Type I/heterotic duality and M-theory amplitudes

This paper investigates relationships between low-energy four-particle scattering amplitudes with external gauge particles and gravitons in the E_8 X E_8 and SO(32) heterotic string theories and the type I and type IA superstring theories by considering a variety of tree level and one-loop Feynman diagrams describing such amplitudes in eleven-dimensional supergravity in a Horava--Witten background compactified on a circle. This accounts for a number of perturbative and non-perturbative aspects of low order higher derivative terms in the low-energy expansion of string theory amplitudes, which are expected to be protected by half maximal supersymmetry from receiving corrections beyond one or two loops. It also suggests the manner in which type I/heterotic duality may be realised for certain higher derivative interactions that are not so obviously protected. For example, our considerations suggest that R**4 interactions (where R is the Riemann curvature) might receive no perturbative corrections beyond one loop by virtue of a conspiracy involving contributions from (non-BPS) Z2 D-instantons in the type I and heterotic SO(32) theories.

1 Overview The dualities of M-theory that relate string theories in different regions of moduli space have been well-studied over the past 20 years. In particular, the interrelationships between various theories with ten-dimensional N = 1 supersymmetry follow from the considerations of Hoȓava and Witten (HW) [1,2]). 1 They understood that the E 8 × E 8 heterotic string (referred to as the HE theory in the following) is equivalent to eleven-dimensional M-theory on an interval in the x 11 direction 2 of length L = π 11 R 11 ( 11 is the eleven-dimensional Planck scale) -in other words in a background space-time with geometry M 10 × S 1 /Z 2 , where M 10 is ten-dimensional Minkowski space. This is equivalent to the compactification of the eleven-dimensional theory on an orbifold of a circle of radius 11 R 11 so that the eleventh dimension is an interval that terminates on tendimensional boundaries. Consistency of eleven-dimensional supergravity in the presence of these boundaries requires boundary degrees of freedom that correspond to an independent N = 1 supersymmetric E 8 gauge theory restricted to each ten-dimensional boundary. The E 8 × E 8 heterotic string coupling constant is g he = R 3/2 11 , and so the limit R 11 → 0 is the weak coupling limit of the HE theory.
When compactifying on an additional spatial circle in x 10 of radius 11 R 10 so that the background is M 9 × S 1 × S 1 /Z 2 Wilson lines may be added, which break the gauge symmetry of the HE theory. Choosing the Wilson lines so that E 8 is broken to SO(16) on each boundary (following [1,2]) the HE theory is related by T-duality to a compactification of the heterotic Spin(32)/Z 2 (referred to as the HO theory in the following) to 9 dimensions in the presence of an HO Wilson line that breaks SO (32) to SO(16) × SO (16). The HO theory has a coupling constant g ho = R 11 /R 10 . This, in turn is related by a weak/strong duality transformation (which will be referred to as the "S transformation" in the following) to the type I theory compactified on 1 We will refer to theories with half-maximal supersymmetry as N = 1 theories, which reflects their tendimensional supersymmetry. 2 The coordinates of eleven-dimensional Minkowski space will be dlabelled x µ with µ = 1, 2, . . . , x 11 and with x 1 chosen to be the time coordinate.
a circle, where the circle direction is x 11 and an appropriate Wilson line again breaks SO (32) to SO(16) × SO (16). The type I coupling constant is g I = R 10 /R 11 . Finally, T-duality of the type I theory along the x 11 direction relates it to the type IA theory with coupling constant g IA = R 3/2 10 in a configruration where there are eight D8-branes and their mirror images coincident with each of the two orientifold 8-planes. We see from this circle of dualities, which is illustrated in figure 1, that the type IA and HE theories are related by interchanging R 10 and R 11 . The precise correspondence between the parameters of the various N = 1 superstring theories and the parameters of the HW theory is reviewed in appendix A.
M theory on M 9 × S 1 /Z 2 × S 1 Type IA SO(32) Heterotic/Type I In this paper we will investigate these duality relationships further by considering certain scattering amplitudes in M-theory determined by tree-level and one-loop Feynman diagrams of eleven-dimensional supergravity in the Hoȓava -Witten background. We will consider amplitudes involving the scattering of four gauge particles as well as those involving four gravitons (for economy of space we will not consider mixed gauge particle/graviton amplitudes or amplitudes involving dilaton fluctuations).
Of course, we do not expect the Feynman diagram approximation to capture the detailed behaviour of M-theory, but it should provide some information about the low-lying terms in its low-energy approximation. Since we will discuss scattering amplitudes of massless states in D = 9 dimensions, much of the rich structure of the theory, such as that associated with the effects of M2-branes and M5-branes will not enter in this perturbative approximation. Furthermore, we will treat the Hoȓava-Witten background geometry as rigid, ignoring in particular the fluctuations of the boundaries, which are associated with the dynamics of the HE dilaton.
The perturbative approximation to supergravity should be valid at momentum scales k << −1 11 , where k is a characteristic momentum in the scattering amplitude and 11 is the elevendimensional Planck scale. In this low-energy regime we would expect supergravity to reproduce terms in the low-energy expansions of scattering amplitudes, which are insensitive to a Planckscale cutoff. However, we will be considering the transformation of frames involved in discussing the low-energy approximations to the dual heterotic and type I string theories (the relationships between these frames are reviewed in appendix A). These dualities involve compactification on length scales << 11 , where perturbation theory cannot in general be justified. However, just as in the analogous discussion of four-graviton scattering in the type II superstring [3], we expect that the BPS properties of the low order terms in the low-energy expansion should justify these approximations.
The precise pattern of non-renormalisation conditions in theories with half-maximal supersymmetry (ten-dimensional N = 1 supersymmetry) is not completely understood. Following [4] and references therein, single-trace contributions to the low-energy expansion of the four-gluon scattering amplitude, which contribute terms of order s n t 8 trF 4 to the effective action, 3 are expected to get contributions from all orders in perturbation theory, apart from the cases n = −2 (which is the Yang-Mills pole term) and n = 0, which should receive no corrections beyond one loop. Double-trace contributions of order s n t 8 (trF 2 ) 2 are expected to get contributions from all orders in perturbation theory apart from the cases n = −1 (which is the gravitational pole term), n = 0, which should receive no contributions beyond one loop, and n = 1, which should receive no contributions beyond two loops. 4 The non-renormalisation conditions on terms in the low-energy expansion of the four-graviton amplitude are naively expected to parallel those of the double-trace terms of the same dimension in the Yang-Mills amplitudes. However, this would suggest that t 8 t 8 R 4 (where R is the Riemann curvature and t 8 is an eighth-rank tensor that will be discussed later) gets contributions from all loops, whereas explicit multi-loop calculations in supergravity [5] suggest that t 8 t 8 R 4 gets no contributions beyond one loop (see also [6]). Additional non-renormalisation conditions apply to those parity-conserving interactions that are related by supersymmetry to the parity-violating anomaly-cancelling terms, which only get one-loop contributions [7,8].

Outline
In section 2 we will discuss the Feynman rules of relevance to the calculations that follow. These differ from conventional bulk Feynman rules by virtue of the presence of space-like boundaries, which correspond to the fixed points of the orbifold of S 1 . 5 The discussion of the propagators will follow that given in [9], while the vertices for bulk fields coupling to the boundary fields was given in [1,2]. In order to clarify the discussion, in appendix B we will give a brief review of the action and Feynman rules in the HW background. We will also introduce a streamlined notation for terms quartic in field strengths and curvatures that arise in the low-energy tendimensional N = 1 superstring actions. Terms in the action that are quadratic in gravitational field strengths, would vanish on shell in ordinary gravity theories, but in the context of the Hoȓava-Witten background such interactions are localised on the boundaries and they give rise to two-point and three-point amplitudes involving Kaluza-Klein modes of the bulk fields, that will be discussed in section 2.3 3 Here, the symbol tr denotes the trace in the fundamental representation of SO (16). 4 More precisely, these statements apply to the heterotic theories, while the rules in the type I and type IA theories are slightly modified from the ones described in [4] . 5 For the most part we will use the language of the "downstairs" formalism in this paper. This is the description in which x 11 is restricted to the interval with two boundaries, 0 ≤ x 11 ≤ π 11R11. In the "upstairs" formalism x 11 spans the circle, 0 ≤ x 11 ≤ 2π 11R11 with fixed points at x 11 = 0 and x 11 = π 11R11 implied by the orbifold condition.
In order to set the scene for the discussion of four-particle Yang-Mills amplitudes in supergravity in the Hoȓava-Witten background, in section 3 we summarise a number of detailed properties of these amplitudes that emerge in the various N = 1 perturbative string theories. Many of these features are well known from earlier work, but some details of the more subtle properties will be presented elsewhere [10].
Section 4 is concerned with the low-energy expansion of four-particle supergravity tree amplitudes with external gauge particles in the M 9 × S 1 × S 1 /Z 2 background. The simplest example is the standard Yang-Mills tree amplitude localised in either boundary, which simply translates into the leading low-energy contribution (the massless Yang-Mills pole contribution) in the various N = 1 string theories.
The contribution of the Yang-Mills tree with a gravitational propagator will then be considered. This is a generalisation of the gauge boson amplitude considered in [9] (which also considered the scattering of massive SO (16) spinors, which are not considered here). The low-energy expansion of this term leads to double-trace contributions of the schematic form s n t 8 (tr i F 2 i ) (tr j F 2 j ), where i, j = 1, 2 and t 8 is a standard eighth rank tensor, and the subscripts 1 and 2 label the different SO (16) subgroups. These effective interactions arise in very different ways in the various string theories. For example, the n = 0 term arises as a purely tree-level effect in the heterotic theories but arises at two loops (spherical world-sheets with three boundaries) in the type I and IA theories. This term is expected to be protected from higher order renormalisation. More generally, such terms are tree-level contributions in the HE and HO theories but arise from world-sheet with n + 2 boundaries (and no handles) in the type I theories. However, only the interactions with n = −1, 0, 1 (where n = −1 denotes the term with the gravitational pole) are protected against renormalisation by higher loop effects.
In section 5 we will consider the Yang-Mills loop amplitude localised on either boundary and compactified on a circle of radius 11 R 10 and with a Wilson line breaking the symmetry, which generates Kaluza-Klein towers of massless and massive SO(16) adjoint states and massive SO (16) spinor states circulating in the loop. After expressing this as a sum over windings of the loop we will argue that the ultraviolet divergent zero winding term must vanish after renormalisation. The sum of non-zero windings gives a finite coefficient for the t 8 (tr 1 F 4 1 + tr 2 F 4 2 ) interaction that contributes at one loop to the decompactified HO theory and at disk level to the type I and IA theories, but does not contribute to the decompactified HE theory. Here the trace is in the fundamental representation for each SO (16) subgroup, whereas the naive expectation based on conventional gauge theory would be for the trace to be in the adjoint representation. These observations are in accord with string theory expectations. We will also comment on the relation of this interaction to the chiral gauge anomaly cancelling terms of the HO and type I theories.
In section 6 we will summarise some detailed properties four-graviton amplitudes in N = 1 perturbative string theories. This will include a review of the relationships between the different kinds of R 4 terms that arise as higher derivative interactions. Those that are related by N = 1 supersymmetry to the anomaly cancelling terms are again not expected to be renormalised. However, the interaction of the form t 8 t 8 R 4 is known to get contributions at tree level and one loop in all the theories. Based on naive dimensional analysis this is not a BPS interaction, but there is a possibility that it has special features since the expected three-loop divergence in D = 4, N = 4 supergravity is known to be absent [5] (see also [6]). It is of interest to understand how this can be consistent with S-duality between the HO and type I theories since the individual perturbative contributions do not transform into each other.
In section 7 we will consider tree-level graviton scattering amplitudes in supergravity in the Hoȓava-Witten orbifold background. Apart from the bulk tree-level supergravity amplitude, there are tree amplitudes with one of the vertices localised on one or on both boundaries. These generate terms of the form t 8 trR 4 and t 8 (trR 2 ) 2 (where the trace is in the fundamental representation of the tangent-space group.), which are the gravitational analogues of the gauge interactions of section 4. These parity-conserving interactions combine with the familiar parityviolating terms needed for chiral anomaly cancellation in the HE theory (as discussed in [2]) to form a sum of N = 1 superinvariants.
One-loop contributions to the four-graviton amplitude that generate terms in the effective action of order R 4 will be considered in section 8. In section 8.1 we will consider the oneloop amplitude in which the external gravitons interact with a supermultiplet of gauge particles localised in either boundary and compactified on S 1 . This is the supergravity analogue of the four gauge particle loop considered earlier. It gives a contribution to the R 4 interaction that is a linear combination of t 8 trR 4 and t 8 (trR 2 ) 2 . In order to evaluate the amplitude in which the external gravitons couple to bulk supergravity states circulating in the loop we will introduce an adaptation of the eleven-dimensional world-line superparticle formalism that was used to describe one loop in eleven-dimensional supergravity compactified on S 1 in [3,12]. This formalism will be reviewed briefly in section 8.2 and extended to implement the Z 2 orbifold of the Horăva-Witten background. We will see that the component of the loop with a circulating superparticle carrying zero Kaluza-Klein mode (m = 0) in the x 11 interval gives the one-loop supergravity contribution to the N = 1 theory that combines with the gauge loops of section 8.1 to complete the parity conserving part of the superinvariants that also contain the parity-violating anomaly cancelling interactions. It is expected that this is not renormalised by higher supergravity or string theory loop contributions.
The piece of the supergravity loop in which the circulating particles carry non-zero Kaluza-Klein charges in the interval gives rise to an effective interaction of the form t 8 t 8 R 4 , much as in the type II theories. This has a coefficient that is a function of the ratio R 11 /R 10 = g ho = g −1 I . This coefficient, which is the discussed in section 8.3, is simply the non-holomorphic Eisenstein series E 3/2 (i/g ho ) of the type that arises in the nine-dimensional type II theories, with the important distinction that the pseudsocalar field (the real part of the type IIB coupling constant) is set to zero since it is projected out by the orientifold that takes type II to type I. As remarked above, even though naive dimensional analysis suggests that this interaction is not protected by supersymmetry, it is known not to have the three-loop R 4 divergence in four-dimensional N = 4 supergravity that would have been expected for an unprotected interaction.. This function, which is invariant under S-duality, has some interesting features. Firstly, it correctly reproduces the tree-level and one-loop coefficients in the heterotic and type I/IA theories and has no further perturbative corrections. Furthermore, it possesses an infinite set of D-instanton contributions in both the HO and type I theories but not in the HE and type IA theories. These non-BPS objects are identified with the effects of wrapped euclidean world-lines of D0-branes. One might be skeptical that these non-BPS effects can be predicted accurately, although it has been forcibly argued [11] that the presence of Z 2 instantons associated with the homotopy relation π 9 (SO(32)) = Z 2 is an essential feature in the type I theory. This homotopy condition would also allow instantons in the HO theory, but their origin is much more questionable. The fact that π 9 (E 8 ) = 0 means such instantons must be absent in the HE theory, which is indeed a property of the amplitude presented in section 8.3.
Higher order terms in the low-energy expansion can arise from many sources. Firstly, there are higher order contributions from the low-energy expansion of the tree and loop terms described above. In addition there are many further loop diagrams that are very complicated to analyse. Since these correspond to non-BPS protected interactions it is unclear to what extent they illustrate genuine features of the string theories. Nevertheless, in section 9 we will consider the most intriguing example of such a diagram, that contributes to the four gauge particle amplitude at order s t 8 tr i F 4 i . This is a loop of gauge particles localised in a boundary, but with one gauge propagator replaced by a gravitational propagator. It is associated with a function that transforms in a non-trivial manner under S-duality. Although we do not expect that it is the complete story, since we are ignoring intrinsically "stringy" effects that we do not have control over, it is of interest that this diagram contains perturbative and instanton contributions in the HO and type I theories with sensible powers of the coupling constants.
A summary and discussion of these results are given in section 10.

