Deformations, Moduli Stabilisation and Gauge Couplings at One-Loop

We investigate deformations of $\mathbb{Z}_2$ orbifold singularities on the toroidal orbifold $T^6/(\mathbb{Z}_2\times\mathbb{Z}_6)$ with discrete torsion in the framework of Type IIA orientifold model building with intersecting D6-branes wrapping special Lagrangian cycles. To this aim, we employ the hypersurface formalism developed previously for the orbifold $T^6/(\mathbb{Z}_2\times\mathbb{Z}_2)$ with discrete torsion and adapt it to the $\mathbb{Z}_2\times\mathbb{Z}_6\times\Omega\mathcal{R}$ point group by modding out the remaining $\mathbb{Z}_3$ subsymmetry and the orientifold projection $\Omega\mathcal{R}$. We first study the local behaviour of the $\mathbb{Z}_3\times\Omega\mathcal{R}$ invariant deformation orbits under non-zero deformation and then develop methods to assess the deformation effects on the fractional three-cycle volumes globally. We confirm that D6-branes supporting USp(2N) or SO(2N) gauge groups do not constrain any deformation, while deformation parameters associated to cycles wrapped by D6-branes with U(N) gauge groups are constrained by D-term supersymmetry breaking. These features are exposed in global prototype MSSM, Left-Right symmetric and Pati-Salam models first constructed in arXiv:1509.00048 and arXiv:1409.1236, for which we here count the number of stabilised moduli and study flat directions changing the values of some gauge couplings. Finally, we confront the behaviour of tree-level gauge couplings under non-vanishing deformations along flat directions with the one-loop gauge threshold corrections at the orbifold point and discuss phenomenological implications, in particular on possible LARGE volume scenarios and the corresponding value of the string scale $M_{\text{string}}$, for the same global D6-brane models.


Introduction
Since the dawn of string phenomenology, toroidal orbifolds have played a prominent rôle in string model building [3][4][5][6][7][8][9]: they provide for exactly solvable conformal field theories, allow for supersymmetric compactifications and are capable of accommodating the necessary ingredients to construct chiral gauge theories. In the context of Type IIA orientifold model building with intersecting D6-branes, factorisable toroidal orbifolds come with factorisable special Lagrangian (sLag) three-cycles as underlying building blocks for such chiral gauge theories, see e.g. [10-17, 6, 18-25, 2, 1]. 1 More precisely, these fractional three-cycles are wrapped by (stacks of coincident) D6-branes, which support (non)-Abelian gauge theories on their worldvolumes. Consequently, the parameters characterising the gauge theory are related to geometric data associated to the three-cycles. The square of the tree-level gauge coupling for instance scales inversely proportional to the volume of the three-cycle wrapped by the corresponding D6-brane [31,24].
At the singular orbifold point, all exceptional three-cycles located at orbifold singularities have vanishing volumes, and the volume of a fractional three-cycle is simply (a fraction of) the volume of the bulk three-cycle inherited from the ambient six-torus. However, a thorough study of the four-dimensional effective field theory emerging from a Type IIA orientifold compactification requires to consider a region in moduli space where the orbifold singularities have been resolved or deformed. 2 The resolution or deformation of such singular points will have undeniable geometric and physical consequences for the D6-branes wrapping them. In first instance, one has to verify whether the sLag condition of the corresponding fractional three-cycle is preserved under the deformation or not. Whenever the deformation violates the sLag condition, supersymmetry is broken via the appearance of a Fayet-Iliopoulos D-term in the four-dimensional effective field theory; the deformation modulus is then bound to be stabilised at the singular orbifold point. If a fractional three-cycle remains sLag under a particular deformation, its volume -and thereby also the associated inverse of the tree-level gauge coupling squared at M string -is expected to alter with an increasing deformation along this flat direction. One can of course also deform a singularity in the toroidal orbifold at which none of the D6-branes are located, in which case the associated deformation modulus can take any vacuum expectation value (vev) without affecting the physics of the chiral gauge theory at leading order.
When resolving orbifold singularities on a singular Calabi-Yau variety, one usually turns to the toolbox of algebraic geometry and toric geometry, see e.g. [36], which would offer us the necessary techniques to resolve exceptional two-and four-cycles through blow-ups. Toric singularities and blow-up resolutions of divisor four-cycles happen to be part of the modus operandi for constructing chiral gauge theories on the Type IIB side [37][38][39][40][41][42][43][44][45][46][47] using fractional D3-branes located at the singularities or D7-branes wrapping the resolved fourcycles. However, in the case of Type IIA model building with fractional D6-branes on orbifolds with discrete torsion, the orbifold singularities have to be deformed rather than blown up, which forces us to consider different tools from algebraic geometry: by viewing two-tori as elliptic curves in the weighted projective space P 2 112 , a factorisable toroidal orbifold with discrete torsion can be described as a hypersurface in a weighted projective space, with its topology being a double cover of P 1 × P 1 × P 1 . Building on this hypersurface formalism first sketched in [48] for the T 6 /(Z 2 × Z 2 ) orbifold with discrete torsion and extended to its T 6 /(Z 2 × Z 2 × ΩR) and T 6 /(Z 2 × Z 6 × ΩR) orientifold versions with underlying isotropic square [49,50] or hexagonal [51,52,50] two-tori, respectively, we focus here on the so far most fertile patch in the Type IIA orientifold landscape with rigid D6-branes [2,1,53,54], the T 6 /(Z 2 × Z 6 × ΩR) orientifold with discrete torsion and one rectangular and two hexagonal underlying two-tori. In this case, the Z (1) 2 -twisted sector conceptually differs from the Z (2) 2and Z (3) 2 -twisted sectors, necessitating separate discussions for the respective deformations and making the deformations of this toroidal orbifold more intricate than the other previously discussed orbifolds with discrete torsion.
Upon embedding a toroidal orbifold with discrete torsion as a hypersurface in a weighted projective space with carefully chosen weights, (a subset of) sLag three-cycles can be constructed as the fixed loci under the anti-holomorphic involution contained in the orientifold projection ΩR, in a similar spirit as in [55,12,56,57]. The deformations in the hypersurface formalism allow for the description of exceptional and fractional three-cycles, besides the bulk three-cycles, by which the set of sLags three-cycles on the deformed toroidal orbifold can be immensely extended, all corresponding to calibrated submanifolds [58][59][60][61] with respect to the same holomorphic volume three-form Ω 3 . It is exactly the presence of these fractional sLag three-cycles that makes toroidal orbifolds with discrete torsion so appealing for D6-brane model building. Contrarily to a bulk sLag three-cycle, a fractional sLag three-cycle is not necessarily accompanied by an open string moduli space [59], as it is (at least in the absence of an additional Z 3 symmetry completely) projected out by the Z 2 × Z 2 point group. The absence of open string deformation moduli ensures that the non-Abelian gauge group supported by a stack of D6-branes cannot be spontaneously broken by the displacement of a D-brane in that stack. On a more formal level, knowledge about the moduli space of sLag three-cycles is vital in the search for the mirror manifold [62-64, 60, 61, 65] of the deformed toroidal orbifold. The absence of an open string moduli space for fractional three-cycles is expected to complicate this search, which makes studying the geometric characteristics of fractional sLag three-cycles and uncovering their relations to the closed string moduli space all the more essential.
In this article, a first step in revealing those relations for the T 6 /(Z 2 ×Z 6 ×ΩR) orientifold with discrete torsion is taken by studying the functional dependence of the fractional three-cycle volumes on the complex structure (deformation) moduli, whose vevs measure the volumes of the exceptional three-cycles. Through this connection, the viability of a non-zero deformation is assessed by virtue of the preserved sLag conditions of the fractional three-cycles away from the orbifold point, as mentioned before. The physical implications of these deformations for D6-brane model building are discussed in terms of potential Fayet-Iliopoulos terms and/or altering tree-level gauge coupling strength in the effective four-dimensional gauge theories resulting from the orientifold compactifications with D6-branes.
Also Kähler moduli are expected to have a substantial influence on the effective four-dimensional gauge theories, as exhibited through their presence in the one-loop threshold corrections to the gauge couplings at the singular orbifold point, see e.g. [66][67][68][69][70][71] in the context of D6-branes. These gauge threshold corrections can be sizeable for specific anisotropic choices of two-torus volumes [25], given by the vacuum expectation values of the (CP-even part of the) Kähler moduli. In the class of models under consideration, these sizeable gauge threshold corrections are able to lift the degeneracy of the tree-level gauge coupling strengths for distinct fractional D6-brane stacks wrapping the same bulk three-cycle. With a lifted degeneracy of the gauge couplings already at the singular orbifold point at one-loop, it becomes more conceivable to construct global intersecting D6-brane models with e.g. a very strongly coupled hidden gauge group, whose gaugino condensate forms a natural source for spontaneous supersymmetry breaking. Clearly, establishing the full moduli-dependence of the one-loop correction to the gauge coupling represents a conditio sine qua non for string model builders, both at and away from the singular orbifold point.
This article is organised as follows: in section 2, we briefly review the hypersurface formulation for describing local deformations of T 6 /(Z 2 × Z 2 × ΩR) singularities as discussed in [48,49,51,52] and then go on to discuss additional constraints imposed by the extra Z 3 symmetry of the T 6 /(Z 2 × Z 6 × ΩR) orientifold. Special attention will be devoted to the sLag cycles used for particle physics model building. In section 3, additional subtleties in global deformations of T 6 /(Z 2 × Z 6 × ΩR) singularities are discussed, and several prototype examples of global D6-brane models with particle physics spectra are examined. Section 4 is devoted to the computation of the one-loop corrections to the gauge couplings at the orbifold point and the phenomenological implications of their specific geometric moduli dependences. Finally, section 5 contains our conclusions and outlook. Additional technical details useful for the computation of deformations and one-loop corrections are relegated to appendices A, B and C.

