Modular properties of two-loop maximal supergravity and connections with string theory

Modular properties of two-loop maximal supergravity and connections with string theory

Michael B. Green
Department of Applied Mathematics and Theoretical Physics
Wilberforce Road, Cambridge CB3 0WA, UK
M.B.Green@damtp.cam.ac.uk
   Jorge G. Russo
Institució Catalana de Recerca i Estudis Avançats (ICREA)
Department ECM, Facultat de Fisica, University of Barcelona
Av. Diagonal, 647, Barcelona 08028 SPAIN
jrusso@ub.edu
   Pierre Vanhove
Institut de Physique Théorique,
CEA, IPhT, F-91191 Gif-sur-Yvette, France
CNRS, URA 2306, F-91191 Gif-sur-Yvette, France
and
Niels Bohr Institute, University of Copenhagen,
Blegdamsvej 17, DK–2100 Copenhagen Ø, Denmark
pierre.vanhove@cea.fr
\Date
Abstract:

The low-momentum expansion of the two-loop four-graviton scattering amplitude in eleven-dimensional supergravity compactified on a circle and a two-torus is considered up to terms of order (where is a Mandelstam invariant and is the linearized Weyl curvature). In the case of the toroidal compactification the coefficient of each term in the low energy expansion is generically a sum of a number of -invariant functions of the complex structure of the torus. Each such function satisfies a separate Poisson equation on moduli space with particular source terms that are bilinear in coefficients of lower order terms, consistent with qualitative arguments based on supersymmetry. Comparison is made with the low-energy expansion of type II string theories in ten and nine dimensions. Although the detailed behaviour of the string amplitude is not generally expected to be reproduced by supergravity perturbation theory to all orders, for the terms considered here we find agreement with direct results from string perturbation theory. These results point to a fascinating pattern of interrelated Poisson equations for the IIB coefficients at higher orders in the momentum expansion which may have a significance beyond the particular methods by which they were motivated.

Supergravity, Superstring
preprint: DAMTP-2008-54, IPhT-T-08-100, UB-ECM-PF-08/13

1 Introduction

The rich network of string theory dualities provides powerful constraints on the structure of M-theory. These are particularly restrictive for maximally supersymmetric backgrounds although the full power of maximal supersymmetry has proved difficult to exploit. The purpose of this paper is to further investigate the small corner of string theory associated with the low-energy expansion of the four-graviton scattering in nine or ten dimensions and its connection to eleven-dimensional supergravity. In terms of an effective action, this corresponds to an investigation of terms involving derivatives acting on four powers of the linearized Riemann curvature.

More precisely, our aim is to further develop the connections between multi-loop eleven-dimensional supergravity compactified on and and the type II superstring theories, making use of the conjectured relationships between M-theory and type IIA and IIB superstring theories [1, 2, 3, 4]. In earlier work, a number of terms in the low-energy expansion of the type II string theory amplitudes were determined from the compactified one-loop and two-loop supergravity amplitudes [5, 6, 7, 8]. The fact that the full, nonperturbative moduli dependence of the string amplitudes was reproduced is presumably a consequence of the constraints of maximal supersymmetry on ‘protected’ terms. In the absence of a complete understanding of which terms are protected it is of interest to pursue the connections with quantum supergravity further. Here we will develop the low-energy expansion of the two-loop supergravity amplitude in a more systematic fashion and determine several orders beyond those considered previously. We will, furthermore, investigate the extent to which this makes contact with the type II string theories in nine and ten dimensions. We will find that the scalar-field dependent coefficients of the higher-derivative terms in the expansion satisfy a suggestive pattern of differential equations on moduli space. Comparing these coefficients with known ‘data’ from the low-energy expansion of tree-level and genus-one perturbative string theory in nine and ten dimensions [9, 10] shows a surprising degree of agreement. Although it is obvious that there is far more to M-theory than perturbative supergravity, these results suggest patterns that could persist to all orders in the low-energy expansion.

