NC Calabi-Yau Manifolds in Toric Varieties with NC Torus fibration

Using the algebraic geometry method of Berenstein and Leigh (BL), hep-th/0009209 and hep-th/0105229), and considering singular toric varieties ${\cal V}_{d+1}$ with NC irrational torus fibration, we construct NC extensions ${\cal M}_{d}^{(nc)}$ of complex d dimension Calabi-Yau (CY) manifolds embedded in ${\cal V}_{d+1}^{(nc)}$. We give realizations of the NC $\mathbf{C}^{\ast r}$ toric group, derive the constraint eqs for NC Calabi-Yau (NCCY) manifolds ${\cal M}^{nc}_d$ embedded in ${\cal V}_{d+1}^{nc}$ and work out solutions for their generators. We study fractional $D$ branes at singularities and show that, due to the complete reducibility property of $\mathbf{C}^{\ast r}$ group representations, there is an infinite number of non compact fractional branes at fixed points of the NC toric group.


Introduction
Since the original work of Connes et al on Matrix model compactification on non commutative (NC) torii [1], an increasing interest has been devoted to the study of NC spaces in connection with solitons in NC quantum field [2], and string field theories [1,2,3]; in particular in the analysis of D(p − 4)/Dp brane systems (p > 3) of superstrings [6,7] and in the study of tachyon condensation using the GMS approach [8]. However most of the NC spaces considered in these studies involve mainly NC R d θ [9], NC T d θ torii [9,10], some cases of Z n type orbifolds of NC torii [11,12] and some generalizations to NC higher dimensional cycles such as the NC Hizerbruch complex surface F 0 used in [13] and some special CY orbifolds. Quite recently efforts have been devoted to go beyond these particular manifolds by considering particular examples of CY manifolds M, especially those given by homogeneous hypersurfaces with discrete symmetries [14,15,16,17,18] in projective spaces such as the quintic Q [19,20]. Such analysis is important for the stringy resolution of singularities; but also for the study of fractional D branes at these singularities [21,22,23]. The key point in the construction of NCCY orbifolds by help of algebraic geometry method, is based on solving non commutativity in terms of the torsion of discrete isometries of the orbifolds. This idea was successfully applied in [15], for the study ALE spaces and aspects of the NC quintic; then it has been extended in [17,18] for the building of NC orbifolds H nc d of complex d−dimension homogeneous hypersurfaces H d . In this paper, we want to go one step further by extending these results to the large class of complex d dimension CY non homogeneous manifolds M d embedded in singular toric varieties V d+1 [25,26,27]. The organization of this paper is as follows; In section 2, we review general aspects of CY manifolds M d embedded in toric varieties V d+1 and focus on the study of the type IIA geometry. Sections 3 and 4 are mainly devoted to the study of the NC type IIA extension of M d by introducing a NC toric fibration with M d as a base. We first show how NC toric fibrations may be realised and then derive and solve the constraint eqs for the NC structure. In section 5, we study fractional branes at singular points and in section 6, we give our conclusion.