Feynman diagrams in the Hoȓava-Witten geometry
In this section we will review the Feynman rules in the Hoȓava-Witten geometry that enter into the calculations in the subsequent sections. The action described in [1,2] takes the form which is the sum of the bulk eleven-dimensional supergravity action, the ten-dimensional super Yang-Mills action and a boundary contribution that includes the gauge and gravitational Chern-Simons interactions that are required in order to ensure the absence of chiral anomalies, as discussed in [2]). Whereas S sugra and S Y M are second order in derivatives, S boundary includes gravitational interactions of fourth order in derivatives, namely the gravitational Chern-Simons term, a R 2 term (where R is the riemann curvature) and a (∂H) 2 term (where His the field strength of the two-form potential). The action S includes all the interactions that contribute to three-point functions. Ten-dimensional N = 1 supersymmetry guarantees that these interactions are not renormalised by loop corrections.
In discussing the gauge particle and graviton tree diagrams it will be sufficient to use the bosonic components of the Feynman rules obtained from the action (2.1). However, the discussion of loop amplitudes necessarily involves supermultiplets of circulating particles. For the purpose of evaluating one-loop amplitudes in this paper it will prove efficient to make use of an extension of the supersymmetric first-quantised light-cone gauge description of eleven-dimensional supergravity compactified on a circle in [12]. This involves vertices describing the emission of a supergraviton from a superparticle world-line. The extension to include vertices localised in the Hoȓava-Witten boundaries will be briefly described in section 2.3 (its application to a loop amplitude in the bulk will be described in sections 8.2 and 8.3).

The action in the HW background
In writing the various terms in the action below we will only be explicit about the interactions involving bosonic fields -the fermionic terms required by supersymmetry are explicit in many references to earlier papers. The bosonic terms that enter in the total action S in (2.1) are the following. The purely eleven-dimensional supergravity action is given by [13] S sugra = 1 2κ 2

11
M 11 where G 4 = dC 3 is the four-form field strength associated with the three-form potential C 3 and G (D) is the determinant of the D-dimensional space-time metric (with D = 11 in the above case). In our later applications the integration domain M 11 , will be the space M 9 × S 1 × S 1 /Z 2 of the Hoȓava-Witten geometry compatified on S 1 .
The ten-dimensional supersymmetric Yang-Mills action [14] is given in its generally coordinate invariant form by the sum of terms on each boundary where the symbol Tr i indicates a trace in the adjoint representation of (E 8 ) i (which will be broken to its SO(16) subgroup upon compactification on S 1 ) where the subscript i labels the boundary. The Yang-Mills fields depend only on the ten dimensional space-time of the boundaries, but the metric degrees of freedom also depend on x 11 so this interaction includes the vertex coupling bulk gravity to Yang-Mills fields with arbitrary p 11 momentum.
The interactions that we are including in the boundary term in (2.1) comprise the Yang-Mills and Lorentz Chern-Simons interactions, as discussed in [1,2], together with R 2 and (∂ H) 2 interactions, where H = dC, and the only non-zero boundary components of C are C µν11 . So we will write (2.4) The Chern-Simons terms in S CS are obtained by modifying the supergravity action (2.2) by replacing G 4 by where Ω 3 is defined by and we need to impose the condition that only the components C µν11 of the bulk three-form are non-zero at the boundary (and we are again ignoring fermionic contributions). The quantities Ω 3YM and Ω 3L are the standard Yang-Mills and Lorentz Chern-Simons forms. The action now includes the boundary terms where the * operation is with respect to the ten-dimensional boundary space-time. The first term determines the Chern-Simons contribution to on-shell three-point functions (to be discussed in the following section) in which two states may be Yang-Mills gauge bosons or bulk gravitons and the third state is a Kaluza-Klein mode of the bulk C µν11 field. The second term in (2.7) is a contact term that plays an important rôle in the context of the four-particle amplitudes to be considered later. This term is singular since it involves the integral of the product of two δ(x 11 ) (or δ(x 11 −L)) factors, arising from the product of two Ω 3 factors localised on the same boundary, resulting in a term proportional to δ(0) in S CS . Similar divergent terms in the low energy action and their regularisation were briefly discussed in section 4 of [2]. Such divergences cancel in physical amplitudes. For example, in four-particle tree amplitudes beginning and ending on the same boundary the contact term divergence cancels with a similarly divergent contribution from the Feynman diagram in which a propagator joins two three-point vertices located in the same boundary.
Note also that the presence of S CS leads to a modification of the Bianchi identity for the four-form field strength, which has an anomalous boundary contribution of the form which is necessary in order to ensure the cancellation of chiral gauge and gravitational anomalies.
The second higher derivative gravitational term in S boundary in (2.4) is the four-derivative gravitational term that is quadratic in curvature tensors, and gives higher derivative contributions to the two-graviton and three-graviton vertices. This R 2 term has an action of the form (where R is the Riemann curvature scalar and R µν is the Ricci tensor). In writing this expression we have set to zero the coefficients of R 2 and R µν R µν interactions, which vanish on shell and can be removed by field redefinitions. The R 2 interaction in (2.9) does not vanish on shell -it contributes to on-shell graviton two-point functions localised on the boundaries. The gravitons carry arbitrary Kaluza-Klein momentum, p 11 while the tangential momentum k µ is conserved. It also contributes an on-shell three-graviton vertex localised on the boundary. These contributions to the amplitude will be discussed in the next section.
The final boundary interaction between bulk bosons that contributes to two-particle and three-particle amplitudes has the form (see [15]) where H µνρ = ∂ [µ C νρ] 11 . This interaction again contributes non-vanishing on-shell two-point and three-point functions with a pair of external on-shell bulk C-fields, as will also be discussed in the next section.

Bulk and boundary to boundary gravity propagators
Before considering the scalar propagator on an orbifold we recall the form of the momentum space propagator on a circle of radius 11 R 11 , which is given by where the eleven-dimensional momentum (p µ , p 11 ) has components p µ in the ten Minkowski space dimensions and the momentum in the x 11 direction is quantised in units of 11 R 11 p 11 = m R 11 11 = m π L . (2.13) The norm p 2 = p µ p µ (µ = 1, 2, . . . , 10) is with respect to the Minkowski metric diag(−, +, . . . , +).
The propagator between points x 11 and y 11 on S 1 is given by the Fourier sum where the Minkowski space directions 1, 2, . . . , 10 have been left in momentum space.
We are interested in formulating Feynman rules in the presence of the Z 2 orbifold boundary conditions at x 11 = 0 and x 11 + L, as was considered in [9], in which case the momentum conjugate to x 11 is quantised as given in (2.13). The scalar propagator between two points in the orbifold direction is obtained by imposing the additional orbifold boundary conditions, which require the propagator to be invariant under x 11 → −x 11 . This is achieved by identifying the propagator on the orbifold as the combination G(p; x 11 , y 11 ) = G(p; x 11 − y 11 ) + G(p; x 11 + y 11 ) cos p 11 x 11 cos p 11 y 11 .
(2. 15) In considering loop amplitudes later it will be useful to note the following product relations dw 11 G(p 1 ; x 11 , y 11 ) G(p 2 ; y 11 , z 11 ) G(p 3 ; z 11 , w 11 ) G(p 4 ; w 11 , x 11 ) (2.17) We will later be interested in supergraviton propagators that are constrained to begin and end on either boundary, which involve the following two slightly different expressions for the propagator.

Endpoints on the same boundary
Setting x 11 = 0 = y 11 in (2.15) gives where the last step involves a Poisson summation. This transforms the sum over Kaluza-Klein modes of charge m into the sum of windings of the propagator around x 11 with winding number n. The integral can be performed explicitly, giving The propagator G(p M ; L, L) is given by the same expression.

Endpoints on different boundaries
Setting x 11 = 0 and y 11 = L in (2.15) gives where the last step again follows by Poisson summation. Once again, performing the integral gives the expression The propagators of the bosonic supergravity fields, the metric and three-form field, involve extra numerator factors arising from their spin. These were presented in detail in [9] and were important in determining the tree diagram with a propagator joining the boundaries. In subsequent sections we will make use of the expressions in [9] in order to avoid repeating those details here.
It will prove convenient to introduce a matrix notation for the various propagators terminating on the Horava-Witten walls by defining the components of a matrix D ij (i, j = 1, 2) by We may now begin to consider the Feynman diagrams that describe gauge and graviton scattering amplitudes in the Hoȓava-Witten supergravity background. We will begin with the simplest cases of two-point and three-point functions (which only receive contributions from the Chern-Simons and R 2 interactions).

Yang-Mills/graviton three-point function
The Yang-Mills action, S Y M (2.3) that is localised on the boundaries contributes to the usual ten-dimensional interactions between gauge bosons and gravitons, together with the three-point interaction between Yang-Mills states and a graviton carrying non-zero p 11 . The on-shell threepoint function between Yang-Mills states and a bulk graviton polarised in directions parallel to the boundaries is given by In this expression the external gauge bosons have null Minkowski momenta, k 11 ) and symmetric polarisation tensor ζ (2)[µν] , which satisfies (k (1) + k (3) ) µ ζ µν = 0. When viewed from ten dimensions this bulk state has Kaluza-Klein mass given by (mass) 2 = (p (2) 11 ) 2 , which follows from the eleven-dimensional massless condition, (p (2) ) 2 = 0, Before compactification the matrix T (1) is a generator in the adjoint of the E 8 gauge group associated with the gauge particle labelled r. After compactification on S 1 with the insertion of an appropriate Wilson line, the gauge group is broken to SO (16). In that case the external Yang-Mills states are either massless SO(16) gauge particles or massive SO(16) spinor states, as will be discussed in more detail in section 4.

Yang-Mills Chern-Simons three-point function
The linearised Yang-Mills Chern-Simons interaction in S CS (2.7) is given by 2 11 Recall that the ten-dimensional antisymmetric tensor potential is identified with the zero Kaluza-Klein charge of the three-form, B µν = C µν11 | p 11 =0 . This gives rise to a three-point function of the same form as (2.23) but with an antisymmetric polarisation tensor wave function for the C µν11 field, so that Although knowledge of the bosonic components of the propagators and vertices is sufficient to construct the gauge and graviton tree diagrams, when we discuss loop diagrams we will make implicit use of a supersymmetric formalism. This will be based on an extension of the firstquantised light-cone formalism used to describe the S 1 compactification of eleven-dimensional supergravity [12] to the Hoȓava-Witten background. The linearised forms of the interactions in S (2.1) arise in this formalism as expectation values of vertex operators that describe the emission of single particle states from a world-line in a manner that is modelled on the vertex operator construction of light-cone gauge closed superstring theory. For present purposes we need to extend the formalism to include vertex operators localised in either boundary, acting on states that may be in either the boundary Yang-Mills supermultiplet or the components of the bulk graviton that couple to the boundary. For example, the three-point interactions between a pair of Yang-Mills particles and the bulk graviton or C-field are given by a matrix element of the form where V bulk (k (2) , ζ (2) ) is a vertex operator describing the emission of an on-shell bulk state from the world-line of a Yang-Mills superparticle embedded in light-cone superspace. The emitted state is a graviton when ζ (2)µν is symmetric and an on-shell C µν11 state when it is antisymmetric. This reproduces the three-particle interactions in (2.23) and (2.25). This vertex operator is a function of bosonic and fermionic light-cone superspace coordinates x i and S A , which are similar to the zero modes of the coordinates that enter the light-cone description of the heterotic string. Since we will only use very general features of this formalism we will not present the details, which can be reconstructed from [12].
Lorentz Chern-Simons, R 2 and (∂H) 2 interactions The on-shell amplitude for three bulk tensor bosons (the graviton or C-field) interacting on a boundary is given by where t ν 1 ν 2 ν 3 is given by and we have specialised to the situation in which the only non-zero components of both the metric tensor and the C µν11 field are those parallel to the boundaries.

The Lorentz Chern-Simons interaction
When one of the three polarisation tensors is antisymmetric in (µ, ν) the t ... t ... term in (2.27) does not contribute and this expression describes the linearised gravitational Chern-Simons interaction in S CS in (2.7). This is given by where state number 2 is the C-field and the other two states are gravitons (and ζ {µν} denotes a symmetrised tensor).
This interaction is reproduced by replacing the Yang-Mills states in (2.25) by graviton states (which entails replacing the gauge states and their colour factors by the tensor graviton states) and once again using an antisymmetric polarisation tensor for particle 2. The resulting matrix element is (2.30) The R 2 and (∂ H) 2 interactions The two-point functions for gravitons may be obtained by linearising the curvature tensors in (2.9). The two gravitons have momenta (k µ , p 11 ) and (−k µ , p 11 ), where we have used conservation of the ten-dimensional momentum (k (1) = −k (2) = k). The mass-shell condition requires that k 2 + (p 11 . This leads to a two-particle on-shell vertex given by where we have used the physical state condition k (r) µ ζ (r)µν = 0 and ζ (r)µν is symmetric for external gravitons. The (∂ H) 2 interaction gives rise to the same on-shell two-point function with the graviton polarisations replaced by antisymmetric ζ (1)µν and ζ (2)µν , which are the C µν11 polarisation tensors.
The three-point functions for three gravitons or two C-states and one graviton can be extracted from (2.27) by a suitable choice of polarisation tensors. The t ... t ... term simply reproduces the gravitational interaction of a graviton with two gravitons or with a pair of C's. The remaining part of the interaction in (2.27) is quartic in momenta and corresponds to the three-field terms in the expansions of the effective interactions of the form R 2 and (∂ H) 2 . The on-shell three-point function resembles that discussed in [16][17][18] in the context of the interactions of string ground states in the background of the D8-brane.
These three-point functions are reproduced in the world-line first quantised formalism by replacing the antisymmetric polarisation tensor ζ (2) in (2.30) by the symmetric graviton polarisation. Choosing the external states to either be gravitons or C-states leads to the two kind of three-point functions described in the previous paragraph.
While the perturbative rules for constructing amplitudes follow from the local action as described above, the string theory interpretation requires an extrapolation of these rules into a regime in which perturbation theory may be questionable. This is well-illustrated by the interpretation of the amplitudes that involve Kaluza-Klein modes in the x 11 direction (the orbifold direction) that have masses given by p 11 = m/( 11 R 11 ) (m ∈ Z). Such modes decay since p 11 is non conserved because of the orbifold boundary conditions. The spectrum of particle states in the Hoȓava-Witten background compactified on M 9 ×S 1 /Z 2 ×S 1 is briefly reviewed in appendix A.4.
In later sections, we will encounter amplitudes in which these unstable Kaluza-Klein modes contribute to propagators in a manner that accounts for certain low order terms in the effective action that are presumably protected by supersymmetry. In particular, we will encounter interesting instanton terms in the HO and type I amplitudes, where the instantons correspond to winding configurations of euclidean world-lines of these non-perturbative states of the HE and type IA theories.
3 Some features of four-particle Yang-Mills amplitudes in N = 1 string theories Before describing the Yang-Mills amplitude calculations in the Hoȓava -Witten background we will summarise some features of the amplitudes that arise in different string theory limits. The following are some of the features that we expect to reproduce. Although most of these have been noted before (see, in particular, [7,8]) there are some subtleties that will be explained in more detail in [10].
• Upon compacitifcation to nine dimensions on a circle of radius r he = 1/r ho (in the presence of appropriate Wilson lines) the SO(16) × SO(16) tree-level amplitudes in the HO and HE theories are equivalent under T-duality, which equates r he /g 2 he with r ho /g 2 ho . The HO/HE tree level heterotic expression has a Yang-Mills pole g −2 het t 8 trF 4 /st, where the symbol tr again indicates the trace in the fundamental representation of either SO(16) subgroup (and g het is either of the heterotic coupling constants). This is produced by a disk diagram in the type I/IA theories. The tree level terms with a graviton pole in the HO or HE theories have the form t 8 (trF 2 ) 2 /s, which is produced by an annulus (one-loop) diagram in the type I/IA theories.
• The first non-pole term in the low-energy expansion of the tree amplitudes in both the heterotic theories has the form g −2 het t 8 (trF 2 ) 2 . This arises as a three-boundary (i.e. two open-string loop) term proportional to g I in the ten-dimensional type I theory (and vanishes in the ten-dimensonal type IA theory), which is a striking illustration of the way the perturbation expansion of the heterotic theory is reorganised by its type I parameterisation 6 . Note further that the open-string one-loop (annulus) contribution to t 8 (trF 2 ) 2 in the type I theory vanishes at large r I , which is consistent since otherwise S-duality would require a term of order 1/g ho in the HO theory, which does not exist. However the annulus contribution to t 8 (tr i F 2 i ) 2 in the type IA theory (i.e., in the large r IA limit) is non-vanishing and is dual to a one-loop term in the HE theory. 6 The leading term in the low-energy expansion of the three-boundary open string diagram naively has the form s t8(triF 2 i ) 2 but there is closed-string (graviton) propagator that cancels the factor of s, which accounts for the agreement with the heterotic expression.
• The next order in the low-energy expansion of the tree amplitudes in the HO theory contributes a term of the form g −2 ho ζ(3)s t 8 trF 4 (where we have emphasised the occurrence of a ζ(3) coefficient). Likewise, in the type I theory the tree-level disk diagram contributes g −1 I ζ(3)s t 8 trF 4 . These two expressions ought to be related by HO/type I duality, but this cannot act term by term since it would transform the type I coefficient g −1 I into g ho , which is not a possible power of g ho . These tree-level terms must be part of a function of the coupling that is transforms appropriately under the S transformation (S : g I → g ho = g −1 I ), which can be viewed as a remnant of the SL(2, Z) duality of the type IIB theory. Furthermore, the next perturbative contribution to s t 8 trF 4 in the HO theory is at one loop, whereas the one-loop term vanishes in type I and the next contribution is of order g I and is associated with world-sheets with the geometry of a torus with a single boundary. This is the same order in g I as the two-loop disk diagram (which has three boundaries), which contributes to g I t 8 (trF 2 ) 2 as stated earlier (but not to g I t 8 trF 4 or g I s t 8 trF 4 [10]). We will see in section 9 how these general features are reproduced by loop contributions to the four gauge particle amplitude, although since s t 8 trF 4 is not protected from loop corrections, we do not expect our analysis to give the complete expression for its coupling constant dependent coefficient.
• The one-loop amplitudes for the two ten-dimensional heterotic theories are independent of the couplings and are interchanged by the identification r ho = 1/r he . However, the low-energy limits of the theories look rather different. In the large-r he limit the leading behaviour in the HE theory is whereas in the larger ho limit of the HO theory the leading behaviour is tr SO(32) F 4 = tr 1 F 4 1 + tr 2 F 4 2 [19]. In both cases the traces are evaluated in the fundamental representation of either SO(16) sub-group.
The type I theory has a t 8 trF 4 tree (disk) interaction but does not generate a one-loop (annulus) contribution to t 8 trF 4 even in D = 9 (i.e., for finite r I ) [10]. There is a contribution from the annulus diagram to an interaction that is suppressed by a power of r I of the form t 8 ( i=1,2 (tr i F 2 i ) 2 − tr 1 F 2 1 tr 2 F 2 2 )/r I , which leads to a contribution to the type IA theory at large r IA [10] that agrees with the expression in the HE theory.
Whereas in the HE and HO theories the t 8 (trF 2 ) 2 interaction arises at tree level, in the ten-dimensional type I theory it arises at two loops (i.e., at order g I ) and is unrenormalised.
• The preceding description of parity-conserving terms in the low-energy action has its counterpart in the parity-violating sector. These parity-violating terms are important for ensuring the absence of chiral anomalies. The absence of chiral anomalies in the ten-dimensional heterotic theories is attributed to the presence of anomaly cancelling terms [20] of the form B ∧ X (F, R), which arises as one-loop effects associated with the interaction of the Neveu-Schwarz/Neveu-Schwarz antisymmetric tensor with a total of four gauge bosons and gravitons on a toroidal world-sheet. The notation, which is reviewed in appendix B, emphasises that these terms are related by supersymmetry to the parity conserving terms contained in t 8 Y (gs) 8 . As explained in [2], in the Hoȓava-Witten description of the HE theory the eight-form Y (gs) 8 is naturally expressed as the sum of three pieces 7 Y (gs) 8 (R) (implicitly defined by (B.13) and (B.12)) comes from the bulk interaction, and ( 1 30 Tr i F 2 i − 1 2 trR 2 ) 2 arises as an effect of either boundary (labelled i). 8 Upon compactification this has coefficient r he = 1/r ho and therefore vanishes in the large r ho limit. The HO anomaly-cancelling interaction in the large-r ho limit, where the gauge group SO(32) is unbroken, is a one-loop interaction of the form 10 This is related by S-duality to a similar expression in the type I theory which is associated with an amplitude coupling a Ramond-Ramond B-field and a total of four gravitons and gauge bosons to a disk world-sheet.
In the course of indicating how these features arise from the analysis of supergravity coupled to Yang-Mills in the Hoȓava -Witten background we will be led to several insights into possible non-perturbative effects that seem to be required required for their consistency.

