Stabilising the supersymmetric Standard Model on the Z_6' orientifold

Four stacks of intersecting supersymmetric fractional D6-branes on the Z_6' orientifold have previously been used to construct consistent models having the spectrum of the supersymmetric Standard Model, including a single pair of Higgs doublets, plus three right-chiral neutrino singlets. However, various moduli, Kahler moduli and complex-structure moduli, twisted and untwisted, remain unfixed. Further, some of the Yukawa couplings needed to generated quark and lepton masses are forbidden by a residual global symmetry of the model. In this paper we study the stabilisation of moduli using background fluxes, and show that the moduli may be stabilised within the Kahler cone. In principle, missing Yukawa couplings may be restored, albeit with a coupling that is suppressed by non-perturbative effects, by the use Euclidean D2-branes that are pointlike in spacetime, i.e. E2-instantons. However, for the models under investigation, we show that this is not possible.


Introduction
The attraction of using intersecting D6-branes in a bottom-up approach to constructing the Standard Model is by now well known [1], and indeed models having just the spectrum of the Standard Model have been constructed [2,3]. The four stacks of D6-branes wrap 3-cycles of an orientifold T 6 /Ω, where the six extra spatial dimensions are assumed to be compactified on a 6-torus T 6 and Ω is the worldsheet parity operator; the use of an orientifold is essential to avoid the appearance of additional vectorlike matter. However, non-supersymmetric intersecting-brane models lead to flavour-changing neutralcurrent (FCNC) processes induced by stringy instantons that can only be suppressed to levels consistent with current bounds by choosing a high string scale, of order 10 4 TeV, which in turn leads to fine tuning problems [4]. It is therefore natural, and in any case of interest in its own right, to construct intersectingbrane models of the supersymmetric Standard Model. A supersymmetric theory is not obliged to have a low string scale, so the instanton-induced FCNC processes may be reduced to rates well below the experimental bounds by choosing a sufficiently high string scale without inducing fine-tuning problems.
To construct a supersymmetric theory [5,6,7,8,9], instead of T 6 , one starts with an orbifold T 6 /P , where P is a point group which acts as an automorphism of the lattice defining T 6 ; this has the added advantage of fixing (some of) the complex structure moduli. An orientifold is then constructed as before by quotienting the orbifold with the action of the world-sheet parity operator Ω. In previous papers [10,11,12] we have studied orientifolds with the point group P = Z ′ 6 , and derived models having the spectrum of the supersymmetric Standard Model plus three right-chiral neutrinos. The 6-torus factorises into three 2-tori T 6 = T 2 1 × T 2 2 × T 2 3 , with T 2 k (k = 1, 2, 3) parametrised by the complex coordinate z k . Then the generator θ of the point group P = Z ′ 6 acts on z k as where This requires that T 2 1 and T 2 2 are SU (3) root lattices, so that θ is an automorphism, and this in turn fixes the complex structure moduli U 1,2 for T 2 1,2 to be U 1 = U 2 = e iπ/3 ≡ α. (Note that the G 2 and SU (3) root lattices are the same, contrary to our previous assertions.) Since θ acts on T 2 3 as a reflection, θz 3 = −z 3 , its lattice is arbitrary. The embedding R of Ω acts antilinearly on all z k and we may choose the phases so that Rz k =z k (k = 1, 2, 3) This too must be an automorphism of the lattice, and this requires the fundamental domain of each torus T 2 k to be in one of two orientations, denoted A and B, relative to the Re z k axis. In the A orientation of T 2 1 the basis vector e 1 = R 1 is real, whereas in the B orientation e 1 = R 1 e −iπ/6 ; for both orientations the second basis vector e 2 = αe 1 . Similarly for the basis vectors e 3 and e 4 of T 2 2 . For T 2 3 , the basis vector e 5 = R 5 is real in both orientations, but the real part of the complex structure U 3 ≡ e 6 /e 5 satisfies Re U 3 = 0 in the A orientation, and Re U 3 = 1/2 in the B orientation. Thus e 6 = iR 5 Im U 3 in A, and e 6 = R 5 (1/2 + iIm U 3 ) in B.
