Yukawa couplings from magnetized D-brane models on non-factorisable tori

We compute Yukawa couplings in type IIB string theory compactified on a non factorisable six-torus in the presence of D9 branes and fluxes. The setting studied in detail, is obtained by T-dualising an intersecting brane configuration of type IIA theory compactified on a torus generated by the SO(12) root lattice. Particular deformations of such torus are taken into account and provide moduli dependent couplings. Agreement with the type IIA result is found in a non trivial way. The classical type IIB calculation gives also information on a factor accessible only by quantum computations on the type IIA side.


Introduction
One possible extension of the Standard Model of particle physics is to assume the existence of extra dimensions as motivated by string theory. The appeal of such extensions lies in their capability to explain patterns in the Standard Model which are adjusted by hand to match observations. One such example is the hierarchy in the size of Yukawa couplings. In [1] super Yang-Mills theory with extra dimensions was studied in this context. Standard Model fields correspond to zero modes of the extra dimensional Dirac equation. Different fields have different localisations within the extra dimensions. Yukawa couplings arise as overlap integrals of these zero modes; they are large if they are localised near to each other and small otherwise. The authors of [1] mainly focused on the extra dimensions being compactified on a torus which factorises into a product of two-tori. An initial U (N ) gauge group is broken by fluxes to U (N a ) × U (N b ) × U (N c ) which can be further broken by Wilson lines. (The unbroken gauge group could e.g. be the Standard Model gauge group.) Computations in [1] are restricted to the case that N a , N b , N c are mutually coprime. In the present paper, the discussion will be extended to particular non-factorisable tori. This will also make it necessary to abandon the restriction of N a , N b , N c being mutually coprime, and hence the generalisation considered in [2] neither applies.
Restricting considerations to type II string model building, the above setting corresponds to type IIB theory, whereas most of type II string model building has been carried out on the type IIA side in the geometrically intuitive intersecting brane picture, see e.g. . Some constructions have, however, been directly performed on the type IIB side [30][31][32][33][34][35][36][37][38][39]. Computing Yukawa couplings in the type IIB setting is useful also from an intersecting brane model builder's perspective. Type IIA Yukawa couplings have been computed in [40]. There, they are given by sums over exponentials of classical worldsheet instanton actions. A factor in front of this sum cannot be fixed by classical calculations. In [1] also T-duality of intersecting brane models to type IIB flux compactifications is discussed. Couplings do match and further the type IIB calculation fixes the leading (in the small angle limit) contribution to the overall factor. Further discussions on the computation of interactions in type II models, including also quantum corrections, can be found in [41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59].
Usually toroidal constructions are performed on so called factorisable six-tori consisting of three mutually orthogonal two-tori. Generalisations to non factorisable tori are studied in [60][61][62][63][64][65][66][67][68][69]. In particular in [67] Yukawa couplings for intersecting branes on non factorisable six-tori have been computed. The calculations are restricted to cases where the torus is generated by a sublattice of a lattice belonging to a factorisable torus; as a representative example the SO(12) root lattice is considered. In the present paper T-duality of this setup will be carried out. Yukawa couplings are found to match and the leading contribution to the overall factor can be computed in type IIB theory. Some technical details of the calculation are quite appealing. For instance, the SO(12) structure of the type IIA compactification is scrambled in the process of T-duality along some of the lattice vectors. It resurfaces at a later stage when zero modes of the Dirac equation are labelled. As an aside, the methods developed for N a , N b , N c not all beeing mutually coprime can easily be applied to factorisable compactifications. In phenomenological model building such stiuations are not unlikely to arise; for instance an intitial U (N ) gauge symmetry can be broken by fluxes to Pati-Salam which in turn could be broken by Wilson lines to the Standard Model gauge group.
The paper is organised as follows. In the next section T-duality on the configuration of [67] is performed. In section three, chiral fields as zero modes of the Dirac equation are constructed. In section four, Yukawa couplings are computed by integrating the product of three zero modes over compact space. Section five contains some concluding remarks. In an appendix generalisations of the concept of greatest common divisors and lowest common multiples of lattices are reviewed and some examples given.

