hep-th/0701025 Perturbative Anti-Brane Potentials in Heterotic M-theory 7 0 James Gray1, Andr´e Lukas2 and Burt Ovrut3 0 2 n 1Institut d’Astrophysique de Paris and APC, Universit´e de Paris 7, a J 98 bis, Bd. Arago 75014, Paris, France 3 2Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 1 Keble Road, Oxford OX1 3NP, UK v 5 3Department of Physics, University of Pennsylvania, 2 Philadelphia, PA 19104–6395, USA 0 1 0 7 0 / Abstract h t We derive the perturbative four-dimensional effective theory describing heterotic M- - p theory with branes and anti-branes in the bulk space. The back-reaction of both the e h branes and anti-branes is explicitly included. To first order in the heterotic ǫ expansion, S : v we find that the forces on branes and anti-branes vanish and that the KKLT procedure i X of simply adding to the supersymmetric theory the probe approximation to the energy r density of the anti-brane reproduces the correct potential. However, there are additional a non-supersymmetric corrections to the gauge-kinetic functions and matter terms. The new correction to the gauge kinetic functions is important in a discussion of modulistabilization. At second order in the ǫ expansion, we find that the forces on the branes and anti-branes S become non-vanishing. These forces are not precisely in the naive form that one may have anticipated and, being second order in the small parameter ǫ , they are relatively weak. S This suggests that moduli stabilization in heterotic models with anti-branes is achievable. 1email: [email protected] 2email: [email protected] 3email: [email protected] 1 Introduction Recentyearshaveseenconsiderableprogressinvariousaspectsofstringphenomenology,includingabet- ter understandingof modulistabilization. By combining a series of different effects, phenomenologically interesting type II string models with all moduli stabilized have been found. The KKLT procedure [1] first shows that the four- dimensional effective theory associated with D-branes admits a completely stable supersymmetric AdS vacuum. Then, an anti D-brane is added to the compactification. This both breaks supersymmetryand raises the AdS vacuum to a dS one with a small cosmological constant. Many of the details of this procedureare still being worked out in the literature; see, for example [2–7]. However, it seems likely that this is a valid way of obtaining stable de Sitter non-supersymmetricvacua within the context of IIB string theory. Heterotic models, on theother hand,offer anumberof advantages interms of particle physicsmodel building. Forexample,modelswithanunderlyingSO(10)GUTsymmetrycanbeconstructedwhereone right handed neutrino per family occurs naturally in the 16 multiplet and gauge unification is generic due to the universal gauge kinetic functions in heterotic theories. Recent progress in the understanding of non-standard embedding models [8–11] and the associated mathematics of vector bundles on Calabi- Yau spaces [12–15] has led to the construction of effective theories close to theMinimal Supersymmetric Standard Model (MSSM), see [16–21]. This has opened up new avenues for heterotic phenomenology. For example, one can proceed to look at more detailed properties of these models such as µ terms [22], Yukawa couplings [23], the number of moduli [24] and so forth. Many other groups are also making great strides in model building in heterotic, see for example [25–28]. The main motivation of this paper is to combine some of the advantages of both approaches, type II and heterotic, by realizing key features of type IIB moduli stabilization within heterotic theories. Related work on moduli stabilization in heterotic can be found in, for example, [29–35]. Specifically, we would like to study heterotic M-theory in the presence of M five-branes and anti M five-branes. The starting point of our analysis is the five-dimensional supersymmetric effective action of heterotic M-theory [36–38], where the M five-branes appear as three-branes. We will first generalize this five- dimensional theory to include anti three-branes. Our main goal is the derivation of the associated perturbativefour-dimensionaleffectiveaction,includingtheeffectsfromback-reactionofboththebranes and the anti-branes. As a first step, we find the five-dimensional non-supersymmetric domain wall in the presence of anti-branes, a