D6-Brane Model Building on Z(2)xZ(6): MSSM-like and Left-Right Symmetric Models

We perform a systematic search for globally defined MSSM-like and left-right symmetric models on D6-branes on the T6/Z(2)xZ(6)xOR orientifold with discrete torsion. Our search is exhaustive for models that are independent of the value of the one free complex structure modulus. Preliminary investigations suggest that there exists one prototype of visible sector for MSSM-like and another for left-right symmetric models with differences arising from various hidden sector completions to global models. For each prototype, we provide the full matter spectrum, as well as the Yukawa and other three-point couplings needed to render vector-like matter states massive. This provides us with tentative explanations for the mass hierarchies within the quark and lepton sectors. We also observe that the MSSM-like models correspond to explicit realisations of the supersymmetric DFSZ axion model, and that the left-right symmetric models allow for global completions with either completely decoupled hidden sectors or with some messenger states charged under both visible and hidden gauge groups.


Introduction
String theory is arguably the most promising framework for a unified description of all fundamental interactions. However, the question how the experimentally observed particle spectrum and interactions of the Standard Model of Particle Physics or some extension thereof arise from string theory remains open to date, see e.g. [1,2] for early works within the heterotic E 8 × E 8 theory. While a large fraction of today's efforts focusses on the construction of the chiral Standard Model or some GUT spectrum within the non-perturbative F-theory regime (see e.g. the lecture notes [3] and some very recent works [4][5][6][7][8][9][10][11] and references therein), condoning the lack of control over the low-energy effective action, compactifications of Type II string theory at special points in moduli space provide for a very well controlled testing ground in the perturbative regime, where not only the full spectrum but (at least in principle) all interactions are computable. The specific corner in the landscape of string compactifications, namely Type IIA orientifolds on toroidal orbifolds, which this article makes use of, relies on the combined power of using topology and algebraic geometry to describe the positions of D-branes and the chiral matter localised at their intersections, and Conformal Field Theory (CFT) techniques to compute the vector-like spectrum as well as gauge, Yukawa and higher m-point couplings [12,13]. 1 The T 6 /(Z 2 × Z 6 × ΩR) orbifold is chosen in this article since, based on earlier works with similar toroidal orbifold backgrounds, we expect it to be the most fertile one with a plethora of globally defined Type IIA/ΩR string vacua providing the phenomenologically most appealing spectra without flat directions allowing for any continuous gauge symmetry breaking. Most notably, by trial and error we found in earlier works that the existence of some Z 3 subsymmetry is favourable for providing three particle generations. For example, on the T 6 /(Z 4 × ΩR) [15] and T 6 /(Z 2 × Z 4 × ΩR) [16][17][18][19] backgrounds, it is impossible to construct globally defined supersymmetric D6-brane models with three quark generations, while on T 6 /(Z 6 × ΩR) [20,18,21,22] and T 6 /(Z 6 × ΩR) [23][24][25][26][27]22] such models have been obtained. For the latter, diverse investigations of the related low-energy effective field theory were performed in [22,[28][29][30][31], discrete remnants of gauge symmetries were first investigated in [32], and the relation to some Peccei-Quinn symmetry and axions was studied in [33,34]. However, both types of Z ( ) 6 orbifolds face the draw-back of containing matter in the adjoint representation as remnants of the underlying N = 2 supersymmetry in the gauge sector, whose flat directions lead to continuous breakings of the non-Abelian gauge groups to subgroups of equal rank. The sector of Z 2 -twisted three-cycles on the factorisable T 6 /(Z 6 ×ΩR) orientifold on the SU (3) 3 lattice can be viewed as one of three identical Z (i) 2 -twisted sectors on T 6 /(Z 2 ×Z 6 ×ΩR) with discrete torsion. An exhaustive scan for globally defined, phenomenologically appealing, supersymmetric D-brane models on the latter only yielded Pati-Salam models [35]. In a similar way, the Z 2 -twisted sector of the other factorisable T 6 /(Z 6 × ΩR) background can be viewed as occurring twice as subsector of the T 6 /(Z 2 × Z 6 × ΩR) orientifold with discrete torsion, with the third Z 2 -twisted sector different and arising from (T 2 × (T 4 /Z 6 ))/ΩR. In our recent article [36], we were able to exclude SU (5) GUT models with three particle generations (and no chiral exotic 15-representations), and we classified Pati-Salam models. This article is devoted to extending the search to left-right symmetric and MSSM-like spectra. A full systematic search has to be performed for any allowed value of the complex structure parameter on the first two-torus of the SU (2) 2 × SU (3) 2 background, as we will briefly comment on in section 2.2. However, the present search focusses on D-brane configurations that are supersymmetric for arbitrary values of . This article is organized as follows: in section 2, we review basic model building ingredients such as RR tadpole cancellation and supersymmetry on T 6 /(Z 2 × Z 6 × ΩR) with discrete torsion, where we also discuss the most basic phenomenological constraints, such as no exotic matter in the adjoint or symmetric representation of the QCD D6-brane stack, and the resulting severe limitations on the corresponding three-cycle. In section 3, we discuss the missing global consistency conditions, namely the K-theory constraints, in the context of discrete Z n gauge symmetry remnants from massive U (1)s in the low-energy effective field theory, before we present the results of a systematic computer search for MSSM-like and leftright symmetric models in sections 4 and 5, respectively. For each kind of attainable visible spectrum, we provide some prototype example of a globally defined D-brane configuration, for which we first compute the remnant discrete Z n symmetries and/or surviving massless U (1) symmetries and then provide all Yukawa and other three-point couplings needed to render vector-like matter states massive. Our conclusions are given in section 6, and in appendix A we briefly review the method of Chan-Paton labels needed to determine the localisations of matter states for computing Yukawa couplings. Our focus lies here on the rôle of discrete Wilson lines not discussed before in the literature. In appendix B, we present an example of a semi-local model, where all RR tadpoles are cancelled, but some of the K-theory constraints are violated. Last but not least, appendix C contains further prototype matter spectra for globally consistent left-right symmetric models with different hidden sectors.
2 Recollections of the Orientifold T 6 /(Z 2 × Z 6 × ΩR) To fully appreciate the phenomenological aspects of intersecting D6-brane models on the toroidal orientifold T 6 /(Z 2 × Z 6 × ΩR) with discrete torsion, a proper understanding of the background geometry is essential. Our starting point is thus a brief summary of indispensable geometric aspects related to the toroidal orientifold and its fractional three-cycles. For a more detailed account of these aspects we refer to [19,36]. To prepare the systematic search and classification of global MSSM-like and left-right symmetric models in sections 4 and 5, the second part of this section then reviews some results about the search for appropriate local rigid D6-brane configurations allowing for three chiral quark generations with a minimal amount of undesired exotic matter, as first presented in [36].
In this expression, θ m ω n corresponds to a generic element of the point group, where θ generates the Z 2 part of the orbifold group acting on T 2 (1) ×T 2 (2) , and the Z 6 part generated by ω acts only on the four-torus T 2 (2) × T 2 (3) . As an immediate consequence of the Z 6 -action, the lattices of the factorisable four-torus T 2 (2) × T 2 (3) take the shape of SU (3) root lattices, while the complex structure modulus of the first two-torus T 2 (1) remains unconstrained, see figure 1. The T 2 (1) lattice is thus given by a SU (2) 2 root lattice. Various combinations of the generators θ and ω generate additional Z N subgroups with accompanying fixed points and/or fixed lines: Z 6 symmetries are generated by (θω, θω 2 ), a Z 3 symmetry by ω 2 , and Z 2 symmetries by (θ, ω 3 , θω 3 ).
Given that the orbifold group consists of a direct product of two Abelian factors, the Z 2 generator θ can act on the (Z 6 ) ω-twisted sectors with a phase η = ±1 and vice versa as discussed in detail in [19]. For η = −1 we say that the orientifold has 'discrete torsion', and the presence or absence of discrete torsion has a non-trivial impact on the Hodge numbers counting the two-and three-cycles in the twisted sectors, as discussed for T 6 /(Z 2 × Z 6 ) e.g. in [19,36]. For instance, the Hodge numbers associated to the three Z 2 twisted sectors are given by: As we will review later on, the intersecting D6-brane models considered in this article are supposed to wrap exceptional three-cycles stuck at Z 2 fixed points on T 6 /(Z 2 × Z 6 ). The absence of such three-cycles for η = 1 implies that we should focus our attention on the toroidal orbifold with discrete torsion, η = −1, from now onwards.
The orbifold group is extended by an orientifold projection ΩR(−) F L , consisting of the worldsheet parity Ω, the projection involving the left-moving fermion number F L , and the antiholomorphic involution R acting on the coordinates as: Besides reducing the amount of four-dimensional spacetime supersymmetry to N = 1 for Type IIA string theory, the orientifold projection also constrains the shape of the two-torus T 2 (1) to be rectangular (a-lattice) or tilted (b-lattice) and reduces the complex structure parameter on T 2 (1) to one real parameter captured by the ratio ≡ √ 3R 2 /R 1 . The tiltedness of T 2 (1) will be denoted by a discrete parameter b ∈ {0, 1 2 }, where the b-type lattice configuration corresponds to b = 1 2 . The lattices for the two-tori T 2 (2) and T 2 (3) , which are always tilted, only admit two orientations w.r.t. the orientifold invariant direction: an A-type lattice or a B-type lattice orientation as depicted in figure 1. A priori one expects six different ΩR-invariant lattice configurations, a/bAA, a/bAB and a/bBB, but only the first two, aAA and bAA, are truly physically independent as shown in [36]. More explicitly, non-supersymmetric rotations among the lattices relate the lattices aAB and aBB to the lattice aAA on the one hand, and the lattices bAB and bBB to the lattice bAA on the other hand. 2 Hence, it suffices to limit investigations and discussions to the two lattices a/bAA in the remainder of the paper. 2 The full equivalence between the lattices was shown at the level of all explicitly computable quantities, such as the splitting of the Hodge number h11 into orientifold-even and -odd parts (h + 11 , h − 11 ) counting the closed string vectors and Kähler moduli, respectively, the RR tadpole cancellation and supersymmetry conditions, the massless matter spectrum, vacuum amplitudes, and the one-loop gauge threshold corrections.  (1) with complex structure modulus ≡ √ 3R 2 /R 1 on T 2 (1) . The Z 2 fixed points are labeled by the red points (1,2,3,4) on the first two-torus and (1,4,5,6) on the second and third two-torus. The Z 6 action is trivial on the first torus and cyclically permutes three The O6-planes are grouped into four inequivalent orbits under the Z 6 -action, denoted as the ΩR-and ΩRZ (k=1,2,3) 2 -invariant orbits. Each of the four O6-plane orbits carries RR charges, and the sign of their RR charges is denoted by η ΩR and η ΩRZ (k) 2 , respectively. Worldsheet consistency of the Klein bottle amplitude relates [37,19] these charges to the discrete torsion parameter η: where η ΩR , η ΩRZ (k) This relation indicates that one of the O6-plane orbits has to be 'exotic' with positive RR charges (η ΩR(Z (k) 2 ) = −1) in the presence of discrete torsion η = −1. As pointed out in [36], configurations with three exotic O6-plane orbits on T 6 /(Z 2 × Z 6 × ΩR) are excluded based on supersymmetry requirements and bulk RR tadpole cancellation conditions. In order to cancel the RR charges of the O6-planes, we introduce supersymmetric D6-branes whose RR charges compensate those of the O6-planes. For model building purposes we consider D6-branes wrapping fractional three-cycles on the toroidal orientifold. Such fractional three-cycles Π frac x (1) α,a ε (1) α + y (1) α,aε where we decomposed the bulk and exceptional three-cycles with respect to an orbifoldinvariant basis in the second line. More concretely, the integers (P a , Q a , U a , V a ) correspond to the bulk wrapping numbers expressed in terms of the b bulk 3 = 2 + 2h bulk 21 = 4 dimensional basis of bulk three-cycles ρ i∈{1,2,3,4} with non-vanishing intersection numbers [19]: For factorisable three-cycles, the bulk wrapping numbers can be written out explicitly in terms of the torus wrapping numbers (n i a , m i a ) i=1,2,3 : P a ≡ n 1 a n 2 a n 3 a − m 2 a m 3 a , Q a ≡ n 1 a n 2 a m 3 a + m 2 a n 3 a + m 2 a m 3 a , U a ≡ m 1 a n 2 a n 3 a − m 2 a m 3 a , V a ≡ m 1 a n 2 a m 3 a + m 2 a n 3 a + m 2 a m 3 a .
Note that the torus wrapping numbers do transform non-trivially under the Z 6 -action: whereas the bulk wrapping numbers are Z 6 -invariant quantities inherent to a ω-orbit and thus independent of the choice of toroidal representant.
The integers (x (k) α,a , y (k) α,a ) are the so-called exceptional wrapping numbers expressed in terms of the b Z 2 3 = 2h Z 2 21 = 28 dimensional basis of exceptional three-cycles (ε (1) α ,ε (1) α ) α∈{0,...,5} and (ε (l) α ,ε (l) α ) l=2,3 α∈{1,...,4} with intersection form given by [19]: The exceptional wrapping numbers of factorisable three-cycles can be written for each Z (k) 2twisted sector in terms of linear combinations of the torus wrapping numbers (n k a , m k a ) of the one-cycle along the Z (k) 2 -invariant two-torus T 2 (k) , where the linear combination is determined by a set of eight independent discrete parameters, e.g. (x (k) α,a , y (k) α,a ) = (±m k a , ∓(n k a + m k a )) for one of the cases where the given index α receives the sole contribution from a single (Z 6 orbit of a) Z (k) 2 fixed point. The eight discrete parameters can be divided into three types, with each type representing a different geometric characteristic of the exceptional divisor located at the Z (k) (i) three discrete displacement parameters ( σ a ): the 'bulk' cycle can pass through the origin (σ i a = 0) of the two-torus T 2 (i) , or it can be shifted by one-half of a lattice vector (σ i a = 1); (ii) two independent Z (k) : such a parameter indicates the orientation with which the D-brane wraps the exceptional divisor at a reference fixed point on the four-torus T 4 (k) ; we will loosely speaking say that a three-cycle encircles a fixed point 'clockwise' (τ fixed point on the two-torus T 2 (i) , namely with the same orientation (τ i a = 0) or opposite orientation (τ i a = 1) as the divisor at the reference point.
The interplay of displacements ( σ a ) and Wilson lines ( τ a ) is summarised in table 33 of appendix A along the relevant four-torus T 4 (1) ≡ T 2 (2) × T 2 (3) for the -independent global models discussed in this article. More details regarding the construction of the orbifold-invariant basis of three-cycles and the explicit expressions for the exceptional wrapping numbers (x (k) α,a , y (k) α,a ) in terms of the torus wrapping numbers (n k a , m k a ) can be found in [19], and we display the result here in table 1.
Exceptional wrapping numbers (x (z (1) α,a n 1 a , z (1) α,a m 1 a ) (ẑ (1) α,a n 1 a ,ẑ α,a n l a ζ (l) α,a n l a + ζ  Table 1: The exceptional wrapping numbers of type I stem from a single contribution of a fixed point orbit, while those of type II result from a Z 3 orbit contributing twice due to two different Z 2 fixed points on T 2 (2) × T 2 (3) . Details about the sign factor assignments z (1) α,a , ζ α,a ∈ {±1} as well asẑ (1) α,a ∈ {0, ±2} in terms of Z 2 eigenvalues and discrete Wilson lines can be found in section 2.1.3. of [36].
The basic three-cycles used to decompose the fractional three-cycle Π frac a in equation (5) do not correspond to ΩR-even and ΩR-odd three-cycles, which implies that also the bulk wrapping numbers (P a , Q a , U a , V a ) and exceptional wrapping numbers (x (k) α,a , y (k) α,a ) transform under the orientifold projection. Their transformation can be deduced from the transformation properties of the basis three-cycles under the ΩR-projection as summarised in table 2. To simplify the transformation rules for the Z (k) 2 twisted sectors, one introduces the sign factor η (k) : Orientifold images of bulk and exceptional three-cycles on T 6 /(Z2 × Z6 × ΩR) with discrete torsion (η = −1) twisted sectors also on the choice of exotic O6-plane orbit via the sign where the constraint is a simple rewriting of relation (4). Fractional three-cycles with their bulk part parallel to one of the four O6-plane orbits are characterised by ΩR-invariant bulk wrapping numbers, as can be easily checked from table 3. Note that a rectangular lattice  configuration for T 2 (1) also allows an O6-plane displaced by one-half of the lattice vector π 1 or π 2 (see figure 1), hence the factor N O6 = 2(1 − b) to denote the number of identical O6-plane orbits for the lattice configurations a/bAA. The transformation of the torus wrapping numbers under the ΩR-projection, depending on the two-torus lattice orientation, is summarised as: At the intersection points of two fractional three-cycles Π frac a and Π frac b , chiral matter can arise in the bifundamental representation of the gauge groups supported by the corresponding D6brane stacks. The amount of chiral matter is encoded in the net-chirality χ (Na,N b ) , which is computed as follows in the fractional three-cycle language reviewed above: Intersections of a fractional three-cycle Π frac a with its orientifold image and with the O6-planes lead to chiral matter in the symmetric and/or antisymmetric representation, counted by the net-chiralities χ Antia/Sym a : In this expression, the fractional three-cycle Π O6 only contains contributions from the bulk three-cycles of the O6-planes, as they do not carry RR charges coming from twisted sectors.
The formulae (12) and (13) compute the total net-chiralities for intersecting fractional D6branes, but they do not offer a glance at the contributions to the net-chirality per sector a(ω k b) k∈{0,1,2} . To obtain the net-chirality per sector, we can turn to the Z 2 invariant toroidal intersection numbers as introduced in appendix A of [22] for T 6 /Z 2N backgrounds and extended to T 6 /Z 2 × Z 2M orbifolds in [19,28]. For example, the amount of chiral states in the symmetric and antisymmetric representation per sector (ω n a)(ω n a) n=0,1,2 can be computed by the following formulae: (ω n a) represents the intersection number between an orbifold image three-cycle (ω n a) and the O6-plane ΩRZ (k) 2 on the underlying six-torus, with 2(1 − b) the number of parallel O6-planes set by the shape of T 2 (1) .
A complementary approach of calculating the total amount of matter per sector irrespective of its chirality can be taken by computing the beta-function coefficients, as outlined in e.g. table 7 of [28] or table 39 of [35]. This approach includes all vector-like states, in particular for D6branes at some vanishing angle, for which net-chiralities vanish. For the systematic searches of MSSM-like models in section 4 and left-right symmetric models in section 5 of this article, we cross-checked multiplicities of states by combining the three options, namely net-chirality, net-chirality per sector and total counting irrespective of chirality by means of contributions to the beta function coefficients.
Last but not least, the stability and consistency conditions for intersecting D6-brane models rely on the supersymmetric nature of the D6-branes and the cancellation of the RR tadpoles along the internal directions of the string compactification. The supersymmetry conditions for a fractional D6-brane boil down to the requirements that its bulk three-cycle is special Lagrangian on the underlying torus, while its Z 2 twisted sector RR tcc: Z (l) 2 twisted sector with l = 2, 3 a/bAA 2+2b,a ] = 0, When confronting the supersymmetry conditions for the bulk three-cycles with the bulk RR tadpole cancellation conditions, one notices that some choices of exotic O6-plane configuration are a priori ruled out for supersymmetric D6-brane model building, such as the specific choice η ΩRZ (1) 2 = −1 or any combination of three exotic O6-planes. Note that global intersecting D6-brane models are not only characterised by vanishing RR tadpoles, but also satisfy the K-theory constraints, which will be elaborated on in section 3.1 in the context of discrete Z n remnants of massive Abelian gauge symmetries, since the K-theory constraints boil down to the existence of a specific Z 2 symmetry.