D9 branes as T-dualised D6 branes
In this section the T-dual of the setups considered in [67] will be constructed. The dual geometry will be a six-torus whose complex structure matrix has off-diagonal components. D-branes at angles give rise to magnetic flux, whereas multiple intersections with the Tdualised cycle result in constant Wilson lines.

T-dual of T 6 SO(12) : Closed String Sector
Before performing the T-duality, taking one from type IIA to type IIB, the (deformed) six-torus on the type IIA side will be described [67]. The compactification space is chosen to be a six dimensional flat torus T 6 . It is given by the quotient space R 6 /Λ 6 , where Λ 6 is a six dimensional lattice with { α i } i=1,...,6 generating the lattice. Hence, locally the torus looks like R 6 , but points differing by lattice vectors are identified x ∼ x + λ, x ∈ R 6 , λ ∈ Λ 6 .
In the following, the canonical basis of R 6 will be denoted by { e i } i=1,..., 6 with components The metric on flat R 6 is given by where the six canonical coordinates have been combined into three complex coordinates according to At the moment, this choice of pairs is arbitrary. Later D6 branes projecting onto straight lines in each of the complex planes and thus automatically wrapping Lagrangian cycles will be introduced.
A torus is called factorisable if its generators { α i } can be split into three mutually orthogonal pairs of vectors. In this case, one would arrange the choice of complex coordinates such that each of the mutually orthogonal pairs lies within one complex plane. For non factorisable tori this is not possible. As a typical example the root lattice of SO(12), will be considered. Here, vector components are given w.r.t. the canonical basis { e i }. If one was discussing just T 6 compactifications without any further ingredients (such as D branes or envisaged orientifolds) one could change metric and B field components by arbitrary constants. In particular, this allows deforming non factorisable into factorisable tori. Here, additional ingredients allowing deformations only within each of the complex planes will be assumed. This leads to the general metric where K h and τ h are complex parameters with positive imaginary parts. The definition of the complex coordinates in (2) is also deformed In addition, a constant B field of the form ReK h Imτ h du h ∧ dū h will be allowed. For compactifications on a factorisable T 6 the K h 's would be the complexified Kähler moduli of the three T 2 's whereas the τ h 's would form the complex structure moduli. In [67] it was observed that Yukawa couplings of type IIA intersecting branes exponentially depend on these 'would be' complex Kähler moduli even for non factorisable T 6 . Before performing T-duality, it is useful to change coordinates to the lattice basis such that integer shifts in any of the y i coordinates correspond to lattice shifts. Again, expressions for metric and B field can be compressed by means of complex coordinates for which one obtains, Since the y i coordinates are compactified on circles they are particularly useful for performing T-duality. The radii of these circles are taken to be at their selfdual value, R = √ α . In the following α = 1/ 4π 2 will be chosen such that 2πR = 1. The Buscher rules [70] for T-duality along the θ direction readG where i, j label directions other than θ. In addition, there is a shift in the dilaton The T-dual coordinate is again compactified on a circle of selfdual radius. Successively performing T-duality along the y 1 , y 3 and y 5 direction yields type IIB theory. To write the T-dual background the following complex coordinates are introduced (omitting tildes at dual coordinates) T-dual metric and B field can be written as Here, moduli have been suggestively split into complex structure appearing in (9) and the rest. This is not unique. The 6d metric has 21 independent real components whereas complex structure moduli and imaginary part of the Kähler moduli have 18 plus 9 real components. Uniqueness is achieved by imposing the six additional conditions that the B field should have components only along (1, 1) forms [71] 1 . To achieve that, the complex structure will not be modified but instead T-duality will be combined with the gauge transformation which has to be kept in mind when performing T-duality in the open string sector. Notice, that the previously 'would be' Kähler moduli become actual complex structure moduli in the T-dual type IIB theory. Finally, the relation between type IIB and type IIA dilaton is 2.
2 T-dual of T 6 SO(12) : Open String Sector As discussed in [63,67] a D6 brane of type IIA theory spans the following three dimensional subspace of the six dimensional compact space For factorisable tori the wrapping numbers n h and m h should be coprime for each h. In the non factorisable case these conditions are modified. For instance if the pairs are still all coprime, n h + m h has to be even for all h's. Other possibilities are listed in [63,67]. For simplicity, the case that branes pass through the origin will be considered. If one of the wrapping numbers n h is zero the corresponding equation has to be replaced by x 2h−1 = 0. Expressed in y i coordinates (3), equations (14) take the form In the following, the case that any of the N (h) vanishes will be excluded, i.e. T-duality along a D-brane will not be performed. This case has to be treated separately and leads to D7, D5 or D3 branes in the T-dual picture. T duality for open strings has been discussed in e.g. in [72][73][74][75][76][77][78]. Eq. (15) represents Dirichlet conditions on the coordinates with respect to which T-duality will be performed. Dirichlet conditions turn into Neumann conditions, which are obtained by varying the worldsheet action with no boundary conditions on the variation and a gauge field coupling to the boundary. This gauge field is given by minus the right hand sides of (15), As to be discussed shortly, these gauge fields are multiplied by identity matrices whose appearance has been supressed so far. The T-dual fieldstrength is finally computed as (recall (12)) where T-dual complex coordinates are defined in (9). It is consistent that starting with D branes wrapping Lagrangian cycles in type IIA theory the T-dual D9 branes of type IIB carry flux only along (1,1) forms. So far, multiple wrappings of the D9 brane have not been taken into account. The D9 brane wrapping number, N = N D9 , is given by where N D6 is the wrapping number of the D6 brane and the additional multiplicity originates from the intersection number with the cycle along which T-duality has been performed. (Intersection numbers for the type IIA setting are taken from [63].) In the following N D6 = 1 will be considered since for the calculation of Yukawa couplings this number is not relevant.
(Given a gauge group U (A) × U (B) × U (C) the Yukawa coupling of AB BC CA does not depend on A, B, C.) For later convenience, the gauge transformation (12) will be included in a redefinition of the T-dual gauge field. Taking into account multiple wrappings, is chosen, where W is a Wilson line originating from the finite separation of N (1) N (2) N (3) /2 stacks of branes along the T-dualised direction. Although W can be written as g −1 dg with g ∈ SU (N ) it cannot be removed by a globally single valued gauge transformation. On the type IIB side, it breaks the gauge group from U (N ) to U (N D6 = 1). The Wilson line will be discussed more explicitly in the next section.