generalization of the BPS domain wall vacuum of the supersymmetric theory [36–38]. The four-dimensional effective theory is then obtained as a dimensional reduction on this domain wall. Detailed knowledge of this four-dimensional theory is crucial in order to address a number of im- portant problems in heterotic model building. Most notably these are moduli stabilization and, in particular, the stabilization of anti-branes, obtaining a small positive cosmological constant consistent with observation at the stable minimum and the nature of supersymmetry breaking due to the anti- branes. There are also important applications to cosmology analogous to those which have been seen in the case of moving M5 branes, see for example [39–47]. These problems are the main motivation for our work and they will be explicitly addressed in forthcoming publications [48]. In the present paper, we will concentrate on the more formal aspect of deriving the relevant four-dimensional effective theory at the perturbative level. The relative simplicity of our five-dimensional approach allows us to explicitly include the effects of the anti-brane back-reaction, something that is much more difficult to achieve for the complicated geometries in IIB models with anti-branes. Our results are relevant for generic string- 1 and M-theory model building with anti-branes. Let us summarize the main results of this paper. Starting from five-dimensional heterotic M-theory with three-branes and anti three-branes, we calculate the associated bosonic four-dimensional effective theory, including the effects of the back-reaction of the branes and anti-branes. This is done by dimen- sional reduction on a five-dimensional non-supersymmetric domain wall solution which we explicitly 2 determine. The calculation is performed as a systematic expansion in powers of κ3 , where κ is the 11 11 11-dimensional Planck constant. To zeroth and first order we find the effective action is given by the usual supersymmetric result plus an “uplifting potential” from the anti-branes. This potential is first 2 order in the strong coupling parameter ǫ κ3 of heterotic M-theory and, hence, suppressed. Further- S ∝ 11 more, it depends on the dilaton and the Ka¨hler moduli but is, at this order, independent of the brane position moduli. Hence, to order ǫ , the perturbative force on branes and anti-branes vanishes. This S corroborates and justifies the results of [35], where the possibility of a meta-stable vacuum with a small positive cosmological constant was demonstrated within thecontext of a slope-stable heterotic standard model. We have also been able to reliably calculate some contributions to the four-dimensional effective action at order ǫ 2 κ4/3. In particular, we have calculated the complete correction to the brane po- S ∝ 11 tential atthisorder. Thissecondorderpotential describestheexpected “Coulomb-type” forces between the branes and the anti-branes, but also contains an additional unexpected interaction between these objects. This new term can be attributed to the back-reaction of the anti-branes. It is remarkable that these inter-braneforces areall of (ǫ2) andare, therefore, strongly suppressed. We have also calculated O S the (ǫ ) threshold corrections to the gauge kinetic functions in the presence of anti-branes and find S O that they are non-holomorphic due to the supersymmetry breaking by the anti-branes. Furthermore, they depend explicitly on the brane and anti-brane moduli. Our results have significant implications for heterotic compactifications with anti-branes, in partic- ular for the problem of moduli stabilization in such models. The relative weakness of the perturbative forces between the branes means that it is possible that branes and anti-branes can be stabilized by balancing the anti-brane potential against non-perturbative effects. Indeed, given that an anti-brane is typically repelledfromtheboundariesbymembraneinstanton effects whilebeingattracted topositively charged boundaries by Coulomb forces, it seems likely that stabilization can be achieved. In addition, thenon-holomorphicnatureofthegauge-kinetic function andits dependenceon allbranemodulimeans thatnon-perturbativepotentialsduetogauginocondensationwillbedifferentfromwhatonemighthave naively expected. Non-perturbative potentials and moduli stabilization in the presence