Elements of Intersecting D6-brane Model Building
A first step towards intersecting D6-brane model building on T 6 /(Z 2 ×Z 6 ×ΩR) with discrete torsion consists in classifying the fractional three-cycles supporting enhanced SO(2N ) or U Sp(2N ) gauge groups, as the latter can be used to accommodate the SU (2) L left stack in the MSSM gauge factor and/or the SU (2) R right stack in left-right symmetric models. Furthermore, in section 3.1 we will use this classification when discussing the derivation of the K-theory constraints by means of probe U Sp(2) branes and when determining the conditions for the existence of discrete Z n symmetries. In order for a fractional D6-brane Π frac a to support an enhanced SO(2N ) or U Sp(2N ) gauge group, its bulk three-cycle has to be parallel to one of the four O6-plane orbits and its discrete parameters ( σ a , τ a ) have to satisfy a set of topological conditions involving also the individual tiltedness of the two-tori [19]. For the orientifold T 6 /(Z 2 × Z 6 × ΩR) with discrete torsion the topological conditions are written out explicitly in the second column of table 5. The other columns in table 5 review for which combinations of discrete displacements ( σ a ) and discrete Wilson lines ( τ a ) the U Sp(2N ) or SO(2N ) gauge group enhancement occurs in function of the choice of the exotic O6plane. The table also indicates the full amount of matter in the symmetric or antisymmetric representation under the respective gauge group arising in the three sectors (ω k a)(ω k a) a(ω −2k a ) k∈{0,1,2} . By counting the combinations of Z 2 eigenvalues, displacements and Wilson lines we can determine the numbers N U Sp and N SO of configurations giving rise to U Sp(2N ) and SO(2N ) enhancement, respectively: for a rectangular T 2 (1) (b = 0) we have N U Sp = 240 and N SO = 16, whereas N U Sp = 216 and N SO = 40 for a tilted T 2 (1) (b = 1 2 ). For more details concerning gauge group enhancement on T 6 /(Z 2 × Z 6 × ΩR) with discrete torsion, we refer to [36] where the classification was discussed for the first time.
In order to obtain phenomenologically appealing intersecting D6-brane models containing some MSSM-like or left-right symmetric model sector, the QCD stack and the SU (2) L leftstack ought to be constructed with rigid fractional three-cycles free of matter in the adjoint representation. This requirement prevents the corresponding gauge group to be spontaneously broken by (continuous) D-brane displacements or recombinations or Wilson lines. In [36] an exhaustive search for rigid fractional three-cycles without matter in the adjoint representation was presented, from which the following constraints on the fractional three-cycle for the a/bAA lattice can be distilled: (i) the bulk three-cycle is parallel to ΩR or to an orbit of the form (n 1 a , m 1 a ; 1, 0; 1, −1), (ii) the discrete parameters ( σ a , τ a ) satisfy the relation: σ 2 a τ 2 a = σ 3 a τ 3 a ∈ {0, 1}. All other discrete parameters, including the choice of the exotic O6-plane and Z 2 eigenvalues, do not affect the amount of matter in the adjoint representation. The one-cycle wrapping numbers (n 1 a , m 1 a ) are related to the complex structure modulus of the two-torus T 2 (1) through the necessary bulk supersymmetry condition in table 4: and thus cannot be chosen at random. Note, however, that these conditions do not include those fractional three-cycles that are ΩR-invariant and support an enhanced U Sp(2) gauge group accompanied solely by matter in the antisymmetric representation, as listed in table 5. This latter type of fractional three-cycles is well suited to support the SU (2) L and/or SU (2) R gauge group when constructing MSSM-like models or left-right symmetric models, respectively.
The absence of chiral matter in the symmetric representation under the QCD and the U (2) L gauge group requires us also to investigate the intersections between a fractional three-cycle and its orientifold image orbit. More explicitly, only fractional three-cycles with χ Sym a = 0 will be able to serve as candidate three-cycles to support the QCD stack or the U (2) L stack. In the case of the QCD stack, also the amount of chiral matter in the antisymmetric representation has to be constrained by the condition χ Anti ≤ 3. Otherwise, the QCD stack could be characterised by more than three generations of right handed u R (or d R ) quarks.
We can now implement these extra conditions on the two types of bulk orbits, parallel to the ΩR-plane or with representant (n 1 a , m 1 a ; 1, 0; 1, −1), with σ 2 a τ 2 a = σ 3 a τ 3 a specified above as sole configurations without matter in the adjoint representation: • The -independent configurations have their bulk orbit parallel to the ΩR-plane, leading automatically to a vanishing intersection number with the O6-planes, Π a • Π O6 = 0, and thus to the requirement χ Antia ≡ χ Sym a ! = 0.
• For the -dependent configurations with bulk orbit (n 1 a , m 1 a ; 1, 0; 1, −1), the constraint χ Antia ∈ {0, 1, 2, 3} reduces the number of potential tuples (n 1 a , m 1 a ) significantly, as summarised in table 6. The bulk RR tadpole cancellation conditions in table 4 can for -dependent models only be satisfied for the choice η ΩR = −1 of exotic O6-plane. For σ 2 a τ 2 a = σ 3 a τ 3 a = 1, the list with six bulk orbits for the aAA lattice and five bulk orbits for the bAA lattice in table 6 is exhaustive. For η ΩR = −1 and σ 2 a τ 2 a = σ 3 a τ 3 a = 0, only the bulk orbit (1, 1; 1, 0; 1, −1) on the aAA lattice and the bulk orbit (1, 0; 1, 0; 1, −1) on the bAA lattice satisfy all constraints on adjoint and (anti)symmetric representations, cf. table 12 in [36]. 3 3 Also η ΩRZ (2 or 3) 2 = −1 and σ 2 a τ 2 a = σ 3 a τ 3 a = 0 solve the constraints on the matter spectrum, but the second bulk RR tadpole cancellation condition in table 4 cannot be satisfied in a supersymmetric way, see [36] for more details. For this last choice of discrete parameters, we note for completeness that also the bulk orbit 13 All these considerations provide us with a set of candidate fractional three-cycles to support the QCD stack, both for -independent as well as for -dependent configurations.
Once a fractional three-cycle for the QCD stack is identified, we have to determine an appropriate fractional three-cycle for the SU (2) L stack, such that the intersections between the two stacks give rise to three chiral generations of left handed quarks. Thus, another indispensable element of D6-brane model building consists in finding configurations of two fractional three-cycles Π frac a and Π frac b with χ ab + χ ab ! = ±3. In case the left gauge group results from an enhanced U Sp(2) b gauge group on the b-stack, the condition on the total net-chirality reads χ ab ≡ χ ab = ±3 instead. In [36] an exhaustive search revealed a class of -independent D6-brane configurations with three chiral generations, for which the QCD-stack is parallel to the ΩR-plane and the SU (2) L stack is parallel to the ΩRZ 2 -plane is the exotic O6-plane. The fractional three-cycles for the QCD stack are free from matter in the adjoint representation, as well as free from chiral matter in symmetric and antisymmetric representations, while the fractional three-cycles for the SU (2) L stack support enhanced U Sp(2) gauge groups accompanied by five states in the antisymmetric (≡ singlet of U Sp(2)) representation (see table 5).
In order to obtain a full MSSM-like or left-right symmetric spectrum, the ( -independent) combinations of fractional three-cycles with three chiral left-handed quarks have to be completed with an appropriate c-and/or d-stack. For the D6 c -and D6 d -brane stacks, fractional three-cycles with a bulk orbit parallel to any ΩRZ (k) 2 -plane can serve as candidates, since these -independent supersymmetric D6-brane stacks are allowed to be accompanied by matter in the adjoint representation. Regarding chiral matter in the (anti)symmetric representation, one can easily verify that the constraint χ Antia ≡ χ Sym a ! = 0 is satisfied for all fractional three-cycles parallel to one of the ΩRZ (k) 2 -planes, independently of the choice of the exotic O6-plane, the Z (k) 2 eigenvalues, the discrete parameters (σ 1 a , τ 1 a ) or the choice of the lattice orientation. An intensive search for -independent global MSSM-like and left-right symmetric intersecting D6-brane models on the aAA lattice will be presented in sections 4.1 and 5.1, respectively. For these -independent D6-brane configurations, the choice of the exotic O6-plane will be either the ΩRZ (2) 2 -plane or the ΩRZ 2 -plane as dictated by the requirement of having three chiral generations of left-handed leptons. Remember that in addition to the open string matter spectrum, the massless closed string spectrum on the aAA lattice for η ΩRZ (2 or 3) 2 = −1 contains N = 1 supermultiplets with h + 11 = 4 Z 6 vectors, h − 11 = 3 bulk + 8 Z 3 + 4 Z 6 Kähler moduli and h 21 = 1 bulk + 14 Z 2 + 2 Z 3 + 2 Z 6 complex structure moduli as first computed in [19].