The massive version of the effective supergravity describing compactified type IIA string theory in the presence of background fluxes has action [14,15] S IIA = 1 2κ 2 10 d 10 x √ −g e −2φ [R + 4(∂φ) 2 − 1 2 where 2κ 2 10 = (2π) 7 α ′ 4 is the 10-dimensional Newtonian gravitational constant and µ 6 = (2π) −6 α ′ −7/2 is the unit of D6-brane RR charge. The sum over κ is understood to include all D6-brane stacks, their orientifold images κ ′ , and the O6-brane π O6 with charge −4µ 6 , and N κ is the number of D6-branes in the stack wrapping the 3-cycle κ. The field strengths associated with the Kalb-Ramond field B 2 and the RR fields C 1,3 are where H bg 3 and F bg 4 are background fluxes, and the mass m 0 is the background value of F 0 . The presence of the fluxes generally deforms the original metric. The direct product of the four-dimensional Minkowski space and the compactified (Calabi-Yau) space is replaced by a warped product [16,17] which, as we shall see, introduces a potential for (some of) the moduli. dC 1 is the Hodge dual of F 8 , the field strength associated with the 7-form RR gauge field C 7 . One effect of the m 0 term is that a piece of the F 2 ∧ * F 2 term in (9) couples H bg 3 to C 7 so that this term also contributes to the C 7 tadpole equation. The requirement that there are no RR C 7 tadpoles is therefore generalised [18] to where Π m 0 H bg 3 is the 3-cycle of which m 0 H bg 3 is the Poincaré dual. In the models presented in [11,12] tadpole cancellation requires that Π m 0 H bg 3 , and hence m 0 H bg 3 , is non-zero. In general, we must also address the question of whether the total K-theory charge [19] is zero. The presence of K-theory charge may be exhibited by the introduction of a "probe" Sp(2) ≃ SU (2) brane π probe . For a consistent theory we require that there are an even number of chiral fermions in the fundamental representation of Sp (2). Thus the additional constraint [20,8] is that κ N κ κ ∩ π probe = 0 mod 2 (15) where the sum is over all D6-branes, but not including their orientifold images, and π probe is any 3-cycle that is its own orientifold image π probe = π probe ′ (16) although this may be too strong a constraint. It follows that [20,8] π probe = 1 2 Π bulk probe + Π ex probe (17) where, on the AAA lattice, The two independent (supersymmetric) possibilities are with t 0 , t 2 = ±1. In our models, in particular in the model deriving from the fourth entry in Table 1 of reference [12], the contributions to the left-hand side of (15) from the stacks b and c are necessarily even, the former because N b = 2, and the latter because it is zero. For the remaining stacks, we find that a ∩ π probe = −d ∩ π probe for both cases (20) and (21) above. Thus the K-theory constraint (15) is satisfied. The same is true of the other models on the AAA lattice, as well as for the BAA cases too. All of the models that we have considered have the attractive feature that they have the spectrum of the supersymmetric Standard Model, including a single pair of Higgs doublets, plus three right-chiral neutrino singlets. In the presence of these suitably chosen background fields m 0 and H bg 3 the models are consistent string theory vacua. Nevertheless, despite the attraction of having "realistic" spectra, they are deficient. First, there are many unfixed moduli, Kähler moduli, complex structure moduli, axions and the dilaton, all of which have unobserved massless quanta unless they are stabilised. We shall see later that the non-zero background flux H bg 3 required by tadpole cancellation stabilises one linear combination of the (axion) moduli. Tadpole cancellation generally ensures the absence of anomalous U (1) gauge symmetries in the models; the associated gauge boson acquires a string-scale mass via the generalised Green-Schwarz mechanism, and the U (1) survives only as a global symmetry. However, some of the surviving global symmetries forbid the Yukawa couplings required to generate mass terms for some of the quarks and leptons. This is the second deficiency of these models. Further, as noted previously in [11], there is a surviving unwanted U (1) B−L gauge symmetry, associated with baryon number B minus lepton number L. In addition, in all of the models that we constructed, the U (1) stack associated with the fractional 3-cycle c has the property that c = c ′ , where c ′ is the orientifold image of c. This means that the U (1) c gauge symmetry is enhanced to SP (2) = SU (2), so that the models actually have as surviving gauge symmetry group SU (3) colour × SU (2) L × SU (2) R × U (1) B−L . The weak hypercharge is given by Y = 1 2 (B − L) + T 3 R , and the matter is in the following representations (n 3 , n L , In addition, the models we have constructed cannot yield gauge coupling constant unification. A stack κ gives rise to a gauge group factor with coupling constant g κ given [21,22] by where Vol(κ) is the volume of the 3-cycle κ and K κ = 1 for a U (N κ ) stack. The consistency of our treatment with the supergravity approximation requires that the contribution of the bulk part of the fractional 3-cycle 1 2 Vol(Π bulk κ ) to Vol(κ) is large compared to the contribution from the exceptional part 1 2 Vol(Π ex κ ), so we need only consider the former in evaluation g 2 κ . As derived in [10], for a supersymmetric stack, the quantity is real and positive. Here A κ p (p = 1, 3, 4, 6) are the bulk wrapping numbers, e 2k−1 (k = 1, 2, 3) are the basis vectors of T 2 k , and U 3 is the complex structure of T 2 3 ; the complex structure of T 2 1,2 is fixed by the Z ′ 6 orbifold symmetry to be U 1,2 = e iπ/3 . Then The solutions for the AAA lattice given in Table 1 of [12], in which Using equation (27) above, it follows that at the string scale m string the coupling strengths for the SU (3) colour and SU (2) L groups satisfy which is clearly inconsistent with the "observed" unification α 3 = α 2 at the scale m X ≃ 2 × 10 16 GeV. We reach the same conclusion for the solutions on the BAA lattice given in Table 6 of [12], in which U 3 = −i √ 3. Thus, running from the string scale to the TeV scale with the three-generation supersymmetric Standard Model spectrum, none of our solutions can reproduce the measured values of the non-abelian coupling strengths of the SU (3) colour and SU (2) L gauge groups. In fact the only supersymmetric models obtained in [10] yielding three chiral generations 3Q L of quark doublets via (a∩b, a∩b ′ ) = (2, 1) or (1, 2), having no chiral matter in symmetric representations of the gauge groups, and not too much in antisymmetric representations, that also produce non-abelian coupling constant unification, are the two solutions on the BAB lattice given in Table 15 of that paper. We showed in [12] that neither model can have just the required Standard Model spectrum, but it is of interest to see what can be achieved if we relax this constraint and allow additional vector-like matter but not extra chiral exotics. This requires at least two U (1) stacks (both of which must be d-type in the terminology of that paper). The best we can do yields two additional vector-like Higgs doublets 2(H u + H d ) and four additional vector-like charged lepton singlets 4(ℓ c L +l c L ), and in any case the weak hypercharge U (1) Y gauge coupling strength α Y = 3α 3 /5 as required by the "observed" standard-model unification. We have not pursued this any further. The one-loop gauge threshold corrections to (27) have been computed by Gmeiner and Honecker [23]. However, for the models under consideration, these are very small and the above conclusion is unaffected. Another possibility that in principle might yield a realistic model is to start with an SU (3) colour stack a and an SU (2) L stack b satisfying (a∩b, a∩b ′ ) = (3, 0) or (0, 3), and to require gauge coupling constant unification α 3 = α 2 . Following the work of Gmeiner and Honecker [9], we know at the outset that there are no such models that yield the standard-model spectrum and satisfy tadpole cancellation without the introduction of non-zero background flux H bg 3 . However, since we have entertained the presence of such flux, it is of interest to know how far one can get with such models. We have searched for solutions satisfying both of these criteria, but have found none.