of anti-branes will be explicitly studied in separate publications [48]. Theplanofthepaperisasfollows. Inthenextsection, wedescribeourtheoryinfivedimensionsand present the associated five-dimensional action of heterotic M-theory, including anti-branes. Section 3 discusses the various warping effects in this five-dimensional theory within the context of a toy model and explicitly presents the five-dimensional non-supersymmetric domain wall solution. In Section 4, we 2 calculate the four-dimensional bosonic effective theory to first order, that is, to order κ3 , and discuss 11 4 our results. Some results at order κ3 , specifically the complete corrections to the anti-brane potential 11 and the gauge kinetic functions at this order, are presented in Section 5. A simple example of our results is provided in Section 6. We conclude in Section 7. A number of technical Appendices explain the origin of the five-dimensional theory in terms of the underlying 11-dimensional one. In addition, they contain detailed technical results for the five-dimensional gravitational warping which is needed in the reduction to four dimensions. 2 2 The Action The theory we consider is the following. Start with a five-dimensional description of heterotic M- theory [36–38], where six of the dimensions of the underlying Hoˇrava-Witten theory [49] have been compactified on any smooth Calabi-Yau manifold X. We include an arbitrary number of M five-branes which are parallel to the orbifold fixed planes and wrap holomorphic curves in the Calabi-Yau space. In addition,weaddasingleantiMfive-branewhichisassociatedwithananti-holomorphiccurveandisalso taken to beparallel to the orbifold fixed planes. Thetheory is easily generalized to an arbitrary number of anti-branes, but we restrict ourselves to one such object for simplicity. Most of the results which we derive are, in fact, valid for arbitrary numbers of anti-branes, as will be discussed in more detail in the text. M five-branes are chosen parallel to the orbifold planes so as to preserve = 1 supersymmetry in N theeffective theory. Wehavechosenthisconfigurationfortheanti-braneaswellbecauseafteralleffects, both perturbativeand non-perturbative, have been included in thefour-dimensionaleffective theory, we want the vacua obtained to bemaximally symmetric. Choosingthe reduction ansatz to includea slowly varying position modulus for the anti-brane, which describes its displacement from some locus parallel to the fixed planes, ensures that the regime of physical interest is within the regime of validity of the effective theory. We choose the anti-brane to wrap a holomorphic curve for two reasons. First, such a two-cycle is volume minimizing in its homology class and so constitutes a natural choice for a vacuum state. Second, such a choice leads, upon integrating out the Calabi-Yau space, to a supersymmetric theory in five dimensions. The existence of such structure will afford us certain technical advantages in this and future work [48]. Of course, from the perspective of five-dimensional heterotic M-theory, the (anti) M five-branes appear as (anti) three-branes and we will refer to them as such in the following. The general brane configuration is depicted in Fig. 1. The space-time of five-dimensional heterotic M-theory consists of S1/Z ( or, equivalently, an interval) times a four-dimensional space-time with 2 Minkowski signature. Five-dimensional coordinates are denoted by (xα) = (xµ,y) where y [ πρ,πρ] ∈ − isthecoordinatealongtheS1 orbi-circle. Thetwofour-dimensionalorbifoldboundariesarethenlocated at y = 0 and y = πρ. Between those boundaries we have a total of N branes of which N 1 are ± − three-branes andtheremainingoneis theanti three-brane. We willuseindices p,q, = 0,...,N+1to ··· label all these four-dimensional extended objects, where p = 0 and p = N+1 correspond to the orbifold boundaries, p = p¯ corresponds to the anti three-brane and all other values of p refer to three-branes. Also note from Fig. 1 that the region between branep and brane p+1is denoted by (p), a notation that will allow us to easily specify a field configuration in a specific part of the interval. Since the sources on thebranesand boundarieslead tofields which aregenerally notsmooth along theinterval, this notation for the various segments of the interval will be useful. The world-volume