Yang-Mills four-particle tree amplitudes
We will here consider some contributions of tree-level four-particle gauge amplitudes, which are relatively straightforward to evaluate. The external scattering states have null Minkowski momenta, k The linearised field strength for the particle labelled r has the form 9 In the uncompactified ten-dimensional gauge theory T A (A = 1, . . . , 496) are 496 × 496dimensional matrices in the adjoint representation of the Lie algebra of E 8 × E 8 and r Ar is the 7 Note that since anomaly cancellation is only of relevance in ten dimensions where E8 is unbroken, the symbol Tr here refers to trace in the adjoint representation of the unbroken E8 group. 8 The definition of X is a factor of 4 greater than our expression. 9 Here and in the following we indicate a linearised approximation by a hat.
polarization vector in the internal gauge group space, which specifies the quantum numbers of the particle labelled r. This may be written as the sum of the field strength in each E 8 factor of where F 1 and F 2 are 248 × 248 matrices in the adjoint representation of each of the E 8 's (and two-forms in µ, ν). Denoting the generators of the adjoint representation for the and so where we have used for i = 1, 2.
In the following we will discuss dualities that relate amplitudes in the heterotic and type I/IA theories when one direction is compactified on a circle of radius 11 where I is the index labelling the Cartan sub-algebra. The compactified theory then contains a tower of Kaluza-Klein SO(16) adjoint states with square of the masses with integer n, which includes the massless gauge potentials. These correspond to D-particles in type IA theory [22]. There is also a tower of massive SO(16) spinor states with square of the masses given by which correspond to stuck D-particles 10 in type IA theory [22]. In considering the duality between the HE and HO theories we need to break both E 8 subgroups to SO (16), which involves the Wilson line in , which gives masses to the SO(16) spinor states in both subgroups. Although the massive SO(16) spinors will not be relevant to the gauge theory trees with massless external gauge states that we will consider below, they are an essential ingredient in the discussion of loop amplitudes in section 5.
The loop amplitude discussion in appendix D will also involve a discussion of the breaking of the HO theory with SO(32) broken to SO(16) × SO (16). We are here interested in the limit of M-theory of relevance to the HO string, which is the limit in which r ho H = 11 /(R 10 R 11 ) → ∞. In this case the Wilson line is the element of the Cartan subalgebra of SO(32) of the form [21] and gives rise to Kaluza-Klein tower of massive bi-fundamental states of SO(16) × SO(16) with masses given by in addition to the Kaluza-Klein tower of adjoint states with the masses given in (4.8) (with r he replaced by r ho ).
An important point to note in considering the following expressions for scattering amplitudes is that the momentum conservation delta functions will not be explicitly included in the amplitudes. However, in the compactified theory momentum conservation in the compact x 10 dimension involves the replacement of the continuous momentum conservation delta function δ( 4 r=1 k (r) ) by 2πR 10 11 δ 4 r=1 l (r) , where the Kronecker delta imposes conservation of the discrete (Kaluza-Klein) momenta of the external particles. Although in this paper we will be setting l (r) = 0 it is obviously important to keep the volume factor, 2π 11 R 10 . In the following this factor will always be included in the expression for a compactified amplitude.

The Yang-Mills tree amplitude in a single boundary
In the following we will discuss Yang-Mills four-point amplitudes in which pairs of particles may be in either of the Hoȓava -Witten boundaries, and therefore in either of the SO(16) subgroups of the two E 8 's. The amplitude will therefore be written as a matrix, A ij = A ji , where i, j = 1, 2 label the two subgroups.
The simplest example is the sum of tree amplitudes for the scattering of four gauge bosons that are all in a particular SO(16) ⊂ E 8 sub-group of E 8 × E 8 associated with one of the ten-dimensional boundaries, compactified on a circle in the x 10 direction. The lowest order contribution is given by the sum of poles in s, t and u channels, each corresponding to the propagation of an intermediate gauge boson. If the gauge particles are in the E (1) 8 factor (where 10 In type IA theory, a single D-particle is necessarily stuck to the O8 planes and it is sub-threshold BPS bound state. the superscript indicates which boundary we are considering), the amplitude is proportional to where the superscript (r) (r = 1, 2, 3, 4) labels the scattering particle, T (r) 1 = r Ar T Ar 1 encodes the colour dependence and T Ar 1 is a matrix in the 16×16 representation of SO (16). The coefficient is proportional to κ 2 11 /λ 2 together with the factor of 2π R 10 11 to account for the compactification of the tenth dimension. Clearly, the expression for the amplitude localised in the other boundary is A YM-pole 22 is obtained by replacing T and so (4.12) is proportional to the leading term in the low-energy expansion of the Yang-Mills tree amplitude in the HE theory. The fact that this is proportional to r he implies that it has has a sensible ten-dimensional limit as r he → ∞.
Since T-duality implies that r he /g 2 he = r ho /g 2 ho the expression (4.12) transforms consistently to the corresponding expression for the scattering of gauge particles in the SO(16) × SO(16) subgroup of SO(32) in the HO theory.
Similarly, using (A.24) the prefactor in (4.12) is interpreted in terms of the parameters of the type IA theory by noting that −5 11 R 10 = ( I ) −5 g −1 IA , describing the four gauge boson amplitude for scattering in the eight D8 branes and their mirrors that are coincident with one of the orientifold O8 planes. Note, in particular, that this is independent of r IA since the scattering is entirely within one of the orientifold planes and is insensitive to the radius of the eleventh dimension.
Finally, T-duality converts the type IA amplitude into the type I amplitude by the replacement g −1 IA = r I g −1 I using (A.5). This description involves a factor proportional to the radius of the eleventh dimension, r I , since T-duality is non-local along that direction and the type I amplitude depends on r I . We see, therefore, the first (and rather simple) example of an amplitude that has a consistent description in all four versions of the D = 10, N = 1 string theory. The first higher derivative contribution to the four-particle Yang-Mills amplitude arises from tree diagrams with an intermediate graviton or antisymmetric potential propagating between pairs of Yang-Mills particles on the same boundary. In this case the intermediate particle propagates in the eleven-dimensional bulk and so the propagator involves the sum over the quantised momentum p 11 as in (2.19). This is analogous to the case considered in [9], where the amplitude described a pair of scattering gauge particles in each boundary joined by a propagator for a graviton or C field, which will be reviewed later (see (4.23)). In the present case the gauge particles are scattering in a single boundary. The C field couples to the boundary gauge fields via the (two-derivative) Chern-Simons interaction described earlier while the graviton couples via its minimal coupling (which also has two derivatives).

The Yang-Mills/gravity tree amplitude in a single boundary
An important difference from the case considered in [9] is that the boundary conditions require the use of the propagator G(p M ; 0, 0) defined in (2.18) and (2.19). In addition, when all four gauge particles are in a single E 8 it is necessary to include the contact interactions coming from the square of the gauge Chern-Simons three-form that is contained in the |dĈ| 2 term in the boundary action [2]. Applying the Feynman rules to this process, taking into account the factor of κ 2 11 in the propagator and λ −6 for each vertex, and using (2.11) to express these parameters in terms of 11 , gives the Yang-Mills four-particle amplitude with an intermediate graviton propagator and all gauge particles in E We have again inserted a factor of 2π R 10 11 to account for the volume of the compactification of the tenth dimension, as mentioned earlier. The standard kinematic factor t 8F 4 is defined in (B.7). In writing this expression we have used the form of the propagator expressed as a sum over Kaluza-Klein modes in (2.18) since this will be useful for comparison with the analogous expression in type I string theory.
We may now expand the terms in (4.13) in the low-energy limit, sR 2 (4.14) which is an expansion in powers of g 2 he 2 H s. Although we started with the expression in (2.18) as a sum over Kaluza-Klein modes, the same result obviously arises starting from the winding number expression (2.19), where we have with L = πR 11 11 . We will see that the sum over Kaluza-Klein charges makes a direct connection with the form of the one-loop amplitude in the type I description of the amplitude.

The lowest order term in the low-energy expansion
The leading term in the low-energy expansion of (4.13) using (4.14) contributes the pole term, Its coefficient is interpreted in the HE string theory by using the M-theory/string theory dictionary, giving the identification (4.17) Therefore, the leading term in the low-energy expansion reproduces the low-energy limit of the tree-level heterotic E 8 × E 8 theory. This again transforms into the corresponding HO tree amplitude under T-duality, using r he /g 2 he = r ho /g 2 ho . The prefactor in the type I theory is obtained from the relation r ho −3 H g −2 ho = r I ( I ) −3 , which reproduces the fact that the gravity/C-field pole arises in the one-loop amplitude (the annulus diagrams) in the type I theory. T-duality implies that r I = (r IA ) −1 , so the type IA amplitude vanishes in the r IA → ∞ limit. This is the limit in which the non-zero winding numbers of the bulk propagator are suppressed so the implication is that the zero winding number contribution to the amplitude also vanishes.

Higher order terms
The next term in the low-energy expansion of the amplitude in (4.13) is the term of order s 0 in (4.14), so the coefficient (4.17) is multiplied by a factor of −s π 2 R 2 11 /3, giving a contribution to the amplitude (4.13) of the form where we are again only displaying terms that are SO(16) singlet in the s-channel. This is interpreted as a local contribution to the effective action of order t 8 (tr 1 F 2 1 ) 2 (with a similar term involving t 8 (tr 2 F 2 2 ) 2 ). In the parameterisation of the various N = 1 string theories, the coefficient of the amplitude is proportional to We therefore see that this interaction is associated with a one-loop effect in the HE and IA theories in D = 10 (the large r he or r IA limit) but vanishes in HO and type I in D = 10 (as r ho or r I → ∞).
The next term in the low-energy expansion of (4.13) is 20) which is expected to be a protected interaction that has no contributions beyond two loops in the heterotic theories. Using the relations This interaction is interpreted as a two-loop term in the ten-dimensional HE theory, which could be (but has not been) checked by analysing the low-energy limit of the explicit genustwo amplitude in HE perturbation theory [23][24][25][26]. This interaction is not present in the tendimensional limit of the HO theory since it is suppressed by a factor of 1/r 3 ho . It is also interpreted as a one-loop contribution in the the open string theories. We will shortly see (in (4.22)) that the curious-looking dependence of this term on r IA = 1/r I arises explicitly from the expansion of the type I annulus amplitude.
More generally, the low-energy expansion of the propagator given in (4.14) produces a sequence of terms of the form (g 2 that are interpreted as (n + 1)-loop terms of order s n in the low-energy expansion of the HE theory. We do not expect terms with n > 1 to be protected against higher loop contributions and there are sure to be other contributions to these higher order terms in the low-energy expansion. In the type I interpretation this is an expansion in powers of 2 I s r 2 IA = 2 I s /r 2 I , which is independent of the type I coupling constant, g I , and therefore all such terms should originate from the one-loop (annulus) diagram in the type I or IA theory, which we will now describe.
The type I annulus diagram in D = 9 dimensions [10] We can see how the structure of the low-energy expansion changes when n > 1 from the explicit form of the type I annulus contribution to the double-trace terms in the four gauge particle amplitude. Figure 4 represents the amplitude, which has a low-energy expansion proportional to Figure 4. One loop four gauge boson amplitude in type I string theory 2 ) where we are specialising to the case where the quantum numbers of the external states have been chosen so that T contain an SO(16) singlet and therefore couple to the gravitational sector. 11 The sum overn is a sum over the winding numbers around the x 11 circle of the closed type I string propagating in the cylinder channel. This is interpreted as the sum over Kaluza-Klein momentum of the type I open string when the world-sheet is evaluated as an open-string loop (there is no winding number for the type I open string). In the type IA description the open strings satisfy Dirichlet boundary conditions in the x 11 direction so such strings carry no p 11 momentum and the sum translates into a sum over open-string winding modes. This, in turn, transforms into a sum over p 11 Kaluza-Klein momentum modes in the type IA closed-string description (while the Dirichlet boundary conditions on the cylinder boundaries imply that the type IA closed string has no winding around the x 11 circle).
The graviton pole arises as then = 0 term in (4.22). The factor (s 2 I +n 2 r 2 I ) −1 also contains the same infinite sequence of massive poles as the field theory propagator in (4.14), and 11 The expression includes the terms associated with the second SO(16) subgroup as well as the first. In particular, the term proportional to tr1(T expanding it in powers of s 2 I /r 2 I gives the same infinite sequence of higher-derivative terms as in the expansion of the supergravity tree diagram. The higher order terms in the last parentheses arises from the Koba-Nielsen-like factor associated with excited string states. This is a sign that the supergravity expression is not valid for interactions of order s 2 (trF 2 ) 2 and higher (taking into account the fact then = 0 term in the sum cancels one power of s). We see that these interactions, which are not protected by supersymmetry, receive contributions from higher string modes, which are not captured by the supergravity approximation.