CY Manifolds in Toric Varieties
We start by recalling that there are different ways for building complex d dimension CY manifolds M. A tricky way is by embedding M in a toric variety V; that is a complex Kahler manifold with some C * r toric actions. The simplest situation is given by the case where M is described by complex d dimension hypersurface in a complex (d + 1) toric variety V d+1 . To write down algebraic geometry eqs for the CY hypersurfaces which may be singular; one should specify a set of ingredients namely a local holomorphic coordinates patch of the toric manifold V d+1 , the group of toric action and the toric data.
To do so, one should moreover distinguish between two kinds of geometries for the CY manifolds M d ; (1) the so called type IIA geometry, to which we will refer to as M d , and (2) its dual type IIB geometry often denoted as W d . The latter is obtained from M d by exchanging their Kahler and complex structures following from the Hodge identities [28]. In this study, we will mainly focus on the type IIA geometry.
A nice way to introduce the type IIA geometry M d is in terms of 2D N = 2 field theory which is known to describe the propagation of type IIA strings on M d . In this approach, the M background is constructed in terms of N = 2 supersymmetric linear sigma models involving a superfield system {V a , X i } containing r gauge N = 2 abelian multiplets V a σ, θ, θ with gauge group U(1) r and (k + 1) chiral matter superfields X i σ, θ, θ of bosonic components x i . In addition to the usual kinetic terms and gaugematter couplings, the linear sigma model action S [V a , X i ] of these fields may have r Fayet Iliopoulos (FI) D-terms, which in the language of N = 2 superfields read as ζ a d 2 σd 2 θd 2 θ V a σ, θ, θ , with ζ a being the FI coupling constants. The superfields action S [V a , X i ] may also have a holomorphic superpotential W (X 0 , ..., X k ) given by polynomials in the X i 's, which in the infrared limit, is known to describe the CFT 2 of string propagation on the type IIA background [29]. Let us discuss a little bit this special geometry.
In the method of toric geometry, where to each complex bosonic field x i it is associated some toric data {q a i , ν i }, or more generally by taking into account relevant data from the dual geometry q a i , ν i ; p I α , ν * α [19,20,25], with ν i and ν * α being (d + 1) dimension vectors of Z d+1 self dual lattice, one can write down the algebraic geometry equation of the complex d CY manifold M d . This is given by a holomorphic polynomial in the x i 's with some abelian complex symmetries. In the simplest situation where the toric manifold is given by the coset 2 V d+1 = C k+1 /C * r , d = k − r, the complex d dimension CY hypersurface reads as, where the b α 's are complex numbers and where the n αi powers are some positive integers constrained by the C * r invariance. Indeed, under the C * r , x i −→ x i λ q a i a with q a i some integers, invariance of P d [x 0 , ..., x k ] requires the n αi integers to be such that, Eqs i q a i = 0, which ensures the vanishing of the first Chern class of M d , follow from the C * r symmetry of the k i=0 x i monomial while i q a i n αi = 0 come from invariance of k i=0 x n αi i monomials. Setting u α = k i=0 x n αi i , eq (1) can be rewritten as P d [u α ] = α b α u α , where the u α 's are the effective local coordinates of the coset space C k+1 /C * r . As the u α 's are given by u α = k i=0 x n αi i , it may happen that not all of the u α 's are independent variables; some of them, say u α I for I = 1, ...r * , are expressed in terms of the other u α J variables with J = I. In other words; one may have relations type α u p I α α = 1, where p I α are some integers. Substituting u α = k i=0 x n αi i back into α u p I α α = 1, we discover extra constraint eqs on the n αi and p I α integers namely, In toric geometry, the n αi integers are realized as n αi =< ν i , ν * α >= A ν A i ν * αA . In this representation, eqs(2,3) are automatically solved by requiring the toric data of the CY manifold to be such that, Let us illustrate these relations for the case of the assymtotically local euclidean ( ALE ) space with A n−1 singularity. This is a complex two dimension C n+1 /C * (n−1) toric variety with a su(n) singularity u −n 0 u 1 u 2 = 1 at the origin. From this relation, one sees that there are three invariant variables u 0 , u 1 and u 2 ; but only two of them are independent. Since r * = 1, there is only one p I α vector of entries p α = (−n, 1, 1) and three ν * α vectors given by ν * 0 = (1, 0), ν * 1 = (n, −1) and ν * 2 = (0, 1). More generally, we have the following cases: (a) No relation such as α p I α ν * α = 0 exist, that is all the u α 's independent; the algebraic eq of the hypersurface M d is non singular and reads as d Generic cases where there exists r * constraint eqs of type α p I α ν * α = 0, the (d + r * + 1) complex variables u α are not all of them independent; So the algebraic geometry eqs defining the singular hypersurface embedded in V d+1 is where p I α are the integers in eqs (4). Note that this hypersurface has singularities at the u α = 0; but the introduction of NC geometry lifts this degeneracy. In the next section, we will study the NC extension of this geometry.

NC Type IIA Geometry vs NC Torus Fibration
To start note that NCCY manifolds as built in [15] may be viewed as a NC torus fibration with a CY base. Since NC torii have two kinds of realizations namely rational and irrational representations, we have to distinguish two cases of NC torus fibrations. The first kind of these manifolds involve fuzy torii and is exactly the solution obtained in [15]. The second class of NC varieties, which extend naturally the previous one, is the type of solution we want to present here. Before that we will first recall the BL algebraic geometry method; then we present our solution by direct use of NC fibration ideas.

Rational torus
From the algebraic geometry point of view, the NC extension M nc d of the CY manifold M d , embedded in V d+1 , is covered by a finite set of holomorphic operator coordinate patches  [24,18]. This idea was successfully used in the building of NC extensions of the CY homogeneous hypersurface z d+2 where the q a i integers satisfy the CY condition d+2 i=1 q a i = 0, a = 1, ..., d; [15,16]. The NC H nc d extending H d were shown to be given, in the coordinate patch Z d+2 ∼ I id , by the following NC algebra, The θ ij parameters are solved by discrete torsion as θ ij = ω ij ̟ ji with ̟ kl the complex conjugate of ω kl . The ω ij 's are realized in terms of the q a i CY charges and the Z d d+2 discrete group elements ω = exp i 2π j with m ab integers. The Z i 's form a regular representation of the discrete group and are solved as where P =diag 1, ω, ..., ω d+1 and Q satisfy, and generate the NC rational torus. From this solution, one sees that the BL manifold H nc d is just a kind of a NC torus fibration based on the commutative CY hypersurface H d .