Chiral Matter
This section follows closely the strategy of [1] in identifying chiral matter of the effective four dimensional theory. First, Wilson lines are specified. They are viewed as gauge transformations induced by lattice shifts. In the factorisable case these gauge transformations are associated to the direct product of three matrices, or in other words, each of the two group indices on the gauge transformation matrix is conveniently replaced by a triplet of indices. It will be argued that in the non factorisable case the gauge index should be expressed in terms of a vector in a quotient lattice. To really discuss the T-dual of intersecting branes, more than one unitary gauge group factor has to be considered. Zero modes of the Dirac equation in the bifundamental representation will give rise to chiral matter.

Labelling Gauge Indices
Consider a field φ as a function of torus coordinates transforming in the fundamental representation of U (N ). Dependence on uncompactified spacetime is also assumed but suppressed in the notation. It is imposed that this field is periodic under lattice shifts up to gauge transformations, i.e.
The phases are fixed such that a phase originating from A − W is cancelled, e.g.
Non vanishing phases are Notice, that all k ij 's are integers. This is related to conditions that D6 branes should wrap closed cycles in the type IIA geometry [63,67]. One may try to construct the ω i 's by means of two dimensional solutions given in [1]: Consider two matrices w 1 , w 2 ∈ SU (n), where n ∈ Z + . Impose the condition A solution for w 1 and w 2 is For the factorisable torus solutions of the form where matrices with subscript (h) are N (h) × N (h) matrices, solve conditions corresponding to (21) [1]. The resulting ω i are 2N × 2N matrices. A similar overcounting arises on the type IIA side if one just multiplied the intersections numbers in each complex plane [67]. There the overcounting would happen due to an erroneous labelling of intersection points as j (h) ∈ Z N (h) . The resolution advocated in [67] is that the triplet of j (l) 's takes values in a sublattice of 3 h=1 Z N (h) . With triple indices i, j, (23) reads It remains to identify the lattice Λ 3 within which triple indices take values. Wrapping numbers on the type IIA side describe closed cycles if one of the following four cases applies: Whenever g.c.d. (n i , m i ) = 2, N (i) /2 has to be odd. Otherwise the corresponding wrapping numbers have to be divided by two and another case applies. The following quotient lattices, Λ 3 , turn out to yield useful sets for labels: Here, Λ SO(6) denotes the SO(6) root lattice generated by (1, −1, 0), (0, 1, 1), (0, 1, −1). In cases where g.c.d. n h , m h = 2 the corresponding Q matrix has to be replaced by