1.1 Overview of low orders in the momentum expansion

In ten dimensions there is a clear distinction between type IIA and type IIB superstring theories even though it is known that they have identical four-graviton amplitudes at least up to, and including, genus-four in string perturbation theory [11]. The IIA theory has a single real modulus, and at strong coupling this is identified with the radius of a single compact dimension in eleven-dimensional supergravity [2]. The ten-dimensional IIB theory has a complex modulus (a complex scalar coupling constant) that is identified with the complex structure of the torus in the compactification of eleven-dimensional supergravity in the limit in which the torus volume vanishes [12, 4]. Invariance of M-theory under large diffeomorphisms of implies that the IIB theory possesses a duality symmetry [1] that relates strong and weak coupling in a manner that involves both the perturbative and non-perturbative (-instanton) interactions. After compactification to nine dimensions on a circle the two string theories are identified by the action of T-duality, which inverts the radius of the compact dimension and transforms the dilaton appropriately. The nine-dimensional duality group is .

Although the explicit calculations in this paper concern the four-graviton amplitude, maximal supersymmetry ensures that the conclusions apply equally to the scattering of any four states in the supermultiplet. In fact, maximal supersymmetry guarantees that the general type IIA or IIB amplitude has the structure111We are grateful to Nathan Berkovits for emphasizing the generality of this structure.

(1.1)

where we have labeled each external massless particle by its superhelicity , which takes 256 values (the dimensionality of the maximal supergravity multiplet) and its momentum (), where . is a function of the Mandelstam invariants222 The (dimensionless) Mandelstam invariants, , and , are subject to the mass-shell condition and is the string length scale. , , . The kinematical factor in (1.1) is given by (see (7.4.57) of [13])

(1.2)

where the indices on the polarization tensors run over both vector and spinor values (for example, the graviton polarization is , where ) and the tensor is defined in [13]. For the purposes of this paper we will consider the case of external gravitons for which R reduces to the the momentum-space form of the linearized Weyl tensor,

(1.3)

where the symmetric traceless polarization tensor satisfies . The kinematic factor in (1.1) becomes , which denotes the product of four Weyl curvatures contracted into each other by a well-known sixteen-index tensor (often denoted ).

The low-energy expansion of the four-particle amplitude requires the expansion of the function in (1.1) for small . This can be expressed as a complicated mixture of terms that are analytic and nonanalytic functions of the Mandelstam invariants. The analytic terms may be expanded as power series’ in integer powers of , and in a straightforward manner. The lowest-order terms contain the poles and contact terms characteristic of the supergravity tree diagrams. A great deal is also known about higher-order analytic terms up to order . The nonanalytic terms contain massless threshold singularities whose form is determined by unitarity and depends on the number of noncompact space-time dimensions. Generically, there are fractional powers or logarithmic branch points, giving rise to non-integer powers of or factors.

In what follows we shall separate the low-energy expansion of the ten-dimensional amplitude in either type II theory into the sum of an analytic part and a non-analytic part,

(1.4)

where has been normalized to be dimensionless. In the IIB theory the coefficients in the series in the analytic term in (1.4) are -invariant functions of the complex coupling and the series has the form

(1.5)

where

(1.6)

The factors are the most general scalars that are symmetric monomials in , , of order . The functions ’s are modular functions of the complex scalar, , where

(1.7)

and is the Ramond–Ramond scalar, is the type IIB dilaton and is the type IIB coupling constant. The expression (1.5) includes the Born term with its poles and the coefficient . The nonanalytic contribution is a series that contains multi-particle thresholds of symbolic form

(1.8)

where the ’s are modular functions of , which begin with terms that are genus-one or higher.

