the outside (respectively, inside) region of h~. Figures 6b—6d illustrate this.

(a)

-

(c)

(d(

Fig. 6. The behavior of regions under the map h. (a) displays various geometric objects associated with a hyperbolic map h.1h;The andrepulsive the regionand outside attractive C fixed points are z and z~ its h-circle is C5 the region inside Ch is 5 is O~.The points z and z~ are, respectively, inside onto Thesees same when 1,, and In_i; z~ is also inside Op,. In agreement0h with eq. I,,~(d). (2.12), one thatis true h maps C5 the ontodisk C,,-rather (b), upper-half plane is used as the covering surface. that it maps I,, into 0,,-than (c),the and that it maps

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For h in eq. (2.2), the radius, r,,, and center, c,,, of the h-circle, C,, are 2 =

2+(b+c)2 (c2+d2 1)2 (a—d)

ac + bd ch=(2d2l~O).

(2.13)

The intersection of all outside regions, F=

fl

(2.14)

°h’

h E1

h~e

is a fundamental domain for F \ I-I known as the normal polygon centered at z 0 [65, 68—70]. In practice, the intersection in eq. (2.14) may be restricted to a finite number of elements. See sect. 7.1 where an algorithm is provided for finding these elements. Figures 7a, 8a and 9a illustrate some fundamental regions. The second construction applies when c * 0 for all h a- F, h * e, where c is the real number associated with h in eq. (2.2). For Riemann surfaces with boundary, the case of interest for open strings, an SL(2, IF) conjugation can be performed on F to assure this condition. Define the isometric circle of an h of the form in eq. (2.2) by [68,70] C,,

=

{z~Icz

+

dI

=

1)

(2.15)

,

i.e., a circle of radius i/c with center on the real-axis at (—d/c, 0). Let us denote the region outside (respectively, inside) of C,, in H by 0,, (respectively, I,,). Equation (2.12) holds for the newly defined C,,, 0,, and I,,. The region F, defined as in eq. (2.14) but using the new definition of provides a fundamental domain. For figures and other purposes it is sometimes convenient to use the disk lED in lieu of F—F. This is achieved by performing a conformal change of variables °h’

Z—i w=—--—------, z+t

where z is the half-plane variable and eq. (2.2) is transformed to h=

(~~),

i—w z=i

w

,

1+w

(2.16)

is the disk variable. An h of the form in

2A=a+d+i(b—c),

2B=d—a+i(b+c),

where the bars on B and A in eq. (2.17) denote complex conjugation.

(2.17)

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When this transformation is performed the h-circle has a new radius, r1,, and center, c,, rh2_—

2+b2+c2+d2—2 4 _____________________

a

a2+c2—b2—d2 c

=

a2+b2+c2+d2—2

h

—2(cd+ab) ,

a2+b2+c2+d2—2

,

(2.18)

for h of the form in eq. (2.2). The center, c~,lies outside the unit circle. The point z 0 = i in H transforms to 0 in C. For such a choice of z0, the h-circle coincides with the isometric circle of the SL(2,IF) matrix of eq. (2.17): {zI IBZ +AI = 1), and the above two constructions produce the same fundamental region. The relevant normal polygons in string perturbation theory have a finite number of sides. The boundary of the closure of F in li-I consists of two pieces: (a) regions of the real axis and (b) arcs of geodesics. When (a) is empty, the surface is closed; when (a) consists of isolated points, the surface has punctures; and when (a) involves intervals of IF, the surface has boundary. The latter case is the one relevant in the present work because we deal with open strings. The arcs in (b) appear in pairs which are identified by elements of F [65,66, 68,69]. The arcs have the same non-euclidean length and gluing them together produces the surface. We draw arrowed lines labelled by a group element between arcs to indicate the identification. See figs. 7—9 for examples.

//I

z~

zI_

I

//~ C~i

£F/A17~ F

__

(a)

(b(

Fig. 7. The fundamental domain for the B = 2 H = 0 vacuum bubble. (a) shows the fundamental domain, F, relevant for 1-loop computations in the half-plane representation of the covering surface. (b) is the corresponding situation for the disk. In both cases, F is bounded by the boundary of the covering surface and two arcs of circles which are identified under h, as indicated.

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,~

~V/

~

Ch

C

-

cI~.

///~~/

z*

F

_______

z~ (a)

Z~

Z~

(b)

A,/

(C)

Fig. 8. The fundamental domain for the B = 3 H = 0 vacuum diagrams. Shown are the fundamental domains for figs. 2c and 3c in the half-plane (a) and disk (b) representations of the covering surface. When the h-circles are identified under h, and h 2 a surface with three boundaries is produced. The double of the surface is obtained by extending (a) into the lower-half plane as in (c). (d) displays a canonical homological basis.

The elements, h, h * e, of SL(2, IF) are of three types [68—70].Hyperbolic elements have traces greater than 2, implying that the multiplier is greater than one and there are two distinct fixed points on the real-axis, IF, (see eq. (2.5)). Parabolic elements have 2traces equal to 2 z_ andarea common fixed point Because in IF. For <4 and z~and complex conjugates. F elliptic elements, to h, (tr h) fixed points in H, the covering groups F for Riemann is not permitted have surfaces do not contain elliptic elements. The h-circle and the h ‘-circle of a parabolic h are tangent at the unique fixed point. This point must necessarily be —

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~

Z

(a)

~,

///%~~/

(b(

Fig. 9. The fundamental domain for the B = 1 H = 1 vacuum diagram. Shown are the fundamental domains for fig. 4c in the half-plane (a) and disk (b) versions of the covering surface. The h-circles are to be identified under h 1 and h2.

part of the surface boundary and corresponds to a puncture. Furthermore, a parabolic element can be obtained as a limit of hyperbolic elements in which the multiplier goes to one and the fixed points converge. A parabolic element arises when either a boundary shrinks to a puncture or when a handle is pinched. Hence in our present investigations we can restrict ourselves to groups with only hyperbolic elements*. The appearance of a parabolic element takes place as one moves to the certain regions of the boundary of moduli space. There is a theorem [69, 70] that states that, when F consists of only hyperbolic elements, F is discontinuous. Since such a F has fixed points only on the real-axis, it satisfies the constraints of the uniformization theorem and F N H is a Riemann surface. If the closure of F in H has non-zero intersection with IF then a boundary of type (b) is present and a Riemann surface with boundary is generated; the corresponding F is called a horocyclic group or a group of the second kind [68—70].For groups of the first kind, the closure of the set of limit points is IF. These are the criteria we require for the present work: F of the second kind with only hyperbolic elements. In sects. 7 and 8 we use these to discuss the moduli space for such surfaces. 2.4.

THE DOUBLE OF THE SURFACE

For computational reasons it is convenient to work on a closed Riemann surface. Given a Riemann surface, ./, of the type mentioned in the previous paragraph, *

Parabolic elements are also not permitted for closed surfaces.

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E

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>

F

Fig. 10. The double of a surface. When E’ is glued to E and F’ to F the double of fig. 2d is obtained. Its fundamental domain, Ft., is given irs fig. 8c.

the Schottky double, of it can be formed [65, 66]. It is constructed by taking two copies of ~/ and gluing together identical parts of the boundary. When E’ is glued to E and F’ is glued to F, fig. 10 becomes the double of fig. 2d. It is easy to construct .~ when ~/ is represented as F N H. The fundamental domain is joined with its mirror image in the lower-half complex plane along with any boundary on the real axis. Hence ~ = F N C U {cx}. Fig. 8c corresponds to the double of fig. 8a. In open-string string perturbation theory, the gth-loop correction, g ~ 1, is associated with a covering group F generated by g elements. The surface ~/ = F N H can have any values of B and H subject to B + 2H = g + 1, B ~ 1, where B is the number of boundary components and H is the number of handles. For example, at ioop level one, B = 2, H = 0 and at g = 2, one may have B = 3, H = 0 or B = 1, H = 1. The double of the surface has g handles and hence is a closed surface of genus g. On the closed double, P~, a standard homological basis, A1, A2,. Ag, ~,

. .,

B,, B2,..., Bg, can be chosen [64,65]. The cycle or closed curve, A,, wraps around the ith handle. The cycle B. loops along the ith handle. The A, and B, satisfy the intersection property that A, does not intersect any A1, i *1, that B, does not intersect any B1, i *1, and that A, intersects any B1, only for i =1 (see fig. 8d for an example). There is not a unique choice for a standard homological basis. 2.5. THE ONE-LOOP CASE

Let us illustrate the above in the one-loop case for which off-shell computations have been performed in ref. [52]. The cover group is generated by a single

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hyperbolic element: F = (h”, n a- Z, h is hyperbolic). By performing an SL(2, IF) conjugation the fixed points of h can be chosen as 0 and x• Hence h(z) = A2z. For z 1, Im z> 0) and C,,- is (z~Iz~= A. Im z> 0). It is clear 0 i,h C,, is C,, {zI onto Izi = C,,A that maps and that eqs. (2.12) are obeyed. The intersection in eq. (2.14) is saturated by h and h1. Hence F is the annulus region {zIA’ < Izi 0) as shown in fig. 7a. The inner and outer semicircles are identified under F. The surface is AT mentioned in sect. 2.1 for A2 = exp(T). The double of the annulus is a torus: {zlA~< z~
3. The construction of the holomorphic differentials and other quantities associated with the Riemann surface This section provides concrete computation methods for the surface quantities that enter in the open string amplitudes. 3.1. THE HOLOMORPHIC DIFFERENTIALS

Throughout this section we work on the double, ~ = F N C U (cc) of the surface = F N H, where the covering group F is of the second kind* and contains only hyperbolic elements. The double has genus g. Fix a standard homological basis, A,, A 2 Ag, B1, B2,..., Bg. It is a standard result that as a closed surface of genus g, has g independent holomorphic 1-forms, w,, i = 1,. . g [64, 65]. The basis is uniquely determined by requiring ~,

.,

(3.1) To perform the string perturbation expansion, these i-forms must be explicitly calculated. Fortunately for the open string case, a constructive procedure is known. Let x0 be a point on the real-axis IF. Define the Poincaré series, O(z, x0), by [68,70,73,74,81] dh

1

(3.2) ~ h(z)—x0 2 for h of eq. (2.2). For the F we are considering, ref. where dh(z)/dz is (cz + d) [81] has proven that the series in eq. (3.2) converges uniformly if the point cc is not in the closure of the limit points of F. An SU2, IF) conjugation can be performed on F to assure this (then one may undo the conjugation if so desired). *

For the definition of second kind, see the end of sect. 2.3.

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Under a local change of coordinates, O(z, x0) transforms as a i-differential. A q-differential, w, is an object which transforms as

dz

w(w)=w(z)

q ,

(3.3)

when w and z are local coordinates parametrizing the same region of the surface. O(z, x0) is an automorphic form of dimension —2 [68,70,73] meaning that if h a- F, then* dh(z)

dz

‘

6(z,x0).

(3.4)

Equation (3.4) ensures that as a 1-differential 0 is globally well-defined on .~21. Note that 0(z, x0) has no poles as a function of z for z ~ IF: from eq. (3.2), the poles in 0(z, x~)occur when z = h(x0) for any h a- F. By selecting x0 to be a limit point of F, 0(z, x0) becomes a holomorphic 1-form on Sufficiently many holomorphic i-forms, 8(z, x0), can be constructed by varying x0. From these, g-linearly independent ones can be found. Let h, be a covering group element associated with B,; h, is unique up to a conjugation; and it may or may not be one of the original g generators of F. If h, = (a,z + b,)/(c,z + d,), let (3.5)

dz

w~(z)= —0(z,J,,). 2~n

(3.6)

The B, cycle goes from a point, p,, on the h-circle C,,. to the point h(p,) on the h-circle C,,~1(as shown in fig. 8d). Reference [81] has proven that w,, as defined in eq. (3.6), for i = 1, g, is a basis of holomorphic i-forms obeying eq. (3.1). The 0(z, J,,) satisfy . . . ,

0(z,

Jh,h~) =

0(z, .1,,,_~) =

0(z, .~,,.)+ 0(z, j,,1), —0(Z,

J,,.).

(3.7)

From eq. (3.7), conjugation of h. leaves 0(z, J,,) and hence w, unchanged. *For an automorphic form of dimension —2m, the first factor on the right-hand side of eq. (3.4) is 2~0. replaced by (dh(z)/dzY’”

=

(cz

+

d)

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As an example, consider the one-loop case. The group F is generated by a single hyperbolic element, h, with fixed points at z÷and z. The sum in eq. (3.2) gives dz 2irt

1 z—z÷

_____

—

1 z—z

(3.8)

.

It is convenient to partially do the sum over F in eq. (3.2) when constructing w, in eq. (3.6). Let W,,. be the set of words in F which do not begin on the left with h, or ha’. Any element h of F can be written as h~wwhere n a- Z and w a- W,,. The sum in eq. (3.2) can then be performed over n yielding

______

1 =

—

h(z)_z~)

1 ,

dh +

I

—

ZZ÷

ZZ

hEw,,

dz

1

1 ,

h(z)—z~

—

h(z)—z

,

,

(3.9)

h ±e

where z’~and z’. are the fixed points of h. and the second equality is obtained by singling out the contribution when h = e. If it were necessary to numerically compute a g-loop open string amplitude, the can be numerically evaluated by saturating eq. (3.2) or eq. (3.9) with group words. The convergence of these series guarantees that, when a sufficient number of terms are included, any desired accuracy can be achieved. 3.2. THE PERIOD MATRIX, e-FUNCTIONS, PRIME FORM AND OTHER QUANTITIES

The period matrix [64, 65],

Tt

1,

is calculated by doing the integral along B~of w, (3.10)

= jh,(b)

Equation (3.10) is true not only for b a- C,, but for arbitrary b not on the real axis. Line integrals such as in eq. (3.10) can be performed for each term in the sum in eq. (3.2) or eq. (3.9)

f w~=—1 ~ln

h(z) —J,, h(z)—J,,.

z

2~T1hET

~‘

1

~

ln

2~~hnw,,,

Hence,

T~J is

h(z) —z’ .

h(z)—z_

eq. (3.11) with z = h 3(b) and z’

=

b.

(3.11)

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In general, the period matrix is symmetric with positive definite imaginary part [64,65]. In our case has purely imaginary entries. This is seen from fig. 8d. The B3 contour can be deformed onto the real-axis. For z and x0 real, 0(z, x0) is real so that w, in eq. (3.6) is purely imaginary. The integral along a real interval of a purely imaginary integrand is purely imaginary. 9(zIT) by [64,71,721 For z adefine the theta function ~ e(zIT) = ~ exp[ilrntrn + 2’Trintz], (3.12) T~

1

~

n

E

Z~

where the superscript t denotes transpose and matrix multiplication for terms in the square brackets is understood. In eq. (3.12), is the g X g period matrix of ~ and the sum is over n, a g-tuplet of integers, n = (n,,. ng), n, a- Z. Because T has positive-definite imaginary part the infinite sum in eq. (3.12) is highly convergent. Many surface quantities enter through the theta function via the form [77—79] T

. . ,

f(z)

=

~(eo

+

(3.13)

~:WT)~

where e 0 is a fixed vector in and z0 is a fixed point of i.e., of the fundamental domain, F© = (closure of F in C union its complex conjugate) and = (wi, ~ Sending a point, z, on the surface to f~wa- C~is known as the Jacobi map. The object f is a section of a line bundle as opposed to a function because it is not automorphic with respect to F: when z is moved around an A, cycle f(z) returns to its original value; however, when z is moved around a B~ cycle f(h1(z)) *f(z). Instead, the theta function transforms as in table 1. ~

~,

. . .,

TABLE

1

The transformation laws for surface quantities. Here h1 is the group element corresponding to the homological element B, (see figs 8c and 8d). The order of a form is the parameter q in eq. (3.3). The prime form is a — 1/2 form in the variable and in the variable w

Object prime

Symbol E(z, w)

theta function

~

sigma

ff(z)

Order of form —

Transformation law

~

0

E(h~(z),w) ~

f

+

f”~w

if(h~(z)) =

exp( _i~r,i— 2~iiwi)

—

T

=

exp[_i~rii

—

[

1/2

dk(z)

E(z, w)

2~i(fi+

+

/

~

~g

=

(—

T)

w

I)*exp[i~Tii(g — 1) dh,(z) dz

fw

-g/2 if(Z)

+

2~i(g— 1)

jZwj

—

2~i(~~11).J

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[~

The theta function 6~ ~ with characteristic ~ where both a and ~3are g-tuplets with 0 or as entries, is defined by [64,71,721 -~

~ r—a e

=

exp[i1TrtTr+2~Tirt(z+I3)],

(3.14)

so that the sum over r, involves integers or half-integers. The ordinary theta function in eq. (3.12) corresponds to a = f3 = 0. There are 2g such theta functions of which 2g—1(2g 1) are odd, @[1(_zH) = [64,71,72]. Let a=J3=(1/2,0,0,...,0).

t9[~1

—

be any odd non~singular*theta function, e.g.,

Define

g s ~ (z)= ~

E(z,z’)

=

w,(z),

(3.15)

@[I(JzWT)/~s[~J(z)s[1(z~).

(3.16)

It can be shown [72, 78], by the Riemann vanishing theorem, that s ~ as a function of z a- F~,has only double zeroes so that taking the square root in eq. (3.16) makes sense and that E(z, z’) has a single simple zero at z = z’ and no poles: dividing out by the square root factor in eq. (3.16) removes the other zeroes of e[~](f/wIT). Equation (3.16) is independent of the choice of the odd nonsingular characteristic. The object E(z, z’) is a differential in z and in z’ (see eq. (3.3)) and is known as the prime form [72]. It is unchanged when z is moved around an A, cycle but not so under B~cycles (see table 1 for its transformation properties). Define the sigma function, o-(z), by [72] ~‘

—

a-(z) =

-~-

exp[_ ~ ~ w~(z’)lnE(z’, z)]~ i=1

(3.17)

A,

where z’ is the integration variable in the contour integral. It is locally a g/2-differential with no pole and no zeroes and transforms as in table 1. As emphasized in ref. [791, a- carries the conformal anomaly and hence only ratios of a- enter into *

Non-singular means that not all the first derivatives of ~9[] with respect to z, vanish.

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string calculations. The ratio is expressible as [79]

a-(w’)

—

a-(w)

—

~(~f.1f~w

—

f~’,~’W

e(~1J~wJw —

~0IT)l1Y~=lL’(z~,M))

~~0pT)flf1E(Z,,W’)’

3 18

(

where z,, i = 0, 1,..., g, are arbitrary points; the ratio in eq. (3.18) is independent of the choice of these g-points. The g-component object, z1~,,,known as the vector of Riemann constants [71, 721, enters in the Riemann vanishing theorem and depends on a base point z0

(~z,,)j =

-

2

+ i1 ~ ~A w,(z)fw~(w), z0 i +1

(3.19)

where z and w are, respectively, integration variables in the first and second integrals. The results in later sections are expressed in terms of theta functions, the prime form and the a--function. A basic idea is that, although individually they are not automorphic due to changes when going around B, cycles, ratios of products can be constructed to yield the most general function and q-differential on Furthermore, being defined by convergent series and well-defined integrals, they are numerically accessible by computer if need be.

3.3. THE RIEMANN VANISHING THEOREM

In the previous section the zeroes of the prime form and a--function were given. Those of the theta function, e(zIT), are determined by the Riemann vanishing theorem [64, 71,72]. There are different ways of stating the theorem, one of which is the following. Theorem (Riemann): The function f(z) defined in eq. (3.13) either vanishes identically or has g zeroes. In the latter case, the zeroes, z~, satisfy the equation [80] g

~f

i=1 ~o

Cs) =

—e0 + ~

+ fl + Tm,

(3.20)

where is given in eq. (3.19) and where n and m are vectors of integers; Tm means the vector obtained by applying the period matrix to m. Equation (3.20) represents g equations, one for each component of the vector quantities. The term n + ‘rm reflects the ambiguity from the quasi-periodicity of the theta function.

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A simple application is for the function entering in eq. (3.8)

~1 j=1

WJ

IT)

W~

=0,

(3.21)

Z 0

Z0

at z = z, for any i; there are exactly g zeroes. We use the vanishing theorem in sects. 4—6 and appendices B—D.

