Vacuum Constraints for Realistic Strongly Coupled Heterotic M-Theories

The compactification from the 11-dimensional Horava-Witten orbifold to 5-dimensional heterotic M-theory on a Schoen Calabi-Yau threefold is reviewed, as is the specific $SU(4)$ vector bundle leading to the"heterotic standard model"in the observable sector. A generic formalism for a consistent hidden sector gauge bundle, within the context of strongly coupled heterotic M-theory, is presented. Anomaly cancellation and the associated bulk space 5-branes are discussed in this context. The further compactification to a 4-dimensional effective field theory on a linearized BPS double domain wall is then presented to order $\kappa_{11}^{4/3}$. Specifically, the generic constraints required for anomaly cancellation and by the linearized domain wall solution, the constraints imposed by the necessity for positive, perturbative squared gauge couplings to this order and the restrictions on the $D$-terms for preserving or spontaneously breaking $N=1$ supersymmetry are presented.


Contents 1 Introduction
One of the major prerogatives of the Large Hadron Collider (LHC) at CERN is to search for low-energy N = 1 supersymmetry. It has long been known that specific vacua of both the weakly coupled [1,2] and strongly coupled [3][4][5] E 8 × E 8 heterotic superstring can produce effective theories with at least a quasi-realistic particle spectrum exhibiting N = 1 supersymmetry [6][7][8][9][10]. It has also been shown that the various moduli associated with these theories can, in principle, be stabilized [11]. It is of considerable interest, therefore, to examine these low-energy theories in detail and to confront them, and their predictions, with present and up-coming CERN data.
There are several important criterion that such theories should possess. First, they must exhibit a low-energy spectrum in the observable sector that is not immediately in conflict with known phenomenology. In this paper, we will choose the so-called "heterotic standard model" introduced in [12][13][14][15][16]. The observable sector of this theory, associated with the first E 8 gauge factor, contains exactly the matter spectrum of the minimal supersymmetric standard model (MSSM) augmented by three right-handed neutrino chiral superfields, one per family, and a single pair of Higgs-Higgs conjugate superfields. There are no vector-like pairs of superfields or exotic matter of any kind. The low-energy gauge group of this theory is the SU (3) C × SU (2) L × U (1) Y of the standard model enhanced by a single additional gauged U (1) symmetry, which can be associated with B − L. Interestingly, the observable sector of this model-derived same form as in the weakly coupled string with the weak coupling constants replaced by a moduli-dependent expansion parameter. The constraints required for there to be positive squared gauge coupling parameters, as well as the constraints so that the low energy theory be either N = 1 supersymmetric or to exhibit spontaneously broken N = 1 supersymmetry, are then specified. Finally, we consider a specific set of vacua for which the hidden sector vector bundle is restricted to be the Whitney sum of one non-Abelian SU (N ) bundle with a single line bundle, while allowing only one fivebrane in the bulk space. Under these circumstances, the constraint equations greatly simplify and are explicitly presented. We demonstrate how the parameters associated with these bundles are computed for a specific choice of these objects. 2 The Compactification Vacuum N = 1 supersymmetric heterotic M-theory on four-dimensional Minkowski space M 4 is obtained from eleven-dimensional Horava-Witten theory via two sequential dimensional reductions: First with respect to a Calabi-Yau threefold X whose radius is assumed to be smaller than that of the S 1 /Z 2 orbifold and second on a "linearized" BPS double domain wall solution of the effective five-dimensional theory. Let us present the relevant information for each of these within the context of the heterotic standard model.

