Deformations, Moduli Stabilisation and Gauge Couplings at One-Loop
Gabriele Honecker Isabel Koltermann Wieland Staessens





MITP/16-103
 
IFT-UAM/CSIC-17-010

December 30, 2023








Deformations, Moduli Stabilisation and Gauge Couplings at One-Loop



Gabriele Honecker^a,, Isabel Koltermann^a,,  and Wieland Staessens^b,c,





^aPRISMA Cluster of Excellence, MITP 
 & Institut für Physik  (WA THEP), 
Johannes Gutenberg-Universität, 55099 Mainz, Germany

^b Instituto de Física Teórica UAM-CSIC, Cantoblanco, 28049 Madrid, Spain

^c Departamento de Física Teórica, 
Universidad Autónoma de Madrid, 
28049 Madrid, Spain
 ^Gabriele.Honecker@uni-mainz.de,  ^kolterma@uni-mainz.de
 ^wieland.staessens@csic.es




Abstract
140mm
We investigate deformations of _2 orbifold singularities on the toroidal orbifold T^6/(_2×_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/(_2×_2) with discrete torsion and adapt it to the (_2×_6×) point group by modding out the remaining _3 subsymmetry and the  orientifold projection . We first study the local behaviour of the _3 × 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 <cit.>, 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_string, for the same global D6-brane models. 






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§ INTRODUCTION


Since the dawn of string phenomenology, toroidal orbifolds have played a prominent rôle in string model building <cit.>: 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. <cit.>.[Type IIA orientifold compactifications on non-factorisable toroidal orbifolds <cit.> have only recently been considered for model building purposes <cit.>.] 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 <cit.>. 

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.[In addition to phenomenological considerations, the known prescriptions for identifying dual string theoretic descriptions via mirror symmetry to Type IIB orientifolds or via M-theory to E_8 × E_8 heterotic compactifications, see e.g. <cit.>,  are to our best knowledge only valid for smooth Calabi-Yau backgrounds. 
] 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. <cit.>, 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 <cit.> using fractional D3-branes located at the singularities or D7-branes wrapping the resolved four-cycles. 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 _112^2, 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 ^1 ×^1 ×^1. Building on this hypersurface formalism first sketched in <cit.> for the T^6/(_2 ×_2) orbifold with discrete torsion and extended to its T^6/(_2 ×_2 ×) and T^6/(_2 ×_6' ×) orientifold versions with underlying isotropic square <cit.>  or hexagonal <cit.>  two-tori, respectively,
we focus here on the so far most fertile patch in the Type IIA orientifold landscape with rigid D6-branes <cit.>, the T^6/(_2 ×_6 ×) orientifold with discrete torsion and  one rectangular and two hexagonal underlying two-tori. In this case, the _2^(1)-twisted sector conceptually differs from the _2^(2)- and _2^(3)-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 , in a similar spirit as in <cit.>. 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 <cit.> 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 <cit.>, as it is (at least in the absence of an additional _3 symmetry completely) projected out by the _2×_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 <cit.> 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/(_2 ×_6 ×) 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. <cit.> in the context of D6-branes. These gauge threshold corrections can be sizeable for specific anisotropic choices of two-torus volumes <cit.>, 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 <ref>, we briefly review the hypersurface formulation for describing local deformations of T^6/(_2 ×_2 ×) 
singularities as discussed in <cit.> and then go on to discuss additional constraints imposed by the extra _3 symmetry of the T^6/(_2 ×_6 ×) orientifold.  Special attention will be devoted to the sLag cycles used for particle physics model building.
In section <ref>, additional subtleties in global deformations of T^6/(_2 ×_6 ×) singularities are discussed, and several 
prototype examples of global D6-brane models with particle physics spectra are examined.
Section <ref> 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 <ref> contains our conclusions and outlook.
Additional technical details useful for the computation of deformations and one-loop corrections are relegated to appendices <ref>, <ref> and <ref>.








§ 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 <ref>,
in particular on the orientifold of phenomenological interest T^6/(_2 ×_6 ×). 
Then, we move on to reviewing  Lagrangian (Lag) lines on two-tori of rectangular and hexagonal shape in the hypersurface formalism in section <ref>. 
In section <ref>, we first discuss deformations of _2 ×_2 singularities 
on  and afterwards impose relations among deformations due to the specific additional _3 symmetry of the _2 ×_6 action. As a final element,
in section <ref> we discuss the general procedures allowing for the quantitative study of  special Lagrangian (sLag) three-cycles on deformations of
 T^6/(_2 ×_6 ×) in the hypersurface formalism.






 §.§ Reminiscing about three-cycles on the T^6/(_2 ×_6 ×) orientifold


The action of the orbifold group _2 ×_6 on the factorisable six-torus T^6 = T_(1)^2× T_(2)^2 × T_(3)^2 consists of a rotation of the complex coordinates z_k parametrising the respective two-torus T_(k)^2 with k∈{1,2,3}:

    θ^m ω^n: z_k → e^2 π i (m a_k + n b_k) z_k,          with a⃗ = 1/2 (1,-1,0),   b⃗ = 1/6(0,1,-1).

Note that the point group _2 ×_6 is generated by the elements θ and ω, with θ generating the _2-factor acting on the four-torus T_(1)^2× T_(2)^2 and ω generating the _6-part acting on the four-torus T_(2)^2× T_(3)^2. As a direct product of two Abelian factors containing each _2 as a (sub)group, the orbifold group allows for a global discrete torsion factor η=± 1 <cit.>, whose presence alters the amount of two- and three-cycles supported in the orbifold twisted sectors, as indicated in table <ref> listing the Hodge numbers per sector. In the absence of discrete torsion (η=1), the _2^(i) 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 _2^(i) singularities, yielding exceptional three-cycles located at the former _2^(i) fixed loci. The three-cycles in the _2^(i)-twisted sectors turn out to be useful tools with regard to particle physics phenomenology and D6-brane model building <cit.>, encouraging us to focus for the remainder of the article on the orbifold with discrete torsion.  


[ 10|c|Hodge Numbers (h^11,h^21) per sector for the factorisable orbifold T^6/(_2 ×_6);                                                                                2|c||                                                                               _2^(1)                                                                               _2^(2)                                                                               _2^(3)                                                                                   _3                                                                          3|c||_6^(')                                                                        Hodge Numbers;                                                                                    η                                                                            Untwisted                                                                                  ω^3                                                                                 θω^3                                                                                    θ                                                                                  ω^2                                                                                    ω                                                                                   θω                                                                                 θω^2                                                                     ([ h^11; h^21 ]);                                                                                η= +1                                                                          ([ 3; 1 ] )                                                                          ([ 6; 0 ] )                                                                          ([ 8; 0 ] )                                                                          ([ 8; 0 ] )                                                                          ([ 8; 2 ] )                                                                          ([ 2; 0 ] )                                                                          ([ 8; 0 ] )                                                                          ([ 8; 0 ] )                                                                    ([   51; 1+ 2 ] );                                                                                η= -1                                                                          ([ 3; 1 ] )                                                                          ([ 0; 6 ] )                                                                          ([ 0; 4 ] )                                                                          ([ 0; 4 ] )                                                                          ([ 8; 2 ] )                                                                          ([ 0; 2 ] )                                                                          ([ 4; 0 ] )                                                                          ([ 4; 0 ] )                                                          ([        19; 15 + 2 ×2 ] );                                                                                      ]
Z2Z6HodgeNumbersHodge numbers (h^11,h^21) per sector for the factorisable toroidal orbifold T^6/(_2 ×_6) with lattice configuration SU(2)^2× SU(3)× SU(3). In the absence of discrete torsion, the Hodge numbers match those of the orbifold T^6/(_2×_2), namely (h^11,h^21)=(51,3),
but with a different distribution over twisted sectors. Considering the orbifolds with discrete torsion leads to a reduction of the initial 51 three-cycles on T^6/(_2×_2) to only 19 three-cycles on T^6/(_2×_6) due to the additional _3-action.

A first observation regarding the _6-action deals with the shape of the two-tori T_(2)^2 × T_(3)^2 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 _2 ×_6 orbifold. The comparison with the orbifold T^6/(_2×_2) with discrete torsion shows that the additional _3-action reduces the number of three-cycles in the _2^(i) twisted sectors by triple identifications, and thereby also enforces the simultaneous deformation of the associated _2×_2 singularities, as we will discuss in detail in section <ref>. 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 <ref>:


  * Three _2^(i=1,2,3)-twisted sectors generated by (ω^3, ω^3 θ, θ) respectively, where each _2^(i) 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 <ref>. 
In the _2^(1)-twisted sector, the fixed point (11) on T_(2)^2× T_(3)^2 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 _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 _2^(1) fixed points: [(21),(31),(41)], [(12),(14),(13)], [(33),(42),(24)], [(22),(34),(43)] and [(44),(23),(32)]. On the orbifold T^6/(_2×_6) without discrete torsion, these _2^(i) 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_(1)^2. In the _2^(j=2,3)-twisted sectors, the four fixed points (α 1) on T_(1)^2 × T_(k=3,2)^2 are invariant under the _6-action, while the other twelve fixed points recombine into four _6-invariant orbits of three fixed points each: [(α 2),(α 3),(α 4)].  
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 _6-invariant orbits tensored with a one-cycle on T_(j)^2 (which is also rotated under _6) contribute to h^21.


  * One _3-twisted sector generated by ω^2 with nine fixed two-tori labelled by the fixed points (a b) with a,b ∈{1,5,6} on T_(2)^2× T_(3)^2. The fixed points are subject to the _2×_2 orbifold action mapping 5↔ 6 and 1↺, such the _3 fixed points along T^4_(1) recombine into four invariant orbits: (11), [(15),(16)], [(51),(61)] and [(55),(56),(65),(66)]. 
As detailed in <cit.>, the _3-twisted sector does not feel the discrete torsion phase η=± 1, such that in any case, each fixed point orbit supports two two-cycles per T^4_(1)/_3 singularity, and three-cycles arise from tensoring the _2 ×_2 quadruplet [(55),(56),(65),(66)] on T^4_(1) with one-cycles on T^2_(1).


  * One _6-twisted and two _6'-twisted sectors generated by (ω, θω, θω^2), respectively. The _6-twisted sector associated to ω comes with one fixed two-torus or fixed line located at the singularity (11) on T_(2)^2× T_(3)^2. 
As detailed in <cit.>, the discrete torsion phase acts non-trivially in this sector, which accounts for h^11=2 in the case of η=+1 and h^21=2 in the case of η=-1 in analogy to the other _2^(1) twisted sector.

The other two _6^' actions have a different structure: the one generated by θω yields twelve fixed points labelled by (α a 1), and the last one generated by θω^2 comes with twelve fixed points labelled by (α 1 a), where α∈{1,2,3,4} and a∈{1,5,6} for both cases. Under the _2×_2 orbifold action, the twelve _6^' fixed points in the θω sector recombine into eight orbits [(α 1 1)]_θω, [(α 5 1), (α 6 1)]_θω. In the absence of discrete torsion (η=+1), each orbit supports a two-cycle and its dual four-cycle, while in the presence of
non-trivial discrete torsion (η=-1), only the non-trivial orbits [(α 5 1), (α 6 1)]_θω support each one two-cycle and its dual four-cycle, see <cit.> for details. The second _6^'-twisted sector is obtained by permutation of two-torus indices, T^2_(2)↔ T^2_(3).

  

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 three-cycles supported on the orbifold T^6/(_2 ×_6) with discrete torsion:



  (1) Bulk three-cycles are orbifold-invariant products of three one-cycles, where each one-cycle extends along a different two-torus T_(i)^2. 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_(i)^2, see figure <ref> 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  _2×_6 identifications. Generic bulk three-cycles can then be expressed in terms of these four basis bulk three-cycles: 

    [                                                               Π^ bulk                                                                     =       n^1 (n^2 n^3- m^2 m^3) ρ_1 + n^1 (n^2 m^3 + m^2 n^3+m^2m^3) ρ_2;                                                                       +  m^1 (n^2 n^3- m^2 m^3) ρ_3 + m^1 (n^2 m^3 + m^2 n^3+m^2m^3) ρ_4
     . ]


  (2) Exceptional three-cycles are orbifold-invariant products of a one-cycle on the _2^(i)-invariant two-torus T_(i)^2 with an exceptional divisor e_αβ^(i) located at the _2^(i) fixed points (αβ) along the four-torus T^4_(i). The _2^(i) fixed points can be resolved by gluing in a two-sphere per singularity. The _6-invariant  products of the basis one-cycles with the exceptional divisors yield twelve basis exceptional cycles (ϵ_λ^(1), ϵ̃_λ^(1)) in the _2^(1)-twisted sector with λ∈{0,1,2,3,4,5} and 2 × 8 basis exceptional cycles (ϵ_α^(k), ϵ̃_α^(k)) in the _2^(k=2,3)-twisted sectors with α∈{1,2,3,4}. The dimensionality of the full set of basis exceptional three-cycles expected from all _2^(i)-sectors combined matches the Betti-number b^_2_3 = 2 h__2^21 = 2 ·( 6 + 2 × 4  )  = 28.
 Furthermore, the _6- and _3-twisted sectors also yield b^_6+_3_3 = 2 · (2+2)=8 exceptional three-cycles located at the ω and ω^2 fixed points along T_(2)^2× T_(3)^2 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/ orientifold models <cit.>, in particular they expected to contribute to the open string one-loop annulus amplitude <cit.>, 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 _2^(i) 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 _2^(i)-fixed points and represents its own _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

    Π^ frac = 1/4Π^ bulk +  1/4∑_i=1^3 Π^_2^(i) =1/4Π^ bulk + 1/4∑_i=1^3 ∑_λ(  x^(i)_λϵ_λ^(i) + y^(i)_λϵ̃_λ^(i)),
 
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_(i)^2 weighted by sign factors associated to the discrete _2^(i)-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_(i)^4 set by the discrete shift parameter (σ^j, σ^k) ∈{0,1}, as detailed in table 36 of <cit.>. 
The sum over λ runs over at most four different values in the _2^(1)-twisted sector with λ∈{ 0,1,2,3,4,5} and two values in the _2^(2,3)-twisted sectors with λ∈{1,2,3,4 }
for the orbifold T^6/(_2 ×_6).



A detailed discussion of three-cycles on the orbifold T^6/(_2×_6) can be found in <cit.>, while their prospects for intersecting D6-brane model building have been thoroughly investigated in <cit.>. 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 _3-factor in the orbifold group, in analogy to other orbifolds with a _3-factor within the point group <cit.>. 


Type IIA string compactifcations on T^6/(_2×_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 _6-action, corresponding to the O6-planes  and _2^(i=1,2,3), respectively, as listed in table <ref> for the aAA-lattice configuration.[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 <cit.> 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 <cit.> to be the most fruitful background allowing for global three-generation
MSSM, Left-Right (L-R) symmetric and Pati-Salam (PS) models.] Each of the O6-planes carries RR-charge whose sign is denoted by η_ (_2^(i))∈{± 1}, and worldsheet consistency of the Klein-bottle relates them to the discrete torsion parameter <cit.>:

    η = η_∏_i=1^3 η__2^(i).

 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 <ref> and <ref>, we select the _2^(3)-plane as the exotic O6-plane, i.e. .[The choice η__2^(2)=-1 is equivalent upon permutation of T^2_(2)↔ T^2_(3), while there exists a second inequivalent choice of η_=-1 allowing for supersymmetric solutions to the RR tadpole cancellation conditions.]
 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 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 _2^(i)-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 -even and -odd three-cycles under the orientifold projection, depending on the choice of the exotic O6-plane, as can be deduced from table <ref>.
 
 
[ 4|c|R-invariant planes on T^6/(_2×_6×);                               O6-plane                 Torus wrapping numbers                              RR-charge                          Global Models;                                                                 (1,0;1,0;1,0)                                     η_                                    + 1;                                 _2^(1)                        (1,0;-1,2;1,-2)                               η__2^(1)                                     +1;                                 _2^(2)                         (0,1;1,0;1,-2)                               η__2^(2)                                     +1;                                 _2^(3)                         (0,1;1,-2;1,0)                               η__2^(3)                                     -1;                                        ]
O6PlanesZ2Z6O6-planes on the aAA lattice of T^6/(_2×_6×) with discrete torsion (η=-1). The last column indicates the regular/exotic sign of the RR-charges for the global models discussed in sections <ref> and <ref>.
 
 
 


[ 8|c|Orientifold images of basis three-cycles on  T^6/(_2 ×_6×)    with  η= -1;                                                             2|c|| Bulk cycles                                                   3|c|| _2^(1) twisted sector                                      3|c| _2^(l) twisted sector with  l = 2,3;                                                                         (ρ_α)                                                                             α                                                                     (ϵ^(1)_λ)                                                                    (ϵ̃^(1)_λ)                                                                             λ                                                                     (ϵ^(l)_α)                                                                    (ϵ̃^(l)_α)                                                                             α;                                                                           ρ_1                                                                             1                                                               - η_(1) ϵ^(1)_λ                                                              η_(1)   ϵ̃^(1)_λ                                                                       0,1,2,3                                                             - η_(l)   ϵ^(l)_α                                                   η_(l) ( ϵ̃^(l)_α -ϵ^(l)_α )                                                                       1,2,3,4;                                                                     ρ_1 - ρ_2                                                                             2                                                              - η_(1)  ϵ^(1)_5                                                              η_(1)   ϵ̃^(1)_5                                                                             4                                                                                                                                                                                                                                          ;                                                                          -ρ_3                                                                             3                                                               - η_(1) ϵ^(1)_4                                                              η_(1)   ϵ̃^(1)_4                                                                             5                                                                                                                                                                                                                                          ;                                                                       ρ_4-ρ_3                                                                             4                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    ;                                                                               ]
Z2Z6OrientifoldExceptionalCyclesOrientifold images of the basis bulk and _2^(k) exceptional three-cycles for the aAA lattice configuration, depending on the choice of the exotic O6-plane orbit with sign factor η_(k)≡η_η__2^(k).

  


In order for the D6-branes to preserve the same N=1 supersymmetry, they are required to wrap special Lagranigian (sLag) three-cycles Π:

    J_(1,1)|_Π =0, 
           Re(Ω_3)|_Π > 0,      Im(Ω_3)|_Π =0.
  
Three-cycles satisfying condition (<ref>) 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 (<ref>) 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 (<ref>) 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 <ref>. 


For the sake of completeness regarding the discussion of geometric moduli on the  orientifold with discrete torsion, 
we also list the counting of Kähler moduli and closed string vectors on the aAA lattice in table <ref>.
The counting on the inequivalent bAA lattice can be found in table 46 of <cit.>.


[ 10|c|Hodge Numbers (h^11_+,h^11_-) per sector for the factorisable orientifold T^6/(_2 ×_6 ×) with η=-1;                                                                                                   2|c||                                                                                                  _2^(1)                                                                                                  _2^(2)                                                                                                  _2^(3)                                                                                                      _3                                                                                             3|c||_6^(')                                                                                           Hodge Numbers;                                                                                                       η                                                                                               Untwisted                                                                                                     ω^3                                                                                                    θω^3                                                                                                       θ                                                                                                     ω^2                                                                                                       ω                                                                                                      θω                                                                                                    θω^2                                                                                    ([ h^11_+; h^11_- ]);                                                                                                   η= +1                                                                                              ([ 0; 3 ])                                                                                              ([ 1; 5 ])                                                                                              ([ 0; 8 ])                                                                                              ([ 0; 8 ])                                                                                              ([ 0; 8 ])                                                                                              ([ 0; 2 ])                                                                                              ([ 0; 8 ])                                                                                              ([ 0; 8 ])                                                                                            ([  1; 50 ]);                                                                                                   η= -1                                                                                              ([ 0; 3 ])                                                                                              ([ 0; 0 ])                                                                                              ([ 0; 0 ])                                                                                              ([ 0; 0 ])                                                                                              ([ 0; 8 ])                                                                                              ([ 0; 0 ])                                                                            ([ 2(1+η_(2)); 2(1-η_(2)) ])                                                                            ([ 2(1+η_(3)); 2(1-η_(3)) ])                                                          ([  4 + 2(η_(2)+η_(3)); 15 - 2(η_(2)+η_(3)) ]);                                                                                                         ]
Z2Z6HodgeSplittingSplitting of the Hodge number h^11 into -even and -odd part on the aAA lattice.
h^11_- counts the number of Kähler moduli on the orientifold, while h^11_+ counts the number of closed string vectors.
The models in sections <ref> and <ref> obey η_(2)=1=-η_(3).


It is noteworthy that for the phenomenologically interesting choice of exotic _2^(3)-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 _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  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/_2N or T^6/(_2 ×_2M) backgrounds reviewed above formally uses 
exceptional divisors at _2-singularities
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 _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 _112^2 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 _112^2 and describe a two-torus as a hypersurface within _112^2. More explicitly, a two-torus corresponds to an elliptic curve in _112^2, which forms the zero locus of a polynomial f of degree 4:

    f ≡ -y^2 +  F(x,v) = 0,      F(x,v) = 4 v x^3 - g_2 v^3 x - g_3 v^4,

 where we choose the Weierstrass form for the elliptic curve. There exists a _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 ϵ_2, ϵ_3 and ϵ_4,
 
    F(x,v) = 4v ( x - ϵ_2 v) ( x - ϵ_3 v) ( x - ϵ_4 v),
  
the coefficients g_2 and g_3 are easily related to the roots: g_2 = - 4 ( ϵ_2 ϵ_3 + ϵ_2 ϵ_4 + ϵ_3 ϵ_4  ) and g_3 = 4 ϵ_2 ϵ_3 ϵ_4, with the roots satisfying the condition ϵ_2 + ϵ_3 + ϵ_4 = 0. The fourth root ϵ_1 located at x = ∞ (in the v=1 patch) represents the _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/(_2×_6):


  (1) a-type lattices or untilted (rectangular) tori with Re(τ) = 0: generically the roots ϵ_α are all real and can be ordered as ϵ_4<ϵ_3<ϵ_2. A square torus with τ = i represents a special case for which g_3=0, ϵ_2 = - ϵ_4 = 1 and ϵ_3=0. 

  (2) b-type lattices or tilted tori with Re(τ) ≠ 0: generically the roots ϵ_2 and ϵ_4 are related by complex conjugation, ϵ_2 = ϵ_4≡ξ, while ϵ_3 = - 2  Re(ξ) is a real parameter. A hexagonal torus with τ = e^i π/3
(cf. figure <ref> (b))
 forms a special case with g_2 = 0 and ξ = e^2π i/3, for which the elliptic curve exhibits an additional _3 symmetry x/v→ e^2π i/3x/v. This _3 symmetry is in correspondence with a _3 subgroup acting on the hexagonal two-torus lattice, suggesting that the two-tori T_(2)^2 and T_(3)^2 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 <ref>. 