3.4. THE ONE-LOOP CASE

This subsection illustrates the computation of surfaced quantities for the oneloop or (g = 1) case. The single holomorphic i-form, to, is given in eq. (3.8). To make contact with the results in ref. [52] we take the limits z—* cc and z÷—’0 dz (3.22)

2 iT IZ Consequently, the integral which appears in eq. (3.11) is

(3.23) and the 1 x 1 period matrix is 2 = iT (3.24) 2ii- A 2ir —In when the generating element of F is h, where h~(z)=A2z = exp(T)z. Equation (3.24) is eq. (2.2) of ref. [52]. For g = 1, the theta function in eq. (3.12) becomes the widely studied [71] theta function of a single variable. Straightforward algebra yields ~,

T=

s[~~](Z

=(1/2iTiZ)e’[~](0)~

and one finds that

E(z,w)

=

2iTi~I~91/2 ~~~~_in(±-) 1/2 2iT1 \W 1 2 e’[ i/2](0~T)

T

(3.25)

is the prime form. The logarithm of eq. (3.25) entered as a piece of the propagator in the one-loop computations of ref. [83] (see sect. 4 of ref. [83]).

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Evaluating the cr-function using eqs. (3.17) and (3.25), we get (3.26)

a-(z)=i,

for our choice of F. The vector of Riemann constants is (3.27) since the second term in eq. (3.19) does not contribute. Unlike the g ~ 2 case, it is independent of z0. The Riemann vanishing theorem says 1 @ —ln 2iT1

Z z,

iT 2 +T

2

=0,

(3.28)

when z =z 1, which is clearly true since 9(~

—

~TIT)

=

0 [71].

4. The conformal map The conformal map takes a canonical domain of C onto the Witten configuration. For the canonical domain we use a fundamental region, F, of the Riemann surface, ~/ = F N H, where the covering group, F, is to be determined. We choose for F a hyperbolic polygon of the type described in sect. 2 so that F is a connected region of the upper half plane bounded by the real-axis and by 2n arcs of circles with centers on the real-axis. The arcs are identified pairwise by hyperbolic SL(2, IF) maps, h,, i = 1,..., n. The image of the conformal map, p(z), as z varies in F is a configuration of the Witten string theory: three strings interact by overlapping halves, asymptotic strings are semi-infinite rectangular strips of width and intermediate propagatars are finite-length rectangular world sheets of width As F is continuously varied, subject to certain constraints (eqs. (4.3), (4.5) and (4.6) below), the lengths of the propagators vary. When we take the Schottky double, of the surface, the range of z is extended to F~.When arcs of F~are identified, a closed surface of genus g is obtained, for the gth loop amplitude. The conformal map, p(z), taking F onto the string configuration is determined from the quadratic differential, iT,

iT.

~,

~t

(~)

g

2

~(z)=

72

Ef

2N+I—1

W+fW~T)

2

[a-(z)J~,

fl~iE(Z,ZrH

(4.1) by taking the square root of eq. (4.1) and integrating with respect to z. The

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parameters in eq. (4.1) are fixed by imposing constraints. The rest of this section explains these and discusses The notation is as follows. In eq. (4.1), N is the number of external states and I = 3(g 1). The variable Zr is the asymptotic positions of the rth string located somewhere on the real-axis part of the boundary of F. The quantities, E1( I )~ E( ), ~ L1~ and are, respectively, the theta function, the prime form, the period matrix, the vector of Riemann constants, and the vector of holomorphic differentials, to = to 2,. . tog), where to,, i = 1,..., g is a basis for the holomor~i.

—

,

to

(to,,

.,

phic differentials on They and a-(z) were given in sect. 3. The point, z0 a- F, is arbitraryandthepoints,P,,i=i,...,g,andw0,a=1,...,2N+I—1,areallinF~, and fixed by conditions specified below in eqs. (4.3), (4.5), (4.6) and (4.7). Finally, .~Kis a real normalization constant. Define ~.

Ym_=W2m_i, WlN±Jl±,~Pj,

YU,~W2m,

i= 1

m=i,...,2g—2+N,

g,

(4.2)

where we require W2m -, and W2m to be complex conjugates of each other and hence V,,, a- F is the complex conjugate of ~m a- F. Equation (4.2) says that the Wa are the I’m and Ym and that the last g Wa are also called P~.We use the redundant P, for notational and expository convenience. The first constraint on the map is 2N±!—)+g ~

N

~ f “to1—2~f ‘to~=4(z1~,,).—2,

a=1

Z0

r=1

(4.3)

Z0

that is, we require the w, and Zr to satisfy eq. (4.3) for j = 1,. g. Note that the argument, 4,~,— f~to+ f~to,of the e-function in eq. (4.1) can be written as ~ + ~ 2~iJ~z,,~to + f~tousing eq. (4.3). Equation (4.3) is needed to ensure that p is periodic when arcs of the domain are identified under the h1, j=i,...,n. For the i-loop case, eq. (4.3) reduces to the constraint in eq. (2.3) of ref. [52] found previously for g = 1. This is seen by using eqs. (3.19) and (4.2). Furthermore, eq. (4.1) becomes eq. (2.1) of ref. [52]. Using the formula for in eq. (3.19), eq. (4.3) is seen to be independent of z0. We can thus let z0 be a boundary point of F on the real axis. The construction of the holomorphic differentials in sect. 3 reveals that f~,’w.,and f2”°~ + f~”to3are purely imaginary for z0 and Zr real, and when Ym and I’,,, are complex conjugates of each other. From eq. (3.19), (~Z~)j is purely imaginary. Hence when multiplied by i, eq. (4.3) represents g real constraints. . . ,

~—

—

—

-~-

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For to be a quadratic differential, (a) it must correctly transform under local coordinate change of variables and (b) it must transform properly under the identifications of the h,, i = 1 n. Under local transformations, E(z, w) transforms as a /~-differentialin z (and in w if w is transformed), 0 is locally a 0-differential, i.e., a function, and a- transforms as a g/2 differential (see table 1). Since a p-differential times a q-differential is a (p + q)-differential, the order of ~ is ~(2N + I 1 2N) + 3g/2 = 2. Hence, is a quadratic differential and (a) is verified. Since the h, are generated by the h 3, j = 1,.. g, which cover the ~t

—

—

—

—

~i

.,

cycles, B1, it suffices to check (b) for these h3. Using the transformation properties in table 1 and eq. (4.3), one sees that

( ~) dh-(z)

=

—2

~(z).

(4.4)

This is the correct global transformation to guarantee that p is single-valued when going around the I3~cycles. The quadratic differential, p., is uniquely determined up to a normalization constant by its poles and zeroes. The prime form E(z, w) has a single zero at z = w and no poles, a-(z) has neither poles nor zeroes and ~ E~=1f~’to+ f~to)has, according to eq. (3.21), g zeroes at z = P, = W2N+l i = 1,. g. Hence, p. has double poles at z = Zr,..., z = ZN and simple zeroes at z = W1,.. Z = W2N+j_ I Let p.’(z) be any other quadratic differential with double poles at the Zr and simple poles at the w1. Then p.(z)/p.’(z) is a function defined on the compact surface ~ and hence a constant [64—66].Using p.’(z) in place of p.(z) simply redefines ~K. The argument in the previous paragraph shows that it does not matter which g of the I’m and I’,,, we choose for the P, in eq. (4.1). A different choice is compensated by 1,. a different 1gm’ m = V, are ~V. the interaction points, that is, the preimages under p The of the midpoints where three strings join. Here V = N + 2(g — 1) is the number of vertices in the Feynman graph. To correspond to a Witten configuration, the quadratic differential p. is required to have zeroes at the interaction points and double poles at the asymptotic positions, Zr [28].This is the case for p. in eq. (4.1). After integrating with respect to z, p(z) has logarithmic singularities at the Zr and behaves like const X (z ~m)3”2 and const X (z ~m)3’~’2 near ~m and ~‘m~ The width of rth external state is in a Witten configuration. This is the case provided —

~

. . ,

.,

. . ,

—

—

iT

tim where a

+ (respectively,

dp

(Z~Zr)~~ =±1,

dz

(4.5)

—) sign occurs for a strip coming in from the left

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541

(respectively, right). That the ath internal propagator string have width

I

dz(~-~) =

iT

requires (4.6)

±11T,

where the integration path, Ca, goes from 1~m to I’,,, within the region corresponding to the ath propagator strip. Here, I’,,, is either one of the two vertex points at the end of the propagator. An example is illustrated in fig. Ii for the surface in fig. 2. There are two Y,,, located at A and D. The widths of the strip on the left, the middle strip and the strip on the right in fig. 2 are fixed by doing the integral in eq. (4.6) over the paths, CL, CM and CR in fig. ii. The length of the ath propagator strip is given by

7, ~

(4.7)

for any integration path residing in the propagator strip between the two interaction vertices, I’m and I’,. For example, these paths can be taken along the mid-lines of strips where Im(p(z)) = ‘ir/2. The absolute value in eq. (4.7) assures that Ta is positive. The differential p. should only be a function of the ratios of a--functions. Although this does not manifestly appear to be the case in eq. (4.1), the condition in eq. (4.6) induces a 1/a-3 factor in the normalization constant, .A7. As a check, let us count the number of parameters in p(z) versus the number of constraints. There should be three more parameters than constraints because the map is determined only up to an SL(2, IF) transformation. For F, it takes g elements to generate the group and three parameters for each generator; the two fixed points and the multiplier or a, b,c in d being fixed by ad bc = i~. The number of real parameters associated with the W~, the Zr, the degrees of freedom in F and the normalization constant, ~ are respectively 2N + 4g 4, N, 3g and 1, yielding a total of 3N + 7g 3 real free parameters in the map. The number of constraints is as follows. There are I + N internal and N external width = equations (eqs. (4.5) and (4.6)); there are the I + N equations for the lengths of the internal propagators, eq. (4.7); and there are the g real constraints from eq. (4.3). Using I = 3(g 1), we get a total of 3N + 7g 6 constraint equations. The counting works. The image of p is a configuration with N + 2(g — 1) interaction vertices, N external string strips and N + 3(g 1) internal propagator rectangles, all joined in the Witten manner. Figures 2—6 of ref. [49], fig. 2 of ref. [52] and 2e display some

(~~),

—

—

—

iT

—

—

—

*

Alternatively, SL(2, t4) invariance can be used to fix three parameters in F and then the number of free parameters is reduced by three and should equal the number of constraint equations.

S. Samuel

542

/

/

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/

Open bosonic string

/

I

QO~JK(LO

HQM(

E

(b)

Fig. 11. Fundamental domain—string configuration correspondence. (a) and (b) show, respectively, the string configuration and the fundamental domain for fig. 2, the latter consisting of the upper-half plane outside the four circles. The covering group F identifies the two circles on the left and the two circles on the right in (b) thereby creating the two outside loops in (a). The middle string region is shaded in both figures. Points in (a) are shown in (b) by the same letter or a primed letter. Primed and unprimed letters are identified under F. Dotted lines indicate Im(p) = ir/2 (a) or the values of z which produce Im(p) = 2r/2 (b). When the dotted lines in (b) are joined under F a picture of the Feynman graph is obtained. This construction is used in sect. 9. Also displayed are the paths of integration in eq. (4.6) used to determine the width = cr/2 condition on string strips. The paths CL, CM and CR are for the left, middle and right propagator strips. The paths go from a singular vertex point (A or D) in the upper-half plane to its mirror image (= complex conjugate of A or D = B or E) in the double.

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examples of the p-plane image of the map. The map has cut singularities atYm and 17m and 17m’ m = ~‘m In the z-plane, draw the branch cuts between each pair, 1,..., N + 2(g 1), so that, in the p-plane, a physical cut appears at the interac—

tion point along a line of constant Re(p(z)). There are three possibilities for drawing the cut, corresponding to choosing which pair of strips to sever. Pick a case that severs the maximum number of external legs. In the p-plane, these cuts are to be identified. 5. The correlation functions This section computes the correlation functions for fixed values of Ta by transforming from the Witten configuration to the fundamental domain, F. In transforming, computable conformal factors are generated. Finally, the correlation functions on F are calculated using the results in refs. [78,79, 108]. 5.1. TRANSFORMING TO F

The relevant correlation functions contain tachyon operators to produce the external states and b 0 insertions in the propagators. The tachyon operator for the rth string is c(tor)exp(k~.X)(to~~), where k~is the external momentum and w’= exp(~(z))is the rth asymptotic position. The string variables ~(Z) are defined in terms of p(z) (see refs. [85,42, 48]). The b0 insertion on the ath propagator strip is represented as bø=~(d~la/2iT~)7lab(37a),

where the contour is taken across the strip and its double. When it is transformed to F, it becomes ~

rth7a 2iri

r<,2iT1 dWa

dp(Wa) dWa

—

b(Wa),

(5.1)

where ~ is the corresponding closed contour in F~. The transformation law for a primary field, O(tor), of dimension d, located at the asymptotic position to’~, is O(tor) —±exp(dN~)O(Z~),

(5.2)

where N~is the zero—zero Neumann function for the rth string. The dimensions of exp(kr X) and c are respectively a~kr. kC and 1, so that the conformal factor generated by the rth tachyon state is exp[(a’k’~ k~ —

.

5.2.

—

i)N~].

CORRELATION FUNCTION IN F

The amplitude factorizes into a correlation function for the X’~-systemand the bc-system. The bc-correlator can be computed using bosonization via the methods in refs. [78, 79, 108].

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Let us compute the correlation function on F for an arbitrary boson system with e = 1 and background charge Q [19] using the axioms in ref. [108]. The boson, 4’, is decomposed into classical and quantum pieces

4’(z) =4’~~(z) +cbqu(z),

(5.3)

where

&i(z)

=

2iTi

g ~ q~f Wi. i=1

(5.4)

Z 0

The classical background solution depends on g momenta, q,, compactified on a lattice, A. To compute a correlation function, the decomposition in eq. (5.3) is substituted for 4’(z) and the classical piece is averaged over [108]

(i-~i~ ,1

)exP[iiT~ ~ q~T~1q1—2iTiQ ~ i=) j=)

q,wA

~t(~5)i]~

(5.5)

i=1

where is the period matrix and 45) is the vector of Riemann constants (see sect. 3). The quantum piece, 4’qu(2), is treated as a free field so that Wick’s theorem is used in conjunction with the two-point correlator i-,1

(4’qu(Z)41qu(14~’))=ln[E(z,w)J

,

(5.6)

where E(z,w) is the prime form in eq. (3.16). When A = Z, the integers, the above rules give ~exp(A14’)(Z1)...exp(A~4’)(Z~)) =<1)~~~e( ~ArfZ~W_Q4zo)flE(Zr,Zs)ArA~fl[a-(Zr)1QAr, r=)

Z0

r
(5.7) r—1

where, with the exception of ~ the quantities on the RHS of eq. (5.7) are given in sect. 3; (1)~~~ is the partition function for the oscillator part of 4’ with zero modes excluded. beenin computed (there, it =is 2) andIt ishasgiven eq. (B.9). inForrefs. g = [78,79] 1, (i>~=f(TY’ denoted byexp(—mT)Y’, Z~” where T= 2iTiT and eq. (5.7) agrees with eq. (4.13) of ref. [83] after eqs. (3.14), (3.25) and (3.26) are used. Equation (5.7) is non-zero only when )~_iAr=Q(g— 1). —

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Open bosonic string

For the bc-system, Q = —3, c = exp(4’) Using eqs. (5.1), (5.2) and (5.7), we get

=

(c(wi)

. . .

C(toN)

=

1, and b

=

exp(—4’)

~—‘

A

=

—1.

1~

~

)~1(N~J~:)

><

fl

fl

E(Zr,Zs)

1+r
fl

E(Wa,Wb)

(~a
x fl [a(Z~)] when

A

d

i+N ~bc

~-*

545

3fl[a-(w

E(Zr,Wa)~’

1~r~N (~a~I+N

3, 0)]

Z

1 >Z2> >ZN> 1w11> W21> > IWI+NI. The points Wl,...,WN in the string configuration correspond to Z1,. . ., ZN in F. Likewise, ~ correspond to w1 WJ+N, I = 3(g 1). The X-correlation function can be computed by the same 26p~/(2iT)26: methods, except Q = 0 and the momenta become continuous, q, —s p,, ~q —s fd ...

...

—

,~

K ~

.exp(kN .x)(toN))

=

(exp(k1 .X)(w’)

=

fl exp( a~kr krN~)[Ki~

. .

N

d26

g

.

]26

~f (2iT)~

i=1

XeXP[(2a’)(iiT~P~TiJP~J + 2iTi i,j

~ ~ r=(

i=I

Z 0

X

fl

E(Zr, ~

(5.9)

I ~
when Z1 > Z2> ... > ZN. For g = 1, eq. (5.9) reduces to eq. (3.1) of ref. [52] when eqs. (3.22)—(3.25) are used. Equation (5.9) does not depend on z0 due to momentum conservation, ~ 1k’~= 0. 5.3. THE ZERO-ZERO NEUMANN FUNCTIONS

Formulae for the g-loop N~have been presented in eqs. (2.8)—(2.li) of ref. [49] and in eqs. (3.6)—(3.9) of ref. [52] using the methods in ref. [42]. One simply substitutes the map in eq. (4.1) into these formulas. To save space, we do not repeat the formulae here but we refer the reader to refs. [49,521.

S. Samuel / Open bosonic string

546

TABLE 2

The combinatorial factor for vacuum graphs. The second column displays the graph. In the third column is c 0, the combinatorial factor from field theory including the number of ways of joining propagators to produce the graph. The fourth column contains O~,the number of times a Feynman graph overcounts a Riemann surface. The total combinatorial factor is given in column 5 and should equal l/2t’, according to eq. (J.l), where V is the number of vertices in a graph Entry

Graph

c0

0~

Total Entry

Graph

g=1

B

=

2

H

c0

0~

Total

g=3

1

= 5)

B =

4

~I~—~D

H

=

0 (continued)

I

g=2 B—3

H=0 8

2

Q(3

-~

2

~:-~~i::~

_.~

22

-~

I

3

—3! 3!22

~ 22

g=2 B=I

9

(0

I

(2

~~‘1’~

H=l

—~----

3

—

2~

3!2~

g=3 3~2

3!

B =2

B=4H=0

11

5~_~-~

6

O~O—O

H= I

2

-~

2

112

-~

13

(:)

2

1

-~

2

-~

22

24

1

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TABLE

Entry

Graph

c

0

0~

2

(continued)

Total Entry

Graph

g=3 B

=

2

c0

00

Total

g=g

H = I (continued)

14

547

B =g

~2

2

+

I

~g

(8

H

=

(1

1

g-

~-2

tadpoles

22~2

attached

g— (5

_22 4)8

l 2~ 19

00

0 gclrcles

g=4 B=3

5.4.

2 2253

225_2

H=l

1

16

l7

I

2(1

x

2g

—

2

1

THE OFF-SHELL RESULT IN T,~-SPACE

The contribution from an individual Feynman graph, G, is I±N

AG(kl,...,kN)

=gVa~Pc0

fl a=1

f

dTaK )~( >bc’

(5.10)

0

where K )~and K )bc are given in eqs. (5.8) and (5.9), g,~ is the on-shell three-point tree-level tachyon coupling, a’ is the Regge slope parameter and CG is the combinatorial factor associated with a Feynman graph (see tables 2 and 3 for examples of cG). The term g~”in eq. (5.10) arises because each vertex has a factor of the coupling constant g~.Here V= N + 2g 2 is the number of vertices. —

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3 The combinatorial factor for graphs with external legs. For notation, see the caption for table 2. The factor l/F~ is the number of times a Feynman graph is overcounted by integrating the Zr freely over the boundary. The total combinatorial factor, according to eq. (J.2), should be l/ 2V. TABLE

Entry

Graph g=2

cG

F~

1

1

O~

N=I

1

1

i_—~—Q

2

2

1—~--O

2

3

0T0

2

4

i—Ill

5

i—c

I

W

g=2

9

8

1 —

6

1 3!

—

2~

3!

N=2

6

8

Total

-

-

2

-~

-~

~

-

2

~

-~

2

1 1—Q—Q_2

-~

1

1 2

S. Samuel / Open bosonic string

549

TABLE 3 (coNTINUED)

Entry

Graph

c

0

1

10

g=3

g=3

13

1

2

—~

1

-~

1

1

1

N=2

12

g=4

3!