Bifundamentals
Consider two D6 branes wrapping cycles labelled by a and b on the type IIA side. Assume that neither cycle has zero intersection number with the T-dualised cycle. On the type IIB side this corresponds to N a + N b D9 branes wrapping the T-dual six torus where N a and N b are the respective intersection numbers with the T-dualised cycle. These D9 branes accommodate a U (N a + N b ) gauge symmetry which is broken to U (N a ) × U (N b ) by magnetic fluxes and finally to U (1) × U (1) by Wilson lines. The magnetic flux is given by the following non vanishing fieldstrength components where χ ab i = χ a i − χ b i denotes the difference between the two phases (20). Defining , the phase differences can be written as Inserting the explicit representations for the Wilson lines (24) one finds Notice that ω 2 , ω 4 and ω 6 act as shifts by SO(6) roots on the gauge indices k a and k b . Therefore the convention to label the gauge group elements by a subset of SO(6) roots is consistent with gauge transformations. In [1] it is demonstrated that replacing the double index at matrix components by a single index is very useful. The details can be summarised as follows. Focusing on just one T 2 factor the expression corresponding to the first line in (28) reads (ϕ replaces φ for the case of two extra dimensions) where in the last step the double index has been replaced by a single index from which it is obtained by This means that there is a pair of integers (s, t) such that implying that the difference k a − k b has to be an integer multiple of For this reason the discussion in [1] is restricted to the case d = 1. For general d, the intersection number I ab = n a N a − n b N b is a multiple of d. Hence, (30) should be raplaced by providing not enough labels. In addition, one should introduce another label The distribution of δ among individual shifts of k a and k b is carried out as follows. First, one chooses a solution (p, q) of the linear Diophantine equation Then (33) is compatible with Summarising, the correspondence between (k a , k b ) and ( , δ) is Then, the second identity in (29) generalises to i.e. there is an additional phase taking values in Z d .
Returning to the non factorisable torus, discussed previously, it was noticed that where Γ a , Γ b are sublattices of Λ SO (6) . The lattice Γ d is defined as follows: Γ a and Γ b are sublattices of Γ d and there is no proper sublattice of Γ d containing Γ a and Γ b as sublattices. In other words, Γ d is the coarsest lattice containing Γ a and Γ b .
The number of inequivalent index combinations (k a , k b ) is given in terms of indices of quotient lattices 2 The second equality with a reference to its proof is discussed further in appendix A. These observations suggest replacing the index pair (k a , k b ) by two lattice valued labels The pair (k a , k b ) can be again obtained by shifting values of l and modding out by lattices Γ a , respectively Γ b . The details are as follows. There are classes of differences Throughout the paper three dimensional lattice vectors are viewed as a column with three entries corresponding to the components with respect to a given basis (mostly (1)). Let ) be the generators of the lattices Γ a , Γ b , Γ d , respectively. It turns out to be convinient to combine these into three by three matrices The requirement that Γ a and Γ b are sublattices of Γ d is equivalent to the existence of three by three integral matrices 3 M a and M b such that Let (P, Q) be two three by three integral matrices satisfying 4 A natural generalisation of (34) would be There is however a problem with that. The partitions AP D −1 δ and BQD −1 δ are not always in Λ SO (6) . A way out is to give up invariance under equivalence shifts of δ. So, in the following δ will be taken from a finite set consisting of one representative for each equivalence class. Then it makes sence to assign since now shifts of k b by elements of Γ d cannot be absorbed by picking another δ from the same equivalence class. The non factorisable version of (35) reads 3 Matrices with integer components are called integral matrices. 4 These matrices exist, reference to a proof is given in appendix A.