The coefficients in the expansion are known up to terms of order ,

(1.9)

The terms in (1.9) are analytic in and and translate into local higher-derivative interactions in a -invariant effective action,

(1.10)

where is the curvature scalar, the ten-dimensional type IIB string metric and the coefficients and will be described below. The derivatives in (1.10) are contracted so that the four-point amplitude contributions arise in a manner that is defined by the pattern of Mandelstam invariants in (1.9). From (1.9) it follows that the coefficients in (1.5) are given by

(1.11)

The quantities in (1.9) and (1.10) are Eisenstein series that solve the Laplace eigenvalue equations on the fundamental domain of ,

(1.12)

Given the fact that the is a function that can have no worse than power growth as (which is required for consistency with string perturbation theory at weak coupling) the solution of this equation is uniquely given by

(1.13)

which can be expanded at weak coupling in the form

(1.14)

The two power-behaved terms in this expansion correspond to the tree-level and genus- contributions in string theory333In order to avoid confusion, we will refer to the number of ‘loops’ (denoted by ) in the context of the supergravity Feynman rules, and the ‘genus’ (denoted by ) in the context of the string theory perturbative expansion., as can be seen by taking into account the powers of in (1.9) and identifying with the IIB string coupling, . The exponential terms correspond to the infinite set of -instanton contributions.

The fact that is the coefficient associated with the term in (1.9) was initially deduced via indirect arguments [14, 5]. One of these made use of properties of loop amplitudes of eleven-dimensional supergravity compactified on a circle or on a two-torus, combined with dualities that relate M-theory to type II string theory in nine dimensions. In this way the function describes the dependence of the low-energy limit of the one-loop () four-graviton scattering amplitude on the modulus of the compactification torus [5]. The ultraviolet divergence, which behaves as , where is a momentum cutoff, is independent of and can be subtracted by a local counterterm. The coefficient of this counterterm is fixed by requiring the IIA and IIB amplitudes to be equal, as they are known to be. The modular function can also be derived as a consequence of supersymmetry combined with -duality [15]. Although it is suspected that the other modular functions appearing in higher derivative terms (at least up to the order shown in (1.9)) should also be determined by supersymmetry combined with non-perturbative dualities, there is no systematic procedure for doing this (a sketchy outline is given in section 5 of this paper).

Expanding the supergravity amplitude in powers444 The dimensionless Mandelstam invariants of eleven-dimensional supergravity are denoted by upper case letters , , , where is the eleven-dimensional Planck length, and related to the invariants in the ten-dimensional string frame by , where is the radius of the eleventh dimension. of , and leads to higher-order terms in the derivative expansion of the form [6, 7]. This results in an infinite set of analytic terms that are interpreted in IIB string coordinates as modular invariant coefficients multiplying powers of order ,

(1.15)

where the ellipsis stand for the non-analytic contributions [7] and are simple constants and is a polynomial in and of order in the Mandelstam invariants. All contributions with vanish in the ten-dimensional type IIB limit where the two-torus volume, , vanishes. So we see that in the ten-dimensional limit the compactified one-loop () eleven-dimensional supergravity amplitude contributes only at order . In order to obtain higher-derivative interactions one has to consider eleven-dimensional supergravity at higher loops (). The coefficient of the ten-dimensional IIB theory indeed arises from a one-loop subdivergence of the low-energy limit of the two-loop amplitude of eleven-dimensional supergravity compactified on a two-torus in the limit in the limit [7, 8].

The function in (1.5) is obtained by expanding the two-loop supergravity amplitude to the next order in , , and compactifying on a two-torus [8]. It satisfies the Poisson equation

(1.16)

in which the source term on the right-hand side is quadratic in the modular function . We will denote the solution to this equation by , as in [8]. This source term makes the equation quite different from the Laplace eigenfunction equation (1.12). Its structure was argued in [8] to follow, at least qualitatively, from the constraints of supersymmetry. The solution of (1.16) is complicated, but the zero-mode of , which contains the perturbative terms, is found to have the form