Deforming Orbifold Singularities in the Hypersurface Formalism
To start, we first briefly review the construction of fractional three-cycles as sums of toroidal and exceptional three-cycles stuck at orbifold singularities in section 2.1, in particular on the orientifold of phenomenological interest T 6 /(Z 2 × Z 6 × ΩR). Then, we move on to reviewing Lagrangian (Lag) lines on two-tori of rectangular and hexagonal shape in the hypersurface formalism in section 2.2. In section 2.3, we first discuss deformations of Z 2 ×Z 2 singularities on T 6 = (T 2 ) 3 and afterwards impose relations among deformations due to the specific additional Z 3 symmetry of the Z 2 × Z 6 action. As a final element, in section 2.4 we discuss the general procedures allowing for the quantitative study of special Lagrangian (sLag) three-cycles on deformations of T 6 /(Z 2 × Z 6 × ΩR) in the hypersurface formalism.
Note that the point group Z 2 × Z 6 is generated by the elements θ and ω, with θ generating the Z 2 -factor acting on the four-torus T 2 (1) × T 2 (2) and ω generating the Z 6 -part acting on the four-torus T 2 (2) × T 2 (3) . As a direct product of two Abelian factors containing each Z 2 as a (sub)group, the orbifold group allows for a global discrete torsion factor η = ±1 [48,18,23], whose presence alters the amount of two-and three-cycles supported in the orbifold twisted sectors, as indicated in table 1 listing the Hodge numbers per sector. In the absence of discrete torsion (η = 1), the Z (i) 2 singularities can be resolved through a blow-up in the respective twisted sector. In the presence of discrete torsion (η = −1), one has to resort to deformations of the Z (i) 2 singularities, yielding exceptional three-cycles located at the former Z (i) 2 fixed loci. The three-cycles in the Z (i) 2 -twisted sectors turn out to be useful tools with regard to particle physics phenomenology and D6-brane model building [2,1], encouraging us to focus for the remainder of the article on the orbifold with discrete torsion.
A first observation regarding the Z 6 -action deals with the shape of the two-tori T 2 (2) × T 2 (3) whose underlying lattice is constrained to be (up to overall rescaling per two-torus) the root lattice of SU (3) × SU (3), i.e. both lattices are hexagonal, and the complex structures of these two-tori are fixed. Only the first two-torus, whose lattice configuration corresponds to (up to overall scaling) the root lattice of SU (2) 2 , has an unfrozen complex structure modulus, matching the Hodge number h 21 bulk = 1 for the Z 2 × Z 6 orbifold. The comparison with the orbifold T 6 /(Z 2 × Z 2 ) with discrete torsion shows that the additional Z 3 -action reduces the number of three-cycles in the Z (i) 2 twisted sectors by triple identifications, and thereby also enforces the simultaneous deformation of the associated Z 2 × Z 2 singularities, as we will discuss in detail in section 2.3.2. For now, we restrict ourselves to counting the (orbits of) singularities appearing in the various twisted sectors of the orbifold and to indicating how they relate to the Hodge numbers in table 1: • Three Z (i=1,2,3) 2 -twisted sectors generated by (ω 3 , ω 3 θ, θ) respectively, where each Z (i) 2 sector comes with 16 fixed two-tori or fixed lines labelled by the points (αβ) with α, β ∈ {1, 2, 3, 4} along T 4 (i) ≡ T 2 (j) × T 2 (k) as depicted in figure 1. In the Z 2 -twisted sector, the fixed point (11) on T 2 (2) × T 2 (3) remains invariant under the orbifold action by ω, while the other fixed points recombine into orbifold-invariant orbits consisting of three fixed points each. More explicitly, the Z 6 -action rotates the fixed points as 2 → 3 → 4 → 2 on T 2 (2) and as 2 → 4 → 3 → 2 on T 2 (3) , which implies the following five orbits of Z (1) 2 fixed points: [(21), (31), (41)], [(12), (14), (13)], [(33), (42), (24)], [(22), (34), (43)] and [(44), (23), (32)]. On the orbifold T 6 /(Z 2 × Z 6 ) without discrete torsion, these Z (i) 2 fixed point orbits contribute to the h 11 Kähler moduli, while they contribute to the h 21 complex structure moduli for the orbifold set-up with discrete torsion when tensored with a one-cycle on T 2 (1) . In the Z (j=2,3) 2 -twisted sectors, the four fixed points (α1) on T 2 (1) × T 2 (k=3, 2) are invariant under the Z 6 -action, while the other twelve fixed points recombine into four Z 6 -invariant orbits of three fixed points each: [(α2), (α3), (α4)]. On the orbifold without discrete torsion, all fixed point orbits contribute to the counting of h 11 , while on the orbifold with discrete torsion only the non-trivial Z 6 -invariant orbits tensored with a one-cycle on T 2 (j) (which is also rotated under Z 6 ) contribute to h 21 .
Now that we have a clear understanding of untwisted and twisted sectors and how they contribute to the Hodge numbers, we can infer the different types of orbifold-invariant threecycles supported on the orbifold T 6 /(Z 2 × Z 6 ) with discrete torsion: (1) Bulk three-cycles are orbifold-invariant products of three one-cycles, where each onecycle extends along a different two-torus T 2 (i) . The homology class of each one-cycle is specified by two integer-valued co-prime torus wrapping numbers (n i , m i ) w.r.t. the basis one-cycles π 2i−1 , π 2i of each two-torus T 2 (i) , see figure 1 for the conventional choice of basis used in this article. The orbifold-invariant products of the basis three-cycles combine into four basis bulk three-cycles (ρ 1 , ρ 2 , ρ 3 , ρ 4 ), matching the Betti-number b bulk 3 = 2(h 21 bulk + 1) = 4 counting the number of basis three-cycles inherited from the factorisable six-torus (T 2 ) 3 after Z 2 × Z 6 identifications. Generic bulk three-cycles can then be expressed in terms of these four basis bulk three-cycles: (2) Exceptional three-cycles are orbifold-invariant products of a one-cycle on the Z αβ located at the Z (i) 2 fixed points (αβ) along the four-torus T 4 (i) . The Z (i) 2 fixed points can be resolved by gluing in a two-sphere per singularity. The Z 6 -invariant products of the basis one-cycles with the exceptional divisors yield twelve basis exceptional cycles ( 2 -twisted sector with λ ∈ {0, 1, 2, 3, 4, 5} and 2 × 8 basis exceptional cycles ( -twisted sectors with α ∈ {1, 2, 3, 4}. The dimensionality of the full set of basis exceptional three-cycles expected from all Z (i) 2 -sectors combined matches the Bettinumber b Z 2 3 = 2h 21 Z 2 = 2 · (6 + 2 × 4) = 28. Furthermore, the Z 6 -and Z 3 -twisted sectors also yield b Z 6 +Z 3 3 = 2 · (2 + 2) = 8 exceptional three-cycles located at the ω and ω 2 fixed points along T 2 (2) × T 2 (3) as detailed above. These latter basis three-cycles need to be taken into account when searching for a unimodular basis of the full three-cycle lattice, but they do not contribute to the standard CFT constructions of Type IIA/ΩR orientifold models [72][73][74], in particular they expected to contribute to the open string one-loop annulus amplitude [12,20], such that they require no further attention from our part, and we shall only focus on the exceptional three-cycles that can be expressed in terms of the Z (i) 2 exceptional basis three-cycles.
(3) Fractional three-cycles are linear combinations of some bulk three-cycle and several exceptional three-cycles. When a bulk three-cycle passes through the Z (i) 2 -fixed points and represents its own Z 2 -orbifold image, one has to add the appropriate set of exceptional three-cycles (weighted with appropriate sign factors) in order to form a closed fractional three-cycle. As such, a fractional three-cycle can be expressed as where the integer-valued exceptional wrapping numbers x (i) , y (i) are constructed from the torus wrapping numbers (n i , m i ) along the two-torus T 2 (i) weighted by sign factors associated to the discrete Z (i) 2 -eigenvalues ±1 and to (-1) exponentiated by the discrete Wilson-lines (τ j , τ k ) ∈ {0, 1}. The explicit form of x (i) , y (i) is constrained by the position of the two-cycle on T 4 (i) set by the discrete shift parameter (σ j , σ k ) ∈ {0, 1}, as detailed in table 36 of [2]. The sum over λ runs over at most four different values in the Z (1) 2 -twisted sector with λ ∈ {0, 1, 2, 3, 4, 5} and two values in the Z (2,3) 2 -twisted sectors with λ ∈ {1, 2, 3, 4} for the orbifold T 6 /(Z 2 × Z 6 ).
A detailed discussion of three-cycles on the orbifold T 6 /(Z 2 × Z 6 ) can be found in [23], while their prospects for intersecting D6-brane model building have been thoroughly investigated in [2,1]. For instance, phenomenologically viable models with three chiral generations were identified in abundance on this orbifold -a feature that can be traced back to the Z 3 -factor in the orbifold group, in analogy to other orbifolds with a Z 3 -factor within the point group [15,21,22,25,75].
Type IIA string compactifcations on T 6 /(Z 2 × Z 6 ) preserve N = 2 supersymmetry in the closed string sector, which can be broken to a phenomenologically more appealing N = 1 supersymmetry by including an orientifold projection consisting of the worldsheet parity Ω, a left-moving fermion number projection (−) F L and an anti-holomorphic involution R. The fixed planes under the involution R combine into four inequivalent orbits under the Z 6 -action, corresponding to the O6-planes ΩR and ΩRZ , respectively, as listed in table 2 for the aAA-lattice configuration. 3 Each of the O6-planes carries RR-charge whose sign is denoted by η ΩR(Z (i) 2 ) ∈ {±1}, and worldsheet consistency of the Klein-bottle relates them to the discrete torsion parameter [18,23]: This implies that at least one of the O6-planes is exotic, with the sign of the RR-charges opposite w.r.t. the other O6-planes. Anticipating the phenomenologically appealing global models discussed in sections 3 and 4, we select the ΩRZ 2 -plane as the exotic O6-plane, i.e. η ΩRZ (3) 2 = −1. 4 The absence of twisted sector contributions in the tree-channel for the Klein bottle and Möbius strip amplitudes indicates that the sum over all O6-planes corresponds topologically to a (fraction of a) pure bulk three-cycle. As a consequence, D6-branes wrapping 3 The anti-holomorphic involution R also constrains the shape of the two-torus lattices and limits the orientation of each lattice w.r.t. the orientifold-invariant direction to two invariant orientations: A or B for a hexagonal lattice and a or b for a rectangular lattice. Through a non-supersymmetric rotation of the lattices, the a priori six independent lattice configurations can be reduced [2] to two physically distinct ones: the aAA and bAA lattices. From the model building perspective with intersecting D6-branes, the aAA-lattice configuration turned out [2,1] to be the most fruitful background allowing for global three-generation MSSM, Left-Right (L-R) symmetric and Pati-Salam (PS) models. 4 The choice η ΩRZ (2) 2 = −1 is equivalent upon permutation of T 2 (2) ↔ T 2 (3) , while there exists a second inequivalent choice of ηΩR = −1 allowing for supersymmetric solutions to the RR tadpole cancellation conditions.
fractional three-cycles should be chosen such that the sum of their bulk three-cycles cancel the RR-charges of the O6-planes, while the sum of the Z (i) 2 -twisted exceptional three-cycle part should vanish among itself for each i ∈ {1, 2, 3}, in order to ensure vanishing RR tadpoles. Note that the basis bulk and exceptional three-cycles decompose into ΩR-even and ΩR-odd three-cycles under the orientifold projection, depending on the choice of the exotic O6-plane, as can be deduced from table 3.
O6-plane Torus wrapping numbers RR-charge Global Models Orientifold images of basis three-cycles on exceptional three-cycles for the aAA lattice configuration, depending on the choice of the exotic O6-plane orbit with sign factor η (k) ≡ η ΩR η ΩRZ (k) 2 .
In order for the D6-branes to preserve the same N = 1 supersymmetry, they are required to wrap special Lagranigian (sLag) three-cycles Π: Three-cycles satisfying condition (5) where the pullback of the Kähler (1,1)-form J (1,1) w.r.t. the three-cycle worldvolume vanishes, are called Lagrangian (Lag) cycles. It is straightforward to check that the (factorisable) bulk three-cycles satisfy this condition. Three-cycles satisfying condition (6) are calibrated w.r.t. the (real part of the) holomorphic volume form Ω 3 , deserving the epithet special. At the orbifold point, the condition (6) reduces to constraints on the torus wrapping numbers and the bulk complex structure moduli. Deforming the background away from the orbifold point can yield an exceptional three-cycle with non-vanishing volume, which no longer satisfies the special condition, implying that supersymmetry can only be maintained when the volume of such an exceptional three-cycle vanishes; in other words the twisted complex structure modulus is stabilised at vanishing vacuum expectation value (vev). Explicit examples of this phenomenon will be discussed in section 3.
For the sake of completeness regarding the discussion of geometric moduli on the T 6 /(Z 2 × Z 6 × ΩR) orientifold with discrete torsion, we also list the counting of Kähler moduli and closed string vectors on the aAA lattice in Hodge Numbers 2 -plane, i.e. η (2) = −η (3) = 1, the θω-twisted sector does not contain any Kähler moduli, i.e. h 11 − = 0. The orientifold projection thus removes the geometric moduli in this sector required for resolving the Z 6 singularities, and a full resolution and deformation of the toroidal orbifold background to a smooth Calabi-Yau threefold is not possible in the presence of the ΩR orientifold action on Type IIA string theory.

Lagrangian lines on the elliptic curve in the hypersurface formalism
At the orbifold point, the geometric engineering method for D6-brane models on T 6 /Z 2N or T 6 /(Z 2 × Z 2M ) backgrounds reviewed above formally uses exceptional divisors at Z 2singularities and their topological intersection numbers, even though their volumes are set to zero, or in other words the associated twisted complex structure moduli have vanishing vevs. When moving away from the orbifold point into the Calabi-Yau moduli space by deforming the Z 2 -singularities, we have to use an extended toolbox of algebraic geometry and embed the orbifold as a hypersurface in an ambient toric space. The first step in this process consists in reformulating the two-tori as elliptic curves in the weighted complex projective space P 2 112 and describing Lag lines on the elliptic curves.
Thus, we introduce the coordinates (x, v, y) with weights (1, 1, 2) as the homogeneous coordinates of the projective space P 2 112 and describe a two-torus as a hypersurface within P 2 112 . More explicitly, a two-torus corresponds to an elliptic curve in P 2 112 , which forms the zero locus of a polynomial f of degree 4: where we choose the Weierstrass form for the elliptic curve. There exists a Z 2 reflection symmetry acting only on y → −y, yet its fixed points correspond to the roots of the polynomial F (x, v). By expanding F (x, v) in terms of its (finite) roots 2 , 3 and 4 , the coefficients g 2 and g 3 are easily related to the roots: g 2 = −4 ( 2 3 + 2 4 + 3 4 ) and g 3 = 4 2 3 4 , with the roots satisfying the condition 2 + 3 + 4 = 0. The fourth root 1 located at x = ∞ (in the v = 1 patch) represents the Z 2 fixed point at the origin. The coefficients g 2 and g 3 are on the other hand uniquely determined by the torus lattice and its complex structure parameter τ , such that we can limit ourselves to those torus lattices relevant for the orbifold T 6 /(Z 2 × Z 6 ): (1) a-type lattices or untilted (rectangular) tori with Re(τ ) = 0: generically the roots α are all real and can be ordered as 4 < 3 < 2 . A square torus with τ = i represents a special case for which g 3 = 0, 2 = − 4 = 1 and 3 = 0.
(2) b-type lattices or tilted tori with Re(τ ) = 0: generically the roots 2 and 4 are related by complex conjugation, 2 = 4 ≡ ξ, while 3 = −2Re(ξ) is a real parameter. A hexagonal torus with τ = e i π 3 (cf. figure 1 (b)) forms a special case with g 2 = 0 and ξ = e 2πi/3 , for which the elliptic curve exhibits an additional Z 3 symmetry x v → e 2πi/3 x v . This Z 3 symmetry is in correspondence with a Z 3 subgroup acting on the hexagonal two-torus lattice, suggesting that the two-tori T 2 (2) and T 2 (3) are perfectly described by this type of elliptic curve.
A pictorial representation of a square untilted and a hexagonal torus lattice with their respective roots is given in figure 1. The two-torus T 2 (1) is not affected by the Z 3 subgroup of the Z 2 × Z 6 point group, hence its torus lattice can in principle be either untilted (a-type lattice) or tilted (b-type lattice). As a tilted two-torus T 2 (1) does not provide for any (known) phenomenologically appealing global intersecting D6-brane models [2,1], we confine ourselves to an untilted T 2 (1) and simplify the set-up even more by choosing a square two-torus when studying deformations. This simplification is justified by the fact that for the choice of an exotic ΩRZ (3) 2 -plane, the (bulk) RR tadpole cancellation conditions can only be solved in a supersymmetric way if all D6-branes extend along π 1 on the a-type lattice, see section 3 for several examples. Such configurations are supersymmetric for any value of the complex structure parameter Im(τ ) > 0 on T 2 (1) . Figure 1: Lattice configuration for a square (a) and hexagonal (b) two-torus T 2 (i) . The grey areas indicate a fundamental cell in the two-torus lattice, while the coloured vectors represent the basis onecycles π 2i−1 , π 2i spanning the fundamental cell (with the two-torus index i suppressed in the following). The red points correspond to Z 2 fixed points on T 2 , whose labels match the indices of the roots of F (x, v) in equation (8). An anti-holomorphic involution keeping the one-cycle π 1 fixed also leaves all fixed points on the square two-torus invariant, while such an involution permutes the fixed points 2 and 4 on the hexagonal lattice and leaves the other two points 1 and 3 invariant.
The full map between a two-torus and an elliptic curve is given by Weierstrass' elliptic function ℘(z), mapping bijectively the holomorphic coordinate z on the two-torus with modular parameter τ to the elliptic curve with coefficients g 2 and g 3 . It is easy to see that the Weierstrass' elliptic function ℘(z) satisfies the hypersurface equation (7) through the identification ℘(z) = x/v, ℘ (z) = y/v 2 , which reduces to a differential equation on ℘(z).
One-cycles on a two-torus T 2 (i) were introduced in the previous section parameterised by the torus wrapping numbers (n i , m i ) w.r.t. the basis one-cycles. In order to discuss Lag lines on an elliptic curve, we introduce an anti-holomorphic involution σ acting on the homogeneous coordinates as follows: with A ∈ GL 2 (C). For this action to be an involution, the matrix A has to satisfy the condition AA = 1 I. The involution also has to be a symmetry of the elliptic curve, which boils down to the following condition σ (F (x, v)) = e 2iβ F (x, v). Solving both conditions allows to extract the unequivocal forms of the various anti-holomorphic involutions [49]. Afterwards, one can determine the fixed loci for each individual anti-holomorphic involution, which will constitute only a subset of all Lag lines on the elliptic curve [49,51,52]. Fortunately for us, the Lag lines defined as fixed loci under σ are in one-to-one correspondence with the torus one-cycles used as building blocks for global intersecting D6-brane models, as can be checked explicitly by virtue of the Weierstrass' elliptic function ℘(z). Distinguishing between square and hexagonal lattices leads to the following classification of Lag lines: (1) untilted square torus: we distinguish between four one-cycles aX (with X = I, II, III, IV) passing through two roots α and one-cycles cX (with X = I, II) not passing through any of the roots. The first type of one-cycles will serve as fractional cycles, while the latter type of one-cycles remain bulk cycles once the orbifold T 6 /(Z 2 × Z 6 ) is modelled as a hypersurface in an ambient toric space. A full overview of these Lag lines on the square torus and the relations to the roots α is offered in table 5; their positions in the x-plane are depicted in figure 2 (a) for the v = 1 coordinate patch.
Lagrangian lines on square untilted torus T 2 Table 5: Overview of Lag lines on an elliptic curve corresponding to a square untilted two-torus. The first column lists the torus wrapping numbers (n i , m i ) describing how the one-cycles wrap on the two-torus lattice, while the second column indicates a potential displacement along one of the basis one-cycles (we distinguish between no displacement, a displacement over one-half of a basis one-cycle or a continuous displacement in between the latter two options). The third column presents the equation for the Lag line in the homogenous coordinate x (considering the v = 1 patch) in line with figure 2 (a). The last column gives a graphical representation of the Lag one-cycle on the two-torus lattice.
(2) hexagonal torus: here we can identify the one-cycles bX (with X = I, II, III, IV) passing through two roots α and corresponding to σ-involution invariant directions. Due to the Z 3 symmetry, we also find the by ± 2π 3 rotated images of these one-cycles, tripling the number of individual one-cycles. A full list of Lag lines is given in table 6, while figure 2 (b) shows their position in the x-plane for the v = 1 coordinate patch and clearly exhibits the Z 3 symmetry.
The holomorphic one-form Ω 1 defined on the elliptic curve descends from the following holomorphic two-form defined on the ambient space P 2 112 : cI cII aII aI aIV aIII 2 3 4 Im(x) Figure 2: Graphic representation of Lag lines on an elliptic curve embedded in P 2 112 corresponding to a square untilted two-torus (a) or a hexagonal two-torus (b). The Lag lines are drawn in the x-plane with v = 1, following the notation in the overview tables 5 and 6.
where we divided by the polynomial f defined on the l.h.s. of equation (7) to obtain a welldefined scale-invariant two-form on P 2 112 . The integral is taken over a curve γ around the singular region f = 0, such that we can apply Cauchy's residue theorem in a suitable patch to obtain the expression: The expression for y in terms of x and v follows by imposing the hypersurface equation f = 0 and choosing one branch of the square root. With this prescription, the holomorphic oneform (11) allows us to uncover the calibration form for each of the Lag lines identified above. Roughly speaking, Lag cycles with X=even are calibrated w.r.t. Re(Ω 1 ), while Lag lines with X=odd have Im(Ω 1 ) as calibration form.
Lagrangian lines on hexagonal torus T 2 Table 6: Overview of Lag one-cycles on an elliptic curve corresponding to a hexagonal two-torus. The first column lists the torus wrapping numbers (n i , m i ) describing how the one-cycles wrap on the two-torus lattice, while the second column indicates a potential displacement along one of the basis one-cycles. The third column presents the equation for the Lag line in the homogenous coordinate x (considering the v = 1 patch) in line with figure 2 (b). The last column gives a graphical representation of the Lag one-cycle on the two-torus lattice.
2.3 T 6 /(Z 2 × Z 6 × ΩR) in the hypersurface formalism Describing deformations of the orbifold T 6 /(Z 2 × Z 6 ) requires us to adopt a hypersurface formalism in which T 6 /(Z 2 × Z 6 ) with discrete torsion is embedded into an appropriate toric space. In a first step, we will review in section 2.3.1 how the deformations of T 6 /(Z 2 × Z 2 ) with discrete torsion thrive in the hypersurface formalism, after which we mod out the remaining Z 3 symmetry within the Z 2 × Z 6 point group in section 2.3.2 to end up with the hypersurface formalism describing the deformations of T 6 /(Z 2 × Z 6 ). Finally, we impose additional constraints due to the orientifold involution ΩR.