N=l

‘—0—0-—C

11

Total

Fz

-

2

1

1

N=1 —i

—~

Likewise the propagator in eq. (1.2) produces a factor of a’ for each internal line. The number of propagators is P = I + N = 3g 3 + N. The total amplitude is the sum over graphs G each of which is given by eq. (5.10). —

6. The integrand for the g-loop amplitude For the four-point [33] and N-point tree amplitudes [49], it was discovered that most of the ghost correlation function is cancelled by the Jacobian in going from I~ variables to Koba—Nielsen variables, Xa~This ghost-Jacobi identity was extended to the one-loop level in ref. [52]. Because the b0 associated with each propagator is represented as a line integral (see eq. (5.1)) the ghost correlator involves an 1+ N-dimensional integral where I + N is the number of propagators. The goal is to show that .1K )bc is equal to ordinary functions associated with the surface and does not involve an I + Ndimensional integral. Here N = the number of external particles, I = 3(g 1), g = loop order and J is the jacobian —

J=det(Jab),

(6.1)

5. Samuel / Open bosonic string

550

where

“ab

is the I + N by I + N matrix aT ~ab

=

(6.2)

~

and j arises from I+N

fl a=I

f

d1~

=

0

f ~G

I+N

fl

(6.3)

dXa J.

a=1

The region 0 to cc for all the Ta corresponds to some region, ~G’ in the variables for the Feynman graph, G. This section derives the general g-loop (g 2) ghost-Jacobi identity. N of the Ta are traded for x1~, Z~,. ZN, the asymptotic positions of the states in

Xa~

~‘

..

n

,

F~, IF. The remaining I of the Ta are exchanged for a certain set of moduli parameters, x1,.. ,x,,, which is specified below and which describes the geometry of the Riemann surface, ~/. It is only for a judicial choice of x1,. .,x1 that we are able to derived the g-loop ghost-Jacobi identity and hence eliminate the I + N integrals associated with the bc-system. Once the identity is established for this choice, one can convert to any other set of moduli variables. In short, this section obtains the integrand of the Witten field theory in terms of the variables used in the first-quantized or Polyakov approach [63]. .

.

6.1. THE MODULI PARAMETERS

The x1,. x1 are determined as follows. In sect. 3, g linearly independent holomorphic differentials, to,, i = 1,. g, were constructed. A consequence of the Riemann—Roch theorem (see refs. [64, 65]) is that the number of independent holomorphic 2-differentials is I = 3(g 1). Denote such a set by 1t-~,a = 1,..., I. Candidates for the 1It’~are the g(g + 1)/2 different products, to,to1. For g ~ 2, g(g + 1)/2 ~ I, so one might expect that all lpa can be constructed from such products. This is almost the case. There is a theorem of Noether (see ref. [64]) which states that for g ~ 3 the products to,to3 fail to span the space of holomorphic 2-differentials only for hyperelliptic surfaces, compact surfaces represented as a double cover of the sphere via the Riemann surface of the function f(z) = ,/(z a1)(z a2) (z a2g+2) [64,651. For g ~ 3, the dimension of the space of hyperelliptic surfaces is 2g 1 which is less than the dimension, 3(g 1), of moduli space. Hence, almost everywhere in moduli space, the ~p~a can be written as a linear combination of to,to~ . . ,

. .,

—

—

—

...

—

—

~I~a=

~

c~(2iTito~)(2iTito1), a

—

= i,...,I,

1 +i+j+g

where c,~ are constants and the factors of 2iri are put in for convenience.

(6.4)

S. Samuel / Open bosonic string

551

Given the i = 1, g, it is straightforward to determine a linearly independent set of bilinear products and hence a set of coefficients, c’1,1. The 11,11 can then 2 3 3 be defined by eq. (6.4). 2, one 2can take = 1, c 22 = 1 and c 12 = c 21 = 2, For ~I~2g= =(2iTito and sJ,3c’11 = (2iTiw,)(2iTiw so that ~P’= (2iriw,) 2) 2). The parameters, x1,. x~,are defined in terms of the same constants to,,

. . .,

. .,

~

Xa =

I ~i 2iT

a

Cat.,

=

1,..., I,

ej ~g Im(T~~),

(6.5)

t,1

where

Im(T~ 3)is

6.2.

the imaginary part of the periodic matrix.

THE GHOST-JACOBI IDENTITY

The derivation of the ghost-jacobi identity employs the following strategy. Consider correlation involving operators the of asymptotic positions 1,. to” a and possibly function other points. Denote it by f. at Think it as a function of to the Xa or of the Ta . . ,

f(T

(6.6)

1,...,TJ±~)=g(x1,...,x,+~).

Differentiate the correlation function with respect to Ta and use the chain rule to obtain an equation involving the inverse of the matrix, tab’ in eq. (6.2), then

aT~

b=1

3Ta

(6.7)

öXb

The right-hand side of eq. (6.7) is computed by explicitly differentiating with respect to Xb. For the left-hand side, note that the derivative, af/aTa, inserts —L0 in the ath propagator strip since the Ta dependence in the Feynman rules comes from f~dTa exp(— TaL0). In other words, af/aTa is the correlation function f with an extra —L0 insertion and can be computed using the off-shell conformal methods. At the tree [49] and one-loop level [521, the correlation function, f, was taken to be

f= (e~i4(toI)eA24(w2)

. . .

,

eA~v4(w~~))

(6.8)

where 4’ is an arbitrary boson compactified on a lattice, A, and A,. are arbitrary boson compactified on a lattice, A, and A,. are arbitrary momenta subject to a conservation constraint. At tree level the derivatives, axb/aTa, were obtained by comparing the coefficients of Ar A5, 1 ~ r
—

S. Samuel / Open bosonic string

552

level, not enough equations were generated from these invariants. Instead, q A,. and q q dot products sufficed, where, q, is the loop momentum in the Feynman graph. It takes discrete values on the lattice A. Because A can be freely varied the coefficients of q A,. and q q on both sides of eq. (6.7) must be equal. This generated equations for ax6/aT~.In the final step, the determinant of axb/aTa was constructed and after much algebra the ghost-Jacobi identity was derived [52]. At g-loop, g ~ 2, we find that the correlation function in eq. (6.8) does not generate enough equations using the above methods. Instead, we take 4(w2)

. . .

e~~(wN)eA 14(toN~))

,

(6.9)

f=(e~(to1)e52

where to”~1 is an arbitrary point on the string configuration corresponding to ZN÷lin F~and AN±,is a free parameter. ZN+, is taken to depend on the Zr in a specified way. Equation (6.7) is used and the coefficients of q. Ar and q q, are compared, where r ranges from 1 to N and i and j range from 1 to g. The q, are g-independent loop momenta. Combining the above method with computation of K1> 05~from refs. [78,79], the following identity is derived i

26K

)bc~

[Ki>05~1 N

(2~i)~cr(w)detg~g y

rr

to~(

1)fl~,E(y~,w) =

flexp(—N00)

x

0(~iJ~W

—

~

_4z0IT)fli~j
detJXJlII’1(wb)

—

34zoIT)hhi~a
3

(6 10)

Wa)]

where z 0, w, yr,. Yg~ w1, and W1 are arbitrary distinct points in F~. In appendices C and D, it is shown that eq. (6.10) does not depend on the choices of these points. The vertical bars in eq. (6.10) stand for the absolute value. detgxgtoj(yj) is the determinant of the gxg matrix whose i,j entry is to~(y,); likewise for det1~1~P’1(w6).Formulae for the prime form, E, theta function, 0, etc. were given in sect. 3. The holomorphic 2-differentials, ~I”1, are given in eq. (6.4). Note that a change of basis for ~P’1results simultaneously in a change of det1~1~I”1(W6) as well as a change in the definition of the x~in eq. (6.5) and thus a change in the jacobian, .1. These two effects compensate for each other leaving eq. (6.10) unchanged. Equation (6.10) is derived in appendix B. ..,

...

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553

6.3. THE INTEGRAND

Combining eqs. (5.8)—(5.i0), (6.3) and (6.10), the following result is obtained for the N-point tachyon g-loop (g ~ 2) integrand, I,~

I~({Xa}) =a’P(~)

2

(nf j=i

d26P;6)Ki)~~exP((a~kr.kr_l)Nor~) (2iT) r=1 T

X

exP[(2a’)(iiT~P~TiJP~J + 2iTi r=l ~ i=l ~ k i,j

Z 0

where Ki)’ stands for 1er
1~I

(6.11) )b~J/~~2at~~k~), E(Zr, Z5)( 1exp( —N~),i.e., the second two factors

in eq. (6.10). In eq. (6.11), the constant (i/2Y’ has two sources: c~in eq. (5.10) and the fact that certain field theory diagrams overcount Riemann surfaces. This is explained in appendix J. Here V= 2(g 1) + N is the number of vertices and P = 3(g 1) + N is the number of propagators. The overall sign and normalization of eq. (6.11) was determined by world-sheet loop factorization. For certain limiting Riemann surfaces in which all Ta —p cc, the holomorphic differentials, prime form, period matrix, etc. can all be computed analytically. In this limit, factorization can be used so that the result from field theory can be directly compared to eq. (6.11). (This is carried out in appendix E.) —

—

7. Moduli space: the integration region for low loops This section and the next two discuss the final ingredient in the string amplitude: the integration region, commonly known as moduli space. We choose for integration variables the parameters in the elements of the generating set, i.e., their multipliers and fixed points. This type of parametrization of Teichmüller space has been studied in mathematics [144]. At loop level g, there are g generators and hence 3g real parameters since each element has one multiplier and 1 twoforfixed any points. Since F and yield Riemann if F’ =for hFh h a- SL(2, IF), two three (two F’for g =the 1) same parameters aresurface redundant g ~s 2. This correctly gives 3g 3 for the dimension of moduli space for g ~ 2 for vacuum bubble diagrams. When N external states are present, we sum over all possible ways of assigning the Zr to a location one of the B-boundaries regions and then integrate the Zr throughout the entire boundary. Hence the integration region for the Zr is —

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Open bosonic string

fl~1f~dZ,. where

.~ is the union of the boundary regions on the real axis. Some overcounting of Riemann surfaces occurs and contributes to the (i/2)’~’factor in front of eq. (6.11) (see appendix J). To complete the integration region, we must determine moduli space for those degrees of freedom describing the geometry of the surface. In using the uniformization theorem, it is convenient, for aesthetic reasons, to display “pictures” of the Riemann surfaces using the unit disk, lii, rather than the half-plane, 0-I. The two representations are related by eq. (2.16). Although the values of fixed points in sects. 7 and 8 are given numerically in the half-plane representation, i.e., as numbers on the real-axis, we use as the same symbols to label the corresponding points on the unit circle. This abuse of notation is unlikely to lead to confusion. This section computes algebraically the moduli space for low g. An explicit construction is given for the case when there are B boundaries and no handles, H, and when there is one boundary and one handle: B = 1 and H = 1. Section 8 discusses the general case. The group F is constructed as the set of words whose “letters” are the generating elements as discussed in sect. 2.2. Not every such F is discontinuous and produces a Riemann surface as explained in sect. 2.1. It is therefore important to have a simple procedure to decide this question. This is provided in sect. 7.1.

7.1. CONSTRUCTION OF THE GROUP LIST AND TESTING FOR WHETHER THE GENERATORS YIELD A RIEMANN SURFACE

Beginning with the g generators, h,,. hg, we shall construct a set of n elements {h~,.. h~,}which we call the word list. These elements saturate the multiple intersection of F = ~ ~ ~ O~~ °h~,~’in eq. (2.14) and satisfy the property that arcs of their h-circles border F. The construction procedure is an algorithm for adding and deleting elements from an initial word list. If during the process an elliptic element is generated then hg do not generate a group F satisfying the requirements of the uniformization theorem and F N H is not a Riemann surface. If the entire boundary of H is covered then F yields a closed surface which is irrelevant since we are interested in open string processes. If all elements of the word list are hyperbolic then, as demonstrated in appendix F, F N H is a bonafide Riemann surface. The algorithm thus provides a means of deciding when g arbitrary elements generate a relevant covering group. Begin with {h1,.. hg) as the initial word list. Examine the h-circle of each h, and its inverse in turn. Depending on how h-circles overlap elements are added to or deleted from the initial word list. There are many cases. Case A: If the interiors of no pair of h-circles overlap, ~‘ = I,,, ~ = = ~ ‘67k = 0 for i
.,

. ,

‘h~

~

~

. ,

‘6,

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555

F is a Schottky group (see appendix F). When ‘h Ih, 1hz’ 0 we call this case A for elements i and j. Figures 8b and 9b are ~ examples with i 1 and j 2. Case B: It occurs when the two interiors of two h-circles partially overlap and no other h-circles among these elements and their inverses overlap (see fig. 12b), e.g., ~ Ih, * 0 and neither interior is contain in the other, ~ ~ ‘h,~ ‘h, ~ ‘h 1 is added 1’ to and the I/i, verified ‘h[’ ~ using ‘h7’ 0. definitions For case B,in h’eq. h~h1 word list. It‘hI’ can be the (2.11) that ‘h covers a segment of Ch-1, namely the image under h, of the arc of C,, which is covered by I~~• We indicate this in fig. 12B by dotted lines. Likewise I,~-icovers the segment of C,,-~which is the image of h 1 of the arc of C,, which is covered by I,,. In case B it may be that the h-circles of hI’ and h~,or h1 and hj’, or hI’ and h7’ overlap rather than h1 and h1. One then should replace h, by hI’ or h3 by h;’ or replace both in the discussion in the previous paragraph. The same holds for the cases discussed below. Sometimes h, and h. should be interchanged as well. Cases BB: They involve the partial overlaps of two h-circles. They are treated as two versions of case B. There are two generic situations, BB.1 and BB.2 (see figs. 12BB.1 and 12BB.2). Of these, BB.1 has two subcases depending on whether the h-circles overlap on the same side h” (BB.la) 1and hr ‘hor on opposite sides (BB.lb). For BB.1 add the elements h’ h~hI 1 to the word list. In appendix G, it is shown however that h’ (and h”) are elliptic for case BB.la and hence if this situation arises the original g generators do not produce an appropriate F. In case BB.2 one h-circle is overlapped on two sides (see fig. 12BB.2). The elements h~hI’and h” h; ‘hT~are added to the word list. Arcs of the h-circles of C,,,, C,,-, C,,. and C,,- are covered by the h-circles of h’ and h’ as illustrated in fig. 12BB.2 and these are the images under hIt, h5’ and h~of the previously covered arcs. Case BBB: It involves three overlaps (see fig. 12BBB). The elements h’ h1hI ~ h’ h~h1and h” h7 ‘h~are appended to the word list. Case BBBB: There are four overlaps (fig. 12BBBB). Since no boundary is left, this is not an open surface and the generators do not produce a covering group relevant for open strings. In fact, elliptic elements are present in F. Case C: One h-circle is covered by the interior of another while the remaining two h-circles do not overlap, e.g., I,, ~ ~ and I,, I~ I,,~,~ I,,, ~ ~ 0 (see fig. 12C). In this situation, h~ is dropped from the word list and is added. It can be verified from eqs. (2.10) and (2.12) that I,,. completely covers I~_ so that the h1- and h; ‘-circles are completely covered. Cases CC: They occur when two h-circles are covered (see figs. 12CC.la, 12CC.lb and 12CC.2. In CC.lb, I,, and I,,-, are covered. This situation cannot happen as is shown in appendix H. For the other CC cases the deletion and addition procedure is similar to case C. For CC.la, h~is dropped from the word ~

=

~

=

~

=

=

~

=

=

~

=

=

=

=

=

=

=

=

=

=

~

=

=

=

=

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add: h~h~1

~

(B)

add: h~h~’and

There are elliptic elements (BB.la)

Odd:

(BB.lb)

h~h~’and (BB.2) Fig. 12. See p. 558 for explanation.

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add: h~h~’,h,h~andh~’h,

557

not relevant

(BBB)

(BBBB)

Oeplac e

h~by h, (C)

0

~E~’
1hi~’

impossible

(CC.la)

(CC.lb) Fig. 12 (continued).

9

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1

replaceh~ byh,h~ (CC.2)

___~-÷—-\

7— droph, and add h~h~’ and hkh~’ (D) Fig. 12. Cases concerning the construction of the word list. Case B occurs when the interior circles of two group elements, denoted by h. and h~,overlap. Cases BB, BBB and BBBB involve respectively two, three and four overlapping interiors. Case C occurs when one interior circle is inside another. Cases CC, BC, BBC and CCB involve multiple overlaps and coverups. Case D occurs when the interior of one group element h, is covered by the interiors of two group elements and hk. Beneath each figure is written what needs to be done to the word list. In cases B, BB.2, C and D we also show the h-circle of one or more of the new elements added to the word list. Although the cases are generic, the figures are specific outputs generated in computer simulations. The values of the fixed points and multipliers are given on the next page.

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replace h~byh,h~’

replace h~byh,h~’

(BC.1)

(BC.2)

1

impossible

replace h~by h,h~ (BBC) CASE

z’

B BB.la BB.lb BB.2 BBB BBBB C CC.la CC.2 BC.l BC.2 BBC

—0.25 —0.5 0 0

D

—0.25

—10 0 0 0 0 —10 0 —0.15

559

(BCC) z’~.

—2 —4 —6 —10 0.25

0 —2 —10 —10 —10 0 0.5 0.5

A

z~

4 3 4 4 3 2 4 4 3 4 4 2.1

0 0 —0.25 —0.25 0 —1.1 0.15 0.1 —0.2 —0.1 —8 —0.05

6

0

2 4 6 0.32 1 1.1 2 —8 0.2 0.25 0.25 0.2 —10

Fig. 12 (continued).

6 4 6 4 3 2 10 8 8 6 10 2.02 4

z~

4

—0.35

—1

3

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list and h’ h.hT’ is added; for CC.2, h~is dropped and either h’ h~h/’ or is added. Cases BC and BBC: This is the situation simultaneously involving overlaps and coverups (see figs. 12BC.1, 12BC.2 and 12BBC). When one of each occurs, situations BC.1 and BC.2 take place. When two overlaps take place, case BBC arises. The deletion and addition procedure is similar to case C. When I~,is covered up as in figs. 12BC.1, 12BC.2 and 12BBC, we drop h3 and add h’ h,h;’. 1-circles are completely The interior In-imay is covered by I,, so the h3- and covered. There be overlapping of that the h’-circle withh; the h-circle or hI ‘-circle but this is taken care of when the procedure of examining h-circles is repeated. Case BCC. It is shown in fig. 12BCC and cannot occur for the same reason as in case CC1b. Final Case: Finally, it may happen that I,,. is covered by I,,. and I,,,,: L~,U ~ ~ (as shown in fig. 12D). It is easily verified that if several ‘h cover I,,. that two suffice. Let h’ h 1. It can be demonstrated that ‘h’ U 1~~ ‘hI’ and h” arehkhl so that the h, and 1h[’ h~’circles completely covered. For this case we drop h, and add h’ and h” to the word list. Note that h, can be written as a word using h~,h, h’ and h”: h,=h’~h~or h~=h”’hk. The algorithm to construct the word list consists in going through the elements of the list and seeing which of the above cases occurs and appropriately adding and deleting element. One repeats the process until no h-circle of any element in the list is covered and if the h-circles of two elements h, and h 3 overlap, L~, ~ I,,, * 0, then h’ h.h[’ or its inverse is in the word list. If, at any stage, a parabolic or elliptic element is generated or the entire unit circle boundary is covered then the g generators do not produce a relevant covering group F. If only hyperbolic elements appear in the final list (hi, h,} then F \ H is a Riemann surface (see appendix F). Note that when an element is deleted, it is reconstructible from the other elements in the list. Hence (hi, h,} generates the same group as the original g generators, h1,.. .,hg. The above algorithm can be implemented in a computer. In fact, to gain insight into moduli space we wrote such a program and used it extensively. Figs. 13 and 14, respectively, show examples of a set of generators surviving and failing the algorithm test. =

=

=

=

=

=

. . . ,

. . . ,

7.2. THE MODULI SPACE FOR THE PANTS SURFACE

Our goal is to determine the moduli space, or integration region, for surfaces topologically equivalent to a sphere with g + 1 holes like those in fig. 15a. In sect. 2.4, we found B+2H—g+1,

(7.1)

where B is the number of boundary components and H is the number of handles

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~01 (a)

(b)

(c)

Fig. 13. A set of generators surviving the algorithm test. (a) displays the h-circles of the two generators, h, and h 2, of the covering group r. The notation here and in other figures is as follows. A positive number “i” marks the center of Ci,,; a negative number ‘—i” marks the center of C,,-~,e.g., in (a) h1 takes the larger arc on the left onto the larger arc on the right. In (a) C,,1., is covered by C,,-i and case (BC.1) in fig. 12 is relevant. Thus, we replace h2 by h3 hj’h2. The h-circle of h3 overlaps with the one of h~’ so that case (B) 1h in fig. 12 occurs. We add h4 = h1h, to produce the final word list th,, h1, h4} = {h1, h~‘h2, h, 2h1}. (c) is the final configuration. It corresponds to a B = 1 H = 1 Riemann surface. Fig. 13 is output from 1~,A a computer implementation of the algorithm in sect. 7.1. The parameters are (z~,z 1) = (— 10,0,3) and (z~, A2) = (—0.3, 0.1,5).