Intermezzo: towards three generations in -dependent configurations
The D6-brane configurations with the QCD stack wrapping a fractional three-cycle parallel to the ΩR-plane and the left stack parallel to the ΩRZ (1) 2 -plane are the only -independent configurations which yield three chiral generations of left-handed quarks without exotic matter as specified above (i.e. no matter in the adjoint representation nor chiral matter in the (anti)symmetric representation). Indeed, considering a left stack parallel to one of the other O6-planes does not offer the right amount of chiral left-handed quarks, as shown in table 14 of [36]. This prompts us to consider the alternative roads of -dependent D6-brane configurations consisting of two distinct choices, as expressed in tables 15 and 16 of [36]: (1) Choice 1: the QCD stack remains parallel to the ΩR-plane, but the SU (2) L stack has a bulk orbit (1, m b ; 1, 0; 1, −1) with m b ∈ {1, 3, 5, 7, 9, 11, 13, 15} for the aAA lattice and m b ∈ {0, 1, 2, 3, 4, 5, 6, 7} for the bAA lattice; (2) Choice 2: the QCD stack is characterised by a bulk orbit (1, m a ; 1, 0; 1, −1) with m a ∈ {1, 3, 4, 5, 6} for the aAA lattice and m a ∈ {0, 1, 2, 3, 4} for the bAA lattice, while the SU (2) L stack can be parallel to the ΩR-plane or to a bulk orbit (1, m b ; 1, 0; 1, −1) with m b ∈ {1, 3} for the aAA lattice and m b ∈ {0, 1, 4} for the bAA lattice, see table 16 in [36] for the exact configurations.
Other choices of (m 1 a , m 1 b ) are excluded by requiring the existence of three chiral quark generations.
The -dependence of the D6-brane configurations, deducible from equation (16), excludes the exotic O6-plane choices η ΩRZ (2) 2 = −1 and η ΩRZ (3) 2 = −1. That is to say, the bulk orbits preserving supersymmetry for specific -values are characterised by a bulk wrapping number V + bQ = 0, implying that the bulk RR tadpole cancellation conditions in table 4 are only satisfied when the ΩR-plane plays the rôle of the exotic O6-plane. A more detailed account of the search for -dependent configurations yielding three chiral left-handed quark generations is offered in section 3.4.2 of [36], accompanied by a precise counting of the number of consistent D6-brane configurations. This exhaustive scan for three generations of lefthanded quarks, however, still needs to be supplemented by the requirement of three righthanded quark generations and three lepton generations, after which global completions will have to be investigated. Such a systematic scan for global -dependent models is expected to be extremely time-consuming and cumbersome, and at this point we leave it for future work.

K-Theory Constraints and Discrete Symmetries
3.1.1 Basis of ΩR-even three-cycles and K-theory constraints The RR charges of a D6-brane wrapping a fractional three-cycle as in expression (5) are in first instance classified by (co-)homology theory, such that the required vanishing of RR charges on the orientifold T 6 /(Z 2 × Z 6 × ΩR) with discrete torsion can be easily recast into the conditions of table 4 in terms of homology, a N a (Π a + Π a ) = 4 Π O6 . However, not all D6-brane charges are captured by homology, as D-branes carry additional Z 2 valued Ktheory charges [38]. The presence of uncanceled K-theory charges in the compact internal space opens up the worrisome prospect of having an inconsistent compactification, even when the RR tadpoles vanish. We will call string vacua of this type semi-local.
In general, it is rather difficult to directly determine the conditions for vanishing K-theory charges on compact spaces, yet by using a probe brane argument [39] one can deduce necessary conditions for the vanishing of the K-theory charges: This expression requires an even number of states in the fundamental representation of any U Sp(2) probe brane, which is counted by the number of intersections of the set of D6-branes in the model weighted by the corresponding ranks. Violations of condition (17) indicate the presence of a global field-theoretical anomaly in the SU (2) U Sp(2) gauge theory on the probe brane [40]. Thus, for a given D6-brane configuration with vanishing RR tadpoles, global consistency also requires that the K-theory constraints in (17) are satisfied.
To assess the K-theory constraints, one requires the full classification of ΩR-invariant fractional three-cycles supporting an enhanced U Sp(2) gauge group as reviewed in table 5 of section 2.2. Note, however, that not every ΩR-invariant fractional three-cycle with U Sp (2) gauge group leads to an independent constraint. In practice, we expect at most b bulk+Z 2 3 /2 = h bulk+Z 2 21 + 1 = 16 linearly independent conditions associated to the number of linearly independent ΩR-even three-cycles on the orientifold T 6 /(Z 2 × Z 6 × ΩR) with discrete torsion (η = −1). In order to reduce the number of independent constraints resulting from equation (17), we first determine all ΩR-even and ΩR-odd three-cycles, which are either purely of bulk or exceptional type, using table 2. The result is displayed in table 7 for the choice η ΩRZ (3) 2 = −1 of exotic O6-plane on the aAA lattice. 4 In the next step, we can express the 4 The constraints for the bAA lattice can be obtained in an analogous manner by (i) replacing m 1 a →m 1 a = m 1 a + b n 1 a which amounts to (Ua, Va; y 2 )-plane on T 2 (1) now pass through fixed points {1, 4} σ 1 a =0 and {2, 3} σ 1 a =0 , see figure 1. However, we anticipate here that we only find three-generation models with cancelled RR tadpoles on the aAA lattice as detailed in sections 4 and 5.
ΩR-invariant fractional three-cycles with U Sp(2) gauge group in terms of the basis of purely bulk/exceptional ΩR-even three-cycles and deduce which fractional three-cycles are truly linearly independent. More explicitly, by choosing different combinations of discrete parameters, one can easily show that various fractional three-cycles can be written as linear combinations of other fractional three-cycles, allowing us to reduce the 240 (216) ΩR-invariant fractional three-cycles with U Sp(2) gauge group on the aAA (bAA) lattice to only sixteen linearly independent combinations.
ΩR-even and -odd three-cycles on Table 7: Overview of the ΩR-even and ΩR-odd pure bulk or exceptional three-cycles on for the aAA lattice. The right column lists all intersections number of the symplectic lattice (vanishing intersection numbers are omitted).
As concrete example we consider the aAA lattice with the choice of exotic O6-plane η ΩRZ (3) 2 = −1 and the fractional three-cycles with bulk orbit parallel to the ΩRZ (1) 2 -plane and show how they can be written as linear combinations of the fractional three-cycles parallel to the ΩR-plane. As indicated in table 5, fractional three-cycles parallel to the ΩRZ (1) 2 -plane can support an enhanced U Sp(2) gauge group for σ 2 a τ 2 a = 1. For the displacement parameters ( σ a ) = (σ 1 a , 1, 1), the discrete Wilson lines have to be chosen as ( τ a ) = (τ 1 a , 1, 0) in order to guarantee an enhanced U Sp(2) gauge group. For this explicit choice of discrete parameters, the fractional three-cycles parallel to the ΩRZ (1) 2 -plane read as follows in terms of the basis of purely bulk/exceptional ΩR-even three-cycles: 5 Π frac,(σ 1 a ,1,1) a↑↑ΩRZ By looking at the fractional three-cycles parallel to the ΩR-plane with the following choice of discrete parameters ( σ a ) = (σ 1 a , 1, 1) and ( τ a ) = (τ 1 a , 0, 1) allowing for an enhanced U Sp(2N ) gauge group (see table 5): Π frac,(σ 1 a ,1,1) a↑↑ΩR we can easily deduce the following relation among the fractional three-cycles: This relation explicitly shows that the fractional three-cycles parallel to the ΩRZ eigenvalues. Hence, the fractional three-cycles parallel to the ΩRZ (1) 2 -plane do not lead to independent K-theory constraints. Applying such reasonings we can reduce the number of Ktheory constraints to the maximally possible number of 16 linearly independent constraints, associated to fractional three-cycles parallel to the ΩR-plane or ΩRZ Note that we have already used the RR tadpole cancellation conditions here to simplify the K-theory constraints, and that three conditions, namely in rows 2, 4 and 8, are now trivially satisfied.

Massless Abelian Symmetries and Discrete Gauge Symmetries
Even though the RR tadpole cancellation conditions often do not suffice to guarantee the global consistency of Type IIA orientifold compactifications with intersecting D6-branes, they do guarantee the absence of non-Abelian gauge anomalies in the effective four-dimensional field theory. Mixed Abelian/non-Abelian as well as purely Abelian gauge anomalies vanish due to the generalised Green-Schwarz mechanism. In this process, some U (1) gauge symmetry acquires a mass through Stückelberg couplings to closed string axions. More concretely, the dimensional reduction of the ten-dimensional RR-forms C (3) and C (5) along the basis of ΩReven and ΩR-odd three-cycles, provides a set of closed string axions φ i and their Hodge-dual two-forms B i (2) in four dimensions, with i ∈ {0, 1, . . . , h 21 }. The reduction of the Chern-Simons action for the stack of D6 a -branes provides a set of Stückelberg couplings to the U (1) a ⊂ U (N a ) gauge group with field strength F a and a set of couplings to the SU (N a ) × U (1) a field strengths G a involving the closed string axions: These two types of terms combined provide the Green-Schwarz couplings necessary to cancel the mixed gauge anomalies of the purely Abelian type U (1) a − U (1) 2 b and the Abelian/non-Abelian type U (1) a − SU (N b ) 2 . The (rational) wrapping numbers r i a and s i a follow from decomposing the fractional three-cycle Π a with respect to the basis of ΩR-even and ΩR-odd three-cycles: A linear combination of U (1)'s, say U (1) X = a q a U (1) a with q a ∈ Q, remains as massless anomaly-free U (1) gauge symmetry if all its associated Stückelberg couplings in equation (23) vanish. The vanishing of the Stückelberg couplings can be rewritten in terms of the following set of topological conditions: Massive Abelian U (1) symmetries obviously do not satisfy this condition. Instead they couple to (some linear combination of) a closed string axion φ i and acquire mass through the Stückelberg mechanism. At energies below the Stückelberg mass scale, these U (1) symmetries behave as perturbative global symmetries that are broken further to discrete Z n symmetries [41] by non-perturbative corrections. The existence conditions for discrete Z n symmetries can also be written through a set of topological conditions: In order to unambiguously determine the correct value of n, the coefficients k a ∈ Z are chosen such that they lie within the interval 0 ≤ k a < n and satisfy the condition gcd(k a , k b , . . . , n) = 1. In case all the coefficients satisfy k a ≡ 1 (∀a), we reproduce the K-theory constraint equations in (17), which in turn imply the existence of a discrete Z 2 symmetry. Note that the interpretation of the K-theory constraint as Z 2 symmetry is only valid if the full lattice of ΩR-even three-cycles can be spanned by cycles, which support U Sp(2) gauge factors (and not SO(2) gauge groups) as in the present situation.
In order to clarify the conditions (26) on the existence of discrete Z n symmetries in the lowenergy effective field theory, we work them out explicitly for the orientifold T 6 /(Z 2 × Z 6 × ΩR) with discrete torsion η = −1. Anticipating the results of our search for global MSSM and left-right symmetric models, we restrict our discussion to the aAA lattice configuration with the ΩRZ 2 -plane as the exotic O6-plane. For this configuration, the basis of ΩR-even and ΩR-odd three-cycles of pure bulk/exceptional type in table 7 does not form a uni-modular lattice, given that it satisfies the relations: As an immediate consequence, the wrapping numbers r i a and s i a for the fractional three-cycles (24) on this lattice are therefore rational numbers taking value in 1 8 Z in agreement with the general form of the expansion displayed in equation (5). It also implies that the discrete Z n symmetry conditions obtained from (26) by using the basic purely bulk or exceptional three-cycles Π even i in table 7 do not provide for all constraints, but rather only provide for a set of sixteen necessary conditions [32,42,34]. Said differently, the cycles Π even i of table 7 only form a sublattice of the full lattice of ΩR-even three-cycles, as clearly suggested by the structure of their intersection form in (27). The necessary conditions on the existence of discrete Z n symmetries can be written out in terms of bulk and exceptional wrapping numbers as follows: These conditions have to be supplemented with a set of sufficient conditions which derive from (26) by taking the set of linearly independent ΩR-even fractional three-cycles supporting an enhanced U Sp(2N ) or SO(2N ) gauge group. For the aAA lattice configuration with η ΩRZ (3) 2 = −1, fractional three-cycles parallel to the ΩRZ 2 -plane support an enhanced SO(2N ) gauge group for the choice of the discrete displacements ( σ a ) = (σ 1 a , 1, 1) and the discrete Wilson lines ( τ a ) = (τ 1 a , 1, 1), see table 5. The corresponding three-cycles can be written down in terms of the ΩR-even basis of table 7: Π frac,(σ 1 a ,1,1) a↑↑ΩRZ (29) Writing out the fractional three-cycles parallel to the ΩRZ 3 -plane and supporting an enhanced U Sp(2N ) gauge group for the choice of discrete parameters ( σ a ) = (σ 1 a , 1, 1) and ( τ a ) = (τ 1 a , 0, 0): allows us to deduce the following relation among the fractional three-cycles: This relation implies that the fractional three-cycles with an enhanced SO(2N ) gauge group do not provide for additional conditions, and the sixteen linearly independent fractional threecycles found when deriving the K-theory constraints suffice -as expected -to derive the sufficient conditions on the existence of some Z n gauge symmetry: 1,a +y 1,a )+y 2,a )+y 1,a +y Each entry in the necessary and sufficient conditions corresponds to an intersection number with some ΩR-even three-cycle and is thus integer-valued. Ultimately, of course only h bulk+Z 2 21 + 1 = 16 conditions on the existence of Z n symmetries are independent, but for practical purposes it is usually convenient to first check the simpler set of necessary conditions and then refine the search by verifying which of the candidate n also obey the sufficient conditions.
Massless Abelian gauge symmetries correspond to those choices of (k a , k b , . . .), for which the entries in each line of equations (28) and (32) add up to exactly zero (without 'mod n'). We will clarify these considerations through the explicit examples in sections 4.2 and 5.2.
As shown in [41,32], discrete Z n symmetries are left unbroken by the non-perturbative effects inherent to string theory, such as Euclidean D-brane instantons. In this respect, gauged Z n symmetries constrain (also) the form of the non-perturbative part of the four-dimensional superpotential, whereas the massive Abelian U (1) symmetries only constrain the perturbative superportential. This observation matches nicely the field theoretic motivation for the existence of discrete symmetries to explain the absence of dangerous lepton/baryon-number violating operators in supersymmetric field theories [43,44]. Any appropriate discrete symmetry should allow for the presence of the traditional Yukawa couplings, such that a generic discrete Z n symmetry in the MSSM with generator g n can be decomposed [43] in terms of three independent generators R n = e i2πR/n , L n = e i2πL/n and A n = e i2πA/n : under which the MSSM states are charged as follows, The charges of the MSSM fields are chosen such that the standard Yukawa couplings are allowed by the discrete Z n symmetry generated by g n . In section 4.2 we will investigate the discrete symmetries in a global five-stack intersecting D6-brane model with a MSSMlike gauge group and spectrum, and compare the discrete symmetries to the decomposition in (33). Taking into account the anomaly constraints concerning the discrete Z n symmetries eliminates all but three discrete symmetries compatible with the MSSM: matter parity R 2 , baryon-triality B 3 ≡ R 3 L 3 and proton-hexality P 6 ≡ R 5 6 L 2 6 . A recurring example of two states whose identification is not always straightforward is the candidate left-handed leptons L versus the candidate down-type HiggsinosH d in MSSM-like D6-brane models. Furthermore, intersecting D6-brane models also come with various massless singlet fermions under the visible gauge factor, which at first sight are all able to serve as candidate right-handed neutrinos ν R . To identify the matter on massless open string states unambiguously, we have to determine the Yukawa and other three-point couplings involving the left-handed leptons, the Higgses H d and/or the right-handed neutrinos ν R . Apart from a correct identification of the chiral spectrum, the computation of the Yukawa couplings also forms an essential litmus test to assess how close a consistent string theory model comes to real-world physics. The Yukawa couplings arising from a string compactification should for instance be able to exhibit the mass hierarchies among the different quark and lepton generations.