The two-torus T_(1)^2 is not affected by the _3 subgroup of the _2×_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_(1)^2 does not provide for any (known) phenomenologically appealing global intersecting D6-brane models <cit.>, we confine ourselves to an untilted T_(1)^2 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 _2^(3)-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 <ref> for several examples. Such configurations are supersymmetric for any value of the complex structure parameter Im(τ)>0 on T^2_(1).  
 
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 (<ref>) 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_(i)^2 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:

    σ:   ([ x; v ]) ⟼  A ([ x; v ]),          y ⟼ e^i βy,

 with A ∈ GL_2(). For this action to be an involution, the matrix A has to satisfy the condition A  A =. The involution also has to be a symmetry of the elliptic curve, which boils down to the following condition σ( F(x,v) ) = e^2i β F( x,  v). Solving both conditions allows to extract the unequivocal forms of the various anti-holomorphic involutions <cit.>. 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 <cit.>. 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 ϵ_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/(_2×_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 ϵ_α is offered in table <ref>; their positions in the x-plane are depicted in figure <ref> (a) for the v=1 coordinate patch.   
 
   
 
 
  (2) hexagonal torus: here we can identify the one-cycles  bX (with X = I, II, III, IV) passing through two roots ϵ_α and corresponding to σ-involution invariant directions. Due to the _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 <ref>, while figure <ref> (b) shows their position in the x-plane for the v=1 coordinate patch and clearly exhibits the _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 ^2_112:
 
    Ω_1 = 1/π i∫_γx dv ∧ dy - v dx ∧ dy  + y dx∧ dv/f,
  
where we divided by the polynomial f defined on the l.h.s. of equation (<ref>)
to obtain a well-defined scale-invariant two-form on ^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:

    Ω_1 = vdx - xdv/y|_f=0.

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 one-form (<ref>) 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.
 






 §.§ T^6/(_2 ×_6 ×) in the hypersurface formalism

Describing deformations of the orbifold T^6/(_2 ×_6) requires us to adopt a hypersurface formalism in which T^6/(_2 ×_6) with discrete torsion is embedded into an appropriate toric space. In a first step, we will review in section <ref> how the deformations of T^6/(_2 ×_2) with discrete torsion thrive in the hypersurface formalism, after which we mod out the remaining _3 symmetry within the _2 ×_6 point group in section <ref> to end up with the hypersurface formalism describing the deformations of T^6/(_2 ×_6). Finally, we impose additional constraints due to the orientifold involution .



  §.§.§ Hypersurface formalism for T^6/(_2 ×_2) with discrete torsion

The factorisable orbifold T^6/(_2 ×_2) with discrete torsion can be mapped <cit.> to the direct product of three distinct elliptic curves modded out by the orbifold group _2 ×_2, whose action only keeps the _2 ×_2-invariant subring of the ring of polynomials. The _2 ×_2-invariant “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 <ref>. 


Indeed, the orbifold T^6/(_2 ×_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:

    f ≡    -y^2 + F_(1)(x_1,v_1) F_(2)(x_2,v_2) F_(3)(x_3,v_3) 
       - ∑_i≠ j ≠ k ≠ i∑_α,β=1^4 ε^(i)_αβ F_(i)(x_i,v_i) δ F_(j)^α (x_j,v_j) δ F_(k)^β (x_k,v_k) 
       + ∑_α,β,γ =1^4 ε_αβγδ F_(1)^α (x_1,v_1) δ F_(2)^β (x_2,v_2) δ F_(3)^γ (x_3,v_3) 
      .

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_(i)^2 as in equation (<ref>) 
or its rewritten form (<ref>)
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 _3 symmetry, cf. section <ref>).  
Setting all the deformation parameters to zero, ε_αβ^(i) = 0 = ε_αβγ, corresponds to the orbifold point in the complex structure moduli space, with the roots of F_(i)(x_i,v_i) determining the positions of the _2^(k≠ i) fixed points. 

  * The deformation polynomials δ F_(i)^α(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, δ F_(i)^α(x_i,v_i) allows to deform the _2-fixed point associated to the α^ th root.

  * Deformations of the form δ F_(i)^α F_(j) F_(k) with (ijk) a cyclic permutation of (123) are not explicitly considered in equation (<ref>) as they correspond to deformations of the complex structure for two-torus T_(i)^2, up to PSL(2,) transformations acting on (x_i,v_i).[The PSL(2,) transformations are able to eliminate three complex parameters, leaving exactly one independent parameter per two-torus. 
] 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 <cit.>.

  * The parameter ε_αβ^(i) allows for the deformation of the _2^(i) singularity with index (αβ) on the four-torus T^4_(i)≡ T^2_(j)× T^2_(k) with (ijk) some permutation of (123). Counting the number of distinct deformation parameters, one finds 3 × 4 × 4 = 48 parameters in total, one for each _2^(i) singularity. The number 48 matches exactly the number of truly marginal operators in the _2^(i) twisted sectors of the corresponding N=(2,2) SCFT.

  * Within the set of possible deformations of T^6/(_2 ×_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 _2×_2 fixed points on the orbifold. However, these parameters do not represent independent deformations, but rather depend on the complex structure deformations ε_αβ^(i) from the _2^(i) twisted sectors, such that (at most) 64 conifold singularities remain and cannot be deformed away. This reflection is supported <cit.> by the absence of truly marginal operators in the N=(2,2) SCFT corresponding to ε_αβγ-deformations.[Contrary to the blow-up procedure, where blowing up the co-dimension two singularities in the _2^(i) twisted sectors also eliminates the co-dimension three singularities on T^6/(_2×_2) without discrete torsion, the deformation procedure does not automatically lead to the resolution of the 64 _2×_2 fixed points on T^6/(_2×_2) with discrete torsion.] 
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_αβ^(i).  


In order to determine the holomorphic volume form Ω_3 on T^6/(_2 ×_2) with discrete torsion,
 we extend the philosophy from above that allowed us to identify the appropriate hypersurface equation (<ref>). More explicitly, we consider the wedge product of three one-forms Ω_1, one for each two-torus T_(i)^2 as defined in equation (<ref>), and mod out the _2 ×_2 symmetry to obtain the following (simplified) expression (in the v_i=1 patch):

    Ω_3 = dx_1∧ dx_2 ∧ dx_3/y(x_i)|_f=0,

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 (<ref>) and fixing a branch cut for the square root. 

A thorough analysis of Lag lines on the (deformed) orbifold T^6/(_2 ×_2) on square tori was the subject of <cit.> and further comments 
and extensions to products of three hexagonal tori can be found in <cit.>. As is well known, the orbifold T^6/(_2 ×_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 _2^(i) singularities rather than deform them, one ought to look at the version of the orbifold without discrete torsion, see e.g. <cit.>. Physical implications of resolving the orbifold singularities through blow-ups have been investigated in <cit.>. 






  §.§.§ Hypersurface formulation for orbifolds with additional _3 × action

Setting up the hypersurface formalism for T^6/(_2×_6) with discrete torsion now consists in acting with the _3-subgroup 
generated by 2b⃗=1/3(0,1,-1) on the hypersurface formulation of T^6/(_2×_2) with discrete torsion, such that the resulting hypersurface polynomial is invariant under the _3-symmetry.[Notice that in <cit.> a different T^6/(_2 ×_6') orbifold with the _6' factor generated by 
b⃗^ '=1/6(-2,1,1) was considered. The analysis in that case with _2 ×_6' point group was simpler due to all three two-tori being of hexagonal shape and all three _2^(i)-twisted sectors being equivalent (up to a relative sign factor in the orientifold projection if one of the _2^(j)-planes is chosen as the exotic one).
] 
The _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 (<ref>). Anticipating the torus lattice configurations for the global models in section <ref>, we consider the first two-torus to be a square untilted two-torus, such that the homogeneous polynomials are given by: 

    [            F_(1) (x_1, v_1)                           = 4 v_1 x_1 (x_1^2 -  v_1^2),;            F_(2) (x_2, v_2)                           =      4 v_2 (x_2^3 - v_2^3),;            F_(3) (x_3, v_3)                           =      4 v_3 (x_3^3 - v_3^3).;                             ]
 
Evidently, the _3-action also constrains the form of the deformation polynomials δ F_(j=2,3)^α, while the deformation polynomials δ F_(1)^α are shaped by the untilted square lattice choice for the two-torus T_(1)^2:

    [                                δ F_(1)^1                                        =                   x_1^2 (x_1^2 - v_1^2),                                δ F_(j)^1                                        = x_j (x_j^3 - v_j^3),         for   j=2,3;                                δ F_(1)^2                                        =                    v_1 x_1(x_1 + v_1)^2,                                δ F_(j)^2                                        =        v_j^2 (v_j - x_j) (v_j - ξ x_j) ,;                                δ F_(1)^3                                        =                 - v_1^2 (x_1^2 - v_1^2),                                δ F_(j)^3                                        =    v_j^2 (v_j - ξ x_j) (v_j - ξ^2 x_j) ,;                                δ F_(1)^4                                        =               -  v_1 x_1 (x_1 -  v_1)^2,                                δ F_(j)^4                                        =      v_j^2 (v_j - x_j) (v_j - ξ^2 x_j) .;                                          ]

By virtue of the Weierstrass' elliptic function, one can easily deduce that the _3-subgroup also acts on the homogeneous coordinates x_i as follows:[Note that the holomorphic three-form defined in equation (<ref>) remains invariant under the _3-symmetry.]

    ω^2: (x_1,x_2,x_3) ↦ (x_1, ξ x_2, ξ^2 x_3),      with ξ = e^2π i/3,

which leaves the homogeneous polynomials F_(i)(x_i,v_i) invariant, but forces the deformation polynomials to transform as follows:

    [                                                                    ω^2:                                   δ F_(1)^α↦δ F_(1)^α    ∀ α∈{1,2,3,4},;                                                                              δ F_(2)^1 ↦ξδ F_(2)^1, δ F_(2)^2 ↦δ F_(2)^3 ↦δ F_(2)^4 ↦δ F_(2)^2,;                                                                         δ F_(3)^1 ↦ξ^2 δ F_(3)^1,   δ F_(3)^2 ↦δ F_(3)^4 ↦δ F_(3)^3 ↦δ F_(3)^2.;                                                                         ]

Keeping in mind these transformation properties, we can deduce that the linear combination δ F_(j)^2+δ F_(j)^3+δ F_(j)^4 = 3 v_j^4 represents a  _3-invariant polynomial for j=2,3. However, a simple coordinate transformation on x_j eliminates any deformation of the type 3 v_j^4 F_(1) F_(k≠ j), leaving the complex structure of the two-torus T_(j=2,3)^2 unaltered. The non-invariance of δ F_(j=2,3)^1 under _3 also excludes any type of deformation of the form δ F_(j=2,3)^1 F_(1) F_(k=3,2). This observation agrees with the considerations in section <ref> that the complex structures of the two-tori T_(j=2,3)^2 are frozen to hexagonal shape by the _3-action. 
The deformation polynomials δ F_(1)^α are left invariant by the _3-action, such that deformations of the type δ F_(1)^α F_(2) F_(3) do exist, up to PSL(2,) 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 <ref> on page Ref:PSLT2 that the three complex parameters of the PSL(2,) symmetry allow to reduce the four deformations to a single independent deformation.


Deformations of the _2^(i) singularities are performed through polynomials of the form F_(i)δ F_(j)^αδ F_(k)^β, which should also be _3-invariant for consistency. In this sense, the _3-action will put restrictions on the deformation parameters ε_αβ^(i) and reduce the number of independent deformation moduli. For the _2^(1)-twisted sector, we find six independent deformation parameters which concur with the Hodge number h^21__2^(1) = 6 from table <ref>:


  * 
 ε_11^(1)≡ε_0^(1) is left untouched and deforms the singularity (11) on the four-torus T^2_(2)× T^2_(3);

  * ξ^2 ε_21^(1) =ε_31^(1) = ξ ε_41^(1)≡ε_1^(1) deforms the singular orbit [(21),(31),(41)] on T^2_(2)× T^2_(3);

  * ξ^2 ε_12^(1) =ε_13^(1) = ξ ε_14^(1)≡ε_2^(1) deforms the singular orbit [(12),(13),(14)] on T^2_(2)× T^2_(3);

  * ε_33^(1) =ε_42^(1) = ε_24^(1)≡ε_3^(1) deforms the singular orbit [(33),(42),(24)] on T^2_(2)× T^2_(3);

  * ε_22^(1) =ε_34^(1) = ε_43^(1)≡ε_4^(1) deforms the singular orbit [(22),(34),(43)] on T^2_(2)× T^2_(3);

  * ε_44^(1) =ε_23^(1) = ε_32^(1)≡ε_5^(1) deforms the singular orbit [(44),(23),(32)] on T^2_(2)× T^2_(3).


The _2^(j=2,3) sectors are equivalent as a result of the T_(2)^2↔ T_(3)^2 exchange symmetry, such that we can treat them jointly. In the _2^(j)-sector we find four independent deformation parameters, which match the Hodge numbers h^21__2^(2) = h^21__2^(3)=4 in table <ref>:


  * 
 ∀ α∈{1,2,3,4} we find ε_α 1^(j) = 0, which excludes any type of deformation of the fixed points (α 1) on T_(1)^2 × T_(k≠ j)^2; 

  * ε_1 2^(j) = ε_1 3^(j) = ε_1 4^(j)≡ε_1^(j) deforms the singular orbit [(12),(13),(14)] on T^2_(1)× T^2_(k);

  * ε_2 2^(j) = ε_2 3^(j) = ε_2 4^(j)≡ε_2^(j) deforms the singular orbit [(22),(23),(24)] on T^2_(1)× T^2_(k);

  * ε_3 2^(j) = ε_3 3^(j) = ε_3 4^(j)≡ε_3^(j) deforms the singular orbit [(32),(33),(34)] on T^2_(1)× T^2_(k);

  * ε_4 2^(j) = ε_4 3^(j) = ε_4 4^(j)≡ε_4^(j) deforms the singular orbit [(42),(43),(44)] on T^2_(1)× T^2_(k).


The skeptical reader might object that there exists a certain freedom in choosing the forms of the deformation polynomials, yet any consistent _3-invariant choice of the polynomials δ F_(j)^α should yield the same numbers, as the pairing of the _2^(i) fixed points into orbifold-invariant orbits is an eternal consequence of the _3-action. After all, the amount of independent _2^(i) deformations has to match the Hodge numbers h^21__2 in the _2^(i)-twisted sectors discussed in section <ref>.


The last type of deformations to consider have the form δ F_(1)^αδ F_(2)^βδ F_(3)^γ and are also subject to the _3-action. The invariance of δ F_(1)^α under the _3-action suggests that we should only worry about the remaining two deformation polynomials and make sure they recombine into _3-invariant combinations with the following relations:


  * 
 ε_α11 remains unconstrained (∀ α∈{1,2,3,4});

  * ξ^2 ε_α 21 = ε_α 31 = ξ ε_α 41 (∀ α∈{1,2,3,4});

  * ξ^2 ε_α 12 = ε_α 13 = ξ ε_α 14 (∀ α∈{1,2,3,4});

  * ε_α 33 = ε_α 24 = ε_α 42 (∀ α∈{1,2,3,4});

  * ε_α 22= ε_α 34 = ε_α 43 (∀ α∈{1,2,3,4});

  * ε_α 44 = ε_α 23 = ε_α 32 (∀ α∈{1,2,3,4}).


 Hence, we obtain in total 6× 4 = 24 _3-invariant deformations of the type δ F_(1)^αδ F_(2)^βδ F_(3)^γ on the T^6/(_2 ×_6) orbifold, which should be contrasted with the 64 parameters on the  orbifold. Observe that the initial 64 _2×_2 fixed points split up into four fixed points, which are left invariant under the _3 action, and 60 fixed points, which are regrouped into _3-invariant triplets. This simple counting explains the 24 allowed deformations δ F_(1)^αδ F_(2)^βδ F_(3)^γ. Similar to the _2 ×_2 orbifold, the ε_αβγ-deformations depend on the twisted complex structure deformation parameters ε_αβ^(i), such that at most 24 conifold singularities remain upon deformation by ε_αβ^(i). 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:

    σ_ R: (x_i,v_i,y) ⟼ ( x_i,  v_i,  y).
 
This anti-holomorphic involution,  constructed from the involution σ defined in equation (<ref>) by choosing A=_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 A- and B-orientation for hexagonal lattices)
as discussed in section <ref> and ensures that the orientifold involution is an automorphism of the torus-lattices. The orientifold involution also acts on the deformation polynomials (<ref>) as follows:

    [                                                                     δ F_(1)^α↦δ F_(1)^α                                                                           ∀ α∈{1,2,3,4};                                                                   δ F_(j)^1 ↦δ F_(j)^1, δ F_(j)^2 ↦δ F_(j)^4,     δ F_(j)^3 ↦δ F_(j)^3,     δ F_(j)^4 ↦δ F_(j)^2   for  j=2,3
    , ]
 
which concurs with the -action on the _2 fixed points of the aAA lattice, whose individual two-torus positions are depicted in figure <ref>. 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_4^(1) and ε_5^(1), for which we can introduce ε_4|5^(1) = 1/2( ε_4+5^(1)± i ε_4-5^(1)) with ε_4±5^(1)∈. All the other deformation parameters  ε_λ=0,1,2,3^(1) and ε_α=1,2,3,4^(2,3) are constrained to be real. A summary of the independent deformation parameters for the orientifold T^6/(_2×_6 ×)
with discrete torsion and aAA lattice is given in table <ref>.


[ 5|c|Independent deformation parameters for T^6/(_2×_6 ×) with discrete torsion;                                                                         _2^(i)                                                                                                                                      Parameter identification                                                                Parameter range                                                   Exceptional wrapping numbers;                                                                                                                                                       ε_0^(1)                                                                       ε_11^(1)                                                                                                                                             (x_0^(1),y_0^(1));                                                                                                                                                       ε_1^(1)                                           ξ^2 ε_21^(1) =ε_31^(1) = ξ  ε_41^(1)                                                                                                                                             (x_1^(1),y_1^(1));                                                                         _2^(1)                                                                        ε_2^(1)                                           ξ^2 ε_12^(1) =ε_13^(1) = ξ  ε_14^(1)                                                                                                                                             (x_2^(1),y_2^(1));                                                                                                                                                       ε_3^(1)                                                  ε_33^(1) =ε_42^(1) = ε_24^(1)                                                                                                                                             (x_3^(1),y_3^(1));                                                                                                                                              ε_4^(1), ε_5^(1)                   ε_22^(1) =ε_34^(1) = ε_43^(1) =ε_44^(1) = ε_23^(1) =ε_32^(1)                                                                                                                            (x_4^(1),y_4^(1), x_5^(1),y_5^(1));                                                                         _2^(2)                                                                ε_α=1,2,3,4^(2)                                                 ε_α2^(2) = ε_α3^(2) = ε_α4^(2)                                                                                                                                             (x_α^(2),y_α^(2));                                                                         _2^(3)                                                                ε_α=1,2,3,4^(3)                                                 ε_α2^(3) = ε_α3^(3) = ε_α4^(3)                                                                                                                                             (x_α^(3),y_α^(3));                                                                                ]
DefParZ2Z6CompleteOverview of the independent deformation parameters per _2^(i)-twiste sector for  with discrete torsion (η = -1) and the restrictions following from the _3-action and the -action. The last column relates the deformation parameters to the relevant exceptional wrapping numbers introduced in equation (<ref>)
of section <ref>.


In conclusion, the T^6/(_2×_6 ×) orientifold with discrete torsion corresponds to the zero locus of the following polynomial f:

    f=    - y^2 +  v_1 x_1 (x_1^2 -  v_1^2) v_2 (x_2^3 - v_2^3) v_3 (x_3^3 - v_3^3) 
           - F_(1){ε_0^(1)· x_2 (x_2^3 - v_2^3) · x_3  (x_3^3 - v_3^3) + ε_1^(1)· 3 v_2^3 x_2  · x_3  (x_3^3 - v_3^3) .
                  + ε_2^(1)· x_2  (x_2^3 - v_2^3) · 3 v_3^3 x_3 + ε_3^(1)· 3 v_2^2 v_3^2 (v_2^2 v_3^2 + v_2 v_3 x_2 x_3 + x_2^2 x_3^2)
                 .+ ε_4+5^(1)· 3 v_2^2 v_3^2 (v_2 v_3 - x_2 x_3) (2 v_2 v_3 + x_2 x_3) + ε_4-5^(1)· 3 √(3) v_2^2 v_3^2 x_2 x_3 (v_2 v_3 - x_2 x_3 ) }
           - ∑_j,k∈{2,3},  j≠ k F_(j){ε_1^(j)· x_1^2 (x_1^2- v_1^2) · 3 v_k^4 +  ε_2^(j)· v_1 x_1 (x_1 + v_1)^2 · 3 v_k^4 . 
                         .+  ε_3^(j)· v_1^2  (x_1^2 - v_1^2) · 3 v_k^4 + ε_4^(j)· v_1 x_1  (x_1 - v_1)^2 · 3 v_k^4 } .

In this polynomial expression, one clearly notices the difference in the _2^(1) deformations on the one hand and the _2^(2,3) deformations on the other hand, in analogy with the difference in exceptional cycles from the respective _2^(i)-twisted sectors as reviewed in section <ref>. This will obviously imply that the effect of _2^(1) deformations on sLag three-cycles has to be studied separately from the effect of _2^(2,3) deformations. The latter deformations on the other hand are expected to follow a similar pattern due to the two-torus exchange symmetry , which is reflected in the coordinate permutation (x_2,v_2)↔ (x_3,v_3) accompanied by a permutation of the twisted parameters ε_1^(1)↔ε_2^(1).






 §.§ Deforming special Lagrangian cycles on T^6/(_2 ×_6 ×)

Having formulated the consistent hypersurface formalism to discuss deformations of  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 <ref> 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 (<ref>). Due to the additional _3-symmetry in equation (<ref>) 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×_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 _3 element ω^2 maps the O6-planes from the σ_ Rω^2-and σ_ Rω^4-sectors to this one.
In a coordinate patch where v_i≠0, we can use the (^⋆)^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 (<ref>). 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 <ref> 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 <ref> and three-dimensional subspaces on the x_i-planes. 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 <ref> and M_2, M_3 Lag lines from table <ref> (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.[The relative angles w.r.t. the - invariant plane can be inferred from the torus wrapping numbers: ϕ^(1) = arctan( m^1/n^1) determines the angle (mod π) on an A-type square T_(1)^2 and ϕ^(j) = arctan( √(3) m^j/2 n^j + m^j) on an
A-type hexagonal T_(j)^2 with j=2,3, and (n^i) encodes the orientation to arrive at (mod 2π).] The factorisable bulk three-cycles of section <ref> are then constructed from Lag lines which do not cross the _2^(i) fixed points, such as lines cI and cII on T_(1)^2 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 <ref>, 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 _2^(i) 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.