The Calabi-Yau Threefold
The Calabi-Yau manifold X is chosen to be a torus-fibered threefold with fundamental group π 1 (X) = Z 3 × Z 3 . Specifically, it is a fiber product of two rational elliptic dP 9 surfaces, that is, a self-mirror Schoen threefold [13,59], quotiented with respect to a freely acting Z 3 × Z 3 isometry. Its Hodge data is h 1,1 = h 1,2 = 3 and, hence, there are three Kähler and three complex structure moduli. The complex structure moduli will play no role in the present paper. Relevant here is the degree-two Dolbeault cohomology group H 1,1 X, C = span C {ω 1 , ω 2 , ω 3 } (2.1) where ω i = ω iab are harmonic (1, 1)-forms on X with the property Defining the intersection numbers as where v is a reference volume of dimension (length) 6 , it follows from (2.2) that The (i, j)-th entry in the matrix corresponds to the triplet (d ijk ) k=1,2,3 . Our analysis will require the Chern classes of the tangent bundle T X. Noting that the associated structure group is SU (3) ⊂ SO (6), it follows that rank(T X) = 3 and c 1 (T X) = 0. Furthermore, the self-mirror property of this specific threefold implies c 3 (T X) = 0. Finally, we find that (2.5) We will use the fact that if one chooses the generators of SU (3) to be hermitian, then the second Chern class of the tangent bundle can be written as where R is the Lie algebra valued curvature two-form.
Note that H 2,0 = H 0,2 = 0 on a Calabi-Yau threefold. It follows that H 1,1 (X, C) = H 2 (X, R) and, hence, ω i , i = 1, 2, 3 span the real vector space H 2 (X, R). Furthermore, it was shown in [14] that the curve Poincare dual to each two-form ω i is effective. It follows that the Kähler cone is the positive octant The Kähler form, defined to be ω ab = ig ab where g ab is the Calabi-Yau metric, can be any element of K. That is, suppressing the Calabi-Yau indices, the Kähler form can be expanded as ω = a i ω i , a i > 0, i = 1, 2, 3. (2.8) The real, positive coefficients a i are the three (1, 1) Kähler moduli of the Calabi-Yau threefold. Here, and throughout this paper, upper and lower H 1,1 indices are summed unless otherwise stated. The dimensionless volume modulus is defined by and, hence, the dimensionful Calabi-Yau volume is V = vV . Using the definition of the Kähler form and (2.3), V can be written as It is useful to express the three (1, 1) moduli in terms of V and two additional independent moduli. This can be accomplished by defining the scaled shape moduli It follows from (2.10) that they satisfy the constraint and, hence, represent only two degrees of freedom. Finally, note that all moduli defined thus far, that is, the a i , V and b i are functions of the five coordinates x α , α = 0, . . . , 3, 11 of M 4 × S 1 /Z 2 , where x 11 ∈ [0, πρ].

The Observable Sector Gauge Bundle
On the observable orbifold plane, the vector bundle V (1) on X is chosen to be holomorphic with structure group SU (4) ⊂ E 8 , thus breaking (10). (2.13) To preserve N = 1 supersymmmetry in four-dimensions, V (1) must be both slopestable and have vanishing slope [14,15]. In the context of this paper, these constraints are most easily examined in the d = 4 effective theory and, hence, will be discussed in Section 3 below. Finally, when two flat Wilson lines are turned on, each generating a different Z 3 factor of the Z 3 × Z 3 holonomy of X, the observable gauge group can be further broken to * (2.14) Our analysis will require the Chern classes of V (1) . Since the structure group is SU (4), it follows immediately that rank(V (1) ) = 4 and c 1 (V (1) ) = 0. The heterotic standard model is constructed so as to have the observed three chiral families of quarks/leptons and, hence, V (1) is constructed so that c 3 (V (1) ) = 3. Finally, we found in [12,14] that (2.15) Here, and below, it will be useful to note the following. Let V be an arbitrary vector bundle on X with structure group G, and F V the associated Lie algebra valued twoform gauge field strength. If the generators of G are chosen to be hermitian, then where ch 2 (V) is the second Chern character of V. Furthermore, we denote by tr G the trace in the fundamental representation of the structure group G of the bundle. When applied to the vector bundle V (1) in the observable sector, it follows from c 1 (V (1) ) = 0 that where F (1) is the gauge field strength for the visible sector bundle V (1) and tr E 8 indicates the trace is over the fundamental 248 representation of E 8 . Note that the conventional normalization of the trace tr E 8 includes a factor of 1 30 , the inverse of the dual Coxeter number of E 8 . We have expressed c 2 (V (1) ) in terms of tr E 8 since the fundamental SU (4) representation must be embedded into the adjoint representation of E 8 in the observable sector.
For the visible sector bundle V (1) with structure group SU (4), the group-theoretic embedding is simply the standard SU (4) ⊂ SU (9) → E 8 . * As discussed in [17,18], the two U(1) factor groups depend on the explicit choice of Wilson lines. For the renormalization group analysis of the low-energy d=4 theory, it is more convenient to choose U (1)T 3R × U (1)B−L. However, since this is not our concern in this paper, we present the more canonical choice U (1)Y × U (1)B−L.