coordinates of the p-th brane µ are denoted by σ and its embedding into five-dimensional space-time is given by (p) xµ = σµ , y = y (σµ ). (1) (p) (p) That is, the position of the p-th brane in the orbifold direction1 is determined by y , where p = (p) 1,...,N. Even though the orbifold boundaries are non-dynamical, it is convenient to introduce trivial embeddingcoordinates y = y = 0 and y =y = πρ for them. For ease of notation, we also denote (0) (N+1) the embedding coordinate of the anti three-brane as y¯ = y . We will also frequently use normalized (p¯) 1If we work in the “upstairs” picture, that is, with the full orbi-circle, we will have to include “mirror branes” at y=−y ,where p=1,...,N in order to havea Z symmetric configuration. (p) 2 3 Figure 1: The brane configuration in five-dimensional heterotic M-theory orbifold coordinates z = y/πρ [0,1] and z = y /πρ [0,1], where p = 0,...,N +1, to simplify (p) (p) ∈ ∈ our notation. Finally, weshouldbrieflydiscussthecharges andtensionsoftheorbifoldboundariesandbranes. For this purpose, it is useful to introduce an integral basis Ck, where k,l, = 1,...,h1,1(X), of the second ··· homology of the internal Calabi-Yau space X. Suppose that the p-th M five-brane, where p = 1,...,N, wraps the cycle C(p) given by C(p) = β(p)Ck . (2) k (p) Then the integer coefficients (β ) represent the charge vector of the brane p. The charges on the k orbifold boundaries are determined by the second Chern classes c (X), c (V ) and c (V ) of the 2 2 (0) 2 (N+1) Calabi-Yau spaceX andthetwointernalvector bundlesV andV ontheboundaries,respectively. (0) (N+1) (0) (N+1) More explicitly, we can define the charge vectors (β ) and (β ) of the two boundaries by k k 1 1 c (V ) c (X) = β(0)Ck , c (V ) c (X) = β(N+1)Ck . (3) 2 (0) − 2 2 k 2 (N+1) − 2 2 k Heterotic anomaly cancellation dictates that N+1 (p) β =0 (4) k p=0 X for each k = 1,...,h1,1(X). For the orbifold boundaries and the three-branes the tensions τ(p) are k (p) (p) equal to the charges, so we have τ = β for p = p¯. For the anti three-brane, on the other hand, k k 6 (p¯) (p¯) tensions and charges have opposite signs, so τ = β . It should be noted that the tension of the k − k anti three-brane is positive. In subsequent equations it will often be instructive to single out terms 4 related to the anti three-brane and, when doing so, we will simply write the tension and charge of the anti three-brane as τ¯ τ(p¯) and β¯ β(p¯). It is also useful to introduce the step-functions k ≡ k k ≡ k βˆ (y) = β(p) (5) k k p,y <y X(p) which represent the sum of the charges to the left of a given point y in the interval. 2.1 The Five-Dimensional Theory Our starting point is the action of five-dimensional heterotic M-theory [36–38], obtained from 11- dimensional Hoˇrava-Witten theory [49] by compactifying on a Calabi-Yau manifold X in the presence of both M5 and anti-M5 branes. In this subsection, we will describe the field content of this theory and then present the action itself. Many of the detailed definitions of quantities which appear here can be found in the Appendix. We start by describing the bulk field content, focusing on the bosonic degrees of freedom. In addition to the five-dimensional metric g , the bulk fields consist of a Ka¨hler sector, labeled by indices αβ k,l, = 1,...,h1,1(X) and a complex structure sector, labeled by indices a,b, = 1,...,h2,1(X) ··· ··· or A,B, = 0,...,h2,1(X). In the Ka¨hler sector, we have h1,1(X) Abelian vector fields k with ··· Aα field strengths k which descend from the M-theory three-form, a real scalar field V which measures Fαβ the Calabi-Yau volume and h1,1(X) metric Ka¨hler moduli bk which obey the condition d bibjbk = 6 ijk and measure the relative size of the Calabi-Yau two-cycles. In the complex structure sector, we have h2,1(X) metric complex structure moduli za and 2(h +1) scalar fields ξA and ξ˜ which descend from 2,1 B the M-theory three-form. This three-form also gives rise to a five-dimensional three-form C and its αβγ field strength G . Of these bulk fields, V, za, g , g , bk, C and k are even under the Z action αβγδ µν yy µνy Ay 2 of the S1/Z orbifold, while g , ξA, ξ˜ , k, C are odd. The 11-dimensional origin of these fields is 2 µy B Aµ µνγ explained in Appendix A. The bulk theory is a d = 5, = 1 supergravity and, hence, the various fields above should form N the