Hypersurface formalism for
The factorisable orbifold T 6 /(Z 2 × Z 2 ) with discrete torsion can be mapped [48] to the direct product of three distinct elliptic curves modded out by the orbifold group Z 2 × Z 2 , whose action only keeps the Z 2 × Z 2 -invariant subring of the ring of polynomials. The Z 2 × Z 2invariant "monomials" are subject to a single equation describing a hypersurface in the toric space parametrised by the coordinates (x i , v i , y) i=1,2,3 with weights q i according to the weight diagram in table 7.
Scaling charges q i for T 6 /(Z 2 × Z 2 ) toric space Table 7: Overview of the weights q i for the coordinates (x i , v i , y) i=1,2,3 parametrising the toric space, as well as for the polynomial f allowing for the embedding of T 6 /(Z 2 × Z 2 ) as a hypersurface in this toric space.
Indeed, the orbifold T 6 /(Z 2 × Z 2 ) with discrete torsion or its deformations are described by the zero locus of the polynomial f (x i , v i , y) whose most general form reads: A few comments regarding this polynomial are in order: • The polynomials F (i) (x i , v i ) correspond to the homogeneous polynomials of degree four defining the two-torus T 2 (i) as in equation (7) or its rewritten form (8) and encode information about the complex structure of T 2 (i) . As each two-torus has its own complex structure, we have a priori three bulk complex structure moduli in total (with the number later on being reduced by imposing an extra Z 3 symmetry, cf. section 2.3.2). Setting all the deformation parameters to zero, ε • The deformation polynomials δF α (i) (x i , v i ) are also homogeneous polynomials of degree four and have the same roots as F (i) (x i , v i ) up to the α th zero. Thus, δF α (i) (x i , v i ) allows to deform the Z 2 -fixed point associated to the α th root.
• Deformations of the form δF α (i) F (j) F (k) with (ijk) a cyclic permutation of (123) are not explicitly considered in equation (12) as they correspond to deformations of the complex structure for two-torus T 2 (i) , up to P SL(2, C) transformations acting on (x i , v i ). 5 Hence, their coefficients correspond to untwisted moduli and their CFT counter-parts are given by the three truly marginal operators from the untwisted sector in the associated N = (2, 2) super-conformal field theory, following the construction prescriptions in [76].
• The parameter ε • Within the set of possible deformations of T 6 /(Z 2 × Z 2 ), one also observes the deformations associated to the parameters ε αβγ . The total number of these parameters amounts to 4 × 4 × 4 = 64 and is in one-to-one correspondence with the number of Z 2 × Z 2 fixed points on the orbifold. However, these parameters do not represent independent deformations, but rather depend on the complex structure deformations ε (i) αβ from the Z (i) 2 twisted sectors, such that (at most) 64 conifold singularities remain and cannot be deformed away. This reflection is supported [48] by the absence of truly marginal operators in the N = (2, 2) SCFT corresponding to ε αβγ -deformations. 6 Thus, determining whether or not conifold singularities are present can only be assessed through the geometric description of the deformed orbifold in the hypersurface formalism in terms of the independent deformation parameters ε (i) αβ . 5 The P SL(2, C) transformations are able to eliminate three complex parameters, leaving exactly one independent parameter per two-torus. 6 Contrary to the blow-up procedure, where blowing up the co-dimension two singularities in the Z (i) 2 twisted sectors also eliminates the co-dimension three singularities on T 6 /(Z2 × Z2) without discrete torsion, the deformation procedure does not automatically lead to the resolution of the 64 Z2 × Z2 fixed points on T 6 /(Z2 × Z2) with discrete torsion.
In order to determine the holomorphic volume form Ω 3 on T 6 /(Z 2 × Z 2 ) with discrete torsion, we extend the philosophy from above that allowed us to identify the appropriate hypersurface equation (12). More explicitly, we consider the wedge product of three one-forms Ω 1 , one for each two-torus T 2 (i) as defined in equation (11), and mod out the Z 2 × Z 2 symmetry to obtain the following (simplified) expression (in the v i = 1 patch): up to an overall normalisation constant and possible phase. The expression for y in terms of x i follows by imposing the hypersurface equation f ! = 0 on the defining equation (12) and fixing a branch cut for the square root.
A thorough analysis of Lag lines on the (deformed) orbifold T 6 /(Z 2 ×Z 2 ) on square tori was the subject of [49] and further comments and extensions to products of three hexagonal tori can be found in [51,52,50]. As is well known, the orbifold T 6 /(Z 2 ×Z 2 ) with discrete torsion does not support exceptional two-cycles, implying the absence of twisted sector blow-up modes. If one wants to blow up the Z (i) 2 singularities rather than deform them, one ought to look at the version of the orbifold without discrete torsion, see e.g. [77,36]. Physical implications of resolving the orbifold singularities through blow-ups have been investigated in [78][79][80][81].

Hypersurface formulation for orbifolds with additional Z 3 × ΩR action
Setting up the hypersurface formalism for T 6 /(Z 2 × Z 6 ) with discrete torsion now consists in acting with the Z 3 -subgroup generated by 2 b = 1 3 (0, 1, −1) on the hypersurface formulation of T 6 /(Z 2 × Z 2 ) with discrete torsion, such that the resulting hypersurface polynomial is invariant under the Z 3 -symmetry. 7 The Z 3 -action generated by ω 2 will in the first place restrict the form of the homogeneous polynomials F 2 and F 3 , while the form of F 1 remains generic as in equation (8). Anticipating the torus lattice configurations for the global models in section 3, we consider the first two-torus to be a square untilted two-torus, such that the homogeneous polynomials are given by: Evidently, the Z 3 -action also constrains the form of the deformation polynomials δF α (j=2,3) , while the deformation polynomials δF α (1) are shaped by the untilted square lattice choice for 7 Notice that in [51,52] a different T 6 /(Z2 × Z 6 ) orbifold with the Z 6 factor generated by b = 1 6 (−2, 1, 1) was considered. The analysis in that case with Z2 × Z 6 point group was simpler due to all three two-tori being of hexagonal shape and all three Z the two-torus T 2 (1) : By virtue of the Weierstrass' elliptic function, one can easily deduce that the Z 3 -subgroup also acts on the homogeneous coordinates x i as follows: 8 which leaves the homogeneous polynomials F (i) (x i , v i ) invariant, but forces the deformation polynomials to transform as follows: Keeping in mind these transformation properties, we can deduce that the linear combination δF 2 (j) + δF 3 (j) + δF 4 (j) = 3v 4 j represents a Z 3 -invariant polynomial for j = 2, 3. However, a simple coordinate transformation on x j eliminates any deformation of the type 3v 4 j F (1) F (k =j) , leaving the complex structure of the two-torus T 2 (j=2, 3) unaltered. The non-invariance of δF 1 (j=2, 3) under Z 3 also excludes any type of deformation of the form δF 1 (j=2,3) F (1) F (k=3,2) . This observation agrees with the considerations in section 2.1 that the complex structures of the two-tori T 2 (j=2,3) are frozen to hexagonal shape by the Z 3 -action. The deformation polynomials δF α (1) are left invariant by the Z 3 -action, such that deformations of the type δF α (1) F (2) F (3) do exist, up to P SL(2, C) transformations acting on (x 1 , v 1 ), and they represent one untwisted complex structure modulus, in line with h 21 bulk = 1. Recall from footnote 5 on page 18 that the three complex parameters of the P SL(2, C) symmetry allow to reduce the four deformations to a single independent deformation.
Deformations of the Z (i) 2 singularities are performed through polynomials of the form F (i) δF α (j) δF β (k) , which should also be Z 3 -invariant for consistency. In this sense, the Z 3 -action will put restrictions on the deformation parameters ε     sectors are equivalent as a result of the T 2 (2) ↔ T 2 (3) exchange symmetry, such that we can treat them jointly. In the Z (j) 2 -sector we find four independent deformation parameters, which match the Hodge numbers h 21 • ∀ α ∈ {1, 2, 3, 4} we find ε (j) α1 = 0, which excludes any type of deformation of the fixed The last type of deformations to consider have the form δF α (1) δF β (2) δF γ (3) and are also subject to the Z 3 -action. The invariance of δF α (1) under the Z 3 -action suggests that we should only worry about the remaining two deformation polynomials and make sure they recombine into Z 3 -invariant combinations with the following relations: • ε α11 remains unconstrained (∀ α ∈ {1, 2, 3, 4}); Hence, we obtain in total 6 × 4 = 24 Z 3 -invariant deformations of the type δF α (1) δF β (2) δF γ on the T 6 /(Z 2 × Z 6 ) orbifold, which should be contrasted with the 64 parameters on the T 6 /(Z 2 × Z 2 ) orbifold. Observe that the initial 64 Z 2 × Z 2 fixed points split up into four fixed points, which are left invariant under the Z 3 action, and 60 fixed points, which are regrouped into Z 3 -invariant triplets. This simple counting explains the 24 allowed deformations δF α (1) δF β (2) δF γ (3) . Similar to the Z 2 × Z 2 orbifold, the ε αβγ -deformations depend on the twisted complex structure deformation parameters ε (i) αβ , such that at most 24 conifold singularities remain upon deformation by ε (i) αβ . In CFT language this would imply that the truly marginal operators associated to the ε αβγ -deformations do not exist in the associated N = (2, 2) SCFT. Hence, also for this orbifold the potential presence of conifold singularities can only be assessed by investigating the hypersurface equation algebraically in the hypersurface formalism.
The last element missing to describe sLags in this hypersurface set-up is the orientifold involution σ R which acts on the homogeneous coordinates as follows: This anti-holomorphic involution, constructed from the involution σ defined in equation (9) by choosing A = 1 2 on each two-torus T 2 (i) , has to be a symmetry of the hypersurface, which boils down to the condition σ R (f ) = f . At the orbifold point, this latter condition constrains the shape of the three two-tori to be either of a-type or b-type (the latter corresponding to both Aand B-orientation for hexagonal lattices) as discussed in section 2.2 and ensures that the orientifold involution is an automorphism of the torus-lattices. The orientifold involution also acts on the deformation polynomials (15) as follows: which concurs with the ΩR-action on the Z 2 fixed points of the aAA lattice, whose individual two-torus positions are depicted in figure 1. Taking into account the action of the involution, we observe that the deformation parameters are even further reduced: the a priori complex deformation parameters are either constrained to be real, or two complex deformation parameters are identified, leaving only one independent complex deformation parameter. The latter occurs for the deformation parameters ε λ=0,1,2,3 and ε (2,3) α=1,2,3,4 are constrained to be real. A summary of the independent deformation parameters for the orientifold T 6 /(Z 2 ×Z 6 ×ΩR) with discrete torsion and aAA lattice is given in table 8.