(1.17)

so it contains tree-level, genus-one, genus-two and genus-three perturbative string theory terms as well an infinite series of -instanton – anti--instanton pairs. The tree-level and genus-one terms agree precisely with direct string theory calculations, while the genus-two term has not yet been extracted directly from string theory. The genus-three term cannot yet be computed in string perturbation theory but it is gratifying that the value of its coefficient agrees, as it should, with that of the genus-three term in the IIA theory that is predicted from one loop in eleven-dimensional supergravity compactified on . This agreement is striking since extracting the coefficient in the IIB theory from the amplitude involves the use of a Ramanujan identity (see appendix D.1), whereas the coefficient in the IIA theory obtained from the amplitude arises from a simple integral. Although there is no proof that is the exact modular function, these agreements strongly suggest that it is. It is notable that the terms in the expression (1.17) are not of uniform transcendental weight. Whereas, there is a correlation of the power of and the weight of the values for the first three terms, this breaks down for the genus-three term. We will see an analogous lack of transcendentality in many of the examples to be described later in this paper.

The first nonanalytic term beyond the Born (pole) term arises at order and comes from the ten-dimensional supergravity one-loop diagrams. It has the symbolic form given by the first term on the right-hand side of (1.8). Its precise expression, reviewed in [10], has a much more complicated threshold structure but it has the notable property that the scale of the logarithm cancels, using .

Obviously the analysis of Feynman diagrams of eleven-dimensional supergravity has limited use since it does not capture the full content of quantum string theory, or M-theory. To begin with, eleven-dimensional supergravity is not renormalizable. Our procedure is to regulate the ultraviolet divergences by introducing a momentum cutoff and subtracting the divergences with counterterms. The result is finite but the counterterms contribute arbitrary coefficients that parameterize our ignorance of the short-distance physics. However, at low orders the values of some of these coefficients are known to be determined by supersymmetry if we also assume the result should be in accord with string dualities. One of the aims here is to investigate the extent to which this continues at higher orders.

A related issue is that the Feynman diagrams describe a semi-classical approximation to the theory in a particular classical background space-time. This can only be motivated in the limit in which the radii of the compact dimensions are much larger than the eleven-dimensional Planck length. This means for the compactification (where is the dimensionless radius of the eleventh dimension in Planck units). This is the limit of large IIA string coupling, . Bearing in mind that this is far from the regime of string perturbation theory, we will see to what extent there is agreement between the compactified two-loop Feynman diagrams and corresponding perturbative string theory results. For the compactification the analogous condition is (where is the dimensionless volume of in Planck units). In IIA string theory compactified to nine dimensions this is the limit in which , where is the radius of the compact dimension (in string units). Nevertheless, the coefficients of the , and terms reviewed above give the correct values in the limit – presumably the extrapolation from small to large works because these terms are protected by supersymmetry. The low energy limit we are considering is one in which . Since the supergravity loop diagrams are ultraviolet divergent we will also introduce a dimensionless momentum cutoff measured in units of the eleven-dimensional Planck length. We will see that the low energy expansion of the Feynman diagrams possesses a very rich structure. In particular, the coefficients that depend on the scalar fields satisfy a series of mathematically intriguing Poisson equations that are nontrivial extensions of (1.16) satisfied by , as we will see.

1.2 Outline of paper

In this paper we will consider the higher-order terms in the low-energy expansion of the four-graviton amplitude that are obtained by expanding the two-loop amplitude of eleven-dimensional supergravity, compactified to ten dimensions on and nine dimensions on to several higher orders in the Mandelstam invariants.

The four-graviton amplitude (1.4) at two loops () in maximal supergravity has the form [16]

(1.18)

where the scalar function has the structure

(1.19)

The terms in brackets are sums of scalar field theory two-loop planar and non-planar ladder diagrams,

(1.20)

with analogous expressions for and . The expression (1.18) has an overall prefactor of , which has eight powers of the external momenta, together with four more powers from the factors of , or . This means that the loop integrals, and , are much less divergent than they would naively appear. We will be interested in the compactified amplitude, so that is a function of the moduli of the compact space. Ignoring for the moment the nonanalytic pieces, we shall expand the analytic part of in a power series,

(1.21)

where is a function of the moduli that will be defined by the integral (2.56) and the constant coefficients can be read off from (2.55). Note that since the well-known term only arises at one loop. The dependence on the Mandelstam invariants in (1.21) is contained in the and , which are defined by

(1.22)

(whereas in the string variables we used the symbol in (1.6)). The coefficients in (1.21) depend on the moduli in a manner to be determined. The infrared massless threshold effects give rise to nonanalytic terms that we will also need to discuss.