bosonic parts of five-dimensional supermultiplets. The bosonic field content of the five-dimensional supergravity multiplet consists of the five-dimensional metric g and an Abelian gauge field, which αβ can be identified as the linear combination b k. The remaining h1,1(X) 1 vectors, together with the kAα − h1,1(X) 1 Ka¨hler moduli bk, form the bosonic parts of h1,1(X) 1 vector multiplets. The remaining − − scalar fields, that is, the Calabi-Yau volume modulus V, the dual of the three-form C , the complex αβγ structure moduli za and the axions ξA, ξ˜ account for the bosonic parts of h2,1 +1 hypermultiplets, B each of which contains four real scalars. As usual, we have additional degrees of freedom which live on the orbifold boundaries and branes. On the four-dimensional orbifold boundaries, labeled by p = 0,N +1, we have = 1 gauge theories N with gauge fields A transforming in the adjoint of the gauge groups E and gauge matter (p)µ (p) 8 H ⊆ fields in = 1 chiral multiplets with scalar components CIx. They transform in representations of N (p) which we shall denote by R , with I,J,... labeling the different representations and x,y... the (p) (p)I H states within each representation. More details on the origin and structure of the matter sector can be found in Appendix B. Theworld-volumefieldsassociated withthethree-braneswhichdescendfromwrappinganM5brane onaholomorphic(oranti-holomorphic)curveintheCalabi-Yau spacearethefollowing. Theembedding coordinate (brane position) y together with the world volume scalar s which descends from the (p) (p) 5 two-form on the M five-brane world-volume, pair together to form the bosonic content of an = 1 N chiral multiplet, (y ,s ). In addition, we have = 1 gauge multiplets with the associated field (p) (p) N strengths denoted by Eu . Here u,v = 1,...,g , where g is the genus of the curve C(p) wrapped (p) ··· (p) (p) by the p-th M5 brane. In general, there will be additional chiral multiplets describing the moduli space of the five brane curves and non-Abelian generalizations of the gauge field degrees of freedom when M5 branes are stacked. These are not vital to our discussion and we will not explicitly take them into account. A similar selection of four-dimensional fields appears on the anti-brane world volume. Given this field content, the following is the bosonic part of the five-dimensional action describing Hoˇrava-Witten theory [49] compactified on an arbitrary Calabi-Yau manifold [36–38, 50, 51] in the presence of M5 and anti-M5 branes. 1 1 1 1 1 S = d5x√ g R+ G (b)∂bk∂bl+ G (b) k lαβ + V−2(∂V)2+λ(d bibjbk 6) −2κ2 − 2 4 kl 2 kl FαβF 4 ijk − 5 Z (cid:20) 1 + (z)∂za∂¯z¯b V−1( ˜ M¯ (z) B)([Im(M(z))]−1)AC( ˜α M (z) Dα) 4Ka¯b − XAα− AB Xα XC − CD X 1 + V2G Gαβγδ +m2V−2Gkl(b)βˆ βˆ αβγδ k l 4! (cid:21) 1 2 d k l m+2G ((ξA ˜ ξ˜ A) 2mβˆ k) −2κ2 3 klmA ∧F ∧F ∧ XA− AX − kA 5 Z (cid:18) (cid:19) m 1 d5x δ(y) h V−1bkτ(0)+ Vtr(F2 )+G D CIx DµC¯J (6) − − (0) κ2 k 16πα (0) (0)IJ µ (0) (0)x Z q (cid:20) 5 GUT ∂W ∂W¯ +V−1GIJ (0) (0) +tr(D2 ) (0) ∂CIx ∂C¯J (0) # (0) (0)x m 1 d5x δ(y πρ) h V−1bkτ(N+1)+ Vtr(F2 ) − − − (N+1) κ2 k 16πα (N+1) Z q (cid:20) 5 GUT ∂W ∂W¯ +G D CIx DµC¯J +V−1GIJ (N+1) (N+1) +tr(D2 ) (N+1)IJ µ (N+1) (N+1)x (N+1)∂CIx ∂C¯J (N+1) # (N+1) (N+1)x 1 N 2m(nk τ(p))2 d5x (δ(y y )+δ(y +y )) h mV−1τ(p)bk + (p) k j jµ −2κ25 Z Xp=1 − (p) (p) q− (p)  k V(τl(p)bl) (p)µ (p) +[ImΠ] Eu Ewµν 4mCˆ τ(p)d(nk s ) 2[ReΠ] Eu Ew  (p)uw (p)µν (p) − (p)∧ k (p) (p) − (p)uw (p)∧ (p) i o Let us briefly discuss the various quantities in this action. In the previous section, we defined the charges β(p), the tensions τ(p) and the charge step-functions βˆ . To introduce the remaining objects, k k k we start with the bulk theory, the first part of the above action. Of course, κ is the five-dimensional 5 Planck constant, related to its 11-dimensional counterpart κ by κ2 = κ2 /v where v is the Calabi-Yau 11 5 11 reference volume. The constant m is given by 2π κ 2 m = 11 3 (7) v32 4π (cid:16) (cid:17) 6 and represents a reference mass scale of the Calabi-Yau space. The quantity λ is a Lagrange multiplier enforcing the constraint on the bk moduli. The Ka¨hler and complex structure moduli metrics G and kl are definedin AppendixA. Adefinition of theCalabi-Yau intersection numbersd andthespecial Ka¯b ijk geometry quantity M can also be found in this Appendix. The various bulk form