Finally, the presence of a non-zero flux H bg 3 means that there may also arise a Freed-Witten anomaly [24]. In the presence of D6-branes the localised Bianchi identity associated with the stack κ imposes the constraint [25] H bg where [κ] is the 3-form that is the Poincaré dual of κ. Since H bg 3 is odd under the orientifold action R, only the R-even part of [κ], deriving from the R-odd part of κ, can contribute to the anomaly. We have studied this in Appendix A. Our conclusion in all cases is that there is a non-zero anomaly deriving from the SU (3) stack a and also from one of the U (1) stacks.
The deficiencies detailed above mean that our models can only be considered as semi-realistic. Nevertheless, it is of interest to see the extent to which the first two deficiencies can be remedied in models with a realistic spectrum. In this paper we study the fixing of moduli using background fluxes, the stability of these solutions and their consistency with the supergravity approximation in which they are derived. We also investigate the utility of non-perturbative effects, so-called E2-instantons, to stabilise axion moduli and to repair the missing Yukawa couplings.

Moduli stabilisation
In this and the following section we parallel the the treatment given by DeWolfe et al. [14] of the Z 3 ×Z 3 orientifold. It has been shown by Grimm and Louis [26] that the effective four-dimensional theory deriving from type IIA supergravity compactified on a Calabi-Yau 3-fold is an N = 2 supergravity theory. The moduli space is the product of two factors, one containing the vector multiplets (which include the Kähler moduli), and the other the hypermutiplets (which include the complex structure moduli and dilaton). The metric on each space is derived from a Kähler potential, K K and K cs respectively. The orientifold projection R to an N = 1 supergravity reduces the size of each moduli space.
Consider first the Kähler moduli. The complexified Kähler form is odd under the action of R and can therefore be expanded in terms of the R-odd (1, 1)-forms. In our case, on the Z ′ 6 orbifold, we have three untwisted, invariant (1, 1)-forms w k (k = 1, 2, 3) defined by There are also eight θ 3 -twisted sector invariant harmonic (1, 1)-forms e (1,j) ,ŵ j , (j = 1, 4, 5, 6), defined as follows. Associated with each of the 16 fixed points f i,j , defined in (7), is a localised (1, 1)-form After blowing up the fixed point using the Eguchi-Hanson EH 2 metric [27], ω k,l has the form when the fixed point The functions a(u) and b(u) are given by with λ the blow-up parameter and Under the action of the point group generator θ these (1, 1) forms transform as Thus the eight invariant θ 3 -twisted (1, 1) forms are e (1,j) and We denote the blow-up parameter associated with e (1,j) by λ j . Point-group invariance (41) requires that e (4,j) , e (5,j) and e (6,j) all have the same blow-up parameter, which we denote byλ j . All of the invariant θ 3 -twisted (1, 1) forms given above are odd under the action of R, so in general we may expand the complexified Kähler form as where b k , B j ,B j are associated with the Kalb-Ramond field B 2 , and the Kähler moduli v k , V j ,V j with the Kähler form J. The Kähler potential K K for the Kähler moduli is given by where Vol 6,2 are the coordinate volumes of T 6 and T 2 2 respectively. Thus where As previously noted, the SU (3) lattice used for T 2 1,2 has U 1 = α = U 2 , so that ImU 1 = √ 3/2 = ImU 2 . For the models found in [11,12], ImU 3 = −1/ √ 3 on the AAA lattice and − √ 3 on the BAA lattice. It is convenient to absorb the coordinate volumes into the moduli, so we make the redefinitions and then Note that, unlike in the Z 3 × Z 3 case discussed in [14], the twisted moduli V j andV j are inextricably coupled to the untwisted modulus v 2 . The complex structure moduli are obtained by expanding the holomorphic (3, 0)-form Ω in terms of the basis 3-forms. There are four Z ′ 6 -invariant untwisted 3-forms, defined as in [12] by The invariant θ 3 -twisted 3-forms ω j ,ω j (j = 1, 4, 5, 6) are also as defined in [12] as As above, it is convenient to factorise out coordinate volumes, so that the Kähler potential K cs for the complex structure moduli is independent of them. Then on the AAA lattice we may expand the holomorphic 3-form as On the BAA lattice ω j andω j are interchanged. In both cases Z 0,1 and Y j are associated with the R-even forms, and g 0,1 , f j with the R-odd ones. It is easy to show that the complex conjugates of the twisted 3-forms are given byω The orientifold constraint requires that RΩ =Ω which gives The required Kähler potential is The R projection projects out half of the moduli of the N = 2 theory, including half of the universal hypermultiplet; the dilaton and one axion survive. The surviving moduli are all contained in the complexified 3-form where C 3 is the RR 3-form gauge potential, and C is the "compensator" that incorporates the dilaton dependence with the four-dimensional dilaton D defined by Since C 3 is even under the action of R we may expand it as on the AAA lattice; as before, in the BAA case we interchange ω j andω j . Expanding Ω c as in (65), on the AAA lattice with the usual interchange for the BAA case. Then the surviving moduli are the expansion of Ω c in H 3 + , i.e. the R-even states with moduli in both cases. The potential V arising after dimensionally reducing the massive type IIA supergravity is where the Kähler potential It follows from (65) that on the AAA lattice Also, since Ω is the holomorphic (3, 0)-form, * Ω = −iΩ and * Re(CΩ) = Re where K cs is given in (71), and the last equality follows from the definition (73). The same result follows on the BAA lattice. Like the Kähler moduli t k , the complex structure moduli N 0,1 , M j enter the Kähler potential only via their imaginary parts. The superpotential [28,29,30] and We note that W Q depends only on the NS-NS flux H bg 3 and W K only on the RR fluxes F bg n (n = 0, 2, 4, 6). H 3 is odd under the action of R, so that, analogously to (75), we may expand its background value as on the AAA lattice, with ω ↔ω on BAA. As shown in [12], flux quantisation requires that the coefficients are quantised. On the AAA lattice where n 3,6 andn j respectively are associated with the flux of H bg 3 through the 3-cycles ρ 3,6 and ǫ j ; note that p 0,1 , P j are independent of the coordinate scales R 1,3,5 . For the solution discussed in §5.1 of reference [12], relating to the fourth solution in Table 1 of that paper, the exceptional part of the tadpole cancellation condition (14) requires that |n 0nj | = 12 for j = 4, 6. Thus |n 0 | = 1, 2, 3, 4, 6, 12. For j = 1, we get |n 0n1 | = 12|1 − t c 1 | = 0, 24, which is always consistent with these values of n 0 . Cancellation of the untwisted part proportional to ρ 4 + 2ρ 6 requires that the corresponding values of n 3 satisfy |n 3 | = 1296, 648, 432, 324, 216, 108, and of n 6 satisfy |n 6 | = (1 + t c 1 )(144, 72, 48, 36, 24, 12). (t c 1 = ±1 is one of the Wilson lines associated with the stack c.) Alternatively, on the BAA lattice where n 1,4 andñ j respectively are associated with the flux of H bg 3 through the 3-cycles ρ 1,4 andǫ j . In this case tadpole cancellation of the exceptional parts requires that |n 0ñj | = 12, so that |n 0 | = 1, 2, 3, 4, 6, 12. Then, n 4 = 0 and the corresponding values of n 1 satisfy |n 1 | = 432, 216, 144, 108, 72, 36.