4.

(see fig. 15b). The sphere with g + 1 holes corresponds to H 0, B g + 1. We use one assumption. Assumption A: Given a set of fixed points and multipliers of maps, corresponding to a valid Riemann surface and hence a point of moduli space, we assume that all other Riemann surfaces are obtainable by continuous deformation of the fixed points and multipliers. In other words, we assume that the parametrization of moduli space via the parameters in the generating elements covers moduli space for fixed B and H. Below we find evidence that this assumption is correct. If moduli space with our parametrization is a connected set then assumption A is true. It is interesting to address whether there are several components. Our construction of moduli space for the B g + 1 proceeds by induction on g. It is necessary to define a region, denoted by ~B=g+1,H~.O in addition to the moduli space, =

=

=

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(a)

/

Open bosonic string

(b)

1~

(c)

(d)

Fig. 14. A set of generators failing the algorithm test. For notation see the caption of fig. 13. Case (B) occurs for (a) so we add h 3 = h2h, to the word list. The h-circles of h3 are displayed in (b). Cases (B) and (BB), involving the h-circles of h3 and its inverse, occur. We add h4 = h3hj to the word list. Its h-circles are displayed in (c). Add h5 =h4h~’ to the list. Notice that its and its inverses h-circles intersect in (d) so that h5 is an elliptic element. A Riemann surface is not generated. Fig. 14 is the computer output starting with (z~,z~,A1)=(—10,0,2.5)and (z~,z~,A2)=(—1,4,2.5).One finds Tr(h5) = 1.92... <2.

The case g 1 corresponds to the annulus family. As explained in sect. 2.5, the cover group is generated by a single hyperbolic element: F (hi’, n E Z, h is hyperbolic) whose fixed points can be chosen be for andthe 0 by performing 2z. The moduli tospace annulus family an is SL(2, ll~)conjugation: h(z) A {A > 1). When A2 exp(T), the moduli region corresponds to 0 < T < This is the integration region used in ref. [52] for the one-loop computation. The next simplest case is g 2, a sphere with three holes. It is topologically equivalent to the pants surface [76].The pants Riemann surfaces play an important role in the Frenchel—Nielsen representation of Teichmüller space [76, 136]. Because we wish to integrate over each surface precisely once [631,we are interested in moduli space rather than Teichmüller space. =

=

=

=

~.

=

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Open bosonic string

563

011

O

:c~ (a)

(b)

Fig. 15. Typical open Riemann surfaces (a) is a Riemann surface with six boundaries and no handles. (b) has two handles and five boundaries.

Let h1, h2 and h3 h0 be the elements of F which cover the three holes*. These elements are unique up to a conjugation. We may fix the SL(2, )~)ambiguity by imposing conditions. Let h, correspond to the hole with the smallest multiplier and h2 have the next smallest multiplier: 1 z~..In any case, after fixing h,, it is only the ratio =

—

~c

~.

=

()22/2

(7.2)

that matters. A typically configuration is shown in fig. 8b. As long as the h-circles of h1 and h2 do not overlap, the covering group F, generated by these elements, satisfies the requirements of the uniformization theorem and the half-plane modded out by this group is a genuine Riemann surface, as discussed in appendix F. * For notation reasons it is convenient to denote the element which covers the (g + 1)th boundary

by h0.

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(a)

(b)

Fig. 16. The parabolic limit of fig. 8b. For notation concerning the labelling of h-circles see the caption of fig. 13. When the h-circles of h 2 in fig. 8b first touch those of h, as in (a), h0 = hj’h~’ becomes parabolic as in (b). The third 1÷, A,) boundary ~(— 1000000,0,3) has shrunk toand a puncture. (z~, AThe parameters producing fig. 16 are (z~,z 2) (0.5,2.5,5).

4~

The map h0 is not independent of h1 and h2. We can take h0=hI’h~’,

(7.3)

to be the element covering theregion third hole. 3’~’° to be the of parameters continuously connected to the Define S~ configuration in fig. 8b with the restriction A,

3’~°=j(h~,h S

2)~z~=—00~z~=0,z~z~= 1,0
* There is only one point at infinity so

unit circle when eq. (2.16) is used.

—~

is

}.

(7.4)

the same as ~ and both correspond to (— 1,0) on the

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Algebraically

3’~°_~

—00,

S

z~=0, z~z~= 1,0
+ A,A

2 1
A,

+

~,

A2

(7.5)

where y is defined eq. (7.2). (1 condition + A,A2)/(A1 + A2) < 1cannot then (h1, h2) is not 3”°inbecause the Iflast in eq. (7.5) be satisfied. included in S~ To obtain RB3,F~Owe require A 2
=

3,~°=

{(h,~h

2÷z~= 1,0


2)iz~=00~ z~=0,z

R

1
2 1
where again the condition (1 constraint can be satisfied.

+

< 00,

1
1 <

+

A 2 1A2

A,+A

~

(7.6)

2 ~

A,A~)/(AI+ A~)>1 is implicit so that the last

7.3. THE MODULI SPACE FOR B = g + 1, H = 0

The general strategy for constructing RB =g + 1, H 0 is inductive. From SB g, H— 0 we construct SB=g+1,H=0 Then we impose the condition Ag
cc

...

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Open bosonic string

(a)

(hi)

(b.2)

-22

(c) Fig. 17. The reason for the additional constraints. For notation concerning the labelling h-circles 84,~° obtained fromofR~93,”~° see the caption of fig. 13. (a) is a typical starting configuration in R by adding a generator h 3. The initial word list is {h,, h2, h1}. When the h-circles of h3 and h~ overlap (b.1), the element h4=h3h2 added (b.2). When I,,, is covered by ‘hi” I,,~-icovers ‘~2 and h2 removed from the word list = (h,, h2, h4). As is moved towards z~, the h-circle of h~’ overlaps with the one of h~ (dl) and h5 =h~’h4 becomes important (d.2). When ‘h1.’ is covered by Isi-!, I,,, covers the h-circle of 3~h4 until (e)the andh-circle h4 is of removed h from the word list making it {h,, h2, h5}. Continue decreasing z~ and z 5 overlaps with the one of h1’ (f.1) and h,, =h5h, comes into play (f.2). When I,,, is covered by I,,-, (g) results. As soon as the h-circle of h,, overlaps with the one of h~’(hi), the element h7=h~’h,,=(h2h,Y’h3(h2h,)=h0h3h~’ appears (h.2). Continuing, is covered by ‘h1~’(i) and the final word list is {h,,h2,h7).

is

4

(c)

is

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iiIi~-i

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567

4~2

(d.1)

2

(d.2) 2 54

0-I -2

(e)

2 5

~Ii655 (fl)

(f.2) Fig. 17 (continued).

n I,,~

‘h~~

=

0

and I,,,

fl

‘h

0 where 1...h;’, h0=hI’h~

0h1.’h,3’

*

(7.7)

is a map which covers the (g + 1)th hole. Here we are treating the g 3 case. The above defines S~94’5t0 R84,”0 is obtained as those points in =

5B4.H0

satisfying A3
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i~II~.i 1

(h.1)

-1

(h.2) ~-22

(I)

Fig. 17 (continued).

The next three paragraphs explain the need for the two additional constraints. We show why it is not necessary to insert z~ and z~ in other regions of the (g + 1)th boundary so that it is sufficient to have z~< z~< z~< cc~Finally, insight into the correctness of assumption A is gained. Figure 17 reveals what happens as z~ and z~are pushed towards z~.First decrease z~keeping z~fixed. Eventually {h,, h2, h3} do not generate the covering group F of a relevant Riemann surface for the following reason: as z~decreases it

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569

becomes equal to the fixed point of some h E F. Although, a priori, this point could be z~,there is always some other fixed point between z~and z~(see sect. 8.6). There is a theorem [681 which states that if two hyperbolic maps, h and h’, have one common fixed point then hh’h - th’ -, is parabolic. Since we do not permit parabolic elements the checking procedure of sect. 7.1 will reveal a difficulty. Thus, the range of z~ is restricted. Likewise z~cannot be increased all the way to infinity (or (— 1, 0) in a disk representation of the covering space). Figure 17b illustrates what happens when I~~3 and ~ overlap. The element h 4 h3h2 becomes important and is added to the word list. Its h-circle and its inverse h-circle, as shown in fig. 17b.1, overlap with I/~2 and ‘h3.’• If as decreases, I/~3becomes covered by I,,~-,i.e., I,,2~iD I/~3then I(,,3,,2)_1 covers I,,~-ias shown in fig. 17c. At this point the h-circles of h3 and its inverse are completely covered and h3 is removed from the word list. If we continue decreasing z~, eventually a parabolic element is generated as explained in the previous paragraph. Instead, it is interesting to decrease z~until I,,~-ioverlaps with I/~2~1 (see fig. 17d). Then the element h5 h~h3h2becomes important and is added to the word list. If, as z~decreases, I~~4~I becomes covered by I,,~-i,then I,,~is covered by I,,~ and h4 is replaced by h5 in the word list. Notice that the h-circle of h5 and its inverse are located between z’~and z~.Being a conjugate of h3 it covers the third hole. This explains why one should not place the fixed points of h3 between z~ and z~.:moduli space would be overcounted. Start with the configuration in fig. 17e and decrease z~until I,,~overlaps with This is shown in fig. 17f. The element h6 h5h1 h~h3h2h, is added to the word list. By decreasing z~ further, it may happen that I/,5 is covered by ‘h~’ in which case I,,~-iis covered by I,,,,-, as in fig. 17g so that h5 is removed from the word list. Now decrease z~ so that ~ overlaps with ‘h1~’ (see fig. 17h). The element h7 (h2h1Y’h3(h2h1) h0h3h~ is added to the word list. It covers the third hole and its fixed points are located between z~and z’. If z~is decreased further so that I~~-i covers I,,,,~then fig. 17i is obtained. Now comes the key point. There is a danger that moduli space is overcounted if both configurations a and i (and h.2) of fig. 17 are included. A simple way to avoid this is to require 1hh-’h~’ * 0 so that that I,,, be covered by or overlap with I~, i.e., ‘h, n configurations h.2 and i do not double count configuration a. Then, in configuration a, we do not allow I,,~-ito overlap with I,,,. This explains the condition =

=

=

=

=

=

=

I~

3Ifl I,,~ 0. Note that there is always some of the (g + 1)th boundary (here g 3) between I,,-~and I,, so that in going to the next higher value of g there is room to insert the h-circles of hg+1 and its inverse. Although the above discussion depended on having certain sizes for the h-circles, the conclusion is more general as will be shown in sect. 8.6. Without further detailed explanation we present the inductive procedure for getting 5B=g+l,H=0 from ~B=g.H=0 Start with a point in ~Bg~H0 Add a gth generator, hg, to ~B=gH.~0 so that its fixed points are located close together in the =

=

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gap between I,,-i, and I,,~.This requires z~’Agi• We vary the fixed points and multiplier of hg under the condition that the test in sect. 7.1 is satisfied, i.e., a Riemann surface is generated, and we impose I,,_~n I,,~ 0 and I,,, ~ ~ * 0 (where h 0 is given in eq. (7.7)) to avoid double =

counting. Although these conditions can be worked out algebraically, as the g 2 case in sect. 7.2 illustrates, it is even simpler to program a computer to generate ~B=g±1,H=0 if a numerical evaluation of an amplitude is needed. Finally, ~ is obtained from 5B=g+1, H=0 by imposing on the latter the condition A0 > Ag. This completes the construction of moduli space for the case of a sphere with an arbitrary number of holes. =

7.4. THE MODULI SPACE FOR B = 1, H =

1,”’, is the second and final case (the first is RB3,~~lo R’~ level, g 2. It is topologically a torus with a disk of The sect.moduli 7.2) atregion, the two-loop removed. The corresponding covering group, F, is generated by two elements. If the disk is filled in, the surface becomes a torus and the two generating elements are associated with a standard homological basis. The difficulty which arises in obtaining RB t,H=1 that homological basis is not unique and hence there is arbitrariness in the selection of the two generators, h 1 and h2, of F. To fix the ambiguity we require h, and h2 to be a standard homological basis for the disk-filled-in surface (the torus) and we require them to 2 have minimum properties. More precisely, wehomological demand thatbasis (Tr(h,)) <(Tr(h 2 andtrace thatsquared if h~and h~is any other standard then (Tr(h 2)) 2 ~ (Tr(h~))2and (Tr(h 2 ~ (Tr(h~))2. 1)) 2))used to set z~to — cc and z~to zero. As in sect. The SL(2, 11) invariance can be 7.2 the residual invariance can be used to fix z~< 0, z~z~=—1, where the minus sign occurs because the fixed points alternate around the circle (see fig. 9b). The moduli parameters are A,, A 2 and y> 0 where =

=

—z~/z~,

()2

(7.8)

that is, z?~=—y and z~=1/y. Assume that h, has minimum trace. with 0, Candidates z~>0, we 2 ~ the (Tr(h~))2 for any h~EGiven F withh2z~ < 0, z~< z~>0. demand that (Tr(h2)) h’ 2 are of the form h~ (h,Y’~h2(h1)”where n and p are integers. Hence one must impose 2> (Tr(h 2, (Tr(h?h2)) 2)) =

(Tr(hj~h

2>(Tr(h 2))

2, 2))

(7.9)

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for n ~ 1. It is shown in appendix I that the n 1 case implies the n> 1 case so that eq. (7.9) need be satisfied only for n 1. Given the representation of SL(2, 10 elements in eq. (2.7), the n 1 constraints in eq. (7.9) reduce to =

=

=

______
A1A~—1 <

(7.10)

.

Finally, it is necessary to have some boundary. The element h0=h2h1h~’hI’,

(7.11) 2> 4 or

covers the only hole. Since it must be hyperbolic, (Tr(h0)) (A,_~._)(A 2_~_) ~2(~+_)~

(7.12)

as a little algebra reveals. Summarizing 1, RB

H~t

=

{(h 1 h2)Iz~= —cc, z’~=0, z~z~=—1, z~<0
2< A1A~—1 A~—A1
A,—— 1 A,

A

1 2—— A2

1 ~2 y+— y

~,


(7.13)

where z~=—y and z~=l/y. Summarizing, the conditions on the fixed points and multipliers in eqs. (7.5), (7.13) and sect. 7.3 (and sect. 8 below) are imposed to mod out TeichmUller space by the action of the mapping class group, thereby producing moduli space [64, 76, 771.

8. General results on higher order moduli space This section presents methods to determine the moduli space for the higher ioop cases not considered in sect. 7, namely ~ for B ~ 1 and H>e 2 and RB.J~~ for B ~ 2.

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8.1. CANDIDATE CANONICAL STARTING CONFIGURATIONS

Since RB, H enters at loop level g B — 1 + 2H, B — 1 + 2H generators are needed. We choose B — 1 of them to be associated with boundaries and 2H of them to be associated with handles: h,,h2,.. ~ ..,h~H, i.e., primed =

generators are the handle generators. We also use primes on the corresponding fixed points and multipliers. Define such a set of h. to be canonically ordered if in going counterclockwise around the unit circle the order of the fixed points is ~

1

2

2

—,

+,

—,

+,...,

B—i — ,

B—t

,1

,2

,i

,2

+

—,

—,

±,

+,...,

,

,2H—i

—

,

,2H

,2H—1

—

±

,

,2H ,

+

We call a generating set a candidate canonical starting configuration if it is canonically ordered and no two interiors of any h-circles of the generators or their inverses intersect. Denote by F the group generated by h,, h2, he,, h, h~,. h~. By appendix F, F \ H is a Riemann surface. The generators of the covering group F are not unique. Two sets of generators may lead to the same F. Both are not to be included in the moduli integration region. One must impose conditions to avoid this overcounting problem. . . . ,

. . ,

8.2. THE FIRST TWO CONSTRAINTS

The first set of constraints we require is 1
...

1

(8.la) (8.lb)


A’,
(8.lc)

Equation (8.la) is similar to what was done in sect. 7.3. Below we provide a raison d’être for eqs. (8.lb) and (8.lc). Recall that two F produce the same Riemann surface if they are related by conjugation with any h E SL(2, 110: F \ H F’ \ H F’ hFh To eliminate this ambiguity we fix three of the fixed points: =

—cc,

z~=—cc, —cc,

=

4=0,

z~z~.= 1,

4=0,

z~z~=1, ifB=2,

z~=0,

z~=z~0, ifB=1,

‘.

if B~3,

(8.2)

where z~0is a constant and where we assume g ~ 2 so that if B ~ 2 then H ~ 1. The value of z~0is not too important but must be positive and must be chosen sufficiently small so that I,,.~- n I,,~ 0, i.e., part of the EI~-axisbetween z~and z~ =

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is uncovered. This can be achieved by starting with the conditions z — cc, z~=0, z’~z~=—1 and performing a conjugation with h(A) h(z1~’,z~’,A) for an appropriate value of A. Conditions other than those in eq. (8.2) can also be chosen. Equation (8.2) is the second constraint. A candidate starting canonical configuration satisfying eqs. (8.la)—(8.lc) and (8.2) is called a starting canonical configuration. Given a candidate configuration, a starting canonical configuration is obtained by using the SL(2, Il) invariance to satisfy eq. (8.2), and then by making the A1 and A’3 appropriately =

=

larger to satisfy eqs. (8.la)—(8.lc) and the non-overlap condition*. Below, use of the Bth boundary generator is needed. It is given by —

““~ ~ 2

j,~

B—i ~ i

2

~

1

2

“~ 2H—1 “ 2H

h’2H—1 j,,—i 2H

83

The boundary generators, including h0, are unique up to conjugation.

8.3. ON THE CANONICAL ORDERING

At this stage it is instructive to explain why orderings of fixed points other than the canonical one are not needed. Basically, the reason is that such orderings are covered when varying the fixed points of the canonically ordered generators. A different ordering corresponds to conjugating one or more generators by other elements of the group. We showed this for in sect. 7.3. The rest of this subsection discusses the situation when B ~ 1 and H ~ 1. Section 7.3 showed the ordering of the ~ is changed by moving the fixed points and considering conjugate elements. Let h, be a boundary generator with fixed points z~ and z~ (see fig. 18). Push z~ and z’÷ toward the h-circle of some element ha e F. If the h-circles of h, and its inverse are smaller than the h-circle of ha then the h-circles of h1 h; ‘hiha and its inverse appear on the opposite of the h-circle of h;’ as shown in fig. 18e. In general, if the fixed points are ordered as z~,z~,z~,z~then the fixed points, ~ and of hj=h~hjha are ordered as z~z~,z~,za÷. When ha is a boundary generator, the above procedure alters the order of the fixed points of a pair of boundary generators. Repeating the process permits the ordering of ~ to be arbitrary. When ha is a handle generator, the ordering of the fixed points of boundary generators and handle generators is altered. It remains to be shown that the order of the fixed points ofz~’,is handle generators 1,z~, accomplished can be changed. The interchange, z~, z~, z~,z~—~ z’.~, z ~ as follows. Push z~ towards z~’(see fig. 19a). Then the fixed points of h~‘h~ alternate with those of h~.Next push the attractive fixed point of h~h~toward 5B,~~’=0

=

~,

* Recall that the h-circles shrink in size as the corresponding A is increased.