Tree stretching between distinct boundaries
When particles 1 and 2 are in one E 8 subgroup and particles 3 and 4 are in the other, the ends of the tree are on distinct Hoȓava-Witten boundaries, which was the example considered in [9]. The amplitude again consists of the sum of Feynman diagrams with a graviton or the third-rank potential, C propagating between the boundaries, (but with no contact term). The resulting amplitude has the form ( [9]) where we have again included a factor of 2π R 10 11 to account for the volume of the compactification of the tenth dimension. The low-energy expansion can be obtained by using The lowest order term in the low-energy expansion The leading behaviour of the amplitude in the low-energy limit 2 H s, 2 H t, 2 H u 1 reduces, after using the M-theory/heterotic string theory dictionary to 2 ) which again agrees with the corresponding term in the HE tree-level amplitude as noted in [9]. The correspondence with the HO, type I and type IA theories follows as in the discussion following (4.17).

Higher order terms
The next term in the expansion of (4.23) following from (4.24) 4 11 tr(T which differs from that of (4.18) by a factor of −1/2. This relative factor of −1/2 between the coefficient of tr(T and A Y M −gravity (1,2) arises from the factor of (−1) m in (2.20) as can be seen from the identity (4.27) The relative factor of −1/2 is in accord with the computation of the annulus loop diagram in type I string perturbation theory, where the coefficient (4.27) is obtained from the factor n =0 (−1)n (s 2 I +n 2 r 2 I ) −1 in (4.22) in the s → 0 limit. This relative factor is also in accord with the analysis given in equation (3.16) of [27], where, in the low-energy limit, the one-loop effective action in HE string perturbation theory was found to have the form This expression is the parity conserving partner of the parity-violating interaction that serves to cancel the chiral gauge anomalies when decompactified to the ten-dimensional heterotic E 8 × E 8 limit. By contrast, the tree-level HE effective action for the double-trace terms has the form [15,28] r he which we will obtain from supergravity in the Hoȓava-Witten background in section 4.4.
The terms that arise at the next order in the expansion of the propagator in powers of (R 2 11 2 11 s) following (4.28) include the H g 2 he s t 8 (tr 1 F 2 ) 2 interaction in (4.20) together with terms related by permutations of the external particles and involving both SO(16) subgroups. These are described by a two-loop effective action of the form which should agree with the low-energy limit of the genus-two contribution to HE superstring theory.
To summarise, the tree amplitudes illustrated in figures 3 and 5 that have a single gravitational propagator capture the low order terms of the form s n t 8 (trF 2 ) 2 in the low-energy expansion of the N = 1 superstring four-point amplitude compactified on a circle. Indeed the field theoretic amplitudes in (4.13) and (4.23) precisely reproduce the corresponding factors associated with closed-string ground states with arbitrary winding numbers in the contribution of the annulus (one open string loop) diagram to the four-point function of the type I string theory (4.22). For n < 2 the agreement is exact, whereas stringy corrections enter into the low-energy expansion of (4.22) at order s 2 t 8 (trF 2 ) 2 . Other arguments (for example, see [4]) suggest that this is a non-BPS interaction that is not protected against higher loop corrections.

"Iterated" Yang-Mills tree diagrams
We will now consider an infinite class of generalised tree diagrams that are illustrated in figures 6 and 7. The black dots in these diagrams represent two-point functions induced by the presence of the trR 2 and (∂H) 2 terms that enter as boundary interactions in (2.9). These vertices have the form C s 2 −6 11 , where the dimensionless constant C can be determined from (2.31) to have the value C = 1/2(2π) 7 . At any given order, the low-energy expansion of the amplitude involves the sum over all ways of arranging a given number of propagators to form a chain linking the pairs of external gauge particles. Each propagator (D) introduces a factor of κ 2 11 D = (2π) 8 9 11 D/2. The pairs of gauge particle at the ends of the chains may lie in the same SO(16) subgroup or in distinct subgroups, as is represented by the two figures. and each chain of given length may have both endpoints on the same boundary (and is of the form of the form D 11 or D 22 defined in (2.22a) or it may stretch between boundaries (and is of the form D 12 or D 21 ). All such possibilities are to be summed over. In figure 6 there is an even number of propagators linking the boundaries while in figure 7 there is an odd number. The amplitude obtained by summing over all possible ways of joining the initial and final pairs of gauge particles is given by the matrix The indices i and j are not to be summed on the right-hand side of the above equation. This expression can be written as an expansion in powers of π 3 11 s 2 D/4. Each factor of D can itself be expanded as a power series in R 2 11 2 11 s (with the leading term ∼ 1/(πR 11 11 s)). In terms of string theory parameters, the former is an expansion in powers of g IA 2 I s/r IA and the latter is an expansion in g 2 he 2 H s. The term of zeroth order in the two-graviton vertex manifestly reproduces the result obtained earlier due to the exchange of a single D ij between the pairs of external gauge particles, which we previously related to the contribution of closed-string winding number states to the annulus diagram in type IA string theory. If we keep the leading term in the expansion of D ij , which is −(πR 11 11 s) −1 (independent of i, j), we have Thus, keeping only the massless Kaluza-Klein modes in D ij gives an expression that is interpreted as a tree-level expression in the HE theory.
We may compare (4.32) with the expression for the tree-level four gauge particle amplitude in the heterotic string given in equation (4.4) of [15]. This is proportional to 12 +non − cyclic perms.
We see that the supergravity amplitude in (4.32) is the component of the double-trace part of this string theory tree amplitude that involves the 1/(stu) term in the last parenthesis 13 . The dependence on string theory factors begins at order s 2 (trF 2 ) 2 , which is not protected by supersymmetry. and is the order at which the ζ(3) factor in the last parenthesis enters in the expansion. Similarly, the single-trace term of order s trF 4 is not expected to be protected by supersymmetry and also has a prefactor proportional to ζ(3). We will see in section 9 how these ζ(3) terms can be motivated from the effect of loop amplitudes in supergravity in the Hoȓava-Witten background -although in these cases we do not expect to reproduce the exact coefficients.
Note that the apparent pole at 2 H s = −4 in (4.33) is cancelled by stringy corrections that are subsumed in the terms of O(s 2 H ) in the last parenthesis. Such a cancellation is not captured by the Feynman diagram expression (4.31). This is consistent with the fact that we do not expect to reproduce the exact expressions for interactions of order s 2 t 8 (trF 2 ) 2 and beyond.
Furthermore, the expansion of (4.32) in powers of 2 H s can be interpreted in the type I theory as an expansion in powers of g I 2 I s (using (A.5)). Each power of g I is interpreted as the insertion of a boundary or cross-cap in the open string world-sheet. In this way we see that a contribution to the tree-level HE amplitude is associated with an infinite series of higher order terms in type I perturbation theory. Thus, we have described the HE and HO tree level contributions to the interaction t 8 (tr 1 F 2 1 + tr 2 F 2 2 ) 2 and the one-loop contribution in the HE theory of the form which has yet to be verified by a direct string theory calculation).

Yang-Mills one-loop amplitudes
We will now turn to consider one-loop Feynman integrals for supergravity in the Hoȓava-Witten background compactified on a circle. We need to include the complete supermultiplet of states circulating in the loop, which can be expressed in a simple manner by using the world-line lightcone superspace procedure described earlier. In the following we will be interested in determining local terms induced by the loop amplitudes and will not discuss the non-local effects associated 12 In [15] the heterotic string scale was chosen to be 2 H = 1/2. 13 We have not attempted to compare the overall normalisations of these expressions, but the agreement of the residue at s = 0 in (4.32) with the HE string pole term, as noted earlier, guarantees the agreement of the rest of the expression.
with non-analytic parts of the amplitude, which can be separated from the analytic terms in unambiguous fashion 14 . We will first consider the four gauge boson one-loop amplitude in E 8 gauge theory compactfied to nine dimensions on a circle of radius 11 R 10 in the presence of a Wilson line that breaks the gauge symmetry to SO (16). 15 We will consider the loop amplitude with external nine-dimensional massless states that transform in the SO (16)

A loop of gauge particles on one boundary
The contribution of the SO(16) adjoint states in the loop is given by

2)
14 It is far from obvious that such a separation of analytic and non-analytic parts of the amplitude is possible at higher orders in the low-energy expansion. 15 This amplitude can be determined in an efficient manner by means of a first-quantised world-line formalism (modelled on string theory calculations), in which vertex operators describe the emission of massless gauge particles from a circulating N = 1 gauge supermultiplet. We omit the details here, but see section 8.2 for a discussion of the vertex operator construction of the gravitational loop amplitude.
where C adj is the colour factor for the loop amplitude of SO(16) adjoint states (and the overall factor of t 8F 4 is determined by maximal Yang-Mills supersymmetry). This is given by setting N = 16 in the SO(N ) colour factor (where N is even) that has the following form, for a particular colour ordering: We have left N as a free parameter in order to emphasise later the special features of the value N = 16. The dynamical part of the loop amplitude is contained in I adj (s, t, u; R 10 ) that is given by a scalar box Feynman diagram compactified on the circle of radius 11 R 10 with the loop momentum p 10 replaced by the sum over integer Kaluza-Klein charges, as will be discussed below.
The other piece of the four gauge boson amplitude, where only the SO(16) spinor states circulate in the loop, is given by where C spin is the colour factor for the loop of spinor states and is given, for any SO(N ) group (with even N ), and as before, for a particular colour ordering, by The quantity I spin (s, t, u; R 10 ) is the dynamical part of the amplitude, which is again given by a Feynman box diagram with the integral over the p 10 component of the loop momentum replaced by a sum over half-integer Kaluza-Klein charges. This will also be discussed below.
It is an important fact that when N = 16 the sum of the adjoint colour factor and the spinor colour factor satisfies C adj + C spin = 3 tr(T a 1 T a 2 ) tr(T a 3 T a 4 ) + perms. , (5.6) and therefore does not contain any fourth order Casimir invariant in the SO(16) fundamental representation. This, of course, is connected with the fact that the amplitude is inherited from the ten-dimensional amplitude in which the circulating states are the massless states in the adjoint of E 8 , which has no independent fourth order Casimir. Another combination of the colour factors that will prove important below is which is a purely fourth order invariant.

Evaluation of the lowest order terms in the loop amplitude
In the following we will make use of the standard Poisson summation formulae As mentioned above, the Feynman integral reduces to a kinematic prefactor t 8F 4 C adj multiplying the compactified scalar box diagram with integer Kaluza-Klein charges. The box diagram evaluated in nine dimensions contains non-analytic threshold terms of order √ s, which do not concern us here and are, in any case, subleading in the low-energy expansion. The lowest term in the low-energy expansion is obtained by setting s = t = u = 0, in which case it is simple to show that where the sum in the lattice factor is over the Kaluza-Klein charge, m. Each term in the m sum obviously possesses the ultraviolet divergence of the ten-dimensional theory. This will be dealt with by defining the loop in the winding number basis by performing a Poisson summation (using (5.8)) that converts (5.9) into 10 11 ζ (2) , (5.10) whereτ = 1/τ . Here we have separated the divergent zero winding (m = 0) term, that depends on the cut-off and is represented by C 1 = Λ 0 dτ , where Λ is an arbitrary dimensionless constant that we need not specify. The non-zero winding terms have precisely determined coefficients.
The leading term in the low-energy expansion of I spin is again obtained by setting s = t = u = 0 and evaluating the box diagram with the appropriate lattice factor to describe the circulating spinor states, giving, where we have used (5.8) and the fact that m>0 1/m 2 = −2 m>0 (−1)m/m 2 = ζ(2). The dimensionless constant, C 2 , again arises from the zero winding mode (m = 0) and is cut-off dependent.
The cut-off dependent terms proportional to C 1 R 10 / 11 and C 2 R 10 / 11 should be renormalised by the addition of counterterms. However, since R 10 / 11 = r he /g

2/3
he H there is no consistent perturbative string theory interpretation of such terms so we will choose the counterterms so that C 1 = C 2 = 0. In that case the total contribution to the amplitude in the low-energy limit is given by adding (5.10) and (5.11)), which gives We see therefore that the amplitude vanishes in the R 10 → ∞ limit, which is consistent with fact that there is no t 8 trF 4 term in the E 8 gauge theory.
The preceding results translate into the following effective action in the various string theories. (assuming that the gauge potentials are in the SO(16) subgroup labelled 1 and ignoring the overall normalisation) From the expressions on the first line we see that the ten-dimensional limit of the HE theory (r he → ∞) has no one-loop contribution to t 8 tr 1 F 4 1 whereas the r ho → ∞ limit of the HO theory is non-zero. Recall that we earlier found an explanation for a ten-dimensional contribution to the one-loop t 8 (tr 1 F 2 1 ) 2 interaction in the HE theory (in terms of a tree-level supergravity amplitude), which vanished in the ten-dimensional HO limit. From the second line of (5.13) we see that both the type IA and type I theories have tree-level contributions to t 8 tr 1 F 4 1 in the ten-dimensional limit. In the type I case the presence of the requisite volume factor of r I indicates that the limit involves the distance between the Hoȓava -Witten walls. In the type IA theory there is no volume factor because the interaction is localised in one wall independent of the value of r IA . These type I contributions come from disk diagrams, which only generate the single-trace t 8 trF 4 .
We have thus accounted for the string theory result that the one-loop amplitude in the HO theory is proportional to tr 1 F 4 1 +tr 2 F 4 2 with the trace in the fundamental representation of either SO(16) subgroup, whereas the standard (UV divergent) SO(16) × SO(16) gauge theory loop amplitude would be proportional to Tr 1 F 4 1 + Tr 2 F 4 2 , with the trace in the adjoint representation of either SO (16). This is also in agreement with the form of the type I open-string tree amplitude (defined on a world-sheet disk) that has a group theory Chan-Paton factor.
One-loop amplitude in the compactified SO(32) gauge theory It is of interest to see how the above results are complemented by starting from a one-loop fourparticle amplitude in the compactified SO(32) theory. However, since this is not a Feynman diagram that arises in supergravity in the Hoȓava -Witten background we have relegated the detailed argument to appendix D. There we consider the one-loop amplitude compactified on a circle of radius 11 /(R 10 R 11 ) = H r ho with SO(32) gauge symmetry broken to SO(16) × SO (16). The discussion parallels that of the E 8 gauge theory loop in section 5.1. In this case we find that when the non-zero winding number configurations of the loop around the dual x 10 direction are interpreted as Kaluza-Klein modes of the HE theory, and with the identification r he = 1/r ho , the loop amplitude reproduces the correct effective action involving a combination of t 8 (trF 2 ) 2 interactions obtained earlier (see (4.28)) by considering tree amplitudes in supergravity in the Hoȓava-Witten background.
6 Some features of four-graviton amplitudes in N = 1 string theories As in the case of the Yang-Mills amplitude it is useful to describe some features that arise in considering four-graviton amplitudes in the various kinds of N = 1 supersymmetric string theory amplitudes before describing the corresponding features in supergravity in the Hoȓava-Witten background.
Recall that in the maximally supersymmetric N = 2 case the only kinematic structures that contribute to the low-energy expansion of the four-graviton scattering amplitude have the form of derivatives acting on t 8 t 8 R 4 and 10 10 R 4 . In the type IIA theory there is a single parityviolating term (the Vafa-Witten term [29]) of the form B ∧ R ∧ R ∧ R ∧ R (or 10 BY (vw) 8 in the notation of (B.12) in appendix B.5). This structure is a consequence of the very strong constraints of maximal supersymmetry. However, in N = 1 theories two other structures arise. These are parity conserving terms of the form t 8 trR 4 and t 8 (trR 2 ) 2 , where the trace is over the ten-dimensional tangent-space group, SO(9, 1). These parity conserving interactions are not independent since they satisfy the identity [30] This linear relationship means there is an ambiguity in the choice of basis for these terms. We will see below that a natural basis is defined in the Hoȓava-Witten description since t 8 t 8 R 4 is produced entirely by a bulk effect while a particular combination of the other two terms is localised on the boundaries. As reviewed in appendix B.5 these two localised R 4 interactions are related by supersymmetry to corresponding parity-violating interactions terms, 10 BtrR 4 and 10 B(trR 2 ) 2 . In the following it will be useful to recall the origin of R 4 terms in N = 1 perturbative superstring theory, which we will now summarise.
• The HE and HO theories have leading tree-level pole contributions of the form t 8 t 8 R 4 /stu, which is simply a compact way of expressing the sum of all the tree level four-graviton amplitudes in Einstein gravity. There are also poles in amplitudes of higher order in the low-energy expansion that correspond to terms of the form t 8 trR 4 /st and t 8 (trR 2 ) 2 /s.
• At the next order of the tree-level expansion of the heterotic theories there are terms of the form ζ(3) t 8 t 8 R 4 and t 8 (trR 2 ) 2 , where we have again explicitly indicated the occurrence of a notable factor of ζ(3). Although (6.1) implies an ambiguity in how the combination of R 4 terms is expressed, the coefficient of ζ(3) in the ζ(3) t 8 t 8 R 4 term suggests that this particular term is singled out from the other one in an unambiguous fashion.
In the type I theory, the term ζ(3) t 8 t 8 R 4 again arises from the spherical world-sheet diagram and is of order 1/g 2 I , but t 8 (trR 2 ) 2 comes from diagrams with two open-string loops (three boundaries/cross-caps) and are of order g I (which is analogous to the origin of the (trF 2 ) 2 terms considered earlier).
• The one-loop HE or HO contributions again have form t 8 t 8 R 4 and t 8 (trR 2 ) 2 . But it is now natural to express this (using (6.1)) as the sum of t 8 t 8 R 4 and t 8 Y . We will describe later how these observations fit with HO -type I duality. N = 1 effective R 4 actions at tree-level and one loop We will here summarise the duality relationships between the R 4 interactions in the various N = 1 string theories in terms of their effective actions compactified to nine dimensions. These can be expressed with the help of the N = 1 superinvariants J 0 (defined in (B.17)), I 2 (defined in (B.18)) and X 1 and X 2 (defined in (B.21)), where the notation is based on [8]. The string frame effective action for the terms of order R 4 in the nine-dimensional HO theory at tree-level and one loop, obtained by combining expressions in [15,19,27], is 16 Here the contributions of the massless Kaluza-Klein scalar associated with the compact x 10 direction should be included in the definition of the superinvariants, although they have been ignored in the first two lines of this equation.
For later comparison with the type I theory it is useful to write the HO one-loop contribution to the parity-conserving terms in the last line of (6.2) using the identities in appendix B.5 3) The combination of terms in parentheses will be identified with disk-level contributions to the amplitude in type I theory whereas the first term arises from the torus diagram. Further understanding of these points will emerge from the analysis in section 8.
The following comments concerning the effective action are of note: • The parity-violating anomaly cancelling term −12 10 BY (gs) 8 in the second line is contained in the combination of the invariants X 1 , X 2 and I 2 in the fourth line. These terms are oneloop exact in the HE and HO theories. The tree-level interaction in the first line t 8 (trR 2 ) 2 is part of the expression t 8 (trF 2 − trR 2 ) 2 when the gauge fields are included. This receives no loop corrections since it is related by supersymmetry to the three-point interactions in t 8 (trF 2 − trR 2 ) and these are unrenormalised beyond tree level. The relationship (6.1) means that there is an ambiguity in the coefficients of t 8 t 8 R 4 , t 8 trR 4 and t 8 (trR 2 ) 2 in the one-loop terms in the second line of (6.2). However, there is no ambiguity in the expression written in terms of superinvariants, which is exhibited in the last two lines of the equation.
• The effective action for the HE theory is obtained simply by performing the heterotic Tduality relation (A.1). The expression (6.2) is invariant under this transformation since the four-graviton amplitudes in HO and HE theories on a spherical or a toroidal world-sheet are identical -they are insensitive to the details of the gauge group 17 .
• The nine-dimensional type I superstring effective action is proportional to As expected the interactions that are tree-level or one-loop exact, translate straightforwardly from the HO theory using the S-duality relations. The (2ζ(3)/g 2 I + 2π 2 /3)J 0 terms arise from the diagrams with spherical and toroidal world-sheets, just as in the HO theory. S-duality cannot simply act on these terms in isolation, since they would transform into terms of order g ho and 1/g ho , respectively, which are powers that do not make sense in the HO theory. This is a signal that the coupling constant-dependent coefficient of t 8 t 8 R 4 in the HO/type I theories is a non-trivial function of the coupling constant that transforms under S-duality in a manner that preserves these first two perturbative terms. We will later find a candidate for such a function motivated by supergravity in the Horava-Witten background, in a manner analogous to the modular function that enters as the coefficient of t 8 t 8 R 4 in the type IIB theory.
• Whereas in the HO expression (6.2) the 10 10 R 4 interaction only arises at tree level, in the type I expression (6.4) there are three distinct terms containing this interaction. Two of these arise from the spherical and toroidal world-sheets in the same manner as in the type IIB theory (and are proportional to 1/g 2 I and g 0 I ). According to (6.4) there should also be a disk contribution proportional to 1/g I , which has not been determined directly from the string theory. The 10 10 R 4 interaction does not contribute to the graviton four-point function so it would be necessary to evaluate an amplitude with N gravitons coupling to a disk, with N ≥ 5, in order to verify its presence.
• A certain amount is known concerning higher derivative terms in N = 1 supersymmetric string perturbation theory that will also be discussed in the context of supergravity in the Hoȓava-Witten background.