The Hidden Sector Gauge Bundle
On the hidden orbifold plane, we will consider more general vector bundles and group embeddings. Specifically, in this paper, we will restrict any choice of hidden sector bundle to have the generic form of a Whitney sum where V N is a slope-stable, non-Abelian bundle and each L r , r = 1, . . . , R is a holomorphic line bundle with structure group U (1). Note that a subset of hidden sector vector bundles might have no non-Abelian factor at all, being composed entirely of the sum of one or more line bundles. On the other hand, one could choose the hidden sector bundle to be composed entirely of a non-Abelian vector bundle, that is, with no line bundle factors. Should the hidden sector bundle contain a non-Abelian factor, one could generically choose it to possess an arbitrary structure group. However, in this paper, for specificity, we will assume that the structure group of the non-Abelian factor is SU (N ) for some N . The explicit embeddings of the SU (N ) and individual U (1) structure groups into the hidden sector E 8 gauge group will be discussed below. Finally, to preserve N = 1 supersymmmetry in four-dimensions, V (2) , being a Whitney sum of vector bundles, must be poly-stable-generically with vanishing slope (but, importantly, see Section 3.2.1 below). As with the observable sector vector bundle, these constraints are most easily examined in the d = 4 effective theory and, hence, will be discussed in Section 3 below. Let us first examine the non-Abelian factor.
• Hidden Sector SU (N ) Factor Since the structure group of V N is SU (N ), it follows immediately that rank(V N ) = N and c 1 (V N ) = 0. The precise form of the second Chern class depends on the type of V N bundle one chooses. Since this bundle is no longer constrained to give any particular spectrum it, and its associated second Chern class, can be quite general. The generic form for the second Chern class is given by where c ij N are, a priori, arbitrary real coefficients. Finally, note from (2.16) that since Let us now consider the line bundle factors.

• Hidden Sector Line Bundles
Let us briefly review the properties of holomorphic line bundles on our specific geometry. Line bundles are classified by the divisors of X and, hence, equivalently by the elements of the integral cohomology (2.21) It is conventional to denote the line bundle associated with the element aω 1 +bω 2 +cω 3 of H 2 (X, Z) as O X (a, b, c).
Furthermore, in order for these bundles to arise from Z 3 × Z 3 equivariant line bundles on the covering space of X, they must satisfy the additional constraint that Finally, as discussed in [47], for the purposes of constructing a heterotic gauge bundle from O X (a, b, c), (2.23) is the only constraint required on the integers a, b, c. Specifically, it is not necessary to impose that these integers be even for there to exist a spin structure on V (2) . We will choose the Abelian factor of the hidden bundle to be where ( 1 r + 2 r ) mod 3 = 0, r = 1, . . . , R (2.25) for any positive integer R. The structure group is U (1) R , where each U (1) factor has a specific embedding into the hidden sector E 8 gauge group. It follows from the definition that rank(L) = R and that the first Chern class is Note that since L is a sum of holomorphic line bundles, c 2 (L) = c 3 (L) = 0. However, the relevant quantity for the hidden sector vacuum is the second Chern character defined in (2.16). For L this becomes Since c 2 (L r ) = 0, it follows that with Q r the generator of the r-th U (1) factor embedded into the 248 representation of the hidden sector E 8 . Computation of this a coefficient depends on the choice of the hidden sector and will be discussed in more detail in Section 3.3.
The relevant topological object in the analysis of this paper will be the second Chern character of the complete hidden sector bundle (2.30) Using (2.20) and (2.27),(2.28) this becomes with a r given in (2.29). Note from (2.16), the explicit embedding of the structure group of V (2) into E 8 and (2.31) that (2.32)

Bulk Space Five-Branes
In addition to the holomorphic vector bundles on the observable and hidden orbifold planes, the bulk space between these planes can contain five-branes wrapped on twocycles C (n) 2 , n = 1, . . . , N in X. Cohomologically, each such five-brane is described by the (2, 2)-form Poincare dual to C (n) 2 , which we denote by W (n) . Note that to preserves N = 1 supersymmetry in the four-dimensional theory, these curves must be holomorphic and, hence, each W (n) is an effective class.

Anomaly Cancellation
As discussed in [60,61], anomaly cancellation in heterotic M-theory requires that where Note that the indices n = 0 and n = N + 1 denote the observable and hidden sector domain walls respectively, and not the location of a five-brane. Using (2.6), (2.17) and (2.32), the anomaly cancellation condition can be expressed as is the total five-brane class. Condition (2.35) is expressed in terms of four-forms in H 4 (X, R). We find it easier to analyze its consequences by writing it in the dual homology space H 2 (X, R). In this case, the coefficient of the i-th vector in the basis dual to (ω 1 , ω 2 , ω 3 ) is given by wedging each term in (2.35) with ω i and integrating over X. Using (2.5), (2.15) and the intersection numbers (2.3), (2.4) gives it follows that the anomaly condition (2.35) can be expressed as The positivity constraint on W follows from the requirement that it be an effective class to preserve N = 1 supersymmetry. Finally, it is useful to define the charges Note that the anomaly condition (2.33) can now be expressed as