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~ (a)

hahi

~ __

a

3-~

/hahi

(c)

(b)

a,a

~hah,

(d)

(e)

(a), a new

Fig. 18. The basic interchange operation for fixed points. When z’. is moved toward z~ element h,h1 becomes important as shown in (b). Moving z’÷toward z~ (c), the element h-= h~h1h;’ comes into play (d). Compared to h., the fixed points of h1 appear on the opposite of those of h~.Since h- is a conjugate of h,, it is in the same homology class. The conjugation operation permits the ordering of a pair of hyperbolic elements to be interchanged.

z~’ (see fig. 19b). Define =h’h~’h~. Then the ordering ~ is achieved. By using a similar procedure the orderings of two sets of handle generators is altered. Suppose the order is z~’,z~,z~’,z~,z~,z~,z~,z~(see fig. 20a). Move zi~, z~,z~and z~toward z~One concludes that the ordering is z~,z~,z~,ii~,i~, ~ where h~=h~h~’hy’ and h~=h’3h’2h’4’ (see fig. 20b). Next,

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575

(a)

~

(b)

(c) 1.

Fig. 19. Interchanging the fixed points of a handle set of generators. When z~ is moved toward z (a), a new element h~’h 1becomes important as shown in (b). Moving 4 toward z~ (b), the element h1 =hj’h~’h, It h2 and the ordering of its fixed points are interchanged with those of h, compared with those of h2 in (a).

arises (c). is a conjugate of as

2~ toward z~ to find the ordering z~,z~ z~,z’ where h~=h~’h~h’h’~h~and ~ (cf. fig. 20c). Move 2~, i~,~ toward z~to establish the ordering z~,2~, E~,2~,~ z~,z~ where hç ~ and ~ h~h~ (see fig. 20d). Finally, move 2~, ~ toward z’~to find the ordering z~,z~,z~, z~,1’, 2’2, 2~, 2~ where h~~ and h~,= h~h~ ‘h~‘h’~h~hy ‘h~,h~h~’. The conjugating element h~‘h~h~h~’ is that piece of the last boundary element h0 which “takes one around the 3—4 handle”. Although certain size constraints on the h-circles are needed to obtain the results move

~,

~,

~,

~,

~,

~,

~,

,

~,

~,

~,

~,

* There should be double tilde, on some of the quantities here but to avoid cumbersome notation we

use only one tilde.

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Open bosonic string

(a)

(b)

~

(c) 4

-)

-3

(d)

(e)

sets of

Fig. 20. Interchanging the fixed points of two handle generators. The first set is moved toward 4 (a). Conjugate elements, denoted by a tilde, appear between z~.. and z~ shown in (b). When 4÷and z~ as shown in (c). Moving these are moved toward z~,conjugate elements appear between z these toward z~ produces conjugate elements between 4 and 4 as displayed in (d). Finally, moving these toward yields conjugate elements on the other side of z~ as shown in (e). The order of the fixed points of the two sets is interchanged.

4

as

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577

in this paragraph, sect. 8.6 shows they are independently valid of the sizes of h-circles. In short, using the conjugation operation, ha h~’hah,for some h E F, where ha denotes a generic generator, the canonical order may be changed. The conjugating element h may vary from generator to generator. 8.4. THE THIRD CONSTRAINT AND THE INDUCTIVE PROCEDURE FOR CONSTRUCTING 5B.H tt we first construct a bigger space, To obtain R” 5B, H• It is obtained by fixing a starting canonical configuration and varying the multipliers and the fixed points subject to ITrh~>2 for all h ~F and such that eqs. (8.1) and (8.2) and a third constraint are satisfied. This third constraint is the topic of this subsection. Note that the group, F, associated with each point in SB, H is a Riemann surface: F is discontinuous because all the elements are hyperbolic. This follows because the starting configuration contains only hyperbolic elements and we are varying parameters subject to ITrhI > 2. In sect. 7.3, when discussing ~B=~±iH~.0 we found that by pushing the fixed points of hg sufficiently toward ~ that the points hof the conjugate 1,appeared between z~ andfixed z~,where element, hg=h0hgh~ 0 is given in eq. (7.7). This means that the substitution hg hg does not change the canonical ordering of the g boundary generators and overcounting occurs if the range of z~ is not restricted. This led to the constraint I,,-i 11I~, 0 and I,,, fl Ih[hg~1h~~* O~ A similar condition must be imposed here. Conjugation of generators by a version of the Bth boundary operator preserves the canonical ordering. As in sect. 7.3, it is convenient to deal with this third constraint inductively. If H o, 5B~H=0 is given in sect. 7.3. If B> 2 and H> 1, first construct SB!10 by using the methods in sect. 7.3. This construction leaves uncovered a gap of the boundary between I,,, and I,,~’,. The insertion of the first handle is carried out by placing z~,z~,z~’and z~in the gap region and choosing the multipliers A’, and A~sufficiently large so that ~ ~ I/~ I,,~fl I,,~ I,,~fl I,,~_i I~~— ‘h, 0and andA~are so thatvaried eqs. 1,z~,z~1,z~,A’, (8.1) and (8.2) are satisfied. The values of z~ while maintaining eqs. (8.lb) and (8.lc), making sure that the test in sect. 7.1 is satisfied and that I,,i~. n I,, 0 and I,,, fl Ih)h2_hi~ * 0 (where ho is given in eq. (8.3) for H 0, i.e., the products involving the handle generators is absent). The last condition, ‘h~~’ ~h, 0 and ‘h, ~ ‘h,h~—’h~’* 0, is the third constraint. It eliminates the overcounting discussed in the previous paragraph. The conditions in this paragraph define S’~’~’for B >2. If B 1 5B~,Ki is constructed by varying zi~,A~and A~,imposing the constraints in eqs. (8.lc) and (8.2) and satisfying the test in sect. 7.1. When the first handle is added there is an uncovered region between z~and z~ (or z~’if B 1) for placing the next handle. =

=

=

=

=

=

=

~

=

=

=

~

~

=

‘h~—’

~

=

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Assume that

5B,51 has been constructed such that an uncovered region exists between z~” and z~(or z~..’if B 1) is obtained from SB,H by adding 2~~’+lzP2H+2z~2H+1 two newingenerators, h~H+,and h~H±2, points, zP z~’~2, that order (counterclockwise) inwith the fixed uncovered region between z~”and z~.Choose the multipliers A~H+,and A’ 2H±2sufficiently large so that eqs. (8.lb) and (8.lc) (with H — H + 1) are satisfied and so that I n I I fl I I fl I I nI 0. Varyn Iz’2”~’ Iz’211~2, =

/220+2

=

z’”~’

, ‘~2H±2

-i 1’2H+,

z’~”~2, ~

=

~‘2H+I

and

1’2K+2

5B,H-~-i

=

,-, 1’2H+2

“I

,-, ~‘2H

=

1’2H+, ,

=

—

“2H4-I ,

—

‘

A’

2H2 subject to eqs. (8.lb) and (8.1.c) making sure that the test in sect. 7.1 is satisfied and that I,,,— n I,, , 0 and I,, , n I,, ,,,~ /2—’ 2H±2 0, where h0 is given in eq. (8.3). This defines 5B,H+t =

0

8.5. REDUCING

5B,H

2H+2

0

TO RB,~~’: CONSTRAINTS FOUR AND FIVE

Like in sect. 7, we make the following reasonable but unproved assumption. Assumption A: R’~,H is contained in 5B, H that is any point in RB, H is continuously connected to the canonical starting configuration by varying fixed points and multipliers while maintaining F \ H to be a Riemann surface. To obtain RB,H from 5B,H we need to make sure the generators are unique. This requires two more constraints. We require the set of generators to obey minimum trace conditions. The following fourth constraint determines the boundary generators: AB,
(8.4)

where A0 is the multiplier of the generator associated with the Bth boundary operator given in eq. (8.3). Now consider the uniqueness problem for the handle generators. Let g,, g2,.

. .

ga—,, g{, g~,.. g~ be any canonically ordered set of generators in .,

RB, H which produces the same F as the h1. We demand that Trg~I> ITrh~I, orif~Trgfl=~Trh~I then

~

orif~Trgfl=~Trh~ and

Trg~I=~Trg~I then

orif~Trg;~ Trh;I =

fori=lto2H—1

then

Trg~I>jTrh’3I,

ITrg~~~ > Trh~H~,(8.5)

where equality in the last line of eq. (8.4) means equality of all traces and hence each g is a conjugate of the h*. Equation (8.5) is the fifth constraint. Imposing eqs. (8.4) and (8.5) on SB,H gives ~ * There may be accidental equality in which elements are not related by conjugation. This is not a

problem in practice because as soon as fixed points and/or multipliers are changed slightly the accidental equality is broken.

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579

8.6. THE UNIVERSAL ORDERING OF FIXED POINTS

To check whether any other set of generators g1, satisfying eqs. (8.4) and (8.5), is canonically ordered, it is necessary to know the order of the fixed points of elements of F. This problem is not as hard as might first appear because there is a universal ordering of fixed points in 5B,~~• This follows because, in varying parameters to generate 5B, H from the canonical starting configuration, fixed points cannot cross. If two fixed points become equal, a parabolic element is present and eqs. (8.1), (8.4) and (8.5) are not all satisfied. Since the ordering is universal, a particular configuration in 5B, H can be selected and examined to decide the orderings of all configurations in S’~,’~’. We have carried this out numerically for non-overlapping I~for g ~ 4 and “experimentally” deduced a rule for determining the universal order. Express each h ~ F as a word in the generators and their inverses (see eq. (2.8)). Define the “decimal” of h, D(h), to be the word obtained “by raising h to the infinite power”. For example, if h h,h2 then D(h) h,h2h,h2h1h2 and D(h~)=~ We always begin at the right and construct the infinite series to the left. If a generator and its inverse appear next to each other they are to be cancelled, e.g., D(h~’h,h2) h,h,h,h2. The ordering of z~ is determined from D(h). Label the generators and their inverses from 1 to 2g, h~h2...h2g. We know the order of the z~’,i= 1 2g. Starting from the right of the decimal, the ordering of two elements, h and h’, is determined by examining where they first differ. If h h1h,~.and h’ h.h,~, where h~is a word and h, and h3 are distinct, the ordering of z~and z~is the same as z~.and z~.For example, suppose h h1h~,h h~h,,,and h” ...hkhw, with h~,hJ,hk distinct. If one has the ordering Z~,Z~,Zhk then the ordering of the repulsive fixed points of h, h’, and h” is z~,z~,z~.iLet h h~h~’, where the first letter not equal to h1 is h3. If in going counterclockwise (respectively, clockwise) the ordering is z~’,z~J,z”J’ then in going counterclockwise (respectively, clockwise) the ordering is zhj, z~,z~ Since D(h) D(h”) for n E z~and z~”should have the same position. This is the case since h and h” have the same fixed points. Examples of fixed points orderings are presented in figs. 21 and 22. =

=

=

. ..

=

=

..

. . .

.

..

=

.

=

..

.

=

=

=

. . .

...

~,

8.7. SOME EXAMPLES

Sects. 8.1—8.5 gave a mathematical construction of RB, H~ It is theoretical rather than practical at least by the criterion we have imposed on ourselves: that everything be programmable in a computer. The difficulty is with the fifth constraint in eq. (8.5): there are an infinite number of conditions to be checked. In a brute force approach, a computer would need an infinite amount of time to compute R’~H

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Open bosonic string

—2—2—2 —2—2 1_2_22

222 22 2 —122

1—2 ~ ll—2..’ 21—2 2-1-2/

-~—1—I2 \212 ~ 12

—2—1—27 —1—2

21

—2—1

I21 221 —121 —1—21 —2—21 —22 1—21 211 —211 11

—1—2—1 —2—2—1 1—2—1 12—1 22—I 2—1 —12—1 —2—1—1 2—1—I

111

—I—I—i

3~ H—U for words up to three letters. The ordering Fig. 21. The universal ordering of fixed points of R~ of the fixed points for fig. 8b is displayed. A number list indicates the element in the following manner. The number dictates which generator and a minus sign means an inverse, e.g., — 121 stands for hj’h 2h,. Because some fixed points appear quite close to one another making the display difficult, the locations of fixed points are not shown but simply the ordering.

exact

— —21

1—2—2 ..~

—2—2—2 —2—2 —2 —1—2—2 2 I 2 1-2

11—2

—1—1—2

21—2

2—1—2

—1—21 —2—21 —21 1—21 —211

1—2—1 —2—2—1 —2—1 —1—2—I —2—1—1

111 11 I 21! 121 21 221

—1 —1—1 2—1—1 —12—I 2—1 22—1

—121

12—1

—2—12 —1—12

—222 112

—12 12 212 122

2

—122 2—12

22 222

1,

Fig. 22. The same as fig. 21 for

R~

H—i

—1—1—1

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58)

(a)

2_it’

~

21p

—1

_1q 2I~’

(d(

(hi

~ 2_lp

(e)

~ (0

Fig. 23. Candidate replacements for h2. (a) shows the fixed points of fig. 9b. There are eight generic possible replacements for h2. In each box the candidate is indicated using the notation in the caption of fig. 21: hence (b) considers the replacement h2 —*h2hj~’where p is a positive integer. The location of the fixed points of the candidate element are displayed in the diagram; the tail (respectively, head) of the arrowed line corresponds to z (respectively, z~).

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For the cases considered in sect. 2, a finite number of conditions was found. It is useful to reconsider these from the point of view of sects. 8.1—8.6. When H 0, the fifth constraint is not applicable and a finite number of equations need to be checked. This is why the H 0 case is solvable (see sect. 7.3). Next, consider the B 1, H 1 case. Suppose {h,, h2) is the generating set and =

=

=

=

4, 4

z~,z~, is the ordering (see fig. 23a). Consider trying to replace h2 with an alternative generator h~:{h,, h2} {h,, h~j.We require the ordering to be canonical, z~, z~, z~,and {h,, h~}must generate F. The latter implies that h~, —‘

4,

__Q;2

____________________________________________(al

O

p

q

—2

2

02 2

(b)

p

2 3—2

p

2~2 —1 —, - . - - -- —i3 2313

P2

p

2213—2 q —1q - - - 3—2

Q

____________________________________________

~

p

2213 —13 (gJ

.- -- —13 23 2~’I 2~’i...i3

—2

2

—‘p2...—13

(c)

—1 ~_1p2...,3 I~2 I~~2 -- - --. --- 23 —13 23 13

— —,

—3

—,

—

~

—1

q

3—2

“

~,

2.3—2

p

q

02 2

O

P

(d) —I —2

2

2~i

(e(

q

p

q

P

2 —1.32

2 i...32

3—2 -

-- - - -

~

32”

—23

—3

—

2

q

1 2—23

—

—, —i

p

q

232

—2—23

(0

U)

p

2..

.32

q —i —2

p

-

- - 32

Iq 32 p

~

82 H—I case. Notation is as in fig. 23. A series of dots Fig. 24. indicates Candidate anreplacements arbitrary word forconstructed h3 the Rout of h 1 and h2 p and q are positive integers.

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considered a word, contains only one h2 letter. Using the rules in sect. 8.6 for determining the ordering, the possibilities for h~are h~ h~h2h~,

(8.6)

=

where n E Z and m E 1. The locations of the fixed points of h~is given in figs. 23b—i. The fifth constraint requires

I> ITr(h2)

Tr(h~1h2hr)

.

(8.7)

Using the cyclicity of the trace eq. (8.7) is equivalent to eq. (7.9). Similar considerations for h, lead to eq. (8.7) with h2 h,, but this case is already covered in eq. (8.7). By appendix I ITr(h,h2)I > ITr(h2)I > Tr(h,)I implies Tr(h1h~)~ > ITr(h,)I and Tr(h,h~’)~ ITr(hj~h2)I> ITr(h2)~> ITr(h,)I implies Tr(h,h~)I > Tr(h,)I. The case B 2, H 1 is the lowest g example illustrating that eq. (8.5) leads to an infinite number of conditions. Let h1, h2, and h3 be the generators with h, a boundary generator and h2—h3 a handle pair (see fig. 24a). Suppose one looks for a replacement h3 —* h~preserving the canonical order. The candidates for h’3 are given in figs. 24b—k. Since in cases b, d, e, f, g, h, i and j an arbitrary string of letters is involved, an infinite number of conditions must be analyzed. The replacement h2 —~ h~ also leads to an infinite number of constraint equations which we omit to save space. Of course, it might be that the constraints are redundant and that only a finite number of conditions need to be checked. If so, then RB,H can be obtained with a finite algorithm. A solution would yield a practical construction of moduli space for open Riemann surfaces. This is an interesting area of mathematical research: to see if there are certain important group elements which must be considered in eq. (8.5). ~-‘

=

=

=

9. Constructing moduli space via string field theory This section uses string field theory to obtain the integration region for those degrees of freedom associated with the surface geometry, i.e., moduli space. The key assumption is that string field theory provides a single covering of moduli space. An algorithm is provided to construct R~!Hfor B> 1, where B is the number of boundaries and H is the number of handles. Although awkward to implement, the algorithm can be programmed into a computer and hence satisfies

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our computability criterion. The amount of computer time required to produce ~B~I increases with B and H.

9.1. THE ALGORITHM

Fix the topology, i.e., B and H with B> 1. Set g B + 2H — 1 and I 3g — 3. Choose a starting canonical configuration using sects. 8.1 and 8.2. This involves selecting g SU2, 110 generators. They depend on 3g parameters of which three are fixed by eq. (8.2). Let y1, i 1,..., 1, denote generically these parameters. The =

=

=

generators determine the covering group, F, and the fundamental region, F; see sects. 2.2 and 8.3. Step 1. Enumerate all the Feynman vacuum graphs which, via the method of sect. 1.2, produce a Riemann surface with B boundaries and H handles. Step 1 is used to make sure that one does not overcount or miss a region of moduli space. Step 2. Since the generators are fixed, the surface is determined. Compute the holomorphic 1-forms using eqs. (3.5), (3.6) and (3.9). Obtain the period matrix, the ~9-function, the prime form, the o—function and the vector of Riemann constants using eqs. (3.10)—(3.12), (3.16), (3.17) and (3.19). Step 3. Construct ~t in eq. (4.1) with N 0, determining, ~ and 1~,m 1,. 2g — 2, by requiring eqs. (4.3) and (4.6). This renders ~V, and ~m implicit functions of the y1. We indicate this by ~K((y~}) and }‘m({YiD~ Step 4. Compute the Ta, a 1,. I, from eq. (4.7). This makes the Ta explicit functions of the y1. Step 5. Determine lines of Im(p(z)) ‘~r/2in the fundamental region, F. If a line terminates on an h-circle, Ch, there is a corresponding line terminating on the h-circle, C~-I. Identify the ends of two such lines. At a ~ three lines of Im(p(z)) ~/2 meet. The lines from a graph involving vertices of order 3. This graph is the string field theory Feynman graph, G, associated with this region of moduli space. Checkoff G in the graph list in step 1. Figure lib exemplifies stepS. When the dotted lines in the upper-half plane are joined, the graph in fig. 2a is obtained. Step 6. Vary the y1 slightly and repeat steps 2—5 making sure that the same graph instep 5 is produced. Continue varying the y until one of the following occurs: (a) I~,({y~)) —~cc for some a; (b) 1~({y~}) —‘ 0 for some a. Varying the y, under step 6 in all possible ways under the restrictions in (a) and (b) produces a region which we denote by RG where G denotes the graph obtained in stepS: (0< 1~
. . ,

=

. .,

=

=

=

=

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Open bosonic string

585

17m separate producing the same graph or a new graph. In the case of the latter, we repeat the procedure. Step 7. For boundaries of R 0 associated with restriction (b), vary the parameters to produce a graph not checked off on the list in step 1, if possible. Repeat steps 2—6 to produce another R0. Step 8. Repeat step 7 until all graphs are checked off in step 1. We define

R’~,’~’= ~R0.

(9.1)

G

We conjecture that all Feynman graphs can be generated by successively shrinking a propagator line to zero and stretching it in the other direction (see fig. 25a). This was found to be the case at tree-level [49] and at the one-loop level 3”° [ 52]. Figure 25b illustrates this for the vacuum diagrams 2a and 3a. Hence R~ involves two terms in eq. (9.1). Since the vacuum graph 4a goes into itself when a propagator line shrinks to zero length, one term appears in eq. (9.1) for R’~”111 Assuming the conjecture, we can produce all the RG via step 7. A proof of the

(a)

(b) Fig. 25. Switching channels. (a) illustrates the process locally: a propagator line is shrunk to zero length and then expanded in the other direction. (b) shows how figs. 2a and 3a are transformed into each other.