Parameter identification
Parameter range Exceptional wrapping numbers 2 -twiste sector for T 6 /(Z 2 × Z 6 × ΩR) with discrete torsion (η = −1) and the restrictions following from the Z 3 -action and the ΩR-action. The last column relates the deformation parameters to the relevant exceptional wrapping numbers introduced in equation (3)  locus of the following polynomial f : In this polynomial expression, one clearly notices the difference in the Z 2 deformations on the one hand and the Z deformations. The latter deformations on the other hand are expected to follow a similar pattern due to the two-torus exchange symmetry

Deforming special Lagrangian cycles on
Having formulated the consistent hypersurface formalism to discuss deformations of T 6 /(Z 2 × Z 6 × ΩR) with discrete torsion, we can now turn our attention to the geometric properties of sLag three-cycles away from the orbifold point, after providing a concise translation of the three-cycles introduced in section 2.1 into the hypersurface formalism.

sLags at the Orbifold Point
A minimal set of sLags is defined as the fixed loci under the orientifold involution σ R , introduced in equation (18). Due to the additional Z 3 -symmetry in equation (16) on the x 2,3 coordinates, this set of sLags can be extended by demanding that they be invariant under the action of σ R × Z 3 , a group isomorphic to the symmetric group S 3 . To describe the location of the O6-planes in the hypersurface formalism, it suffices to determine the invariant solutions for the σ R -sector, as the Z 3 element ω 2 maps the O6-planes from the σ R ω 2 -and σ R ω 4 -sectors to this one. In a coordinate patch where v i = 0, we can use the (C ) 3 -scaling symmetry of the ambient space to set v i = 1, such that the O6-planes form a three-dimensional subspace spanned by {Re(x 1 ), Re(x 2 ), Re(x 3 )} within the complex three-dimensional space parametrised by the coordinates {x 1 , x 2 , x 3 }. Furthermore, this three-dimensional subspace corresponds to the region y 2 (x i ) ≥ 0, implying that the O6-planes are calibrated w.r.t. Re(Ω 3 ), the real part of the holomorphic three-form Ω 3 defined in equation (13). This identification of the O6-planes as a real three-dimensional subspace of the x i -planes matches the geometric description in terms of the torus wrapping numbers provided by table 2 and will allow us to verify which sLag three-cycles are calibrated w.r.t. the same holomorphic three-form and therefore preserve the same N = 1 supersymmetry.
When it comes to the sLag three-cycles, we should first offer a clear dictionary between the three types of three-cycles defined in section 2.1 and three-dimensional subspaces on the x iplanes. In order to construct (factorisable) three-cycles on the x i -planes, we can consider the product N 1 ⊗ M 2 ⊗ M 3 consisting of Lag lines on each of the two-tori, with N 1 one of the Lag lines from table 5 and M 2 , M 3 Lag lines from table 6 (or displacements thereof). A necessary condition for the product N 1 ⊗ M 2 ⊗ M 3 to be a supersymmetric three-cycle is that their relative angles w.r.t. the O6-planes add up to 0 modulo 2π. 9 The factorisable bulk three-cycles of section 2.1 are then constructed from Lag lines which do not cross the Z (i) 2 fixed points, such as lines cI and cII on T 2 (1) and continuous displacements of bX 0,± with X ∈ {I, II, III, IV}. This type of Lag lines forms curves (circles) on each x i -plane separately, as for instance shown in figure 2, indicating that a bulk three-cycle has typically the topology of a three-torus T 3 . Only bulk three-cycles that lie sufficiently close to a deformed Z (i) 2 singularity will experience alterations to their overall three-dimensional volume, yet they will always keep their sLag property. This vanilla-like behaviour of bulk three-cycles on deformed orbifolds suggests us to dwell on them no longer than necessary, but rather to focus on the two other types of three-cycles. 9 The relative angles w.r.t. the ΩR-invariant plane can be inferred from the torus wrapping numbers: determines the angle (mod π) on an A-type square T 2 (1) and φ (j) = arctan √ 3m j 2n j +m j on an A-type hexagonal T 2 (j) with j = 2, 3, and sgn(n i ) encodes the orientation to arrive at (mod 2π).
As a matter of fact, the interesting phenomena occur for exceptional and fractional threecycles passing through deformed singularities. At the orbifold point, there is no possibility to express the exceptional divisors e (i) αβ in terms of the homogeneous coordinates x i , as their volumes are shrunk to zero. Hence, we relegate the discussion of the exceptional three-cycles to the next two subsections, where we will discuss in detail the geometry of exceptional divisors located at deformed Z (i) 2 fixed points for the various distinct deformation parameters. In the meantime, we develop a dictionary for the fractional three-cycles associated to the Z (i) 2twisted sectors at the orbifold point and assume that for only one twisted sector at a time the deformations will be turned on. In that case, we can describe the fractional three-cycles for the Z (i) 2 -twisted sector as a direct product of a one-cycle on T 2 (i) and a two-cycle on in later sections). If we limit ourselves to the cycles aI, aII, aIII and aIV on T 2 (1) , the total number of two-cycles on T 2 (2) × T 2 (3) based on table 6 is 12 2 = 144. The sum of the relative angles for these two-cycles w.r.t. the O6-planes adds up to {0, ± π 6 , ± π 3 , π 2 }, suggesting six different calibration angles. The 24 two-cycles with calibration angle 0 combine with the one-cycles aI or aIII to form three-cycles calibrated w.r.t. the same three-form Ω 3 as the O6-planes, while the 24 two-cycles with calibration angle π 2 combine with one-cycles aII or aIV to form supersymmetric three-cycles. This leads to 96 three-cycles which are further reduced to 32 independent ones as a result of the identification of the two-cycles on Taking into account the possibility of turning on discrete Wilson lines or discrete Z (i) 2 -eigenvalues at the singularities offers a large enough class of fractional three-cycles to construct a variety of phenomenologically interesting global intersecting D6-brane models [2,1].
If we look closer at the fractional three-cycles constructed from aI, aII, aIII, aIV, bI 0 and bII 0 , we notice that these one-cycles all lie along the real axis Re(x i ) in the complex x i -planes in figure 2 and have Z 2 fixed points as boundaries. For the aI, aII, bI 0 and bII 0 cycles one should imagine the point 1 at infinity x i = ±∞ as the second boundary, representing the singularity at the origin of the two-tori T 2 (i) . For a fractional three-cycle associated to e.g. the Z (1) 2 -twisted sector, the one-cycle on T 2 (1) has a topology of a circle S 1 , while the two-cycle on T 4 (1) /Z 2 corresponds to a two-torus pinched down at the boundary points, i.e. at the Z (1) 2 fixed points lying on the zero locus y = 0. Hence, the topology of a fractional threecycle is simply S 1 × T 2 /Z 2 . A more pictorial representation will be given in sections 2.4.2 and 2.4.3. For these kinds of fractional three-cycles, the holomorphic three-form Ω 3 factorises as Ω 3 = Ω 2 ∧ Ω 1 , such that we can compute the integrals of Ω 2 over the two-cycles on separately from the integrals of Ω 1 over the one-cycles on T 2 (i) .
In the next two subsections we discuss the effects of deformations in the Z -twisted sectors on the exceptional and fractional three-cycles and investigate how their volumes increase or decrease due to the deformation. Due to the exchange symmetry T 2 (2) ↔ T 2 (3) it suffices to discuss only one of the Z (2,3) 2 -twisted sectors, as the other one will yield the same results. Hence, we can choose to focus on deformations in the Z 2 -twisted sector.

sLag s in the deformed Z
(1)

-twisted sector
For a qualitative appreciation of the deformation effects in the Z (1) 2 -twisted sector, we switch to the x i = 1 patch and describe the sLags in terms of the homogeneous coordinates v i . The cycles bI 0 and bII 0 are still given by real hypersurfaces at the orbifold point in terms of the homogeneous coordinates v i=2,3 , though in comparison to figure 2 their regional conditions are changed: for the Lag line bI 0 we find the constraint 0 Figure 3 depicts the T 2 /Z 2 topology for the two-cycles constructed from bI 0 and bII 0 on Def , with the blue-shaded regions representing sLag two-cycles calibrated with respect to Re(Ω 2 ). The blue contourlines correspond to the zero locus y = 0 in the R-projected plane (v 2 , v 3 ), and these lines intersect at the Z (1) 2 fixed points (11), (13), (31) and (33). The fact that we are able to depict graphically the behaviour of the aforementioned singularities follows immediately from the choice of the coordinate patch x i = 1. Other singularities correspond to complex roots and therefore do not lie in the R-restricted plane (v 2 , v 3 ), such that they are not depicted in figure 3. The two-cycles bI 0 ×bI 0 and bII 0 ×bII 0 should be paired with a one-cycle aI or aIII on T 2 (1) to form a sLag three-cycle calibrated with respect to Re(Ω 3 ). The white regions in figure 3 on the other hand represent sLag two-cycles calibrated with respect to Im(Ω 2 ), namely the two-cycles bI 0 ×bII 0 and bII 0 ×bI 0 .
Anticipating the examples later on, the hidden stacks h 1,2 in the Left-Right (L-R) symmetric model I of section 3.4 belong to the three-cycle type aI×bI 0 ×bI 0 , while the hidden stacks of the Pati-Salam (PS) II model of section 3.5 and of the L-R symmetric II model of section 3.4 are of the type aI×bII 0 ×bII 0 , and the hidden stacks of the L-R symmetric IIb model are of the type aIII×bII 0 ×bII 0 .