The Linearized Double Domain Wall
The five-dimensional effective theory of heterotic M-theory, obtained by dimensionally reducing Horava-Witten theory on the above Calabi-Yau threefold, admits a BPS double domain wall solution with five-branes in the bulk space [54,56,57,60,62,63]. This solution depends on the previously defined moduli V and b i as well as the a, b functions of the five-dimensional metric all of which are dependent on the five coordinates Denoting the reference radius of S 1 by ρ, then x 11 ∈ [0, πρ]. These moduli can all be expressed in terms of functions f i , i = 1, 2, 3 satisfying the equations where each H i is a linear function of z = x 11 πρ with z ∈ [0, 1] and whose exact form depends on the number and position of five-branes in the bulk space. As a simple, and relevant, example, let us consider the case when there are no five-branes in the vacuum. Then and the charge β (0) i is given in (2.42). The k, k i are independent of z, but otherwise arbitrary functions of the four-dimensional moduli.
We are unable to give an exact analytic solution of (2.45) and (2.46). However, one can obtain an approximate solution by expanding to linear order in S β (0) i z − 1 2 . It is clear from (2.46) that this approximation will be valid under the conditions that for each i = 1, 2, 3. This linearized solution was discussed in detail in [54,56,57]. Here we present only the results required in this paper. For an arbitrary dimensionless function f of the five M 4 × S 1 /Z 2 coordinates, define its average over the S 1 /Z 2 orbifold interval as where ρ is a reference length. Then f 11 is a function of the four coordinates x µ , µ = 0, . . . , 3 of M 4 only. The linearized solution is expressed in terms of orbifold average functions We have defined R 0 2 = b 11 to conform to specific normalization later in the paper. One then finds that the linearized solution specifies that In terms of these averaged moduli, the conditions (2.48) for the validity of the linearized approximation can be written as where we have removed the absolute value of the right-hand side since all elements of d ijk given in (2.4) and each field b i are non-negative. Equivalently, one can write for each i = 1, 2, 3. These conditions are actually much simpler than they first appear. This can be seen by writing out each equation fori = 1, 2, 3 respectively. Using (2.42), we find that Clearly, if the equation for i = 3 is satisfied then equations i = 1, 2 are automatically satisfied as well. Hence, one can replace the constraint (2.53) by the simpler requirement that Assuming that these conditions are fulfilled, the linearized solution for V , b i , a and b can be determined in terms of the orbifold average functions. For example, assuming there are no five-branes in the bulk space, the linearized solution for V is given by (2.58) The linearized expressions for b i and a, b are similar expansions in the moduli dependent quantity ( S i . It follows that another check on the validity of these expansions is that However, it follows from (2.4) for d ijk that where we have used expression (2.12). However, it is clear from (2.61) that and therefore Hence, condition (2.59) is automatically satisfied if constraint (2.52) is. Thus far, we have considered the case when there are no five-branes in the bulk space. Including an arbitrary number of five-branes in the linearized BPS solution is straightforward and was presented in [60,62,63]. Here, it will suffice to generalize the above discussion to the case of one five-brane located at z 1 ∈ [0, 1]. The conditions for the validity of the linear approximation then break into two parts. Written in terms of the averaged moduli, these are Assuming these conditions are satisfied, the linearized solution for V , b i and a, b can be determined in each region. For example, the linearized solution for V is given by It follows that the conditions for the validity of this linearized solution for V are given by When dimensionally reduced on this linearized BPS solution, the four-dimensional functions a i 0 , V 0 , b i 0 and R 0 will become moduli of the d = 4 effective heterotic Mtheory. The geometric role of a i 0 and V 0 , b i 0 will remain the same as above-now, however, for the averaged Calabi-Yau threefold. For example, the dimensionful volume of the averaged Calabi-Yau manifold will be given by vV 0 . The new dimensionless quantity R 0 will be the length modulus of the orbifold. The dimensionful length of S 1 /Z 2 is given by πρ R 0 . Finally, since the remainder of this paper will be within the context of the d = 4 effective theory, we will, for simplicity, drop the subscript "0" on all moduli henceforth.