As a matter of fact, the interesting phenomena occur for exceptional and fractional three-cycles 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 _2^(i) fixed points for the various distinct deformation parameters. In the meantime, we develop a dictionary for the fractional three-cycles associated to the _2^(i)-twisted 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 _2^(i)-twisted sector as a direct product of a one-cycle on T_(i)^2 and a two-cycle on T_(i)^4/_2^(i) (or on Def(T_(i)^4/_2^(i)) in later sections). If we limit ourselves to the cycles aI, aII, aIII and aIV on T_(1)^2, the total number of two-cycles on T_(2)^2× T_(3)^2 based on table <ref> 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 T^4_(1)≡ T_(2)^2× T_(3)^2 under the _3-action. Taking into account the possibility of turning on discrete Wilson lines or discrete _2^(i)-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 <cit.>.

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 <ref> and have _2 fixed points as boundaries. For the aI, aII, bI^0 and bII^0 cycles one should imagine the point ϵ_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 _2^(1)-twisted sector, the one-cycle on T_(1)^2 has a topology of a circle S^1, while the two-cycle on T^4_(1)/_2^(1) corresponds to a two-torus pinched down at the boundary points, i.e. at the _2^(1) fixed points lying on the zero locus y=0. Hence, the topology of a fractional three-cycle is simply S^1× T^2/_2. A more pictorial representation will be given in
sections <ref> and <ref>. 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 T^4_(i)/_2^(i) separately from the integrals of Ω_1 over the one-cycles on T_(i)^2.



In the next two subsections we discuss the effects of deformations in the _2^(1)- and _2^(2,3)-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_(3)^2 it suffices to discuss only one of the _2^(2,3)-twisted sectors, as the other one will yield the same results. Hence, we can choose to focus on deformations in the _2^(3)-twisted sector.








  §.§.§ sLags in the deformed _2^(1)-twisted sector

For a qualitative appreciation of the deformation effects in the _2^(1)-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 <ref> their regional conditions are changed: for the Lag line bI^0 we find the constraint 0 ≤ v_i≤ 1, while the Lag line bII^0 consists of the union {-∞≤ v_i≤ 0}∪{ 1≤ v_i ≤ + ∞}. Figure <ref> 



depicts the T^2/_2 topology for the two-cycles constructed from bI^0 and bII^0 on Def(T_(1)^4/_2^(1)), with the blue-shaded regions representing sLag two-cycles calibrated with respect to Re(Ω_2). The blue contour-lines correspond to the zero locus y=0 in the -projected plane (v_2, v_3), and these lines intersect at the _2^(1) 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 -restricted plane (v_2, v_3), such that they are not depicted in figure <ref>.
The two-cycles bI^0 ×bI^0 and bII^0 ×bII^0 should be paired with a one-cycle aI or aIII on T_(1)^2 to form a sLag three-cycle calibrated with respect to Re(Ω_3). 
The white regions in figure <ref> 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 <ref> belong to the three-cycle type
aI×bI^0 ×bI^0, while the hidden stacks of the Pati-Salam (PS) II model of section <ref> and of the L-R symmetric II model of section <ref> 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 _2^(1) deformation parameter ε^(1)_λ separately in figure <ref>, 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 <ref>, which results from determining the singular points of the hypersurface equation (<ref>). At the deformed singularity, an exceptional cycle with non-vanishing volume emerges, which is indicated by a red dashed line in figure <ref>. An interesting observation is that the point (33) gets deformed too when turning on the deformation parameters ε_4+5^(1) and ε_4-5^(1). This phenomenon was also observed <cit.> for specific deformations of complex co-dimension 2 singularities on the subspace T^4_(i)/_2 of the orbifold T^6/(_2×_6') with discrete torsion and can be resolved by turning on a correction-term ε_3^(1) depending on the respective deformation parameter, as depicted in the lower diagrams of figures <ref> (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_3^(1)≠ 0. Further details about the counter-terms can be found in appendix <ref>.      


[ 4|c|Behaviour of _2^(i) fixed points under deformations of T^6/(_2×_6 ×) with discrete torsion;                                                  4|c|Deformations in the _2^(1)-twisted sector;                                                                                        modulus                                                                        deformed singular orbit                                                                      additional deformed orbit                                                                                displaced orbit;                                                                                        ε_0^(1)                                                                                  e_0^(1) ≡(11)                                                                                                                                                                             e_1^(1) , e_2^(1);                                                                                        ε_1^(1)                                                                      e_1^(1) ≡[(31),(41),(21)]                                                                                                                                                                     e_3^(1), e_4^(1), e_5^(1);                                                                                        ε_2^(1)                                                                      e_2^(1) ≡[(13),(12),(14)]                                                                                                                                                                     e_3^(1), e_4^(1), e_5^(1);                                                                                        ε_3^(1)                                                                      e_3^(1) ≡[(33),(42),(24)]                                                                             e_4^(1),   e_5^(1)                                                                                               ;                                                                     ε_4+5^(1)  &     ε_4-5^(1)                                          e_4^(1) ≡[(22),(34),(43)], e_5^(1)  ≡[(44),(32),(23)]                                                                                        e_3^(1)                                                                                               ;                                                  4|c|Deformations in the _2^(3)-twisted sector;                                                                                        ε_1^(3)                                                                      e_1^(3) ≡[(13),(14),(12)]                                                                                                                                                                              e_2^(3), e_4^(3);                                                                                        ε_2^(3)                                                                      e_2^(3) ≡[(23),(24),(22)]                                                                                                                                                                              e_1^(3), e_3^(3);                                                                                        ε_3^(3)                                                                      e_3^(3) ≡[(33),(34),(32)]                                                                                                                                                                              e_2^(3), e_4^(3);                                                                                        ε_4^(3)                                                                      e_4^(3) ≡[(43),(44),(42)]                                                                                                                                                                              e_1^(3), e_3^(3);                                                                                                ]
OverviewDeformperZ2SectorOverview of the deformed singular _3-orbits composed of the _2^(i) fixed points of  with discrete torsion per deformation parameter ε_λ=0,…,4± 5^(1) and ε_α=1,…,4^(3). The third column indicates which other singular orbits are deformed, while the last column lists which orbits remain singular but are displaced.
Figure <ref> provides a graphical representation of how the _2^(1) singularities (11), (13), (31) and (33) on T_(1)^4/_2^(1) behave under deformations. The other singular points belonging to the respective _3-invariant orbits are not depicted as they do not lie in the -restricted plane. To depict these latter singularities, one has to perform an appropriate _3-rotation on the coordinates. Similar considerations hold for the _2^(3) singularities depicted in figure <ref>.




For a more quantitative picture of the deformation effects, we go back to the coordinate patch in which v_i=1 
where we can rewrite the hypersurface equation in the vicinity of a singular point on T_(1)^4/_2^(1) as a ^2/_2 (or A_1-type) singularity.
In first instance, we look at the zero locus of the hypersurface equation (<ref>), turn on each _2^(1) 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_3^(1), which deforms the exceptional cycles ϵ_3^(1) and ϵ̃_3^(1) at the singularity (33). For the other deformations  (ε_0^(1), ε_1^(1), ε_2^(1)), we have to perform a Möbius transformation as explained in appendix <ref>, or a complex rotation (ε_4+5^(1), ε_4-5^(1)) to extract the proper local structure by placing the singularity at the point (33). 
More explicitly, there exists a Möbius transformation λ_3 <cit.> acting on the homogeneous coordinate x_i that allows to map the _2 fixed point ϵ_1 situated at the origin of a two-torus T^2_(i) to the fixed point ϵ_3 located on the real axis in the new coordinate x̃_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 _2 fixed points ϵ_2 and ϵ_4 on the other hand are mapped to the point ϵ_3 by a _3-transformation as can be seen in figure <ref> (b). 
With an appropriate rescaling of the homogeneous coordinates, we then find that the singularity is locally described by the following hypersurface equation:

    ỹ^2 = x̃_2 x̃_ 3 - c_λε_λ^(1),           c_a = {[          9   (λ = 0),;          3 (λ = 1,2),;          1   (λ = 3),;        1/2 (λ = 4±5). ].

A first observation is of course that the co-dimension two singularity locally takes the form of a ^2/_2 singularity in the (x̃_2, x̃_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_(3)^2, respectively, pass through the _2^(1) fixed points affected by a non-vanishing deformation parameter ε_λ=0,1,2,3^(1).  A full fractional three-cycle calibrated with respect to Re(Ω_3) is then constructed as described in section <ref> by combining these two-cycles with e.g. the one-cycle aI on T_(1)^2, such that the _2^(1) exceptional  contributions to a fractional three-cycle can be expressed in terms of the basis three-cycles (ϵ_λ^(1), ϵ̃_λ^(1)) ∼orbits of (π_1 ⊗ e_λ^(1), π_2 ⊗ e_λ^(1)) as follows: 

    Π^_2^(1) = (-)^τ^_2^(1)( ϵ_0^(1) + (-)^τ^2ϵ_1^(1) + (-)^τ^3ϵ_2^(1) +(-)^τ^2 + τ^3ϵ_3^(1)).

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)/_6, and it clear from this construction that the volume of an associated fractional three-cycle 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)/_6 can be represented by a set of 
real two-dimensional regions x̃_2 ·x̃_3 ≥ 0 
in the (x̃_2, x̃_3)-plane. When turning on a deformation parameter ε_λ=0,1,2,3^(1), the volume of these two-cycles will shrink or grow depending on the sign of the deformation parameter:


  * 
 ε_λ^(1) > 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_λ^(1) grows out of the singularity (33) in the region ỹ^2 <0. This situation is represented in the upper diagram of figure <ref> (b)-(c)-(d)-(e). 
The exceptional two-cycle satisfies the algebraic condition x̃_2 = x̃_3, by which the hypersurface equation (<ref>) reduces to the equation for a two-sphere S^2 with radius √(c_λε_λ^(1)). As the two-cycles bI^0 ×bI^0 and bII^0 ×bII^0 remain two separate sLags, the exceptional two-cycle e_λ^(1) 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_λ^(1) Re(Ω_2) of the exceptional two-cycle as a function of the parameter ε_λ^(1) in the hypersurface formalism and taking into account that ỹ^2 <0 and x̃_2 = x̃_3. Extending these considerations to the three-cycles on (T^2_(1)× T^4_(1)/_6)/_2^(3), 
we conclude that the exceptional three-cycle ϵ_λ^(1) 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^_2^(1) in order for the volume of the fractional cycle to decrease upon deformation of the singularity, in line with figure <ref> (e).

The cycles bI^0 ×bII^0 and bII^0 ×bI^0 on the other hand have merged into one big two-cycle 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_λ^(1) is eliminated, such that the union two-cycle remains sLag with respect to Im(Ω_2).   



  * ε_λ^(1) < 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 <ref> (e), while an exceptional two-cycle e_λ^(1) grows out of the singularity (33) in the region ỹ^2 >0. The hypersurface equation (<ref>) reproduces the topology of a S^2 for the algebraic condition x̃_2 = - x̃_3, which implies that the exceptional two-cycle e_λ^(1) 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_λ^(1) 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_λ^(1)|. Combining them with e.g. the one-cycle aII 
on T_(1)^2 allows for the construction of fractional three-cycles calibrated with respect to  Re(Ω_3), and bulk three-cycles parallel to the _2^(2)-and _2^(3)-plane, respectively. Their _2^(1) exceptional contributions Π^_2^(1) can thus be decomposed in terms of the basis three-cycles ϵ̃_λ^(1):

    Π^_2^(1) = (-)^τ^_2^(1)( ϵ̃_0^(1) + (-)^τ^2ϵ̃_1^(1) + (-)^τ^3ϵ̃_2^(1) +(-)^τ^2 + τ^3ϵ̃_3^(1)),

implying that the basis three-cycles ϵ̃_λ=0,1,2,3^(1) 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 <ref> (e), there should be a relative minus sign between Π^ bulk and the contribution to Π^_2^(1).  
For instance, for ε^(1)_3<0 equation (<ref>) describes a fractional three-cycle with τ^_2^(1) + τ^2 + τ^3 = 1 mod 2.

The situation for the deformations ε_4+5^(1) and ε_4-5^(1) 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^(1) or ε_4-5^(1), which explains the required non-vanishing correction term ε_3^(1)(ε_4+5^(1)) or ε_3^(1)(ε_4-5^(1)), respectively, as depicted in the lower diagrams of figures <ref> (f) and (g). A brief discussion on how to obtain these corrections terms is given in appendix <ref>. 
The (local) discussion of the singularities deformed by a non-vanishing parameter ε_4+5^(1) and ε_4-5^(1) follows the same pattern as the one conducted above for the other deformations ε^(1)_λ∈{0,1,2,3}. Nonetheless, there is an important 
difference, as the exceptional three-cycles (ϵ^(1)_4, ϵ̃^(1)_4 ) are not mapped to (linear combinations of) themselves under the orientifold projection, but to 
(linear combinations of) the three-cycles (ϵ^(1)_5, ϵ̃^(1)_5 ) and vice versa, as indicated in table <ref>. This implies that,
within the context of Type IIA/ orientifolds,  the _3-invariant orbits e_4^(1) and e_5^(1) from table <ref> are always deformed simultaneously for a single non-zero deformation ε^(1)_4+5 or ε^(1)_4-5.  


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 (<ref>) 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_(1)^4. Consistency with the plots in figure <ref> 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_λ^(1))≤ 0, which allows to define the integration domain for the volume of the respective exceptional three-cycles. Consider first the deformation ε_0^(1) of the complex co-dimension 2 singularity at the origin (11) on T_(1)^4/_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 _2-singularity on T^4/_2, implying that the _3-action does not influence the geometrical properties of this exceptional three-cycle. Depicting the volume of the exceptional cycle e_0^(1) as a function of the deformation parameter ε_0^(1) fully confirms this statement, as can be seen explicitly from the left plot of figure <ref>. For small deformations, the exceptional cycle volume exhibits a square-root like dependence on ε_0^(1), characteristic for deformed exceptional two-cycles on ^2/_2. For larger values of the deformation parameter, the exceptional cycle volume goes over into a more linear-like behaviour, before it evolves into a quadratic dependence for very large values of ε_0^(1), enforced by the topology of the ambient T^4.[The volumes of the exceptional cycle and the fractional three-cycles are normalised to the volume of the fractional cycle at the orbifold point, i.e. Vol(Π^ frac)=1 for vanishing deformation ε_λ^(1) with , throughout the paper. In this section we compute the volumes for the fractional cycles with bulk orbit parallel to aI×bI^0×bI^0, such that the integration contours lie completely along the real lines Re(x_i)≥ 1, with i∈{1,2,3}.]  The middle panel of figure <ref> shows the ε_0^(1)-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_- = 1/2( Π^ bulk - ϵ_0^(1)). On the right panel of figure <ref>, we depict the ε_0^(1)-dependence of the volume of the fractional three-cycle Π^ frac_+ = 1/2( Π^ bulk + ϵ_0^(1)), 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.



Next, we focus on the deformation ε_3^(1) for which it suffices to impose the exceptional cycle condition x_2 =  x_3 on equation (<ref>) to extract the real hypersurface equation describing the exceptional cycle volume. The points (33), (24) and (42) in the _3-invariant orbit e_3^(1) are simultaneously deformed for a non-vanishing ε_3^(1), such that the exceptional cycle consists initially of three distinct S^2's resolving each of the three _2^(1) singularities, as shown in the left plot of figure <ref>. 
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 <ref>, for which we observe a similar qualitative behaviour as for the exceptional cycle e_0^(1). More precisely, we notice a square-root type functional dependence of Vol(e_3^(1)) 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_0^(1) is the region of validity for the parameter ε_3^(1). For values of ε_3^(1)∼ 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 <ref>, 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^(1)≥ 0.37 in the left plot of figure <ref>. The other two plots in figure <ref> represent the (normalised) volumes of the fractional three-cycles Π^frac_± = 1/2( Π^ bulk±ϵ_3^(1)) as a function of ε_3^(1) 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_(3)^2. 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_3^(1) 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 <ref>, 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 <cit.>.     








Discussing the global aspects of the exceptional cycle e_1^(1) on T^4_(1)/_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_0^(1) and e_3^(1) allows us, nonetheless, to express the functional dependence of e_1^(1) on the deformation parameter ε_1^(1) through a small detour: we first compute the normalised volume of the bulk cycle aI× C^0×bIII^0 as a function of the deformation parameter ε_1^(1) and then subtract the normalised volume of the fractional cycle with integration contours completely along the real lines Re(x_i=1,2,3)≥ 1. The result of that computation is depicted in the left panel of figure <ref>, from which we can extract the square-root like functional dependence Vol_ norm(e_1^(1))∼√(ε_1^(1)).  
The plot does not contain information about a potential quadratic dependence on ε_1^(1) for large deformations, as was the case for the exceptional cycles e_0^(1) and e_3^(1). 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 _2^(1) singularities (31), (21) and (41) for non-vanishing ε_1^(1), which merge together for larger deformation parameters analogously to the behaviour of the deformed exceptional cycle e_3^(1) depicted in figure <ref>. In this respect, the _3-action and the T^4 topology do qualitatively constrain the behaviour of the exceptional cycle e_1^(1), even though their full effects cannot be extracted more quantitatively due to an indisputable imaginary component of the hypersurface equation for the exceptional cycle. 
The functional dependence of the (normalised) volumes for the fractional three-cycles Π^ frac_± = 1/2( Π^ bulk±ϵ_1^(1)) is given in the middle and right plot of figure <ref>, respectively. As expected, the volume of the fractional cycle Π^ frac_- shrinks with growing deformation ε_1^(1), while the volume of Π^ frac_+ grows with increasing deformation ε_1^(1). Due to the exchange symmetry T_(2)^2 ↔ T_(3)^2, the discussion of the global description of the exceptional cycle e_2^(1) is completely analogous to the one for e_1^(1).  



This brings us finally to the global description of the exceptional cycles e_4^(1) and e_5^(1) on T^4_(1)/_6, which are related to each other through the orientifold projection. In the hypersurface equation (<ref>) this relation under the -projection is manifestly built in, such that a non-zero deformation parameter ε_4+5^(1) resolves both e_4^(1) and e_5^(1) simultaneously and similarly for a non-zero deformation parameter ε_4-5^(1). To extract the hypersurface equation for the exceptional cycles, we have to rotate the x_i=2,3-coordinates over an angle ξ or ξ^2 (or use the Möbius transformation λ_2 or λ_4), after which we can impose the algebraic condition x_2 = ± x_3. Unfortunately, the resulting hypersurface equation does not correspond to a fixed set under the orientifold involution, which is manifested by a purely imaginary contribution to the hypersurface equation. Hence, similarly to the exceptional cycle e_1^(1), we are not able to directly access the exceptional cycles e_4^(1) and e_5^(1).[One can, however, focus on the real part of the hypersurface equation for e_4^(1) and e_5^(1) and compute the volume as a function of the respective deformation parameter. This offers a qualitative understanding of the geometry of e_4^(1) and e_5^(1) and shows that these orbits have a similar behaviour under deformation as the orbit e_3^(1): 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_3^(1), e_4^(1) and e_5^(1) on the parent toroidal orbifold T^4_(1)/_6.]  The situation is even more complicated in this case as the fractional three-cycles wrapping one or more of the _2^(1)-fixed points in the orbits e_4^(1) and e_5^(1) do not lie along the real axes in the x_2- and x_3-planes, such that we are not able to directly compute the fractional cycle volume as a function of ε_4+5^(1) or ε_4-5^(1) either. To understand the impact of the deformation ε_4+5^(1) on the volume of a fractional cycle, one first has to 
establish that the resolved orbits e_4^(1) and e_5^(1) on the parent toroidal orbifold T^6/(_2×_6) with discrete torsion have the exact same structure as the resolved orbit e_3^(1) discussed above, upon respectively considering non-zero complex deformation parameters ε_4^(1) and ε_5^(1) individually. Taking afterwards the orientifold projection into account  implies - based on the calibration properties with respect to the volume three-form Ω_3 - that the exceptional three-cycles ϵ_4^(1) + ϵ_5^(1) and ϵ̃_4^(1) + ϵ̃_5^(1) are resolved by a non-zero deformation parameter ε_4+5^(1), while the exceptional three-cycles ϵ_4^(1) - ϵ_5^(1) and ϵ̃_4^(1) - ϵ̃_5^(1) are resolved by a non-zero deformation parameter ε_4-5^(1). To assess the impact of the deformation ε_4+5^(1) on the volume of a fractional cycle wrapping _2^(1) singularities in the orbits e_4^(1) and e_5^(1), we exploit our intuition obtained from the other deformations in the _2^(1)-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 <ref>, as a function of the deformation parameter ε_4+5^(1);

  * Consider the volume of a single two-sphere S^2 obtained by deforming the exceptional cycle e_3^(1), as presented in the left panel of figure <ref>, and re-interpret[This identification of the exceptional cycle volumes is supported by the _3-symmetry among the orbits e_3^(1), e_4^(1) and e_5^(1) on the ambient toroidal orbifold.] this volume as the volume of the resolved exceptional two-cycle in the _3- and -invariant exceptional three-cycle ϵ_4^(1)+ϵ_5^(1);

  * 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:

    Vol_ norm(Π^ frac_±) =  Vol_ norm(Π^ bulk) ± Vol_ norm(ϵ_4^(1)+ϵ_5^(1)).

     
The proposed method does not allow us to obtain any quantitative information about the fractional cycle volume for a given deformation ε_4+5^(1)≠ 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 <ref>. There, we see that the volumes of the fractional three-cycles Π_±^ frac exhibit the expected behaviour under deformation: the volume Vol(Π_-^ frac) decreases with growing ε_4+5^(1), while the volume Vol(Π_+^ frac) increases for growing deformation ε_4+5^(1). The numerical noise for large deformations (ε_4+5^(1)≥ 0.37) is a reflection of the merging of the two-spheres S^2 at the resolved singularities into one large exceptional two-cycle.  






One can easily repeat this method for fractional three-cycles wrapping any of the other _2^(1) singularities in the orbits e_4^(1) and e_5^(1), or apply this method to probe the effect of a non-vanishing deformation ε_4-5^(1) 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 <ref>, namely Vol(Π^ frac_±) ∼ Vol(Π^ bulk) ±√(ε_i^(1)) with i=4+5 or i=4-5. 







  §.§.§ sLags in the deformed _2^(3)-twisted sector

To investigate the deformation effects in the  _2^(3)-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 <ref>. 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_(1)^2 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/_2 on Def(T_(3)^4/_2^(3)) 
and calibrated with respect to Re(Ω_2), represented by the blue-coloured regions in figure (<ref>) (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 -projected (x_1, x_2) plane represent the zero locus y=0 and intersect at the real _2^(3) fixed points (23), (33) and (43). In order to obtain fractional three-cycles calibrated w.r.t. Re(Ω_3) on (T^4/_2^(3)× T^2_(3))/_6, the two-cycles calibrated w.r.t. Re(Ω_2) and Im(Ω_2)
on T^4_(3)/_2^(3) should be paired with a one-cycle bI^0/bIII^0 and bII^0/bIV^0, respectively, on T_(3)^2.