Graviton tree amplitudes in the Hoȓava-Witten background
We will now turn to the explicit calculations of graviton amplitudes in the Hoȓava-Witten background. This will extend the earlier analysis of the gauge theory amplitudes. The graviton polarization tensor satisfies k µ ζ µν = k ν ζ µν = 0. The symmetric part of ζ µν describes the graviton polarisation while the antisymmetric part describes the polarisation of an antisymmetric tensor potential. The curvature tensor linearised around Minkowski space has the form R µνρσ = k µ k ρ ζ νσ . (7.1) 17 The group theory lattice factor is same for the two gauge groups.
We will again only consider amplitudes in which the external momenta and polarization tensors are oriented in the nine non-compact dimensions, so we will not consider the scattering of the scalar states arising from Kaluza-Klein compactification. Since much of the analysis is a simple extension of the analysis of the scattering of gauge particles the following exposition will be brief.

Graviton tree amplitudes
Four graviton tree amplitudes in the Hoȓava-Witten background arise from the four-graviton vertex operator on sphere in the heterotic theories. But they arise at different orders of type I perturbation theory.
There are tree amplitudes with conventional gravitational vertices in the eleven-dimensional bulk, as well as trees with either one or both vertices localised in the boundary induced by the boundary Chern-Simons and R 2 interactions. In the following we will adapt the notation used to label the Yang-Mills tree amplitudes by denoting the gravitational amplitudes by where i, j = 1, 2 labels the boundary in which a vertex is localised and 0 denotes a bulk vertex operator.

The bulk tree
h µν Figure 9. Tree level four-graviton amplitude with graviton exchange in the bulk The supergravity tree amplitude (together with the standard four-graviton contact interaction) has the form where we have displayed the interpretation of the supergravity tree amplitude in all this N = 1 string theory. This amplitude is the tree level amplitude of ordinary Einstein gravity where the intermediate particle can only be graviton. Since the external states are assumed to have p 11 = 0 this is identical to the expected tree-level interaction of ten-dimensional gravity coupled to a dilaton. This amplitude arises from a spherical world-sheet in any of the N = 1 string perturbation expansions.

Tree amplitude with one vertex on a boundary
h µν Figure 10. Graviton tree amplitude with graviton exchange coupling to one vertex localised on a boundary This amplitude is obtained by joining a bulk graviton vertex to the cubic contribution to the R 2 vertex localised on either boundary as in figure 10, together with the four-point contact term that arises from R 2 . This is a higher derivative pole contribution of order s 2 H relative to the Einstein gravity tree. This gives a contribution to the amplitude that is independent of R 11 since it is localised on only one of the boundary, and has the following form where we have translated the amplitude into the heterotic, type I and type IA parmeterisations. The fact that the amplitude is localised on a single Hoȓava-Witten boundary means that its type IA description is independent of r IA , although it is proportional to r I in the T-dual type I description.
While this amplitude arises as a higher order term in the expansion of the tree amplitude in the HE and HO theories, in type I perturbation theory, it arises from the sum of the amplitudes on the disk and RP 2 world-sheets with four closed string vertex operators.
The amplitude A G 20 has an identical form and the total amplitude is A G 10 + A G 20 .

Tree amplitude with both vertices on boundaries
In this case the vertices are either the R 2 vertex or the gravitational Chern-Simons vertex, both of which are localised on the boundaries. The exchanged particle can now be either a graviton or the potential C µν11 . This case is analogous to the Yang-Mills amplitudes depicted in figure 3 and figure 5. The form of the amplitude is

5)
h µν + C µν11 Figure 11. Graviton tree amplitude with graviton and C exchange with vertices localised on one or both boundaries where the numerator factor is 1 when the vertices are on the same boundary or (−1) m when they are on different boundaries. The term with m = 0 gives the pole contribution The next term in the low energy expansion is given by setting s = 0 in (7.5). Combining the contributions from A G 11 , A G 22 , A G 12 and A G 21 gives 3 R 10 2 4 (2π) 6 This corresponds to a t 8 (trR 2 ) 2 contribution to the one-loop HE effective action that is the parityconserving partner of the parity-violating term 10 B (trR 2 ) 2 , which is part of the GS anomaly cancelling term. As we will see in section 8 the other part of the parity-conserving partner of the anomaly-cancelling term in the HE theory, which is proportional to t 8 t 8 R 4 , emerges from the contribution of a graviton loop propagating in the bulk in the Hoȓava-Witten background and has the coefficientĈ in (8.19) and (8.21).
The discussion of the expansion of the non-zero KK terms in powers of R 2 11 2 11 s is similar to the discussion of the higher derivative t 8 (trF 2 ) 2 interactions in section 4. As in that case all the terms in the expansion arise from contributions of order g 0 I in the type I description, which are associated with a world-sheet cylinder, Möbius strip and Klein bottle. The integer m is the type I closed string winding number around the compact dimension.

"Iterated" graviton tree diagrams
Similarly there are gravitational tree amplitudes with the same structure as the Yang-Mills amplitudes discussed in subsection 4.4, in which there are chains of propagators joining the vertices with the external particles. These are gravitational analogues of the Yang-Mills processes shown in figures 6 and 7. These possibilities generate higher derivative contributions to the tree processes described in section 7, in much the same way as that discussed in the context of the Yang-Mills interactions earlier.

Graviton one-loop amplitudes
We turn now to consider the one-loop four-graviton amplitude. There are two kinds of loop amplitudes that contribute to leading terms in the low-energy expansion. In section 8.1 we will consider the loop with circulating super-gauge particles localised in either ten-dimensional boundary and compactified on a circle of radius R 10 . This can be constructed by use of a lightcone gauge world-line vertex operator formalism based on the vertex in (2.26), which describes the emission of a graviton from a super Yang-Mills world-line. The resulting loop amplitude is identical to the contribution that arises from the gauge loop in ten-dimensional N = 1 supergravity compactified to nine dimensions. Much as in the case of the Yang-Mills four-particle loop amplitude discussed in section 5, after transforming to the winding number basis by performing a Poisson summation over the Kaluza-Klein modes in the x 10 direction, we are able to make contact with various R 4 terms in the string theory effective action.
The other loop contribution is the "bulk" gravity loop with circulating supergravity particles propagating in the eleven-dimensional space compactified on the interval of length L = πR 11 11 and a circle of radius R 10 11 , which is the subject of sections 8.2 and 8.3. In section 8.2 we will make use of an extension of the light-cone gauge vertex operator construction used in the description of the four-graviton loop in eleven-dimensional supergravity compactified on S 1 [3,12]. Implementing the Z 2 orbifold condition that defines the Hoȓava-Witten background raises some subtleties connected with the breaking of supersymmetry. The amplitude that results from this construction is discussed in section 8.3. Its low energy limit contains a t 8 t 8 R 4 contribution that manifests the strong coupling duality relating the HO and type I theories in an interesting manner. The one-loop amplitude with gravitons coupled to a circulating supermultiplet of N = 1 gauge fields in one boundary is depicted in figure 12. Since we are restricting the graviton polarisations and momenta to lie in the boundary directions the bulk propagation plays no rôle in this calculation and the amplitude could, in principle, be obtained from the Feynman rules of ten-dimensional N = 1 supergravity compactified to nine dimensions in the x 10 direction. The expression for this amplitude could also be determined by making use of a world-line formalism analogous of that described in the case of the bulk loop in sections 8.2 and 8.3. This involves the product of four vertex operators of the form V bulk (2.26) that couple the external gravitons to a N = 1 gauge supermultiplet. We will denote the low energy limit of this amplitude by A gauge loop i , where the subscript i = 1, 2 labels the boundary. The complete ten-dimensional N = 1 one-loop supergravity amplitude also involves a contribution from the supergravity multiplet circulating in the loop. As we will see in section 8.3 this contribution is naturally thought of as a p 11 = 0 contribution to the "bulk loop". We will here denote it by A gravity loop

Four gravitons coupled to gauge particle loop on the boundary
Although we have not performed the explicit gauge loop calculations in detail we know that when added to A gravity loop p 11 =0 the result will contribute to the parity conserving part of a linear sum of N = 1 invariants, which also contains parity violating pieces of these R 4 terms. Therefore, the precise combination of parity conserving R 4 interactions is determined from knowledge of the anomaly cancelling terms. So we conclude that the amplitude must be proportional to the oneloop kinematic factor t 8 trR 4 + (trR 2 ) 2 /4 = t 8 Y (gs) 8 (R, 0) multiplying a scalar box integral. In other words, after adjusting the normalisation to agree with (6.2), the low energy limit of the loop amplitude is given by As in the case of the gauge theory loop amplitude, we will set the coefficient of the renormalised divergence to zero,Ĉ = 0, since the quantity R 10 / 11 translates (using the relations (A.25)) into (r ho )/ H in the HO description, which involves a nonsensical power of the string coupling. The expression (8.2) corresponds to an effective action in terms of the HO string theory parameters of the form The analogous expression in the HE theory is proportional to 1/r he . In type I string theory this interaction is of order 1/g I and arises from four graviton vertex operators coupled to a worldsheet disk and to the projective plane RP 2 (a sphere with a cross-cap). However, as pointed out following (6.3), the type I theory also has a term proportional to t 8 t 8 R 4 (contained in the N = 2 invariant J 0 ), which is crucial for understanding how HO/type I duality is realised, as we will see following the discussion of the bulk loop in section 8.3.
Comments on supersymmetry connection with chiral anomaly cancelling terms. The above argument gave the parity-conserving part of the combination of superinvariants (−I 2 + 24X 1 + 18X 2 ) in the HO theory (where the invariants are defined (B .21) and (B.23)). This combination also contains the parity-violating anomaly-cancelling term B ∧ X (gs) 8 (R) and is protected from higher loop corrections. There is no ten-dimensional HE contribution from (8.3) in the r he → ∞ limit. However, we earlier found the HE one-loop contributions to t 8 (trR 2 ) 2 arising from the tree-level supergravity graphs obtained in (7.7) in the previous section. This is part of the same superinvariant as the parity-violating ten-form, 10 B(trR 2 ) 2 . It was argued in [2] that the remaining part of the anomaly cancelling term in the HE theory is provided by the bulk Vafa-Witten ten-form of the type IIA theory 10 B Y In other words, the tree amplitudes of supergravity in the Hoȓava-Witten background combine with the Vafa-Witten interaction to give the anomaly cancelling terms in the HE theory, whereas in the HO theory these terms arise from the loop of gauge particles localised in either boundary.