Irrational torus
To build the NC type IIA geometry extending eq(1), we will mainly adopt the same method as in [15,17] and look for realizations of the NC variables Z i as where r a i and s a i are positive integers to be determined later. In this fibration, U a and V b are the generators of a complex r dimension NC torus satisfying, amongst others, the familiar relation with γ ab some given irrational C-numbers. One of the main differences between this fibration and the rational torus one of [15]; see also eqs (7,8,9), is that here γ ab is no longer a root of unity and so the Z i 's have infinite dimensional representations which make such kind of NC extension very special as we will see when we discuss fractional D branes. For the moment let us expose our result by first giving the constraint eqs, their solutions and the Z i 's regular representations.

• Constraint Eqs
Extending naively the algebraic geometry method used for H nc d to our present case by associating to each x i variable the operator Z i , then taking q a k = 0 and working in the coordinate patch Z k = I id , the NC type IIA geometry M nc d may be defined as, Since M d should be in the centre of M nc d , it follows that the Z i generators and the θ ij parameters should satisfy the constraint eqs, Actually these relations constitute the defining conditions of NC type IIA geometry M nc d . While the constraint relation θ ij θ ji = 1 shows that θ ij = θ −1 ji , non trivial solution of the

• Solving the Constraint Eqs
First of all note that since the θ ij 's are non zero parameters, one may set and solve the constraint eqs(13) by introducing torsion for the C * r toric actions. Putting these parameterizations back into eqs (13), one gets the following constraint on the J ab ij 's, Moreover as we are looking for solutions to the Z i operators as in eq (10), let us explore what one may call NC C * r toric actions. NC C * r Toric group: Recall the C * r toric group as used in toric geometry is a complex abelian group which reduce to U (1) r once the group elements exp iψ a T a , with parameters ψ a = α a − iρ a ∈ C, are chosen as exp iα a Q a where the α a 's are now real numbers. To have NC toric actions, there are two ways to do: (1) either through the use quantum symmetries as in case of discrete torsion or (2) by introducing CDS non commutative torii [1]. Let us comment briefly these two realizations.
1. Quantum toric symmetry: To illustrate the idea, we consider the simple case r = 1. Since C * is an abelian continuous group and its representations have very special features, we have to distinguish the usual cases; (a) the discrete infinite dimensional spectrum and (b) the continuous one. Both of these realizations are important for the present study and should be thought of as extensions of the rational and irrational representations of NC real torii [10,12]. a) Discrete Spectrum: Let R dis (C * ) = {U = exp iψT } denote a representation of C * on an infinite dimensional space E dis with a discrete spectrum generated by the orthonormal vector basis |n >; n = (n 1 , n 2 ) ∈ Z × Z ∼ Z 2 3 . Here ψ is a complex parameter, ψ ∈ C; and T is the complex generator of C * ; the ψT combination may be split as ψT = ρK + iαQ, where K is the generator of dilatations and Q is the generator of phases. For ψ = α real, R dis (C * ) reduces to R dis (U (1)) = {U = exp iαQ}. The generator T of R dis (C * ) acts diagonally on the vector basis {|n >}; i.e T |n >= n|n > and so U|n >= (exp iψn) |n > or equivalently U = n χ n (ψ) π n , with characters χ n (ψ) = exp (iψn). and π n = |n >< n|, the projectors on |n > π n . Since R dis (C * ) is completely reducible, its I id identity operator may be decomposed in a series of π n as, Like for U (1) and Z N subgroups, C * has also a complex shift operator V τ acting on {|n >; n ∈ Z + iZ} as an automorphism exchanging the characters χ n (ψ). This translation operator which operate as V τ |n >= |n + τ >; with τ = 1 + i, may also be defined by help of the a + (n 1 ,n 2 ) = | (n 1 + 1) + in 2 >< n 1 + in 2 | and b + (n 1 ,n 2 ) = |n 1 + i (n 2 + 1) >< n 1 + in 2 | step operators as V τ = V 1 ⊗ V i with, Due to the remarkable properties a + (n 1 ,n 2 ) π (n 1 ,n 2 ) = π (n 1 +1,n 2 ) a + (n 1 ,n 2 ) , b + (n 1 ,n 2 ) π (n 1 ,n 2 ) = π (n 1 ,n 2 +1) b + (n 1 ,n 2 ) and b + (n 1 ,n 2 ) a + (n 1 ,n 2 ) π (n 1 ,n 2 ) = π (n 1 +1,n 2 +1) b + (n 1 ,n 2 ) a + (n 1 ,n 2 ) , it follows that U and V satisfy the following non commutative algebra, describing the complex extension of the CDS torus [1]; to which we shall refer to as the NC complex two torus. Since ψ is an arbitrary complex parameter, eqs(18) define an irrational NC complex two torus. b) Continuous Case: In this case, the generator T of R (con,con) (C * ) has a continuous spectrum with a vector basis {|z >, z ∈ C; <z ′ |z>= δ (z − z ′ )}. It acts diagonally as < z|T | = z < z| and so < z|U = (exp iψz) < z| which imply in turns, U = dz χ (ψ, z) π (z) with χ (ψ, z) = exp (iψz) and π (z) = |z >< z|; π (z) π (z ′ ) = δ (z − z ′ ) π (z). Since R (con,con) (C * ) is completely reducible, the I id identity operator may be decomposed as, The shift operator by an ǫ amount, denoted as V (ǫ), acts on {|z >, z ∈ C} as < z|V (ǫ) =< z + ǫ|. It may be defined, by help of a + (z, ǫ) = |z >< z + ǫ| operators, as, These operators satisfy similar relations as in eqs (17,18) namely UV = exp (−iψτ ) V U and may also be realized on the space of holomorphic functions F (z) = <z|F > as We give details on the differential realization of U and V operators in the next section.
2. NC C * toric cycles If one forgets for a while about the quantum symmetry generated by V a and focus on the C * toric generators U a only, one may also build representations where the r complex cycles of the C * r group are non commuting. This is achieved by demanding [T a , T b ] = im [ab] = 0 which is ensured by introducing torsion among the C * factors. Here also we should distinguish between discrete and continuous spectrums. In the continuous case for instance, the algebra of NC toric cycles is, where m ab is a r × r matrix. A realization of T a on the space F of holomorphic functions f (y 1 , ..., y r ) with r arguments, is given by, [T a , f (y 1 , ..., y r )] = (∂ a − im ac y c ) f.
Quantum symmetries may be also considered by allowing the f (y 1 , ..., y r ) functions to depend on extra variables z a so that we have new function F (z a ; y a ) satisfying, Having studied the main lines of NC toric actions, we turn now to solve the constraint