By turning on the _2^(3) deformation parameters ε_α^(3) one by one, figure <ref> shows exactly which singularities (or singular orbits under the _3 ⊂_6 symmetry) are deformed and which singularities are displaced, in agreement with the lower part of table <ref>. 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 <ref>. We can study the effects of the deformation parameters more qualitatively by studying the zero locus of the hypersurface equation (<ref>) in a local patch around the singular point (23), for which the hypersurface equation reduces locally to the form (after rescalings):

    ỹ^2 = x̃_1 x̃_2 - 2 ε_α^(3),         α = 1,2,3,4.

For the deformations ε_1^(3), ε_3^(3) and ε_4^(3) we have to perform an appropriate Möbius transformation from appendix <ref> to mould the hypersurface equation (<ref>) into this specific form, corresponding locally to a ^2/_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_(1)^2× T_(2)^2, respectively. Combining for instance the first two-cycle  aI×bI^0
with a one-cycle bI^0 on T_(3)^2 yields a fractional three-cycle as defined in section <ref>, calibrated w.r.t. Re(Ω_3). Its overall _2^(3) exceptional three-cycle is given  in terms of the basis exceptional three-cycles as:

    Π^_2^(3) = (-)^τ^_2^(3)( (-)^τ^3ϵ_1^(3) + (-)^τ^2 + τ^3ϵ_2^(3)).
 
By turning on the deformation parameter ε_2^(3), an exceptional two-cycle e_2^(3) with non-vanishing volume grows out of the singular point (23) on T^4_(3)/_2^(3), 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:
 

  * 
 ε_2^(3)>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 region ỹ^2 <0, as depicted in the upper diagram of figure <ref> (b). The local hypersurface equation (<ref>) reduces to the equation of a two sphere S^2 with radius √(2 ε_2^(3)), when we impose the algebraic condition x̃_1 = x̃_2 for the exceptional cycle. The exceptional two-cycle e_2^(3) 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 (<ref>), we find that the exceptional three-cycle ϵ_2^(3) 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^_2^(3), i.e. (-)^τ^_2^(3) + 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^(3) has been eliminated. 

  * ε_2^(3)<0: we observe the opposite picture for negative deformations, namely the two-cycles aI×bI^0 and aIV×bII^0 have merged together to one big two-cycle as shown in the lower diagram of figure <ref> (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^(3) is now calibrated by the same two-form as both two-cycles. The S^2 topology of the exceptional cycle with radius √(-2ε_2^(3)) follows from equation (<ref>) by restricting to the slice x̃_1 = - x̃_̃2̃. Combining the two-cycles with the one-cycle bII^0 on T_(3)^2 allows for the construction of fractional three-cycles, whose volumes are now decreasing for increasing |ε_2^(3)|. This implies a relative minus sign between Π^ bulk and 
the contribution of ϵ_2^(3)
to Π^_2^(3) 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 (<ref>) 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 _2^(3) deformations globally is an immediate consequence of the different complex structures on T_(1)^2 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_2^(3) by using the following strategy: compute the (normalised) volume of the bulk three-cycle aIII× C^0×bI^0 under non-vanishing deformation ε_2^(3) and subtract it from the (normalised) volume of the fractional three-cycle with integration contours completely along the real lines Re(x_i)≥ 1 in the complex x_i=1,2,3-plane. The result of this computation is shown in the left panel of figure <ref> and exhibits a square-root like dependence on the parameter ε_2^(3) for the (normalised) exceptional cycle volume. The behaviour of the (normalised) fractional three-cycle volumes Vol(Π^ frac_±) under non-zero deformation ε_2^(3) is shown in the middle and right panel of figure <ref>. As expected, the three-cycle Π^ frac_- is characterised by a shrinking volume for increasing ε_2^(3), while the volume of the three-cycle Π^ frac_+ increases for growing ε_2^(3). Given that the other (_3-orbits of)
_2^(3) exceptional divisors are related to e_2^(3) by virtue of a Möbius transformation, the volumes of the other exceptional cycles reproduce the same structure under their respective deformation as the one presented in figure <ref>.     





Due to the exchange symmetry T_(2)^2↔ T_(3)^2, we can straightforwardly transpose the entire analysis into the _2^(2) sector, where the same conclusions can be drawn for the _2^(2) exceptional and fractional three-cycle volumes. One subtle difference arises between the _2^(2)- and _2^(3)-twisted sector when the choice of the exotic O6-plane is taken into account. If either the _2^(2)- or _2^(3)-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 -even and -odd cycles for both sectors, according to table <ref>, 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 <ref> 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 <ref>, the different types of three-cycles on the T^6/(_2 ×_6 ×) orientifold with discrete torsion were briefly reviewed.   
While a D6-brane a by itself wraps a fractional three-cycle Π_a of the form (<ref>), it is generically accompanied by its orientifold image a' with associated three-cycle Π_a'. The global model remains inert under the exchange  when simultaneously changing the gauge representations for their conjugates, e.g. .
The scalar potential only depends on the sum of the two <cit.>,


    V_scalar^NS-NS⊃∑_a N_a [ Vol(Π_a) +  Vol(Π_a') ] - Vol(Π_O6)
      {[                        =0 all D6_a-branes are  sLag;                       > 0                      else ].
    ,

which leads to the following qualitative situations observed first in the context of  and  models 
with discrete torsion in <cit.>, see also <cit.>:


  * The D6-brane a couples to the _2^(i)-twisted deformation modulus ζ^(i)_α such that the sLag condition is violated for a non-vanishing
vev ⟨ζ^(i)_α⟩∼√(ε^(i)_α).
 The deformation modulus ζ^(i)_α itself can be seen as the period associated to the _2^(i) exceptional three-cycle δ̃_α^(i):

    ζ^(i)_α = ∫_δ̃_α^(i)Ω_3,
 
where the Calabi-Yau three-form is defined in equation (<ref>) and the three-cycle δ̃_α^(i) is an -odd linear combination of the exceptional three-cycles (ϵ_α^(i),ϵ̃_α^(i)) in line with table <ref>.

  * The D6-brane a couples to the _2^(i)-twisted deformation modulus ζ^(i)_α and stays sLag for arbitrary values of ⟨ζ^(i)_α⟩∼√(ε^(i)_α).  In this case the deformation modulus ζ^(i)_α corresponds to the period associated to an -even linear combination of the  exceptional three-cycles (ϵ_α^(i),ϵ̃_α^(i)) following table <ref>. 

  * The D6-brane a does not couple directly to the _2^(i)-twisted deformation modulus ζ^(i)_α.

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 <ref> for T^6/(_2 ×_6 ×) with discrete torsion.[
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 volume-form as the bulk three-cycle (and thereby also the orientifold fixed planes). According to equation (<ref>) 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 FI-parameter 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(Ω_3), the positive contribution to the NS-NS scalar potential will scale as ( ∫_Π_a Im(Ω_3) )^2 for small deformations, which is understood from a four-dimensional perspective as (the square of) a FI-parameter <cit.>.]
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.[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^(i)_α⟩, but due to the 
 form of the scalar potential V_scalar∼∑_z D_z^2, such a vev can only atone for 
 the stabilisation of the deformation modulus if the two gauge factors have equal rank, i.e. instead 
 a gauge symmetry breaking SU(N) × SU(N) ⟨(,)⟩ or ⟨(,)⟩≠ 0⟶ SU(N)_diag
 occurs <cit.>. ]
 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 (<ref>) 
possesses a flat direction in the deformation modulus ζ^(i)_α if for all D6-brane stacks in a given global model the second or third case applies,
as we will first demonstrate further in a global model with USp(2)^4 gauge group in section <ref> .


The four-dimensional gauge couplings at tree-level are obtained by dimensionally reducing the (6+1)-dimensional Dirac-Born-Infeld (DBI) action along 
the compact cycle Π_a and its orientifold image Π_a' of the D6_a-brane worldvolume <cit.>,

    4π/g_a,tree^2 = 1/16 √(2) k_aM_Planck/M_string Vol(Π_a + Π_a')/√(Vol_6)  and  k_a = {[            1      SU(N_a);            2 SO/USp(2N_a) ].
    ,

with the toroidal three-cycle volume at the orbifold point of the aAA lattice of  given by:

    Vol(Π_a)/√(Vol_6)|_orb =
    Vol(Π_a')/√(Vol_6)|_orb =   ∏_i=1^3 L_a^(i) = 
     √(R_1^(1)/R_2^(1) (n^1)^2 + R_2^(1)/R_1^(1) (m^1)^2)∏_i=2^3 √(2/√(3)((n^i)^2 + n^i m^i + (m^i)^2 ) )
    
    =   √(R_1^(1)/R_2^(1))×{[      2/√(3)        a ||;      2 √(3) a || _2^(1) ].
    ,

where R_i^(1) 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 (<ref>).


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 non-perturbative) backreactions of the deformation on the toroidal cycles.


In sections <ref>, <ref> and <ref>, 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/(_2 ×_6 ×) orientifold with discrete torsion, aAA lattice background and the choice of _2^(3) as the exotic O6-plane orbit, 
the orientifold-even and -odd wrapping numbers  are obtained from

    ,

and

    ,

where (P_a,Q_a,U_a,V_a) correspond to the four different products of toroidal wrapping numbers appearing in equation (<ref>) and (x^(i)_α,a, y^(i)_α,a) to the exceptional wrapping numbers in equation (<ref>).
The latter have also been associated to the deformation parameters ε^(i)_α in table <ref>.  



At the singular orbifold point, only relative _2 ×_2 eigenvalues among pairs of D6-brane stacks a and b are of physical importance; e.g.
for the counting of chiral fermions the intersection numbers 
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 (_2^(1), _2^(2), _2^(3)) eigenvalues (+++).
The details of a prototype MSSM example, four different L-R symmetric examples and two PS examples are provided in sections <ref>, <ref> and <ref>, respectively. 
The D6-brane models presented in these sections form prototypes for the respective supersymmetric extensions of the Standard Model with three chiral generations of quarks and leptons and a minimal amount of (chiral) exotic matter (in the adjoint, symmetric and/or antisymmetric representation under the strong and weak interactions) constructed with ϱ-independent - i.e. all D6-branes along the one-cycle π_1 of T^2_(1) -  supersymmetric sLags at the orbifold point of T^6/(_2 ×_6 ×) 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 _2^(i) 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 ϱ-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 - or the _2^(1)-invariant (orbit of) O6-plane(s), and orientifold image branes
- such as a and a'- thus only differ in their _2 ×_2 eigenvalues, with the precise relation depending on the discrete Wilson lines and displacements (τ⃗,σ⃗), 
see table <ref> for a compact summary.



Further anticipating 
the decomposition of bulk and exceptional wrapping numbers into orientifold-even and -odd parts as summarised in table <ref>,



we can summarise the counting of stabilised moduli per model as well as flat directions affecting the gauge couplings according to the relation (<ref>)
in table <ref>.




Details of deformations of the global MSSM, L-R symmetric and PS models will be discussed in sections <ref>, <ref> and <ref>, respectively.





 §.§ Deformations in a global USp(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 USp(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 <cit.> for the choice of the _2^(3)-orbit
as exotic O6-plane, the bulk part of each fractional three-cycle has to be either parallel to the - or the _2^(1)-invariant orbit.
In either case, only gauge group enhancement of the type U(N) ↪ USp(2N) occurs for arbitrary choices
of _2 ×_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 <cit.> for details.
With the discrete Wilson lines and displacements (2τ^2τ^3;2σ^2σ^3) along T^4_(1) listed as lower index,
such three-cycles have the form:

    Π^|| _(201;211) =    ρ_1/4 
    + (-1)^τ^_2^(1)/4(ϵ^(1)_5  - ϵ^(1)_4  )
     + (-1)^τ^_2^(2)/4(  [ ϵ^(2)_κ_1  -2  ϵ̃^(2)_κ_1 ]
    + (-1)^τ^1   [ ϵ^(2)_κ_2  -2  ϵ̃^(2)_κ_2 ])
     - (-1)^τ^_2^(3)/4(  ϵ^(3)_κ_1   + (-1)^τ^1 ϵ^(3)_κ_2)
     ,
    
    Π^|| _2^(1)_(21τ^3;210) =   3  ρ_1/4 
    + (-1)^τ^_2^(1)+τ^3/4(  ϵ^(1)_4 - ϵ^(1)_5 )
    -  (-1)^τ^_2^(2)+τ^3/4(
     [ϵ^(2)_κ_1 - 2 ϵ̃^(2)_κ_1 ]
    + (-1)^τ^1 [ϵ^(2)_κ_2 - 2 ϵ̃^(2)_κ_2 ]
    )
    - 3   (-1)^τ^_2^(3)/4(  ϵ^(3)_κ_1 +(-1)^τ^1 ϵ^(3)_κ_2)
    ,


and the remaining two possibilities of gauge group enhancement, Π^|| _(2τ^21;201) with arbitary discrete Wilson line 
 and Π^|| _2^(1)_(210;211), 
are obtained from those by choosing identical values of (τ⃗;σ⃗) along (T^2)^3 and replacing the _2 ×_2 eigenvalues
(τ⃗^_2) as follows:

    (τ^_2^(1)+τ^2+1, τ^_2^(2) ,τ^_2^(3)+τ^2+1 )^|| _(2τ^21;201)   ↔ (τ⃗^_2)^|| _(201;211)
    ,
     
    
    (τ^_2^(1)+ τ^3 + 1, τ^_2^(2) + τ^3 +1 , τ^_2^(3) )^|| _2^(1)_(21τ^3;210)   ↔ (τ⃗^_2)^|| _2^(1)_(210;211)
    .


Π^|| _(201;211) with the choice of _2 ×_2 eigenvalues (+++) and discrete data τ^1=0=σ^1 on T^2_(1)
only differs from the QCD stack a of all phenomenologically appealing models discussed later on in this article
in the choice of one discrete Wilson line, τ^2, cf. e.g. the MSSM D6-brane configuration in table <ref>. 
Π^|| _2^(1)_(21τ^3;210) with the choice τ^3=0 on the other hand corresponds to the left- and right-symmetric
D6-branes b and c, respectively, of all L-R symmetric and 
the PS prototype II model, cf. tables <ref> and <ref>, 
for the choice of _2 ×_2 eigenvalues (+++) and (-+-), respectively.
Moreover,  Π^|| _2^(1)_(21τ^3;210) with the choice τ^3=0 corresponds to
stack b (and c) of the MSSM (and the PS prototype I), cf. table <ref> (and <ref>),
for a different choice of _2 ×_2 eigenvalues (–+) (and (+–)).

The D6-brane data of a global model with USp(2)^4 gauge group enhancement
satisfying all bulk and twisted RR tadpole cancellation conditions[Notice that the K-theory constraints are trivially satisfied in this model.] 
is presented in table <ref>,


[ 7|c|D6-brane configuration of a global USp(2)^4 model on the aAA lattice of T^6/(_2 ×_6 ×)_η=-1;                                                                                                                                                                                wrapping numbers                                                                                         Angle/π                                                                              _2^(i) eigenvalues                                                                                            (τ⃗)                                                                                            (σ⃗)                                                                                     gauge group;                                                                                               a                                                                                   (1,0;1,0;1,0)                                                                                         (0,0,0)                                                                                           (+++)                                                                                         (0,0,1)                                                                                         (0,1,1)                                                                                          USp(2);                                                                                               d                                                                                   (1,0;1,0;1,0)                                                                                         (0,0,0)                                                                                           (-+-)                                                                                         (0,0,1)                                                                                         (0,1,1)                                                                                          USp(2);                                                                                               b                                                                                 (1,0;-1,2;1,-2)                                                                                    (0,1/2,-1/2)                                                                                           (+++)                                                                                         (0,1,0)                                                                                         (0,1,0)                                                                                          USp(2);                                                                                               c                                                                                 (1,0;-1,2;1,-2)                                                                                   (0, 1/2,-1/2)                                                                                           (-+-)                                                                                         (0,1,0)                                                                                         (0,1,0)                                                                                          USp(2);                                                                                                 ]
4stackUSp2-4D6-brane configuration of a global model on the aAA lattice of T^6/(_2 ×_6 ×) with discrete torsion and 
with _2^(3) as the exotic O6-plane orbit. The D6-branes b and c are chosen identical to all L-R symmetric models and the PS II model in tables <ref> and <ref>, respectively.

and the non-vanishing bulk and exceptional wrapping numbers are displayed here in table <ref> for convenience and for comparison with those of the phenomenologically interesting models in table <ref>.



They are obtained from the expansion of the following fractional three-cycles:

    Π_a,d =   ρ_1/4∓1/4(ϵ^(1)_4  - ϵ^(1)_5  )
     + 1/4(  [ ϵ^(2)_1  -2  ϵ̃^(2)_1 ] +   [ ϵ^(2)_2  -2  ϵ̃^(2)_2 ])
     ∓1/4( ϵ^(3)_1  + ϵ^(3)_2)
     ,
    
    Π_b,c =   3  ρ_1/4±1/4(  ϵ^(1)_4 - ϵ^(1)_5 )
     -  1/4( [ϵ^(2)_1 - 2 ϵ̃^(2)_1 ] +  [ϵ^(2)_2 - 2 ϵ̃^(2)_2 ] )
    ∓3/4(  ϵ^(3)_1 +   ϵ^(3)_2)
    .

The massless matter spectrum of this D6-brane configuration is displayed in table <ref> 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 <ref>.




Here, it is noteworthy that at the orbifold point only the relative _2 ×_2 eigenvalues among the D6-branes a, b, c and d are of physical relevance.
The absolute _2 ×_2 eigenvalues in all three _2^(i)-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)_1,z·√(ε^(2)_1) with the absolute sign of y^(2)_1,z =± 2 specified in table <ref> for the choice of discrete D6-brane data in table <ref>.
Exchanging (+++) ↔ (-+-) in the D6-brane configuration of  table <ref> will obviously only pairwise permute the D6-brane labels a ↔ d and b ↔ c,
while the sign flip (+++)(-+-)↔(–+)(+–) provides a physically distinct model once deformations along the directions of non-vanishing exceptional wrapping numbers
listed in table <ref> are switched on. The  D6-branes b and c in this latter case agree with those of the PS prototype I model specified in table <ref>,
whose b-brane in turn agrees with the MSSM stack b of the weak interactions in table <ref>.

Since the non-vanishing wrapping numbers in  table <ref> are only of orientifold-even type - as expected for a model with just USp-gauge factors -
each gauge coupling feels flat directions along the untwisted complex structure ϱ of the a-type two-torus T^2_(1) and the 
twisted complex strucuture moduli ζ^(i)_α associated to the deformation parameters √(ε^(i)_α)∼⟨ζ ^(i)_α⟩
with (i,α) ∈{(1, 4-5),(2,1),(2,2),(3,1),(3,2) }.
All other twisted complex structure moduli only affect the gauge couplings through higher-order, e.g. field redefinitions at loop level, or non-perturbative effects.


It is also important to stress that, while the {b,c} sector of the USp(2)^4 model locally agrees with the L-R symmetric models in section <ref>
and the PS II model  in section <ref> (or the PS I model in section <ref> upon flipping the _2^(1)×_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 _2^(3)-twisted deformation moduli ζ^(3)_1,2, which leads to their stabilisation, i.e.
ε^(3)_1 = 0 = ε^(3)_2. Except for the L-R symmetric model of prototype I, the spectrum does not contain any other U(3) gauge factor, and even in the L-R symmetric prototype I
 the two hidden stacks with U(3)_h_1× U(3)_h_2 symmetry do not possess orientifold-odd wrapping numbers along the directions ζ^(3)_1,2, as can be read off from table <ref>. 
 In all phenomenologically appealing prototypes presented in this article, the Fayet-Iliopoulos term(s) generated by some non-vanishing deformation parameter(s) ε^(3)_1,2 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^(1)_3 and ε^(1)_4+5, from the _2^(1)-twisted sector.
Further stabilisations of deformation moduli involving the remaining stacks with unitary gauge groups will be discussed below on a model-by-model basis.
As can be read off from the summary in table <ref>, all phenomenologically appealing examples share the property that the D6-brane b of weak interactions – and in all L-R symmetric and PS models also the right-symmetric D6-brane c – experience a flat direction in the deformation parameter ε^(1)_4-5 changing the value of the respective gauge coupling(s).



The wrapping numbers and discrete displacements of the D6-brane configuration in table <ref> indicate that the branes a,d are represented by the three-cycle aI×bIII^0×bIII^0 and the branes b,c by the three-cycle aI×bIV^0×bII^0 in the hypersurface formalism, following the dictionary at the singular orbifold point in section <ref>. The deformation ε_4-5^(1) in the _2^(1)-twisted sector allows for resolved exceptional three-cycles calibrated with respect to the same Re(Ω_3) as the bulk parts of the four fractional three-cycles in the parameter region ε_4-5^(1)≤ 0, and similarly the deformations ε_1,2^(i) in the _2^(i=2,3)-twisted sectors yield correctly calibrated (resolved) exceptional three-cycles in the parameter regions ε_1,2^(i)≤ 0. In both cases, this can be verified by applying the Möbius transformation λ_4 on the cycles bIII^0 and bIV^0, after which one can study the (-projected) countour plots that form the equivalent to figures <ref> and <ref> in section <ref>. To obtain the functional behaviour of the fractional three-cycle volumes under the deformations, we first determine which bulk three-cycles and which exceptional cycles at _2^(i) singularities are wrapped by the fractional three-cycles in table <ref>, cf. equation (<ref>). Then, we reconstruct the (normalised) volume of the fractional three-cycles using the results for the exceptional three-cycle volumes from sections <ref> and <ref> as primary building blocks, in the same spirit as the method presented on page It:Method. 
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.







The results of these computations are presented in figures <ref> and <ref> for branes {a,d} and {b,c}, respectively, for
 the deformation parameter ε_4-5^(1) in the _2^(1)-twisted sector and in figure <ref> for all branes {a,b,c,d} for 
  the deformation parameter ε_2^(2) in the _2^(2)-twisted sector. Through the relationship (<ref>) between the tree-level gauge coupling of the gauge theory living on a D6-brane and the volume of the three-cycle wrapped by the D6-brane internally, we can deduce the qualitative behaviour of the gauge coupling when going away from the singular orbifold point by switching on deformations along flat directions. More explicitly, in this toy model we observe in figures <ref> and <ref> that the volumes of the fractional cycles wrapped by branes a and c decrease for increasing deformation parameter |ε^(1)_4-5| in the _2^(1)-twisted sector, implying that the gauge couplings of their respective gauge theories increase when going away from the orbifold singular point. The gauge couplings of branes b and d on the other hand are weaker at points in the moduli space away from the orbifold point, as the fractional cycle volumes wrapped by branes b and d grow for non-zero deformation along the flat direction in the _2^(1)-twisted sector, as depicted in figures <ref> and <ref>. For the _2^(2) deformations, we observe in figure <ref> that branes a and d obtain a stronger gauge coupling, whereas branes b and c have a weaker gauge coupling away from the singular  orbifold point. The computations for the deformation parameter ε_1^(2) in the _2^(2)-twisted sector follow the same logic and reproduce the exactly same respective functional dependence as in figures <ref> for the various D6-brane volumes, as can be inferred from the exceptional wrapping numbers in table <ref>.