The form (87) for H bg 3 gives The background fluxes F bg 2 and F bg 4 that appear in W K have similar expansions. Since F 2 is odd under the action of R and F 4 even Vol k e kwk + j=1,4,5,6 The constant term e 0 in W K , defined in (86), arises from the Hodge dual F bg 6 of F 4 polarised in the non-compact directions. All of these fluxes, including F bg 6 , are quantised, the general constraint being that for any closed (p + 2)-cycle Σ p+2 with µ p = (2π) −p α ′ −(p+1)/2 the electric charge of a Dp-brane. For the present, we set F bg 2 = 0, and then The advantage of this formalism is that we may immediately identify supersymmetric vacua by their vanishing F -terms: for every chiral superfield i. For the complex-structure moduli, taking i = N k , M j , we get As in [14], the imaginary parts of these equations are degenerate. Using (84) and (77) ... (78), they give the single constraint which fixes only one linear combination of the axions x 0 , x 1 and X j This degeneracy derives from the fact that the coeffcients p k , P j that determine H bg 3 are real, and therefore have insufficient degrees of freedom to stabilise both the complex structure moduli and their axionic partners. As we shall discuss later, in §5, E2-instantons can lift the remaining degeneracy. The real parts give where Then (103) determines the moduli g k , f j up to an overall scale fixed by Q 0 . Finally, using (83), (104) gives which fixes the dilaton once the other moduli are all fixed [14]. It follows from (99), (100), (71), (83) and (92) that Thus, using (101), when the complex structure moduli satisfy their field equations 2ImW K + ImW Q = 0 (107) and the vacuum value of the superpotential is determined entirely by the Kähler moduli Vanishing F-terms for the Kähler moduli in (98) give Using (101), the imaginary parts of these equations require that The simplest solution of these is and then the above equations reduce to using (108). They couple the untwisted volume modulus v 2 to the twisted volume moduli V j ,V j . Solving for all moduli in terms of v 2 and X gives Substituting these into the v 2 equation gives Then (54) gives and the definition (117) yields Substituting (118) ... (121) and (113) into (97), it then follows that when the Kähler moduli satisfy their field equations, so that Thus the background fluxes e k , E j ,Ê j , G j ,Ĝ j and m 0 fix v 2 and X, and hence, via equations (118) ...(121), the remaining Kähler moduli. The effective supergravity theory is a justifiable approximation [14] so long as the volumes v k , V j ,V j are large enough that the O(α ′ ) corrections are negligible and the string coupling g s is small enough to neglect corrections. Further, to remain within the Kähler cone we require that the untwisted volumes are large compared with the blow-up volumes, i.e. v k ≫ V j ,V j ≫ 1. Since (the non-zero value of) m 0 is fixed by the RR tadpole cancellation condition (14), and we have set F bg 2 = 0, the question then is whether there are choices of the background 4-form flux F bg 4 for which these constraints are obeyed. It follows from equations (118) ... (121) that v 1,3 /V j = e 1,3 /E j , so that the Kähler cone constraints require that e 1 , e 3 ≫ E j , and similarly forÊ j . Hence F ∼ e 1 e 3 . Then the constraints v k ≫ 1 require that e 1 e 3 ≫ẽ 2 m 0 , e 1ẽ2 ≫ e 3 m 0 and e 3ẽ2 ≫ e 1 m 0 , and these imply that e 1 ,ẽ 2 , e 3 ≫ m 0 . For the blow-up volumes, similarly, the constraints v k ≫ V j ,V j ≫ 1 require that e 1 , e 3 , e 1 e 3 /ẽ 2 ≫ E j ,Ê j ≫ e 1 e 3 /ẽ 2 m 0 . All of these are easily arranged.