Supersymmetry and the bulk one-loop amplitude.
We will here describe the supersymmetric world-line formalism that will be used in the next subsection to determine the properties of the four-graviton loop amplitude in which the circulating particles are bulk supergravitons.
M-theory compactified on a x 11 circle is invariant under eleven-dimensional supersymmetry, associated with a 32-component SO(10, 1) Majorana spinor, Q, which decomposes into two 16-component SO(9, 1) spinors of opposite chirality, Q = (Q 1 , Q 2 ) which satisfy the chirality conditions where r = 1, 2 and Γ 11 = Γ 1 . . . Γ 10 is the product of the gamma matrices of the ten-dimensional theory. The supercharges Q 1,2 are those of the type IIA theory and their anti-commutation relation takes the form {Q r ,Q s } = δ rs Γ µ p µ + rs p 11 µ = 1, ..., 10 (8.5) where the Kaluza-Klein momentum, p 11 , enters as the central extension and is the signal of 1/2-BPS D0-brane states in type IIA string theory. As discussed in [1,2], in the Hoȓava-Witten background the boundary conditions at x 11 = 0 and x 11 = L break the supersymmetry so that only Q 2 survives and the theory possesses N = 1 ten-dimensional supersymmetry.
In constructing the bulk loop amplitude we will adapt the eleven-dimensional light-cone vertex operator formalism [12], which was used to discuss one-loop amplitudes in eleven-dimensional supergravity compactified on a d-torus, T d . In 11-dimensional Minkowski space we choose the light-cone gauge with x ± = (x 1 ± x 2 )/2. The world-line fields comprise the transverse bosonic coordinates x I (I = 3, . . . , 11), which form a SO(9) vector, and the fermionic coordinates, S A , which form a 16-component SO(9) spinor. After compactification on a x 11 circle the light-cone coordinates naturally decompose into x i , x 11 , where x i is a SO(8) vector (i = 3, . . . , 10) and S A = (Sȧ 1 , S a 2 ), where S r (r = 1, 2) are eight-comonent SO(8) spinors of opposite chiralities (indicated by undotted and dotted indices).
Single-particle states are labelled by the vector and spinor indices appropriate for the "leftmoving" and "right-moving" sectors of the type IIA string theory, together with the value of the Kaluza-Klein charge m, |i,j; m , |i,ḃ; m , |a,j; m , |a,ḃ; m .
The states in the massless supermultiplet of the IIA theory are the m = 0 states and the Kaluza-Klein recurrences (the D0-brane states in the type IIA theory) have m = 0. The type IIA supersymmetry generators can be expressed in terms of the SO(8) spinors, S r , which relates fermionic and bosonic states in the following manner (where we have suppressed the vector/spinor labels that are not affected by the action of S r on a state).
In considering the Hoȓava-Witten background we need to identify states under the action of the orbifold Z 2 , which identifies x 11 with −x 11 , which is represented by the action of an operator Ω. This reverses the sign of m and changes the dotted spinor by a minus sign, giving and therefore the dotted space is killed by this projection. The physical states are therefore spanned by |i,j; 0 , |a,j; 0 , (8.11) which are the states of N = 1 supergravity. The matrix elements of the undotted spinor, Sȧ 2 , vanish between these projected states. On the other hand when m = 0 both chiralities contribute with equal weight (the relative minus signs cancel out).
With these preliminaries we can now proceed to evaluate the one-loop four-graviton amplitude in the bulk shown in figure 13.  (8.13) which describes the emission of a graviton with polarisation ζ ij and momentum k i , where we choose i, j = 3, . . . , 9 which are directions transverse to the light-cone directions and to the compact directions, x 10 and x 11 . In the type IIA theory the trace over the eight S 1 and eight S 2 fermionic modes arising from the product of four vertex operators in (8.12) gives a kinematic factor of t 8 t 8 R 4 multiplying a scalar field theory box diagram in M 9 × S 1 × S 1 [3,12]. 18 The orbifold condition of relevance to the Hoȓava-Witten background compactified on a circle, M 9 × S 1 × S 1 /Z 2 , is implemented by inserting a factor of (1 + Ω)/2 between the vertex operators in (8.12). The trace in the m 2 = 0 and m 2 = 0 sectors in the sum over m 2 must be treated separately since the space of states in the m 2 = 0 sector is reduced by the conditions (8.10) to the N = 1 states (8.11).

The bulk one-loop four-graviton amplitude
(i) m 2 = 0 In this case all of the N = 2 states circulating in the loop survive the projection in (8.9). The trace over the components of S 1 and S 2 again leads to the kinematic prefactor t 8 t 8 R 4 of the type IIA theory. The sum over m < 0 is equivalent to the sum over m > 0, which leads to a factor of 1/2 in the overall normalisation of these terms relative to the type IIA case.
(ii) m 2 = 0 In this case the circulating states are those in (8.11) and we can set S 2 = 0 in the vertex operator acting on this projected space, reducing it to (8.14) which is the zero mode piece of the graviton vertex in the heterotic string acting on the supergravity multiplet. The fermionic trace only involves the dotted spinors Sȧ 1 , which produces a factor of t 8 , leading to a prefactor that is a linear combination of t 8 trR 4 and t 8 (tr(R 2 )) 2 . The complete one-loop amplitude is obtained by adding this contribution to that of the boundary gauge loop considered in section 8.1.
The dynamical factors in the amplitude are given in terms of the integral of the product of four Green functions in the orbifold background. From (2.17) we see that, apart from a factor of 1/2 in its normalisation, the p 11 = 0 (m 2 = 0) contribution has precisely the same form as the expression that enters the loop amplitude of eleven-dimensional supergravity compactified on a torus. In order to make heterotic/type I duality manifest it is very useful to write the resulting expression for the sum of the p 11 = 0 terms in the low energy limit of the loop amplitude in the form where we have included a p 11 = 0 contribution to the loop momentum in the box diagram amplitude, I bulk (s, t, u; R 10 , R 11 ), and subtracted it again in the term −I bulk p 11 =0 (s, t, u; R 10 , R 11 ). 18 The 10 10R 4 pieces of the type IIA effective action is not captured by the four-graviton amplitude since it vanishes on shell.

Properties of the bulk loop amplitude
The coefficient of t 8 t 8R 4 in the p 11 = 0 sector, I bulk p 11 =0 (s, t, u; R 10 , R 11 ), has the same form as for the gauge theory loop in (5.9) and is interpreted as a one-loop contribution in the HO theory by the same Poisson summation over m 1 argument that leads from (5.9) to (5.10).
The contribution I bulk (s, t, u; R 10 , R 11 ), which involves the sum of all values of p 11 is very similar to the expression for the loop amplitude of eleven-dimensional supergravity compactified on a torus [3] with purely imaginary complex structure. The leading order term in the low-energy expansion is obtained by setting the momenta to zero in the factors of e ik·x in the vertex operators in (8.12). After performing the integral over the nine-dimensional momentum we obtain I 0 (R 10 , R 11 ) ≡ I bulk (0, 0, 0; R 10 , R 11 ) = 3π 9/2 2 11 R 10 R 11 The integral for each term in the sum over Kaluza-Klein charges is divergent in the ultraviolet region (the t → 0 limit). However, the total integrand may be expressed in terms of the winding The quantity C represents the regulated zero winding piece. The divergence can be subtracted by introducing a one-loop counterterm that leaves a finite but undetermined contribution, C.
The contribution of I 0 (R 10 , R 11 ) to the nine-dimensional low-energy supergravity amplitude can be translated into a term in the effective M-theory action of the form In the limit V = 2π 2 R 10 R 11 → ∞, the expression describes an action in eleven non-compact dimensions and only theĈ term survives, while the first term in parentheses is suppressed by the factor of V − 3 2 .
Relation to the N = 1 string theories The expression for the t 8 t 8 R 4 M-theory action in (8.19) can be translated into the language of string theory by using the dictionary in appendix A, so that the parameters R 10 and R 11 are related to the parameters of the HO theory in D = 9 dimensions via the relations Thus, the contribution to the M-theory effective action in (8.19) translates into the HO effective action The first term in parentheses s proportional to r ho and has a finite ten-dimensional limit as r ho → ∞. This term is closely related to the Eisenstein series E 3 2 (Ω), which is the SL(2, Z)invariant function that arises as the coefficient of the t 8 t 8 R 4 in the ten-dimensional type IIB superstring. Whereas Ω = Ω 1 + iΩ 2 in the type IIB theory, the Ramond-Ramond axial scalar does not arise in the heterotic theories, so Ω 1 = 0. In other words, the first term in parentheses in (8.21) is identified with E 3 2 (g −1 ho ). The second term in parentheses is proportional to the regulated quantityĈ and vanishes as r ho → ∞. We will later argue thatĈ = 4π 2 /3 in order to reproduce the HE one-loop effective action (the term proportional to 1/r ho = r he in (6.2)).
Taking the limit r ho → ∞, it follows that the coefficient of the t 8 t 8 R 4 interaction in the ten-dimensional HO effective action has the form We may now make use of the standard expression for the Fourier modes of the SL(2, Z) Eisenstein series, (8.23) where (see, for example, [31]) the zero mode consists of two power behaved terms, (8.24) and the non-zero modes are proportional to K-Bessel functions, 25) and the divisor sum is defined by σ s (n) = d|n d s .
Using the large-z expansion 1 )), the small g ho (or large Ω 2 ) expansion of (8.23) with s = 3/2 takes the form . (8.26) Substituting this expression in (8.22) gives the perturbative expansion of the t 8 t 8 R 4 interaction in the ten-dimensional HO theory. The following features of the resulting expression are worth noting.
• The expression (8.22) contains two perturbative terms, a tree-level term of order g −2 ho and a one-loop contribution of order g 0 ho . These have the same coefficients as in the type IIB theory. Invariance of the Eisenstein series under the transformation Ω → −1/Ω implies which is a manifestation of HO/type I S-duality. In particular, the perturbative contributions to t 8 t 8 R 4 in the HO theory from spherical and toroidal world-sheets are identical to contributions from spherical and toroidal world-sheets in the type I theory.
• In addition, (8.22) contains an infinite set of non-perturbative terms that appear as Dinstanton contributions proportional to exp(−2π|k|/g ho ) in the HO parameterisation. The instanton action is identified with the action of the euclidean world-line of the m'th Kaluza-Klein mode in the x 11 interval, windingn times around the x 10 circle, where k = mn. In the HE description the contribution of such an object is exp(−2π|k|r he /g he ), which vanishes in the ten-dimensional heterotic limit, r he → ∞. The possible rôle of such D-instantons is intriguing since they do not arise in HO string theory in any obvious manner. However, it is worth recalling that the original argument for the existence of D objects in closed string theories by Shenker [32] was based on a counting argument that applies to any closed-string theory and does not distinguish between heterotic and the type II theories, which allows for the possibility that D-instantons might indeed contribute to heterotic amplitudes. Moreover, the fact that the instantonic contributions might be present in the HO theory, but not the HE theory, is reminiscent of Polchinski's observation [33] concerning the possible rôle of open heterotic strings in the HO theory. We also note the peculiarity that the perturbative expansion around each instanton in (8.26) is an expansion in powers of g ho and not g 2 ho .
• The total contribution to A bulk loop in (8.15) includes the term −I bulk p 11 =0 t 8 t 8R 4 , which subtracts the one-loop contribution from I bulk t 8 t 8R 4 . As a result, there is no one-loop t 8 t 8R 4 contribution in the HO theory and the complete one-loop contribution gives the effective action (8.3) described in section 8.1. This agrees with the expression (6.3). However the −I bulk p 11 =0 contribution is interpreted in the type I theory via HO/type I duality as a disk diagram contribution (of order 1/g I ).
• Since (in the Einstein frame) the coefficient of the t 8 t 8 R 4 contribution contained in (8.22) in the HO theory is identical to the coefficient in the type I theory after the replacement g ho → g −1 I , the type I coefficient also contains effects due to D-instantons. In contrast to the HO description these type I D-instantons are required by symmetry considerations. As argued in [11] the type I theory would have gauge group O(32) were it not for the presence of type I Z 2 D-instantons that break the invariance under the transformation transformations in O(32)/Z 2 to transformations in SO(32) since π 9 (SO(32)) = Z 2 . In our discussion of supergravity in the compactified Hoȓava-Witten background, these non-BPS type I D-instantons have an interpretation, via T-duality, in terms of pairs of euclidean world-lines of type IIA D-particles winding (with opposite orientations) around the x 11 orbifold. 19 Note also that the fact that D-instantons do not contribute to the HE theory in the r he → ∞ limit is consistent with the fact that π 9 (E 8 ) = 0.
• We know that the parity conserving one-loop effective actions of both the heterotic theories are equal, which is consistent with T-duality on the x 10 circle. In this section we have seen how this is obtained in the HO theory by summing boundary and bulk loop contributions to supergravity in the Hoȓava-Witten background. In HE coordinates the one-loop effective action is given by We found in (7.7) that the t 8 (trR 2 ) 2 part of this expression arises from the expansion of a gravity tree diagram and contributed a term to the HE effective action. We also see from (8.21), after replacing r ho by r −1 he , that the bulk loop calculation produces a term of the form r he 2 8 (2π) 6 It follows that in order to ensure that the complete one-loop term in the HE theory has the effective action (8.29) we need to set the renormalised value of theĈ to the valuê • Finally, it is not at all obvious why the expression we have deduced from the bulk supergravity loop should give the exact form of the R 4 interactions. Unlike the type II theories, in which this interaction is 1/2-BPS, in the half-maximally supersymmetric theories the R 4 interactions do not preserve any supersymmetry in any obvious manner. However, the situation is a little murky since this statement also suggests that there should be a three-loop R 4 ultraviolet divergence in four-dimensional N = 4 supergravity. The obvious local counterterm for such a divergence is the volume of superspace, but this was shown to vanish in [35], where an alternative and less obvious counterterm was determined. However, explicit supergravity calculations in [5] demonstrate that this ultraviolet divergence is absent. In addition, as pointed out in [6,36], the absence of a R 4 interaction at two loops in the heterotic string suggests that there is no renormalisation of R 4 beyond one loop.
Even though we have not analysed higher-loop amplitudes in detail we know that these have low energy limits that start with at least two derivatives on R 4 . This adds weight to the suggestion that R 4 is not renormalised beyond one loop.
Since our non-perturbative expression contains the correct perturbative terms for both the heterotic and type I theories it is of interest to further understand the significance of the nonperturbative contributions.

Higher order contributions from other one-loop amplitudes
In the above analysis we have discussed the leading behaviour of boundary and bulk loop amplitudes that contribute, in the limit s, t, u → 0, to low order terms in the low-energy expansion. The expansions of these expressions to higher orders in s, t, u is straightforward since the field theory box diagram has a simple expansion. At least at low orders in this expansion the amplitude can be separated into an analytic part that and the part that contains non-analytic threshold behaviour. The first of these thresholds gives contributions of order t 8F 4 s log s or t 8 t 8 R 4 s log s in the ten-dimensional theory, although the nature of the singularity changes when compactified. For example, in nine dimensions the gauge theory amplitude has a threshold term of the form t 8F 4 s 1 2 . As stressed earlier, only the first few terms in the low-energy expansion are likely to be protected by supersymmetry against receiving corrections beyond those exhibited by the Feynman diagrams we are considering.
In addition to the higher order terms obtained by expanding the loop diagrams we have already considered in powers of s, t and u, there are other one-loop Feynman diagrams that contribute to Yang-Mills amplitudes in supergravity in the Hoȓava-Witten background. The low-energy limit of these diagrams starts with higher powers of the Mandelstam invariants than those we have considered so they do not affect the terms that we expect to be protected by supersymmetry, but the systematics of their contributions may nevertheless be of some interest. The first of these diagrams is illustrated in figure 14. In this contribution to the four gauge particle amplitude one of the propagators in figure 8 is replaced by a gravitational propagator. Two vertices now have an extra power of momentum and the low-energy limit contributes to the d 2 trF 4 interaction in the low-energy expansion of the type I four gauge particle disk amplitude.
Before discussing the details of the diagram, we note that S-duality must act in a non-trivial manner -much as we saw with the t 8 t 8 R 4 interaction, it cannot act term by term. This can be seen from the fact that there is a disk amplitude in the type I theory that gives a contribution of the form ζ(3) d 2 trF 4 /g I . Transforming to the Einstein frame produces no powers of the dilaton and therefore, applying the type I/heterotic S-duality transformation, g I → g −1 ho , results in a HO interaction of order g ho . This does not make sense, which means that the coefficient of this interaction must be a non-trivial duality invariant function of the coupling constant, which gives a HO tree contribution of the same form as the type I disk contribution. Another feature to note for this interaction is that it vanishes when two F s belong to each E 8 subgroup of E 8 × E 8 . This follows since d 2 trF 4 is then a total derivative (equivalently, s + t + u = 0).
As before, we will consider the amplitude compactified on S 1 to nine dimensions in the presence of the Wilson line that breaks the symmetry to SO(16) × SO (16). In this case only adjoint SO(16) gauge particles can propagate in the loop in figure 14 since the external states are in the adjoint representation and the graviton is a SO(16) singlet. A straightforward extension of earlier arguments leads to the expression for the low-energy limit of this amplitude of the form 48 (2π) 10 C(R 10 , R 11 ) d 2 trF 4 . (9.1) Here C(R 10 , R 11 ) is the s, t, u → 0 limit of a ten-dimensional scalar box diagram compactified on the x 10 circle and with the unusual feature that one propagator is of the form (2.18), which involves a sum over the Kaluza-Klein momentum in the x 11 direction, which is m/( 11 R 11 ) in the following expressions (whereas the Kaluza-Klein momentum in the x 10 is n/( 11 R 10 ). Including the volume factor 2π 11 R 10 and ignoring an overall normalisation constant we have C(R 10 , R 11 ) = . We may now analyze the expansion of this expression in the perturbative HO limit in which g ho = R 11 /R 10 → 0 or the perturbative type I limit in which g I = R 10 /R 11 → 0.
As in the previous examples, the ultraviolet divergence of this Feynman diagram is contained in the zero winding,m =n = 0 term. A high momentum cut-off at a momentum scale −1 11 , which regularises this divergence, translates into a cut-off at the lower endpoint of the σ 1 and σ 3 integrations. Substituting in (9.6), the renormalised value of this contribution is whereC is a dimensionless constant. Its value is arbitrary, but since we know that there is no t 8 trF 4 interaction in the HE theory (where the trace is in the fundamental representation of a SO(16) subgroup of E 8 ), the only consistent value isC = 0.
The remaining non-zero winding terms in (9.6) are finite and can be interpreted in the string parameterisation in the following manner.