Massless Dirac Zero Modes
First, consider open strings on a stack of branes with gauge symmetry U (N a + N b ). The fields Ψ corresponding to the open string states transform in the adjoint representation of the gauge group. Massless fermions in four dimensions arise from massless fermionic states in ten dimensions satisfying the Dirac equation with Γ l and Γ l being elements of the six dimensional Clifford algebra and Here After turning on magnetic flux as in (25), the states in ψ l decompose into the adjoint representation of U (N a ) and U (N b ) and bifundamentals of U (N a ) × U (N b ) [1]. The bifundamentals will be denoted by φ . For e.g. l = + the corresponding equation in (42) reads As in [1], normalisable solutions to (43) for fermions in the N a , N b will be considered. Normalisability leads to the condition The chirality of the resulting four dimensional massless fermion is fixed by the sign of 1 2 3 , i.e. by sign Ĩ (1) ab . Apart from that the solutions depend only on the absolute values I (i) ab . In the following vertical bars will be dropped and postive I (i) ab 's will be assumed since negative values can be accomodated easily by changing the chirality. The following ansatz solves (43) The ξ kak b are holomorphic functions of the z i . Plugging this ansatz into the boundary conditions (28) yields First, focus on boundary conditions (46), (48), (50), resulting in the general solution where On the right-hand sides of (46), (48), (50) there will be additional, trivial phase factors of the form exp [2πiI] with the integer I given by (M a P δ) 1 , − (M a P δ) 2 , − (M a P δ) 3 , respectively. The insertion of these factors of one will be helpful in mapping zero mode labels to intersection labels on the type IIA side, shortly. The last term, ρ n l stands for z independent factors which will be further fixed by solving the remaining boundary conditions. Imposing conditions (47), (49), (51) and comparing coefficients at coinciding powers of exp z i leads to These conditions are solved by Independent normalisation constants N n indicate independent zero modes. Imposing invariance under shifts of l by elements of Γ a ∩ Γ b identifies some constants. For λ ∈ Γ a ∩ Γ b this leads to N n = N n , for n i = n i +Ĩ Possible lattices Γ a ∩ Γ b are listed in appendix A. For all cases one finds number of independent constants = I For the final counting of zero modes it is worthwhile noticing that boundary conditions relate different pairs k a , k b of identical k a − k b (see (46)- (51)). In (55) As expected, this equals the intersection number in the T-dual type IIA configuration. It will be useful to detail the relation between intersections and zero modes by identifying their labellings. In [67] intersections in type IIA theory are labelled by a triplet j (1) , j (2) , j (3) of the following form This is subject to equivalence relations which will not be further discussed since matching of the overall numbers has already been established. The relation (59) is taken into account by renaming the summation index where k is a fixed label and λ ∈ Γ a ∩ Γ b is summed over. Combining (52) and (58) one obtains for one zero mode which is now labelled by the pair k, δ. To make contact with the type IIA labelling one notices that the solution depends only on the combination k + φ which can be broought into the form where j is the type IIA label (61) with That this is really in Λ SO (12) can be seen with (39), (40) and the fact that δ ∈ Λ SO (6) . Expressing the type IIB label in terms of the type IIA label via (62) leads finally to where the notation has been changed to remove a redundancy in specifying the δ dependence of the zero mode. The notation for the original zero mode (45) will be changed accordingly φ kak b → φ i l . Keep in mind that in (45) and (64) one should actually replace, I h ab → I h ab (see discussion after (44)).