Non-supersymmetric vacua
In general, besides the supersymmetric vacua identified in the previous section, we expect there to be additional vacua that are non-supersymmetric. To identify these we should find the effective potential in the four-dimensional Einstein frame, in which the four-dimensional Einstein-Hilbert action has the standard normalisation. However, the axion fields x k and X j , defined in (75), enter the ten-dimensional action (9) only via the C 3 ∧H bg 3 ∧dC 3 term in the Chern-Simons piece. This term is only non-zero if dC 3 is "polarised" in the four-dimensional spacetime directions, i.e. dC 3 = f d 4 x ≡ F 0 ; it has no physical degrees of freedom and can be treated as a Lagrange multiplier. The part of the action involving F 0 has the form where The equation of motion for F 0 gives * Then subsituting back gives which is stationary when X = 0. The equation that stabilises the axion follows from Using (43), (75), (87) and (94) this gives This fixes the same linear combination of the axions x 0 , x 1 and X j as in (102), and indeed, using (97), the value agrees with that found in the supersymmetric treatment when the Kalb-Ramond fields b k , B j andB j have the values given in (113). The remaining moduli are stabilised by minimising the effective potential V in the Einstein frame with metric g E µν . We pass to this frame by redefining the four-dimensional metric with g 6 the determinant of the 6-dimensional metric. Invariance of the 6-dimensional Kähler metric under the action of the point group and the orientifold projection R requires that where the γ i (i = 1, 2, 3) are real and positive. In the θ 3 -twisted sector there are 16 Z 2 fixed points f i,j ∈ T 2 1 × T 2 3 with i, j = 1, 4, 5, 6, defined in (7) and (8). These fixed points are blown up using the Eguchi-Hanson EH 2 metric ds 2 = g k,l dz k dz ℓ where k, ℓ = 1, 3 and The functions A(u) and B(u) are given by with λ the blow-up parameter and u as defined in (39). In general, both the twisted modulus Γ and the blow-up parameter λ depend on the fixed point f i,j with which they are associated. However, the transformation property (41) of the twisted 2-forms, or rather the analogous property of the twisted 2cycles, shows thatΓ j andλ j , associated with f 4,j , f 5,j and f 6,j , are independent of the T 2 1 fixed point i = 4, 5, 6; the corresponding parameters for f 1,j are denoted by Γ j and λ j . In the untwisted sector there are then three real moduli and where Vol 6 is defined in (49) and (50). The (4-dimensional) volume of the blow-up is taking 0 ≤ u λ 2 . The local analysis that we carry out here is valid provided that the volume of the blow-up modes is small compared with the untwisted volume Vol( . Blowing up f i,j in this manner removes a volume Vol(f i,j ) from the untwisted volume Vol(T 2 1 )Vol(T 2 3 ). With g E µν as given in (134), the effective potential V is defined by Taking F bg 2 = 0, as in (97), there are four contributions to V deriving respectively from the |H 3 | 2 , |F 4 | 2 , m 2 0 and the Born-Infeld terms in (9). With H bg 3 given by (87), we find where Vol(T 2 k ) = γ k Vol k for k = 1, 2, 3 with Vol k defined in (50). Likewise with µ = 1/2. As in [14], the only terms relevant to the stabilisation of the twisted moduli are V F and V m 0 , since the former dominates as Vol(f i,j ) → 0 and the latter as Vol(M) → ∞. In equation (149) we may write where Vol 0 (M) = Vol(T 2 1 )Vol(T 2 2 )Vol(T 2 3 )/6 is the volume with no blow up. Then, minimising the potential gives and we are justified in using this local treatment provided that the F bg 4 fluxes are chosen so that With these values for the blow-up volume The Born-Infeld term gives and using the (bulk part of the) tadpole cancellation condition given in (14), we can rewrite this as where Π m 0 H bg 3 is the 3-cycle of which the field m 0 H bg 3 is the Poincaré dual; H bg 3 is given in (87). For the two cases of interest, as shown in [12], for AAA and BAA respectively. To calculate the integral in (158), we use the result [31] quoted in [14], since Π m 0 H bg 3 is a special Lagrangian 3-cycle. The holomorphic 3-form Ω, defined in (65), is normalised by demanding that Then, according to the calibration formula So for both lattices. The various contributions to V are homogeneous in Vol(T 2 k ). Hence at the stationary point Also, we require that ∂V /∂Vol(T 2 k ) = 0, which gives (with y > 0). It follows that The requirement (153) that justifies the local treatment gives so that η ≪ 27(m 0 y) 2 We may also write Vol(M) in terms of y: so that Defining x ≡ e φ Vol(M) it follows from (168) that which fixes x as a function of y. Hence |m 0 |x = 3b 32(µ + 2η and at the stationary point, we may eliminate the dilaton and express the potential in terms of y alone: where Vol(M) is given by (175) and x 4 y 2 (1 + ǫ) 1 + ηy 2 3(36|e 1 e 2 e 3 | 2 + ηy 2 ) (182) with x given by (178). Since B > 0, it is easy to see that the potential V → +∞ as y → 0+. Similarly, V → 0 as y → ∞. The limit is approached from above or below depending upon the sign of A in this region. If A < 0, then there is certainly an anti-de-Sitter minimum at a finite value of y; otherwise, no conclusion can be reached without a more detailed consideration of the parameters. It follows from (181) and (178) that In the same limit, (179) gives To proceed further, we need to know the dependence of the moduli g 0,1 , f j that appear in (161) on Z 0,1 , Y j . For simplicity, we consider only the bulk contributions g 0,1 and assume that these derive from a homegeneous quadratic prepotential G, defined in (162), of the form with α, β and γ (real) constants (not functions of Z 0 /Z 1 ). Then the moduli g 0,1 are given by The question we address is whether G may be chosen so that (185) is always satisfied. Keeping only the bulk contributions, the minimum value of subject to the constraint (161) that G(Z 0 , Z 1 ) = 3/32 is Evidently, we may ensure that (185) is satisfied by choosing α, β, γ sufficiently small. On the AAA lattice, so that the minimum value of b 2 and 2h are for t c 1 = +1, −1 respectively. On the BAA lattice, since p 0 = p 1 in this case, b 2 and 2h have the same values as in the t c 1 = −1 case for the AAA lattice. The untwisted part of the 4-form flux F bg 4 given in equation (94). It is specified by the quantities e k Vol k /Vol 6 (k = 1, 2, 3). Using (169), the ratios of the metric moduli γ i /γ j = e j Vol j /e i Vol i are specified for a given value of F bg 4 . The minimisation of V fixes y 2 /|e 1 e 2 e 3 | 2 , and F bg 4 also specifies the combination |e 1 e 2 e 3 |/Vol 2 6 . Thus, the stabilisation fixes the overall scale of the metric moduli (γ 1 γ 2 γ 3 ) 2 = (y 2 /|e 1 e 2 e 3 |) 3 (|e 1 e 2 e 3 |/Vol 2 6 ) in terms of the specified background fluxes. Similarly, the (untwisted) background flux H bg 3 , defined in equation (87), is specified by p 0,1 / √ Vol 6 . Thus equation (179) fixes x/ √ Vol 6 in terms of the background fluxes m 0 and H bg 3 . With x defined in (177), it follows that x/ √ Vol 6 ≃ e φ √ γ 1 γ 2 γ 3 , and since the moduli γ 1,2,3 have already been fixed, this result stabilises the dilaton φ in terms of the background fluxes. The argument may be extended to include the twisted moduli.