Yukawa and Other Cubic Couplings
Generally, determining the allowed Yukawa and three-point couplings consists of two steps. First of all, a three-point coupling is allowed whenever it satisfies the charge selection rule: a set of three massless open string states φ x ab ∈ Π frac combines into a three-point coupling in the perturbative part of the superpotential W, provided that the total three-point coupling forms a singlet representation under the full gauge factor (including hidden gauge groups). In this expression the subscripts a, b and c refer to the fractional three-cycles Π frac a , Π frac b and Π frac c of the corresponding D6-branes whose intersections provide for the massless states, while the superscripts x, y, z are related to the multiplicity or generation of the respective massless state. Invariance under the full gauge group also implies invariance under massless Abelian gauge symmetries and gauged discrete Z n symmetries. In this respect, non-trivial discrete Z n symmetries, which are not homomorphic to the centre of some non-Abelian gauge factor, are able to rule out nonperturbative m-point couplings, analogously to their field theoretic "raison d'être" discussed at the end of the previous section. An explicit example of a non-trivial discrete Z 3 symmetry is presented below in section 4.2 for a prototype global five-stack MSSM-like D6-brane model, which is characterised by an abundant collection of up-type Higgses (H u ,H u ) and downtype Higgses (H d ,H d ). The Higgs doubletsH u andH d carry different Z 3 -charges from their untilted counterparts, from which one can immediately deduce that the Yukawa coupling Other examples of Z 3 -forbidden and -allowed couplings will be discussed in section 4.2.
A second criterium for the existence of the three-point coupling in (35)  (i) , whose apexes correspond to the D6-brane intersections at which the massless states φ x ab , φ y bc and φ z ca are located. In a more formal language [45,46], the one-cycles of the factorisable bulk three-cycles Π bulk where A (i) xyz represents the area of the closed triangle [a, b, c] on the two-torus T 2 (i) . In case the three-cycles a, b and c intersect in a single point on a two-torus, the corresponding contribution to the amplitude W xyz is of the order O(1). When the three-cycles a, b and c do not form a closed sequence (on at least one of the three two-tori), the coefficient W xyz vanishes. Notice that while charge selection and stringy selection coincide on the mere six-torus, for orbifolds the stringy selection rule plays a vital rule due to the existence of orbifold image cycles (ω k a) on the underlying torus. In all cases with a closed triangle of non-vanishing size, the amplitude W xyz is exponentially suppressed by the area A (i) xyz which scales with the Kähler modulus v i measuring the area of the two-torus T 2 (i) . The expression in (36) corresponds to the worldsheet instanton at leading order, and an infinite set of copies with larger areas will refine the size of the coupling [46].
A first consideration to take into account is that the form of the amplitude in (36) is valid for the ambient space T 6 , thus neglecting a potential overall numerical factor 1/(2 · 6) accounting for the Z 2 × Z 6 orbifold geometry. In this respect, expression (36) should be considered as a reasonable order of magnitude for the three-point coupling, such that it allows for instance to identify hierarchies among the Yukawa couplings in intersecting D6-brane models on T 6 /(Z 2 × Z 6 ×ΩR). In the absence of cubic couplings, one can conceive perturbative non-renormalisable higher m-point couplings which are string mass scale suppressed with the appropriate power M 3−m string and where the worldsheet instanton takes the shape of an m-polygon, in the same spirit as the construction for the cubic couplings outlined above. Next, we also point out that the expressions in (35) and (36) only contain the classical part of the coupling. The quantum contribution to the Yukawa coupling takes into account the proper normalisation of the matter fields φ x ab , φ y bc and φ z ca and can be deduced by computing four-point scattering amplitudes involving the matter fields as external legs [47][48][49]. The normalisation of a matter field is in principle proportional to its Kähler metric upon dimensional reduction to four dimensions, and the Kähler metrics can contribute to establishing mass hierarchies among the different particle generations [30,31]. The Kähler metrics for the matter fields on the orbifold T 6 /(Z 2 × Z 6 × ΩR) can also be deduced from the one-loop computation of the running gauge couplings [50,28,29], offering an alternative (and often simpler) method to obtain the proper normalisation of the matter fields. We end our list of considerations with a specific feature of the toroidal orientifold T 6 /(Z 2 × Z 6 × ΩR) regarding various three-point couplings: The invariance of the first two-torus T 2 (1) under the Z 6 orbifold action in equation (8) indicates that a bulk three-cycle a will have orbifold images (ωa) and (ω 2 a) parallel to the original bulk three-cycle on T 2 (1) . This immediately implies that three bulk three-cycles a, b and c with identical torus wrapping numbers (n 1 , m 1 ) along T 2 (1) will have a vanishing three-point coupling on the ambient space T 6 , which might possibly be subsequently generalised to threepoint couplings involving their orbifold images. For such cases we will nevertheless compute the (classical) contributions to the amplitude associated to the ambient space T 2 (2) × T 2 (3) . This approach is motivated by the fact that the vanishing of the Yukawa coupling on just the six-torus is related to the extended N = 2 supersymmetry if one angle vanishes, while on T 6 /(Z 2 × Z 2M × ΩR) only N = 1 supersymmetry is preserved due to the Z 2 symmetries, and m-point couplings on such orbifolds containing Z 2 symmetries have to our best knowledge not been computed so far -in particular the option to have a non-vanishing classical contribution remains. To clarify some of the points discussed in this section, we will compute various Yukawa and other cubic couplings for the global five-stack MSSM-like D6-brane model in section 4.3 and for the global six-stack left-right symmetric D6-brane models in section 5.3.

Global -independent configurations
As shown in [36] and reviewed in section 2.2, -independent D6-brane configurations yielding three left-handed quark generations without excessive exotic matter can only be realised for the following bulk three-cycles: provided that either the ΩRZ = −1). In a next step, we complete the MSSM gauge groups and chiral spectrum by embedding additional U (1) gauge factors on fractional three-cycles that are supersymmetric for all values of the complex structure modulus . 6 The three generations of right-handed quarks and left-handed leptons then ought to be realised at the intersections between these U (1) D6-brane stacks, the QCD stack and the SU (2) L stack, according to table 8 for threestack and four-stack D6-brane models. The explicit construction of the chiral MSSM-like spectrum with three-stack and four-stack D6-brane models is further constrained [51,52] by the realisation of the U (1) Y hypercharge as a linear combination of the U (1) gauge groups: where x c , x d ∈ {±1}. In first instance, one notices that -if at all -only the relative sign between the charges Q c and Q d might provide distinguishable physical situations under the assumed D6-brane set-up in equation (37), for which none of the right handed d R quarks are realised as chiral states in the antisymmetric representation of the QCD gauge group. 7 For the three-stack models we can pick x c = 1, as the other sign choice reproduces the same chiral spectrum upon exchanging c ↔ c . Hence, by including the orientifold images of all fractional three-cycles in the set of candidate c-stacks to complete the three-stack MSSM-like model, we cover both choices for x c . Similarly, the choices (x c , x d ) = −(1, ±1) are equivalent 6 It was argued in [36], based on the bulk RR tadpole conditions in table 4 with η ΩRZ (2 or 3) 2 = −1, that the supersymmetric D6-brane configurations in (37) can only be completed consistently using fractional D6-branes with bulk wrapping number V + bQ = 0. From the classification of supersymmetric three-cycles in appendix A of [36], one can then deduce the four candidate bulk orbits listed in table 9, which happen to be supersymmetric for all values of the complex structure modulus . These considerations thus exclude ab initio the possibility to use -dependent supersymmetric fractional three-cycles to account for missing U (1) gauge factors when completing the MSSM gauge group. Also any potential hidden sector will consist of -independent D6-branes. 7 More exotic expressions [52] for the hypercharge, such as QY = 1 6 Qa + 1 2 Qc ± 3 2 Q d , are also excluded based on the consideration that here the dR quarks cannot be realised through chiral states in the antisymmetric representation located in the aa sector of the QCD stack.   (37) is (x c , x d ) = (1, 1), for which the phenomenological constraints for MSSMlike spectra on the topological intersection numbers are listed in table 8. When identifying suitable bulk orbits, first intuition is provided by the bulk RR tadpole cancellation conditions for η ΩRZ (2 or 3)

Overview of topological intersection # for chiral MSSM spectrum
which help us to exclude various options. More precisely, in order not to overshoot the bulk RR tadpole cancellation conditions, the bulk wrapping numbers have to satisfy 2P x + Q x ≤ 20 (8) and V x + b Q x = 0 for x ∈ {c, d} on the aAA (bAA) lattice. 8 The bulk orbits that are su- 8 Observe that the bulk wrapping numbers of supersymmetric D6-branes always satisfy the conditions 2Px + Qx 0 and −(Vx + bQx) 0, resulting from the bulk supersymmetry conditions in table 4 upon using the expansionss in one-cycle wrapping numbers in eq. (7). persymmetric irrespective of the -value and satisfy the latter constraint are listed in table 9 for both lattices a/bAA.
Overview of SUSY bulk three-cycles in compliance with eq. (39) ∀ aAA lattice bAA lattice bulk wrapping numbers Table 9: The bulk wrapping numbers of -independent supersymmetric three-cycles on the a/bAA lattices with V +bQ = 0. The last bulk orbit on the bAA lattice does not play a rôle in supersymmetric model building as it overshoots the first bulk RR tadpole cancellation condition in equation (39).
Based on the list in table 9 we can speculate which combinations of bulk orbits for the c-stack and the d-stack would allow for favourable MSSM-like configurations. We have to make sure that the choice of the bulk orbits does not lead to a violation of the first bulk RR tadpole condition in (39). Thus, given the implied constraint 2P x + Q x < 8 for x ∈ {c, d} on the bAA lattice, this boils down to considering the three-and four-stack configurations as listed in table 10. Hence, the c-stack and the d-stack can only have bulk orbits parallel to the ΩR-plane. Table 10: Combinations of supersymmetric bulk orbits for three-stack and four-stack models aiming at -independent configurations of the MSSM spectrum on the bAA lattice for T 6 /(Z 2 × Z 6 × ΩR) with discrete torsion (η = −1) and exotic O6-plane η ΩRZ (2 or 3) 2 = −1. The second and third column indicate the bulk orbit for the c-stack and d-stack, respectively, the fourth and fifth column test whether the bulk RR tadpole cancellation conditions (39) are not violated, the second-to-last column verifies if three right-handed quark generations can be realised as prescribed by table 8, and the last column does the same for three left-handed lepton generations, with the subscript indicating the number of combinatorial possibilities of ( σ x ), ( τ x ) and relative (−) ∆τ Z (k) Once the bulk RR tadpole cancellation conditions are verified, we also have to check whether the intersections between the c-stack (and d-stack) and the QCD stack in the set-up of equation (37) can provide for three chiral generations of right-handed quarks d R and u R , i.e. |χ ac (+χ ad )| = 3 = |χ ac (+χ ad )| where the sign of the net-chirality has to be chosen opposite to the one of the net-chirality χ ab U Sp(2) b ≡ χ ab , as indicated in table 8. As the corresponding fractional three-cycles for the c-stack (and d-stack) should at this point not support an enhanced U Sp(2) gauge group, the discrete displacement parameters ( σ) and discrete Wilson lines ( τ ) have to be chosen accordingly for the respective fractional three-cycles. And by verifying the topological intersection numbers for all candidate fractional three-cycles on the bAA lattice, we end up with the results in table 10, from which we conclude that three-stack (and four-stack) configurations with three chiral right-handed quark generations cannot be found for the cases where the c-stack (and the d-stack) is (are) parallel to the ΩR-plane. In summary, three-stack and four-stack intersecting D6-brane models on the bAA lattice do not allow for -independent global MSSM-like models.

3-or 4-stack combinations with gauge group
Next, we turn our attention to the aAA lattice and repeat the same reasoning as above. Upon identifying which bulk orbits do not overshoot the first bulk RR tadpole cancellation condition in equation (39), we can list all potential combinations of bulk orbits for the c-stack and the d-stack in table 11 to identify potential three-stack and four-stack configurations, with the definition of the hypercharge given in equation (38). All but one of the nine combinations comply with the bulk RR tadpole cancellation conditions, but only three combinations of four-stack D6-brane models give rise to three chiral generations of right-handed quarks u R and d R . In the last column of table 11 we also indicate whether the three-stack and four-stack configurations yield three chiral generations of left-handed leptons. In this way we end up with the combinations (5,6,8) for which three generation intersecting D6-brane models with chiral quarks and left-handed leptons can be constructed.
Looking further into these three combinations of table 11, we find that the combinations 6 and 8 allow for D6-brane configurations with three generations of right-handed quarks and/or three generations of left-handed leptons, and the bulk RR tadpoles are saturated by just the four stacks required to engineer the MSSM gauge group. Note that only a fraction of the fractional D6-brane configurations represented by the combinations 6 and 8 allow for three generations of right-handed quarks and left-handed leptons simultaneously. More explicitly, for both combinations 6 and 8 we found 576 D6-brane configurations with three generations of right-handed quarks and 201024 configurations with three generations of lefthanded leptons. Yet there exist only 144 configurations where the three generations of righthanded quarks u R and d R are compatible with the three generations of left-handed leptons. 9 The identical counting of models in configurations 6 and 8 agrees with the expectation that they yield physically equivalent models upon exchanging (c, d) ↔ (d, c). An insurmountable obstruction to completing these D6-brane configurations into global intersecting D6-brane models, however, is the observation that bulk RR tadpoles for these cases are always saturated, while the twisted RR tadpoles are never cancelled, regardless of the specific fractional D6brane configuration under consideration. We remark that the configuration counting reflects 9 The numbers are given here for the exotic O6-plane choice η ΩRZ (3) 2 = −1, but the same numbers are also valid for the choice η ΩRZ (2) 2 = −1, as expected from the permutation symmetry T 2 (2) ↔ T 2 (3) for the aAA lattice. Table 11: Combinations of supersymmetric bulk orbits for the c-and d-stacks aiming at -independent three-stack and four-stack D6-brane configurations of the MSSM on the aAA lattice for T 6 /(Z 2 × Z 6 × ΩR) with discrete torsion (η = −1) and exotic O6-plane η ΩRZ (2 or 3) 2 = −1. The second and third column indicate the bulk orbit for the c-stack and d-stack, respectively, the fourth and fifth column test whether the bulk RR tadpole conditions (39) are not violated, the second-to-last column verifies if three right-handed quark generations can be realised through |χ ac (+χ ad )| = 3 = |χ ac (+χ ad )|, and the last column does the same for three left-handed lepton generations with |χ bc (+χ bd )| = 3, with the proper relative sign among the net-chiralities as dictated by table 8. For combinations (5,6,8), three chiral generations of both q R and L can be realised, and the compatibility between the two constraints is indicated by the symbol in parenthesis in the last column. The subscript indicates the number of combinatorial possibilities of ( σ x ), ( τ x ) and relative (−) ∆τ Z (k) 2 xy for x, y ∈ {a, b, c, d} and one given choice of exotic O6-plane η ΩRZ (2 or 3)