The deformations in the _2^(3)-twisted sector require the exactly same method as the one used for the _2^(2)-twisted sector, which leads (qualitatively) to similar functional dependences of the fractional cycle volumes on the parameters ε_1,2^(3) as presented in figure <ref>: the volumes of branes a and b exhibit the functional behaviour of the left panel, while the volumes of the branes c and d increase for growing deformations |ε_1,2^(3)| as in the middle and right panel of figure <ref>. Hence, the gauge groups USp(2)_a and USp(2)_b acquire a stronger gauge coupling at the string scale M_string, whereas the gauge groups USp(2)_c and USp(2)_d a weaker gauge coupling under deformations along the two flat directions in the _2^(3)-twisted sector.








 §.§ Deformations in a global MSSM model


In <cit.>, we had performed a systematic search for MSSM-like models on T^6/(_2 ×_6 ×) with discrete torsion
and found that there is a unique choice of bulk parts supporting the
Standard Model gauge group SU(3)_a × SU(2)_b × U(1)_Y ⊂ U(3)_a × USp(2)_b × U(1)_c × U(1)_d, and that all seemingly different 
choices of discrete (Wilson line, displacement and _2 ×_2 eigenvalue)
parameters lead to the same massless matter spectrum when completing to a global five-stack model with one `hidden' stack, see the discussion in section 4.1 of <cit.> for details.
The D6-brane configuration displayed in table <ref> agrees with the previous example from <cit.> in all relative _2 ×_2
eigenvalues and absolute values of Wilson line and displacement parameters and is thus identical at the orbifold point. For the sake of comparing the different phenomenologically appealing models, we choose here, however, the absolute _2 ×_2 eigenvalues (+++) for the (up to rank) common QCD stack a.


[ 7|c|D6-brane configuration of a global 5-stack MSSM model on the aAA lattice;                                                                                                                                          wrapping numbers                                                                      Angle/π                                                           _2^(i) eigenvalues                                                                         (τ⃗)                                                                         (σ⃗)                                                                  gauge group;                                                                            a                                                                (1,0;1,0;1,0)                                                                      (0,0,0)                                                                        (+++)                                                                      (0,1,1)                                                                      (0,1,1)                                                                         U(3);                                                                            b                                                              (1,0;-1,2;1,-2)                                                                 (0,1/2,-1/2)                                                                         (–+)                                                                      (0,1,0)                                                                      (0,1,0)                                                                       USp(2);                                                                            c                                                              (1,0;-1,2;1,-2)                                                                (0, 1/2,-1/2)                                                                        (-+-)                                                                      (0,1,1)                                                                      (0,1,1)                                                                         U(1);                                                                            d                                                              (1,0;-1,2;1,-2)                                                                (0, 1/2,-1/2)                                                                         (+–)                                                                      (0,0,1)                                                                      (0,0,1)                                                                         U(1);                                                                            h                                                                (1,0;1,0;1,0)                                                                      (0,0,0)                                                                         (–+)                                                                      (0,1,1)                                                                      (0,1,1)                                                                         U(4);                                                                              ]
5stackMSSMaAAPrototypeID6-brane configuration of a global five-stack model with gauge group SU(3)_a × SU(2)_b × U(1)_Y × SU(4)_h ×_3 after Green-Schwarz mechanism, which leads to the orientifold-even and -odd wrapping numbers labelled by MSSM in table <ref>.

The massless open string spectrum of this MSSM example is for convenience displayed in table <ref>.


[ 8|c|Overview of the massless matter spectrum of a global 5-stack MSSM on the aAA lattice of  T^6/(_2 ×_6 ×);                                                                                                      sector                                                                    (U(3)_a ×USp(2)_b ×U(4)_h)_U(1)_c×U(1)_d                                                                                                         Q_Y                                                                                                          _3                                                                                                      sector                                                                    (U(3)_a ×USp(2)_b ×U(4)_h)_U(1)_c×U(1)_d                                                                                                         Q_Y                                                                                                          _3;                                                                                                          ab                                                                                             3 ×(, , )_(0,0)                                                                                                         1/6                                                                                                           0                                                                                                         aa'                                                                                     2×[ (_,,)_(0,0) + h.c.]                                                                                                        ±1/3                                                                                                           0;                                                                                                          ac                                                                                              6 ×(, ,)_(1,0)                                                                                                         1/3                                                                                                           1                                                                                                          bb                                                                                              5 ×(,_,)_(0,0)                                                                                                           0                                                                                                           0;                                                                                                          ad                                                                                            3 ×(, , )_(0,-1)                                                                                                        -1/3                                                                                                           1                                                                                                          cc                                                                                               4 ×(,,)_(0,0)                                                                                                           0                                                                                                           0;                                                                                                         ad'                                                                                            3 ×(, , )_(0,-1)                                                                                                        -2/3                                                                                                           1                                                                                                          dd                                                                                               5 ×(,,)_(0,0)                                                                                                           0                                                                                                           0;                                                                                                          bc                                                                  3 ×(, ,)_(1,0) + 3 ×[ (, ,)_(1,0) + h.c. ]                                                                                                    1/2,±1/2                                                                                                      1,1||2                                                                                                         dd'                                                                                         [(,,)_(0,2) + h.c.]                                                                                                          ±1                                                                                                        1||2;                                                                                                   5-8
   bd                                                                    6 ×(,,)_(0,-1)+2 ×[ (,,)_(0,1)  + h.c. ]                                                                                                  -1/2, ±1/2                                                                                                      1,2||1                                                                                                          bh                                                                                               3 ×(,,)_(0,0)                                                                                                           0                                                                                                           2;                                                                                                          cd                                                                   3 ×(,,)_(-1,1) +3 ×[ (,,)_(-1,1) + h.c. ]                                                                                                           0                                                                                                      1,1||2                                                                                                         ch'                                                                                              6 ×(,,)_(-1,0)                                                                                                        -1/2                                                                                                           0;                                                                                                         cd'                                                                      3 ×(,,)_(1,1)+3 ×[ (,,)_(1,1) + h.c. ]                                                                                                       1, ±1                                                                                                           0                                                                                                          dh                                                                                               3 ×(,,)_(0,1)                                                                                                         1/2                                                                                                           0;                                                                                               1-4  
     ah                                                                                     2 ×[ (,,)_(0,0)  +h.c.]                                                                                                        ±1/6                                                                                                        1||2                                                                                                         dh'                                                                                               3 ×(,,)_(0,1)                                                                                                         1/2                                                                                                           1;                                                                                                         ah'                                                                                         [(,,)_(0,0)  +h.c.]                                                                                                        ±1/6                                                                                                       2 ||1                                                                                                         hh'                                                                                   2 ×[ (,,6_)_(0,0) + h.c.]                                                                                                           0                                                                                                        1||2;                                                                                                             ]
5stackMSSMaAAPrototypeISpectrumChiral and non-chiral massless (open string) matter spectrum of the five-stack D6-brane model from table <ref>
with gauge group SU(3)_a× USp(2)_b × U(1)_Y × SU(4)_h ×_3
after the Green-Schwarz mechanism has been taken into account.
For vector-like states, different charges under the discrete _3 ⊂ U(1)_c + 2 U(1)_d + 2 U(1)_h symmetry  are denoted using the logic symbol ||.
The closed string sector for this model - as well as all other explicit examples discussed in this article -  contains (h^11_+,h^11_-,h^21)=(4,15,19) vectors, Kähler moduli and complex structure moduli, respectively.



Ignoring the possibility of vevs for charged scalars as suggested in footnote <ref> on page Footnote:SUNSUNFI for the moment, we can read off from table <ref>
that the QCD stack a and the `hidden' stack h each couple with orientifold-odd wrapping numbers to the deformation moduli ζ^(1)_3,4+5 
and ζ^(3)_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 d to ζ^(2)_1,2 and ζ^(3)_1,2. This MSSM example thus allows to stabilise at most six of the 14 _2-twisted deformation moduli 
as displayed in table <ref>. Out of the remaining eight _2-twisted deformation moduli, only ζ^(1)_4-5 couples in an orientifold-even 
way to the two D6-branes b and d. The associated gauge couplings of SU(2)_b ≃ USp(2)_b and U(1)_Y = U(1)_a/6 + [U(1)_c+U(1)_d]/2 thus experience one flat direction at tree-level.
The remaining seven _2-twisted deformation moduli ζ^(1)_0,1,2 and ζ^(2,3)_(3,4) 
on the other hand do not couple directly to any D6-brane in this global model and can at most change the couplings in the low-energy effective MSSM Lagrangian via 
non-perturbative or higher order corrections. 
Let us emphasise here that at the singular orbifold point, the volume of a given fractional three-cycle and thereby the tree-level
gauge coupling only depends on the untwisted complex structure modulus ϱ∝R^(1)_2/R^(1)_1 as stated in equations (<ref>) and (<ref>), while the untwisted Kähler moduli v_i,i∈{1,2,3} defined in equation (<ref>) influence the gauge couplings once the one-loop gauge threshold corrections are taken into account, as will be discussed in section <ref>.

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 <ref> provides all charged (open string) scalars that might possibly 
trigger some D6-brane 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 <ref> 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 <ref> only. The only states in bifundamental representations of two unitary gauge factors of equal rank
are the right-handed selectrons and sneutrinos[Notice that U(1)_massive = U(1)_c - U(1)_d in this example acts as perturbative global Peccei-Quinn symmetry in the low-energy effective action with the right-handed sneutrinos naturally identified as QCD axions of a generalised `stringy' DFSZ model <cit.>.] in the cd' and cd sectors, respectively, charged under U(1)_c × U(1)_d. 
As summarised in table <ref>, both D6-branes c and d couple to the twisted deformation moduli ζ^(2)_1,2, and table <ref> shows that the corresponding 
orientifold-odd exceptional wrapping numbers are opposite, [2 x^(2)_α,c + y^(2)_α,c ]_α=1,2 = - [2 x^(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 <cit.>, 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 = 2  Π^bulk_c=d + Π^_2^(1),odd_c + Π^_2^(1),even_d + Π^_2^(3),even_c + Π^_2^(3),odd_d/4 as a merging of cycles in the _2^(2)-twisted sector is contrasted by the naive merging of orientifold image cycles Π_c + Π_c'= Π^bulk_c + Π^_2^(3)_c/2 and Π_d + Π_d'= Π^bulk_d + Π^_2^(1)_d/2 leaving only one _2-twisted sector.

One might wonder if taking the vector-like matter states into consideration as well can produce additional flat directions, in particular if (some of) the antisymmetric representations of SU(3)_a or SU(4)_h
receive a vev. The former is clearly undesirable since it would break the part SU(3)_a × U(1)_Y of the Standard Model gauge group. On the other hand, if the vector-like pairs
in the aa' and hh' sectors of table <ref> were originating from N=2
supersymmetric sectors, their vevs would not be protected by the N=1 SU(N) D-term argument of <cit.> but would instead be expected to constitute flat directions
associated to the recombination of orientifold image cycles of the type Π_z+ Π_z' = Π^bulk_z + Π^_2^(2)_z /2 for z ∈{a,h} such that the deformation moduli ζ^(1)_3,4+5
and ζ^(3)_1,2 in the _2^(1)- and _2^(3)-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 <ref> 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  _2 ×_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 <cit.> rely on N=2 supersymmetry on the factorisable six-torus T^6=(T^2)^3 or its _2 ×_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 _2^(1)-twisted deformation ε_4-5^(1), to which the D6-branes b and d couple through the orientifold-even exceptional three-cycle. Hence, this deformation represents a _2^(1)-twisted modulus with a flat direction as indicated in table <ref>. A closer look at the three-cycle configuration in table <ref> for branes b and d reveals that the functional dependence of their volumes in terms of ε_4-5^(1) is equivalent to the branes supporting the gauge groups USp(2)_c and USp(2)_b, respectively, in the toy model of the previous section: the b-brane volume behaves as the righthand part of figure <ref> under a deformation by ε_4-5^(1)≠ 0, while the d-brane cycle exhibits the behaviour of the lefthand side of figure <ref> under the deformation parameter ε_4-5^(1). This implies that the weak gauge coupling becomes stronger when the _2^(1)-twisted modulus ζ_4-5^(1) acquires a non-zero vev and we move away from the singular orbifold point along the flat direction. The massless hypercharge on the other hand is characterised by a smaller gauge coupling away from the orbifold point. Compatibility between the calibration forms for the bulk and (resolved) exceptional three-cycles limits the parameter space of the deformations to the half-line ε_4-5^(1)≤ 0, similarly to the toy model of the previous section.
 





 §.§ 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 <cit.>.
The common local D6-brane configuration of the observable sector is displayed in the upper part of table <ref>, with the four different types of global completion by two `hidden' sector D6-branes provided in the lower part of the same table.


[ 7|c|D6-brane configurations of global L-R symmetric models on T^6/(_2 ×_6 ×);                                                                                                                                          wrapping numbers                                                                      Angle/π                                                           _2^(i) eigenvalues                                                                         (τ⃗)                                                                         (σ⃗)                                                                  gauge group;                                              7|c|Universal observable sector;                                                                            a                                                                (1,0;1,0;1,0)                                                                      (0,0,0)                                                                        (+++)                                                                      (0,1,1)                                                                      (0,1,1)                                                                         U(3);                                                                            b                                                              (1,0;-1,2;1,-2)                                                                 (0,1/2,-1/2)                                                                        (+++)                                                                      (0,1,0)                                                                      (0,1,0)                                                                       USp(2);                                                                            c                                                              (1,0;-1,2;1,-2)                                                                (0, 1/2,-1/2)                                                                        (-+-)                                                                      (0,1,0)                                                                      (0,1,0)                                                                       USp(2);                                                                            d                                                                (1,0;1,0;1,0)                                                                     (0, 0,0)                                                                         (+–)                                                                      (0,1,1)                                                                      (0,1,1)                                                                         U(1);                                         7|c|Global completion of prototype I;                                                                          h_1                                                                (1,0;1,0;1,0)                                                                      (0,0,0)                                                                        (+++)                                                                      (0,0,0)                                                                      (0,0,0)                                                                         U(3);                                                                          h_2                                                                (1,0;1,0;1,0)                                                                      (0,0,0)                                                                         (+–)                                                                      (0,0,0)                                                                      (0,0,0)                                                                         U(3);                                        7|c|Global completion of prototype II;                                                                          h_1                                                              (1,0;-1,2;1,-2)                                                                 (0,1/2,-1/2)                                                                        (+++)                                                                      (0,0,0)                                                                      (0,0,0)                                                                         U(1);                                                                          h_2                                                              (1,0;-1,2;1,-2)                                                                 (0,1/2,-1/2)                                                                         (+–)                                                                      (0,0,0)                                                                      (0,0,0)                                                                         U(1);                                       7|c|Global completion of prototype IIb;                                                                          h_1                                                              (1,0;-1,2;1,-2)                                                                 (0,1/2,-1/2)                                                                        (-+-)                                                                      (0,0,0)                                                                      (1,0,0)                                                                         U(1);                                                                          h_2                                                              (1,0;-1,2;1,-2)                                                                 (0,1/2,-1/2)                                                                         (–+)                                                                      (0,0,0)                                                                      (1,0,0)                                                                         U(1);                                       7|c|Global completion of prototype IIc;                                                                          h_1                                                              (1,0;-1,2;1,-2)                                                                 (0,1/2,-1/2)                                                                        (+++)                                                                      (0,0,1)                                                                      (0,0,1)                                                                         U(1);                                                                          h_2                                                              (1,0;-1,2;1,-2)                                                                 (0,1/2,-1/2)                                                                        (+++)                                                                      (0,0,1)                                                                      (1,1,1)                                                                         U(1);                                                                              ]
6stackLRSaAALocal observable L-R symmetric D6-brane sector in the first block with four different global completions by two `hidden' branes h_1, h_2 listed below.
Observe that for prototype IIb, we have (+++)_h_1' and (+–)_h_2', and thus the only difference w.r.t. prototype II is the different choice of the discrete displacement parameter σ^1_h_i^ IIb=1.


The common observable part of the matter spectrum is displayed in table <ref>,

 
[ 8|c|Visible spectrum of all L-R symmetric & the PS I models on T^6/(_2 ×_6 ×);                                                                        sector                                          (U(3)_a ×USp(2)_b ×USp(2)_c )_U(1)_d                                                                          _3^I                                                           U(1)_B-L^II+IIb+IIc                                                                        sector                                          (U(3)_a ×USp(2)_b ×USp(2)_c )_U(1)_d                                                                          _3^I                                                           U(1)_B-L^II+IIb+IIc;     8|c|L-R symmetric models & PS I model (with →, 3_→ 6_ of U(3)_a → U(4)_a);                                                                            ab                                                                   3 ×(,,)_(0)                                                                             1                                                                           1/3                                                                           aa'                                                         2×[ (_,,)_(0) + h.c.]                                                                          2||1                                                                          ±2/3;                                                                            ac                                                                   3 ×(,,)_(0)                                                                             2                                                                          -1/3                                                                            bb                                                                  5 ×(,_,)_(0)                                                                             0                                                                             0;                                                                            bc                                                                  10 ×(,,)_(0)                                                                             0                                                                             0                                                                            cc                                                                 5 ×(,,_A)_(0)                                                                             0                                                                             0;                                                 8|c|L-R symmetric models only;                                                                            ad                                                              (,,)_(-1) + h.c.                                                                          1||2                                                                          ±4/3                                                                            bd                                                                  3 ×(,,)_(-1)                                                                             0                                                                             1;                                                                           ad'                                                         2 ×[ (,,)_(1) + h.c.]                                                                          1||2                                                                          ∓2/3                                                                            cd                                                                   3 ×(,,)_(1)                                                                             0                                                                            -1;                                                                               ]
6stackLRSMaAAVisibleSpectrumCommon visible spectrum of the L-R symmetric models. Prototype I has the low-energy gauge group SU(3)_a × USp(2)_b × USp(2)_c × SU(3)_h_1× SU(3)_h_2×_3 
with the _3 charge displayed in the third and seventh column. The prototype II model has the low-energy gauge group SU(3)_a × USp(2)_b × USp(2)_c ×U(1)_B-L with the 
U(1)_B-L≡1/3 U(1)_a - U(1)_d - U(1)_h_1 + U(1)_h_2 charge displayed in the fourth and eighth column. For the prototype IIb & IIc models, the U(1)_B-L symmetry is massive and thus only a perturbative global symmetry.
The massless matter spectrum is completed by the `hidden' sectors displayed in table <ref> for each prototype I, II, IIb and IIc.
The massless closed string sector is for each case identical to the MSSM-like model, i.e.  it contains (h^11_+,h^11_-,h^21)=(4,15,19) vectors, Kähler moduli and complex structure moduli, respectively.


and the individual `hidden' spectra are given in  table <ref>.


[ 6|c|Overview of the `hidden' spectra for the L-R symmetric models on T^6/(_2 ×_6 ×);                                                                     6|c|Prototype I;                                                                              sector                            (U(3)_a ×USp(2)_b ×USp(2)_c ×U(3)_h_1 ×U(3)_h_2 )_U(1)_d                                                                                  _3                                                                              sector                            (U(3)_a ×USp(2)_b ×USp(2)_c ×U(3)_h_1 ×U(3)_h_2 )_U(1)_d                                                                                  _3;                                                                                ah_1                                                                       2 ×(,,,,)_(0)                                                                                   0                                                                                dh_1                                                                       2 ×(,,,,)_(1)                                                                                   2;                                                                                ah_2                                                                       2 ×(,,,,)_(0)                                                                                   0                                                                                dh_2                                                                      2 ×(,,,,)_(-1)                                                                                   1;                                                                                bh_1                                                   (,,,,)_(0)  +[ (,,,,)_(0) +h.c. ]                                                                             2, 2||1                                                                             h_1 h_2                                                                   (,,,,)_(0) + h.c.                                                                                   0;                                                                                bh_2                                                   (,,,,)_(0) + [(,,,,)_(0) + h.c. ]                                                                             1, 1||2                                                                            h_1 h_2'                                                            2×[ (,,,,)_(0) + h.c.  ]                                                                                2||1;                                                                                ch_1                                                    (,,,,)_(0) + [(,,,,)_(0) +h.c. ]                                                                            2 , 2||1                                                                            h_1 h_1'                                                             2×[ (,,,_,)_(0) + h.c.]                                                                                2||1;                                                                                ch_2                                                    (,,,,)_(0) + [ ,,,,)_(0) + h.c.]                                                                              1,1||2                                                                            h_2 h_2'                                                             2×[ (,,,,_)_(0) + h.c.]                                                                                2||1;                                                                    6|c|Prototype II;                                                                              sector                              (U(3)_a ×USp(2)_b ×USp(2)_c)_U(1)_d×U(1)_h_1 ×U(1)_h_2                                                                            U(1)_B-L                                                                              sector                              (U(3)_a ×USp(2)_b ×USp(2)_c)_U(1)_d×U(1)_h_1 ×U(1)_h_2                                                                            U(1)_B-L;                                                                                ah_1                                                         2 ×[ (,,)_(0,-1,0) + h.c. ]                                                                                ±4/3                                                                                dh_1                                                         2 ×[ (,,)_(1,-1,0) + h.c. ]                                                                                   0;                                                                               ah_1'                                                                 (,,)_(0,1,0) + h.c.                                                                                ∓2/3                                                                               dh_1'                                                                 (,,)_(1,1,0) + h.c.                                                                                  ∓2;                                                                                ah_2                                                         2 ×[ (,,)_(0,0,-1) + h.c. ]                                                                                ∓2/3                                                                                dh_2                                                           2×[(,,)_(1,0,-1) + h.c. ]                                                                                  ∓2;                                                                               ah_2'                                                                 (,,)_(0,0,1) + h.c.                                                                                ±4/3                                                                               dh_2'                                                                 (,,)_(1,0,1) + h.c.                                                                                   0;                                                                                bh_1                                        3 ×(,,)_(0,1,0) + 3 ×[ (,,)_(0,-1,0) +h.c. ]                                                                              -1, ±1                                                                             h_1 h_2                                                         5 ×[ (,,)_(0,1,-1) + h.c. ]                                                                                  ∓2;                                                                                bh_2                                       3 ×(,,)_(0,0,-1) + 3 [ ×(,,)_(0,0,1) + h.c. ]                                                                             -1,  ±1                                                                            h_1 h_2'                                                           6×[ (,,)_(0,1,1) + h.c. ]                                                                                   0;                                                                                ch_1                                       3 ×(,,)_(0,-1,0) + 3 ×[  (,,)_(0,1,0) +h.c. ]                                                                               1, ∓1                                                                              h_1h_1                                                                     4 ×(,,)_(0,0,0)                                                                                   0;                                                                                ch_2                                        3 ×(,,)_(0,0,1) + 3 [ ×(,,)_(0,0,1) + h.c. ]                                                                               1, ±1                                                                              h_2h_2                                                                     4 ×(,,)_(0,0,0)                                                                                   0;                                                                   6|c|Prototype IIb;                                                                             h_1 h_2                                                         5 ×[ (,,)_(0,1,-1) + h.c. ]                                                                                  ∓2                                                                              h_1h_1                                                                     4 ×(,,)_(0,0,0)                                                                                   0;                                                                            h_1 h_2'                                                           6×[ (,,)_(0,1,1) + h.c. ]                                                                                   0                                                                              h_2h_2                                                                     4 ×(,,)_(0,0,0)                                                                                   0;                                                                   6|c|Prototype IIc;                                                                                ah_1                                                                     3 ×(,,)_(0,1,0)                                                                                -4/3                                                                                dh_1                                                                     3×(,,)_(1,-1,0)                                                                                   0;                                                                               ah_1'                                                                     3 ×(,,)_(0,1,0)                                                                                -2/3                                                                               dh_1'                                                                    3×(,,)_(-1,-1,0)                                                                                  -2;                                                                                bh_1                                                          4 ×[ (,,)_(0,-1,0) +h.c. ]                                                                                  ±1                                                                              h_1h_1                                                                     5 ×(,,)_(0,0,0)                                                                                   0;                                                                                ch_1                                        6 ×(,,)_(0,-1,0) +2 ×[  (,,)_(0,1,0) +h.c. ]                                                                             1,   ∓1                                                                              h_2h_2                                                                     5 ×(,,)_(0,0,0)                                                                                   0;                                                                                     ]
6stackLRSMaAAHiddenSpectrum`Hidden' massless spectrum per L-R symmetric model completing the common observable sector  displayed in table <ref>.