Stability
Since we have taken F bg 2 = 0, the |F 2 | 2 and |F 4 | 2 terms in the the action S IIA , given in (9), are at least quadratic in the fields B 2 , there being no Z After eliminating the Lagrange multiplier F 0 ≡ dC 3 , the effective action deriving from this field is given in (131) with X in (132). With the B 2 -moduli set to zero, the stabilised linear combination of the axions given in (102) reduces to The B 2 ∧F bg 4 +C 3 ∧H bg 3 piece in X is linear in the fluctuation fields and the above stabilised combination of axion fields. Hence the action (131) mixes them and we need to consider the quadratic terms, including kinetic terms, for both sets of fields simultaneously. The unstabilised (orthogonal) axion fields are, of course, massless.
The kinetic terms for the B 2 field fluctuations derive from the contribution where the kinetic Lagrangian density is with ∂ µb k = g µν E ∂ νbk etc., and the fieldsb k ,B j ,B j defined so that they are canonically normalised: Quadratic terms in these fields arise from where s j ,ŝ j are respectively the signs of G j /m 0 ,Ĝ j /m 0 , and the last term follows when we substitute the stabilised values (151) and (152) of the blow-up volumes. The kinetic terms for the C 3 fluctuations arise from and the canonically normalised fieldsx 1,2 andX j are given bỹ Quadratic terms in b k and x 0,1 arise from (131) As noted previously, the only coupled combination of axion fields corresponds to the stabilised axion, whose normalised fieldã is given in terms of the rescaled fieldsx 1,2 ,X j by We shall consider only the untwisted contibutions. Then the quadratic terms deriving from (210) are Stability requires that the eigenvalues of the mass matrix are all positive. However the uncoupled axion is massless, so the best we can hope for is that the remaining four mass eigenstates are non-tachyonic. The mass matrix deriving from (213) may be written in the form and s 1,2,3 = ±1 are the signs of e 1,2,3 . The general expressions for the eigenvalues are too large to be tractable, but positive-definiteness is ensured provided that the following quantities are all positive: where a 2 = 1 4γ 2 1 + 1 + 8γ 2 , a 3 = 1 for γ 2 > 1 (225) Note that a 1 is always negative, and a 2,3 positive. In the special case that γ = 0, the function d(a) = β 2 (1 − a 2 ) is positive only in the range −1 < a < 1.
We also require that d 4 (a) is positive. Evidently this is always the case for a > 0, so we need only consider whether negative values of a lead to stronger constraints than those already derived. According to (223), the most negative value that we need to consider is a = a 1 , which satisfies d(a 1 ) = 0. For this value of a it follows that Further, d 4 (a) has only a single real (negative) root, so the positivity of i,j (m 2 ) i (m 2 ) j gives no further constraints. Finally, we require also that d 6 (a) is positive. It is convenient to write The special case in which γ = 0 is easy to analyse. In this case N (a) = 1 − a 4 is positive only in the range −1 < a < 1 in which d(a) is also positive. Since D(a) = a 2 − 3 is negative throughout this range, it is only for values of a in this range that we have positive definiteness. The case in which γ 2 = 1/4 is also easy to analyse. Positivity of d(a) requires that either 1 − √ 3 = a 1 < a < a 2 = 1 or a > 1 + √ 3. The function N (a) = 1 + a 3 is positive only when a > −1. Thus N (a) is positive in both of these ranges, while D(a) = 3(a 2 − 2)/2 is negative in the region a 1 < a < a 2 , but positive in a > a 3 . It follows that m 2 is positive definite for any value of β 2 when a is in the range a 1 < a < a 2 , but only for values of β 2 < N (a)/D(a) in the range a > a 3 .
The Although we have been discussing the conditions under which the (untwisted) mass eigenstates are non-tachyonic, in principle this is too strong a requirement in the anti-de Sitter space of our vacuum solutions. Tachyonic mass eigenstates are stable provided that they satisfy the Breitenlohner-Freedman bound [32,33] where −|V min | is the value of the potential at the anti-de Sitter minimum. The massless uncoupled axion obviously satisfies the bound, so it will not generate instability. However, determining which values of a lead to other mass eigenstates that satisfy this weaker constraint is something that can only be done when V min has actually been calculated, and this in turn requires a detailed consideration of the parameters, as already noted. The expectation or, more accurately, the hope is that when the anti-de Sitter minimum is lifted to Minkowski, in the manner of KKLT [34], then the tachyonic states will be lifted too. However, as Conlon has noted [35], it is not clear that all tachyons will be lifted by this mechanism. The uplifting is generally rather poorly controlled, and it is at least plausible that there may remain tachyons in the Minkowski space.