Tree coefficient in heterotic string theory
The tree-level term arises as the most singular contribution in the small-g he (or small-R 11 /R 10 ) limit, which comes by settingn = 0 in (9.6) C(R 10 , R 11 ) HO tree = π 11/2 2 This has the right form, including the presence of the ζ(3) factor, to correspond to the tree-level terms compactified to nine dimensions in both the heterotic theories (although we have not kept track of rational prefactors and powers of π in the overall coefficient).

Loop contributions to heterotic string theory
In order to extract the loop contributions to C(R 10 , R 11 ) we need to consider the intermediate summation given in (9.5). Firstly consider the m = 0 term. This gives C(R 10 , R 11 ) HO loop = π 5 2 3 11 R 10 R 11 n =0 Further perturbative terms arise from m = 0 terms in (9.5). The integral can be evaluated explicitly for m,n = 0. Converting to the HO parameterisation, these terms give where k = mn and σ −2 (k) = d|k 1/d 2 . In the weakly coupled HO limit the terms in the first line of this equation are contributions of the form expected for two, three and four loop HO string contributions. The last term containing the Bessel function gives rise to instantonic contributions, as we will see shortly.
The complete contribution of perturbative terms to the HO amplitude arising from the diagram in figure 14 therefore has the form where we have used T-duality to relate the HO and HE amplitudes in the last step.

Instanton contribution in the HO theory
The last line of (9.10) gives rise, in the g ho → 0 limit, to an infinite set of instanton contributions to the interaction (9.1) of the form g ho (1 + O(g ho )) , (9.12) where the instanton number is k = |nm|. So, we have an indication that, ss in the case of the t 8 t 8 R 4 interaction discussed in section 8.3, the ten-dimensional HO amplitude contains the contribution of an infinite sequence of D-instantons. As commented earlier, we do not have an explanation of the origin of such instantons witihn the HO string theory.

Tree coefficient in type I string theory
We may now consider the weakly coupled type I limit, in which R 11 >> R 10 . In that case the tree coefficient is obtained by settingm = 0 in (9.6) giving C(R 10 , R 11 ) I tree = π 11/2 4 11 R 10 n =0 The presence of the ζ(3) is again in qualitative agreement with the expression obtained by expanding the type I tree-level amplitude. However, the ratio of the heterotic tree-level coefficient in (9.8) to the type I coefficient in (9.13) does not correspond to the result obtained by explicit calculation in string perturbation theory, which is not surprising since this is not expected to be a protected process.

Two-loop (and absence of one-loop) coefficient in type I string theory
Higher order perturbative terms in the type I theory can be obtained by performing a Poisson summation over the integer m in (9.4) instead of over n. In that case we are led to the expression C(R 10 , R 11 ) = π 5 3 11 n m =0 14) which is analogous to (9.5), but with sums overm and n so the rôles of Kaluza-Klein momentum and winding number reversed. The term in this expression that corresponds to the lowest order perturbative loop term in type I string theory is obtained by setting n = 0 and is a two-loop contribution. In this case, changing integration variables toσ i = (π 2 R 2 11 ) −1 σ i , gives C(R 10 , R 11 ) 16 105 ζ(2) = π 3 r I I g I 16 105 ζ (2) . (9.15) There are undoubtedly higher order perturbative terms arising from terms in the integral (9.14) with n = 0, but since we do not have a useful closed form expression for the integral we have not extracted them. The perturbative contributions in the weakly coupled type I and IA limits limit that we have extracted are summarised in the type I and IA theories by

Instanton contribution in type I string theory
The D-instanton contribution in the type I theory also arises from the terms in (9.14) with n = 0. Although there appears to be no closed-form expression in terms of Bessel functions for this integral, it is easy to make use of a saddle point analysis to find the terms that are exponentially damped when R 10 /R 11 = g I << 1. They give a series of the form where k =mn and c k is a constant that can be determined by standard saddle point methods.

Comments and summary of features of the s t 8 trF 4 calculation
We do not expect that the loop diagram in figure 14 should generate the exact coefficient of the d 2 trF 4 interaction, but we have included it because it does generate a coefficient that demonstrates several of the expected features of string theory. A summary of these is as follows.
• The expression for the coefficient of the s trF 4 interaction contains perturbative terms up to four loops in the HO theory. Strikingly, the one-loop term is absent in the weakly coupled type I limit -this agrees with the explicit string calculation [10]. Since we expect that there are further contributions to this interaction from other sources, it is quite possible that there is an infinite number of contributions in the full perturbation expansion.
• The origin of the type IA two-loop term in string perturbation theory can be traced to the world-sheet diagram shown in figure 15, which is a torus with a boundary localised on a stack of eight-branes coincident with one of the orientifold planes. The four gauge particles are attached to these eight-branes. This contribution is of the same order as the disk world-sheets with two holes or cross-caps inserted, which contribute to g IA t 8 (trF 2 ) 2 as we discussed earlier. • Just as we saw for the t 8 t 8 R 4 interaction that was obtained from the bulk supergravity loop, the coefficient function C(R 10 , R 11 ) contains an infinite sequence of non-perturbative instantonic terms. These make exponentially small contributions of order e −2πk/g ho in perturbative HO theory and e −2πk/g I in perturbative type I theory.

Discussion
The arguments of this paper, based on perturbative supergravity in the Hoȓava-Witten background, M 9 × S 1 × S 1 /Z 2 , lead to an interpretation of a number of features of the low-energy expansion of scattering amplitudes for gauge particles and gravitons in N = 1 string theories in D = 9 and D = 10 dimensions (although we did not discuss mixed gauge/gravity amplitudes). Several of these are known features of superstring perturbation theory that would seem mysterious without such an interpretation and some of them are indications of non-perturbative features, such as non-renormalisation theorems and the contributions of instantons. The interpretation of supergravity Feynman amplitudes in terms of string theory has some unusual features, the most striking of which are summarised here.
• The gravitational propagator in the Hoȓava-Witten bulk depends of the interval length, 11 R 11 . As a result, the gauge boson tree diagrams in section 4 and graviton tree diagrams in section 7, depend on the string coupling constants induced by the orbifold geometry. One consequence is that the low-energy expansion of these tree amplitudes contains a power series in s R 2 11 2 11 = g 2 he s 2 H = s 2 I /r 2 I multiplying (trF 2 ) 2 . This infinite series of terms is therefore interpreted in terms of contributions to loop amplitudes in HE string theory to all orders in the coupling constant g 2 he . In the type I description, this sum of this series reproduces the factor (4.22) in the one-loop cylinder amplitude (figure 4) that comes from the sum of the ground states of closed string winding modes. These modes, which correspond to the Kaluza-Klein modes in the Hoȓava-Witten interval, are unstable. However, the agreement of the supergravity and string amplitudes up to order sg 2 he t 8 (trF 2 ) 2 is in accord with expectations based on supersymmetry and suggests that the effects of this instability enter at higher order in the low-energy expansion.
• The tree amplitudes were generalised in section 4.4 to tree contributions with "iterated" propagators induced by the R 2 and (∂H) 2 interactions localised on the boundaries. This generated another infinite series of powers of s 2 11 /R 11 = s 2 H = g I s 2 I . This corresponds an infinite sequence of tree-level contributions in the HE and HO theories that reproduces the explicit factor in the heterotic tree amplitude displayed in (4.33). In the type I theory interpretation this low-energy expansion of a tree level HO factor is interpreted as an infinite series of higher order terms corresponding to world-sheets with arbitrary numbers of boundaries.
• In section 5 we gave a novel analysis of the ten-dimensional The sum over Kaluza-Klein modes circulating in the loop was transformed into a sum over windings of the loop around the x 10 circle by means of a Poisson summation and the ultraviolet divergence was thereby isolated in the zero winding term. We argued that, after renormalisation, this zero-winding term gives a contribution that has no sensible string theory interpretation and so its renormalised value should be taken to vanish. The nonzero winding terms give a sum of finite contributions proportional to 1/r he = r ho that were interpreted as contributions to the S 1 compactification of the HO theory. Making use of the conspiracy between adjoint and spinor traces in (5.7), which is a special feature of SO(16) these non-zero winding terms contribute to t 8 tr i F 4 i (i = 1, 2), where the traces are in the fundamental representation of either SO (16), which agrees with the result in HO perturbation theory. Strikingly, this would not be the structure of the one-loop amplitude in conventional SO(16) × SO(16) gauge theory, where the traces would be in the adjoint representation of either SO(16) (and the loop would be ultraviolet divergent). The trace has to be in the fundamental representation in order to agree with HO/type I duality, since the type I amplitude arises from a disk diagram with a Chan-Paton factor, which obviously gives single trace in the fundamental representation.
• The arguments concerning the contributions of one-loop four-graviton amplitudes considered in section 8 provided suggestive illustrations of HO/type I duality in the M 9 × S 1 × S 1 /Z 2 background. There were two kinds of loops. The first was one in which the external gravitons coupled to a loop of gauge particles localised on either boundary, which generalised the gauge theory loop amplitude summarised in the previous item and led to a combination of t 8 trR 4 and t 8 (trR 2 ) 2 interactions.
The second contribution came from a loop in which the external gravitons coupled to a loop of gravitons propagating in the bulk. In this case the amplitude involved a sum over the Kaluza-Klein momentum, p 11 , in the eleventh dimension. We argued that the p 11 = 0 contribution to this loop integral added a term to the gauge loop contribution, leading to a total one-loop effective action in the HO theory proportional to t 8 Y (gs) 8 (R). This is the parity conserving partner of the anomaly cancelling ten-form 10 B Y (gs) 8 (R) and is expected to be one-loop exact.
The p 11 = 0 terms in the loop integral generated a t 8 t 8 R 4 interaction, analogous to that of the type II theories, but with a coupling constant dependence described by is an Eisenstein series that has a weak coupling expansion containing just two perturbative terms (tree-level and one-loop) and an infinite series of Z 2 D-instanton contributions The presence of −2ζ(2) subtracts the oneloop term of order g 0 ho from the Eisenstein series, which accounts for the absence of the p 11 = 0 term in this loop contribution. It follows that the t 8 t 8 R 4 interaction in the HO theory only has a perturbative tree-level contribution together with D-instanton contributions. In addition there is a one-loop contribution proportional to t 8 Y (gs) 8 (R, F ), which is the parity-conserving partner of the anomaly-cancelling ten-form (the curvature dependent piece was obtained in (8.3)). After applying HO/type I duality we conclude that the type I theory has contributions to t 8 t 8 R 4 of order 1/g 2 I and g 0 I , corresponding to spherical and toroidal world-sheets.
This analysis hints at the non-renormalisation of the R 4 interactions beyond one loop in any of the N = 1 string theories -a feature that ties in with expectations based on perturbative supergravity and string theory calculations [5,6].
• The presence of D-instanton terms in the expansion of the function E 3 2 (i/g ho ) that multiplies the t 8 t 8 R 4 interaction in (8.22) is crucial in ensuring its invariance under HO/type I duality, g ho → g I = 1/g ho . Although the necessity of type I D-instanton contributions is well documented [11], there is much less evidence that there should be well-defined contributions of HO D-instantons. After all, these objects originate from world-lines of unstable D0-branes in the HE theory wound around the x 10 direction. Such unstable heterotic Dbranes are generally not expected to play a preferred rôle, so while the coupling constant dependence in (8.22) is interesting, it is by no means proven to be exact.
• Similar considerations determine the structure of the parity-violating gauge and gravitational effective interactions that are necessary to ensure the absence of chiral gauge, gravitational and mixed anomalies. These are contained in the ten-form B ∧ X (gs) 8 and were discussed in the context of the HE theory in [2]. The expression for X (gs) 8 is given in (3.1) as a sum of the bulk "Vafa-Witten" term, X (vw) 8 , and (tr i F 2 i −trR 2 /2) 2 terms localised on the two boundaries. In this paper we have seen how the parity-conserving partners of these interactions arise as the sum of the t 8 t 8 R 4 interaction induced by a gravitational loop and the combination t 8 (tr i F 2 i − trR 2 /2) 2 induced by a sum of tree amplitudes with vertices localised in each boundary.
In the HO case the expression for X is given in (B.15)). We found that the corresponding parity-conserving terms of the form t 8 Y (gs) 8 arise in a rather different fashion since they originate from a loop of gauge particles localised in either compactified boundary, coupling to external gauge particles (as in section 5), or gravitons (as in section 8), or a mixture of both gauge particles and gravitons (which we have not explicitly considered). The anomaly-cancelling terms in the HO theory defined by (3.2) are clearly determined by analogous parity-violating loop amplitudes with an external B-field coupling to four gauge or gravitational particles, although we have not explicitly evaluated these amplitudes in this paper. As in the parity-conserving case, such an analysis would explain the occurrence of the fundamental trace in the trF 4 term in (3.1) and (B.15).
More generally, it is obvious that the Feynman diagram approximation is not adequate for understanding the physics of M-theory beyond the low-energy approximation. Even in the context of the low-energy expansion in terms of Feynman diagrams, there are many other sources of contributions to higher derivative interactions of the form d 2k F 4 , d 2k R 4 and mixed gauge/gravity amplitudes, that are not generally expected to be protected against receiving higher order corrections. These include contributions from higher order terms in the low-energy expansion of tree-level and one-loop Feynman amplitudes considered in this paper, as well as from higherloop amplitudes that we have not considered. Clearly, understanding the dynamics of M-theory beyond the first few terms in the low-energy expansion requires a deeper understanding of intrinsically stringy effects that are not probed by the supergravity approximation. It is nevertheless of interest to probe the extent to which low order terms are determined by supersymmetry.

T-duality of type I and type IA theories
When compactified to nine dimensions on a circle of radius r I (in string units) the type I theory is T-dual to the type IA theory compactified on a circle of radius r IA . This is the image of the transformation that relates the type IIA and type IIB closed string theories compactified on a circle. The radii and the coupling constants are related by The details of this duality are again particularly simple in the situation in which the symmetry group is SO(16) × SO (16), which is the case in which the dilaton charge of the orientifold planes is locally screened.

5.
Heterotic Spin(32)/Z 2 /Type I strong coupling duality Heterotic Spin(32)/Z 2 theory and type I SO (32) theory are conjecturd to be related by S-duality [41,42] with the following relationships between the patrameters of the theories According to S-duality the D-string of the type I theory can be identified with fundamental heterotic string [41]. It was noted in [11] that in order for this duality to be satisfied there have to be non-perturbative Z 2 instanton effects in the type I theory that are associated with the breaking of O(32) to SO(32). We will comment on these in an explicit calculation later in this paper.

An important comment:
A term of fixed loop number in the HO theory has a low-energy expansion in powers of 2 H s. Since these powers translate into powers of g I 2 I s in the type I theory this expansion may be reinterpreted as a sum of terms of higher order in type I perturbation theory, with the number of world-sheet boundaries increasing as the power of s increases. However, this identification of individual terms in the heterotic and type I/IA low-energy effective actions only applies to special terms. More generally the strong/weak coupling duality does not allow the identification of specific terms in the expansion of the HO theory with specific terms in the type I theory, as is seen explicitly in the body of the paper.
A.2 Relationships between N = 1 and type II string theories.
We will here review the relations between the parameters in maximally supersymmetric (N = 2) string theories (type II theories) compactified on a circle and the N = 1 theories compactified on S 1 . The objective is to clarify certain factors of √ 2 that arise in passing from the type II theories to the heterotic and type I theories.