The d=4 E 8 × E 8 Effective Theory
When d = 5 heterotic M-theory is dimensionally reduced to four dimensions on the linearized BPS double domain wall with five-branes, the result is an N = 1 supersymmetric effective four-dimensional theory with (potentially spontaneously broken) E 8 × E 8 gauge group. The Lagrangian will break into two distinct parts. The first contains terms of order κ

The κ 2/3 11 Lagrangian
This Lagrangian is well-known and was presented in [57]. Here we discuss only those properties required in this paper. In four dimensions, the moduli must be organized into the lowest components of chiral supermultiplets. Here, we need only consider the real part of these components. Additionally, one specifies that these chiral multiplets have canonical Kähler potentials in the effective Lagrangian. The dilaton is simply given by However, neither a i nor b i have canonical kinetic energy. To obtain this, one must define the rescaled moduli where we have used (2.11), and choose the complex Kähler moduli T i so that Denote the real modulus specifying the location of the n-th five-brane in the bulk space by z n = x 11 n πρ where n = 1, . . . , N . As with the Kähler moduli, it is necessary to define the fields These rescaled Z n five-brane moduli have canonical kinetic energy. The gauge group of the d = 4 theory has two E 8 factors, the first associated with the observable sector and the second with the hidden sector. As discussed previously, both vector bundles must be chosen so as to preserve N = 1 supersymmetry in fourdimensions. We now explicitly discuss the conditions under which this will be true. We begin with the observable sector.

• Stability of the Observable Sector Vector Bundle
To preserve N = 1 supersymmmetry in four-dimensions the holomorphic SU (4) vector bundle V (1) associated with the observable E 8 gauge group must be both slopestable and have vanishing slope [64][65][66]. The slope of any bundle or sub-bundle F is defined as where ω is the Kähler form in (2.8)-now, however, written in terms of the a i moduli averaged over S 1 /Z 2 . Since c 1 (V (1) ) = 0, V (1) has vanishing slope. But, is it slopestable? As proven in detail in [15], this will be the case in a subspace of the Kähler cone defined by seven inequalities required for all sub-bundles of V (1) to have negative slope. These can be slightly simplified into the statement that the moduli a i , i = 1, 2, 3 must satisfy at least one of the two inequalities a 1 < a 2 ≤ 5 2 a 1 and a 3 < −(a 1 ) 2 − 3a 1 a 2 + (a 2 ) 2 6a 1 − 6a 2 or 5 2 a 1 < a 2 < 2a 1 and The subspace K s satisfying (3.6) is a full-dimensional subcone of the Kähler cone K defined in (2.7). It is a cone because the inequalities are homogeneous. In other  words, only the angular part of the Kähler moduli are constrained, but not the overall volume. Hence, it is best displayed as a two-dimensional "star map" as seen by an observer at the origin. This is shown in Figure 1. For Kähler moduli restricted to this subcone, the four-dimensional low energy theory in the observable sector is N = 1 supersymmetric.

• Poly-Stability of the Hidden Sector Vector Bundle
To preserve N = 1 supersymmmetry in four-dimensions, the hidden sector vector bundle must satisfy two conditions, First, since it is generically a Whitney sum, the vector bundle must be poly-stable. That is, each factor of the Whitney sum must be slope-stable and, in addition, all factors in the sum must have the same slope. Second, generically, this slope must vanish identically-but with one important caveat discussed in Section 3.3.2. In order to make this more concrete, we now present three non-trivial examples to illustrate the these two conditions. As a first example, let us choose First, in this case, since for a single vector bundle slope-stability implies polystability, one need only check that V N is slope-stable. For example, one could choose V N to be identical to the SU (4) bundle in the observable sector, V (1) , presented above. Note that, since we are restricting all hidden sector non-Abelian bundles to have structure group SU (N ), it follows that µ(V N ) must vanish, thus satisfying the second condition for N = 1 supersymmetry. As with the observable sector bundle SU (4) bundle, stability of a generic non-Abelian vector bundle will only occur within a specific region of Kähler moduli space.
2. V (2) = L: As in the previous case, one need only check that the line bundle L is slopestable, which will imply poly-stability. Fortunately, every line bundle is trivially slope-stable, so any line bundle can be used. It is important to note that the slope of a line bundle which appears as a lone factor in the Whitney sum has, a priori, no further constraints-that is, µ(L) need not vanish. Using (3.5), (2.26) and (2.4), it follows that the slope of an arbitrary line bundle specified by L = O X ( 1 , 2 , 3 ) is given by That is, its value is a highly specific function of the Kähler moduli. We will discuss the requirements that such a line bundle lead to four-dimensional N = 1 supersymmetry in Section 3.2.2 below.
3. V (2) = V N ⊕ L : As specified above, the non-Abelian vector bundle V N must be slope-stable in a region of Kähler moduli space. Furthermore, since we are restricting the structure group in our discussion to be SU (N ), it follows that µ(V N ) = 0. As we just indicated, any line bundle L will be slope-stable everywhere in Kähler moduli space. However, the full Whitney sum V (2) = V N ⊕L will be poly-stableand, hence, preserve N = 1 supersymmetry-if and only if µ(L) = µ(V N ) = 0. That is, because of the existence of a non-Abelian SU (N ) factor, the line bundle L now has the additional constraint that its slope vanish identically. It is clear from (3.7) that this will be the case only in a restricted region of Kähler moduli space. It follows that the full Whitney sum V (2) = V N ⊕ L will only be a viable hidden sector bundle if the region of stability of V N has a non-vanishing intersection with the region where the slope of L vanishes. This is a very nontrivial requirement. To give a concrete example, let us choose is the SU (4) observable sector bundle specified above. Recall that the region of slope-stability of this bundle in Kähler moduli space is delineated by the inequalities in (3.6) and shown in Figure 1. Plotted in 3-dimensions, this region of slope-stability over a limited region of Kähler moduli space is shown in Figure 2(a). Furthermore, let us specify, for example, that L = O X (1, 2, −3). Note that L satisfies condition (2.23), as it must. It follows from (3.7) that the region of moduli space in which µ(L) = 0 is given by the equation Plotted over a limited region of Kähler moduli space in 3-dimensions, the region where µ(L) = 0 is shown in Figure 2(b). Figure 2(c) then shows that these two regions have a substantial overlap in Kähler moduli space. Furthermore, since V N was chosen to be V (1) , it follows that Figure 2(c) also represents the overlap with the stability region of the observable sector vector bundle. We conclude that the specific choice of V (2) = V (1) ⊕ O X (1, 2, −3) is, potentially, a suitable choice for a poly-stable hidden sector vector bundle. These three examples give the rules for constructing specific poly-stable vector bundles. They can easily be generalized to construct generic poly-stable Whitney sum hidden sector vector bundles.