Three-stack combinations with gauge group
x , but identical absolute displacements ( σ x ) and Wilson lines ( τ x ) have been counted as one independent configuration only, as they all provide identical chiral and non-chiral massless spectra and field theoretical results at the current state-of-the-art, i.e. gauge couplings and Kähler metrics with formal expressions collected in [28]. Further identifications might exist for combinations 6 and 8, but due to the local character of these models, we will not pursue this issue here but only explore it further in the context of global models, where all RR tapdoles are cancelled and the K-theory constraints are satisfied.
Combination n • 5 on the other hand leads to a class of global five-stack MSSM-like models with initial gauge group and as required by the permutation symmetry of T 2 (2) ↔ T 2 (3) the same number for the choice η ΩRZ (3) 2 = −1, as we checked explicitly. Out of the 576 local four-stack models in table 11, only 288 can account for three generations of right-handed electrons e R and satisfy all RR tadpoles for the maximal hidden gauge group U (4) h . The K-theory constraints are then automatically satisfied, as we explicitly checked. Again, the number 288 counts fractional D6-brane configurations with different combinations of relative Z (i) 2 eigenvalues and absolute discrete displacements and Wilson lines. Thus, the 288 D6-brane configurations correspond to the maximal set of physically inequivalent D6-brane configurations. One can nevertheless show that the chiral and non-chiral massless spectra for the 288 D6-brane configurations are all identical (upon a potential exchange of c ↔ d and h ↔ h ), suggesting a further reduction to a smaller set (maybe even a unique version) of physically inequivalent D6-brane configurations by virtue of to date unknown additional maps between non-identical relative discrete parameters.
An explicit sample of fractional D6-branes providing such a global five-stack MSSM-like model is given in table 12 for η ΩRZ (3) 2 = −1, and the resulting massless spectrum is summarised in table 13. In the next section we will determine the massless U (1) symmetries and the discrete Z n symmetries for this model, yet the charges under the massless hypercharge U (1) Y and the discrete Z 3 symmetry are already indicated in table 13 for all massless states. For later reference, we also list the charges under the massive Peccei-Quinn symmetry, Q P Q ≡ Q c − Q d . Note that the absence of a massless U (1) B−L symmetry slightly complicates the proper identification of the chiral MSSM states as it prevents an unambiguous distinction between the chiral states corresponding to the left-handed lepton multiplets L and those corresponding to the down-type Higgs multiplets H d andH d .
Furthermore, a closer look at the chiral spectrum shows an abundance of right-handed downquarks d R from the ac sector and left-handed leptons L from the bd sector. The proper identification of the first follows by looking at possible Yukawa couplings among the quarks. More precisely, charge conservation only allows the following types of Yukawa couplings: where we indicated explicitly the xy sectors from which the states emerge as a superscript. With respect to the MSSM gauge group (SU (3) a × U Sp(2) b ) U (1) Y the three chiral states d R from the ad sector form hermitian conjugates to the right handed down-quarks from the ac sector. In order for these states to be heavy, we consider cubic couplings involving three of D6-brane configuration of a global 5-stack MSSM configuration on the aAA lattice wrapping numbers with discrete torsion (η = −1) and the ΩRZ the six down-quarks d R from the ac sector and some Standard Model singlet states Σ cd : where Σ cd can receive a non-vanishing vacuum expectation value. A similar consideration is valid for the Higgses from the bc sector and the three surplus left-handed leptons from the bd sector, which can be combined into cubic couplings of the form: Using the argument of charge conservation, we can schematically write down cubic couplings which are expected to lift the abundant d R -quarks, H u Higgses and three of the six leptons L upon giving a vev to the Standard Model singlet states Σ cd . A more in-depth analysis involving the stringy selection rules will be performed in section 4.3, where we will verify explicitly whether such mechanisms can be invoked to give masses to the abundant vectorlike pairs of matter states in table 13 and effectively obtain a three-generation MSSM-like model with continuous gauge group Overview of the massless matter spectrum for global 5-stack MSSM on the aAA lattice

Discrete Symmetries
Next, we focus on the phenomenological aspects of the global five-stack MSSM-like model presented in table 12 starting with revealing the presence of discrete symmetries. The main motivation to discuss discrete symmetries for this model consists in potentially prohibiting undesired cubic and/or baryon/lepton-number violating couplings and in reinforcing the interpretation of the chiral spectrum presented in table 13 by virtue of non-trivially acting gauged Z n symmetries. To this end, we first write down the necessary existence conditions (28) for the D6-brane configuration given in table 12: A row-by-row comparison clearly shows that ten of the conditions are trivially satisfied, and that the remaining six conditions correspond to only three independent conditions: 37 These relations have to be supplemented with the sufficient existence conditions (32): where various rows turn out to be linearly dependent of each other, and some of the rows (i.e. rows 2, 4, 6 and 14) yield the same conditions as the necessary conditions in (45). Moreover, the last sufficient condition in (46) is a linear combination of the first and fifth sufficient condition in the third block, such that the sufficient conditions only provide three linearly independent constraints: The sufficient conditions allow to further reduce the number of linearly independent necessary conditions: more explicitly, subtracting twice the second condition in (47) from the third condition in (47) corresponds to the second constraint in (45). Adding the first condition in (45) to two times the third condition in (47) reproduces the third constraint in (45). Hence, there are effectively four linearly independent constraints, i.e. the first condition in (45) and the three conditions in (47), which agrees with the existence of four Abelian gauge factors U (1) a × U (1) c × U (1) d × U (1) h as starting point. In order to identify the Abelian massless and the massive Z n symmetries for the global five-stack MSSM-like model, the four linearly independent conditions from (45) and (47) have to be satisfied simultaneously. A first observation is that the linear combination Q Y = 1 6 Q a + 1 2 Q c + 1 2 Q d satisfies the constraints 38 exactly, for any value of n, implying that this linear combination of U (1)'s corresponds to the massless hypercharge, in line with the discussion surrounding equation (38). In our search for discrete Z n symmetries, this massless hypercharge can be used to set the Z n charges of the left-handed quarks to zero.
The full set of solutions to the constraints (45) and (47) can then be summarised as: • The configuration (k a , k c , k d , k h ) = (1, 0, 0, 0) is a discrete Z 3 symmetry homomorphic to the centre of the SU (3) a gauge symmetry, playing the rôle of a baryon-like discrete symmetry. Upon a massless hypercharge rotation this discrete symmetry acts trivially on the visible and hidden sector.
• The configuration (k a , k c , k d , k h ) = (0, 0, 0, 1) corresponds to the discrete Z 4 symmetry homomorphic to the centre of the 'hidden' gauge group SU (4) h and acts only nontrivially on exotic states charged under the hidden gauge group, reproducing the same charge selection rule as the non-Abelian 'hidden' SU (4) h .
• The linear combination (k a , k c , k d , k h ) = (1, 1, 1, 1) corresponds to the discrete Z 2 symmetry guaranteed by the K-theory constraints, which acts trivially on the massless spectrum upon a rotation over the massless hypercharge. Note that this Z 2 symmetry corresponds to a linear combination of the Z 2 symmetry hiding within the massless hypercharge and the Z 2 symmetry within the former Z 4 symmetry. As such, the discrete Z 2 symmetry associated to (k a , k c , k d , k h ) = (1, 1, 1, 1) should not be considered as an independent discrete symmetry.
• Finally, we also find a discrete Z 6 symmetry for the combination (k a , k c , k d , k h ) = (0, 2, 4, 1), for which the charges of the massless open string states are listed in the last column of table 13. Nonetheless, the order 6 does not correspond to a viable discrete symmetry in the low-energy effective field theory, as this Z 6 can be reduced to a discrete Z 3 symmetry. More explicitly, in order to identify the discrete Z n symmetry acting independently from the centres of the non-Abelian gauge factors, we have to mod out those centres from the independent discrete Z n symmetries found above, being the discrete Z 3 , Z 4 and Z 6 symmetry. Thus, when we consider the quotient group we notice that it is homomorphic to the discrete Z 3 gauge symmetry arising from (k a , k c , k d , k h ) = (0, 1, 2, 2) and acting non-trivially on the massless spectrum as indicated in the second-to-last column of table 13. The Z 6 charges are mapped to the Z 3 charges as follows: Hence, this global five-stack MSSM-like model contains a non-trivial discrete Z 3 symmetry, which can however not be decomposed according to (33) when comparing the Z 3 charges in table 13 to the generic expressions for the charges in (34). This conundrum can be traced back to the appearance of two up-type Higgses H bc u andH bd u , where the first ones are required to compose the Yukawa couplings for the right-handed neutrinos and the latter ones to compose the Yukawa couplings for the right-handed down-quarks d R . If we relax the required existence of the Yukawa couplings for the right-handed neutrinos, then they do not have to be identified with the singlet states from the cd sector and we could identify the right-handed neutrinos with the singlet states in the bb, cc or dd sectors. Under these assumptions, the discrete Z 3 symmetry can be reinterpreted as the Z 3 symmetry R 2 3 A 3 L 3 . This simple example of a discrete Z 3 symmetry exhibits the intimate rôle between the assumed existence of Yukawa couplings and the classification of a discrete symmetry. At the same time, it also shows that (global) intersecting D-brane models can realise discrete Z n symmetries which do not appear in the purely field theoretic set-up of the MSSM, due to the presence of extended Higgs sectors in the massless spectrum of intersecting D-brane models. For the extended Higgs sector listed in table 13 one can clearly see that the up-type Higgses H u andH u have different charges under the Z 3 symmetry, which forbids Yukawa couplings to H u for the up-quarks. A similar consideration for the down-type Higgses H d andH d teaches that the discrete Z 3 symmetry also forbids Yukawa couplings toH d for the down-quarks, consistent with the observations surrounding equation (40).
In the same way, one can use the discrete Z 3 symmetry to verify that the up-type Higgs H u allows for Yukawa-type couplings (42) involving the left-handed leptons L and the neutral states located in the chiral cd sector. Other neutral states under the Standard Model gauge group, as listed in table 13 for the five-stack MSSM-like model, require the other up-type HiggsesH u to participate in the respective three-point couplings.
In the next section, we will investigate in more detail which three-point couplings are allowed from the stringy selection rules of closed polygons. This will allow us to verify which Yukawa couplings are present in the perturbative superpotential, and to justify our identification of the chiral states in the cd sector as the right-handed neutrinos. At this point, we point out that the neutral states in the non-chiral cd sector seem to be suitable candidates to construct supersymmetric versions of the DFSZ axion model, through the Z 3 preserving couplings of the form H u ·H d Σ cd andH u ·H dΣ cd and with the Peccei-Quinn symmetry identified as one of the two natural options, for open string axion models [33,34]. In the next section, we will devote more attention to this consideration and derive the associated scalar Higgs-axion potential in full detail.
In summary, the full gauge group for the five-stack MSSM-like model is given by below the string mass scale.