field strengths are AB defined in the usual way as G = dC, k = d k and A = dξA, ˜A = dξ˜A away from the boundaries, F A X X but are subject to boundary source terms specified by the relations (dG) = 4κ2(J(0) δ(y)+J(N+1)δ(y πρ)) (8) yµνγρ − 5 4µνγρ 4µνγρ − (d k) = 4κ2(J(0)kδ(y)+J(N+1)kδ(y πρ)) (9) F yµν − 5 2µν 2µν − (d A d ˜ B) = 4κ2(J(0)δ(y)+J(N+1)δ(y πρ)) (10) X GA− XBZ yµ − 5 1µ 1µ − where 1 (p) J = tr(F F ) (11) 4µνγρ 16πα (p) ∧ (p) µνγρ GUT J(p)k = i Γk (D CIxD C¯J D C¯I D CJx) (12) 2µν − (p)IJ µ (p) ν (p)x− µ (p)x ν (p) I,J X −K e J(p) = λ f(IJK)CIxCJyD CKz (13) 1µ 2V IJK xyz (p) (p) µ (p) I,J,K X for p = 0,N +1. The various matter field objects in these sources are defined in Appendix B. One important observation from these Bianchi identities is that the three-branes and anti three-branes do not contribute any source terms. This fact will be crucial in our later analysis. The second and third parts of the above action are the theories on the two orbifold boundaries respectively. They are written in terms of the matter field Ka¨hler metrics G , the matter field superpotentials W and the D- (p)MN (p) terms D . Definitions for these quantities can also be found in Appendix B. The (reference) gauge (p) coupling constant α is given by α = (4πκ2 )2/3/v. GUT GUT 11 We move on to discuss the three-brane world volume theories, the last part of the above action. The quantities nk = β(p)/ N β(p)2 are a normalized version of the three-brane charges and the axionic (p) k l=1 l currents j are defined by (p)µ P (p) β j = k (d(nk s ) ˆk ) , (14) (p)µ nl β(p) (p) (p) −A(p) µ (p) l where Cˆ and ˆk denote the pull-backs of the bulk forms C and k to the p-th brane. The gauge (p) A(p) A kinetic functions Π of the three-brane gauge fields are defined in Appendix C. (p)uv Finally, we need to mention that the induced metrics h on the orbifold boundaries and branes (p)µν are explicitly given by h = g +g ∂ y +g ∂ y +g ∂ y ∂ y , (15) (p)µν µν µy ν (p) yν µ (p) yy µ (p) ν (p) where the embedding (1) has been used. Recall that the boundaries are non-dynamical with associated static embeddings y = 0 and y = πρ. Hence, the induced boundary metrics h and h are (0) (N+1) (0) (N+1) simply equal to g , the four-dimensional part of the bulk space-time metric. The action described in µν this section must be supplemented by the usual Gibbons-Hawking boundary terms. A careful analysis reveals that form-field boundary terms are not required in this case. 7 Having described the five-dimensional theory, our starting point, we proceed in the next section to discuss the appropriate reduction ansatz in the presence of anti-branes. The dimensional reduction to four-dimensions will be performed in section 4. 3 The Five-Dimensional Domain Wall with Anti-Branes In this section, we illustrate the main features of the five-dimensional reduction ansatz in the context of a simple scalar field toy model. We will then explicitly work out the essential part of this reduction ansatz, the five-dimensional non-supersymmetric domain wall. This is a generalization of the BPS domain wall solution of Ref. [36, 37] and includes the back-reaction effects of the anti three-brane. The key new point for us will be to discover how the back-reaction on the bulk fields due to the presence of the branes and, in particular, the anti-brane is taken into account in the reduction ansatz. While this is technically complicated for five-dimensional heterotic M-theory, the basic ideas can be explained in a simple setting. Before dealing with the full problem, we will, therefore, discuss a scalar field toy model [52] to illustrate the key features involved. The structure of space-time and branes for this model is precisely as described above and illustrated in Fig. 1. The action is given by N S d5x ∂ Φ∂αΦ δ(y)S Φ δ(y πρ)S Φ (δ(y y )+δ(y+y ))S Φ , (16) α (0) (N+1) (p) (p) (p) ∼  − − − − −  Z p=1 X   where S are sources on the boundaries and branes (which can depend on other fields) and Φ is a (p) Z even scalar field. What we want to discuss in this model is the warped background solution which 2 arises due to the presence of the source terms and the four-dimensional effective theory associated with it. To this end, it is useful to split the scalar field as Φ = φ+φ , where φ is a function of the four- 0 0 dimensional coordinates only and is the quantity that