E2-instantons and Yukawa couplings
We have so far fixed only one linear combination of the axion fields. As noted previously, we may use non-perturbative effects to stabilise the remaining axions. The non-perturbative effects under discussion are Dp-branes that wrap non-trivial cycles in the compactification space M 6 , and that are pointlike in M 4 . Their world-volume is (p + 1)-dimensional and spacelike, so they are Euclidean Dp-branes, called Ep-branes or Ep-instantons for short. In type IIA string theory, p is even and p+1 ≤ 6. Hence p = 0, 2, 4.
where y c = ± 1 2 . Using equations (63) and (66) of that paper, the intersection numbers of a with the SU (2) L stack b and its orientifold image b ′ are given by arises at the intersection with its orientifold image c ′ : The quark singlets arise at the intersections of a with c and c ′ (c ∩ a, c ′ ∩ a) = (3, 3) the former giving 3u c L and the latter 3d c L . First, consider the case described by (257). u-quark mass terms arising from the two Q L states at a ∩ b require the coupling of the states at a ∩ b, b ∩ c and c ∩ a, which is allowed by the conservation of Q a , Q b and Q c . However, the u-quark mass term arising from the Q L state at a ∩ b ′ requires the coupling of the states at a ∩ b ′ , b ∩ c and c ∩ a, which is allowed by the conservation of Q a and Q c , but not by Q b , since the product has ∆Q b = 2. Similarly, only two d-quark mass terms are allowed by conservation of Q b . The alternative choice described by (256) allows only one quark mass term for both uand d-type quarks.
We also have which generate the required total of 3L lepton doublets (with Y = − 1 2 ), while the lepton singlets arise from the former giving the 3ℓ c L charged lepton singlets, and the latter the 3ν c L the neutrino singlet states. For the case (257) under consideration, equation (261) gives the location of the lepton doublets. The charged lepton mass term arising from the lepton doublet at d ′ ∩ b require the coupling of the states at d ′ ∩ b, b ∩ c ′ and c ′ ∩ d ′ , which is allowed by the conservation of Q b , Q c and Q d . However, the charged lepton mass terms arising from the two lepton doublets at d ′ ∩b ′ require couplings that again have ∆Q b = 2. Similarly, only one neutrino mass term is allowed by conservation of Q b . The alternative choice described by (262) allows two lepton mass term for both charged leptons and neutrinos. Thus, at the perturbative level, after electroweak symmetry breaking, we either have two massive quark generations and one massive lepton generation, or vice versa. In the model discussed in reference [11], the same correlation is obtained.
In both cases the missing couplings can only be provided by non-perturbative instanton effects. These generate terms in the superpotential W of the form that violate the global U (1) symmetries that survive after the Green-Schwarz mechanism breaks any anomalous U (1) gauge symmetry [36]; here Φ i are the (generally charged) matter superfields and S inst is the action of the non-perturbative instanton. Such a term is allowed if and only if the gauge transformation of the matter field product i Φ i under an anomalous U (1) gauge transformation is cancelled by the transformation of the exponential factor induced by the shift of Im S inst under the U (1) transformation [37]. Under a U (1) κ gauge transformation, associated with the stack κ, parametrised by Λ κ , in which the 1-form vector potential A κ 1 is shifted by the imaginary part Im S inst of the instanton action (233) is shifted by where Q κ (E 2 ) is the U (1) κ charge of the instanton, given by To repair the missing Yukawa couplings we require that The general form of Ξ is given in equation (234) Then, using our solution for the SU (2) L stack on the AAA lattice given in Table 1 and equation (66) of reference [12], it follows from (268) above that so that z 1 or z 5 , but not both, are odd. We also require, as in (269), that the instanton has zero charge with respect to the other U (1) charges. For Q c this is guaranteed, since c = c ′ . Further, since a − a ′ = d ′ − d in our solution, Q a (E2) = 0 ensures that Q d (E2) = 0. Thus, there is just one further constraint, which yields It follows from (270) that Ξ is of d 1 -type, as defined in equation (241), and it is easy to find solutions with all of the desired properties. For example with The above solution gives Thus the required total instanton charge (268) derives from one (massless) particle at the intersection of Ξ with b, and two at the intersections of Ξ with b ′ . To repair the missing u-quark Yukawa, for example, we need a 5-point coupling in which both b and b ′ intersect the fractional 3-cycle Ξ of the instanton: Since Ξ ∩ b = −1, we should interpret it as one intersection with Q b = +1, rather than -1 intersections with Q b = −1. However, since b ′ ∩ Ξ = 2 is positive, the coupling (275) does not then conserve Q b , and we conclude that we cannot repair the Yukawa with this E2-instanton. Further, equation (268) requires that Ξ ∩ b − Ξ ∩ b ′ = 1 which can only be satisfied with non-zero Ξ ∩ b and Ξ ∩ b ′ when they have the same sign, as in the above solution. Consequently Ξ ∩ b and b ′ ∩ Ξ cannot have the same sign in any of the solutions, and they therefore contribute zero to the total Q b charge in (275). Hence we cannot repair the Yukawa with any of the single E2-instanton solutions of the constraints. The same conclusion follows for the model discussed in reference [11], as well as for the models on the BAA lattice given in Table 6 of reference [12].

Conclusions
All of the models that we have considered have the attractive feature that they have the spectrum of the supersymmetric Standard Model, including a single pair of Higgs doublets, plus three right-chiral neutrino singlets. In the presence of the previously derived non-zero background field strength m 0 H bg 3 they are also free of RR tadpoles, and therefore constitute consistent string-theory models. We showed in §2 that this background field also stabilises one of the axion moduli. Further, we found that it is easy to choose a non-zero background field strength F bg 4 that stabilises the Kähler and complex-structure moduli associated with the supersymmetric mininima at values within the Kähler cone in which the supergravity approximation is valid. In §3 we showed that there are also non-supersymmetric stationary points of the effective potential, and in §4 we determined the parameter ranges in which these are stable minima. The stabilisation of all of the axion moduli can only be achieved by the use of non-perturbative instanton effects, and these were discussed in §5. In principle, such effects might also restore the missing quark and lepton Yukawa couplings to the Higgs doublets that are needed to generate masses when the electroweak symmetry is spontaneously broken. However, we also showed that this does not happen for the particular models of interest here.