The κ 4/3 11 Lagrangian
The terms in the BPS double domain wall solution proportional to S lead to order κ 4/3 11 additions to the d = 4 Lagrangian. These have several effects. The simplest is that the five-brane location moduli now contribute to the definition of the dilaton, which becomes where the fields t i are defined in (3.2). More profoundly, these κ 4/3 11 terms lead, first, to threshold corrections to the gauge coupling parameters and, second, to additions to the Fayet-Iliopoulos (FI) term associated with any anomalous U (1) factor in the low energy gauge group. Let us analyze these in turn.

Gauge Threshold Corrections
The gauge couplings of the non-anomalous components of the d = 4 gauge group, in both the observable and hidden sectors, have been computed to order κ 4/3 11 in [54]. Written in terms of the fields b i defined in (2.11) and including five-branes in the bulk space, these are given by respectively. The positive definite constant of proportionality is identical for both gauge couplings and is not relevant to the present discussion. It is important to note that the effective parameter of the κ 2/3 11 expansion in (3.10) and (3.11), namely S R V , is identical to 1) the parameter appearing in (2.52) (and its five-brane extension (2.66) and (2.67)) for the validity of the linearized approximation, as well as 2) the κ 2/3 11 expansion parameter for V presented in (2.58) (and its five-brane extension (2.68) and (2.70)). That is, the effective strong coupling parameter of the κ 2/3 11 expansion is given by We point out that this is, up to a constant factor of order one, precisely the strong coupling parameter presented in equation (1.3) of [67]. Recall that n = 0 and n = N + 1 correspond to the observable and hidden sector domain walls-not to five-branes. Therefore, z 0 = 0 and z N +1 = 1. Using (2.34) and (2.41), one can evaluate the β (n) i coefficients in terms of the the a i , i = 1, 2, 3 Kähler moduli defined in (2.8). Rewritting the above expressions in terms of these moduli using (2.6), (2.8), (2.10), (2.11), (2.27), (2.28), as well as redefining the five-brane moduli to be λ n = z n − 1 2 , λ n ∈ − 1 2 , 1 2 , (3.13) we find that and where a r is given in (2.29). The first term on the right-hand side, that is, the volume V defined in (2.10), is the order κ 2/3 11 result. The remaining terms are the κ 4/3 11 M-theory corrections first presented in [54].
Clearly, consistency of the d = 4 effective theory requires both (g (1) ) 2 and (g (2) ) 2 to be positive. It follows that the moduli of the four-dimensional theory are constrained to satisfy respectively. It is of interest to compare the (g (1) ) 2 , (g (2) ) 2 > 0 conditions calculated to order κ 4/3 11 in strongly coupled heterotic M-theory, that is, (3.16) and (3.17), to the one-loop corrected conditions computed in the weakly coupled heterotic string [51]. Assuming the same observable and hidden sector vector bundles used in this paper, we find that the weakly coupled conditions for (g (1) ) 2 , (g (2) ) 2 > 0, derived using equation (3.103) in [51], are identical to (3.16) and (3.17) if one replaces in the weak coupling formulas, where g s and l s = 2π √ α are the weak coupling parameter and the string length respectively and S is defined in (2.47).