The QCD stack a agrees by construction with the one of the MSSM example discussed in section <ref>, while the left- and right-symmetric groups USp(2)_b × USp(2)_c 
have identical bulk cycles and discrete Wilson line and displacement parameters, but differ in their _2 ×_2 eigenvalues from the D6-brane b of the MSSM example.
The D6-brane d only differs from the QCD stack a in the _2 ×_2 eigenvalues and stack size N_d=1 vs. N_a=3.

The orientifold-even and -odd wrapping numbers are summarised table <ref> and lead to the naive counting of the maximal number of stabilised deformation moduli in table <ref>. 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.

  * L-R I is the only model with non-Abelian `hidden' sector and thus states in bifundamental representations of gauge groups of identical rank, SU(3)_a × SU(3)_h_1× SU(3)_h_2. While 
the twisted deformation modulus ⟨ζ^(1)_3 ⟩ =0 is stabilised at the orbifold point by the presence of D6-brane d, vevs of ζ^(1)_0,1,2 and ζ^(2)_1,2 could potentially be compensated by vevs of scalars in the h_1h_2^(') sectors, which would simultaneously break SU(3)_h_1× SU(3)_h_2→ SU(3)_h^diag.[ More explicitly, one can consider e.g. the vector-like pair (,,,,)_(0) + h.c. from the h_1 h_2 sector and solve the Abelian and non-Abelian D-term constraints in terms of non-vanishing vevs of the scalar components in both bifundamental representations of U(3)_h_1× U(3)_h_2. Such a vacuum configuration would break the non-Abelian gauge groups to the diagonal gauge group U(3)_h^ diag, which would correspond geometrically to the recombination of the two D6-brane stacks h_1,h_2 into a single stack h wrapping [Π^bulk_h + Π^_2^(1)_h]/2, cf. also the microscopic origin of the vector-like matter states from the h_1(ω^0 h_2) sector according to table <ref>. ]

  * L-R II, IIb, IIc contain Abelian `hidden' D6-branes and thus three gauge groups of equal rank, U(1)_d × U(1)_h_1× U(1)_h_2. 


In the prototypes II and IIb, some vev of the type ⟨ζ^(1)_0,1,2⟩ might potentially be compensated by vevs associated to charged scalar fields belonging to vector-like pairs in the h_1h_2^(') sector. For prototype IIb, the same considerations apply also to ⟨ζ^(3)_3,4⟩. Details of these potential compensations among vevs of closed and open string scalars depend on the microscopic origin of the latter as detailed in  table <ref>.

  * Prototypes I and IIc have according to the naïve counting in table <ref> only one flat direction, which affects the tree level gauge couplings, in the _2^(1)-twisted sector, while prototypes II and IIb additionally have two flat directions of direct physical consequence in the _2^(2)-twisted sector.

The counting of matter states per intersection sector is displayed in tables <ref> and <ref> 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 _2^(i)-twisted moduli with a flat direction, we notice first of all that the discussion for the USp(2)_b × USp(2)_c sector
can be brought back to the analysis presented in section <ref> for the global USp(2)^4 toy model.
A common calibration w.r.t. Re(Ω_3) for bulk and resolved exceptional three-cycles then constrains the deformation parameter to lie on the half-line ε_4-5^(1)≤ 0. Going away from the orbifold point along the flat direction of the _2^(1)-twisted modulus ζ^(1)_4-5 implies a weaker gauge coupling for the left stack USp(2)_b and a stronger gauge coupling for the right stack USp(2)_c.[For the prototype IIc model, the modulus ζ^(1)_4-5 also couples to the `hidden' D6-brane stacks h_1 and h_2, such that a deformation along its flat direction also affects their respective U(1) gauge couplings 
at the string scale M_string, before these Abelian gauge groups are spontaneously broken at the KK-scale
 by virtue of the Stückelberg mechanism. 
That is to say, the (_2^(1)-twisted sector of the) fractional three-cycle for h_1/h_2 can be brought back to the (_2^(1)-twisted sector of the) fractional cycle supporting the USp(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_4-5^(1). Realising a strongly coupled anomalous U(1) gauge theory at the string scale through these geometric deformations of _2 singularities opens up avenues for D6-brane model building scenarios <cit.> realising Nambu-Jona-Lasinio type models upon integrating out the massive U(1).] 

The deformation parameters ε_1,2^(2) in the _2^(2)-twisted sector allow for a mutually compatible calibration w.r.t. Re(Ω_3) between bulk and exceptional three-cycles provided the parameters lie on the half-line ε_1,2^(2)≤ 0. The (_2^(2) sector of the) fractional three-cycle for the strong D6-brane stack a is identical to the geometry of the three-cycle (in the _2^(2) sector) supporting the USp(2)_a gauge group in the toy model of section <ref>. 
This implies that the QCD gauge group acquires a stronger gauge coupling when either the twisted modulus ζ_1^(2) or ζ_2^(2) acquires a non-zero vev. The opposite occurs for the left stack b and right stack c, whose gauge couplings decrease for non-zero deformations along the flat directions of ζ_1,2^(2) 
as detailed in the context of the USp(2)^4 toy model in section <ref>. 
Also the d-brane couples to the deformation moduli ζ_1,2^(2) along flat directions, yet its geometric properties cannot be reduced to a situation discussed in previous sections. Using the same modus operandi as in section <ref>, we can compute the normalised volume of the fractional three-cycle of the d-brane, which is shown on the left-hand side of figure <ref>. From this figure, the qualitative picture obviously exhibits the U(1)_d gauge coupling decreasing for non-zero deformations in the _2^(2)-twisted sector along the flat directions ζ_1,2^(2). Due to the relative factor 1/3 in the definition of the generalised U(1)_B-L symmetry
of prototype II, the behaviour of the U(1)_d gauge coupling is expected to be dominant with respect to the behaviour of the U(1)_a gauge coupling under _2^(2) deformation, implying that the U(1)_B-L symmetry will also be more weakly coupled for non-zero vevs in the ζ_1,2^(2)-directions.




According to table <ref>, 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 USp(2)_b gauge group in the global toy model of section <ref>. 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 <ref> and that the U(1)_h_1 gauge coupling becomes more weakly coupled for non-zero deformations ⟨ζ_1,2^(2)⟩≠0 in case of prototype II models and ⟨ζ_3,4^(2)⟩≠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 <ref> we can compute its fractional three-cycle volume as a function of the _2^(2) deformations, yielding the plots in the middle and right panel of figure <ref>. Hence, the U(1)_h_2 gauge coupling is expected to increase when going away from the singular orbifold point along the flat directions ζ_1,2^(2) for the prototype II models and along the flat directions ζ_3,4^(2) for the prototype IIb models.






 §.§ Deformations in global Pati-Salam models


In <cit.>, 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 <ref>. The D6-brane configuration is displayed in table <ref> and differs from the original one in <cit.> by a swap of the last two two-tori, T^2_(2)↔ T^2_(3), and by a flip of the absolute _2 ×_2 eigenvalues. 


[ 7|c|D6-brane configuration of a global Pati-Salam model on the aAA lattice: prototype I;                                                                                                                                                                wrapping numbers                                                                                 Angle/π                                                                      _2^(i) eigenvalues                                                                                    (τ⃗)                                                                                    (σ⃗)                                                                             gauge group;                                                                                       a                                                                           (1,0;1,0;1,0)                                                                                 (0,0,0)                                                                                   (+++)                                                                                 (0,1,1)                                                                                 (0,1,1)                                                                                    U(4);                                                                                       b                                                                         (1,0;-1,2;1,-2)                                                                            (0,1/2,-1/2)                                                                                    (–+)                                                                                 (0,1,0)                                                                                 (0,1,0)                                                                                  USp(2);                                                                                       c                                                                         (1,0;-1,2;1,-2)                                                                           (0, 1/2,-1/2)                                                                                    (+–)                                                                                 (0,1,0)                                                                                 (0,1,0)                                                                                  USp(2);                                                                                       h                                                                           (1,0;1,0;1,0)                                                                                 (0,0,0)                                                                                    (–+)                                                                                 (0,1,1)                                                                                 (0,1,1)                                                                                    U(6);                                                                                         ]
PatiSalamaAAPrototypeIPati-Salam model, prototype I, on the T^6/(_2 ×_6 ×) orientifold with discrete torsion. The three-cycles wrapped by the D6-branes a and b are identical to those of the MSSM example in
table <ref>.

The spectrum of prototype I is displayed in table <ref>.


[ 4|c|Matter spectrum of the prototype I global Pati-Salam model on aAA;                                                                sector                                      U(4)_a×USp(2)_b×USp(2)_c ×U(6)_h                                                                sector                                      U(4)_a×USp(2)_b×USp(2)_c ×U(6)_h;                                                                    ab                                                           3 ×(, , ; )                                                                   aa'                                                2 ×[  (6_,,;) + h.c. ];                                                                    ac                                                            3 ×(, , ;)                                                                   bb'                                                            5 ×(,_, ;);                                                                    bc                                                            10 ×(,,; )                                                                   cc'                                                            5 ×(,, _;);                                                                    ah                                                   2 ×[ (,,;) + h.c. ]                                                                    bh                                                           3 ×(, , ;6);                                                                   ah'                                                          (,,;) + h.c.                                                                    ch                                                          3 ×(, , ; 6);                                                                   hh'                                               2 ×[ (,,;15_) + h.c.  ]                                                                                                                                            ;                                                                       ]
PatiSalamaAAPrototypeISpectrumChiral and vector-like massless  (open string) matter spectrum of the prototype I global PS model on the T^6/(_2 ×_6 ×) orientifold with D6-brane configuration given in table <ref>. The universal closed string spectrum of all global models in this article can e.g. be found in the caption of table <ref>.
 Both U(1)_a× U(1)_d gauge factors acquire  masses through the Stückelberg mechanism and survive as perturbative global symmetries at low energies.



The D6-brane configuration of the prototype II in table <ref> is up to the analogous swaps in two-torus indices and absolute choice of _2 ×_2 eigenvalues plus a swap in h ↔ h' also identical to the original one from <cit.> with the massless matter spectrum displayed in table <ref> for the observable part 


[ 7|c|D6-brane configuration of a global Pati-Salam model on the aAA lattice: prototype II;                                                                                                                                                                  wrapping numbers                                                                                  Angle/π                                                                       _2^(i) eigenvalues                                                                                     (τ⃗)                                                                                     (σ⃗)                                                                              gauge group;                                                                                        a                                                                            (1,0;1,0;1,0)                                                                                  (0,0,0)                                                                                    (+++)                                                                                  (0,1,1)                                                                                  (0,1,1)                                                                                     U(4);                                                                                        b                                                                          (1,0;-1,2;1,-2)                                                                             (0,1/2,-1/2)                                                                                    (+++)                                                                                  (0,1,0)                                                                                  (0,1,0)                                                                                   USp(2);                                                                                        c                                                                          (1,0;-1,2;1,-2)                                                                            (0, 1/2,-1/2)                                                                                    (-+-)                                                                                  (0,1,0)                                                                                  (0,1,0)                                                                                   USp(2);                                                                                        h                                                                          (1,0;-1,2;1,-2)                                                                            (0, 1/2,-1/2)                                                                                    (+++)                                                                                  (0,0,0)                                                                                  (0,0,0)                                                                                     U(2);                                                                                          ]
PatiSalamaAAPrototypeIIPati-Salam model, prototype II, on the T^6/(_2 ×_6 ×) orientifold with discete torsion. The three-cycles wrapped by D6-branes {a,b,c} are identical to those of the L-R symmetric models in  table <ref>.
Moreover, the `hidden' brane h of this PS II model wraps the same three-cycle as h_1 of the L-R symmetric II model.

and in table <ref> for the `hidden' part. All discrete data are chosen such that the sector {a,b,c} of the PS II model agrees (up to the stack size N_a) with that of the L-R symmetric models in section <ref>.


[ 4|c|`Hidden' spectrum of the prototype II global Pati-Salam model on aAA;                                                                   sector                                         U(4)_a×USp(2)_b×USp(2)_c ×U(2)_h                                                                   sector                                         U(4)_a×USp(2)_b×USp(2)_c ×U(2)_h;                                                                       ah                                                      2 ×[ (,,;) + h.c. ]                                                                       bh                                        3 ×(, , ;)  + 3×[(, , ;) + h.c. ];                                                                      ah'                                                             (,,;) + h.c.                                                                       ch                                        3 ×(, , ; ) + 3×[(, , ;) + h.c. ];                                                                      hh'                                                    6 ×[ (,,;_) + h.c.  ]                                                                                                                                                  ;                                                                          ]
PatiSalamaAAPrototypeIISpectrumChiral and vector-like massless `hidden' matter spectrum of the prototype II global Pati-Salam model with D6-brane configuration in table <ref>. The observable spectrum is (up to the rank of stack a) identical to the {a,b,c} sector of the L-R symmetric models, cf. 
table <ref>.
Also in this prototype, both U(1)_a× U(1)_d gauge factors acquire a Stückelberg mass, turning them into perturbative global symmetries in the low-energy effective field theory.




The orientifold-odd and -even wrapping numbers are given in table <ref> and lead to the naive counting of the maximal number of stabilised moduli 
of four and seven for the PS I and PS II model, respectively, in table <ref>. Even though the models differ in the sectors {b,c,h}, in both cases
the gauge couplings of branes b,c are sensitive to flat directions in the deformation moduli ζ^(1)_4-5 and ζ^(2)_1,2, with also the gauge couplings of a and h
sensitive to ζ^(2)_1,2. 

Looking more closely at table <ref>, we observe that the deformation moduli ζ_0,1,2^(1) only couple to the hidden stack h in the prototype PS II model. This opens up the possibility to stabilise one of the vevs ⟨ζ_0,1,2^(1)⟩ by compensating it with the vevs of the scalar components in a vector-like pair (, , ;) + h.c. in the bifundamental representation under USp(2)_b× U(2)_h or a vector-like pair (, ,;) + h.c. in the bifundamental representation under USp(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 USp(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 _2^(i) singularities in a single sector i, cf. the overview over the couplings to _2^(i)-twisted sectors in table <ref>. 
 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 <ref> for the prototype I example and in table <ref> for the prototype II examples, from which a field theoretic study of the scalar potential could initiate in terms of the necessary selection rule of the existence of closed polygons.

Let us instead analyse the _2^(i)-twisted moduli representing flat directions for the PS models, as anticipated in table <ref>. In the _2^(1)-twisted sector, the D6-branes b and c couple to the twisted modulus ζ_4-5^(1) and (the relevant part of) their fractional three-cycles can be brought back to the fractional three-cycles supporting the USp(2)_b,c gauge groups in the toy model of section <ref>. 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 <ref> and the one for the c-brane is depicted on the left-hand side of figure <ref>, whereas the prototype II models have the reverse identifications with respect to figure <ref>. This implies that the left-symmetric USp(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_4-5^(1), while the USp(2)_b gauge group in the prototype II models receives a weaker gauge coupling. The gauge coupling of the right-symmetric USp(2)_c gauge group exhibits the exact opposite behaviour w.r.t. the one of the left-symmetric stack for non-zero deformations ⟨ζ_4-5^(1)⟩≠0. This qualitative difference between the two prototype PS models can be traced back to the relative difference in _2×_2 eigenvalues for the b- and c-stack between both prototypes. 

The _2^(2) deformations affect the gauge coupling of the hidden gauge group in the prototype I and II PS models, as the hidden D6-brane stack h in both cases
couples to the twisted deformation moduli ζ_1,2^(2) along flat directions. The (relevant part) of the fractional three-cycle wrapped by the hidden stack h in the prototype I model is characterised by the same geometry as the d-brane in the L-R symmetric model, such that its volume exhibits the behaviour of the left panel of figure <ref> under non-zero deformation parameters ε_1,2^(2). Hence, the hidden U(6)_h gauge group in the prototype I model has a smaller gauge coupling at the string scale M_string for resolved _2^(2) singularities. The _2^(2) exceptional three-cycle of the hidden stack h in the prototype II model on the other hand takes the same form as the one of the USp(2)_b stack in the toy model of section <ref>.
Hence, its fractional three-cycle volume is characterised by the middle and right panel of figure <ref>, and the hidden gauge group U(2)_h has a smaller gauge coupling in the prototype II model as well when we consider non-zero deformations ε_1,2^(2) along flat directions. Note that compatibility of the common calibration w.r.t. Re(Ω_3) of the resolved exceptional three-cycles constrains the parameter space of the deformations to the half-lines ε_4-5^(1)≤0 and ε_1,2^(2)≤0 in both prototype PS models.








§ 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 (<ref>).
At the orbifold point, the volumes stem solely from the toroidal cycles as detailed in equation (<ref>).
For the MSSM example of section <ref>, the relation (<ref>) amounts to the following relation among the tree-level gauge couplings at the orbifold point:

    1/g^2_SU(3)_a = 1/g^2_SU(4)_h = 2/31/g^2_USp(2)_b = 6/191/g^2_U(1)_Y,

which obviously disagrees with the proposal of gauge coupling unification at M_string≃ M_GUT∼ 10^16GeV <cit.>.

As discussed in section <ref>, 
deformations of exceptional cycles along directions of only orientifold-even wrapping numbers change the volumes of fractional cycles such that 
the degeneracy of e.g. the gauge coupling strength of USp(2)_b and USp(2)_c in the L-R symmetric and PS models is lifted, and also the relations in (<ref>) might be ablished. 

On the other hand, the degeneracy of gauge couplings is already at the orbifold point lifted when including one-loop corrections Δ_G_x
 to the gauge couplings of any gauge factor G_x, 

    8 π^2/g^2_G_x(μ) =  8 π^2/g^2_G_x,tree + b_G_x/2lnM^2_string/μ^2 + Δ_G_x/2 ,

which depend on the discrete D6-brane
data such as _2 ×_2 eigenvalues and discrete Wilson lines and displacement parameters. In section <ref>, we will therefore present the 
one-loop gauge thresholds Δ_G_x and discuss in section <ref> impacts on the low-energy phenomenology of the global particle physics vacua
with D6-brane data specified in tables <ref>, <ref>, <ref> and <ref>.
In particular, the class of models at hand shares the unusual feature that all D6-branes wrap the (_2^(1))-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 <cit.>, but the weakness of 
gravity can already be generated at tree level by a large hierarchy between the two radii R_2^(1) and R_1^(1) of the two-torus T^2_(1).