Normalisation Factor
In order to get canonically normalised kinetic terms in four dimensions the zero modes need to satisfy the orthogonality relation [ where the integration is over complex coordinates (9) and the metric is taken from (10). Further, Φ b is the ten dimensional type IIB dilaton which is chosen to be constant. Its exponential in (65) With that the integration in (65) can be expressed as For each zero mode φ i and φ j the parameters δ i , δ j ∈ Λ SO (6) Γ d are fixed according to the definition of the labels in (62) (see also the discussion after (60)). Hence, the sum in the trace in (65) has to be taken only over l ∈ 5 The factor α −3 has been included to match finally the convention of [1] in which the gauge coupling is given by e Φ b /2 α 3/2 . (At the moment α is fixed as in (6)).
where the Kronecker delta δ δ i ,δ j ensures that the sum is indeed a trace. The product of wavefunctions φ i l,l−δ i · φ j l,l−δ j † satisfies the following boundary conditions and therefore integrals of φ i l,l−δ i · φ j l,l−δ j † over T 6 , with different values for l can be related to T 6 lattice shifts in the following way, The relations (69) can be used to replace the sum over l by an enlarged domain of integration. That means instead of integrating all terms, belonging to the trace, over the fundamental domain of T 6 , we just need to integrate one term with a fixed l, for example l = 0, over the enlarged domain of integrationC, whereC is given by the unit cell of the lattice spanned by v 1 = (1, 0, 0) T , v 3 = (0, 1, 0) T and v 5 = (0, 0, 1) T as before, but where e.g. Γ a ∩ Γ b = Γ 1 means that it is of the form Γ 1 in Appendix A.

The explicit expression for
can be deduced by inserting (45) and (64), The y 1 , y 3 , y 5 dependence of the integrand is contained in factors exp (2πiy 2h−1 M h ) with A closer look at (72) reveals that, taking the trace condition δ δ i ,δ j into account, the terms are actually integer, because the potentially non integer part, which is according to (62) given by , vanishes for δ i = δ j . Hence, the integration of (71) over y 1 , y 3 and y 5 yields one if all M h in (72) vanish and zero otherwise. This implies a non vanishing result only for ρ ab = λ ab and i = j, establishing orthogonality of the zero modes. The final result of the y 1 , y 3 , y 5 integration is Similar to (69), the sum over λ can be replaced by an enlarged domain of integration over y 2 , y 4 , y 6 C dy 2 dy 4 dy 6 There are three remaining Gaussian integrals, solved by Plugging the results into (65), one finds the normalisation condition In the next section, normalisation factors will be real solutions of (75).

Yukawa Couplings
The configuration considered in this section will be the T-dual of type IIA with three stacks of intersecting D6 branes. That is, (25) will be amended to 6

Two Extra Dimensions
It will be useful to recapitulate and to generalise the computation of Yukawa couplings in the case of two extra dimensions. This has been dealt with in [1] for the case that all pairs from {N a , N b , N c } are coprime. The computation of the Yukawa coupling boils down to evaluating integrals of the form 7 The φ's denote zero modes in bifundamentals as before. Now, the zero mode label has been supplemented by the intersection number. For all N α 's being coprime the matrix elements are related by shifts by cycles of the T 2 (analogous to e.g. expression (47)). This enabled the authors of [1] to trade the sums (77) for an enlarged integration regionT 2 . Before outlining more details it will be uesful to include also the discussion of non coprime pairs among the N α 's. In this case there are subsets within all matrix elements invariant under shifting zero mode arguments by T 2 cycles. As discussed in (33) these sectors are characterised by differences in row and column number, e.g.
c /2. 7 For simplicity, moduli dependence will be suppressed in the present discussion.
Different δ αβ 's belong to different zero modes. Expression (77) should be modified to Here, the first three δ's are usual Kronecker deltas on Z d ab , e.g. the first is one if (78) holds and zero otherwise. The last δ ensures that the trace is taken and is defined as The following abreviations will be convenient. Similar to the greatest common divisor (79) the lowest common multiple will be denoted as For d abc given by where cases which can be obtained by permuations of (a, b, c) have not been explicitly written. With d abc one can relate the product of three numbers to its lowest common multiple The double index e.g. k a , k b can now be replaced by a single index l as in (35) where it proves useful to change notation slightly. For a fixed δ αβ which is encoded in the label i one replaces φ i,I αβ With that notation, the Yukawa coupling (80) reads Notice, that it has been possible to drop the first three Kronecker deltas of (80). However, the last δ function in (80) translates into a selection rule involving the labels i, j, k. Its explicit form depends on the so far unspecified way δ αβ is encoded in the label. Therefore it has been left out in (83) but should be kept in mind. The gauge indices (summation labels) have been chosen such that the sum implies matrix multiplication, i.e. consecutive row and column indices match (see (34)). Analogous to e.g. (47) l can be shifted by one when replacing z → z + τ , where τ is the complex structure modulus of the compactification T 2 . Any factor induced by such shifts (cf (26)) drops out due to the identitỹ Therefore, one can replace the sum over l by an enlarged integration region leading to whereT 2 has complex structure N abc τ . From hereon one can use the techniques presented in [1] to complete the computation for the generalised configuration with two extra dimensions.