Corrections to a Fayet-Iliopoulos Term
In the heterotic standard model vacuum, the observable sector vector bundle V (1) has structure group SU (4). Hence, it does not lead to an anomalous U (1) gauge factor in the observable sector of the low energy theory. However, the hidden sector bundle V (2) introduced above, in addition to a possible non-Abelian bundle V N , consists of a sum of line bundles with the additional structure group U (1) R . Each U (1) factor leads to an anomalous U (1) gauge group in the four-dimensional effective field theory and, hence, an associated D-term. Let L r be any one of the irreducible line bundles of V (2) . The string one-loop corrected Fayet-Iliopoulos (FI) term for L r was computed in [48] within the context of the weakly coupled heterotic string. Comparing various results in the literature, it is straightforward to show that strong coupling results to order κ 4/3 11 can be obtained from string one-loop weak coupling expressions using the same replacement g 2 s l 4 s −→ SR V 1/3 v 2/3 presented in (3.20). Making this substitution, we find that the expression for the FI term associated with L r in strongly coupled heterotic M-theory to order κ 4/3 11 is given by where µ(L r ) is given in (3.5). We note that the κ 2/3 11 part of this expression is identical to that derived in [58]. Inserting (2.34), (2.6), (2.32) and, following the conventions of [48,51], redefining the five-brane moduli as in (3.13), we find that where a s is given in (2.29). The first term on the right-hand side, that is, the slope of L r , is the order κ it follows that for each L r the associated Fayet-Iliopoulos factor F I r in (3.22) can be written as As discussed in [54], the general form of each D-term in the low energy fourdimensional theory is the sum of 1) the moduli dependent FI parameter (3.24) and 2) terms quadratic in the four-dimensional scalar fields charged under the associated U (1) gauge symmetry weighted by their specific charge. For each line bundle L r , r = 1, . . . , R on the Calabi-Yau threefold, there is an anomalous U (1) r symmetry in the four-dimensional low energy theory on the hidden sector. Written in terms of the simplified notation introduced in [56,58], the associated D-term is given by Here G LM is an hermitian metric on the U (1) r reducible space of all charged, mass dimension one, scalar matter fields C L , which block diagonalizes into the allowed irreducible representations for the r-th line bundle. The indices L and M each run over the full reducible representation, breaking into a sum of the indices over each irreducible sector-each such sector with a unique charge Q. The metric G LM is, in general, a complicated function of the Kähler moduli with positive definite eigenvalues. Note from (3.24) that F I r has mass dimension two-consistent with expression (3.26).
As is well-known, a necessary condition for a static vacuum state of the effective fourdimensional theory to be N = 1 supersymmetric is that the D term associated with each line bundle L r must identically vanish. Generically, this will be the case if These C L r scalars break into two distinct types-1) those that transform only under the Abelian group U (1) r and 2) those which, in addition, transform non-trivially under the non-Abelian gauge factor of the hidden sector low energy theory. This second type of scalar field will also appear in the D-term associated with the non-Abelian group-which cannot contain a FI term. Hence, the demand that the vacuum be supersymmetric generically sets their vacuum expectation values to zero. It follow that one can, henceforth, ignore such fields and restrict the scalars in (3.27) to those that transform under the Abelian U (1) r symmetry only.
In the weakly coupled heterotic case discussed in [47], it was assumed, for simplicity, that the vacuum expectation values C L r all vanish, even for the scalars not transforming under the low energy non-Abelian gauge factor. In that case, each D r will vanish if and only if F I r = 0. This restriction puts very strong constraints on the choice of the hidden sector vector bundle. Be that as it may, the assumption that all C L r vanish and that F I r = 0 remains a valid constraint for strongly coupled vacua. However, in the strongly coupled case we are now considering, an alternative set of constraints can be also be adopted. That is, one can assume that the scalar fields that only transform under the low energy U (1) r groups are, in general, non-vanishing and that each D r is set to zero by the associated vacuum expectation values C L r becoming non-zero. For this to be the case, it is essential to specify the hidden sector vector bundle and to compute the pure U (1) r low energy scalar fields C L r and their associated charges Q L r . This is essential because, should the U (1) r charge be positive, then the associated D r term can vanish if and only if F I r > 0. On the other hand, if the associated charge is negative, then D r can vanish if and only if F I r < 0. That is, the condition one needs to impose on the Fayet-Iliopoulos terms will depend on the sign of the charges in the scalar spectrum.