By turning on each Z
(1) 2 deformation parameter ε (1) λ separately in figure 3, we can explicitly see which singularities are deformed and which singularities are displaced by the respective deformation parameter. This information is also summarised in the upper part of table 9, which results from determining the singular points of the hypersurface equation (20). At the deformed singularity, an exceptional cycle with non-vanishing volume emerges, which is indicated by a red dashed line in figure 3. An interesting observation is that the point (33) gets deformed too when turning on the deformation parameters ε This phenomenon was also observed [51] for specific deformations of complex co-dimension 2 singularities on the subspace T 4 (i) /Z 2 of the orbifold T 6 /(Z 2 × Z 6 ) with discrete torsion and can be resolved by turning on a correction-term ε (1) 3 depending on the respective deformation parameter, as depicted in the lower diagrams of figures 3 (f) and (g). A similar consideration holds for the fixed points (22) and (44) which are also deformed, but now for a non-vanishing deformation parameter ε (1) 3 = 0. Further details about the counter-terms can be found in appendix B.
3 , e 3 , e Deformations in the Z 2 , e 1 , e 1 , e  fixed points of T 6 /(Z 2 × Z 6 × ΩR) with discrete torsion per deformation parameter ε behave under deformations. The other singular points belonging to the respective Z 3 -invariant orbits are not depicted as they do not lie in the Rrestricted plane. To depict these latter singularities, one has to perform an appropriate Z 3 -rotation on the coordinates. Similar considerations hold for the Z as a C 2 /Z 2 (or A 1 -type) singularity. In first instance, we look at the zero locus of the hypersurface equation (20), turn on each Z 2 deformation separately and discuss their effects in a local patch around the singular point (33). This local description can be extracted straightfordwardly for the deformation ε 2 ), we have to perform a Möbius transformation as explained in appendix A, or a complex rotation (ε 4−5 ) to extract the proper local structure by placing the singularity at the point (33). More explicitly, there exists a Möbius transformation λ 3 [51] acting on the homogeneous coordinate x i that allows to map the Z 2 fixed point 1 situated at the origin of a two-torus T 2 (i) to the fixed point 3 located on the real axis in the new coordinatex i , such that a singular point (αβ) with either α = 1 and/or β = 1 can always be mapped by λ 3 to the point (33) in the new coordinates. The Z 2 fixed points 2 and 4 on the other hand are mapped to the point 3 by a Z 3 -transformation as can be seen in figure 2 (b). With an appropriate rescaling of the homogeneous coordinates, we then find that the singularity is locally described by the following hypersurface equation: A first observation is of course that the co-dimension two singularity locally takes the form of a C 2 /Z 2 singularity in the (x 2 ,x 3 )-plane. The two-cycles bI 0 ×bI 0 and bII 0 ×bII 0 , with torus wrapping numbers (n 2 , m 2 ; n 3 , m 3 ) = (1, 0; 1, 0) and (−1, 2; 1, −2) on T 2 (2) × T 2 (3) , respectively, pass through the Z (1) 2 fixed points affected by a non-vanishing deformation parameter ε (1) λ=0,1,2,3 . A full fractional three-cycle calibrated with respect to Re(Ω 3 ) is then constructed as described in section 2.1 by combining these two-cycles with e.g. the one-cycle aI on T 2 (1) , such that the Z (1) 2 exceptional contributions to a fractional three-cycle can be expressed in terms of the basis three-cycles ( λ ) as follows: Now, by switching on one of the associated four deformation parameters, an exceptional two-cycle e (1) λ=0,1,2,3 with non-vanishing volume grows out of the respective fixed point along T 4 (1) /Z 6 , and it clear from this construction that the volume of an associated fractional threecycle is also influenced by the evolution of the volume of the exceptional cycle under deformation. At the orbifold point, the two-cycles bI 0 ×bI 0 and bII 0 ×bII 0 along T 4 (1) /Z 6 can be represented by a set of real two-dimensional regionsx 2 ·x 3 ≥ 0 in the (x 2 ,x 3 )-plane. When turning on a deformation parameter ε (1) λ=0,1,2,3 , the volume of these two-cycles will shrink or grow depending on the sign of the deformation parameter: • ε (1) λ > 0 : bI 0 ×bI 0 and bII 0 ×bII 0 are still two separate two-cycles, both with shrinking sizes as an exceptional two-cycle e (1) λ grows out of the singularity (33) in the regioñ y 2 < 0. This situation is represented in the upper diagram of figure 3 The exceptional two-cycle satisfies the algebraic conditionx 2 =x 3 , by which the hypersurface equation (21) reduces to the equation for a two-sphere S 2 with radius c λ ε (1) λ . As the two-cycles bI 0 ×bI 0 and bII 0 ×bII 0 remain two separate sLags, the exceptional two-cycle e (1) λ has to be calibrated with respect to the same two-form Re(Ω 2 ). This statement can be shown explicitly by computing the (non-vanishing) volume e (1) λ Re(Ω 2 ) of the exceptional two-cycle as a function of the parameter ε (1) λ in the hypersurface formalism and taking into account thatỹ 2 < 0 andx 2 =x 3 . Extending these considerations to the three-cycles on ( 2 , we conclude that the exceptional three-cycle (1) λ is calibrated with respect to the same three-form Re(Ω 3 ) as the bulk three-cycles and that there should be a relative minus sign between the bulk three-cycle Π bulk and Π Z (1) 2 in order for the volume of the fractional cycle to decrease upon deformation of the singularity, in line with figure 3 (e).
The cycles bI 0 ×bII 0 and bII 0 ×bI 0 on the other hand have merged into one big twocycle and are no longer sLag two-cycles separately. As these two-cycles are calibrated with respect to Im(Ω 2 ), we should take a union two-cycle bI 0 ×bII 0 ⊕bII 0 ×bI 0 from which the exceptional cycle e (1) λ is eliminated, such that the union two-cycle remains sLag with respect to Im(Ω 2 ).
• ε (1) λ < 0 : bI 0 ×bI 0 and bII 0 ×bII 0 are no longer separate two-cycles but melt together as shown in the lower diagram of figure 3 (e), while an exceptional two-cycle e (1) λ grows out of the singularity (33) in the regionỹ 2 > 0. The hypersurface equation (21) reproduces the topology of a S 2 for the algebraic conditionx 2 = −x 3 , which implies that the exceptional two-cycle e (1) λ is now calibrated with respect to Im(Ω 2 ). A union two-cycle bI 0 ×bI 0 ⊕bII 0 ×bII 0 from which the exceptional two-cycle e (1) λ is eliminated, will then correspond to one big sLag cycle calibrated with respect to Re(Ω 2 ).
Once again, the two-cycles bI 0 ×bII 0 and bII 0 ×bI 0 both behave differently with respect to the two-cycles bI 0 ×bI 0 and bII 0 ×bII 0 under the deformation as their sizes shrink for increasing |ε (1) λ |. Combining them with e.g. the one-cycle aII on T 2 (1) allows for the construction of fractional three-cycles calibrated with respect to Re(Ω 3 ), and bulk threecycles parallel to the ΩRZ can thus be decomposed in terms of the basis three-cycles˜ (1) λ : implying that the basis three-cycles˜ (1) λ=0,1,2,3 are calibrated with respect to Re(Ω 3 ). As the volumes of these fractional three-cycles shrink for a non-vanishing deformation according to figure 3 (e), there should be a relative minus sign between Π bulk and the contribution to Π Z (1) 2 . For instance, for ε The situation for the deformations ε (1) 4+5 and ε (1) 4−5 is different, as they deform singularities through which the two-cycles bI 0 ×bIII 0 , bIII 0 ×bI 0 , bIII 0 ×bIII 0 , bII 0 ×bIV 0 , bIV 0 ×bII 0 and bIV 0 ×bIV 0 calibrated w.r.t. Re(Ω 2 ) pass. As such the singular point (33) should not be deformed for (at least) a (small) non-vanishing deformation ε 4−5 , respectively, as depicted in the lower diagrams of figures 3 (f) and (g). A brief discussion on how to obtain these corrections terms is given in appendix B. The (local) discussion of the singularities deformed by a non-vanishing parameter ε     Exposing the global behaviour of the exceptional three-cycle volumes for each deformation separately requires us to impose the algebraic condition x 2 = ±x 3 on the full hypersurface equation (20) and to extract a real hypersurface equation allowing for a geometric description of an exceptional three-cycle in the hypersurface formalism. Let us work this out explicitly for three-cycles with a bulk orbit parallel to bI 0 ×bI 0 on T 4 (1) . Consistency with the plots in figure 3 indicates that exceptional three-cycles calibrated w.r.t. Re(Ω 3 ) are subject to the constraint y 2 Re(x 1 ), Re(x 2 ), Im(x 2 ), ε (1) λ ≤ 0, which allows to define the integration domain for the volume of the respective exceptional three-cycles. Consider first the deformation ε (1) 0 of the complex co-dimension 2 singularity at the origin (11) on T 4 (1) /Z 6 , which can be placed along the real axes in the (x 2 , x 3 )-planes by virtue of the Möbius transformation λ 3 . Imposing subsequently the algebraic condition x 2 = x 3 yields a real hypersurface equation reminiscent of a Z 2 -singularity on T 4 /Z 2 , implying that the Z 3 -action does not influence the geometrical properties of this exceptional three-cycle. Depicting the volume of the exceptional cycle e 0 -dependence of the fractional three-cycle volume with bulk orbit parallel to aI×bI 0 ×bI 0 , which shrinks to zero as the deformation parameter goes to one. Hence, this plot depicts the global behaviour of the fractional three-cycle Π frac . On the right panel of figure 4, we depict the ε 0 -dependence of the volume of the fractional threecycle Π frac , where the bulk orbit is once more parallel to aI×bI 0 ×bI 0 . For this latter fractional three-cycle we observe that its volume grows for increasing values of the deformation parameter, with the same functional behaviour as the exceptional cycle volume. Closer inspection of the behaviour of the bulk cycle volume under deformation reveals that the correct representant in the homology class of bulk cycles corresponds to the three-cycle aI×bIII 0 ×bIII 0 , which happens to lie furthest away from the deformed singularity (11), and its volume is therefore the least affected by the deformation. One can confirm this explicitly by adding the exceptional cycle volume to (twice) the volume of the fractional cycle Π frac − and comparing the volume-dependence of the resulting bulk cycle to the volume-dependence of the bulk three-cycle aI×bIII 0 ×bIII 0 under deformation.
0 on T 4 (1) /Z 6 (left) and the fractional cycles (1) /Z 6 (middle and right) as a function of the deformation parameter ε initially of three distinct S 2 's resolving each of the three Z 2 singularities, as shown in the left plot of figure 6. In order to extract the volume-dependence of a single S 2 as a function of the deformation parameter, we depict one third of the exceptional cycle volume in the left plot of figure 5, for which we observe a similar qualitative behaviour as for the exceptional cycle e (1) 0 . More precisely, we notice a square-root type functional dependence of Vol(e 3 ) for small deformations, which goes over into a linear behaviour and ends in a quadratic dependence for larger deformations. A quantitive difference with respect to the cycle e 3 ∼ 0.37 and higher, the three two-spheres S 2 merge together into one large exceptional three-cycle as depicted in the right panel of figure 6, at which point we can no longer reliably describe the exceptional cycle through the hypersurface formalism. This is manifested in the horizontal plateau truncating the exceptional cycle volume for values ε 3 with bulk orbit parallel to aI×bI 0 ×bI 0 . The representant in the bulk homology class is, however, not the factorisable three-cycle aI×bI 0 ×bI 0 itself, but a bulk three-cycle aI×C 0 × C 0 consisting of the union of one-cycles C 0 = bII + ∪ bII − along both two-tori T 2 (2) and T 2 (3) . Once again, it suffices to subtract (twice) the exceptional cycle volume from the fractional cycle volume to uncover the dependence of the bulk cycle on the deformation parameter ε (1) 3 and verify that this functional behaviour matches the one of the three-cycle aI×C 0 × C 0 . A pictorial representation of the one-cycle C 0 is offered in figure 7, from which it is immediately clear that the one-cycle does not represent a sLag cycle, since its pull-back of the Kähler two-form on the two-torus does not vanish. Nonetheless, the three-cycle aI×C 0 × C 0 belongs to the same homology class as the bulk three-cycles parallel to aI×bI 0 ×bI 0 , such that its integrated volumes are equal to each other, as argued in more detail in [51].
Discussing the global aspects of the exceptional cycle e (1) /Z 6 follows a slightly different logic, as the geometric condition x 2 = λ 3 (x 3 ) does not define a fixed set under the orientifold involution, i.e. the resulting hypersurface equation is not real and therefore does not offer the desired direct access to the exceptional sLag. The intuition following from the study of the exceptional cycles e on T 4 (1) /Z 6 and the fractional cycles Π frac 3 , with the representant of the bulk cycle Π bulk lying along the cycle C 0 × C 0 on T 4 (1) /Z 6 defined in the main text. Note that a factor 1/3 has to be taken into account when computing the exceptional cycle volume for the fractional three-cycle with bulk orbit aI×bI 0 ×bI 0 , as the fractional cycle only wraps one of the three Z 3 . It appears that this type of information can only be extracted explicitly from the hypersurface equation for the exceptional cycle e (1) 1 , whose form is constrained by the topology of the ambient T 4 . When restricting to the real part of the hypersurface equation upon imposing the condition x 2 = λ 3 (x 3 ), one can qualitatively see three distinct exceptional cycles growing out of the Z    (1) /Z 6 , which are related to each other through the orientifold projection. In the hypersurface equation (20) this relation under the ΩR-projection is manifestly built in, such that a nonzero deformation parameter ε 2 -twisted sector and propose the following method to compute the volume for e.g. the fractional three-cycle aI×bI + ×bIII − : • Compute the (normalised) volume of the bulk cycle aI×C − × C + , composed of the union one-cycles C − =bII 0 ∪bII − and C + =bII 0 ∪bII + as drawn in figure 9, as a function of the deformation parameter ε 4+5 ; • Consider the volume of a single two-sphere S 2 obtained by deforming the exceptional cycle e 3 , as presented in the left panel of figure 5, and re-interpret 12 this volume as the volume of the resolved exceptional two-cycle in the Z 3 -and ΩR-invariant exceptional three-cycle • Subtract or add the resulting exceptional cycle volume from the bulk cycle volume to obtain the volumes of the fractional cycles Π frac − and Π frac + respectively: The proposed method does not allow us to obtain any quantitative information about the fractional cycle volume for a given deformation ε 4+5 = 0, but it does enable us to envision the qualitative behaviour of the volumes of the fractional cycles Π frac ± parallel to e.g. the three-cycle aI×bI + ×bIII − as presented in figure 10. There, we see that the volumes of the fractional three-cycles Π frac   3 : for small deformations, the exceptional cycle volume exhibits a square-root type functional dependence, while the topology of the ambient T 4 (1) enforces a quadratic behaviour for larger deformations. The two-spheres S 2 at the resolved singularities in the orbit merge together into one large exceptional cycle for a sufficiently large deformation. This common behaviour is inherited from the isotropy between the orbits e  One can easily repeat this method for fractional three-cycles wrapping any of the other Z 5 , or apply this method to probe the effect of a nonvanishing deformation ε (1) 4−5 on the volume of such fractional three-cycles, provided one chooses the appropriate bulk three-cycle. In all of these cases, the qualitative functional behaviour of the fractional three-cycles can be brought back to the case presented in figure 10, namely

-twisted sector
To investigate the deformation effects in the Z 2 -twisted sector, we turn to the v i = 1 patch so that we can describe the Lag lines in terms of the homogeneous coordinates x i as in section 2.2. Real hypersurfaces at the orbifold point are represented in this coordinate by the Lag lines aI, aII, aIII and aIV on the two-torus T 2 (1) and by bI 0 and bII 0 on T 2 (2) . Combining these Lag lines, we can construct a set of sLag two-cycles with topology T 2 /Z 2 on Def T 4 (3) /Z and calibrated with respect to Re(Ω 2 ), represented by the blue-coloured regions in figure (11) (a), i.e. the two-cycles aI×bI 0 , aIV×bII 0 , aIII×bI 0 and aII×bII 0 . The white regions correspond to sLag two-cycles calibrated by Im(Ω 2 ): aI×bII 0 , aIV×bI 0 , aIII×bII 0 and aII×bI 0 . The blue contour-lines in the R-projected (x 1 , x 2 ) plane represent the zero locus y = 0 and intersect at the real Z 2 fixed points (23), (33) and (43). In order to obtain fractional three-cycles calibrated w.r.t. Re(Ω 3 ) on (T 4 /Z (3) )/Z 6 , the two-cycles calibrated w.r.t. Re(Ω 2 ) and Im(Ω 2 ) on T 4 (3) /Z 2 should be paired with a one-cycle bI 0 /bIII 0 and bII 0 /bIV 0 , respectively, on T 2 (3) . α one by one, figure 11 shows exactly which singularities (or singular orbits under the Z 3 ⊂ Z 6 symmetry) are deformed and which singularities are displaced, in agreement with the lower part of table 9. Statements about singularities (1α) cannot be made in this coordinate patch v i = 1 as they are located here at x 1 = ∞, and hence they require us instead to describe the hypersurface equation in terms of the homogeneous coordinates v i in the coordinate patch x i = 1. At the deformed singularities, an exceptional two-cycle with non-vanishing volume appears, as indicated by the red dashed lines in figure 11. We can study the effects of the deformation parameters more qualitatively by studying the zero locus of the hypersurface equation (20) in a local patch around the singular point (23), for which the hypersurface equation reduces locally to the form (after For the deformations ε 3 and ε 4 we have to perform an appropriate Möbius transformation from appendix A to mould the hypersurface equation (20) into this specific form, corresponding locally to a C 2 /Z 2 -type singularity. The two-cycles passing through the singularity (23) are given by aI×bI 0 and aIV×bII 0 , associated to the torus wrapping numbers (n 1 , m 1 ; n 2 , m 2 ) = (1, 0; 1, 0) and (0, 1; 1, −2) on T 2 (1) × T 2 (2) , respectively. Combining for instance the first two-cycle aI×bI 0 with a one-cycle bI 0 on T 2 (3) yields a fractional three-cycle as defined in section 2.1, calibrated w.r.t. Re(Ω 3 ). Its overall Z 2 exceptional three-cycle is given in terms of the basis exceptional three-cycles as: By turning on the deformation parameter ε 2 , an exceptional two-cycle e 2 with non-vanishing volume grows out of the singular point (23) on T 4 (3) /Z 2 , and depending on the sign of the deformation the volumes of the two-cycles aI×bI 0 and aIV×bII 0 will shrink or grow: 2 > 0: aI×bI 0 and aIV×bII 0 still form two separate two-cycles with reduced size as an exceptional two-cycle emerges out of the singular point (23) in the regionỹ 2 < 0, as depicted in the upper diagram of figure 11 (b). The local hypersurface equation (25) reduces to the equation of a two sphere S 2 with radius 2ε (3) 2 , when we impose the algebraic conditionx 1 =x 2 for the exceptional cycle. The exceptional two-cycle e 2 is calibrated with respect to Re(Ω 2 ), a feature supported by the fact that the cycles aI×bI 0 and aIV×bII 0 remain sLag for positive deformations. Translating these considerations to the fractional three-cycle with exceptional part displayed in equation (26), we find that the exceptional three-cycle (3) 2 is calibrated with respect to Re(Ω 3 ) and that there should be a relative minus sign between the bulk three-cycle Π bulk and its contribution to Π Z (3) 2 , i.e. (−) τ Z (3) 2 +τ 2 +τ 3 = −1, in order for the volume of the fractional three-cycle to decrease for positive deformations.
The cycles aI×bII 0 and aIV×bI 0 on the other hand are no longer sLag on their own and melt together to one big two-cycle, still calibrated with respect to Im(Ω 2 ). This bigger cycle is described by the union two-cycle aI×bII 0 ⊕aIV×bI 0 from which the exceptional cycle e 2 has been eliminated.
• ε 2 < 0: we observe the opposite picture for negative deformations, namely the twocycles aI×bI 0 and aIV×bII 0 have merged together to one big two-cycle as shown in the lower diagram of figure 11 (b). This can be traced back to the fact that the exceptional two-cycle is now calibrated with respect to Im(Ω 2 ). By taking the union two-cycle aI×bI 0 ⊕aIV×bII 0 , we can ensure that the exceptional two-cycle drops out, such that the union two-cycle remains a sLag two-cycle.
For negative deformation parameters, the sizes of the two-cycles aI×bII 0 and aIV×bI 0 shrink, in line with the consideration that the exceptional two-cycle e 2 is now calibrated by the same two-form as both two-cycles. The S 2 topology of the exceptional cycle with radius −2ε (3) 2 follows from equation (25) by restricting to the slicex 1 = −x 2 . Combining the two-cycles with the one-cycle bII 0 on T 2 (3) allows for the construction of fractional three-cycles, whose volumes are now decreasing for increasing |ε (3) 2 |. This implies a relative minus sign between Π bulk and the contribution of for the fractional three-cycle.
If we want to turn to a global description of the fractional three-cycles located at the deformed singularities, we have to consider the full hypersurface equation (20) and impose the algebraic conditions x 1 = ±x 2 (beyond a neighbourhood of the original singular point and possibly after acting with an appropriate Möbius transformation on the coordinates), allowing us to determine the fixed loci of the orientifold projection. Note, however, that the resulting equation does not reduce to a real hypersurface equation, not even for vanishing deformations. The inability to describe Z 2 deformations globally is an immediate consequence of the different complex structures on T 2 (1) and T 2 (2) , which prevent the conditions x 1 = ±x 2 to represent the fixed loci of the orientifold projection globally. Nevertheless, we are able to extract information about the functional dependence of the exceptional cycle volume on e.g. the deformation parameter ε is shown in the middle and right panel of figure 12. As expected, the three-cycle Π frac − is characterised by a shrinking volume for increasing ε 2 , while the volume of the three-cycle Π frac + increases for growing ε 2 . Given that the other (Z 3 -orbits of) Z 2 exceptional divisors are related to e 2 .
Due to the exchange symmetry T 2 (2) ↔ T 2 (3) , we can straightforwardly transpose the entire analysis into the Z 2 sector, where the same conclusions can be drawn for the Z 2 exceptional and fractional three-cycle volumes. One subtle difference arises between the Z 2 -plane are taken to be the exotic O6-plane, their respective RR-charges are opposite, i.e. η (2) = −η (3) , resulting in a different decomposition into ΩR-even and -odd cycles for both sectors, according to table 3, as will be discussed in the following section in terms of prototypical global D6-brane models.