Relations between the parameters.
The parameters of the type IIA and IIB theories compactified on S 1 to nine dimensions are related by where r IIA = 1/r IIB and II is the type IIA or IIB string length scale.
The type IA theory is obtained from type IIA by compactifying on S 1 /Z 2 together with a world-sheet orientation reversing operator, Ω. The type I theory is simply obtained by acting on the type IIB theory with Ω and is reduced to nine dimensions by compactifying on S 1 . So the relation (A.6) between type II theories becomes a relation between type IA and IB if the S 1 is replaced by the orbifold S 1 /Z 2 on the left-hand side of the equation, which halves the volume of the compact direction but the right-hand side is unaltered. 20 .
In that case the above relation is replaced by (A.7) The relation between the type I and type IA theories takes the standard form if we define and equate the type I and type II string lengths so that can then be written as 2. Dp-brane tension in type II and type I theories.
The tension of a Dp brane in the type II theory is The D-brane tension in the unoriented theory is smaller by a factor of 1/ √ 2 21 and hence the tension in type I theory is From the relation between the type IIB and type I coupling constants in (A.8a), it follows that the tension in the type I theory is which is consistent with the claim on page 151 of [43]. 20 See the comment below eqn 13.3.30 in page 151 of [43] 21 See footnote 8 in page 21 of [44] 3. Relation between orientable and unorientable closed string loop amplitudes.
An important point of relevance to the interpretation of the graviton loop calculations in section 8 involves the relationship of the type II one-loop amplitude to that of the type I theory. The type II loop amplitudes are defined on orientable world-sheets whileΩ is the orientation reversing operator. Hence, the orientable part of the n-loop diagram in type I theory comes with a factor of (g IIB ) 2(n−1) 2 n , (A.14) using the relation between the type IIB coupling and type I coupling (A.8a) this is simply a factor of 1 2 (g I ) 2(n−1) .
This is relevant to the one-loop calculation in section 8.3, where we suggest that the ratio of tree-level to one-loop amplitudes arising from I bulk (that enters in (8.15)) in the HO theory.is the same as in the type I theory. The factor of 1 2 in (A.15), together with the fact that the closed-string tree-level amplitude (given by four vertex operators attached to a spherical world-sheet) is proportional to 1/g 2 IIB = 1/(2g 2 I ) from (A.8a), is in accord with this suggestion.

Type I T-duality and D-particles in type IA theory
Recall that a D0-brane of the type IA theory moving in the bulk (i.e., in the fundamental domain, 0 < x 11 < π 11 R 11 ) is identical to the D0-brane of the type IIA theory and has a mass This is the description in the downstairs formalism in which x 11 is restricted to an interval with two boundaries. In the upstairs formalism, where x 11 spans the circle, 0 ≤ x 11 ≤ 2π 11 R 11 , the fields are subject to the Z 2 orbifold condition (so that for a scalar field, Φ(x 11 ) = Φ(−x 11 )). In this formalism bulk D0-branes always come in pairs comprising a D0-brane at 0 < x 11 < π 11 R 11 with mass T 0 /(2g IA ) and its mirror image at −x 11 with the same mass. The mass of the pair agrees with the mass of the bulk D0-brane in the downstairs description.
A type IA D0-brane can be its own mirror image if it is "stuck" to either fixed point of the orbifold (x 11 = 0 or x 11 = π 11 R 11 ) and cannot move in the bulk. The mass of such a stuck type IA D0-brane is T-duality along x 11 identifies such a D0-brane with a type I D1-brane wrapping once around the circle of radius I r I . The D1-brane of the type I theory has a tension T 1 2g I , (A. 18) and from (A.11) it is easy to see that its mass agrees with that of the stuck type IA D0-brane in (A.17).

A.3 Relationship between string theory parameters and M-theory parameters
We will here give a brief summary of the relationships between the parameters R 11 and R 10 of the Hoȓava-Witten geometry on a circle (M 9 × S 1 × S 1 /Z 2 ) and the parameters of the various N = 1 string theories compactified to nine dimensions. For each of these theories these parameters consist of the coupling constant and radius of the tenth dimension: (g he , r he ), (g ho , r ho ), (g IA , r IA ), (g I , r I ), for the HE, HO, type IA and type I theories, respectively. Detailed arguments for these relationships can be found in [1,2]. The Hoȓava-Witten geometry is obtained from eleven-dimensional Minkowski space by compactifying on the orbifold of a circle in the eleventh direction, where the generator of the orbifold group acts on the eleventh dimension x 11 by reflection x 11 → −x 11 , as well as acting on the three-form field of eleven-dimensional supergravity, C → −C, i.e, the three form field C is odd under parity reflection Only the components C 11µν are even under reflection and hence survive the Z 2 projection. Similarly the h 11µ components of the graviton are odd under reflection and hence projected out. The gauge fields are vector fields that propagate on the boundary of the space-time.
The radius of the orbifold circle and the other circle, measured in 11-dimensional Planck units, are R 11 and R 10 , respectively 22 . The physical length of the interval in the 11 th direction is then given by L = π R 11 11 .
(A. 19) The metric on the cylinder is given by V Ω The following summarises the relationships between the M-theory pararmeters, R 10 and R 11 and the string theory parameters for each of the N = 1 string theories. It also summarises the relations between the eleven-dimensional Planck length, 11 and the string length in each of the nine-dimensional string theories.
• Heterotic E 8 × E 8 theory The ten-dimensional heterotic E 8 × E 8 theory has a coupling constant that is expressed as while the radius of the spatial circular dimension is given in string units by A.4 Particle states in nine dimensions.
We will here briefly summarise the spectrum of particle states that arise in the M-theory orbifold M 9 × S 1 × S 1 /Z 2 and the corresponding string theories, compactified on S 1 to nine dimensions, as described in [2]. Although the M2-brane states do not enter into the amplitude calculations in the body of this paper, the particle states that arise from wrapping it on S 1 /Z 2 × S 1 enter into a discussion of the multiplets of states in nine-dimensions. These particle states may be obtained starting from the BPS states in the maximally supersymmetric theory obtained from M-theory on M 9 × S 1 × S 1 . Following [2] (with a slight change of notation) the masses of these states are given by where m, n, w ∈ Z. These are the Kaluza-Klein modes in the x 11 and x 10 directions, with charges m and n, respectively, together with the wrapped M2-brane states with wrapping number w. The states of interest to us are those that arise in the M 9 × S 1 /Z 2 × S 1 compactification of M-theory, which must be invariant under the action of the Z 2 orbifold group. This identifies x 11 with −x 11 and so its action on the states is |m, n, → ±| − m, n, . The Kaluza-Klein charge We will illustrate these in subsection B.1 by reviewing the low lying terms in the low-energy expansion of the heterotic and type I SO (32) theories that contribute to on-shell three-point amplitudes. In subsections B.2 and B.5 we will review some notation relating to parity conserving and parity violating terms that are related by supersymmetry and enter the effective action with up to five external on-shell particles.
Here the three-form field strength for the Ramond-Ramond potential, C 2 µν , again includes the presence of the Yang-Mills and Lorenz Chern-Simons terms, and is given bỹ The equivalence of the type I and HO actions is manifest with the identifications together with the relation between the string length scales in the two theories in (A.22).
The corresponding actions for the HE and IA theories have analogous structure.

B.2 Supergravity in the HW background and the Feynman rules
In order to express the Feynman rules in a unified manner in the main text we will express the eleven-dimensional gravitational constant and the ten-dimensional gauge coupling in terms of the eleven-dimensional Planck length, 11 , given in (2.11).
The Feynman rules that follow from this action have the following general features.
• Each vertex coupling three gauge particles contains a single derivative and contributes a factor of 1/ 6 11 .
• Each vertex coupling a pair of gauge particles in a boundary to a gravitational particle (the graviton or the antisymmetric three-form, C, which couples via the Lorentz Chern-Simons term) contains two derivatives and again contributes a factor of 1/ 6 11 .
• Each supergravity bulk interaction vertex is quadratic in derivatives and contributes a factor of 1/ 9 11 .
• Gravitational particles also couple via the Lorentz Chern-Simons term, is localised on a boundary and has four derivatives and contributes a factor of 1/ 6 11 .
• The graviton propagator has a factor of 9 11 while each gauge propagator contributes a factor of 6 11 .
• The expression for an amplitude in Minkowski space-time includes an implicit product of delta functions for momentum conservation in each direction. When a dimension is compactified on a circle of radius R 11 so the conjugate momentum is quantised in units of the inverse radius, the dleta function becomes a Kronecker delta conserving the integer Kaluza-Klein charges multiplied by the volume factor 2π 11 R. As discussed in the paragraph after (4.11) our expressions include this volume factor, but suppress the discrete momentum Kronecker delta.

B.3 Some higher order interactions
In addition to the terms in the actions (B.1), (B.3) and the corresponding HE and IA versions, we will encounter a number of interactions that arise in higher-point on-shell gauge particle and graviton amplitudes in the main part of this paper. Certain of these are parity-violating terms arise that are crucial for understanding the cancellation of gauge and gravitational anomalies. These are components of a D = 10, N = 1 supersymmetry multiplet that also contains analogous parity-conserving terms. These terms arise art one loop in the heterotic theories (so they are independent of the dilaton) and are protected from renormalisation beyond one loop. It is convenient to introduce a notation that highlights this relationship.

B.4 The gauge sector
The eighth rank tensor t 8 is defined by its contractions with the gauge field strength, F µν , which is a matrix in some representation of E 8 × E 8 or SO(32), In order to distinguish the single and double traces on the group theory indices we will use the notation t 8 TrF 4 and t 8 (TrF 2 ) 2 , where the capital Tr indicates that trace is in the adjoint representation. We will use the lower case tr symbol to indicate a trace in the fundamental representation, where this is appropriate.

B.5 The gravitational sector
We will again introduce a notation that emphasises the relationship between terms in the tendimensional effective action that are integrals of ten-forms with analogous scalar expressions. For example, the Vafa-Witten term in the type IIA theory will be denoted B Y (R) is an eight-form (that is inherited from a ten-dimensional characteristic class). The eleven-dimensional version of this term [46] is the eleven-form, C ∧ X (vw) 8 (R). We have introduced the notation X The third line follows from the earlier definition of t 8 and uses the notation in which the curvature is viewed as a matrix in the fundamental representation of the tangent space Lorentz group, SO(9, 1). As with the corresponding term in the gauge sector, the parity-violating term is replaced by the parity-conserving term simply by exchanging a factor of B for a factor of t 8 .
The same combination of four powers of the curvature, Y (vw) 8 (R), enters the N = 1 theories as the N = 2 theories. In addition, a different combination involving the fourth power of curvatures and Yang-Mills field strengths arises in the N = 1 D = 10 theories, which is a key ingredient necessary for the absence of anomalies. The parity-violating piece is the Green-Schwarz ten-form given by B Y (this value differs from the normalisation in [2] because of the different definition of the anomaly cancelling term, as stressed in footnote 8).
Again the low-energy expansion involves a parity-conserving partner of this ten-form, which can be written as t 8 Y (gs) 8 (R, F ), which includes the t 8 trF 4 term that follows from (B.6).
It is notable that the ratio of the coefficient of the trR 4 term to that of the (trR 2 ) 2 term in (B.13) has the opposite sign to the ratio of these coefficients in (B.15), so that The (trR 2 ) 2 /2 term is a boundary contribution that was explained in the context of M-theory in the Hoȓava-Witten background in [2].

R 4 superinvariants
It is useful for the discussion in sections 7 and 8 to identify which combinations of the above R 4 terms are bosonic components of superinvariants. We will here summarise the discussion of Tseytlin in [8], adapted to our present conventions. We will also make use of the detailed analysis in [47,48].

Type II invariants
The type II theories have N = 2 supersymmetry. The following combination of bosonic terms involving the Riemann curvature is a superinvariant that enters into the tree-level effective action of both the IIA and IIB theories. The two ten-dimensional epsilon tensors contract into the sixteen indices of R 4 leaving two pairs to contract into each other. The eight-dimensional analogue of the 10 10 R 4 term is proportional to the Euler invariant.
At one loop the type IIA effective action receives an extra contribution. This is proportional to the superinvariant, I 2 defined by I 2 = − 1 8 10 10 R 4 + 6 10 BY (R) is the odd-parity Vafa-Witten term, which is here seen to be related by supersymmetry to 10 10 R 4 . The R 4 terms in the type IIA effective action only arise at tree-level and one loop and can be summarised by an effective action proportional to where II is the type IIA or IIB string length scale.
It is notable that in the type IIA theory the combination of t 8 t 8 R 4 and 10 10 R 4 arises at one loop with the opposite relative sign to the tree-level combination. The one-loop odd parity Vafa-Witten term in the type IIA theory is protected against renormalisation at higher loops. It follows that 10 10 R 4 is also protected against higher loop corrections. In the type IIA theory these tree-level and one-loop contributions are the only contributions to 10 10 R 4 .
In the type IIB theory there is no Vafa-Witten term and the only invariant containing R 4 is J 0 . In this case the relative signs of the t 8 t 8 R 4 and 10 10 R 4 terms are the same in both the tree-level and one-loop terms. There is no reason to expect J 0 to be protected against getting higher loop or D-instanton corrections, and in fact the dilaton dependence of the coefficient of J 0 enters as a modular invariant function of the complex scalar, Ω. The type IIB theory effective R 4 action is proportional to where E s (Ω) is a non-holomorphic Eisenstein series of weight s that is modular function of the complex scalar Ω = C (0) +i/g IIB and is discussed in section 6. The coupling constant dependence of E 3 2 (Ω) shows that that this t 8 t 8 R 4 interaction has contributions from tree level and one loop in string perturbation theory and from an infinite set of D-instantons.

Heterotic and type I invariants
The heterotic or type I effective actions contain N = 1 superinvariants. These can be chosen to be J 0 (the N = 2 invariant defined in (B.17), and X 1 and X 2 , defined as follows The combination that contains the N = 1 odd-parity anomaly-cancelling term is the N = 1 invariant

C Relations between traces
In the body of the paper we make use of a number of well-known identities between traces of matrices in various representations of SO(N ) and E 8 , which we summarise in this appendix. Note that the ratio of coefficients is independent of M , while the overall normalization is proportional to 2 M . The same relations also hold for the spinor representation of SO(2M − 1).

C.2 E 8 traces
The adjoint representation of E 8 is 248 dimensional and it is given by the sum of the adjoint and spinor representations of SO (16) 248 = 120 + 128 . (C. 3) The

D Loop amplitude in compactified SO(32) gauge theory
We will here consider the one-loop amplitude in the SO(32) gauge theory with gauge group broken to SO(16) × SO(16) by the Wilson line in (4.10). This is analogous to the loop amplitude in the E 8 gauge theory considered in section 5.1. However, as emphasised earlier, this loop contribution does not arise in supergravity in the Hoȓava-Witten background so it is not directly relevant to the bulk of this paper, but the structure of the following argument complements that of the E 8 case.
We will again consider the loop where all the external states are massless in nine dimension and are in the adjoint representation of the SO(16)×SO(16) gauge group. The states circulating in the loop can either be SO(16) × SO(16) adjoint states with masses given by (4.7), or the SO(16) × SO(16) bi-fundamental states with masses given by (4.11).
The amplitude is the sum of contributions of the circulating massless gauge states and their massive Kaluza-Klein recurrences in the adjoint of SO(16) × SO (16) and of the tower of massive bi-fundamental states propagating around the loop, so that The other piece of the four gauge boson loop amplitude, in which SO(16) × SO(16) bifundamental states are circulating, has the form A bif un = 2 3(2π) 10 t 8F 4 C bif un I bif un (s, t, u; r ho ) , (D. 5) where C bif un is the colour factor for the loop of bi-fundamental states and is given by tr i (T a 1 T a 2 T a 3 T a 4 ) + tr 1 (T a 1 T a 2 )tr 2 (T a 3 T a 4 ) . (D.6) The low-energy limit is obtained by setting s, t, u = 0 in the dynamical factor I bif un , which leads to where, as before we have used a cut-off in 11 units, and we again need to set the arbitrary renormalised coefficient of the zero winding number term to zero (Ĉ 2 = 0) since it multiplies an unphysical power of the string coupling. The total amplitude is given by adding (D.4) and (D.7). We see that setting N = 16 in the effective action gives and so the double-trace amplitude survives in the ten-dimensional limit of the HE theory (r he → ∞). The result is identical to the expression we obtained earlier by expanding tree amplitudes in (4.28), and agrees with the direct evaluation of the low-energy limit of the HE loop amplitude in string perturbation theory. As noted above, the renormalisation procedure used to obtain the values of the zero winding terms in (D.4) and (D.7), which are proportional to r ho , is sensitive to our choice of renormalisation procedure. This is not determined by present considerations since the SO(32) gauge theory does not originate from a local action analogous to that of supergravity in the Hoȓava-Witten background. With the choice of cut-off in M-theory Planck units used in (D.4) and (D.7) we are led to the expression (D.8), which vanishes in the limit r ho → ∞. Therefore the ten-dimensional HO expression is not apparent from this perspective, although it was determined from the E 8 gauge theory loop as the r ho → ∞ limit of (5.13) in section 5.1.