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conjecture would demonstrate the connectedness of proof of assumption A in sect. 8.5.

RB. H

and would constitute a

9.2. THE FULL INTEGRATION REGION

The first I variables in eq. (6.11) namely the x1, i 1,.. I, defined in eq. (6.5) are not the same as the group parameter y,. Although one can convert from x1 to y1, the change of variable jacobian would have to be computed. It is simpler to determine the integration region for the x1 directly. This can be done in step 2 by finding the coefficients ca~~ which produce the I holomorphic 2-differentials in eq. (6.4) and keeping a record of the values of the x1 as functions of the y1. Define {x0((y~))~{y1) E R0}. Define =

.,

=

(9.2) G

Then the region over which to integrate

f

I~({Xa})in

fldx~f

RBHj=I

eq. (6.11) is

lldZr,

(9.3)

~&r=i

where .~6is F n I~,that is, the union of the boundary regions. Here F denotes the closure of F in the complex plane.

10. Conclusion We have computed the general N-point g-loop (g> 2) tachyon amplitude in the open bosonic string. The final result is the product of eq. (9.3) with eq. (6.11). The former is the measure and integration region; the latter is the integrand. Equations (9.3) and (6.11) along with their derivation are the main result. Although we have treated only tachyon states, it is straightforward to obtain the result for other external states. The integrand must be computed using the appropriate vertex operators at the asymptotic positions. This produces a different correlation function which is easily computed. Equation (9.3) for the integration region remains unchanged. We have deliberately separated the integration region and integrand. When put together, infrared infinities arise from various sources. The presence of a tachyon in the open bosonic string produces divergences in the region where the propagator2 + in 1)T) a loop becomes large. In this region, the integrand behaves where T is the length of the intermediate strip and k is like the exp((—a’k loop momentum. For a’k2 < 1 and large T the integrand is exponentially large. There is also a milder divergence due to massless states. Additional divergences

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arise from closed string intermediate states in various channels. Open strings naturally produce closed strings at the one-loop level and beyond and the tachyon in the latter produces another exponentially large integrand. When off-shell external states are present, other singularities from closed strings can arise [51]. The above-mentioned infinities are not expected in an anomaly-free superstring theory, at least on-shell. In such a theory, tachyons are absent and supersymmetry assures a cancellation between bosons and fermions for the infrared divergences due to massless intermediate states. We have found a good way of parametrizing open Riemann surfaces using the uniformization theorem. Our surfaces are domains in the upper-half plane bordered by arcs of circles identified pairwise. Such a concrete realization allows us to calculate in a concrete manner. Each calculational step can be programmed into a computer to obtain a numerical evaluation, if need be. If we were able to generalize to the superstring, finite perturbative corrections to physical processes in that theory would be obtainable. One of the key ingredients was the conformal map taking the domain in the upper-half plane onto the string configuration produced by the Witten string field theory. We were able to determine this map from the zeroes and poles of its derivative using Riemann surface theory. The general N-point g-loop (g> 1) map, p, was presented in sect. 4. With p it was straightforward to compute correlation functions on the worldsheet generated by covariant string field theory. We transformed to the upper-half plane domain and used existing formulas. In the process, computability conformal factors were generated. The integrand, at this stage, was the product of two correlation functions, a b—c one and an X’~ one. We then made a change of variables. The resulting jacobian cancelled much of the b—c correlation function. A similar cancellation was observed by Giddings in his verification that the Witten string field theory reproduced on-shell the Veneziano formula [33]. The ghost-jacobi identity at the tree and the one-loop levels for N external states was derived in refs. [33,49,52]. In this work, we have obtained the general g-loop ghost-jacobi identity for g> 2. The selection of integration variables was crucial. Without the judicial choice in eq. (6.5), we would not have been able to derive the identity. With the correlation functions and the ghost-jacobi identity we obtained the integrand for the scattering process of N tachyons at the g-loop level, both on-shell and off-shell. We then turned to the task of determining the integration region. This was done analytically for the H 0 case, the case in which the surface has no handles but an arbitrary number of boundaries. Some general features about moduli space were derived which might have some interest to mathematicians. In particular, we found a universal ordering along the real-axis of the fixed points of the elements of the covering group. An algorithm was presented providing the order. We were greatly aided in these studies by computer production of sample Riemann surfaces. Finally, in sect. 9, string field theory itself was used to =

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compute the moduli space for the generic open Riemann surface. This represents the solution of a very difficult and old problem. Most of our results were expressible in terms of functions, quantities and differentials defined on the surface. Crucial in the computability of these was the convergence of the Poincaré series for the g holomorphic differentials, a result obtained by Burnside in ref. [81]. After the holomorphic functions are computed, it is straightforward to obtain the periodic matrix, the e-functions, etc. In one limiting situation we were able to proceed via ordinary algebra. This was for surfaces whose double involved long thin handles. Such surfaces offer an interesting laboratory to performing and checking string calculations. Let us explain why we are unable to treat theories other than the open bosonic string. For closed stings, supersymmetric or not, we are confronted with two difficulties. Firstly, the Burnside—Poincard series fails to convergence. This means we do not have a concrete way to obtain the holomorphic differentials (and hence almost all surface quantities) that satisfies our computability criterion. Higher loop results in closed string theories are often expressed in terms of theta functions. How to actually calculate them needs to be specified. The second problem concerns moduli space. We were able to find it using covariant string field theory. For closed strings, a complete field-theory formulation does not yet exist*. The computation of moduli space for closed Riemann surfaces is a notoriously difficult problem. There is some chance that our methods can be extended to the open superstring. The original superstring field theory [150—153]had difficulties reproducing amplitudes involving Neveu—Schwarz states due to infinities [154—156].A modified theory [157] produces finite on-shell amplitudes but appears to have difficulties with off-shell processes [158]. It is unclear what happens at the one-loop level and beyond. This line of research needs further investigation. For recent progress see refs. [159—161]. The extension of our methods to other string theories is an interesting area of research. Another area is to put in Chan—Paton factors and consider the large N limit. Such a limit dampens the contributions to surfaces with handles leaving only surfaces with boundaries. The moduli space for this restricted set was determined analytically in sect. 7. Hence the large N limit of string theory is much more accessible. Our work has made progress in computing the perturbative series in string theory. Many higher ioop string results are formal in that they involve abstract quantities which cannot actually be computed. To avoid this, we have imposed upon ourselves the criterion of computability by computer algorithm. Some day a standard model string theory might be found which reproduces the known physics.

*FOr

recent progress see refs. [146—149].

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Our concrete results will hopefully be useful in obtaining the perturbative corrections in such a theory. I would like to thank Robert Bluhm and Alan Kosteleck~for discussions. I acknowledge short conversations with Joaquim Gomis, Irwin Kra, and Olaf Lechtenfeld. I am particularly grateful to Glenn Schober who provided me with a solid foundation in Riemann surface theory. This work is supported in part by the US Department of Energy under grants DE-ACO2-83ER40107 and DE-ACO282ER40125 and by a NATO Collaborative Research Grant 0763/87. Appendix A COMPUTATION OF PARTIAL DERIVATIVE FOR THE GHOST-JACOBI IDENTITY

This appendix derives equations for 3Xb/aTa and 3Zr/0Ta which are used in the next appendix to prove the ghost-Jacobi identity. Consider the correlation function f({Ta})

=g({xa})

=

(A.1)

(~5’),

where ~

(A.2)

When <~> is thought of as a function of Ta we call it f, whereas when it is thought of as a function of Xa we call it g. The ~r r 1,2,. N + 1, in the string configuration correspond to Zr on the real axis. We choose ZN+, to be a function of the other Zr, r=1,...,N, we take 4 to be a Q=0, e= 1 boson [19]and we require =

. .,

N+ i

(A.3)

~ A,.=0, r= i

in order that f be non-zero. The following integrand factor, I, is common to many of the subsequent equations do

N

1= flexp(h~N~) r=i

x =

~ATAT

—hN+I

g

(ZN+,)exp ilr>.

fl

[E(Zr,Zs)IA~A~exP[2~i

g

~q~r,1q3

j=1 1=i

Z

i~r
where hr

-~—

Eqj~ArfZ~wj], i=i

is the conformal weight of exp(A4).

r=i

z1~

(A.4)

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It is necessary to compute af/~91~in 2 eq. (6.7). Using the technique described in sect. 6 and the methods in sect. 5, one gets dw {dp

—

1’

öT~q~fl~~2 ‘ir,dw —(w)lJ F

N+i

2

g

xI(2~i~qiwi(w))+

L\

I

i1

g

~

~ r=i

i=i

(A.5) where the terms omitted do not involve q1q3 or q,A,. factors. Further computation gives

g

g

at1~

0Zr

where

~ —

q.EA

—

—

—

t~is

N-o-1

II— ~q~q1 + 2~ik=1 ~ q~~

r=1

L

I

g

L

i=1

~ II2~i~ q1nA

A

f

Z,190)kl —

Z0

8ti~

jI

(A.6)

+...,

/

q~(A~w1(Z~) +

ôZr AN+lwI(ZN+l))]

+...,

(A.7)

defined in eq. (6.5). From the chain rule 9f

ôx”3g

‘ =

b=1

~a3X

N

(A.8)

+ r~i aT~aZr’

(A.9)

at1’

Using eqs. (A.5)—(A.9) and comparing coefficients of q,q, and qA,. in eq. (A.8) one finds dwfdp (A.10)

~b(w)

~ 3Zr( w1(Z)— r

_~

azN±l azr

J

dw I dp

=

~[~(w)J

{w1(w)w(z~zN+I)(w) +

~

b=i

~b(w)f~

3WI

ZN±!aXb}

(A.11)

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where

E(w,x) 0)(x, _

5)(W)

=

3~ln

E(w,

(A.12)

~

is the unique meromorphic 1-differential with simple poles at z x and z y with residues + 1 and — 1, respectively, and having zero a1-periods [64,72]. In eq. (A.10) b ranges from 1 to I and in eq. (A.11) r goes from 1 to N. In eqs. (A.10) and (A.11) a ranges from 1 to I + N. Equation (A.11) gives g equations for aZr/aTa since i may be 1,.. ., g. Furthermore, the point ZN+i may be an arbitrary function of the Z,.. To consolidate the redundant information for aZr/aTa do the following. Let w E~1c1w1,where c1 are constants. Choose the c1 so that w vanishes at ZN±,.This can always be done for g>2. For example, if w~vanishes at ZN+, use it, otherwise let w(z) — w2(z). Multiply eq. (A.11) by c1 and sum over i. This eliminates the second term on the left-hand-side of eq. (A.11) =

=

=

=

—1

3Zr =

dw

dp

£O(Zr) ~

-,

~(w)

{w(w)~(z ~z++)(w)

+

z b=31ZN*lax} (A.13)

Equations (A.10) and (A.13) constitute the main results of this appendix.

Appendix B DERIVATION OF THE GHOST-JACOBI IDENTITY

From eqs. (A.10) and (A.13), the inverse jacobian can be constructed as dw

dp

-‘

J’=~~~~4 ~—(W~) ~

‘IT!

detM’,

W~

(B.1)

where fora=1,...,I,

M~b=~(wb) Mj±rb =

W(Zr)

{w(wb)w(Z

forr=1,...,N,

forb=1,...,I+N,

ZN+I)(wb) + a~i ~a(Wb)fZr~}

forb=1,...,I+N,

(B.2)

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and w is a holomorphic differential constructed in appendix A which vanishes at ZN±,.

By multiplying row a of M’ by

J~

1(ôw/8x’O/w(Z,.), summing from a and subtracting the outcome from row I + r, one sees that

=

detM’=detM,

1 to I

(B.3)

where fora=1,...,I,

MQb=I/Ja(Wb),

—z MI+rb=

forb=1,...,I+N,

)(wb)

~~)N÷1

,

Now consider the main piece, function (see eq. (5.9))

forr=1,...,N,

forb=1,...,I+N.

of the integrand in the ghost correlation

‘bc’

I+N

fl [E(Zr,Zs)] i-~r
‘bc

~

JZ,

r=i ~o

(B.4)

N

fl E(Wa,Wb)flfl[E(Wa,Zr)1~’ )~a
~ a=i

3.

(B.5)

[~(Zr)] Z 0

a=~i

r=l

In what follows we show ‘bc det M, where is a surface-dependent constant. It depends neither on N nor on Wa nor on Zr. It straightforward to verify from table 1 that both Ii,,. and det M are 2-differentials in each each of the variables, Wa~ a 1,. .., I + N. Our proof that ‘be det M uses induction on N. For N 0, appendix D shows that =

K

=

K

=

K

=

K

~

(B6)

det1~1~a(W)

—

is independent of the Wa~ i.e., a constant depending on geometry of the surface. For N 1, I,~,.has, as a function of Wa~ a single pole at Z,. Likewise, det M has a single pole at Wa Z~coming from °~(wi,_ZN+l)(Wa) in the (I + 1, a) entry of M. The potential pole at Wa ZN±,in det M (see eq. (B.4)) is absent because 0. Focus on the last variable WN±,. The residue of ‘bc at WN±, Z, is ‘be for N 0, i.e., the numerator in eq. (B.6). The residue of det M at WN± 1 is the determinant of the first I 1< I piece of M, i.e., det M for N 0. Hence the residue at WN+, Z~of ‘bc — det M for N 1 is ‘bc — det M for N 0 and vanishes. Because both ‘bc and det M are antisymmetric in the Wa~ the vanishing of the residue at WN+, Z~implies the vanishing of the residues in Wa at Wa Z, for =

=

=

=

=

=

=

=

=

K

=

=

K

=

=

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all a

=

1,

. .

. ,

N + 1. Hence

‘be —

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Open bosonic string

593

det M is a holomorphic 2-differential in each of

w

1,.. w1.~1 and can be expanded in terms of a complete set of I holomorphic 2-differentials . ,

1+11 IbC—KdetM=

3(g

=

—

1)

1

1~~II ~ c~t/i°’(w~) =0,

(B.7)

a=1 (b=1

where cb are constants. To show the last equality in eq. (B.7). note that the left-hand side of eq. (B.7) is antisymmetric in w1, . . . , w1~1so that the same must be true of the right-hand side. However, there are only I independent cli”; consequently the right-hand side vanishes. The situation is like that of a fermion system. We have N + 1 fermions to be placed in N states. We have completed the induction process for N 1. The above analysis shows that the constant for N 1 is the same as N 0. Assume the identity when N L and consider the N L + 1 case. The ghost integrand factor I~ has poles at Wa Z~from the third term in eq. (B.5). Likewise, det M~’~’~° has poles at Wa Zr from ~wr, _zN+I)(Wa) in the (I + r, a) entry but not at Wa ZN+i due to the presence of W(Wa) (here N + 1 L + 2). Both ~ and detM~’~~ are 2-differentials in each of the Wa~ a 1,...,L + 1 and antisymmetric in the Wa and antisymmetric in the Zr. By inspection the residue in I~,~t)at WJ+L±t ZN+i is I~ and the residue in det ~ at W1+L +1 ZN+i is det M~. By the induction assumption, I~ K det M~, with K given in eq. (B.6), so that the residue of I~ —KdetM~~’~ at W1+L±1 =ZN+i vanishes. By antisymmetric in the Wa and in the Zr, all residues of ~ 1) — KdetMU+’) at Wa =Zr vanish. The difference I —KdetM~’~’ can again be expanded as in eq. (B.7) except that the product over a goes from 4t~0. 1 to I + N. The reasoning below (B.7) stillfactor applies M°~ Comparing theeq. integrand in so eq.i~~—Kdet (B.5) with the exact integrand for the ghost system in eq. (5.8) and using eqs. (B.1), (B.3) and (B.7), the jacobi-ghost identity is obtained =

=

K

=

=

=

=

=

=

=

=

=

=

=

=

N

K

)b,.J=Kllexp(—Noo)K ~

(B.8)

r=)

where K is given in eq. (B.6). To arrive at eq. (6.10), ( )~ must be computed. Up to a pure numerical constant (independent of the geometry of the surface), this has been accomplished by Verlinde and Verlinde [78,79]

(K )~Y~

—

=

2’IT!) (

f~w zolT)l1 0(W)detgxg w~(y,)fl~

1~i
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where W and y, are arbitrary points on the surface, i.e., in F. Appendix C demonstrates that eq. (B.9) is independent of the choice of w and y1. Employing world-sheet factorization, appendix E computes the overall normalization constant in eq. (B.9). Equation (B.9) differs from the result in ref. [78,79] by a factor of (2’rjy~• The one power of K ) from the ghosts in eq. (5.8) combines with 26 powers of K )~ from the X~L(z) in eq. (5.9) to give (K ))27 in the amplitude. This generates the right-hand side of eq. (B.9) to the minus ninth power and is the second factor in eq. (6.10). The first and third factors in eq. (6.10) come, respectively, from the second and first terms in eq. (B.8). The overall sign in eq. (6.10) was determined by world-sheet factorization and results in the absolute values in eq. (6.10). This completes the derivation of eq. (6.10).

Appendix C INDEPENDENCE OF EQ. (B.9) ON VARIOUS VARIABLES

Let .,Yg,W)

f(y3,y2,..

— fz~j°~

—

z0I~)lhi~j
(Cl)

u(W)fl~1E(y1,w)

—

This appendix demonstrates that f/detgxg a~(y1)is independent of y1, y2,..., Yg and w. First consider the variable Using table 1, one sees that f is locally a 0-differential in and globally periodic about the A1 and B, cycles. Thus f is a function of w. By the Riemann vanishing theorem, eq. (3.21), ~ has simple zeroes at y1, w y2,..., W Yg—t and w Yg~These zeroes however are removed by the E(y,, factor in the denominator of eq. (C.1) which also has simple zeroes at w y.. The function f is defined on the double of ~/ which is a compact surface. However, such a function, if not constant, must have poles and zeroes. Consequently, f is independent of Now consider the dependence on y,, i fixed. From table 1 it is seen that f is a globally defined 1-differential in y1. The potential pole at y,=w in E(y1,W) is cancelled by the y1 W zero in 6~by the Riemann vanishing theorem. The 1-differential f is holomorphic in each of the y1. Since the w~,j 1, . . , g, are a complete basis for holomorphic 1-forms, f is proportional to a linear combination W.

W

W =

=

=

=

W)

=

W.

=

=

.

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595

of w,(y,) in each y,

fl

f=

~ a11,o~(y,),

(C.2)

1=1

where a,, are constants independent of the y,. Since

f=

1 —,-

g.

g

~

(—

~ (1)fT g. a’wSg 1

=

—,-

g.

g

g

~

aff1(,),,w,~(y,)

j,1

I~1

g

det( a)

~ a111w1( Y,.(t))

1i

1

=

is antisymmetric in the y,

g

fT

1)~

o’~Sg

f

g

11 E

~JJ2f’ , . ~

Yt)

~‘~J2~

y2)

. .

o,,~(Yg)

,~

1 —~-det(a)detgxg.ü~(y,).

(C.3)

Hence f/detgxgw,(y,) is (1/g!)det(a) and independent of the y,.

Appendix D INDEPENDENCE OF EQ. (B.6) OF THE

Wa

Let

f(w1,w2,...,W1)=O

a=1 ~f”w_3~~0T) Z 0

3.

fT

E(Wa,Wb)fl[~(Wa)] 1,~a
(D.l)

This appendix demonstrates that f/detj~jtIi”(Wa) is independent of W11 and W1. The method of proof is similar to that of the previous appendix. Consider the variable Wa• From table 1, one finds that f is locally a 2-differential in Wa and globally periodic about the A, and B, cycles. Since f has no poles, f is holomorphic 2-differential in all the Wa~ It may be expanded as W1,

I

f=

U a

=

W2,.