Representations of the Z i 's
The constraint eqs(13) may be solved in different ways depending on whether quantum symmetries are taken into account or not. In general, τ ab torsion between U a and V a generators is introduced through the relations U a V b = Ω τ ab ab V b U a . The solution for the Z i 's reads as, where the T a and S a operators may be realized, for the case of a continuous spectrum, as and where x i are complex moduli, which we shall interpret as just the commutative coordinates of the toric manifold V d+1 . The q a i s are shifted CY charges having extra contributions coming from the toric data, q a i = q a i + d A=1 k A Q aA i + α k α N a iα with k A and k α are numbers; they still obey k i=0 q a i = 0 which follow from the identities k i=0 Q aA i = k i=0 N a iα = 0 eqs (4). The θ ij parameters of the NC type IIA geometry we get are, where now J ab ij = m [ab] q a i q b j and K ab ij ∼ τ [ab] q a i q b j . By appropriate choices of Λ ab , Ω ab , m [ab] and τ [ab] , one recovers, as special cases, the representations involving discrete torsions obtained in [15,17]. According to the nature of spectrums of T a and S a , the Z i operators will have two sectors; discrete and continuous. To illustrate the previous analysis, we consider the NC extension of a CY manifold with a conic singularity. This manifold is defined by a hypersurface M 2 embedded in the toric variety V 3 ⊂ C 6 /C * 2 with C * 2 actions x i −→ x i exp i (ψ a q a i ) where q 1 i = (1, −1, 1, −1, 0, 0) , q 2 i = (1, 0, −1, 1, −1, 0) and The four u α gauge invariants read as u α = 6 i=1 x n αi i , i = 1, ..., 6 with n αi integers as, They satisfy the constraint eq n 0i + n 2i = n 1i + n 3i , showing that V 3 is described by u 0 u 2 = u 1 u 3 with a conic singularity at the origin. The complex two dimension CY hypersurface embedded in V 3 , reads, in terms of the x i local coordinates, as The NC extension M nc 2 of this holomorphic hypersurface is directly obtained. For the special case where the L ij antisymmetric matrix is restricted to L ij = m q 1 i q 2 j − q 1 j q 2 i , with L ij = −L ji and L i6 = 0 and, the NC complex surface M nc 2 is then given by a one parameter algebra generated by following relations, where Λ m is given by Λ m = exp (−imψ 1 ψ 2 ). Since ψ a = ρ a − iα a ; it follows that mψ 1 ψ 2 = m (ρ 1 ρ 2 − α 1 α 2 ) − im (α 1 ρ 2 + α 2 ρ 1 ) which we set as Λ m = exp (κ + iφ) for simplicity. This is a one complex parameter algebra enclosing various special situations corresponding to: (1) Hyperbolic representation described by (κ, φ) ≡ (κ, 0). (2) Periodic representations corresponding to (κ, φ) ≡ (0, φ + 2π) where |Λ m | = 1. (3) Discrete periodic representations (κ, φ) ≡ (0, Nφ + 2π) with |Λ m | = 1 but moreover (Λ m ) N = 1. This last case is naturally a subspace of the periodic representation and it is precisely the kind of situation that happens in the building of BL NC manifolds with discrete torsion.