 §.§ 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 <cit.> before applying these general results to the stringy particle physics vacua on the  
 background with exotic O6-plane orbit _2^(3). The Kähler metrics and contributions to the SU(N_x) beta function coefficients,

    b_SU(N_a) =   N_a (-3+φ^_a)+ N_a/2 ( φ^_a + φ^_a)
     + ( φ^_a -φ^_a)
     +∑_b≠ aN_b/2( φ^ab + φ^ab') 
    ≡       ∑_k=0^2
     b_a(ω^k a)^ A   +      ∑_k=0^2 b_a(ω^k a')^ A       
     +   ∑_k=0^2 b_(ω^k a)^ M     +  ∑_b≠ a∑_k=0^2 (b_a(ω^k b)^ A + b_a(ω^k b')^ A) 
    ,

 per open string sector are displayed in table <ref>


[                                                                            4|c|Kähler metrics and SU(N_x) beta function coefficients on T^6/(_2 ×_2M×) with discrete torsion for vanishing angle  ϕ_xy^(1)=0;                                                                                                                                                                                 (ϕ^(1)_xy,ϕ^(2)_xy,ϕ^(3)_xy)                                                                                                                                                                                                        K_R_x                                                                                          b^A_xy  = 
[ ∑_i=1^3 δ_σ^i_x^σ^i_y   δ_τ^i_x^τ^i_y b̃^A,(i)_xy;      or    δ_σ^1_x^σ^1_y   δ_τ^1_x^τ^1_y  b̃^A_xy ]                                                                                                                            b^M_x (= ∑_i=1^3 b^M,(i)_x  for  (ϕ⃗)=(0⃗))
    for  η__2^(3)=-1  (only for y=x');                                                                                                                                                                                                      (0,0,0)                                                                                                                                                                           g_string/v_1v_2v_3   √(2π) L_x^(i)                                                                                                                                        ∑_i=1^3 δ_σ^i_x^σ^i_y   δ_τ^i_x^τ^i_y  ( -N_y)   I_xy^_2^(i),(j ·k)/4 [ -2   ( 1 + (-1)^τ^2_x σ^2_x  - (-1)^τ^3_x σ^3_x )                                            for  x; -2   ( 1 - (-1)^τ^2_x σ^2_x  + (-1)^τ^3_x σ^3_x )                                     for  x _2^(1) ];                                                                                                                                                                                                     (0,ϕ,-ϕ)                                                                                                                                                                           g_string/v_1v_2v_3   √(2π) L_x^(1)                                                                                                                             δ_σ^1_x^σ^1_y   δ_τ^1_x^τ^1_y   (- N_y )   (I_xy^(2 ·3) + I_xy^_2^(1),(2 ·3) )/4                                                                                                                                                               - 1/2 ( |Ĩ_x^, (2, ·3)| + |Ĩ_x^_2^(1),(2 ·3)|);                                                                                                                                                                                                              ]
beta_coeffs_Kaehler_metricsKähler metrics K_ R_x with two-torus volumes v_i defined in (<ref>) and one-cycle lengths L_x^(i)in (<ref>)
 for matter representations R_x ∈{ (_x,_y), (_x,_y), (_x), (_x) }
and contributions to the beta function coefficients from annulus and Möbius strip topologies, b^ A_xy , b^ M_x. 
For details on the computation of _2-invariant intersection numbers, the interested reader is referred to e.g. <cit.>.
The factor (-1)^2 b_i  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 <cit.>, 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 <cit.> for an extended discussion. 


for the cases at hand. For the generic case with three non-vanishing angles and/or non-rigid D6-branes, the interested reader is referred to <cit.>.
The associated one-loop gauge threshold corrections per open string sector with at least one vanishing angle
are collected in table <ref>,

 [ 3|c|One-loop corrections to gauge couplings on T^6/(_2 ×_2M×) with discrete torsion for  ϕ_xy^(1)=0;                                                                        (ϕ^(1)_xy,ϕ^(2)_xy,ϕ^(3)_xy)                                                                  Δ^A_xy = N_y Δ̃^A_xy  ⊂Δ^A_SU(N_x)                                                                                  Δ^M_x ⊂Δ^M_SU(N_x);                                                                                             (0,0,0)                                                         ∑_i=1^3 b̃^A,(i)_xy   Λ_τ^i_xy,σ^i_xy (v_i)                                                        ∑_i=1^3 b̃^M,(i)_x   Λ_b_i,τ^i_x,σ^i_x (v_i);                                                                                            (0,ϕ,-ϕ)         b̃^A_xy   Λ_τ^1_xy,σ^1_xy (v_1) 
+ N_y   ln2/2 ( I_xy^_2^(2) -  I_xy^_2^(3) ) (sgn(ϕ)/2 -ϕ)                              b̃^M_x   Λ_0,τ^1_x,σ^1_x (v_1)
+ ( |I^_2^(2)_x| -  |I^_2^(3)_x|
)ln2/2;                                                                                                     ]
1-loop-thresholdsOne-loop corrections to the gauge couplings of fractional D6-branes at some vanishing angle on T^6/(_2 ×_2M×) orientifolds with discrete torsion. 
The annulus lattice sums only depend on relative Wilson lines  τ^i_xy≡ |τ^i_x - τ^i_y| ∈{0, 1} and displacements σ^i_xy = |σ^i_x - σ^i_y|∈{0, 1}.
To complete the picture, the gauge thresholds for rigid D6-branes at three non-vanishing angles can be found in <cit.>, where also the conversion from Δ_G_x to the 
holomorphic gauge kinetic function f_G_x using the Kähler metrics from table <ref> is discussed in detail. 
For all D6-brane examples parallel to the (_2^(1))-invariant planes on the aAA lattice of T^6/(_2 ×_6 ×) discussed in this article,
the constant contribution from the Möbius strip topology vanishes due to  |I^_2^(2)_x| =  |I^_2^(3)_x|. The standard Annulus lattice sums are defined 
in (<ref>), while the
hatted lattice sums in the one-loop Möbius corrections are defined in equation (<ref>).


where the following abbreviations of lattice sums for the annulus topology are used,

    Λ_0,0(v) =    - ln( 2 π L^2 η^4 (i v) ),
    
    Λ_τ,σ≠ 0,0(v) =     -  ln( e^-π (σ)^2 v/4| ϑ_1 (τ - i σ  v/2, iv) |/η(iv))^2 ,

with the two-torus volume v defined by

    v = {[    R_1R_2/α'      _2 ( a); √(3)/2r^2/α'      _3 ( A) ].
    ,


and the one-cycle length L as given in equation (<ref>).

For later use in section <ref>, we already provide the asymptotic behaviour of the lattice sums here for two-torus volumina larger than α':

    Λ_τ,σ(v)  v ≫ 1⟶   {[                             π  v/3 -ln (2π L^2)                                     (τ,σ)=(0,0); [3 (1-σ)^2 -1]  π v/6 - 2 δ_σ,0 ln[2 sin(πτ/2)]                                         ≠ (0,0) ]}τ,σ∈{0,1}={[ π v/3 - ln (2π L^2)               (0,0);        π v/3 -2 ln2               (1,0);              -π v/6               (τ,1) ].
    ,

which turns out to be already an excellent approximation for v ≳ 1, cf. figure 2 of <cit.>.

Let us discuss for example the MSSM model with D6-brane data specified in table <ref>. The massless matter spectrum per sector is provided in table <ref>, from which the beta function coefficients b^ A,(i)
– or whenever vanishing the reduced numbers b̃^ A,(i) – can be read off. 
Combining this information with the discrete D6-brane data in table <ref>, we can derive 
the full annulus contributions to the gauge thresholds, which are  given by (with R_1 ≡ R_1^(1)):


    Δ_SU(3)_a^A,  MSSM =     3 ×( Δ̃_aa + Δ̃_aa')
    + 2 ×Δ̃_ab
    + ( Δ̃_ac +  Δ̃_ac')
    + ( Δ̃_ad +  Δ̃_ad')
    + 4 ×(  Δ̃_ah +  Δ̃_ah')
    
    
    =    16  Λ_0,0^|| (_2^(1)) (v_1) + 2  Λ_0,0^||  (v_2) - 4/3 ln 2
    
       v ≫ 1⟶ 16 π/3 v_1 + 2 π/3 v_2 - ln(2^4/3 (2π)^18( R_1^2/α')^16( r_2^2/α')^2 )
    ,
    
    
    Δ_USp(2)_b^A,  MSSM =    3 ×Δ̃_ab 
    + 2 ×Δ̃_bb 
    + Δ̃_bc
    + Δ̃_bd
    + 4 ×Δ̃_bh
    
    =    28  Λ_0,0^|| (_2^(1))(v_1) -2  Λ_0,0^|| _2^(1)(v_2) -2  Λ_0,0^|| _2^(1)(v_3) +  Λ_1,1(v_3) + 11/3 ln 2
    
       v ≫ 1⟶ 28 π/3 v_1  - 2π/3 v_2 - 5π/6 v_3 - ln(2^-11/3 (2π)^24( R_1^2/α')^28(r_2^2/α')^-2(r_3^2/α')^-2)
    ,
    
    
    Δ_U(1)_Y^A,  MSSM =   1/36 Δ_U(1)_a^A,  MSSM 
    + 1/4 Δ_U(1)_c^A,  MSSM
    + 1/4 Δ_U(1)_d^A,  MSSM
    + ( - Δ̃_ac +  Δ̃_ac')
    +  ( - Δ̃_ad +  Δ̃_ad')
    + ( - Δ̃_cd + Δ̃_cd')
    
    
    =   152/3 Λ_0,0^|| (_2^(1)) (v_1)  + 1/3 Λ_0,0^||   (v_2) + 2  Λ_0,0^|| _2^(1)  (v_2) -2  Λ_1,1(v_2) + Λ_0,0^||   (v_3) +Λ_1,1(v_3)
    + 47/18 ln 2
    
       v ≫ 1⟶ 152 π/3 v_1 + 10 π/9 v_2 + π/6 v_3
    - ln( 9 · 2^-47/18 (2π)^54( R_1^2/α')^152/3(r_2^2/α')^7/3( r_3^2/α')
    )
    ,
    
    
    Δ_SU(4)_h^A,  MSSM =    3 ×(  Δ̃_ah +  Δ̃_ah')
    + 2 ×Δ̃_bh
    + (  Δ̃_ch + Δ̃_ch')
    + ( Δ̃_dh +  Δ̃_dh')
    + 4 ×(  Δ̃_hh +  Δ̃_hh')
    
    
    =    16  Λ_0,0^|| (_2^(1))(v_1)  - 2  Λ_0,0^|| (v_2) + 2  ln 2
    
       v ≫ 1⟶16 π/3 v_1 - 2 π/3 v_2 
    - ln( 2^-2 (2π)^14( R_1^2/α')^16(r_2^2/α')^-2)
    ,


where we have made use of the same notation Δ̃_xy=Δ̃_yx as in e.g. <cit.>.
We will come back to the impact of these one-loop correction in section <ref> 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 USp(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 <ref> and the resulting state-per-sector counting in tables <ref> and <ref>
with the following results:


    Δ_SU(3)_a^A,  L-R =   {[                 16  Λ_0,0^|| (_2^(1))(v_1)                                 L-R I & II; 10  Λ_0,0^|| (_2^(1))(v_1) + 6  Λ_0,1(v_1)                                    L-R IIb; 13  Λ_0,0^|| (_2^(1))(v_1) + 3  Λ_0,1(v_1)                                    L-R IIc ]}
        -4  Λ_0,0^|| (v_2) - 10/3 ln 2
          ,
     
    
    Δ_USp(2)_b^A,  L-R =   {[                33  Λ_0,0^|| (_2^(1))(v_1)                                L-R I & II; 24 Λ_0,0^|| (_2^(1))(v_1) + 9  Λ_0,1(v_1)                                   L-R IIb; 28 Λ_0,0^|| (_2^(1))(v_1) + 5  Λ_0,1(v_1)                                   L-R IIc ]}
    - 4  Λ_0,0^|| _2^(1)(v_2) -20/3 ln 2
          ,
    
    
    Δ_USp(2)_c^A,  L-R =   {[                33  Λ_0,0(v_1)^|| (_2^(1))                                L-R I & II; 24 Λ_0,0(v_1)^|| (_2^(1)) + 9  Λ_0,1(v_1)                                   L-R IIb; 29 Λ_0,0(v_1)^|| (_2^(1)) + 4  Λ_0,1(v_1)                                   L-R IIc ]}
    - 4  Λ_0,0^|| _2^(1)(v_2) -20/3 ln 2
          .

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,


    Δ_SU(3)_h_1/h_2^A,  L-R I only =     16  Λ_0,0^|| (_2^(1))(v_1) ∓20/3 ln 2
      ,

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:


    Δ_U(1)_B-L^A,  L-R II =   704/3 Λ_0,0^|| (_2^(1))(v_1) 
    + 16/3 Λ_0,0^||  (v_2)
    + 16  Λ_0,0^|| (v_3) 
    + 16   Λ_0,0^|| _2^(1)(v_3)  
    + 674/27 ln 2
      .



Finally, for the PS models, the analogous computations using the D6-brane data in tables <ref> and <ref> and the state-per-sector counting in tables <ref> and <ref> leads to:


    Δ_SU(4)_a^A,  PS =    16  Λ_0,0^|| (_2^(1))(v_1) + {[   4  Λ_0,0^|| (v_2) - 4/3 ln 2                           PS I; -8  Λ_0,0^|| (v_2) - 16/3 ln 2                          PS II ].
      ,
    
    
    Δ_USp(2)_b^A,  PS = Δ_USp(2)_c^A,  PS =     33  Λ_0,0^|| (_2^(1))(v_1) - 4 Λ_0,0^|| _2^(1)(v_2) +  {[                   8/3 ln 2                       PS I; -2  Λ_1,1(v_2) - 26/3 ln 2                      PS II ].
      ,
    
    
    Δ_SU(6_ I/2_ II)_h^A,  PS =   {[                         16  Λ_0,0^|| (_2^(1))(v_1) - 4  Λ_0,0^|| (v_2) + 128  ln 2                                                                               PS I; 48  Λ_0,0^|| (_2^(1))(v_1)  - 4  Λ_0,0^|| _2^(1)(v_2)  -4  Λ_1,1(v_2)- 1280/3 ln 2                                                                              PS II ].
      .

As for the MSSM and L-R symmetric examples, any of these gauge threshold contributions grows (asymptotically) linearly with v_1.


For the Möbius strip topology, modified lattice sums,

    Λ_b,τ,σ(v̂)
        with    v̂≡v/1-b,

with b ∈{0,1/2} appear. 
For the untilted torus with b=0, i.e. the two-torus T^2_(1) in the T^6/(_2 ×_6 ×) examples at hand, and only discrete Wilson lines and displacements τ,σ∈{0,1},  
the lattice sum on T^2_(1) is simply given by

    Λ_0,τ,σ(v̂) τ,σ∈{0,1}≡Λ_0(v̂) =Λ_0,0(v) - 2  ln 2
    ,


where the constant term -2  ln 2 stems from the replacement <cit.> L^2 → 2  L̂^2 with L̂^2=2   L^2 in the first line of equation (<ref>).

However, for (τ,σ)=(1,1) on tilted tori b=1/2, the sign factor (-1)^τσ in the beta function coefficients in table <ref>
indicates that also the lattice sum for the hexagonal two-tori  T^2_(2)× T^2_(3) needs to be modified in a yet unknown way.
We will therefore write the formal expression (<ref>) for the lattice sums throughout the computation and only estimate their size via the ansatz for the asymptotics Λ_1/2,τ,σ(v̂)
v ≫ 1⟶π c^1/2_τ,σ/3 v with coefficients c^1/2_τ,σ =  O(1) when discussing hierarchies among compact direction in section <ref>.

Using the same D6-brane data as for the annulus amplitudes, we obtain as joint expressions for all models,


    Δ^ M,  all_SU(3/4)_a = Δ^ M,  MSSM_SU(4)_h = Δ^ M,  PS I_SU(6)_h
    =    -4   Λ_0,0,0^|| (_2^(1))(v̂_1)
     +   2   Λ_1/2,1,1 (v̂_2)
    -2  Λ_1/2,1,1 (v̂_3)
       ,
    
    Δ^ M,  all_USp(2)_b = Δ^ M,  PS + L-R_USp(2)_c 
    =    - 6  Λ_0,0,0^|| (_2^(1))(v̂_1)
    -  Λ_1/2,1,1 (v̂_2)
    -   Λ_1/2,0,0^|| _2^(1) (v̂_3)
      ,

since Möbius strip contributions are independent of the _2 ×_2 eigenvalues.
The remaining massless and anomaly-free gauge symmetries of the MSSM, PS II and L-R symmetric II  model receive the following 
Möbius strip contributions to the one-loop gauge thresholds,


    Δ^ M,  MSSM_U(1)_Y =    - 38/3 Λ_0,0,0^|| (_2^(1))(v̂_1)
     +   Λ_1/2,0,0^_2^(1) (v̂_2) -   2/3 Λ_1/2,1,1 (v̂_2)
     +    5/3 Λ_1/2,1,1 (v̂_3)
       ,
    
    Δ^ M,  L-R II_U(1)_B-L =   
    - 176/3 Λ_0,0,0^|| (_2^(1))(v̂_1) 
     + 8   Λ_1/2,0,0^_2^(1)  (v̂_2)
    + 16/3 Λ_1/2,1,1 (v̂_2)
     -8  Λ_1/2,0,0^_2^(1)  (v̂_3)
    - 16/3 Λ_1/2,1,1 (v̂_3)
      ,
    
    Δ^ M,  PS II_SU(2)_h =    
    -12  Λ_0,0,0^|| (_2^(1))(v̂_1)
     + 2  Λ_1/2,0,0^_2^(1) (v̂_2)
    -2  Λ_1/2,0,0^_2^(1)  (v̂_3)
      .

At this point, it is noteworthy that the absolute values of the negative coefficients  in equations (<ref>) and (<ref>)
of the lattice sum Λ_0,0,0^|| (_2^(1))(v̂_1) on T^2_(1) are always smaller than the positive coefficients in the corresponding Annulus contributions, and thus the one-loop threshold correction of each gauge group still grows (asymptotically) linearly with the two-torus volume v_1 in any model considered in this article.
Moreover, the coefficients of Λ_1/2,1,1 (v̂_2) and Λ_1/2,1,1 (v̂_3) have identical absolute value and opposite sign for SU(3/4)_a^ all, SU(4)_h^ MSSM, SU(6_ I/2_ II)_h^ PS. When choosing isotropic torus volumes v_2=v_3, these yet unknown contributions thus cancel.





 §.§ 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,

    M_Planck^2/M_string^2 = 4π/g^2_string  v_1v_2v_3
      with 
    g_string≡ e^ϕ_10,


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 <cit.>, 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 _2^(3) on T^6/(_2 ×_6 ×) with discrete torsion 
 enforces all D6-branes to lie along the (_2^(1))-invariant direction on the first two-torus T^2_(1) with length scale R_1^(1), while gravitational forces also propagate along the perpendicular direction and are thus sensitive to v_1 =R_1^(1) R_2^(1)/α'.
The tree-level gauge coupling in equation (<ref>) of SU(3)_a and SU(4)_h of the MSSM example can e.g. be rewritten using the relation (<ref>),

    8π^2/g^2_SU(3)_a/SU(4)_h,tree = π/2√(6)√(R_1^(1)/R_2^(1))M_Planck/M_string
    ,

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^(1) and R_1^(1) to arrive at some phenomenologically appealing order of magnitude of the gauge couplings as already briefly sketched in <cit.>. 
Such large hierarchies in the D6-brane models at hand would provide explicit examples for the low-string scale scenario, see e.g. <cit.>.

We will now also take into account the one-loop gauge threshold corrections, which for the MSSM example of section <ref>
 have the following asymptotic behaviour,

    Δ^ A+ M,  MSSM_SU(3)_a/2v ≫ 1⟶      
    2π  v_1 +π/3(v_2 + c^1/2_1,1(v_2 - v_3 )  ) - ln( (R_1^(1)/R_2^(1)  v_1 )^6  v_2 ) - 12
    ,
    
    Δ^ A+ M,  MSSM_USp(2)_b/2v ≫ 1⟶     11 π/3  v_1 - π/3(1 + c^1/2_1,1/2)  v_2 - 3π/4  v_3
    - ln((R_1/R_2  v_1 )^11 (v_2)^-1 v_3)^-3/2) - 11
    ,
    
    Δ^ A+ M,  MSSM_U(1)_Y/2v ≫ 1⟶     209 π/9  v_1 + (8-  c^1/2_1,1 )π/9  v_2
    +  (3 + 10   c^1/2_1,1) π/36  v_3
    - ln( ( R_1/R_2  v_1)^19   v_2^5/3 v_3^1/2)
     - 36
    ,
    
    Δ^ A+ M,  MSSM_SU(4)_h/2v ≫ 1⟶      
    2π  v_1 +π/3( - v_2 + c^1/2_1,1(v_2 - v_3 )  )
     - ln( ( R_1/R_2  v_1 )^6 v_2^-1) - 7 
    ,

where we  have rewritten the logarithmic terms in terms of the complex structure parameter R_2^(1)/R_1^(1)
 on T^2_(1) and the Kähler moduli v_i using the definition (<ref>) and evaluated the constant contributions, e.g. ln(2^4/3 (2π)^18( 2/√(3))^2 ) ≈ 34.29 in the annulus contribution Δ^ A,  MSSM_SU(3)_a.


Let us stress here again that in all MSSM, L-R symmetric and PS models of section <ref>,
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 orientifold-invariant 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 (<ref>) are classified as reliable:


  * 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.

  * 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 <ref>.

Let us discuss each of these two cases further: in the first case of two isotropic tori , 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 (<ref>), and at least for  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 USp(2)_b × U(1)_Y × SU(4)_h charges in the 
last block of table <ref>.
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 (<ref>)
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 USp(2)_b × U(1)_Y 
the contributions of v_1 cancel the negative constant contributions. To achieve v_1 ≳ 1, we further assume R_1^(1)∼ R_2^(1)∼√(α'), and equation (<ref>)
favours a high string scale M_string. Such a choice of scales is in turn consistent with very small volumes v_i ∼ O(1) in (<ref>) and a not too weak string coupling  g_string.


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 (<ref>) leading to v_1v_2v_3 ∼ 10^11. In order to achieve α^-1_QCD∼ O(1), equation (<ref>) then requires R_1^(1)/R_2^(1)∼ 10^-12, which is problematic if we require v_i ∈{1,2,3}≳ 1 and R_1,2^(1)/√(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 (<ref>) 
- as well as the analogous expressions for all L-R symmetric and PS models -
receive 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 (<ref>) and discuss the possibility of gauge coupling unification.
The tree-level relation (<ref>) together with the fact that in the MSSM example only USp(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 USp(2)_b ≃ SU(2)_weak and by roughly a factor of three for U(1)_Y.
Figure <ref> shows that the volume Vol(Π_b) ∝1/g_USp(2)_b^2 decreases by at most 20 % when reaching the upper bound |ε^(1)_4-5| ≈ 0.4, while Vol(Π_d)  increases at the same time by about 10 %.
Thus, the one-loop corrections of USp(2)_b × U(1)_Y both have to be sufficiently negative compared to the one-loop correction of SU(3)_a. 
Using equation (<ref>),  we find that for the unknown constant c^1/2_1,1 < - 3/22 at least the prefactor of v_3 is negative for both gauge threshold differences
[Δ^ A+ M,  MSSM_USp(2)_b - Δ^ A+ M,  MSSM_SU(3)_a]/2
and [Δ^ A+ M,  MSSM_U(1)_Y - Δ^ A+ M,  MSSM_SU(3)_a]/2 under consideration, while for 
5/4 < c^1/2_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_iτ^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 _2 singularities of toroidal orbifolds T^6/(_2×_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/(_2 ×_6 ×) orientifold. To study the deformations of such singularities, we employ the techniques of the hypersurface formalism developed for the _2 ×_2 toroidal orbifold with discrete torsion and extend them to the _2 ×_6 toroidal orbifold, in a similar spirit as was done for the _2 ×_6' toroidal orbifold before. The extension boils down to modding out an additional _3 ⊂_6 action along a four-torus in the hypersurface formalism for the _2 ×_2 toroidal orbifold. 
  Since the _6 action does not constrain the geometry of the first two-torus, the deformations in the _2^(1)-twisted sector are structurally different from the ones in the _2^(i=2, 3)-twisted sectors. This finding agrees with the difference in the relevant Hodge numbers as follows:
 we can identify six deformation moduli (four real and one complex) in the _2^(1)-twisted sector and four (real) deformation moduli in the _2^(i=2,3)-twisted sector each, as presented in the hypersurface equation (<ref>). This inherent difference among the _2^(i)-twisted sectors presents new challenges that where absent for the orbifold groups _2 ×_2 and _2 ×_6', which act isotropically on all three two-tori. It also forces us to discuss the deformations in the _2^(1)-twisted sector in section <ref>
 separately from the ones in the _2^(i=2,3)-twisted sectors in section <ref>.  

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 three-cycles, 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) ^2/_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 _2^(2  or  3)-twisted sector. Only the exceptional three-cycles associated to the _2^(1) deformation parameters ε_0^(1) and ε_3^(1) can be fully described globally, with the first deforming only the singularity at the origin of T^4_(1)/_2^(1) and the latter deforming a _3-invariant orbit of three singularities on T^4_(1)/_6. By computing the volume-dependence on these deformation parameters for bulk and fractional three-cycles parallel to the -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_0^(1) and ε_3^(1) 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 It:Method.    