In section 2 we will show how the expression for can be reexpressed in a useful form that would also arise naturally in a world-line functional integral describing the two-loop process. This involves attaching vertex operators for external states of momentum () to points on the three world-lines, of length (), of the two-loop vacuum diagram. The amplitude involves the the usual factor of , where is the Green function connecting pairs of points on these world-lines, as discussed in [17]. This provides a very compact expression for the sum of all diagrams as an integral over all insertion points and over the lengths of the three world-lines, with an appropriate measure. The low energy expansion is obtained, formally, by expanding the integrand in powers of the Green function

(1.23)

which are to be integrated over the positions with a specific measure.

We will discuss a ‘hidden’ modular invariance that acts on the three Schwinger parameters, . This symmetry is particularly useful in evaluating the compactification of the amplitude on a spatial -torus and was used in [7, 8] in evaluating terms of order and . This becomes more explicit after a change of variables from the Schwinger parameters, , to variables , and . The quantity enters in a manner analogous to the modulus of a world-sheet torus embedded in the target space in genus-one string theory. After the above redefinition of variables we will see that the coefficient in (1.21) has the schematic form (the precise coefficients will be included later)

(1.24)

where is a lattice factor that contains the information about the compactified target space with metric (). It will be important that the integrand is invariant under , when suitably extended outside the fundamental domain. This integral has ultraviolet and infrared divergences, depending on the values of and . These will require a careful treatment of the integration limits, which will be discussed in detail in section 2.3.

An important property of the coefficients, in the integrand is that they can be written as sums of components ,

(1.25)

where and the components satisfy Green function equations in of the form

(1.26)

where , is a polynomial in of degree (see appendix A for details)555We would like to thank Don Zagier for explaining the mathematical significance of this decomposition. This property will be used extensively to determine .

The compactification to ten dimensions will be described in section 3, together with appendix B. This will lead to coefficients for higher-momentum terms in the type IIA theory up to order . Although this reproduces the terms considered in earlier work, important new issues are encountered at order () where further non-analytic terms arise. Such nonanalytic behaviour arises from infrared threshold effects that are not captured by the power series expansion (1.23), so we will be careful to regulate the infrared limit of the integrals. In ten dimensions unitarity implies that such thresholds are logarithmic and arise at this order in at genus-one and genus-two. Further logarithmic singularities arise at genus-two at order , and at genus-one and genus-three at order , with a complicated pattern of thresholds at all orders in thereafter. Unlike in the case of the lowest-order nonanalytic term (1.8), the scales of the logarithms, which we will not evaluate, do not cancel. The translation of these supergravity results into the language of type IIA superstring theory is summarized in section 3.3.

Compactification to nine dimensions on a two-torus will be considered in section 4. The coefficients in the expansion now have a richer structure since they depend on the three moduli of , or the complex coupling, , and the radius of the compact dimension, , in the type IIB string theory language. Each term with a distinct kinematic structure must have a coefficient that is an independent function that is invariant under the nine-dimensional duality group, . We will determine certain analytic terms in the double expansion of the amplitude up to order that are associated with particular inverse powers of . In order for the Feynman diagram approximation to have a chance of being a sensible approximation it is necessary that , or . The coefficients will be modular functions of . In fact, we will see that each coefficient is generally a sum of a number of modular functions that satisfy independent Poisson equations analogous to (1.16). The structure of these equations, which generalizes (1.12), is summarized by (4.15), which is one of the most intriguing results of this paper.