I

~

I b~=

Cab~IJ””(Wa), I

(D.2)

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where cab are constants. The asymmetry of f in the Wa then implies f a det1~1~Ii”(Wa). The reasoning is similar to that use in appendix C (see eq. (C.3)). Hence we have obtained the desired result: f/det1~jl//’(Wa) is independent of the Wa~

Appendix E LIMITING RIEMANN SURFACES AND WORLD-SHEET FACTORIZATION

This appendix discusses a set of limiting Riemann surfaces in which the quantities entering in the amplitude can be computed quite explicitly. They are the surfaces for which the multipliers, A, of the SL(2, 10 elements of F are large. Since the h-circles shrink in size as A increases, the circles of the g generating elements do not overlap in this limit. The double of the surface is that of the Schottky type [66,731 (see appendix F) and corresponds to a sphere of g very thin handles. This is conformally equivalent to the situation in which the length of each handle becomes very long (see fig. 26). The process in fig. 26 is computable by world-sheet factorization. The contribution of a particle masst, is m the flowing the ith like 2)t,) of where length through of the handle and handle p, is thebehaves correspondexp(—a’(p~ +m ing loop momentum. The state of minimum m2, i.e., the tachyon, dominates. Each handle can be replaced by local tachyon operators at the source and sink of the

(a)

‘N.~

(b)

Fig. 26. Limiting Riemann surfaces and world-sheet factorization. (a) displays a typical surface in which the handles become long and thin. Such surfaces can be treated algebraically. The loop momentum flowing through the ith handle is p,. (b) illustrates world-sheet factorization. The handles may be clipped and replaced by pairs of local tachyon operators producing and absorbing the loop momenta. The length of the ith handle is t,.

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Open bosonic string

handle times the factor exp(—ct’(p7 — 1/a’)t,) since m2 1/a’ for tachyons. The momentum of the source tachyon operator is p, whereas that of the sink is —p, and as A, —~cc the source and sink are located at the fixed points of the map, z~ and z’.... When such a replacement is done for all handles, an easily computable tree-type correlation function of tachyon operators remains. These limiting surfaces form an attractive laboratory for ioop computations. They serve as a check on our computation since the precise momentum dependence of the loop amplitude must coincide with the factorized tree-like tachyon process. It is by these means that the sign of the amplitude as well as the normalization constant in (K )~)~ in eq. (B.9) is determined. Let us first compute some surface quantities, such as the period matrix, the vector of Riemann constants, the prime form, etc. The holomorphic differentials are given by eqs. (3.6) and (3.9). The first term in second equation of eq. (3.9) usually dominates as A, —~cc (an exception is ‘r,, for i * j for which one word in the sum over W~, must be taken into account). Using eq. (3.10), we obtain for the period matrix =

T 11

=

~ln(A1) 1

T1~—Tlfl

2~t

(E.1)

+...,

(Z~—Z~)(Z—Z~) J\I ~Z+—Z)~Z—Z~)

~

+...

,

t*j,

(E.2)

where A~,z~and z’ are, respectively, the multiplier, the attractive fixed point and the repulsive fixed point of the ith group generator (see eq. (2.4)). It covers the homological cycle B,. The omitted terms in eq. (E.1) and (E.2) and below represent non-leading effects in the A, —* cc limit. From eq. (3.19), the vector of Riemann constants is calculated to be

(z’—Z3+)(~0—~~)

g (~ZO)

=

—

ln (z—zi)(z0 —z~) .

~ 2 + — 2~z j1

+...,

(E.3)

i.5~j

and the prime form, away from the fixed points, becomes, as expected, E(z,W)=(z-W)+....

(E.4)

Equation (E.4) is obtained from eq. (3.16). Finally a-(z) from eq. (3.17) is 1

g

u(z)=fl 1=)

i

Z_~Z

+....

(E.5)

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Our goal is to calculate the various factors in eq. (6.11) using eqs. (E.1)—(E.5). Throughout this appendix we take z~> z~> ... > z~i> z~. In what follows z~., z~ and are treated differently from the other fixed points. We first consider the vacuum process so that N = 0. We work on-shell since this suffices for our purposes and renders equations simpler: the zero—zero Neumann function factors are absent. In the rest of this appendix we set a’ 1/2. To compute (K ~ i.e., the inverse of the right-hand side of eq. (B.9), we take advantage of the fact that it is independent of the Y1 (see appendix C) so that the Y1 1 for i 1,. . . , g. Straightforward but may be algebra set to any desired lengthy reveals thatvalues. Let Y, —~z

4>

4

4>

=

=

(K

))3

=

( —

1)/2

1)g(g—

~[]~(z’

—

1 ~i
z~)+

(E.6)

To compute K, i.e., the right-hand side of eq. (B.6), a similar strategy is followed. First, however, a set of holmorphic 2-differentials must be selected. Denoting ~fi’~ by42 (2’ITiw1)(2’ITiw,), we choose for this set ~ ~22 ,j,u ~~12 ~,3t ~32 ~41 . ~, ~ cl’ 1~g2, i.e., using the notation of sect. 6, c’1, 1, i 1, . . . , g, c~’ 1, and ~ 1, cf~f~ 1, for i 1,..., g — 2 and all other c,~are zero. Denote the Xa corresponding to ,01i” by x,,. Notice there are I such i,li” and I such x,~as should be the case. Because K in eq. (B.6) is independent of the Wa (see appendix D) they can be set to specific values. We choose the limit w- —‘z’, for i 1,.. .,g, Wg~~ =

=

=

=

=

=

=

Wg+2

t,,

~

Wg~3 ~

Wg~4 ~

Wg~5 ~

W11

Wj~Z~.

~

To compare to the result from world-sheet factorization, it is convenient to use i 1, . ., g, where =

.

A,

4,

exp(t,)

=

4,

and the fixed points z~., z~, z~.,..., This introduces the additional jacobian —

dx11

—

A

dx22

A

dt1 A dt2

A

... ...

A A

dxgg dtg

A

A

+ z~

dx12

...

and

A

(E.7)

,

z~.

as the integration variables.

dx3~A dx32

A

dz~Ad4A dz~A...

After a large amount of algebra, we find for the combination 1z1 —z2 ~ —z2 ~ KJ’ exp ~g t~ (_ 2±ik + +J~ + 2

fT

=

i~1

1)g

fl

A dXg2

...

A

E 8)

dz~

KJ’

(z~—z~)9

+

...

1(z~—z~..)

(E.9)

/

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Open bosonic string

599

Combining eqs. (E.6) and (E.9), the partition-ghost-Jacobi factor,

.1’[K1)03~]26K

J’Kl)’

=

>b~L becomes

J’K1~’=

(z~-z~)(z~-z~)(z’-zfl

(E.10)

+....

fI~

1(z’~— z~)2

26p The (6.11)measure, is thus d.d’, for the momentum integrations, fl~1fd

26, in eq. 1/(2~r)

d4’= dt 1 dt2

. . .

dz~J’K1)”

1 ~

fT

x

dz~.dz~dz~

. dtg

. .

~ —~‘ \(Z~ (~

g

exp ~(—~p~+i)t

—Z

1

1~i
+...,

(E.11)

i=1

2iriLt
..

.

dtg

dz~dz~dz~

. . .

dz~

=exP(~(_~P7+ 1)t~) x ~ =

exp

..

Eg ~/

—

1 2 \ ~p 1 + 1)t1

1 —z2~ ~(~‘ ±—z2—I’. “(i’—

‘~ +

(~

.

fl~,(z~—z)

1~1 I

x

—z2+

I

fT

j\I

J\

I

P~~PJ

~

(E.12)

1~i
4,

4

where we have chosen z~and as the c-type vertices [191.Equations (E.11) and (E.12) agree on-shell when p, .p 1/a’ 2 and when use of eqs. (E.10) is made. This concludes our check of the vacuum case. When N tachyon operators are present at Z1,. ZN, the correlation function contains the additional product of operators, 1 exp(k’~ This_z1)~’1)i generates 11~-. t~rXXZ,.). +~“(14 the additional factor f1±r
=

..,

.

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Open bosonic string

eq. (E.4) is used. The second factor comes from exp(2~i

11kg ~p1f~’w1) in eq. (6.11) when the first term in the second equation in eq. (3.9) is used for w,. Thus agreement is obtained when N external tachyon states are present.

Appendix F DEMONSTRATION ThAT THE GENERATORS YIELD A RIEMANN SURFACE

This appendix shows that when F generates a word list consisting only of hyperbolic elements then no two points in the interior of the fundamental domain are identified. This is the key ingredient in proving [66] that F \ 11-I is a Riemann surface and hence showing that F is a group satisfying the conditions of the uniformization theorem. Fl. THE SCHOTTKY CASE

We start with a Schottky group [66,731 and generalize. It is useful to review this case in preparation for the more general situation. When the h-circles of the g generators of F in sect. 2 do not overlap, our covering group is a Schottky group. Let F be a free group on g generators, h1, hg, such that none of the interiors of the h-circles of the generators overlap: ‘h ~ 0 and Ih~’ 0 for all i and j, and ‘h, I~i 0, for i *j. Such a F is called a Schottky group and it is known [66,73] that F \ C is a Riemann surface. The key step [661in the proof is to show that no two points in the region, F,~, O~ ~ ~ 0h/” exterior to all h-circles are identified under any h E F, h * e. When the centers of the h-circles are on the real axis, the case of interest in this ~ work, it is possible to show that, in addition to F \ C, F \ H is a Riemann surface. Let us demonstrate that no h E F, h * e, identifies points in F~.Such an h is a word constructed from the h,: h ha. . h,, 1. Let w E F~. Then since W E 1 and ‘h~’~ ha~W)E hai ha maps °h, into In-i. Because ha * h, ha(W) E O,~.since Hence 2ha~W) ~ I,,,_~. ~y continuing the process, one sees that h(w) ha. ha(W) ~ Ih~.Since Ia-i 11 F~ h(W) cannot equal W. . . . ,

=

~

~

=

=

=

Ohg

=

.

=

=

. .

=

0,

F.2. THE GENERAL SITUATION

Let F be a free group on g generators such that the construction in sect. 7.1 leads to the group list L {h,,. h~)*.Since {h1,.. h,,} generates the group, any h E F is expressible (not necessarily uniquely) as a word in the h,, i 1, . m. Let h =ha ...hai, m>l, where the ha are selected from {h1,hjt,...,h~,h~) and nearest neighbor letters are not inverses. Let W E F~ ~ ~ ~ °h2 ~ =

. . ,

.,

=

=

*

For notational convenience we drop the primes on the h- here.

.

.

,

S. Samuel

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Open bosonic string

601

Look at the left part of h: h ha ha Find the longest subword of h beginning with the left-most letter which is in the word list. Replace this subword by this element. If the subword has q letters then h hA ha,,, q~• haj, where hA1 ha,,,.. . ha,,,*i q E L but ha,,,.. ha,,,~q~ L. Repeat the procedure beginning =

~••••

=

=

.

‘~

with ha,,,_q~••hai. Repeat further until h is represented by h hA and no hA hA i 1,.. . p — 1, is in the word list. Note that I~,, IhAI for if this were not the case then hA. hA,hA,±~ would be in’ the ~vord list by the construction procedure in sect. 7.1, contradicting the fact that hAhA ~ L. Examination of the proof for the Schottky case in sect. F.1 reveals that if h=hA...hA~ and ‘h4 flI~~i =0 then h(W)*w for wEF~. The fact that h(w) * W f~h E F, h * e, can be folded into the proof in ref. [661 to demonstrate that F \ C and F \ H are Riemann surfaces. In short, if the procedure in sect. 7.1 produces only hyperbolic elements in the word list then F satisfies the conditions of the uniformization theorem and F \ H is a Riemann surface. =

,

=

,

j+1

=

hA1

..

=

.

0

Appendix G AN IMPOSSIBLE OVERLAP SITUATION

This appendix proves the following. Result: If Ch and C,,2 overlap and C~1-iand Ch2-1 overlap on the same side (see fig. 12BB.la), that is, the order of the h-circles in going around the unit circle is C~,Ch, Ch-I and C,,1-, then h3 h2h~’is elliptic. There is probably a way to make our proof more efficient but we proceed by brute force. We consider three situations and work in Ill rather than in H. Since corresponding maps are related by an SL(2, C) conjugation, elliptic in 1111 implies elliptic in H*. Recall that when C,,1 and C,,2 overlap, the arc of Ch_1 from A to h,(C) is covered by IJ~3and the arc of C~2-i from B to h2(C) is covered by I~3~1 (see fig. 12.B). Situation 1. Suppose C,,1- and C,,2-1 are tangent at one point, A, on the boundary (see fig. 27a), then h3 is elliptic. Proof~A is inside 1/73 ~ and by continuity 1/73 ‘ I,,~ * 0 in Ill so the h3-circle appears as in fig. 26b and h3 is elliptic. Situation 2. Suppose C,, and C,,2 and C,,1-~ and C,,2- overlap by the same amount (see fig. 27a). By “same amount” we mean that the euclidean length of the arc of C,,~covered by ~. is the same as the length of arc of C,,1-i covered by I,,2I. Then, the points C and D in fig. 27c are related by h1(C) D and hj~(D) h~(D) C. Consequently, D is an interior fixed point of h3 and h3 is elliptic. =

=

=

*

=

By elliptic, we mean in this appendix that —2
h2(C)

=

602

5. Samuel

/

Open bosonic string

(a)

(b)

(c)

(d)

Fig. 27. Cases in the proof of the impossible overlap. (a) shows the case when two h-circles touch at a point and their inverse h-circles overlap. The element h 3 = h2h~1 is elliptic as can be seen in (b): the h-circles of h3 and its inverse intersect at a point in III. This point is an interior fixed of h3. (c) is the case when the two h-circles overlap by the same amount. Point D is an interior fixed point of h3 as (d) shows. The parameters producing fig. 27a are (z~,z~,A1)~(—l000000,0,3)and (z~,z~.,A2)~ (0.4,1.95,5). Those for fig. 27c are (z~,4, A,) = (0.2,5,3) and (z~, z~÷, A2) = (0.5,2,5).

point

Situation 3. Suppose C,, and C,,2 overlap by an amount different from the amount that Chj-1 and C,,1-1 overlap. Without loss of generality we may assume that1) C,,1 C,,2h(A) overlap more h(z~, than C,,1-iA),and as h(z~,z~,A and and define by h(A) i.e.,Ch2-I. h(A)Representing has the same h1fixed points as but a different multiplier. Perform a conjugation of h 1, h2 and t(A)=h h3 by h(A): h~(A)=h(A)h3h’(A), h’2(A)=h(A)h2h~(A) and h~(A)=h(A)h1h 1 because h1 and h(A) commute. At A 1, we have the situation with which we started. As A cc, the fixed points of h~(A)move towards and the size of the h-circle of h’2(A) decreases. So by increasing A, the overlap of C,,1 and C,,2 can be made to vanish. Let A0 be the minimum value of A for which C,,1 and C,,2 no longer overlap. As A is increased from 1 to A0 either the overlap of C,,1-i with C,,1-i =

4,

=

—‘

4

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603

Open bosonic string

vanishes or becomes equal to the amount by which C,,1 and C,,2. The first case corresponds to situation 1; the second to situation 2. In either case, for some value of A, h~(A)is elliptic. Since h3 is elliptic if and only if h~(A)is, the proof is completed.

Appendix H THE IMPOSSIBLE COVERUP

This appendix shows that the situation in fig. 12CC.lb, namely I,, ~ I,,. and I~-~ I,,_i, cannot take place. Define r,, minWEC,, d(w, z0). Suppose this is achieved for 1(zw,,: r,, d(wh, z0). Then h(w,,) E C,,-i and since w,, E C,,, d(h(W,,), z0) d(w,,, h 0)) d(W,,, z0) r~.This implies rn-i r~ and r,,_1 > ~ the latter inequality being equivalent to r~> r,,., so a contradiction arises. =

=

=

=

=

=

Appendix I SOME TRACE INEQUALITIES

This appendix proves a result which is used in sect. 7.4 to determine the moduli space for the one-hole one-handle 2 > (Tr(h case. 2 then (Tr(h~’h 2> (Tr(h~’h 2 for n > 1, Theorem: If (Tr(h1h2)) 2)) 2)) 2)) where h 1 and h2 are hyperbolic elements of SL(2, 110. When we normalize SL(2, 10 elements so that their traces are positive, the above result holds without squaring the traces. We were motivated to look for such a relation for physical reasons: Since F is homeomorphic to each element h E F is associated with an h E Such an h is a closed curve and hence in a homological class [hi. Tr(h) is related to the infinitum of the Poincaré lengths of the elements in [hi (see refs. [68, 76])*. A value of Tr(h) slightly larger than 2 means that the length of the minimum geodesic homologically equivalent to h is small. When h becomes parabolic, Tr(h) 2, and the geodesic shrinks to zero (either a handle has been pinched or a hole has shrunk to a point). As Tr(h) increases, the length of this minimum geodesic increases. The product h1h2 represents a path which first goes around h2 and then goes around h1. If the length of the minimum geodesic associated with h1h2 is bigger than that associated with h2 then one suspects that going around h2 ~

=

*

The relation is as follows: Let L = the length of the minimum geodesic. Then L = 2 log(A) when Tr(h)=A +A~.

604

5. Samuel

/

Open bosonic string

and then going around h1 more than once makes the minimum geodesic even longer. Proof. Since the trace of a product of elements is invariant under conjugation of the group, perform an SL(2, 110 transformation to make the attractive and 2z. repulsive h1, respectively, zero and minus infinity: h1(z) A1 There are fixed threepoints genericofcases. Case 1: z~..<0 Using the representation in eq. (2.7), one finds that =

<4.

Tr(hTh 2)

[A~(z~A~’ -z~A2)+ A~(z~A2 _z~A~1)]. (1.1)

=

Consider Tr(h~h2) — Tr(h~’h2) — —

z+ z_ A1—1 2

2

n—lI 1

2

—t_ 2

2

\

Z_

2!

—n±1 t

z~A2—z~A~ A1

.

)

On the right-hand side of eq. (1.1), the two terms in round parentheses are positive. The premise of the result is that eq. (1.2) is positive for n 1. Hence the first term in round parentheses is larger than the second term in round parentheses. As one increases n beyond 1, one makes the first term in square brackets even larger than the second term. In other words if Tr(h1h2) — Tr(h2)> 0 then Tr(h~h2)— Tr(h~’h2) > 0. Case 2: z~0since A2> 1. There are two subcases. 1h Subcase 2.A. In addition, assume 4A2 —z~A~ >0. Then Tr(h’1 2)> 0 and eqs. (1.1) and (1.2) hold. The assumption that eq. (1.2) is true for n 1 implies that 1(z~A~’—z~A ~ which implies that A 2)> 1)/A A~[(z~A2—z?iA~ 1]and that Tr(h~h2)— Tr(h~’h2)> 0 for n >2. Subcase 2.B. In addition, assume <0. Note that (z~A~’ — 2Al 2 )(A z~.A2—z~A~ z 2)—(z~A~ —4A2) =(4—z 2 + Ar)> 0 so that A~(z~A~’ —z~A2)+ A(z~A2—z~A~’)>0 for n>0. Hence Tr(h~’h2)>0and eq. (1.2) holds. In this equation the first terms in round brackets is positive while the second is negative, so that Tr(h~h2) — Tr(h~ - ‘h2)> 0 for n>It1. isNote that one didofnot to use 2 > (Tr(h 2 for this subcase. a consequence thehave additional (Tr(h1h2)) 2)) assumptions. Case 3: 0 0. There are two subcases. Subcase 3.A. In addition, assume z~A~’ — z~A2>0. Then the proof proceeds as in Case 1 above. The n 1 result implies the n > 2 result in eq. (1.2). =

=

<4.

=

S. Samuel

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Open bosonic string

605

Subcase 3.B.

In addition, assume z~A~— z~A2<0. There are two subsubcases. 2A~’)> 0 then Subsubcase 3.B.1. If AI(4A~’ — z~A2) + A~(z~A2— z Tr(h 1h2)> 0 and eq. (1.2) holds for n 1. The first term in round brackets is negative while the second is positive so the right-hand side is negative. Hence Tr(h1h2) — Tr(h2) < 0 and the hypothesis of our theorem is not satisfied and no conclusion need be drawn. 2÷A~ — z~.A 1)< 0 then 2)+ Aj~(z~A2 — z~A~ ~ Subsubcase 3.B.2. If A1(z because by increasing n we are making the first summand more negative and the second one less positive. Thus Tr(hç’h 2) < 0 for n > 1. One finds that 1h Tr(—h~h2)—Tr(—h~ 2) =

—

A1 —1 2

2

n—ti 1

2

—

2

—i~

Z+2!+

~

—n+t 4A2 —z~A~ t

A1

‘

(.)

which is positive for n > 2 since both terms in round brackets are positive. The conclusion of the theorem is true independent of the hypothesis. The cases not covered in 1, 2 and 3 reduce to the above when the transformation z —~ —z is performed. When h1 and h2 are conjugated by this transformation, the fixed points of h1 are unchanged while those of h2 transform as and z~— —z~.All possibilities are taken into account and the theorem is proven.