The analysis of N (a) = (4γ 2 − 1)a 4 + 4γ 2 a 3 + 1, defined in equation (229), is more complicated. For all values of γ 2 it has a root at a = −1, a saddle point at a = 0, and one other stationary point at When 0 < γ 2 < 1/4 this stationary point is at a positive value of a and is a maximum. In this case, N (a) is positive for a 4 < a < a 5 , and negative elsewhere; here −1 = a 4 < a 1 < 0 and a 5 > a D > 0 is the (positive) root α(γ 2 ) of the cubic factor in equation (230); in the special case γ = 0, for example, a 5 = 1.
It is easy to verify that N (a − ) ≤ 0 (actually for all values of γ 2 , with equality only when γ 2 = 1); thus a − < a 4 < a 1 . Further, N (a + ) is negative for 0 < γ 2 0.1255, vanishes when γ 2 ≃ 0.1255, and is positive for all other values of γ 2 ; it follows that a 5 < a + for 0 < γ 2 0.1255, but a + < a 5 for 0.1255 γ 2 < 1/4. Alternatively, when γ 2 > 1/4, a D is negative and N (a D ) is a minimum. In this case, N (a) is negative for a 4 < a < a 5 , and positive elsewhere, and now both a 4 and a 5 are negative roots of N (a), with a 4 < a D < a 5 ; for γ 2 < 1 the position of the stationary point satisfies a D < −1, whereas for γ 2 > 1 we find a D > −1. Thus when γ 2 < 1, −1 = a 5 < a 1 and a 4 is the root of the cubic in equation (230), whereas for γ 2 > 1, −1 = a 4 < a 1 and a 5 is the root of the cubic. Obviously, a + > a 5 for values of γ 2 in this range. These considerations lead us to consider three ranges of values for γ 2 , with the signs of the functions given in the associated Tables. I a < a − -+ -II a − < a < −1 --III −1 < a < a 5 + -+ IV a 5 < a < a + --V a + < a -+ - Table 1: Signs of the functions N (a), D(a), d 6 (a) when 0 < γ 2 < 0.1255 We need to identify the regions in which both d(a) and d 6 (a) are positive. With a 1 defined in equation (223), we note that D(a 1 ) is negative for any value of γ 2 . Similarly, N (a 1 ) is positive (actually for any value of γ 2 ). It follows that a 1 is in region III of Table 1; this is consistent with the observation above that a 4 < a 1 in this case. From equation (226) we see that a 2 = 1 for values of γ 2 in this range. Since D(1) = 2(γ 2 − 1) < 0, in this case, and N (1) = 8γ 2 > 0, it follows that a 2 is also in region III of Table 1. Finally, using the value of a 3 given in (226), we find that D(a 3 ) is positive, (actually for any γ 2 < 1; it vanishes when γ 2 = 1, and is negative for all other values.) For values of γ 2 0.1915, N (a 3 ) is negative; (it vanishes when γ 2 ≃ 0.1955, and is positive for all other values.) It follows that a 3 is in region V of Table 1 in which d 6 (a) is negative. Thus the only region in which both d(a) and d 6 (a) are positive is a 1 < a < a 2 = 1, and this is the case for all values of β 2 ; note that this range does not require the solution of the cubic.
• 0.1255 < γ 2 < 1 4 The properties of the functions given above show that in this case a 1 is in region III of Table 2, as is a 2 , and if γ 2 0.1955 then a 3 is in region V; otherwise it is region IV. Thus, if γ 2 0.1955, the only region in which both d(a) and d 6 (a) are positive is again a 1 < a < a 2 = 1, and as before this is the case for all values of β 2 . However, in the case γ 2 0.1955, N (a) and D(a) are both positive in the region a 3 < a < a 5 , so that both d(a) and d 6 (a) are positive here too, provided that β 2 < N (a)/D(a). The determination of a 5 requires the solution of the cubic, which we discuss below.
Region a N (a) D(a) d 6 (a) I a < a − -+ -II a − < a < −1 --III −1 < a < a + + -+ IV a + < a < a 5 + + V a 5 < a -+ -  In this case we conclude that a 1 is in region IV of Table 3, as is a 2 , and a 3 is in region V. Thus again both d(a) and d 6 (a) are positive in the region a 1 < a < a 2 , and this is the case for all values of β 2 . As noted previously, there is a further region a > a 3 in which positive-definitenesss is assured provided that β 2 < N (a)/D(a). Since N (a) grows with a more rapidly than D(a), the upper bound on β 2 grows with a.
The following two examples illustrate the procedure.
• γ 2 = 0.2 Since γ 2 > 0.1955, we expect positive-definiteness for some values of a in region IV of Table 2, besides those in −0.7656 = a 1 < a < a 2 = 1 in region III. For this value of γ 2 equations (226), In this range the upper bound on β 2 is approximately linear, starting at β 2 0.51 when a = a 3 and decreasing to zero at a = a 5 , where N (a) vanishes.
• γ 2 = 2 The solution of the cubic is not needed in this case. Besides the range −0.3904 = a 1 < a < a 2 = 0.6404 in region IV of Table 3, both d 6 (a) and d(a) are positive in the range a > a 3 = 1 in region V, provided that β 2 < 7a 4 +8a 3 +1 5a 2 −3 . Positive-definiteness is assured for all values of a in this range when β 2 6.985, whereas larger values of β 2 are only allowed for larger values of a.