Yukawa Couplings for the T-dual of T 6 SO(12)
Now the Yukawa coupling is determined via computing 8 where z has been introduced in (9), the prefactor comes from √ G with the metric taken from (10). The region of integration is a parallelepiped C ⊂ C 3 whose edges are given by the following vectors Again, the sum over gauge indicies l can be replaced by an enlarged integration region since shifts by l 2 , l 4 or l 6 induce index shifts according to (47), (49), (51). To be more specific, one needs to identify Γ a ∩ Γ b ∩ Γ c . Repeating the analysis given in appendix A one finds that Γ a ∩ Γ b ∩ Γ c is given by either Γ 1 , Γ 2 or Γ 3 with Here, d abc are defined as in respectively (79), (82) for each l ∈ {1, 2, 3}. The summation over l ∈ Λ SO (6) Γa∩Γ b ∩Γc can be traded for an integration over a larger parallelepipedC ∈ C 3 , The edges ofC are l 1 , l 3 and l 5 as in (86) but l 2 , l 4 and l 6 replaced by the generators of one of the lattices Γ 1 , Γ 2 or Γ 3 from appendix A with Next, the integration variables are replaced by {y 1 , . . . , y 6 } as in (9), The range for the {y 1 , y 3 , y 5 } integration is the cube spanned by l 1 , l 3 and l 5 . The range for {y 2 , y 4 , y 6 } is a parallelepiped whose form depends on whether Γ a ∩ Γ b ∩ Γ c is of the form Γ 1 , Γ 2 , or Γ 3 . One finds for the edges of the parallelepiped Inserting the expressions from (45) and (64) into φ j,Ica 0 φ i,I ab −δca φ k,I cb δ cb , and using the relation (84), with the choice the explicit expression for the product of wavefunctions is given by where x, y ∈ {a, b, c} and the wavefunction φ k,I cb k b ,kc has been relabelled such that c . (This corresponds to swapping the label of ξ kc,k b with minus the label of ξ k b ,kc , see (64).) Before performing the integration, a closer look at the terms reveals them to be integers. From the way the labels i, j, k in (62) where introduced, it can be deduced that the potentially non integer part in (89) is However, when considering the trace of φ j,Ica l φ i,I ab l−δca φ k,I cb l+δ cb , only terms with δ ca + δ ab = δ cb contribute and hence the terms in (90) vanish, the expression in (89) is indeed integer ∀ {λ ab , λ ca , λ cb } and ∀ h ∈ {1, 2, 3}. Therefore the integration over y 1 , y 3 and y 5 leads to Kronecker deltas, which imply the following Diophantine equations h ∈ {1, 2, 3} In [67] Diophantine equations arose from the requirement that projections of the intersecting D6 branes form closed triangles in each plane. These equations could be solved in terms of wrapping numbers after performing a relabelling of the intersection points, where e.g. d ac . It can happen that intersection points lose their label. The corresponding Yukawa couplings are equal to others for which no label is lost [67].
Performing the same relabelling (93) on the type IIB side, the solutions to (92) are given by where p (h) and q (h) are components of three dimensional lattice vectors to be specified shortly, and M After integrating φ j,Ica φ i,I ab φ k,I cb over y 1 , y 3 and y 5 and evaluating the condition (94), one gets where Λ 3 p and Λ 3 q are three dimensional lattices, with a lattice structure such that λ ab , λ ca and λ cb in (94) belong to the lattices Γ a ∩ Γ b , Γ c ∩ Γ a and Γ c ∩ Γ b , respectively. The components p (h) have to be chosen, such that the vectors (N (1) abc p (1) , N (2) abc p (2) , N abc p (3) ) T belong to the SO(6) lattice. Therefore Λ 3 p takes the form and the terms N matching the definition of [67]. In [67] dimensionful Kähler moduli have been used. The explicit relation is (see (6)) where g (h) is the determinant of the metric in the h th complex plane in the coordinates (1), as it was used in [67]. Up to now the lattice, to which the summation index q belongs, has not been specified. It can be deduced from (94): The p dependence in (94) is eliminated in linear combinations of the three Diophantine equations The left-hand sides of (98) are components of Λ SO(6) lattice vectors. Renaming the summation index 2q (h) → (h) , this leads to the following conditions (h ∈ {1, 2, 3}) Taking into account also the dilaton shift (13) and using (6) to obtain a manifestly dimensionless coupling, one finds . This result looks exactly as the one reported in [1] for factorisable tori. Here, however the definition of θ