4—’ —4

Appendix J COMBINATORIAL FACTORS

The combinatorial factor, cG, in eq. (5.10) varies from grapht”/2”, to graph. In the independent final in eq. (6.11), normalization constant a”°g Feynman graphs of theintegrand graph. The reason forthe this independence is that iscertain cover the same region of moduli space several times. Let O~denote the overcounting factor for the graph G. For vacuum bubble diagrams we shall show that CGOG

=

1/2~’.

(J.1)

Here V 2(g — 1) + N is the number of vertices, P 3(g — 1) + N is the number of propagators and N is the number of external states. For vacuum graphs, N 0. In a ç~scalar field theory, it is well known that 1/c 0 is [order of the symmetry group] x [2]numberof tadpole Ioops~ A tadpole loop occurs when two legs at the same vertex join to form a loop. The graph in fig. 2a has two tadpoles and the symmetry group has two elements so c0 1/8. After labelling vertices and edges, the =

=

=

=

606

S. Samuel

/

Open bosonic string

symmetry group is the set of maps taking vertices into vertices and edges into edges leaving the graph invariant. For fig. 2a, the group consists of an identity element and the map which interchanges the two vertices, the two tadpole loops and leaves the middle edge in place. Whenever n lines span two vertices there is a factor of 1/n! in c0 because the n lines may be permutated among themselves. For the f 1P*~P*1I1theory we have rigid vertices. A Feynman diagram in the q~ theory decomposes into one or more rigid-vertex graphs. As a consequence, c0 is reduced by a factor related to the restricted number of ways of joining legs. The third and fourth entries in table 2 are the same in ~ theory but different in string field theory. Treating them as separate, we divide c0 by an additional factor of 2. A given Witten string configuration may count the same Riemann surface several times. For example, the graph in fig. 2a covers its portion of moduli space twice as T1 and T3 vary from 0 to cc• Here T1 and T3 are the lengths of the left and right loops in fig. 2. The surface with (T1, T2, T3) is the same as (T3, T2, T1) as can be seen by rotating the configuration by 180°.The field theory counts this region of moduli space twice; 0~ 2 for fig. 2a; an extra factor of two must be included in the integrand when the measure in eq. (9.3) is used. Since cG 1/8 and V 2 for fig. 2a, the normalization constant cGOG I We do not have a proof of eq. (J.l) but have checked it with many examples. Table 2 displays some of these. We have discovered that 0~ equals the order of the rigid symmetry group. The rigid symmetry group is the subgroup of the symmetry group of a graph which preserves, around each vertex, the cyclic ordering of labelled edges. When external legs are present, the computation of the normalization constant is slightly more complicated. In integrating Zr throughout the same string configuration is sometimes produced. If we denote F~the factor that compensates for this, then one needs to show that =

=

=

=

=

=

,~

c0F~,00

=

(J.2)

J/2V

where °G is the overcounting factor for the Riemann surface, i.e., the same factor for the graph in which external legs are removed. Table 3 displays many examples. In the first entry in table 3, F~ 1/2 since as Z1 is integrated over the boundary it will appear on the other tadpole loop, thereby double counting the Feynman graph. Although we do not have a proof, all cases for g> 2 which we have examined satisfy eq. (J.2). =

References El] J.H. Schwarz, ed., Superstrings, vols. I and II, (World Scientific, Singapore, 1985) [2] MB. Green, J.H. Schwarz and E. Witten, Superstring Theoiy, vols. I and II (Cambridge University Press, Cambridge, 1987) [3] MB. Green and DJ. Gross, eds, Unified String Theories (World Scientific, Singapore, 1986)

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607

[4] M. Kaku, Introduction to Superstrings (Springer, Berlin, 1988) [5] MB. Green and J.H. Schwarz, Phys. Lett. B149 (1984) 117; B151 (1985) 21; NucI. Phys. B225 (1985) 93 [6] D.J. Gross, iA. Harvey, E.J. Martinec and R. Rohm, Nucl. Phys. B256 (1985) 253; B267 (1986) 75 [7] P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, NucI. Phys. B258 (1985) 46 [8] S. Weinberg, Phys. Rev. Lett. 19, (1967) 1264; A. Salam, in Elementary Particle Theory, ed. N. Svartholm (Almquist and Forlag, Stockholm, 1968); S.L. Glashow, NucI. Phys. 22 (1961) 579 [9] CS. Hseu, B. Sakita and MA. Virasoro, Phys. Rev. D2 (1971) 2857 [10] J.-L. Gervais and B. Sakita, Phys. Rev. D4 (1971) 2291; Nucl. Phys. B34 (1971) 477; Phys. Rev. Lett. 30 (1973) 716 [11] L. Brink, DI. Olive and J. Scherk, Nuci. Phys. B61 (1973) 173 [12] A.A. Belavin, AM. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B241 (1984) 333 [13] VS. Dotsenko and V. Fateev, NucI. Phys. B240 (1984) 312 [14] D. Friedan, Z. Qiu and S. Shenker, in Vertex Operators in Mathematics and Physics, ed. J. Lepowsky, S. Mandelstam and I.M. Singer (Springer, Berlin, 1984) p. 419 [15] V.G. Knizhnik and A.B. Zamolodchikov, Nucl. Phys. B247 (1984) 83 [16] MA. Bershadsky, V.G. Knizhnik and MG. Teitelman, Phys. Lett. B151 (1985) 31 [17] D. Friedan, Z. Qiu and S. Shenker, Phys. Lett. B151 (1985) 37 [18] V.G. Knizhnik, Phys. Lett. B160 (1985) 403 119] D. Friedan, E. Martinec and S. Shenker, Phys. Lett. B160 (1985) 55; Nucl. Phys. B271 (1986) 93 [20] J. Cohen, D. Friedan, Z. Qiu and S. Shenker, Nucl. Phys. B278 (1986) 577 [21] V.G. Knizhnik, Commun. Math. Phys. 112 (1987) 567 [22] V.A. Kosteleck~,0. Lechtenfeld, W. Lerche, S. Samuel and S. Watamura, Nucl. Phys. B288 (1987) 173 [23] D. Friedan, in Unified String Theories, ed. MB. Green and Di. Gross (World Scientific, Singapore, 1986) [24] 5. Shenker, in Unified String Theories, ed. MB. Green and Di. Gross (World Scientific, Singapore, 1986) 125] A. LeClair, ME. Peskin and CR. Preitschopf, Nucl. Phys. B317 (1989) 41 [26] A. LeClair, ME. Peskin and CR. Preitschopf, Nuci. Phys. B317 (1989) 464 [27] E. Witten, NucI. Phys. B268 (1986) 253 [28] 5. Giddings and E. Martinec, NucI. Phys. B278 (1986) 91 [29] E. Martinec, NucI. Phys. B281 (1986) 157 [30] C. Thorn, NucI. Phys. B287 (1987) 61 [311 M. Bochicchio, Phys. Lett. B188 (1987) 330; B193 (1987) 31 [32] S. Giddings, E. Martinec and E. Witten, Phys. Lett. B176 (1986) 362 [33] S. Giddings, NucI. Phys. B278 (1986) 242 [34] D. Gross and A. Jevicki, NucI. Phys. B283 (1987) 1 [35] E. Cremmer, A. Schwimmer and C. Thorn, Phys. Lett. B179 (1986) 57 [36] 5. Samuel, Phys. Lett. B181 (1986) 255 [37] N. Ohta, Phys. Rev. D34 (1986) 3785 [38] D. Gross and A. Jevicki, Nucl. Phys. B287 (1987) 225 [39] C. Thorn, The oscillator representation of Witten’s three open string vertex function, in Proc. XXIIIth Int. Conf. of High Energy Physics (Berkeley, 1986) [40] E. Cremmer, Vertex function in Witten’s formulation of string field theory, in: Proc. Paris-Meudon Colloq. (September 22—26, 1986) [41] 5. Samuel, in Strings and Superstrings (XVIIIth Int. GIFT, El Escorial, Spain, June, 1987) ed. J.P. Mittelbrunn, M. Ramón-Medrano and G.S. Rodero, (World Scientific, Singapore, 1988) [42] S. Samuel, NucI. Phys. B308 (1988) 317 [43] S. Samuel, Phys. Lett. B181 (1986) 249 [44] T. Kugo, H. Kunitomo and K. Suehiro, Kyoto University preprint KUNS 855-HE(TH) 87/06 (May, 1987)

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/

Open bosonic string

[45] K. Itoh, K. Ogawa and K. Suehiro, Nucl. Phys. B289 (1987) 127 [46] H. Kunitomo and K. Suehiro, NucI. Phys. B289 (1987) 157 [47] J.H. Sloan, NucI. Phys. B302 (1988) 349 [48] S. Samuel, NucI. Phys. B308 (1988) 285 [49] R. Bluhm and S. Samuel, NucI. Phys. B323 (1989) 337 [50] 0. Lechtenfeld and S. Samuel, NucI. Phys. B308 (1988) 309 [51] D. Freedman, S. Giddings, J. Shapiro and C. Thorn, NucI. Phys. B298 (1988) 253 [52] R. Bluhm and S. Samuel, NucI. Phys. B325 (1989) 275 [53] V.A. Kosteleck~and S. Samuel, Phys. Lett. B207 (1988) 169 [54] V.A. Kosteleck~iand S. Samuel, Phys. Rev. D39 (1989) 683 [55] V.A. Kosteleck9 and S. Samuel, NucI. Phys. B336 (1990) 263 [56] G. Veneziano, Nuovo Cim. A57 (1968) 190 [571 K. Bardakçi and H. Ruegg, Phys. Rev. 181 (1969) 1884 [58] Z. Koba and H.B. Nielsen, NucI. Phys. BlO (1969) 633; B12 (1969) 517 [59] Ci. Goebel and B. Sakita, Phys. Rev. Lett. 22 (1969) 256 [60] M. Kato and K. Ogawa, NucI. Phys. B212 (1983) 443 [61] 5. Hwang, Phys. Rev. D28 (1983) 2614 [62] W. Siegel, Phys. Left. B142 (1984) 276; B151 (1985) 391; 396. [631 AM. Polyakov, Phys. Lett. B103 (1981) 207, 211 [64] H.M. Farkas and I. Kra, Riemann Surfaces (Springer, New York, 1980) [65] G. Springer, Introduction to Riemann Surfaces (Addison-Wesley, Reading, MA, 1957) [66] A.F. Beardon, A Primer on Riemann Surfaces, London Math. Soc. Lecture Note Series; vol. 78, (Cambridge University Press, Cambridge, 1984) [67] LV. Ahlfors and L. Sario, Riemann Surfaces (Princeton University Press, Princeton, NJ, 1960) [681 J. Lehner, A Short Course in Automorphic Functions (Holt, Rinehart and Winston, New York, 1966) [69] A.F. Beardon, The Geometry of Discrete Groups (Springer, New York, 1980). [70] J. Lehner, Discontinuous Groups and Automorphic Functions (American Math. Soc., Providence, RI, 1964) [711 D. Mumford, Tata Lectures on Theta, vol. I, Progress in Mathematics, vol. 28 (Birkhäuser, Boston, 1983). [72] J. Fay, Theta Functions on Riemann Surfaces, Springer Notes in Mathematics, vol. 352 (Springer, New York, 1973) [73] L.R. Ford, Automorphic Functions, (McGraw-Hill, New York, 1929) [74] I. Kra, Automorphic Functions and Kleinian Groups (Benjamin, Reading, MA, 1972) [751L. Schiffer and D. Spencer, Functionals of Finite Riemann Surfaces (Princeton University Press, Princeton, NJ, 1959) [76] K.W. Abikoff, The Real Analytic Theory of Teichmüller Space, Lecture Notes in Mathematics, vol. 820 (Springer, New York, 1980) [77] L. Alvarez-Gaumé, G. Moore and C. Vafa, Commun. Math. Phys. 106 (1986) 1 [78] E. Verlinde and H. Verlinde, NucI. Phys. B288 (1987) 357 [79] E. Verlinde and H. Verlinde, lAS preprint IASSNS-HEP-88/52 [80] B. Riemann, Journal für die reine und angewandte Mathematik Bd. 54 (1857) [81] W. Burnside, Proc. London Math. Soc. 23 (1891) 49 [82] D.B. Fairlie and H.B. Nielsen, Nucl. Phys. B20 (1970) 637 [83] V.A. Kosteleck~,0. Lechtenfeld and S. Samuel, Nucl. Phys. B298 (1988) 133 [841M. Kaku and K. Kikkawa, Phys. Rev. DiD (1974) 1110; 1823 [85] S. Mandelstam, NucI. Phys. B64 (1973) 205; B69 (1974) 77 [86] E. Cremmer and J.L. Gervais, Nuci. Phys. B76 (1974) 209; B90 (1975) 410 [871MB. Green and J.H. Schwarz, NucI. Phys. B218 (1983) 43; B243 (1983) 475 [88] L. Brink, MB. Green and J.H. Schwarz, Nucl. Phys. B219 (1983) 437 [89] A. Neveu, H. Nicolai and P.C. West, Phys. Left. B167 (1986) 307; NucI. Phys. B264 (1986) 573 [90] H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Rev. D34 (1986) 2360 [91] G.T. Horowitz and S.P. Martin, NucI. Phys. B296 (1988) 220

S. Samuel

/

Open bosonic string

609

[92] S.P. Martin, Noel. Phys. B310 (1988) 428 [93] A. Strominger, Phys. Lett. B187 (1987) 295; Z. Qiu and A. Strominger, Phys. Rev. D36 (1987) 1794 [94] A. Hosoya and H. Itoyama, Nuci. Phys. B313 (1989) 116 [95] C. Lovelace, Phys. Lett. B32 (1970) 703 [96] M. Kaku and L.P. Yu, Phys. Lett. B33 (1970) 166 [97] M. Kaku and L.P. Yu, Phys. Rev. D3 (1971) 2992; 3007; 3020 [98] V. Alessandrini, Nuovo Cim. A2 (1971) 321; V. Alessandrini and D. Amati, Nuovo Cim. A4 (1971) 793 [99] M. Kaku and J. Scherk, Phys. Rev. D3 (1971) 430; 2000 [100] C. Lovelace, Phys. Lett. B34 (1971) 500 [101] G. Moore and P. Nelson, Nucl. Phys. B266 (i986) 58 [102] Yu. Manin, Phys. Lett. B172 (i986) i84 [103] MA. Namazie, KS. Narain and M.H. Sarmadi, Phys. Lett. B177 (1986) 329 [104] G. Moore, Phys. Lett. B176 (1986) 369 [105] H. Sonoda, Phys. Lett. B178 (1986) 390 [106] A. Kato, Y. Matso and S. Odake, Phys. Left. B179 (1986) 241 [107] E. D’Hoker and D. Phong, Nuci. Phys. B278 (1986) 225; B292 (1987) 317 [108] T. Eguchi and H. Ooguri, Phys. Lett. B187 (1987) 127; NucI. Phys. B282 (1987) 308 [109] E. Verlinde and H. Verlinde, Phys. Lett. B192 (1987) 95 [110] K. Mikj, Nucl. Phys. B291 (1987) 349 [111] F. Gliozzi, Phys. Lett. B194 (1987) 30 ~i12] H. Sonoda, Phys. Lett. B184 (1987) 336 [113] J.J. Atick and A. Sen, NucI. Phys. B296 (1988) 157 [114] J.J. Atick, G. Moore and A. Sen, NucI. Phys. B307 (1988) 221 [115] J.J. Atick, G. Moore and A. Sen, NucI. Phys. B308 (1988) 1 [116] E. Gava, R. Iengo and 0. Sotkov, Phys. Lett. B207 (1988) 283 [117] R. lengo and C-i. Zhu, Phys. Left. B212 (1988) 309; Phys. Lett. B212 (1988) 313 [118] C.J. Zhu, Trieste preprint ISAS/88/147 (1988) [119] L. Dixon, D. Friedan, E. Martinec and S.H. Shenker, NucI. Phys. B282 (1987) 13 [120] A.B. Zamolodchikov, Nucl. Phys. B285 (1987) 481 [121] M. Bershadsky and A. Radul, mt. J. Mod. Phys. A2 (1987) 165 [1221 V. Knizhnik, Phys. Lett. B196 (1987) 473 [123] D. Montano, NucI. Phys. B297 (1987) 125 [124] D. Lebedev and A. Morozov, Nuci. Phys. B302 (1988) 163 [125] 0. Moore and A. Morozov, Nucl. Phys. B306 (1988) 387 [126] A. Morozov and A. Perelomov, Phys. Lett. B183 (1987) 296; B197 (1987) 115; B199 (1987) 209 [127] A. Morozov, Phys. Lett. B184 (1987) 177; NucI. Phys. B303 (1988) 343 [128] 0. Lechtenfeld and A. Parkes, Phys. Lett. B202 (1988) 75 [129] 0. Lechtenfeld, NucI. Phys. B309 (1988) 361 [130] A. Morozov and A. Perelomov, preprint ITEP-88-80 (1988) [131] A. Morozov, preprint ITEP-88-i25 (1988) [132] 0. Lechtenfeld, preprint CCNY-HEP-88/13 (1988), Nucl. Phys. B, to be published [1331 A. Parkes, Phys. Lett. B217 (1989) 458 [134] 0. Lechtenfeld and A. Parkes, CCNY preprint CCNY-HEP-89/04—UCD-89-8 [135] 0. Lechtenfeld, CCNY preprint CCNY-HEP-89/09 [136] E. D’Hoker and D.H. Phong, Rev. Mod. Phys. 60 (1988) 917 [137] V. Alessandrini, D. Amati, M. Le Bellac and DI. Olive, Phys. Rep. Ci (1971) 269 [138] J.H. Schwarz, Phys. Rep. C8 (1973) 269 [139]G. Veneziano, Phys. Rep. C9 (1974) 199 [140] C. Rebbi, Phys. Rep. C12 (1974) 1; S. Mandeistam, Phys. Rep. C13 (1974) 259 [141] P. Frampton, Dual Resonance Models (Benjamin, Reading, MA, 1974) [142] J. Scherk, Rev. Mod. Phys. 47 (1975) 123

610

S. Samuel

/

Open bosonic string

[143] J.H. Schwarz, Phys. Rep. C89 (1982) 223 [144] L. Keen, Bulletin Am. Math. Soc. 83 (1977) 1199 and references therein [145] N. Bourbaki, Elements of Mathematics, Algebra (Addison-Wesley, Reading, MA) [146] M. Kaku and J. Lykken, Phys. Rev. D38 (1988) 3067 [147] M. Saadi and B. Zwiebach, Ann. Phys. (N.Y.) 192 (1989) 213 [148] T. Kugo, H. Kunitomo and K. Suehiro, Phys. Lett. B226 (1989) 48 [149] M. Kaku, CCNY preprints CCNY-HEP-89/6, OU-HEP-121 (June, 1989) [150] E. Witten, Nucl. Phys. B276 (1986) 291 [151] D. Gross and A. ievicki, NucI. Phys. B293 (1988) 29 [152] S. Samuel, NucI. Phys. B296 (1988) 187 [153] K. Suehiro, Nuci. Phys. B296 (1988) 333 [154]C. Wendt, NucI. Phys. B314 (1989) 209 [155] 0. Lechtenfeld and S. Samuel, NucI. Phys. B310 (1988) 254 [156] AR. Bogojeviè, A. Jevicki and G-W. Meng, Brown University preprint HET-672 (September 1988) [157] 0. Lechtenfeld and S. Samuel, Phys. Lett. B213 (1988) 431 [158] R. Bluhm and S. Samuel, NucI. Phys. B338 (1990) 38 [159]C.R. Preitschopf, C.B. Thorn and S.A. Yost, University of Florida preprint UFIFT-HEP-89-19 [160] I.Ya. Aref’eva, PB. Medvedev and A.P. Zubarev, Steklov Mathematics Institute preprint SMI-lO1989 (October, 1989) [161] I.Ya. Are?eva, PB. Medvedev and A.P. Zubarev, Steklov Mathematics Institute preprint (December, 1989)