Yukawa Couplings and Higgs-Axion Potential
Focusing on the spectrum associated to the visible sector in table 13, we can easily identify three generations of quarks and leptons, but we are also confronted with an extended Higgssector and various vector-like matter pairs. To probe the phenomenological viability of this global five-stack MSSM-like model, we have to determine the Yukawa couplings and justify why the vector-like states acquire larger masses than the quarks and leptons. As reviewed in section 3.2, the first selection rule for a cubic coupling (composed of three massless open string states) consists in verifying whether it forms a singlet under all gauge symmetries of the model, including the discrete Z 3 gauge symmetry identified in the previous section. Cubic couplings generated through worldsheet instantons also have to form singlets under the global U (1) P Q symmetry from table 13. The massive linear combination U (1) P Q ≡ U (1) c − U (1) d forms an orthogonal direction to the massless hypercharge and acquires its mass through the Stückelberg mechanism (involving a closed string axion). Our interpretation of this massive linear combination of U (1) gauge factors as a Peccei-Quinn U (1) P Q symmetry follows from the charge assignment of the quarks, leptons and Higgses under U (1) P Q , following similar reasoning as the one presented in [33,34]. The second part of our argument is based on the form of the perturbative superpotential, which contains the following three contributions: and where the three contributions can be written (schematically) as: The superpotential contribution (50a) forms the straightforward supersymmetrised version of the DFSZ axion model as proposed in [53,33]. Note that the Standard Model singlets Σ cd and Σ cd couple linearly to the Higgs doublets, which should be contrasted to the quadratic coupling proposed in [54] as a means to solve the µ-problem and the strong CP-problem simultaneously.
Since the singlet fields Σ cd andΣ cd are charged under a Peccei-Quinn symmetry containing the U (1) c factor, this model forms an alternative realisation of the supersymmetric DFSZ axion model within Type IIA string theory with intersecting D6-branes compared to the example discussed in detail in [33,34]. The superpotential contribution (50b) contains the Yukawa couplings for the quarks and leptons, but differs slightly from the usual Yukawa superpotential of the MSSM: the up-type Higgs H u responsible for the Yukawa couplings involving the right-handed neutrinos ν R is here not the same as the up-type HiggsH u appearing in the Yukawa couplings for the right-handed quarks u R . The last renormalisable contribution (50c) to the perturbative superpotential contains cubic couplings for the abundant right-handed quarks and left-handed leptons. These couplings form the key elements to generate the supersymmetric masses for three out of the six right-handed quarks and left-handed leptons by giving a vev to the singlet states Σ cd , such that the model in table 13 effectively becomes the three-generation MSSM (possibly up to additional MSSM-singlet states) at low energies, as suggested at the end of section 4.1.
In order for the parameters of the cubic couplings to be non-vanishing, the corresponding couplings also have to satisfy the stringy selection rule as explained in section 3.2. The first step in determining the closed triangle sequences for the three-point couplings consists in indicating from which sectors x(ω k y) k=0,1,2 the matter states arise. A full overview of the matter states per sector is given in tables 14 and 15 for the global five-stack MSSM-like D6-brane configuration from table 12. Furthermore, to determine the shapes and sizes of the triangles enclosed by the intersecting one-cycles on T 2 (2) and T 2 (3) we also have to pin down at which intersection points the matter states are located. A comprehensive description of the technical precedure using Chan-Paton labels for fractional D-branes is provided in appendix A, where we also clarify the subtlety of discrete Wilson lines. For chiral matter states, one can uniquely identify a Z (i) 2 -invariant point at which an N = 1 supersymmetric chiral multiplet is located. Even when the intersection points correspond to points R and R that are not Z (i) 2 fixed points, one can always form a Z 2 invariant orbit (R, R ) at which the chiral multiplet is located. An explicit example of a closed sequence is presented in figure 2 using the bulk three-cycles a, b and (ω 2 d) . Note that the cycles are all coincident along T 2 (1) , as a consequence of the invariance of the first two-torus under the Z 6 orbifold action, such that we only focus on the intersections along the remaining ambient space T 2 (2) ×T 2 (3) , as anticipated in section 3.2. To the intersections on T 2 (2) ×T 2 (3) between a and b we can allocate two left-handed quarks: Q where v i corresponds to the area (i.e. the real part of the bulk Kähler modulus) of the twotorus T 2 (i) in units of α . The superscripts labelling the generation of the left-handed and right-handed quarks have been chosen in such a way that a realistic pattern of the Yukawa couplings among the different generations can be inferred when taking into account the other allowed cubic couplings listed in tables 16 and 17 as well. The subscripts for the Higgs doublets are chosen with the convention that the non-chiral pairs H  expect, since the exact structure of the cubic couplings turns out to be more involved due to the presence of generation-mixing and of the extended Higgs-sector, expressed through the various superscripts on the parameters y, κ,κ, µ,μ in the last column of tables 16, 17, 18, 19, and 20. Focusing on the quark sector, we notice the absence of a diagonal Yukawa coupling for the up-quark u (2) R , yet the latter does enter in a non-diagonal Yukawa coupling with the left-handed quark Q  which appear in the Yukawa couplings, such that a more elaborate reasoning involving the vevs of the singlets Σ cd(i) has to be developed in order to argue why the masses for the right-handed down-quarks d Before doing so, we consider the cubic couplings (50a)-(50c) involving those states that are uncharged under the strong gauge group and investigate in detail how these couplings can be realised through worldsheet instantons on the ambient space T 2 (2) × T 2 (3) . Recalling that the leptons, Higgses and singlet states Σ cd(i) andΣ cd(i) arise from intersections between the D6-brane stacks b, c and d (including their orbifold and orientifold images), and that these D6brane stacks are characterised by the same bulk wrapping numbers as listed in table 12, matter states are only expected to arise from the intersections b(ω k c) k=1,2 , b(ω k d) k=1,2 , c(ω k d) k=1,2 and c(ω k d) k=1,2 , which has been verified explicitly in tables 14 and 15. Another characteristic of this D6-brane configuration, which has not been encountered in previous studies of Yukawa couplings for fractional intersecting D6-branes [30,35,33], is the potential appearance of both chiral and non-chiral matter in bifundamental representations from the same sector. 10 Following reasonings similar to the ones presented in [26] and in appendix B.1 of [19], one can verify that one non-chiral pair of matter states in the bifundamental representation is located at the Z 2 × Z 2 -invariant quadruplet {(S 2 , S 2 ), (P 3 , P 3 )}, while the chiral matter states -if present -can be allocated to some Z . An interesting observation is that the Yukawa couplings for the right-handed charged leptons e R and right-handed neutrinos ν R involve left-handed leptons from different sectors, namely L (1,2,3) for the righth-handed leptons e R and L (4,5,6) for the right-handed neutrinos ν R . This consideration has non-trivial consequences for the argument establishing three effective generations of left-handed leptons, since the couplings of the right-handed neutrinos to L (i=4,5,6) undermine the provisioned mechanism to make the left-handed leptons L (i=4,5,6) heavier than L (i=1,2,3) by cranking up the vev for the scalar field in the multiplet Σ cd(3) appearing in the couplings L (i=4,5,6) · H (j=2,3) u Σ cd (3) . Considering a large vev for Σ cd(3) would also imply a large supersymmetric mass for the right-handed down-quarks d (1) R and d (2) R , which is phenomenologically unacceptable. More explicitly, the right-handed quarks d (1) R and d (2) R appear in the Yukawa couplings in table 16, where they fulfil the rôle of the down-quark and the strange-quark, respectively, and whose mass cannot be made parametrically large. This last reflection suggests that a proper reasoning arguing for three effective generations of left-handed leptons is intimately connected to the argument for three effective right-handed down-quarks. Moreover, table 20 teaches us that also the other left-handed leptons L (1,2,3) appear in cubic couplings of the form L · H u Σ cd (1,2) . This implies that also the vevs of the singlets Σ cd(1,2) cannot be taken randomly large, as this would suggest a large supersymmetric mass for the left-handed leptons L (1,2,3) . , it can occur that three states are located at points which cannot serve as the apexes of a closed triangle. In this case, the area of the triangle is infinity, and the respective cubic coupling vanishes. We give two explicit examples of such a situation in the first and third row of the table 18. When we take into account the structure of the Yukawa couplings for the quarks and leptons, we observe that the down-type HiggsesH (1) d andH (2) d do not enter at all in the discussion as a consequence of U (1) P Q invariance of the Yukawa interactions. Hence, we can anticipate that the most relevant Higgs-axion couplings to consider are the ones on rows 9, 10, 13 and 14 of table 18, as they are the ones that require the U (1) P Q charged nature of the Higgses appearing in the Yukawa coupling. Due to the appearance of the up-type Higgses H (2,3) u in the Higgs-axion couplings on rows 11 and 12, we should also take these two couplings into account. The other Higgs-axion couplings involve Higgses which do not appear in the Yukawa couplings and given that they are slightly more suppressed, we are able to neglect them to simplify the discussion. The resulting scalar Higgsaxion potential is now expected to have the same structure as the one derived for the T 6 /Z ( ) 6 models in [33,34], namely consisting of four separate contributions: F-term contributions set by the superpotential (49), D-term contributions associated to the U Sp(2) b gauge symmetry, D-term contributions associated to the U (1) P Q symmetry (which acted as a local symmetry before the Stückelberg mechanism) and soft terms added "by hands" at this point, possibly arising from a gaugino condensate in the hidden sector.
Another aspect, which we should turn our attention to, is the presence of additional singlets under the Standard Model gauge group, which can serve as candidate right-handed neutri-nos, namely the five Anti b ≡ (1, 1 A , 1) (0,0,0,0) states in the antisymmetric representation of U Sp(2) b , the four multiplets in the adjoint representation of U (1) c and the five multiplets in the adjoint representation of U (1) d . One might even consider the superpartners of the geometric moduli, but here we focus on open string states. Focusing first on the cubic couplings of the form L ·H (i=1,2) u Adj c , we notice that these couplings are perfectly allowed from the field theory side based on charge conservation arguments. Nevertheless, from the stringy side, we notice that the cubic couplings involving the multiplets Adj c in the adjoint representation of U (1) c are not allowed based on the violation of the stringy selection rule. More explicitly, as both the left-handed leptons and the up-type HiggsesH (i=1,2) u arise from the b(ω k d) k=1,2 sectors, the four multiplets in the adjoint representation of U (1) c located in the c(ω k c) k=1,2 sectors do not allow for closed sequences. The other singlet states do allow for closed sequences, as listed in tables 21 and 22, such that we have to include the following perturbative cubic couplings in the superpotential: with i ∈ {1, 2, 3}, j ∈ {4, 5, 6} and k ∈ {1, 2, 3, 4, 5}. The explicit form of the closed sequences as well as the leading order behaviour of the non-vanishing coupling constants are elaborated in tables 21 and 22, while a pictorial representation of the worldsheet instantons is presented in figure 4 for the third kind of couplings in (52). Apart from representing alternative Yukawa couplings when the rôle of the right-handed neutrinos is played by the states Anti or Adj (k) d , the cubic couplings in (52) can in principle also be useful to lift the masses for the leptons L (4,5,6) with respect to the other three leptons L (1,2,3) by cranking up the vevs for a selected number of matter states in the antisymmetric or adjoint representation. However, as the up-type HiggsH (2) u appears in the Yukawa couplings involving the up-quarks, one cannot give it randomly a large supersymmetric mass to argue for three effective leptons generations without giving a large supersymmetric mass to the left-handed quarks Q   , the sector x(ω k y) comes with a set of non-chiral pairs of matter states, whose multiplicity corresponds to n

Enclosed Area
Parameter and 4 for details. The fourth column presents the corresponding area for the worldsheet instantons expressed in terms of the areas v i of the two-tori 3) , and the last column shows the scaling of the coupling constant corresponding to the considered cubic coupling.

Cubic couplings for the superpotential (49) of a global 5-stack MSSM (part II)
Coupling Sequence Triangles on T 2  3) , and the last column shows the scaling of the coupling constant corresponding to the considered cubic coupling.

Enclosed Area
Parameter

Enclosed Area
Parameter  3) , and the last column shows the scaling of the coupling constant corresponding to the considered cubic coupling.

Enclosed Area
Parameter or [x, y, z] bounded by three branes pairwise intersecting in the indicated apexes x, y, z on T 2 (i=2, 3) for the respective cubic couplings. The fourth column presents the corresponding area for the worldsheet instantons expressed in terms of the areas v i of the two-tori T 2 (i=2, 3) , and the last column shows the scaling of the coupling constant corresponding to the considered cubic coupling.

Enclosed Area
Parameter ) presents the corresponding area for the worldsheet instantons expressed in terms of the areas v i of the two-tori T 2 (i=2, 3) , and the last column shows the scaling of the coupling constant corresponding to the considered cubic coupling.

Enclosed Area
Parameter presents the corresponding area for the worldsheet instantons expressed in terms of the areas v i of the two-tori T 2 (i=2, 3) , and the last column shows the scaling of the coupling constant corresponding to the considered cubic coupling. 55 5 Phenomenology of Global L-R Symmetric Models

Searching for Left-Right Symmetric D6-Brane Models
The starting point for our search of global left-right symmetric models is the same as the one for the MSSM-like models, as formulated in (37). More explicitly, -independent D6-brane configurations with three chiral left-handed quarks are only realisable when the QCD stack is parallel to the ΩR-plane, while the SU (2) L stack wraps a fractional three-cycle parallel to the ΩRZ (1) 2 -plane and supports an enhanced U Sp(2) gauge group. It is also understood that either the ΩRZ (2) 2 -plane or the ΩRZ 2 -plane takes on the rôle of the exotic O6-plane. To obtain leftright symmetric models, the gauge group has to be completed with a right-symmetric SU (2) c and an Abelian U (1) d gauge group with wrapping numbers as classified in table 9, and without overshooting the bulk RR tadpole cancellation conditions (39). Analogously to the last row in table 10, the only possible configuration on the bAA lattice not violating (39) consists in taking the c-stack and d-stack both parallel to the ΩR-plane. The left-right symmetry of the gauge group requires the c-stack to support a U (2) or U Sp(2) non-Abelian gauge group, while intersecting with the QCD stack to yield three chiral generations of right-handed quark doublets Q R ≡ (u R , d R ). In case the c-stack supports a U (2) gauge group, the three chiral generations of right-handed quarks arise for the net-chirality χ ac + χ ac = 3, where the sign has to be opposite to the sign of the net-chirality χ ab ≡ χ ab . For an enhanced U Sp(2) c gauge group, the net-chirality associated to the right handed quarks has to satisfy |χ ac | ≡ |χ ac | = 3 with sgn(χ ac ) = −sgn(χ ab ) instead. Recall from the discussion in section 2.2 that D6-brane configurations, where the c-stack is parallel to the ΩR-plane, do not give rise to three chiral generations of right-handed quarks -neither for a U (2) c nor a U Sp(2) c group, implying that the bAA lattice does not allow for any -independent global left-right symmetric models. Note that the geometric conditions on the fractional three-cycles associated to candidate SU (2) R branes are less stringent in comparison to the ones for the left stack. That is to say, the right stack can be accompanied by (chiral) matter in the symmetric and/or adjoint representation, and the only requirement we impose for the SU (2) R -stack is the existence of three chiral generations of right-handed quarks.
Turning to the aAA lattice configuration, one notices from table 23 that the potential combinations of bulk orbits for the c-stack and d-stack are considerably more numerous than for the bAA lattice. But also here, the requirement to have three chiral generations of right-handed quarks (u R , d R ) eliminates most combinations. On the other hand, the condition to obtain three chiral generations of left-handed leptons, i.e. χ bd = 3 given that the b-stack has to be ΩR-invariant in -independent configurations without adjoint/symmetric representations of SU (2) L in the spectrum, does not constrain any of the bulk three-cycle combinations in table 23. Combining all three requirements (bulk RR tadpoles and three chiral generations of right-handed quarks and left-handed leptons) leaves us with only the combinations (8,10,12) of bulk three-cycles to realise -independent left-right symmetric models on the aAA lattice. Table 23: Combinations of supersymmetric bulk orbits for c-and d-stacks aiming at -independent configurations of left-right symmetric models on the aAA lattice for T 6 /(Z 2 × Z 6 × ΩR) with discrete torsion, η = −1, and exotic O6-plane, η ΩRZ (2 or 3) 2 = −1. The second and third column indicate the bulk orbit of the c-stack and d-stack, respectively, the fourth and fifth column test whether the bulk RR tadpole cancellation conditions (39) are not over-shot, the second-to-last column verifies if three righthanded quark generations can be realised through |χ ac | ≡ |χ ac | = 3 for U Sp(2) c or |χ ac + χ ac | = 3 for U (2) c , and the last column does the same for three left-handed lepton generations with |χ bd | ≡ |χ bd | = 3, in all cases with consistent relative sign choices. For the three bulk orbit combinations (8,10,12) allowing for three chiral generations of right-handed quarks and left-handed leptons simultaneously, we note that the constraints are mutually compatible as indicated in parenthesis in the last column.