In nine dimensions almost all the low-order nonanalytic terms have branch points that are non-integer powers of the Mandelstam invariants rather than logarithms, and so they can be separated from the analytic part unambiguously – the exception is the term of order , which is the contribution from nine-dimensional supergravity and can be obtained by dimensional regularization, as summarized in appendix E.3. However, there are terms that are power-behaved in as well as terms containing, factors such as , which is nonanalytic in , and exponentially suppressed terms of the form . A series of terms that are power behaved in was seen to arise from the expansion of the supergravity amplitude in (1.15). Similarly, we will find that the momentum expansion of the amplitude gives a sum of higher-momentum modular invariant terms,

(1.27)

for various values of that will be specified later. Terms proportional to reproduce the expansion, so that . All contributions with vanish in the ten-dimensional type IIB limit, but they give rise to well defined modular functions in nine dimensions. In addition to terms that are power-behaved in the radius or , there are also terms proportional to or . Such terms arise explicitly at genus-one in nine-dimensional string theory [10]. For example, there is a term of the form , which is intimately related to the presence of the genus-one term in ten dimensions determined in [10]. We will see in the following that this dependence on can also be seen from the reduction of two-loop () eleven-dimensional supergravity. Terms of the form that arise in string theory when are not reproduced by Feynman diagrams at any number of loops.

Perturbative contributions to the string amplitude are obtained from the weak-coupling expansion of these modular functions (making use of the methods described in appendix D). Each term in the momentum expansion derived in this manner is accompanied by a particular inverse power of the radius and the new terms do not contribute in the large- limit. However, after T-duality to the IIA theory, we are able to compare a number of coefficients with those derived explicitly from genus-one in string theory compactified on a circle [10] and find precise agreement. Special issues concerning the terms that contain factors will also be discussed. The issue of the pattern of logarithms is intimately related to the threshold behaviour in maximal supergravity in various dimensions. In appendix E we will evaluate the supergravity amplitude in nine, ten and eleven dimensions, making use of dimensional regularization. These expressions are of relevance to various pieces of the argument in the body of the paper. For example, in ten dimensions the pole term gives rise to a term of order that is identified with a genus-two contribution to in ten-dimensional string theory. In section 5 we will sketch the way in which supersymmetry constrains higher derivative terms and argue that the structure of the Poisson equations satisfied by the coefficients of the terms in the derivative expansion of the nine-dimensional IIB theory can be motivated by supersymmetry.

2 Properties of the two-loop supergravity amplitude

It has been known for a long time that the sum of one-loop Feynman diagrams that contribute to four-graviton scattering in maximal supergravity in any dimension has the form of a box diagram of scalar field theory multiplying , where is the linearized Weyl curvature, as discussed in the introduction. Similarly, the sum of all two-loop diagrams, , is very economically expressed in terms of two particular diagrams of scalar field theory [16]. These are the planar double-box diagram, of figure 1(a), and the non-planar double box diagram, of figure 1(b), together with the other diagrams obtained by permuting the external particles. In addition, one must include the one-loop triangle diagram of figure 1(c) containing a one-loop counterterm at one vertex (indicated by the blob), which subtracts the one-loop sub-divergences from the two-loop diagrams. In addition there are two-loop primitive divergences (that are indicated by the double-blob in figure 1(d)).

Figure 1: The two-loop four-graviton amplitude in eleven dimensions. (a) The -channel planar diagram reduces to multiplying a scalar field theory double-box diagram. (b) The -channel nonplanar diagram reduces to multiplying a nonplanar scalar field theory two-loop diagram. (c) The triangle diagram with a one-loop counterterm at one vertex that subtracts a sub-divergence. (d) A new two-loop primitive divergence.