Deformation Moduli in Global D6-Brane Models
In this section, we apply the findings of geometric deformations in section 2 to global D6-brane models of phenomenological interest and discuss which moduli are stabilised at the orbifold point or constitute flat directions of the global model affecting physical gauge couplings, either by a direct tree-level dependence or only via higher order and/or non-perturbative effects.

Some generic considerations
In section 2.1, the different types of three-cycles on the T 6 /(Z 2 ×Z 6 ×ΩR) orientifold with discrete torsion were briefly reviewed. While a D6-brane a by itself wraps a fractional three-cycle Π a of the form (3), it is generically accompanied by its orientifold image a with associated three-cycle Π a . The global model remains inert under the exchange Π a ↔ Π a when simultaneously changing the gauge representations for their conjugates, e.g. N a ↔ N a . The scalar potential only depends on the sum of the two [12], which leads to the following qualitative situations observed first in the context of T 6 /(Z 2 × Z 2 × ΩR) and T 6 /(Z 2 × Z 6 × ΩR) models with discrete torsion in [49,51], see also [82,52,54]: 1. The D6-brane a couples to the Z where the Calabi-Yau three-form is defined in equation (13) and the three-cycleδ In the first case, it is argued that -from a low-energy field theory point of view -the stack of N a D6 a -branes supports a U (N a ) gauge group, and the U (1) a factor within accounts for a D-term potential with Fayet-Iliopoulos term D a ∝ ε (i) α , whose numerical prefactor is fixed by the associated orientifold-odd combination of the exceptional wrapping numbers (x (i) α,a , y (i) α,a ), cf. table 8 for T 6 /(Z 2 × Z 6 × ΩR) with discrete torsion. 13 The appearance of the Fayet-Iliopoulos D-term leads to the stabilisation of the deformation modulus at the singular orbifold point, i.e. ζ (i) α = 0. 14 In the second case, the D6 a -brane stack has only an orientifold-even exceptional wrapping number, whereas in the last case, both orientifold-even and -odd wrapping numbers of the associated exceptional cycle vanish. The scalar potential (27) with the toroidal three-cycle volume at the orbifold point of the aAA lattice of T 6 /(Z 2 × Z 6 × ΩR) 13 As discussed in the previous section fractional three-cycles loose their sLag property under deformation, when one of its resolved exceptional three-cycles is no longer calibrated with respect to the same volumeform as the bulk three-cycle (and thereby also the orientifold fixed planes). According to equation (27) this results in a positive contribution to the total tension of the D6-branes and O6-planes, upon pursuing the dimensional reduction of the corresponding DBI-actions. Extracting the functional dependence of the FIparameter on the deformation parameter then follows by computing the volume of the D6-brane on the resolved background and subtracting the tension of the O6-planes. In case the O6-planes are calibrated w.r.t. Re(Ω3), the positive contribution to the NS-NS scalar potential will scale as Πa Im(Ω3) 2 for small deformations, which is understood from a four-dimensional perspective as (the square of) a FI-parameter [12,20,24]. 14 There exists in principle the possibility that the vevs of charged scalars belonging to a vector-like pair in the bifundamental representation compensate the Fayet-Iliopoulos term ζ given by: are the radii associated to the one-cycles π i=1,2 of the a-type torus T 2 (1) and r j the length scales of the hexagonal A-type tori T 2 (j),j∈{2,3} appearing in the definition of the two-torus volumes in equation (40). In the second case, the flat direction affects the strength of the gauge coupling, whereas in the third case the gauge coupling only feels the flat direction via (higher-order and nonperturbative) backreactions of the deformation on the toroidal cycles. In sections 3.3, 3.4 and 3.5, we will present how the three different cases appear in several globally consistent protoptype MSSM, L-R symmetric and PS-models, respectively.
For the T 6 /(Z 2 × Z 6 × ΩR) orientifold with discrete torsion, aAA lattice background and the choice of ΩRZ where (P a , Q a , U a , V a ) correspond to the four different products of toroidal wrapping numbers appearing in equation (2) and (x α,a ) to the exceptional wrapping numbers in equation (3). The latter have also been associated to the deformation parameters ε At the singular orbifold point, only relative Z 2 × Z 2 eigenvalues among pairs of D6-brane stacks a and b are of physical importance; e.g. for the counting of chiral fermions the inter- need to be computed. For illustrative purposes, we choose here the D6 a -brane stack supporting the strong interaction SU (3) a (or SU (4) a for PS models) to have (Z 2 , Z 2 )   -supersymmetric sLags at the orbifold point of T 6 /(Z 2 × Z 6 × ΩR) with discrete torsion. The discrete parameter choice in the D6-brane configurations is not unique, as different choices for the discrete Wilson lines, displacements, or Z (i) 2 eigenvalues may also yield global D6-brane models satisfying the same stringent conditions as the prototype D6-brane configurations. Nonetheless, such a global D6-brane model (with different discrete parameters) will always be characterised by the same massless (open and closed) string spectrum as one of the prototype D6-brane models. In other words, all global intersecting D6-brane models with three chiral generations of quarks and leptons and a minimal amount of exotic matter constructed with -independent supersymmetric sLags at the orbifold point will be classifiable by the massless string spectra of one of the prototype models considered here.
At this point, we anticipate that the bulk cycles of all D6-branes in the prototype models are either parallel to the ΩR-or the ΩRZ 4,z + y 1,z +y 3,z + y 4,z + y

4,z ]
Orientifold-odd wrapping numbers 1,z +y 3,z +y    Table 11: Orientifold-even and -odd wrapping numbers of the phenomenologically appealing global MSSM, L-R symmetric and PS models on T 6 /(Z 2 × Z 6 × ΩR) with discrete torsion discussed in sections 3.3, 3.4 and 3.5, respectively. A non-vanishing orientifold-odd wrapping number stabilises the associated deformation modulus according to table 8. If some orientifold-odd wrapping number vanishes, but the orientifold-even number with identical fixed point labels is non-zero, the associated tree-level gauge coupling in equation (29)  Details of deformations of the global MSSM, L-R symmetric and PS models will be discussed in sections 3.3, 3.4 and 3.5, respectively.

Deformations in a global U Sp(2) 4 model
Before discussing phenomenologically interesting particle physics models on D6-branes with all their intricacies of stabilised moduli, here we first consider deformations in a global model with U Sp(2) 4 gauge group, which a priori is expected to contain only flat supersymmetric directions. To obtain gauge group enhancement on all D6-brane stacks and consistency with the bulk RR tadpole cancellation conditions [23,2] for the choice of the ΩRZ 2 -orbit as exotic O6-plane, the bulk part of each fractional three-cycle has to be either parallel to the ΩR-or the ΩRZ (1) 2 -invariant orbit. In either case, only gauge group enhancement of the type U (N ) → U Sp(2N ) occurs for arbitrary choices of Z 2 × Z 2 eigenvalues and arbitrary discrete Wilson line and displacement parameters (τ 1 , σ 1 ) on the a-type torus T 2 (1) , see table 10 of [2] for details. With the discrete Wilson lines and displacements ( τ 2 τ 3 ; σ 2 σ 3 ) along T 4 (1) listed as lower index, such three-cycles have the form: , are obtained from those by choosing identical values of ( τ ; σ) along (T 2 ) 3 and replacing the Z 2 × Z 2 eigenvalues ( τ Z 2 ) as follows: The massless matter spectrum of this D6-brane configuration is displayed in table 15 and agrees in the bb, bc and cc sectors by construction with that of the L-R symmetric models and the PS II model displayed in table 19.
Here, it is noteworthy that at the orbifold point only the relative Z 2 ×Z 2 eigenvalues among the D6-branes a, b, c and d are of physical relevance. The absolute Z 2 × Z 2 eigenvalues in all three Z (i) 2 -twisted sectors will, however, become important when switching on deformations since the gauge couplings scale with the cycle volume, 1 /g 2 z,tree ∝ Vol(Π z ) for Π z = Π z , and the change in Vol(Π z ) is e.g. proportional to y    (2)   2 eigenvalues), the global models differ significantly: All phenomenogically appealing models contain at least two U (N 1 ) × U (N 2 ) gauge factors with N 1 = N 2 . The QCD stack with either U (3) a or U (4) a gauge symmetry e.g. couples by an orientifold-odd exceptional wrapping number to the two Z 1,2 , as can be read off from table 11. In all phenomenologically appealing prototypes presented in this article, the Fayet-Iliopoulos term(s) generated by some non-vanishing deformation parameter(s) ε can thus not be compensated by any supersymmetry preserving vev of some charged scalar matter field. The same argument involving the QCD stack a applies to two deformations, ε singularities are wrapped by the fractional three-cycles in table 13, cf. equation (35). Then, we reconstruct the (normalised) volume of the fractional three-cycles using the results for the exceptional three-cycle volumes from sections 2.4.2 and 2.4.3 as primary building blocks, in the same spirit as the method presented on page 35. More precisely, we compute the fractional three-cycle volumes directly as the sum or difference of (normalised) bulk three-cycle volumes and (normalised) exceptional three-cycle volumes as computed in the aforementioned sections. (1) /Z 6 as argued in section 2.4.2. These plots serve as templates for describing the fractional three-cycle volumes as a function of some deformation along the flat direction in the twisted complex structure modulus ζ (1) 4−5 in the subsequent subsections. In those cases, the plots will represent the normalised volume of a fractional three-cycle The results of these computations are presented in figures 13 and 14 for branes {a, d} and {b, c}, respectively, for the deformation parameter ε

Deformations in a global MSSM model
In [1], we had performed a systematic search for MSSM-like models on T 6 /(Z 2 ×Z 6 ×ΩR) with discrete torsion and found that there is a unique choice of bulk parts supporting the Standard after Green-Schwarz mechanism, which leads to the orientifold-even and -odd wrapping numbers labelled by MSSM in 1,2 , while the D6-brane c couples in an orientifold-odd way to ζ (1) 3,4+5 and ζ (2) 1,2 and the D6-brane Overview of the massless matter spectrum of a global 5-stack MSSM on the aAA lattice of T 6 /(Z 2 × Z 6 × ΩR)   (29) and (30), while the untwisted Kähler moduli v i,i∈{1,2,3} defined in equation (40) influence the gauge couplings once the one-loop gauge threshold corrections are taken into account, as will be discussed in section 4.
Before turning to the technicalities of executing the different deformations, let us briefly discuss possible caveats in the counting of the maximal number of stabilised deformation moduli when allowing for vevs of charged matter fields. While the massless matter spectrum in table 17 provides all charged (open string) scalars that might possibly trigger some D6brane recombination process while compensating the Fayet-Iliopoulos term generated by the vev of some (closed string) deformation modulus, a more detailed analysis of the origin of each matter state per D6-brane intersection sector x(ω k y ( ) ) in table 29 is required to deduce allowed terms in the low-energy effective action, in particular the necessary selection rule of a closed polygon with n edges along the partaking D6-bane directions per n-point coupling.
Let us start by considering the chiral sector of the spectrum in table 17 only. The only states in bifundamental representations of two unitary gauge factors of equal rank are the right-handed selectrons and sneutrinos 16 in the cd and cd sectors, respectively, charged under U (1) c ×U (1) d . As summarised in table 12, both D6-branes c and d couple to the twisted deformation moduli ζ (2) 1,2 , and table 11 shows that the corresponding orientifold-odd exceptional wrapping numbers are opposite, [2x (2) α,c + y (2) α,c ] α=1,2 = −[2x (2) α,d + y (2) α,d ] α=1,2 . The D-terms can therefore only be compensated by some vev of the right-handed sneutrinos due to their opposite U (1) c and U (1) d charges. In [1], three-point functions involving vevs of this type were shown to be suitable for generating mass terms e.g. of the vector-like down-type quark pairs originating from the ac and ad sectors as well as of the vector-like left-handed lepton (or Higgs) pairs from the bc and bd sectors. At this point, we however, notice two caveats: on the one hand, our discussion so far up to now only includes the relative sign among the U (1) c × U (1) d charges, whereas in a more thorough field theoretical study in remains to be seen if the sneutrino/axion or its hermitian conjugate representation are suitable for canceling a Fayet-Iliopoulos term; on the other hand, the naive geometric intuition of Π c + Π d = as a merging of cycles in the Z 2 -twisted sectors were to be at most stabilised by the existence of stacks c and d, respectively. However, the state-per-sector list in table 29 shows that usually a chiral multiplet in one sector x(ω k y) is paired with an anti-chiral multiplet in some x(ω l =k y) sector. Even the vector-like pair of states in the antisymmetric representation in the aa (or hh ) sector with the orientifold image D6-branes parallel along all three two-tori T 2 (i) does not constitute a genuine N = 2 supersymmetric sector due to the different relative Z 2 × Z 2 eigenvalues entering the orbifold projection on the a priori three chiral multiplets containing e.g. the massless scalar ψ i or i − 1 /2 |0 NS . A much more detailed and dedicated computation of n-point couplings appearing in the low-energy effective action is thus needed to completely settle this issue. But this goes way beyond the project discussed here, since not even 3-point Yukawa couplings have been computed for the cases at hand of a vanishing intersection angle along T 2 (1) . The vanishing arguments of [87][88][89][90][91] rely on N = 2 supersymmetry on the factorisable six-torus T 6 = (T 2 ) 3 or its Z 2 ×Z 2 orbifold without discrete torsion and are clearly not applicable for the models with discrete torsion discussed in this article.
The only element left to discuss at this point is the Z