Four-stack combinations with gauge group
The subscript indicates the number of combinatorial possibilities of ( σ x ), ( τ x ) and relative (−) τ Z (k) As the bulk RR tadpoles are saturated for the choice of bulk three-cycles in any combination of type n • 12, there is no room left to add 'hidden' fractional D6-branes in order to compensate the twisted RR charges coming from the four D6-brane stacks with initial gauge group U (3) a × U Sp(2) b × U Sp(2) c × U (1) d . Hence, the resulting D6-brane models associated to combination n • 12 only provide for local left-right symmetric models, given that none of the 288 four-stack fractional D6-brane configurations is characterised by vanishing twisted RR tadpoles. 11 Also here the number of independent fractional D6-brane configurations has been reduced by taking into account that configurations with identical relative Z (i) 2 eigenvalues, but identical absolute discrete displacements and Wilson lines give rise to the same chiral and non-chiral massless spectrum and low-energy effective field theory at the current state-of-the-art.
Combination n • 8 on the other hand allows for the construction of two prototypes of global left-right symmetric models, with the hidden sector gauge group as the defining difference between the prototypes. The hidden D6-brane stacks of the first prototype consist of two stacks of D6-branes wrapping fractional three-cycles parallel to the ΩR-plane supporting the gauge factors U (3) h 1 × U (3) h 2 . An explicit D6-brane configuration for prototype I is given in table 24, with the corresponding massless matter spectrum listed in table 25.
D6-brane configuration for a 6-stack LRS model (prototype I) on the aAA lattice wrapping numbers on the aAA lattice of the orientifold T 6 /(Z 2 × Z 6 × ΩR) with discrete torsion (η = −1) and the ΩRZ 2 -plane as the exotic O6-plane (η ΩRZ (3) Overview of the spectrum for prototype I LRS model on the aAA lattice 1, 1, 1, 1) (1,1,0,0  To obtain prototype II left-right symmetric models, we have to add two D6-branes wrapping fractional three-cycles parallel to the ΩRZ , discrete displacements ( σ x ) and discrete Wilson lines ( τ x ) with x ∈ {h 1 , h 2 }. In the prototype II models, the hidden D6-brane stacks support the Abelian gauge group U (1) h 1 × U (1) h 2 , as indicated in the explicit example in table 26 with the corresponding massless open string  spectrum summarised in table 27. D6-brane configuration for a 6-stack LRS model (prototype II) on the aAA lattice wrapping numbers on the aAA lattice of the orientifold T 6 /(Z 2 × Z 6 × ΩR) with discrete torsion (η = −1) and the ΩRZ 2 -plane as the exotic O6-plane (η ΩRZ (3) Overview of the spectrum for prototype II LRS model on the aAA lattice The D6-brane combination n • 10 can give rise to two types of left-right symmetric models: five-stack models with the hidden D6-brane stack parallel to the ΩR-plane and supporting a U (4) h hidden gauge group, or six-stack models with the two hidden D6-brane stacks parallel to the ΩR-plane and the ΩRZ (1) 2 -plane, respectively, and each supporting a U (1) h i hidden gauge group. A superficial analysis of the chiral and non-chiral massless open string spectrum reveals that various of these global six-stack models correspond to prototype II models as in table 26 or the variants IIb and IIc in tables 37 and 38 of appendix C, where now one of the hidden stacks h i that was parallel to the ΩRZ (1) 2 -plane has been permuted with the dstack which before was parallel to the ΩR-plane. This consideration suggests that the global six-stack models arising from combination n • 10 with hidden gauge group U (1) h 1 × U (1) h 2 might form a subset of the prototype II models identified from combination n • 8. In order to verify this speculative statement, a more thorough analysis of the massless spectra of all 20736 global left-right symmetric six-stack models associated to combination n • 10 has to be performed, which we postpone for future work.
Apart from the hidden gauge factors, there happens to be another appreciable difference between the two prototypes of left-right symmetric models: the absence of a massless U (1) B−L symmetry for prototype I and the presence of a generalized massless U (1) B−L symmetry for prototype II, as we will show in the next section. This observation implies a different approach when identifying the left-handed and right-handed quarks and leptons. For the prototype I model, the absence of a massless U (1) B−L symmetry might entice us to exchange the rôle of one of the hidden stacks h i with the QCD-U (3) a -stack. Indeed, the chiral state in the bh 2 sector can equally be interpreted as a left-handed quark based on its quantum numbers. Nevertheless, the lack of three generations prevents us to exchange the rôles of the U (3)-stacks and provides a solid argument for the identification of the chiral states presented in table 25. In this interpretation, the chiral and non-chiral massless states in the bh 1 , bh 2 , ch 1 and ch 2 sectors form a portal between the visible sector and a dark sector, instead of being inherent to the visible sector.
An argument against exchanging the rôles of the U (1) stacks in the prototype II left-right symmetric models can be made based on the generalised massless B − L symmetry defined in equation (63)   The five-stack models with hidden gauge group U (4) h are new and form an entirely independent prototype for which bulk and twisted RR tadpole cancellation conditions are satisfied.
As explained in appendix B through an explicit example, the K-theory constraints for this prototype of five-stack models are, however, not fulfilled and the models can therefore not be considered as globally consistent models, but rather as semi-local models. An example of a five-stack model can be found in appendix B, more explicitly in table 35    To finish this section, in table 28 we give a summary of the various -independent left-right symmetric models that can be constructed on the aAA lattice. We list the numbers for one particular choice of the exotic O6-plane, namely for the ΩRZ (3) 2 -plane, but remark that we cross-checked that the same summary is valid in case the ΩRZ Secondly, one can verify that the semi-local five-stack and six-stack models do not allow for a massless (generalised) B − L symmetry. This observation implies a subtle difference between semi-local and global prototype II models, whose gauge group and spectrum are fully equivalent, and requires us to define "prototype" more precisely: the term "prototype" captures all fractional D6-brane models with six D6-brane stacks, whose bulk three-cycles are identical to the ones in table 24 (prototype I) or in table 26 (prototype II), and with the same left-right symmetric gauge structure and massless open string spectrum in the purely visible sector of table 25 or table 27, respectively. Within the prototypes, one can find subclasses of global left-right symmetric models whose chiral and non-chiral spectrum in the hidden sector slightly differs. How many physically distinguishable subclasses 12 there exist requires a full comparison of the massless spectrum for all global D6-brane models, which is postponed for future research. In appendix C we provide two other examples of six-stack intersecting D6-brane models fitting within the prototype II models.
Apart from the six-stack left-right symmetric models presented above, we also searched for six-stack D6-brane models associated to combination n • 8 with hidden gauge groups U (3) h 1 × U (1) h 2 , where the h 1 -stack is parallel to the ΩR-plane and the h 2 -stack parallel to the ΩRZ (1) 2plane, and for six-stack D6-brane models associated to combination n • 10 with hidden gauge groups U (2) × U (2), U Sp(4) × U (2) or U Sp(4) × U Sp(4), where the hidden h 1 and h 2 -stack are both parallel to the ΩR-plane. For all those combinations of hidden D6-brane stacks we observed that the resulting six-stack left-right symmetric models are able to satisfy the RR tadpole cancellation conditions, yet always violate some of the K-theory constraints, such that no global six-stack models can be found.

Discrete Symmetries
Also for left-right symmetric models, a classification of discrete Z n symmetries can be useful to constrain the cubic couplings among the massless open string states. At the same time, this computation will determine if the commonly required massless U (1) B−L symmetry exists. That is why this section will be devoted to the search for discrete Z n symmetries for the two prototype I and II examples of global six-stack left-right symmetric models presented in table 24 and 26, respectively. We will briefly comment on the differences of the prototype IIb and IIc examples in tables 37 and 38 compared to the prototype II example of 26 in appendix C. The discrete symmetries for the semi-local five-stack left-right symmetric model will be discussed in appendix B.

Prototype I Left-Right Symmetric Model
Zooming in on the first prototype left-right symmetric model with hidden gauge group we write down the necessary conditions (28) on the existence of some Z n gauge symmetry for the D6-brane configuration listed in table 24: which can be reduced to four linearly independent constraints, since various rows are trivially satisfied or can be related to each other: These four constraints have to be completed with the linearly independent constraints coming from the sufficient conditions (32), which read for the D6-brane configuration in table 24: A closer inspection of the sufficient conditions shows that the first block only leads to two independent conditions (row 1 and row 2), where the second constraint already appeared as one of the necessary conditions. The second block does not impose any additional constraint, as all conditions are trivially satisfied. The third block yields five independent conditions (row 11,13,14,15,16), for which two conditions already appeared before. Note also that the last sufficient condition in (56) forms a linear combination of the first and fifth row of the third block. Hence, there are at most three independent constraints coming from the sufficient conditions: Notice, however, that the last sufficient condition in equation (57) for example can be reduced to 2k d ! = 0 mod n upon inserting the first necessary condition from equation (55), which in turn renders the second and fourth necessary condition identical to 6k a ! = 0 mod n. Continuing along these lines, the set of truly independent conditions can be reduced to match the number U (1) factors in the model. Combining the four independent necessary conditions (55) and three independent sufficient conditions (57), one can easily notice that no non-trivial combination (k a , k d , k h 1 , k h 2 ) can solve them simultaneously for all n, implying that this left-right symmetric model does not come with a massless U (1) B−L symmetry or possible extensions thereof involving the hidden U (1) factors.
Let us thus continue with the classification of discrete Z n symmetries for the prototype I left-right symmetric model: • The combination (k a , k d , k h 1 , k h 2 ) = (1, 1, 1, 1) gives rise to the discrete Z 2 symmetry guaranteed by the K-theory constraints. In first instance, we might feel the urge to see • There exists a set of three discrete Z 3 symmetries, corresponding to the combinations (k a , k d , k h 1 , k h 2 ) = (1, 0, 0, 0), (0, 0, 1, 0), and (0, 0, 0, 1), such that each Z 3 symmetry is homomorphic to the centre of a SU (3) x∈{a,h 1 ,h 2 } gauge symmetry. Hence, these discrete symmetries do not offer any other selection rules for the m-point couplings beyond the ones associated to the non-Abelian gauge symmetries.
• The vector (k a , k d , k h 1 , k h 2 ) = (1, 3, 1, 1) corresponds to a discrete Z 6 symmetry, with the charges for the massless open string states given in the last column of table 25. Note that the K-theory Z 2 symmetry is a subgroup of this discrete symmetry, suggesting that the truly independent discrete symmetry is only a Z 3 symmetry. This Z 3 symmetry also pops up when we mod out the centres of the overall non-Abelian gauge group from the discrete symmetries found as solutions to the necessary and sufficient conditions (55) and (57), i.e. we find that the quotient group (Z 2 ×Z 3 3 ×Z 6 )/(Z 3 3 ×Z 2 2 ) is homomorphic to a Z 3 symmetry with charge assignments listed in table 25, after reduction of Z 6 → Z 3 . In this particular case, the Z 3 symmetry can also be associated to the combination (k a , k d , k h 1 , k h 2 ) = (1, 0, 1, 1), such that the Z 3 symmetry acts effectively as a linear combination of the three discrete Z 3 symmetries identified above. As such, this Z 3 symmetry should not be considered as an independent discrete symmetry and is not expected to yield additional selection rules apart from those associated to the centres of the non-Abelian gauge groups.
In conclusion, the gauge group encountered for prototype I below the string mass scale corresponds to SU (3) a × U Sp(2) b × U Sp(2) c × SU (3) h 1 × SU (3) h 2 × Z 3 with the Z 3 acting trivially on massless matter states.

Prototype II Left-Right Symmetric Model
Next, we discuss the discrete symmetries arising in the second prototype left-right symmetric model through the example presented in table 26 following the same line of thought as for prototype I. Writing down the necessary conditions for the existence of discrete Z n gauge symmetries with respect to the example in table 26: we deduce three linearly independent constraints: from those rows in (58) and (59) that are not trivially satisfied. To these four necessary constraints we have to add the subset of the linearly independent constraints coming from the sufficient conditions (32) written out for the D6-brane configuration in table 26: Clearly, the first block of the sufficient conditions provides one linearly independent constraint, while the second block contains only trivially satisfied conditions. The third block gives three conditions (rows 11, 15, and 16) which have not appeared yet before in the necessary conditions (55). Since row 11 is the sum of rows 15 and 16, we are naively left with three new and linearly independent constraints: The last necessary condition in equation (60) turns out to equal twice the first sufficient condition in equation (62). Since also the second sufficient condition can be expressed as twice the linear combination of the first necessary condition minus the last sufficient condition, only four conditions are truly independent, as expected from the four initial U (1) factors in the model. A closer inspection of the three necessary constraints (60) and the four sufficient constraints (62) teaches us that the non-trivial combination (k a , k d , k h 1 , k h 2 ) = (1, −3, −3, 3) satisfies all seven constraints irrespective of the value of n. This combination points towards the presence of a massless linear combination of U (1)'s: which plays the rôle of a (generalised) B − L symmetry. Turning our attention to the discrete Z n gauge symmetries allowed by the constraints (60) and (62), we obtain the following classification: • Also for this prototype we encounter the discrete Z 2 symmetry guaranteed by the Ktheory constraints for the combination (k a , k d , k h 1 , k h 2 ) = (1, 1, 1, 1), but in this model the Z 2 symmetry is a discrete subgroup of the massless U (1) B−L gauge symmetry. This can be seen explicitly by shifting the charges of the massless open string states under the Z 2 symmetry by virtue of the massless U (1) B−L symmetry, after which all charges are set to zero (modulo 2) simultaneously.
• We only encounter one discrete Z 3 symmetry, namely for (k a , k d , k h 1 , k h 2 ) = (1, 0, 0, 0), which is homomorphic to the centre of the non-Abelian SU (3) a gauge group. This discrete symmetry will thus not provide any new selection rules for cubic and higher order couplings. Also here we can perform a rotation over the massless U (1) B−L symmetry, setting the charges for all open string states to zero (modulo 3), to verify that the action of the Z 3 symmetry is trivial from the effective low-energy perspective.
• Finally, we also encounter a discrete Z 6 symmetry corresponding to the linear combination (k a , k d , k h 1 , k h 2 ) = (1, 3, 3, 3), with the charges of the massless open string states given in the last column of table 27. We must ask ourselves again whether this discrete symmetry should not be reduced to a discrete Z 3 symmetry, given the quotient where the subgroup Z 3 × Z 2 × Z 2 corresponds to the centres of the non-Abelian gauge factors for the prototype II models. Indeed, since the K-theory Z 2 symmetry forms a subgroup of the discrete Z 6 symmetry, the truly independent discrete symmetry is rather the Z 3 symmetry associated with (k a , k d , k h 1 , k h 2 ) = (1, 0, 0, 0) discussed in the previous bullet point. Alternatively, to argue for the triviality of the Z 6 -action we also point out that the Z 6 symmetry forms a discrete subgroup of the massless U (1) B−L gauge symmetry and that we can set the charges of the open string states to zero by virtue of a shift over U (1) B−L .
Hence, the full gauge group for the prototype II left-right symmetric model, exemplified by the D-brane configuration in table 26, is given by between the stacks x, y ∈ {a, b, c, d}, which are all the same for the explicit D6-brane configurations given in tables 24, 26, 37 and 38. Hence, it suffices to discuss the Yukawa couplings for one prototype model to obtain the Yukawa couplings for the other prototype models as well.