The two-loop integrals appearing in the amplitude are sums of planar and non-planar pieces, (1.20). We are interested in compactifying these expressions on the -torus with or . After manipulations that are given in [7] the loop integrals can be expressed as integrals over seven Schwinger parameters, one for each propagator. The integrations over loop momenta in the compact directions are replaced by sums over the Kaluza–Klein integers in each loop and , where . After performing the integration over the continuous -dimensional loop momenta, the planar and non-planar diagrams reduce to

(2.1)

and666 In this section we ignore the ultraviolet and infrared divergences. A treatment of these divergences and a proper definition of the integration limits of the integrals will be discussed in section 2.3.

(2.2)

where

(2.3)

The lattice factor is defined by

(2.4)

where is the inverse metric on and is its volume. The quantities and are given by777The variables in this section are related to those of [7] by , , , , , and in the planar case, , while in the non-planar case, .

and

In writing these expressions we have ignored the ultraviolet divergences, which are manifested as divergences at the endpoints () that will be regulated by a cutoff in subsection 2.3.1 (as in [7]). The complete expression, in (1.19) is obtained by summing the -channel, -channel and -channel diagrams.

2.1 World-line presentation of the two-loop amplitude

The above structure of the two-loop amplitude can, in principle, be deduced by considering the quantum mechanics functional integral associated with the world-lines for the internal propagators in the two-loop diagrams. This has a structure that bears a close resemblance to the world-sheet description of the genus-two string theory amplitude (although that is formulated in ten-dimensional space-time). We will here rewrite the expressions for the two-loop Feynman diagrams of the previous subsection in order to make this explicit. The advantage of this description is that it naturally packages together the planar and nonplanar diagrams of the , and channels.

Figure 2: (a) A planar diagram is represented by the skeleton with a pair of external states connected to each of two internal lines. (b) The nonplanar configuration in which one pair of external states is attached to a single line and the other states are each attached to separate lines. Integrating the positions of the four states over the whole network generates the sum of all Feynman diagrams.

The ‘skeleton’, or vacuum diagram, has three scalar propagators joining the junction to junction in figure 2. The lengths of these lines, (), are moduli that are to be integrated between and . The scattering particles with momenta () are associated with plane-wave vertex operators that are inserted at positions on any of the three lines of the skeleton, as shown in figure 2. These positions are then to be integrated over the whole network. Since there are four vertex operators and only three lines, at any point in the integration domain one pair of vertex operators is attached to one line, say line , while the other two may both be attached to one of the other two lines (line or ), which is the planar situation, or else the other two lines may have only one vertex operator attached, which is the non-planar situation. The labelling of the positions of the vertex operators is arbitrary, but it is convenient to choose coordinates for particle on line such that

(2.7)

where , and , , and coincide at the junction . In other words, the integral over the whole network decomposes into sectors labeled by ,

(2.8)

The expression for the Feynman diagrams can be written in a compact form in terms of the Green function, , between two vertices at points and on the skeleton diagram. Following [17] this is written in terms of two-vectors

(2.9)

where labels the loop and are constant vectors

(2.10)

With this notation the sum of all two-loop contributions to the amplitude defined in (1.18) and (1.19) is given by

(2.11)

where is the one-dimensional Green function for the Laplace operator evaluated between the points and on the skeleton diagram, to be discussed below. The lattice factor is defined in (2.4).

The function appearing in the measure in (2.11) is defined by888The world-line formulation of the two-loop four-graviton amplitude in ten dimensions would arise from a field theory limit of four-graviton genus two amplitude in type II superstring theory. The function , used here, is the field theory limit of the function that enters the string amplitude derived in [18].

(2.12)

where

(2.13)

Note, in particular, that if (i.e., and are on the same line). Furthermore if three of the vertices on the same line (using ), so that the only non-zero contributions come from the planar and non-planar diagrams of figure 1. It is easy to see that in any region in which and are on the same line, , so that

  •    if and/or  ;

  •    if and/or  ;

  •    if and/or   .

This setup makes contact with the discussion in [17], where the Green function for an arbitrary Feynman diagram of scalar field theory was described. Our case differs only due to the presence of a measure factor in (2.11) which encodes the fact that we are discussing maximal supergravity. However, the exponential factor involves the same Green function as in [17], which has the form