Deformations in global Left-Right symmetric models
In this section, we study how deformations affect each of the prototype L-R symmetric models classified in [1]. The common local D6-brane configuration of the observable sector is displayed in the upper part of table 18, with the four different types of global completion by two 'hidden' sector D6-branes provided in the lower part of the same table. The common observable part of the matter spectrum is displayed in table 19, and the individual 'hidden' spectra are given in table 20.
The QCD stack a agrees by construction with the one of the MSSM example discussed in section 3.3, while the left-and right-symmetric groups U Sp(2) b × U Sp(2) c have identical bulk cycles and discrete Wilson line and displacement parameters, but differ in their Z 2 × Z 2 eigenvalues from the D6-brane b of the MSSM example. The D6-brane d only differs from the QCD stack a in the Z 2 × Z 2 eigenvalues and stack size N d = 1 vs. N a = 3.
The orientifold-even and -odd wrapping numbers are summarised table 11 and lead to the naive counting of the maximal number of stabilised deformation moduli in table 12. We observe the following differences: • L-R IIc has the maximal number of ten stabilised deformation moduli due to the different choices of displacements (σ 1 , σ 2 ) = (0, 0) h 1 , (1, 1) h 2 of the two 'hidden' D6-branes.

wrapping numbers
Angle π Universal observable sector a (1, 0; 1, 0; 1, 0) (0, 0, 0) (+ + +) (0, 1, 1) (0, 1, 1) Global completion of prototype I Global completion of prototype II Global completion of prototype IIb Global completion of prototype IIc     • L-R II, IIb, IIc contain Abelian 'hidden' D6-branes and thus three gauge groups of equal rank, In the prototypes II and IIb, some vev of the type ζ The counting of matter states per intersection sector is displayed in tables 31 and 32 for the universal observable and individual 'hidden' sectors, respectively, and -just as for the MSSM example -a dedicated derivation of the low-energy effective action is needed to determine if vevs of matter states can indeed allow for potentially flat directions in the deformation moduli space as stated above.
Looking closer at the Z ] /2, cf. also the microscopic origin of the vector-like matter states from the h1(ω 0 h2) sector according to table 32.
Overview of the 'hidden' spectra for the L-R symmetric models on T 6 /(Z 2 × Z 6 × ΩR) bulk and resolved exceptional three-cycles then constrains the deformation parameter to lie on the half-line ε The deformation parameters ε 2 sector of the) fractional three-cycle for the strong D6- 18 For the prototype IIc model, the modulus ζ (1) 4−5 also couples to the 'hidden' D6-brane stacks h1 and h2, such that a deformation along its flat direction also affects their respective U (1) gauge couplings at the string scale Mstring, before these Abelian gauge groups are spontaneously broken at the KK-scale by virtue of the Stückelberg mechanism. That is to say, the (Z  2 -twisted sector of the) fractional cycle supporting the U Sp(2) b/c gauge group, such that the gauge coupling for U (1) h 1 decreases and the one for U (1) h 2 increases for non-zero deformation ε (1) 4−5 . Realising a strongly coupled anomalous U (1) gauge theory at the string scale through these geometric deformations of Z2 singularities opens up avenues for D6-brane model building scenarios [92] realising Nambu-Jona-Lasinio type models upon integrating out the massive U (1). brane stack a is identical to the geometry of the three-cycle (in the Z (2) 2 sector) supporting the U Sp(2) a gauge group in the toy model of section 3.2. This implies that the QCD gauge group acquires a stronger gauge coupling when either the twisted modulus ζ According to table 12, the hidden D6-brane stacks h 1 and h 2 in the prototype II and IIb models couple to deformations with flat directions too, which requires us to investigate their fractional cycle volume under deformation. In both prototype models the (relevant part of) fractional three-cycle for h 1 can be recast into the fractional three-cycle supporting the U Sp(2) b gauge group in the global toy model of section 3.2. This implies that the fractional cycle volume for hidden stack h 1 exhibits the same behaviour as depicted in the middle and right panel of figure 15 and that the U (1) h 1 gauge coupling becomes more weakly coupled for non-zero deformations ζ (2) 1,2 = 0 in case of prototype II models and ζ (2) 3,4 = 0 in case of prototype IIb models. The (relevant part of the) fractional three-cycle for h 2 has not yet been encountered before in this article, but using the same techniques as in section 3.2 we can compute its fractional three-cycle volume as a function of the Z 1,2 for the prototype II models and along the flat directions ζ (2) 3,4 for the prototype IIb models.

Deformations in global Pati-Salam models
In [2], a systematic computer scan led to two prototype PS models. In order to streamline the discussion of deformations and of one-loop corrections to the gauge couplings, we here present the prototype I with (up to the size N a ) identical stacks {a, b} as in the MSSM example of section 3.3. The D6-brane configuration is displayed in table 21 and differs from the original one in [2] by a swap of the last two two-tori, T 2 (2) ↔ T 2 (3) , and by a flip of the absolute Z 2 × Z 2 eigenvalues. The spectrum of prototype I is displayed in table 22.  Matter spectrum of the prototype I global Pati-Salam model on aAA 1,2 , with also the gauge couplings of a and h sensitive to ζ 1,2 .
Looking more closely at table 12, we observe that the deformation moduli ζ   1, 2; 2) + h.c. in the bifundamental representation under U Sp(2) c × U (2) h . In case that the necessary terms indeed appear in the low-energy effective action, the gauge group will be spontaneously broken from U Sp(2) x∈{b,c} × U (2) to a diagonal SU (2) diag , corresponding to a recombination of two D-brane stacks, which, however, does not bear an interpretation as a new fractional cycle only wrapping Z (i) 2 singularities in a single sector i, cf. the overview over the couplings to Z (i) 2 -twisted sectors in table 12. Whether or not such a recombination can happen, has to be studied in more detail from a geometric perspective in the future and cannot be solely assessed through the study of the D-term equations. Furthermore, a more detailed study of the full scalar potential is required as well for these prototype models, but goes well beyond the scope of this article as well. Yet, we offer the listing of matter states per intersection sector in table 30 for the prototype I example and in table 31  There is a substantial qualitative difference between the prototype I and prototype II models: for the prototype I models the fractional three-cycle volume for the b-brane is given by the right panel of figure 14 and the one for the c-brane is depicted on the left-hand side of figure 14, whereas the prototype II models have the reverse identifications with respect to figure 14. This implies that the left-symmetric U Sp(2) b gauge group in the prototype I models acquires a stronger gauge coupling when we go away from the singular orbifold point along the flat direction ζ

Gauge Couplings at One-Loop, Geometric Moduli and M string
Up to this point, changes in the volumes of D6-branes have been discussed, which are related to the tree-level gauge couplings according to equation (29). At the orbifold point, the volumes stem solely from the toroidal cycles as detailed in equation (30). For the MSSM example of section 3.3, the relation (30) amounts to the following relation among the tree-level gauge couplings at the orbifold point: which obviously disagrees with the proposal of gauge coupling unification at M string M GU T ∼ 10 16 GeV [84].
As discussed in section 3, deformations of exceptional cycles along directions of only orientifoldeven wrapping numbers change the volumes of fractional cycles such that the degeneracy of e.g. the gauge coupling strength of U Sp(2) b and U Sp(2) c in the L-R symmetric and PS models is lifted, and also the relations in (36) might be ablished.
On the other hand, the degeneracy of gauge couplings is already at the orbifold point lifted when including one-loop corrections ∆ Gx to the gauge couplings of any gauge factor G x , which depend on the discrete D6-brane data such as Z 2 × Z 2 eigenvalues and discrete Wilson lines and displacement parameters. In section 4.1, we will therefore present the one-loop gauge thresholds ∆ Gx and discuss in section 4.2 impacts on the low-energy phenomenology of the global particle physics vacua with D6-brane data specified in tables 16, 18, 21 and 23. In particular, the class of models at hand shares the unusual feature that all D6-branes wrap the ΩR(Z 2 )-invariant one-cycle along T 2 (1) such that not only one-loop gauge thresholds can -at least in principle -contribute to lowering the string scale M string well beyond the Planck scale M Planck [25], but the weakness of gravity can already be generated at tree level by a large hierarchy between the two radii R

Gauge couplings at one-loop
Let us briefly summarise the results of the formalities of one-loop gauge threshold corrections for the case at hand of fractional D6-branes parallel along at least one two-torus [68][69][70][71] before applying these general results to the stringy particle physics vacua on the T 6 /(Z 2 × Z 6 × ΩR) background with exotic O6-plane orbit ΩRZ 2 . The Kähler metrics and contributions to the SU (N x ) beta function coefficients,    (40)  For details on the computation of Z 2 -invariant intersection numbers, the interested reader is referred to e.g. [25,2]. The factor (−1) 2bi σ i x τ i x appearing in the Möbius strip contribution to the beta function coefficient is required for consistency of spectra on tilted tori (b i = 1 2 ) as first noted in the caption of table 49 in [23], but the exact shape of the corresponding Möbius strip amplitude with b i σ i x τ i x = 0 -needed for any known phenomenologically appealing model with rigid D6-branes -is to the present day not known, see appendix B.1 of [25] for an extended discussion.
with three non-vanishing angles and/or non-rigid D6-branes, the interested reader is referred to [69][70][71]25]. The associated one-loop gauge threshold corrections per open string sector with at least one vanishing angle are collected in table 26, where the following abbreviations of lattice sums for the annulus topology are used, with the two-torus volume v defined by and the one-cycle length L as given in equation (30).
One-loop corrections to gauge couplings on T 6 /(Z 2 × Z 2M × ΩR) with discrete torsion for φ (1) To complete the picture, the gauge thresholds for rigid D6-branes at three non-vanishing angles can be found in [68,70], where also the conversion from ∆ Gx to the holomorphic gauge kinetic function f Gx using the Kähler metrics from  For later use in section 4.2, we already provide the asymptotic behaviour of the lattice sums here for two-torus volumina larger than α : which turns out to be already an excellent approximation for v 1, cf. figure 2 of [25]. Let us discuss for example the MSSM model with D6-brane data specified in table 16. The massless matter spectrum per sector is provided in table 29, from which the beta function coefficients b A,(i) -or whenever vanishing the reduced numbersb A,(i) -can be read off. Combining this information with the discrete D6-brane data in table 16, we can derive the full annulus contributions to the gauge thresholds, which are given by (with R 1 ≡ R (1) 1 ): 2 ) 0,0 2 ) 0,0 2 ) 0,0 where we have made use of the same notation∆ xy =∆ yx as in e.g. [70]. We will come back to the impact of these one-loop correction in section 4.2 after having also determined the Möbius strip contributions, but already point out here that all threshold contributions have a positive dependence on the two-torus volume v 1 , while v 2 and v 3 appear with negative prefactors in the one-loop correction to U Sp(2) b and SU (4) h .
The annulus contributions to the gauge thresholds of the L-R symmetric models are analogously computed using the D6-brane data in table 18 and the resulting state-per-sector counting in tables 31 and 32 with the following results: 2 ) 0,0 2 ) 0,0 2 ) 0,0 2 ) 0,0 2 ) 0,0 2 ) 0,0 (43) The asymptotic behaviour is again easily extracted, producing again only positive contributions from the two-torus volume v 1 .
For the prototype I L-R symmetric model, the non-Abelian hidden gauge groups experience the following one-loop gauge threshold correction, and finally the generalised massless U (1) B−L gauge group of the prototype II L-R symmetric model receives the following one loop correction: 2 ) 0,0 Finally, for the PS models, the analogous computations using the D6-brane data in tables 21 and 23 and the state-per-sector counting in tables 30 and 31 leads to: 2 ) 0,0 2 ) 0,0 2 ) 0,0 2 ) 0,0

Balancing M string with gravitational and gauge couplings in four dimensions
Dimensional reduction of the ten-dimensional Einstein-Hilbert term leads to the following relation among the four-dimensional Planck scale M Planck , the string scale M string , the string coupling g string and the Kähler moduli v i inherited from the underlying torus T 6 = (T 2 (i) ) 3 , which in conventional D6-brane models on tori or toroidal orbifolds with phenomenologically acceptable sizes of gauge couplings at tree-level is only consistent with a high string scale M string M GUT [83,84], since usually the O6-plane tensions are cancelled by D6-branes at non-trivial angles on all three two-tori. However, the situation in all global D6-brane models discussed in this article is special since the existence of the exotic O6-plane orbit ΩRZ 2 on T 6 /(Z 2 × Z 6 × ΩR) with discrete torsion enforces all D6-branes to lie along the ΩR(Z 2 )invariant direction on the first two-torus T 2 (1) with length scale R 1 , while gravitational forces also propagate along the perpendicular direction and are thus sensitive to v 1 = The tree-level gauge coupling in equation (29) of SU (3) a and SU (4) h of the MSSM example can e.g. be rewritten using the relation (51), which means that a large hierarchy between M Planck and M string can -at least in principle -be compensated by a large hierarchy between R 2 and R 1 to arrive at some phenomenologically appealing order of magnitude of the gauge couplings as already briefly sketched in [54]. Such large hierarchies in the D6-brane models at hand would provide explicit examples for the low-string scale scenario, see e.g. [93][94][95][96][97][98][99][100][101].
We will now also take into account the one-loop gauge threshold corrections, which for the MSSM example of section 3.3 have the following asymptotic behaviour, where we have rewritten the logarithmic terms in terms of the complex structure parameter R Let us stress here again that in all MSSM, L-R symmetric and PS models of section 3, the gauge coupling of the QCD stack is weakened by the positive (asymptotically) linear one-loop contribution of the volume v 1 , at least for v 1 2. Due to the lack of knowledge of the exact shape of the Möbius strip contribution for D6-branes with bulk part along some orientifoldinvariant direction and non-vanishing Wilson line and displacement, (τ, σ) = (1, 1), along some tilted two-torus T 2 (2 or 3) , we have to distinguish two cases, where our (asymptotic) field theory results in equation (53) are classified as reliable: 1. Isotropic volumes of the last two tori, i.e. v 2 = v 3 : in this case, the unknown Möbius strip contributions within the gauge thresholds of SU (3) a and SU (4) h of the MSSM example cancel.