A Specific Class of Examples
The constraint equations listed above are technically rather complicated. Therefore, as we did when discussing poly-stability in subsection 3.1, we now analyze the constraint equations within the context of a specific class of N = 1 supersymmetric hidden sector vector bundles. To do this, one must specify the non-Abelian bundle V N with structure group SU (N ), the number of line bundles L r and their exact embeddings into the hidden E 8 vector bundle. We will, henceforth, consider hidden sector bundles that may, or may not, contain a non-Abelian factor and, for simplicity, are restricted to contain at most a single line bundle In this case, there is only a single a r coefficient-which we denote simply by a. In addition, one must specify the number of five-branes in the bulk space. Again, for simplicity, we assume that there is only one five-brane in this example. It then follows from (2.40), (3.18), (3.19) and (3.24)that the constraints for this restricted class of examples are given by where R is an independent modulus and V satisfies relation (3.25). Note that the expression on the left-hand side of (3.33) is a) > 0 or < 0 if one assumes that some φ α = 0 and that the associated scalar charge q α is positive or negative respectively or b) = 0 if, alternatively, one assumes that all φ α = 0. To proceed, one must specify the the coefficient a, as well as the coefficients c jk N of the second Chern class of V N . We begin with the coefficient a. Recall from (2.29) that a = 1 4 · 30 where Q is the generator of the U (1) structure group of the line bundle. Hence, the value of coefficient a will depend entirely on the explicit embedding of this U (1) into the 248 representation of the hidden sector E 8 . Here, we will present one explicit example of such an embedding, although, as will discussed elsewhere, this specific type of embedding is not unique. First, assume that the hidden sector gauge bundle is the Whitney sum of a non-Abelian bundle V N and a line bundle L. This specifies the exact embedding of U (1) into SU (N + 1). Now choose SU (N + 1) to be a factor of a maximal subgroup of E 8 . The decomposition of the 248 of E 8 with respect to this maximal subgroup, together with (3.36), then determines the generator Q. This is most easily explained by giving a simple explicit example. Let us assume there is no non-Abelian bundle-only a single line bundle. That is, (3.37) The explicit embedding of L into E 8 is chosen as follows. First, recall that is a maximal subgroup. With respect to SU (2) × E 7 , the 248 representation of The generator Q of this embedding of the line bundle can be read off from expression (3.41). Inserting this into (3.34), we find that a = 1. As discussed previously, one may or may not include a non-Abelian factor in the hidden sector vector bundle. If a non-Abelian factor V N is to be included, one must specify it exactly. Generically, there are many possibilities for such a bundle. As an explicit example, let us choose this to be precisely the same SU (4) bundle as in the observable sector described in Subsection 2.1.2. Doing this greatly simplifies the analysis since this V 4 bundle is slope-stable with vanishing slope in the same region of Kähler moduli space as the observable sector bundle V (1) -that is, when the inequalities (3.6) are satisfied. Since N = 4, it follows from (3.43) that coefficient a = 10 (3.44) and, since V 4 is identical to V (1) , it follows from (2.15) that c 2 (V 4 ) = 1 v 2/3 ω 1 ∧ ω 1 + 4 ω 2 ∧ ω 2 + 4 ω 1 ∧ ω 2 . Inserting these coefficients, along with a=10, into (3.30),(3.31),(3.32), (3.33) give the appropriate constraint equations for this class of vacua. As discussed above, if one assumes the VEVs of all scalar fields vanish, then the left-hand side of (3.33) must be zero. However, if not all φ α vanish, then to determine whether the left-hand side of the FI inequality (3.31) should be > 0 or < 0 depends on the sign of the charge of the associated low energy U (1) charged scalars. Since the charge can be different for different choices of the hidden sector bundle, this can only be determined within the context of an explicit example. This will be presented elsewhere. Of course, these constraints have to be solved simultaneously with the condition (3.6) for the slope-stability of both the observable and hidden sector non-Abelian vector bundles; that is a 1 < a 2 ≤ 5 2 a 1 and a 3 < −(a 1 ) 2 − 3a 1 a 2 + (a 2 ) 2 6a 1 − 6a 2 or 5 2 a 1 < a 2 < 2a 1 and 2(a 2 ) 2 − 5(a 1 ) 2 30a 1 − 12a 2 < a 3 < −(a 1 ) 2 − 3a 1 a 2 + (a 2 ) 2 6a 1 − 6a 2 (3.47) Finally, it is essential to implement equations (2.66) and (2.67) for the validity of the linear approximation. These equations depend sensitively on the sign of each component of β where, as defined in (3.13), λ = z 1 − 1 2 .