will become the modulus associated with this degree of freedom in the four-dimensional effective theory. On the other hand, φ represents a function of all five coordinates and contains the warping of the background due to the presence of sources terms. To uniquely define this splitting of Φ, we require that the orbifold average < φ > of φ vanishes. This condition impliesaspecificchoiceofcoordinates onfieldspaceintheresultingfour-dimensionaleffective theory. This choice is particularly useful in finding a clean form for the resulting action, as we will see explicitly throughout this paper. The field equation for Φ, valid in each bulk region indicated in Fig. 1, then reads 2 φ +2 φ+D2φ = 0. (17) 4 0 4 y In addition, Φ is subject to boundary conditions at the edge of each region due to the presence of the sources. For the two orbifold boundaries, these take the form D φ = S , D φ = +S (18) y y=0 (0) y y=πρ (N+1) | − | while, for the branes, we have −Dyφ|y=y(p)++Dyφ|y=y(p)− = S(p) , (19) where p = 1,...,N. The subscript “y = y +” (“y = y ”) indicates that the relevant quantity (p) (p) − should be evaluated approaching the p-th brane from the right (left). 8 We can now take an average of the equation of motion (17) over the orbifold. Using < φ>= 0 and the “boundary conditions” (18) and (19), we obtain 2 φ + S = 0 . (20) 4 0 (p) p X This relation may then be used to eliminate φ in (17) to obtain an equation purely for the warping 0 2 φ+D2φ = S . (21) 4 y (p) p X To pursue the analysis further, we need to know something about the various approximations, and associated expansions, which are made in deriving four-dimensional heterotic M-theory, some of which have already been implicit in our analysis. Two expansions in particular are of central importance at this point. The first of these is simply the usual expansion in four-dimensional derivatives which is always madeindefiningsuchaneffective theory; inother words,thefour-dimensionalfieldsareassumed to be slowly varying relative to the structure in the internal dimensions. The second expansion which we need is in terms of a small parameter ǫ , which controls the size of the source terms. We will S meet this quantity explicitly soon, so let us just state this to be true for now. The zero mode φ is a 0 quantity independent of the warping and is, therefore, zeroth order in the ǫ expansion. By contrast, S φ is precisely the quantity which presents the warping and so is first order ǫ . Looking at (21) which S determines the warping, we see that the first term is both first order in ǫ and second order in four- S dimensional derivatives whereas the remaining terms are simply first order in ǫ . We may therefore, in S a controlled approximation, ignore the first term in (21). This results in the following equation for the warping D2φ= S . (22) y (p) p X Thus, in the end, we need to solve the system of bulk equations and boundary conditions given by (18), (19) and (22). Before moving on to the full calculation, we qualitatively describe such an analysis in various cases by transferring the insight from the above toy example to heterotic M-theory. We start with heterotic M-theory in the absence of anti-branes and proceed to a discussion of the back-reaction of such objects when they are included in the vacuum. a) Zeroth order in sources When all of the sources are set to zero, the warping equation (22) becomes simply D2φ = 0. Since φ must be continuous around the orbifold this, in combination with y the condition < φ>= 0, results in φ = 0. For five-dimensional heterotic M-theory, this implies that the zeroth order vacuum is simply five-dimensional Minkowski space. b) The standard heterotic vacuum In the case of a heterotic vacuum involving only orbifold fixed planesandthree-branes(butnoantithree-branes),thesourcesS arerepresentedbythetensionterms (p) of thebranesandboundariesandthey obeyaparticularly usefulrelation. Sincetheobjects involved are all BPS, their tensions are equal to their charges. The charges, on the other hand, have to sum to zero as aresultof theheterotic anomaly cancellation condition (4). Thus,wehave for such compactifications that S = 0 and, as in the previous case, the equation for the warping (22) reduces to D2φ = 0. p (p) y The boundary conditions (18), (19) are no longer trivial however. Thus, the only solution for φ is a P 9