2.
A much larger first two-torus, i.e. v 1 v 2 , v 3 : the one-loop gauge threshold corrections are expected to be dominated by the asymptotics linear in v 1 , and all gauge couplings will be weakened due to the positive prefactor of v 1 for every single gauge group in each MSSM, L-R symmetric and PS model of section 3.
Let us discuss each of these two cases further: in the first case of two isotropic tori T 2 (2) × T 2 (3) , for generic volumes v 1 > 2 the (asymptotic) linear dependence on v 1 surpasses the negative constant contributions to all four one-loop gauge thresholds in (53), and at least for v 2,3 > 2.5 the QCD stack will have a weaker coupling at one-loop due to the (asymptotic) linear dependence on v 2 , while the coupling of the hidden stack becomes stronger. This behaviour facilitates the formation of a gaugino condensate on the hidden stack, which will in turn lead to supersymmetry breaking mediated to the observable sector on the one hand via gravitational couplings and on the other hand through the messenger particles with U Sp(2) b × U (1) Y × SU (4) h charges in the last block of table 17. Going to the edge of the validity of the geometric regime, v 1 1 and 6 v 2,3 1, one can read off from equation (53) that the one-loop corrections to the inverse of (the squared of) the SU (3) a × SU (4) h gauge couplings will be negative, while for U Sp(2) b × U (1) Y the contributions of v 1 cancel the negative constant contributions. To achieve v 1 1, we further assume R In the second case of one two-torus significantly larger than the other two, v 1 v 2 , v 3 , we can e.g. make the ansatz of M string ∼ 10 12 GeV and g string ∼ 0.1 in equation (51) leading to v 1 v 2 v 3 ∼ 10 11 . In order to achieve α −1 QCD ∼ O(1), equation (52) then requires which is problematic if we require v i∈{1,2,3} 1 and R (1)

1,2/
√ α 1 to be in the geometric regime where the supergravity approximation is expected to be reliable. Even when we choose a larger string coupling of g string ∼ 0.3, v 1 ∼ 10 12 and v 2 = v 3 1, all one-loop corrections in equation (53) -as well as the analogous expressions for all L-R symmetric and PS modelsreceive exponentially large positive contributions from v 1 , thereby exponentially suppressing all gauge couplings.
Let us now assume a high string scale M string ∼ M GUT in equation (51) and discuss the possibility of gauge coupling unification. The tree-level relation (36) together with the fact that in the MSSM example only U Sp(2) b and U (1) d ⊂ U (1) Y possess one flat direction along the deformation modulus ε (1) 4−5 , boils down to the requirement that the combination of a deformation along this direction with the one-loop gauge threshold correction reduces the inverse of the gauge coupling squared by 50% for U Sp(2) b SU (2) weak and by roughly a factor of three for U (1) Y . Figure 14 shows that the volume Vol(Π b ) ∝ 1 /g 2 U Sp(2) b decreases by at most 20% when reaching the upper bound |ε 1,1 , at least the prefactor of v 2 is negative. In both cases, either v 2 or v 3 has to be the largest Kähler modulus, and only the missing computation of the Möbius strip contribution for tilted tori and non-vanishing Wilson line and displacement parameters, i.e. 2b i τ i σ i = 1, can settle the question if gauge coupling unification in the MSSM example is feasible.

Conclusions and Outlook
This article forms the third part in a tryptic about deforming Z 2 singularities of toroidal orbifolds T 6 /(Z 2 × Z 2M ) with discrete torsion in Type IIA superstring theory, focusing here on the -technically most involved but at the same time of greatest phenomenological interest -T 6 /(Z 2 × Z 6 × ΩR) orientifold. To study the deformations of such singularities, we employ the techniques of the hypersurface formalism developed for the Z 2 × Z 2 toroidal orbifold with discrete torsion and extend them to the Z 2 × Z 6 toroidal orbifold, in a similar spirit as was done for the Z 2 × Z 6 toroidal orbifold before. The extension boils down to modding out an additional Z 3 ⊂ Z 6 action along a four-torus in the hypersurface formalism for the Z 2 × Z 2 toroidal orbifold. Since the Z 6 action does not constrain the geometry of the first two-torus, the deformations in the Z Introducing an anti-holomorphic involution -a necessary geometric part of the Tye IIA orientifold projection -in the hypersurface formalism opens the door for identifying sLag threecycles, albeit only a minimal subset of sLag three-cycles defined as the fixed loci under this involution. By virtue of Weierstrass' elliptic function, one can easily divide this subset into bulk three-cycles and fractional three-cycles for vanishing deformations. For non-zero deformations, exceptional three-cycles are characterised in the hypersurface formalim by (real) algebraic equations whose local form reduces to the ones of a (resolved) C 2 /Z 2 singularity. Their global description on the other hand is for most deformations not attainable due to the compact topology of the ambient T 4 and/or different complex structures of the two-torus lattices within the factorisable ambient T 4 (2 or 3) in case of the Z and the latter deforming a Z 3 -invariant orbit of three singularities on T 4 (1) /Z 6 . By computing the volume-dependence on these deformation parameters for bulk and fractional three-cycles parallel to the ΩR-invariant orientifold plane we are able to cross-check and validate the results of the exceptional three-cycle volume. Moreover, by studying the effects of deformations ε (1) 0 and ε (1) 3 on simple fractional three-cycles in detail we obtain the necessary intuition to investigate the effects of other deformations, for which the exceptional three-cycles cannot be accessed directly due to the absence of a global description. This also allows to propose a method for assessing qualitatively any fractional three-cycle volume as a function of a particular deformation parameter, as explained on page 35.
When applying these techniques to global intersecting D6-brane models, we observe in first instance that ΩR-even exceptional three-cycles give rise to deformation moduli with a flat direction, while D6-branes wrapping ΩR-odd exceptional three-cycles couple to deformation moduli which ought to be stabilised at vev to avoid the presence of non-vanishing Fayet-Iliopoulos terms, which in turn would signal a breakdown of supersymmetry at the string or Kaluza-Klein scale. A loophole to this consideration emerges when the massless open string spectrum contains suitably charged states whose scalar components can develop non-zero vevs to compensate the non-zero FI-term(s) in all D-term equations simultaneously. If present, this phenomenon would result in a spontaneous breaking of the gauge group supported by the D6-branes and would correspond geometrically to a recombination of two separate D-brane stacks (or a separation of D-branes within a single stack). In order to fully understand whether the loophole is realisable, it is of uttermost importance to acquire a better handle on the lowenergy effective action from first principles through CFT computations [67,102,68,70] or through dimensional reduction [103]. We hope to resolve and report on this issue in future work.
Bypassing the loophole we can count the maximal number of stabilised deformation moduli ζ (i) λ = 0, i.e. those ζ (i) λ coupling to D6-branes via the respective ΩR-odd exceptional wrapping number. An overview of this counting for the previously constructed and phenomenologically interesting global D6-brane models on T 6 /(Z 2 × Z 6 × ΩR) is provided in table 12, which shows that a maximum of ten out of 14 twisted complex structure moduli can be stabilised in the L-R symmetric prototype IIc model. For the other prototype models, the maximal number of stabilised deformation moduli is lower, with the lowest number being four in case of the global prototype I PS model. In table 12, we also observe a number of twisted moduli with a flat direction affecting the low-energy effective field theory. By going away from the singular orbifold point along these flat directions, the (inverse squared of the) tree-level gauge coupling of some D6-brane acquires a (square root-like) dependence on the non-zero twisted moduli, provided the D6-brane wraps only the associated ΩR-even exceptional three-cycle. Depending on the relative sign between the bulk three-cycle and the exceptional three-cycle, the gauge theory on such a D6-brane can become more weakly or more strongly coupled at the string scale, as discussed e.g. for the left-symmetric gauge group SU (2) L in all global prototype models, for the strong gauge group SU (3) QCD and the generalised B − L symmetry in global L-R symmetric models and for the hidden gauge group in global PS models. An important observation to be made here concerns the Z 2 ×Z 2 eigenvalues of the respective D6-branes: at the orbifold point, the physics only depends on the relative Z 2 × Z 2 eigenvalues among the D6-branes (e.g. in the computation of the particle spectra), whereas the absolute Z 2 × Z 2 eigenvalues enter explicitly in the fractional three-cycle volumes on the deformed toroidal orbifold. This last feature can be useful to improve the matching of the gauge coupling strength at the string scale, or might allow for further subdivisions in the global prototype models. Finally, in each of the global models there exists also a set of deformation moduli to which none of the D6-branes couples, implying that deforming the associated Z 2 singularities is only expected to impact the fractional three-cycle volumes in subleading order through higher order corrections such as field redefinitions of the moduli or instanton corrections.
Additional untwisted moduli-dependent corrections to the tree-level gauge couplings result from the one-loop gauge threshold corrections, which exhibit a linear and logarithmic dependence on the Kähler moduli associated to the two-torus volumes (in units of α ) in the geometric regime (v i 1). In the global D6-branes models considered here, all fractional three-cycles have a bulk one-cycle part along T 2 (1) parallel to the ΩR(Z 2 )-invariant plane, such that the gauge threshold corrections depend in each intersection sector on the Kähler modulus v 1 representing the area of the first two-torus T 2 (1) . This dependence with an overall positive prefactor presents various phenomenological challenges for compactifications with an intermediate string scale M string and LARGE internal volumes as discussed in section 4.2: a large hierarchy between the Planck scale M Planck and the string scale M string can yield a reasonable tree-level coupling when compensated by a hierarchically large complex structure modulus R (1) 2 /R (1) 1 of T 2 (1) . Taking into account the v 1 -dependent one-loop gauge threshold corrections makes it, however, difficult to stay in the geometric regime and implies an exponential suppression of the running gauge coupling, excluding the possibility of a strong coupling regime of the QCD stack. Thus, the phenomenologically interesting, global D6brane models at hand prefer a high string scale, while gauge unification is neither easily realised at tree-level, nor when including the one-loop gauge threshold corrections.
Each global D6-brane prototype model at hand forms a realisation of a supersymmetric Minkowski vacuum with all internal RR-fluxes or NS-NS-fluxes set to zero. Despite the underlying mathematical consistency of these global D6-brane models, there are two important elements missing in these vacuum configurations: a mechanism to stabilise all moduli vevs including the dilaton (and thereby also the moduli masses) and an alternative mechanism for supersymmetry breaking beyond the formation of a gaugino condensate in some hidden sector -which is of particular relevance to those models with only Abelian 'hidden' gauge bosons such as the L-R symmetric prototypes II, IIb and IIc -producing soft supersymmetry terms lifting the mass degeneracy of the massless open string matter states. In Type II superstring theory, one can (at least in principle) kill these two birds with one stone by turning on internal RRand NS-NS-fluxes generating a non-vanishing F-term for one (or more) of the moduli multiplets. This consideration begs the question whether one can consistently switch on twisted internal NS-NS-fluxes supported along resolved exceptional three-cycles and discuss moduli stabilisation for the deformation moduli (or twisted complex structure moduli) perturbatively, in contrast to the moduli stabilisation scheme for Kähler moduli in Type IIB string theory through solely non-perturbative effects as discussed e.g. in [78,104,105]. This question is being addressed in a separate research project and we hope to answer it positively in the near future. Moreover, when switching on twisted NS-NS-fluxes one generally also expects to generate a scalar potential for the axionic (or CP-odd) partners of the twisted complex structure moduli, and it begs the question whether the shape and symmetries of this scalar potential exhibit the generalised Kaloper-Sorbo structure, as recently established [106,107] for Type IIA orientifold compactifications on smooth Calabi-Yau backgrounds. Similar to the potential cancellation of FI-terms by virtue of vevs associated to charged matter states, this question should be settled at tree-level through a dimensional reduction of the Chern-Simons action and the Dirac-Born-Infeld action an effective four-dimensional N = 1 supergravity following the methods in [108,109].

A Möbius Transformations
In section 2.2, a two-torus T 2 was described as an elliptic curve E in the weighted projective space P 2 112 through eq. (7) with a built-in Z 2 symmetry acting on the homogeneous coordinate y of weight 2. The fixed points of this Z 2 action correspond to the roots of the polynomial F (x, v) in equation (8). The position of these roots in the x-coordinate or in the v-coordinate are tied to the complex structure of the two-torus. In table 27 we provide a list of the roots in all coordinate patches for a square two-torus and a hexagonal two-torus. fixed point α x 1 -coordinate v 1 -coordinate fixed point α x l -coordinate v l -coordinate 2 fixed points on the square torus T 2 (1) and the hexagonal tori T 2 (l=2, 3) in the homogeneous coordinates x i (v i = 1 patch) for the second and fifth column, and in the homogeneous coordinates v i (x i = 1 patch) for the third and last column. The labelling of the fixed points matches the one in figure 1.
There exist automorphisms λ α : E → E of the elliptic curve [51,50] interchanging the coordinates x and v through the action: where the α correspond to the roots entering in equation (8). These automorphisms λ α interchange the Z 2 fixed points: in line with the coordinate transformation, and are further constrained by the tiltedness of the torus lattice: untilted: λ α = λ α , tilted: λ 2 = λ 4 , λ 3 = λ 3 .
In a coordinate patch where v = 1 (or x = 1), the automorphisms λ α coincide with Möbius transformations. Recall that for a square two-torus the roots are given by 2 = 1, 3 = 0 and aIII, and finally λ 4 swaps aII ↔ aIII leaving aI and aIV invariant. For a hexagonal T 2 , the Lag lines from table 6 behave as follows under Möbius transformations: λ 3 keeps all the cycles bX invariant, whereas λ 2 and λ 4 exchange the Lag lines bI 0 ↔ bIII 0 and bII 0 ↔ bIV 0 .

B Correction Terms for Z
(1)

-Deformations
In section 2.4.2 we observed that certain deformations in the Z 2 -twisted sector deform too many singular orbits. The deformations with parameters ε (1) 4+5 and ε (1) 4−5 for instance deform the singularity (33), inadvertently. Such a feature was for the first time observed in [51] and appears to be intrinsic to Z 2 -twisted sectors subject to an additional Z 3 action on the full covering four-torus T 4 , as testified in table 9. In order to overcome the unwanted deformation of singular points, which are spatially separated from the one where the deformation parameter is localised, we have to add a counter-term which eliminates the effect of the initial deformation there. Building further on the ε (1) 4+5 example, we have to switch on a correction term ε (1) 3 as a function of ε (1) 4+5 . In order to determine this functional dependence, we consider a sLag two-cycle on T 4 (1) /Z 6 , represented by the algebraic condition x 2 = x 3 , going through the singular point (33) and enforce that there is no exceptional two-cycle emerging from that point for a deformation parameter ε We have to point out that this approach only works for small (initial) deformations ε in , as long as the correction term ε 3 (ε in ) remains smaller than the original deformation. Applying this requirement for the (conservative) assumption ε (1) 3 ∼ 1 10 ε in implies the range ε in 0.3 for the initial deformation.
Similar considerations have to be made for other deformation parameters, as summarised in table 28. In each particular case, we apply the same method as described above and construct a Taylor expansion in terms of the initial deformation. More explicitly, a nonvanishing deformation parameter ε unintentionally, as pointed out in table 9. This effect can be undone by turning on a correction terms ε  3 . Following the same logic as presented above, one finds the following series expansion for ε

C Tables of Matter States per D6-Brane Intersection Sector
In this appendix, we summarise the massless open string spectrum per sector for each of the global prototype D6-brane models with interesting phenomenological features as discussed in section 3. Table 29 contains the massless open string spectrum of the global prototype MSSM model discussed in section 3.3, and table 30 contains the counting of massless open string states per intersection sector for the global PS prototype I model discussed in section 3.5. Table 31   , the sector x(ω k y) comes with a set of vector-like pairs of matter states, whose multiplicity corresponds to n   x = h1 (0, [4] 2 , [ (0, [5] 2 , [5] 2 ) ([2] −2+2+2 , [5], [5] (0,0,0)