Conclusions
In the present paper, Yukawa couplings where computed along the lines of [1]. However, here a particular non factorisable six-torus was considered. This arose as a T-dual of a torus generated by the SO(12) root lattice. For cases in which the SO(12) root lattice is replaced by another sublattice of a factorisable lattice straightforward modifications of the presented calculations are expected. Compared to [1], however, some less straightforward adjustments had to be performed. Gauge indices as well as zero mode labels take values in quotient lattices which appear as generalisations of products of finite sets of integers. On the type IIA side an SO(12) lattice playing a role in labelling the intersection points was directly related to the compactification lattice. In the T-dual description, this SO (12) lattice shows up in a rather indirect way when labelling zero modes. For non coprime flux ranks, not all components of a chiral multiplet are related by boundary conditions and hence expressed by the same set of zero modes.
T-dualising back to type IIA one can identify leading contributions to a factor which can be determined only by a quantum computation on the type IIA side. The result looks exactly as in the factorisable case [1], with some straightforward modifications in the definitions of variables. To confirm the presented result, one could in principle perform T-duality along other cycles. This is expected to be more complicated since the cycles of the presented calculation have been chosen such that they lie within complex planes.
It would be interesting to investigate to what extend the presented type IIB calculation can be generalised to cases being not T-dual to type IIA models of the considered kind. Abelian Wilson lines have not been turned on for simplicity. In the T-dual IIA setting they correspond to an offset from a brane passing through the origin. Their inclusion is expected to be straightforward. Finally, of course, applications to actual model building would be nice. The presented configuration generalises known cases and might help accommodating desirable phenomenological aspects.

A Quotient lattices, divisors and multiples of integral matrices
As discussed in (38) lattices will be associated to integral three by three matrices: Γ a to A, Γ b to B, Γ d to D, and Γ a ∩ Γ b to M . With the following definitions one can establish relations among these matrices. Clearly, the matrix D containing generators of Γ d is a greatest common left divisor of A and B. The greatest common left divisor is unique up to multiplication by unimodular matrices which corresponds to choosing an equivalent set of lattice generators (see e.g. [79]). An explicit construction in terms of matrices taking the three by six matrix (A, B) to its Smith normal form can be found in [80] (proof of Proposition 3.4) where the existence of the matrices P and Q introduced in (40) is proven.

Definition: Let
Similarly one can identify the matrix M with the lowest common right multiple of A and B. Its definition is: Definition: The integral matrix M is a right multiple of the integral matrix A if there is an integral matrix N a such that M = AN a . M is the lowest common right multiple of the integral matrices A and B if it is a right multiple of A and B and any other right multiple of A and B is a right multiple of M .
The lowest common right multiple M and the greatest common left divisor D have been related in theorem 5 of [81], The index of a quotient lattice is related to the integral matrices of generators as follows. Let Λ c be a sublattice of Λ l . Further C and L denote integral matrices of the corresponding generators. Then the index of the quotient lattice is Hence, taking the determinat of (104) proves the second equality in (36).
In the following, examples, relevant for the present paper, will be listed. As discussed in section 3.1 there are three possible lattices for Γ a or Γ b x Z , all N (l) x even, x odd and N (3) x even, , N (2) x and N (3) x odd, where x stands for a or b, respectively. There are six inequivalent configurations corresponding to symmetric pairings of these lattices.