Yukawa and other Cubic Couplings
Let us thus, for instance, consider the Yukawa couplings for the prototype II model with D6-brane configuration in table 26. The first step consists in determining the cubic couplings that are allowed by charge conservation: where we wrote down the Yukawa couplings in a schematic way involving the quarks, leptons and Higgses appearing in the first block of table 27. In order to assess which Yukawa couplings are non-vanishing, we first have to allocate the massless open string states unambiguously to Z 2 × Z 2 invariant intersection points or orbits and then verify that also the stringy selection rules are satisfied. These steps require us to determine from which sectors x(ω k y) k=0,1,2 and x(ω k y) k=0,1,2 with x, y ∈ {a, b, c, d} the massless states arise, as listed in tables 29 and 30, after which we can use the techniques involving the Chan-Paton labels from appendix A to allocate the states explicitly. As a last step, we investigate the area of the closed triangle sequences on T 2 (2) × T 2 (3) with the allocated massless states at their apexes, as explained in section 3.2. An overview of the non-vanishing Yukawa couplings involving the quark sector is given in table 31, while the non-vanishing leptonic Yukawa couplings are listed in table 32. A quick comparison between tables 31 and 32 reveals a subtle symmetry among the Yukawa couplings involving the quarks and leptons: upon exchanging Q (i) L ↔ L (i) and Q (i) R ↔ R (i) we find the same order of magnitude for the corresponding coupling constants. This allows us to discuss solely the quark Yukawa couplings and deduce the same conclusions for the leptonic sector. The numbering of the Higgses (H u , H d ) (i) emerging from the b(ω k c) k=0,1,2 sector follows a normal ordering with i ∈ {1, 2} for k = 0, i ∈ {3, 4, 5, 6} for k = 1 and i ∈ {7, 8, 9, 10} for k = 2. Figure 5 provides a pictorial representation of the perturbatively allowed Yukawa couplings Q By taking a closer look at the Yukawa couplings for the quarks, we notice that all Yukawa couplings are exponentially suppressed. The diagonal Yukawa couplings for the second and third generation only occur for the third Higgs doublet (H u , H d ) (3) and are accidentally equal to each other. We also notice the absence of the diagonal Yukawa coupling for the first generation Q R , yet both chiral states appear in non-diagonal Yukawa couplings to the third and second generation, respectively. For the Yukawa couplings involving the third Higgs doublet (H u , H d ) (3) we observe that the off-diagonal terms are more suppressed than the diagonal Yukawa couplings. For the Yukawa couplings involving the Higgs doublets (H u , H d ) (4,5,6) we notice the opposite pattern, which complicates a clear microscopic explanation of the hierarchies within the CKM matrix. Notice that the off-diagonal Yukawa couplings between the second and third generation proceed according to a separate Higgs-sector from the other off-diagonal Yukawa couplings, which might be a useful observation to explain some hierarchical structure in the CKM matrix entries based on a hierarchy among the vevs for different Higgs sectors. The Higgses (H u , H d ) (1,2) attributed to the bc-sector are somewhat special as they cannot be unambiguously assigned to Z 2 × Z 2 invariant intersection points. This feature can be traced back to the fact that both bulk orbits are fully parallel to each other on all three two-tori. In this respect they give the impression of being a remnant (local) N = 2 supersymmetric multiplet, and it is not entirely clear if the presence of the Z 2 × Z 2 symmetries, which lead to manifestly only N = 1 supersymmetry, will change the existence of cubic couplings involving these states. For that reason, we have not treated Yukawa couplings involving the Higgses (H u , H d ) (1,2) and hope to address this conundrum in future work. To enhance the phenomenological appeal of the massless open string spectrum in table 27, we should also discuss mechanisms to lift the masses of the non-chiral matter pairs from the ad and ad sectors. First of all, observe that cubic couplings involving the non-chiral matter pairs from the ad or ad sectors combined with some Standard Model singlet cannot occur due to the absence of massless states in the adjoint or symmetric representation under the U (1) d gauge group. Hence, we have to look at quartic couplings involving the states from the ad and ad sectors. By virtue of the matter states in the dh 1 , dh 1 , dh 2 and dh 2 sector, we are able to write down the gauge-invariant quartic couplings: with i, j ∈ {1, 2} and M string the string mass scale. Note, however, that this discussion can only be pursued for the prototype II and IIc left-right symmetric models, as the existence of the quartic couplings is tied to the existence of 'messenger' states in the dh 1 , dh 1 , dh 2 and dh 2 sectors. The next step then comprises the compution of the non-vanishing perturbatively allowed quartic couplings, using the same techniques as explained in section 3.2 -including localisations analogous to those in appendix A -generalised to quartic couplings and their associated quadrilateral worldsheet instantons. Looking carefully at the ad sector, we observe that the non-chiral matter pair X ad +X ad arises solely from two D6-brane stacks whose bulk orbits are completely parallel to each other on all three two-tori. In this regard, the non-chiral pair cannot be localised at Z 2 × Z 2 invariant intersection points from a geometric perspective, giving the impression that the non-chiral pair is a remnant (local) N = 2 supermultiplet. Analogously to the Higgses from the bc-sector, it is not entirely clear how the Z 2 × Z 2 symmetries act on the quartic couplings involving the non-chiral pair X ad +X ad . In order to asses whether the quartic couplings are non-vanishing, a better understanding of the CFT computations for m-point couplings on orbifolds with Z 2 factors is required.
Total amount of matter per sector for a 6-stack Left-Right Symmetric model on the aAA lattice (χ xy , χ x(ωy) , χ x(ω 2 y) ) y = a , the sector x(ω k y) comes with a set of non-chiral pairs of matter states,whose multiplicity corresponds to n  x(θ k x) sectors, the upper entries count the numbers of antisymmetric representations and the lower entries the symmetric ones. The ΩR-invariance of the b-stack and c-stack implies b(θ k  [5,4,6], [5, (2, 3), 6]}        3) , and the last column shows the scaling of the coupling constant corresponding to the considered cubic coupling. 76

Conclusions and Outlook
This article proceeds with the study of intersecting D6-brane model building on the fertile T 6 /(Z 2 × Z 6 × ΩR) background with discrete torsion, which was initiated in a previous article by the same authors. The emphasis in this article lies on systematic scans for MSSM-like and left-right symmetric models on the considered toroidal orbifold background, which form consistent Type IIA/ΩR string vacua where the gauge degrees of freedom are attributed to D6-branes wrapping fractional, ideally rigid, three-cycles stuck at Z 2 × Z 2 orbifold singularities. The scans presented in this article are exhaustive for D-brane configurations that are supersymmetric irrespective of the choice of the complex structure parameter on the two-torus that is invariant under the Z 6 and only feels the Z 2 orbifold action.
As starting point for the systematic scans, we considered the requirement that the D6-brane stacks supporting the QCD and the SU (2) L gauge groups are not accompanied by matter states in the adjoint representation. From a physical perspective, this requirement is motivated by ensuring that neither of the two gauge groups can be continuously broken by a non-vanishing vev of a matter state in the adjoint representation under the respective gauge group. A summary of all fractional three-cycles satisfying this constraint on the orbifold T 6 /(Z 2 × Z 6 × ΩR) is offered in section 2.2. In our systematic search for MSSM-like and leftright symmetric models, this requirement has to be supplemented by additional constraints reflecting the correct massless open string spectra with respect to the gauge group configuration under consideration. These latter constraints can be decomposed into two separate requirements: the required absence of chiral matter states in the symmetric representation under the QCD or SU (2) L gauge group on the one hand, and the presence of three chiral generations of quarks and leptons on the other hand. An important observation following from these requirements is the fact that -independent models are only able to satisfy all of the aforementioned restrictions provided that the SU (2) L gauge group is realised as an enhanced U Sp(2) gauge group, and the exotic O6-plane is chosen to be the ΩRZ (2 or 3) 2 -plane. Moreover, by virtue of all these requirements, we can exclude the existence of -independent local MSSM-like and left-right symmetric models on the bAA lattice with three right-handed quark generations and a SU (2) L -stack realised as an enhanced U Sp(2) gauge group. For that reason, our systematic search focused on the only remaining independent aAA lattice configuration of the orientifold T 6 /(Z 2 × Z 6 × ΩR) with discrete torsion, for which we have more room to manoeuvre with respect to the number of D6-brane stacks without overshooting the bulk RR tadpole cancellation conditions.
Regarding MSSM-like D6-brane model searches on the aAA lattice, we noticed the absence of local three-stack MSSM-like configurations, confirmed the existence of local four-stack MSSMlike models and identified a class of global five-stack MSSM-like D-brane configurations. When counting the number of four-and five-stack D6-brane configurations, we took into account the obvious symmetries among the models due to identical relative Z (i) 2 eigenvalues and identical absolute discrete Wilson lines and displacements characterising the fractional three-cycles. Nonetheless, identical massless spectra among different D6-brane configurations might suggest the potential existence of more intricate pairwise symmetries among non-identical relative discrete parameters, such that the number of physically inequivalent models might even be further reduced.
With respect to left-right symmetric D6-brane model searches on the aAA lattice, we confirmed the existence of local four-stack left-right symmetric D-brane configurations, stumbled upon the existence of semi-local five-stack left-right symmetric models and identified two prototypes of six-stack left-right symmetric models based on the ranks of the hidden gauge groups. Both prototype models contain D6-brane configurations yielding semi-local models and D6-brane configurations giving rise to global models. Within the prototype II models, we were also able to identify subclasses IIb and IIc based on the massless open string states in the 'messenger' and 'hidden' sectors. Subclass IIb represents examples of global six-stack left-right symmetric models with both hidden gauge groups completely decoupled from the visible sector, while subclass IIc captures examples of global six-stack left-right symmetric models with one of the hidden gauge groups completely decoupled from the visible sector. Another subtle difference between the various classes of models is the absence of a massless (generalised) B − L symmetry for the prototype I, IIb and IIc models, whereas the other prototype II models -not belonging to IIb or IIc -do come with a massless generalised B − L symmetry. This observation begs the question whether it is possible to identify a massless U (1) Y hypercharge at all upon spontaneous breaking of the SU (2) R gauge group for the prototype models I, IIb and IIc.
should be contrasted to other type II superstring scenarios where the rôle of the QCD axion is played by a closed string axion [64][65][66][67].
In the case of the global six-stack left-right symmetric models, we did not find any discrete Z n symmetry acting non-trivially on the massless open string spectrum that could have provided selection rules beyond the ones associated to the non-Abelian gauge factors. We point out that the Yukawa couplings present a form of universality, due to the fact that the prototype models have an identical visible sector and only differ in the choice of hidden D-branes.
In order to study the related low-energy effective Yukawa and higher order couplings more in-depth, it will be necessary to perform reliable CFT computations for m-point couplings on orbifolds containing Z 2 factors, since the argument of vanishing couplings for some vanishing angle [49] is based on extended N = 2 supersymmetry on the six-torus, which, however is broken here to N = 1 by the Z 2 × Z 2 symmetries.
Other phenomenological aspects to be studied in the future include possible deformations of the exceptional three-cycles away from the singular point in moduli space in analogy to [68,69], which will usually lead to a splitting of previously identical gauge couplings at tree level for some deformations and stabilisation of other twisted moduli at the orbifold point. When also taking one-loop corrections to the gauge couplings into account in analogy to section 5 of [35], it will be interesting to see how low values of the string scale are compatible with the measured strengths of the strong and electro-weak gauge couplings, and if our global models fit into the analysis of low string scale scenarios at the LHC as discussed e.g. in [70][71][72][73][74][75][76][77][78][79].
All models presented here preserve N = 1 supersymmetry at the string scale. Another pressing question thus consists in identifying possible supersymmetry breaking scenarios. While we expect that non-supersymmetric deformations away from the singular orbifold point will predominantly stabilise moduli at the singularity as argued in [68,69], it remains to be seen if the maximal non-Abelian hidden gauge groups SU (4) or SU (3) × SU (3) in the present D6-brane configurations are suitable to generate a gaugino condensate, which breaks supersymmetry, and if so study gauge mediation versus gravity mediation scenarios.
Generalising Chan-Paton labels to arbitrary fractional D-branes boils down to selecting the corresponding N i while setting all other N j =i to zero, e.g. for the fractional branes c and d in the MSSM-like model of table 12 we start with: Using the orbifold image wrapping numbers in equation (8), we obtain I x(ω y) = (−3) · 3 and I x(ω 2 y) = 3 · (−3) for x, y ∈ {b, c, d}, where the signs per two-torus are explicitly shown as a reminder that the angles in these two sectors are exactly opposite. Only one of the nine intersection points is Z 2 × Z 2 invariant, and the signs of sgn(I  (72) The violation of the K-theory constraints in the third and fourth row implies that the example presented in table 35 is semi-local or globally not consistent, a characteristic which was also found to be true for all other 1295 models found within this prototype for the choice of exotic O6-plane η Z (3) 2 = −1. One finds the same amount of models when choosing the ΩRZ (2) 2 -plane as the exotic O6-plane.
Let us now turn to the search for Abelian symmetries associated to the D6-brane configuration given in table 35. First, we compute the necessary conditions (28) for the existence of discrete Z n symmetries, which reduce to the following three linearly independent constraints: These constraints have to be supplemented with the sufficient conditions (32) for the existence of discrete Z n symmetries, which lead to at most three more linearly independent constraints: Further reductions lead to only three truly independent constraints, k d , 3k a , 4k h ! = 0 mod n, as expected from initially three U (1) gauge factors. The first observation we can make is that there does not exist any non-trivial combination (k a , k d , k h ) for which the eight constraint equations are exactly satisfied for all n, indicating that there does not exist any linear combination of U (1)'s which stays massless, in particular no (generalised) U (1) B−L symmetry. Next, we can classify the discrete Z n symmetries, which arise from linear combinations (k a , k d , k h ) satisfying the eight constraints given above: • A discrete Z 3 symmetry homomorphic to the centre of the SU (3) a gauge group appears for the configuration (k a , k d , k h ) = (1, 0, 0), with the charges of the massless states listed in the second-to-last column of table 36. This symmetry acts -as usual -as a baryonlike discrete symmetry, but does not forbid any cubic or higher order coupling which is not already forbidden by the SU (3) a gauge symmetry.
• The combination (k a , k d , k h ) = (0, 0, 1) corresponds to a Z 4 symmetry homomorphic to the centre of the hidden SU (4) h gauge group, and thus does not constrain additional couplings beyond the ones already constrained by the non-Abelian gauge symmetry. For completeness, we list the charges under the Z 4 symmetry for the massless open string spectrum in the last column of table 36, from which we can clearly see that only exotic matter charged under the hidden gauge group carries Z 4 charges as expected.
• A last observation is that the violation of the K-theory constraints forbids the existence of a discrete Z 2 symmetry for the combination (k a , k d , k h ) = (1, 1, 1).
Thus, the full gauge group of the semi-local five-stack left-right symmetric model below the string scale corresponds to SU (3) a × U Sp(2) b × U Sp(2) c × SU (4) h , free of any non-trivial discrete Z n symmetry.
Overview of the Spectrum for 5-stack Left-Right Symm. on the aAA lattice    Another crucial difference between prototype II on the one hand and prototypes IIb and IIc on the other hand concerns the generalised B − L symmetry defined in equation (63). This U (1) B−L symmetry acts as a massless Abelian gauge symmetry for prototype II models, but  turns into a massive chiral global symmetry by virtue of the Stückelberg mechanism for the other two prototype models. This subtle difference among prototype II, IIb and IIc results from solving condition (25) explicitly for appropriate values of q a ∈ Q, using the fractional D6-brane configurations in tables 26, 37 and 38, respectively.