When applying these techniques to global intersecting D6-brane models, we observe in first instance that -even exceptional three-cycles give rise to deformation moduli with a flat direction, while D6-branes wrapping -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 low-energy effective action from first principles through CFT computations <cit.> or through dimensional reduction <cit.>. 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 -odd exceptional wrapping number.
 An overview of this counting for the previously constructed and phenomenologically interesting global D6-brane models on T^6/(_2 ×_6 ×) is provided in table <ref>, 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 <ref>, 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 -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 _2×_2 eigenvalues of the respective D6-branes: at the orbifold point, the physics only depends on the relative _2×_2 eigenvalues among the D6-branes (e.g. in the computation of the particle spectra), whereas the absolute _2×_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 _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_(1)^2 parallel to the (_2^(1))-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_(1)^2.
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 <ref>: 
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_2^(1)/R_1^(1) of T_(1)^2.
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 D6-brane 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 RR- and 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 <cit.>. 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 <cit.> 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 <cit.>. 





Acknowledgements: 
W.S. would like to thank Iñaki García-Etxebarria for enlightening conversations.
This work is partially supported by the Cluster of Excellence `Precision Physics, Fundamental Interactions and Structure of Matter' (PRISMA) DGF no. EXC 1098,
the DFG research grant HO 4166/2-2 and the DFG Research Training Group `Symmetry Breaking in Fundamental Interactions' GRK 1581. W.S. is supported by the ERC Advanced Grant SPLE under contract ERC-2012-ADG-20120216-320421, by the grant FPA2012-32828 from the MINECO, and the grant SEV-2012-0249 of the “Centro de Excelencia Severo Ochoa" Programme.
















§ MÖBIUS TRANSFORMATIONS

In section <ref>, a two-torus T^2 was described as an elliptic curve E in the weighted projective space ^2_112 through eq. (<ref>) with a built-in _2 symmetry acting on the homogeneous coordinate y of weight 2. The fixed points of this _2 action correspond to the roots of the polynomial F(x,v) in equation (<ref>). 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 <ref> we provide a list of the roots in all coordinate patches for a square two-torus and a hexagonal two-torus.


[ 6|c|_2^(i) fixed points per T_(i)^2 in various coordinate patches;                         3|c||_2^(i) fixed point on square T_(1)^2                   3|c|_2^(i) fixed point on hexagonal T_(l=2,3)^2;                                                     fixed point α                                                    x_1-coordinate                                                    v_1-coordinate                                                     fixed point α                                                    x_l-coordinate                                                    v_l-coordinate;                                                                 1                                                           x_1 = ∞                                                           v_1 = 0                                                                 1                                                           x_l = ∞                                                           v_l = 0;                                                                 2                                                           x_1 = 1                                                           v_1 = 1                                                                 3                                                           x_l = 1                                                           v_l = 1;                                                                 3                                                           x_1 = 0                                                           v_1 = ∞                                                                 2                                                           x_l = ξ                                                         v_l = ξ^2;                                                                 4                                                          x_1 = -1                                                          v_1 = -1                                                                 4                                                         x_l = ξ^2                                                           v_l = ξ;                                                                   ]
OverviewZ2FixedPointsPerT2Overview of the _2^(i) fixed points on the square torus T_(1)^2 and the hexagonal tori T_(l=2,3)^2 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 <ref>.


There exist automorphisms λ_α:E→ E of the elliptic curve <cit.> interchanging the coordinates x and v through the action:

    λ_α: ( [ x; v ]) ↦λ_α( [ x; v ]) = 1/√(2 ϵ_α^2 + ϵ_βϵ_γ)( [            ϵ_α ϵ_α^2 + ϵ_βϵ_γ;              1          - ϵ_α ]) ( [ x; v ]),       y ↦ y,
 
where the ϵ_α correspond to the roots entering in equation (<ref>). These automorphisms λ_α interchange the _2 fixed points: 

    λ_2:  1 ↔ 2,   3 ↔ 4,   λ_3: 1 ↔ 3,   2 ↔ 4,   λ_4: 1 ↔ 4,   2 ↔ 3,

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 ϵ_2 = 1, ϵ_3 = 0 and ϵ_4=-1, while a hexagonal two-torus is characterised by the roots ϵ_2 = ξ, ϵ_3=1 and ϵ_4 = ξ^2. The automorphisms λ_α also allow to exchange Lag lines among each other. For a square T^2 the Lag lines aI-aIV from table <ref> behave as follows under Möbius transformations:  λ_2 swaps the Lag lines aI ↔ aIV leaving aII and aIII invariant, λ_3 exchanges aI ↔ aIV and aII ↔ aIII, and finally λ_4 swaps aII ↔ aIII leaving aI and aIV invariant. 
For a hexagonal T^2, the Lag lines from table <ref> 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.






§ CORRECTION TERMS FOR _2^(1)-DEFORMATIONS

In section <ref> we observed that certain deformations in the _2^(1)-twisted sector deform too many singular orbits. The deformations with parameters ε_4+5^(1) and ε_4-5^(1) for instance deform the singularity (33), inadvertently. Such a feature was for the first time observed in <cit.> and appears to be intrinsic to _2-twisted sectors subject to an additional _3 action on the full covering four-torus T^4, as testified in table <ref>. 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_4+5^(1) example, we have to switch on a correction term ε_3^(1) as a function of ε_4+5^(1). In order to determine this functional dependence, we consider a sLag two-cycle on T^4_(1)/_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_4+5^(1)≠ 0 elsewhere. This allows us to find a series expansion for ε_3^(1) as a function of ε_4+5^(1)≡ε_ in:

    ε_3^(1)(ε_ in) =    - 1/4ε_ in^2 - 5/4 × 3!ε_ in^3- 31/4 × (3!)^2ε_ in^4   - 13 × 17/4 × (3!)^3ε_ in^5 - 1753/4 × (3!)^4ε_ in^6   - 11×13×53/2× (3!)^5ε_ in^7  
          - 167×421/2 × (3!)^6ε_ in^8  - 137 × 3331/8 × (3!)^6ε_ in^9 - 13× 17 × 41× 43/4×(3!)^6ε_ in^10    +  O(ε_ in^11 ).



We have to point out that this approach only works for small (initial) deformations ε_ in, as long as the correction term ε_3^(1) (ε_ in) remains smaller than the original deformation. Applying this requirement for the (conservative) assumption |ε_3^(1)| ∼1/10ε_ 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 <ref>. In each particular case, we apply the same method as described above and construct a Taylor expansion in terms of the initial deformation. 

[ 4|c|Deformations & correction terms in the _2^(1) sector;                                                                                        3|c|Deformation parameters;                                         Correction Terms                                                  ε_3^(1)                                                ε_4+5^(1)                                                ε_4-5^(1);                                                  ε_3^(1)                                                        -                                              eq. (<ref>)                                              eq. (<ref>);                                                ε_4+5^(1)                                              eq. (<ref>)                                                        -                                                        -;                                                          ]
SummDefCorrZ21Overview of correction terms for various _2^(1) deformation parameters with the corresponding Taylor expansion indicated by the equations in the main text.

More explicitly, a non-vanishing deformation parameter ε_3^(1)≡ε_ in deforms the _2^(1)-fixed point orbits e_4^(1) and e_5^(1) unintentionally, as pointed out in table <ref>. This effect can be undone by turning on a correction terms ε_4+5^(1), as a function of ε_3^(1). Following the same logic as presented above, one finds the following series expansion for ε_4+5^(1):

    ε_4+5^(1) (ε_ in)    = - 1/2ε_ in^2 -  1/2× 3ε_ in^3  -  19/2^5× 3^2ε_ in^4 -  1/2^5ε_ in^5  -  1/2^6× 3^2ε_ in^6  -  5^2 × 17/2^9× 3^3ε_ in^7 
        -  149 × 463/2^13× 3^5ε_ in^8   -  131 × 223/2^12× 3^5ε_ in^9 -  266401/2^14× 3^6ε_ in^10 +  O( ε_ in^11).

Turning on the deformation parameter ε_4-5^(1)≡ε_ in on the other hand deforms the _2^(1)-fixed point orbit e_3^(1) unwillingly, as can be seen from table <ref>. This effect can be undone by a correction term ε_3^(1) depending on the initial deformation parameter ε_4-5^(1), whose functional dependence follows from repeating the same logic as above: 

    ε_3^(1) (ε_ in)    = - 1/2^2 × 3ε_ in^2 + 1/2^3 × 3 √(3)ε_ in^3 + 1/2^4 × 3^2 ε_ in^4 - 11/2^5 × 3^3 √(3)ε_ in^5 - 7/2^6 × 3^4ε_ in^6 
         + 11/2^3 × 3^5 √(3)ε_ in^7 + 59/2^7 × 3^7ε_ in^8 - 3019/2^9 × 3^6 √(3)ε_ in^9 - 25/2^6 × 3^9ε_ in^10 +  O(ε_ in^11).

After having determined the relevant correction term for each of the deformations, one can also explicitly verify how the correction terms restore the singular nature of the orbits, as depicted in the lower plots of figure <ref> (f) and (g).





§ 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 <ref>. Table <ref> contains the massless open string spectrum of the global prototype MSSM model 
discussed in section <ref>, and
 table <ref> contains the counting of massless open string states per intersection sector for the global PS  prototype I model discussed in section <ref>. 
 Table <ref> displays the counting of the observable massless open string sector of all L-R symmetric models discussed in section <ref> as well as of the PS prototype II model.
 Finally, table <ref> provides the sector-per-state counting for the different `hidden' completions of the massless matter spectrum per L-R symmetric model.
 
 
 The presentation in terms of matter states per sector is indispensable for the computation of the gauge threshold corrections in section <ref> and is useful for the discussion about potentially flat directions for twisted moduli in section <ref> to <ref>.


[h]
21cm



[ 10|c| Total amount of massless matter per sector for a 5-stack MSSM model on the aAA lattice of T^6/(_2 ×_6 ×) with discrete torsion;                                                                                                               (χ^x (ω^k ∈{0,1,2} y))                                                                                                                                  y=a                                                                                                                                 y=a'                                                                                                                                  y=b                                                                                                                                  y=c                                                                                                                                 y=c'                                                                                                                                  y=d                                                                                                                                 y=d'                                                                                                                                  y=h                                                                                                                                 y=h';                                                                                                                                 x= a                                                                                                                              (0,0,0)                                                                                                              ([2]_2-2+2,-1,1)(0,0,0)                                                                                                                              (2,1,0)                                                                                                                           (-4,-1,-1)                                                                                                                              (0,0,0)                                                                                                                              (2,1,0)                                                                                                                            (-2,-1,0)                                                                                                                           ([2],1,-1)                                                                                                                     ([2]_-2+2+2,0,0);                                                                                                                             2-3
x= b                                                                                                                                 2|c|                                                                                                         (0,3 +[2]/2,-3+[2]/2)(0,0,0)                                                                                                                     (0,-3 + [2],[4])                                                                                                                                    -                                                                                                                      (0,3+[2],3+[2])                                                                                                                                    -                                                                                                                            (-2,-1,0)                                                                                                                                    -;                                                                                                                             4-4
x= c                                                                                                                                 3|c|                                                                                                                      (0,[4]/2,[4]/2)                                                                                                      ([2]_2+2-2,3+[2],-3+[2])(0,0,0)                                                                                                                       (0,[4],-3+[2])                                                                                                                        (0,[4],3+[2])                                                                                                                              (0,0,0)                                                                                                                           (-4,-1,-1);                                                                                                                             5-6
x= d                                                                                                                                 5|c|                                                                                                                (0,-3 +|2|/2,3+|2|/2)                                                                                                     (0,3+[2],-3+[2])([2]_-2+2+2,0,0)                                                                                                                              (2,1,0)                                                                                                                              (2,0,1);                                                                                                                             7-8
x= h                                                                                                                                 7|c|                                                                                                                              (0,0,0)                                                                                                              ([2]_2-2+2,-1,1)(0,0,0);                                                                                                                                      ]

Overview of the total amount of chiral and vector-like massless (open string) matter per sector x(ω^k y) for the global five-stack MSSM-like model with fractional D6-brane configuration given in table <ref>. If the net-chirality |χ^x(ω^ky)| < φ^x(ω^ky), the sector x(ω^ky) comes with a set of vector-like pairs of matter states, whose multiplicity corresponds to n^x(ω^ky)_NC≡φ^x(ω^ky) - |χ^x(ω^ky)|, e.g. n^a(ω^0h)_NC=[2] denotes one vector-like pair of bifundamental states in the sector a(ω^0h). 
The diagonal entries φ^x(ω x) = φ^x(ω^2 x) = φ^_x/2 count the number of states in the adjoint representation for the x-stack, e.g. the entry (0,[4]/2,[4]/2)
in the c(ω^k c)_k=0,1,2 sectors corresponds to four multiplets in the adjoint representation arising at intersections of c with its orbifold images (ω^k c)_k=1,2.
The upper and lower entries in the x(ω^k x') sectors count the number of states in the antisymmetric and symmetric representation, respectively. The entries ([2]_2-2+2,-1,1)(0,0,0) for the a(ω^k a') 
sectors e.g. amount to one vector-like pair of antisymmetric reprentations from the aa' sector plus a second pair spread over the a(ω a')+a(ω^2 a') sectors.
The lower index of e.g. 2-2+2 in the aa' sector, moreover, determines the decomposition of ∑_i=1^3 δ^0_σ^i_xyδ^0_τ^i_xyb̃^ A,(i)_xy introduced in table <ref>, which is required as input for the computation of the one-loop gauge threshold Δ_aa'^ A according to 
table <ref>.






 
 

21cm

[ 7|c| Total amount of matter per sector for the Pati-Salam model prototype I on  T^6/(_2 ×_6 ×);                                                                          (χ^x (ω^k∈{0,1,2} y))                                                                                            y=a                                                                                           y=a'                                                                                            y=b                                                                                            y=c                                                                                            y=h                                                                                           y=h';                                                                                           x= a                                                                                        (0,0,0)                                                                        ([2]_2-2+2,-1,1)(0,0,0)                                                                                        (2,1,0)                                                                                      (-2,0,-1)                                                                               ([2]_2+2-2,1,-1)                                                                               ([2]_-2+2+2,0,0);                                                                                       2-3
x= b                                                                                           2|c|                                                                   (0,3 +[2]/2,-3+[2]/2)(0,0,0)                                                                            ([2]_2-2+2,[4],[4])                                                                                      (-2,-1,0)                                                                                              -;                                                                                       4-4
x= c                                                                                           3|c|                                                                    (0,3+[2]/2,-3+[2]/2)(0,0,0)                                                                                        (2,0,1)                                                                                              -;                                                                                       5-5
x= h                                                                                           4|c|                                                                                        (0,0,0)                                                                        ([2]_2-2+2,-1,1)(0,0,0);                                                                                                ]
Overview of the chiral and vector-like matter spectrum per sector x(ω^k y) of the PS I model with D6-brane data specified in table <ref>. 
For details of the notation see the caption of table <ref>.













[h]
21cm



[ 9|c| Total amount of matter per sector: universal visible part for all L-R symmetric models and Pati-Salam II on  T^6/(_2 ×_6 ×);                                                                                                            (χ^x (ω^k∈{0,1,2} y))                                                                                                                              y=a                                                                                                                             y=a'                                                                                                                              y=b                                                                                                                              y=c                                                                                                                         y=d_ L-R                                                                                                                        y=d_ L-R'                                                                                                                       y=h_ PS II                                                                                                                     y=h_ PS II^';                                                                                                                             x= a                                                                                                                          (0,0,0)                                                                                                          ([2]_2-2+2,-1,1)(0,0,0)                                                                                                                          (2,0,1)                                                                                                                        (-2,-1,0)                                                                                                                 ([2]_-2+2+2,0,0)                                                                                                                 ([2]_2+2-2,1,-1)                                                                                                                       ([2],-1,1)                                                                                                                        ([2],0,0);                                                                                                                         2-3
x= b                                                                                                                             2|c|                                                                                                     (0,3 +[2]/2,-3+[2]/2)(0,0,0)                                                                                                             ([2]_2-2+2,[4], [4])                                                                                                                          (2,1,0)                                                                                                                                -                                                                                                                   (0,[4],-3+[2])                                                                                                                                -;                                                                                                                         4-4
x= c                                                                                                                             3|c|                                                                                                     (0,3 +[2]/2,-3+[2]/2)(0,0,0)                                                                                                                        (-2,0,-1)                                                                                                                                -                                                                                                                    (0,3+[2],[4])                                                                                                                                -;                                                                                                                        x= d_ L-R                                                                                                                            4|c||                                                                                                                          (0,0,0)                                                                                                          ([2]_2-2+2,-1,1)(0,0,0)                                                                                                                             2|c|;                                                                                                                       x=h_ PS II                                                                                                                            6|c||                                                                                                                  (0,[4]/2,[4]/2)                                                                                                  ([2]_2-2+2,3+[2],-3+[2])(0,0,0);                                                                                                                                  ]

Overview of the massless visible matter states per intersection sector of the L-R symmetric models and the massless visible plus `hidden' matter of
the PS II model with D6-brane data specified in tables <ref> and <ref>, respectively. 
For the sector-per-state counting of the `hidden' matter in the L-R symmetric models see table <ref>.
Details of the notation are given in the caption of table <ref>.







 
 

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[ 9|c| Total amount of `hidden' matter per sector for  the L-R symmetric models on T^6/(_2 ×_6 ×) with discrete torsion;                                                                                                                                                                                                                                 4|c|| L-R I                                                                                                           4|c| L-R II;                                                                                                         (χ^x (ω^k y))                                                                                                                 y=h_1                                                                                                                y=h_1'                                                                                                                 y=h_2                                                                                                                y=h_2'                                                                                                                 y=h_1                                                                                                                y=h_1'                                                                                                                 y=h_2                                                                                                                y=h_2';                                                                                                                  x= a                                                                                                             (0,-1,-1)                                                                                                               (0,0,0)                                                                                                               (0,1,1)                                                                                                               (0,0,0)                                                                                                            ([2],-1,1)                                                                                                             ([2],0,0)                                                                                                            ([2],1,-1)                                                                                                             ([2],0,0);                                                                                                                   x=b                                                                                                              (2,0,-1)                                                                                                                     -                                                                                                              (-2,0,1)                                                                                                                     -                                                                                                 (0_0,-2,0,[4],-3+[2])                                                                                                                     -                                                                                                   (0_0+2+0,[4],3+[2])                                                                                                                     -;                                                                                                                   x=c                                                                                                              (2,-1,0)                                                                                                                     -                                                                                                              (-2,1,0)                                                                                                                     -                                                                                                  (0_0,-2,0,3+[2],[4])                                                                                                                     -                                                                                                  (0_0+2+0,-3+[2],[4])                                                                                                                     -;                                                                                                                  x= d                                                                                                               (0,1,1)                                                                                                               (0,0,0)                                                                                                             (0,-1,-1)                                                                                                               (0,0,0)                                                                                                            ([2],1,-1)                                                                                                             ([2],0,0)                                                                                                            ([2],-1,1)                                                                                                             ([2],0,0);                                                                                                                x= h_1                                                                                                               (0,0,0)                                                                                                ([2]_2+2-2,-1,1(0,0,0)                                                                                                      ([2]_-2+2+2,0,0)                                                                                                      ([2]_2-2+2,-1,1)                                                                                                       (0,[4]/2,[4]/2)                                                                                       ([2]_2-2+2,3+[2],-3+[2])(0,0,0)                                                                                                  ([2]_-2+2+2,[4],[4])                                                                                              ([2]_2+2-2,-3+[2],3+[2]);                                                                                                         2-36-7
x= h_2                                                                                                                  2|c|                                                                                                               (0,0,0)                                                                                               ([2]_2+2-2,-1,1)(0,0,0)                                                                                                                                                                                                                                                                                                                                                   (0,[4]/2,[4]/2)                                                                                       ([2]_2-2+2,3+[2],-3+[2])(0,0,0);                                                                                                                                                                                                                               4|c|| L-R IIb                                                                                                          4|c| L-R IIc;                                                                                                         (χ^x (ω^k y))                                                                                                                 y=h_1                                                                                                                y=h_1'                                                                                                                 y=h_2                                                                                                                y=h_2'                                                                                                                 y=h_1                                                                                                                y=h_1'                                                                                                                 y=h_2                                                                                                                y=h_2';                                                                                                                  x= a                                                                                                             (0_2,0,0)                                                                                                         (0_2,0_1,0_1)                                                                                                             (0_2,0,0)                                                                                                         (0_2,0_1,0_1)                                                                                                             (-2,-1,0)                                                                                                               (2,1,0)                                                                                                           (0_2,0,0_1)                                                                                                           (0_2,0,0_1);                                                                                                                   x=b                                                                                                           (0,0_5,0_4)                                                                                                                     -                                                                                                           (0,0_5,0_4)                                                                                                                     -                                                                                                           (0,[4],[4])                                                                                                                     -                                                                                                           (0,0_5,0_5)                                                                                                                     -;                                                                                                                   x=c                                                                                                           (0,0_4,0_5)                                                                                                                     -                                                                                                           (0,0_4,0_5)                                                                                                                     -                                                                                                       (0,3+[2],3+[2])                                                                                                                     -                                                                                                           (0,0_4,0_4)                                                                                                                     -;                                                                                                                  x= d                                                                                                             (0_2,0,0)                                                                                                         (0_2,0_1,0_1)                                                                                                             (0_2,0,0)                                                                                                         (0_2,0_1,0_1)                                                                                                               (2,1,0)                                                                                                             (-2,-1,0)                                                                                                             (0_2,0,0)                                                                                                         (0_2,0_1,0_1);                                                                                                                x= h_1                                                                                                       (0,[4]/2,[4]/2)                                                                                      ([2]_2-2+2,3+[2],-3+[2] )(0,0,0)                                                                                                  ([2]_-2+2+2,[4],[4])                                                                                              ([2]_2+2-2,-3+[2],3+[2])                                                                                                       (0,[5]/2,[5]/2)                                                                                                   ([2]_-2+2+2,[5],[5]                                                                                                           (0,0_5,0_5)                                                                                                           (0,0_5,0_5);                                                                                                         2-36-7
x= h_2                                                                                                                  2|c|                                                                                                       (0,[4]/2,[4]/2)                                                                                      ([2]_2-2+2,3+[2],-3+[2] )(0,0,0)                                                                                                                  2|c|                                                                                                       (0,[5]/2,[5]/2)                                                                                           ([2]_-2+2+2, [5],[5](0,0,0);                                                                                                                       ]

Overview of the total amount of `hidden' chiral and vector-like matter per sector x(ω^k h_i^(')) and h_i(ω^k h_j^('))
in the L-R symmetric models with D6-brane data specified in table <ref>.
The counting of the associated visible matter states per sector is displayed in table <ref>, and the notation is explained in the caption of table <ref>.













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