Fractional Branes
The NC type IIA realization we have studied concerns regular points of the algebra.
In this section, we want to complete this analysis by considering the representations at fixed points where live fractional D branes. To do so, we shall first classify the various subspaces S (µ) of stable points under C * r ; then we give the quiver diagrams extending those of Berenstein and Leigh [24].

• Fixed points
The local holomorphic coordinates x i ∈ C k+1 ; 0 ≤ i ≤ k eq(1) are not all of them independent as they are related by the C * r gauge transformations U a : Fixed points of the C * r gauge transformations are given by the solutions of the constraint eq Solutions of this eq depend on the values of q a i ; the x i 's should be zero unless q a i = 0. Fixed points are then given by the S subspace of C k+1 whose x i local coordinates are C * r gauge invariants. To get a more insight into this subspace it is interesting to note that as C k+1 /C * r = C k+1 /C * /C * r−1 ; it is useful to introduce the S (a) = {x i | q a i = 0; i ∈ J ⊂ {0, 1, ..., k}} subspaces that are invariant under the a − th factor of the C * r group. So the manifold S stable under the full C * r is given by the If we denote by x i 0 ; ...; x i k 0 −1 those local coordinates that have non zero q a i charges; J = {i 0 , ..., i k 0 }, and x i k 0 −1 ; ...; x i k the coordinates that are fixed under C * r ; then the manifold S is given by, To get the representation of the Z i variable operators on S, let us first consider what happens on its neighboring space S ǫ = {(ǫ, ..., ǫ, x k 0 + ǫ, ..., x k + ǫ)}, where ǫ is as small as possible. Using the hypothesis q a i k 0 = ... = q a i k = 0 we have made and replacing the x i moduli by their expression on S ǫ , then putting in the realization eqs (20), we get the following result, The representation of the Z i 's on S is then obtained by taking the limit ǫ −→ 0. As such non trivial operators are given by Z i j ∼ x i j I id ; k 0 ≤ j ≤ k; they are proportional to the I id operator of R (C * r ). This an important point since the Z i operators are reducible into an infinite component sum as shown here below, 4 • Brane interpretation The above decomposition of the Z i 's on S has a nice interpretation in the D brane language. Thinking about x i 's as the coordinates of a D p brane, ( p = 2d), wrapping M d , it follows that, due NC torus fibration, D p at singular points fractionate. In addition to the results of [24], which apply as well for the present study, there is a novelty here due to the dimension of the completely reducible R (C * r ) representation. There are infinitely many values for the C * characters and so an infinite number of fractional D (p − 2k 0 ) branes wrapping S. However, this is a unacceptable solution from string compactification point of view. Branes should have finite tensions in string theory background unless there are non compact dimensions. In fact this is exactly what we have since the irrational representation of the NC torus fibration introduces extra non compact directions as shown on the realization eqs (25,26). Such behaviour has no analogue in the BL case.

• Quiver Diagrams
Like for the BL case of CY orbifolds, one can here also describe the varieties of fractional D branes by generalizing the BL quiver diagrams for M nc d at fixed points. One of the basic ingredients in getting these graphs is the identification of the projectors of C * r and the step operators a ± . Since each C * factor has completely reducible representations with four sectors, it is interesting to treat them separately. The various sectors for each C * subsymmetry factor are as follows; (i) (dis, dis) discrete-discrete sector where the C * characters are given by χ n (ψ) = exp iψn; n ∈ Z+iZ, ψ ∈ C; (ii) (dis, con) discrete-continuous and (con, dis) continuous-discrete sectors and finally (iii) (con, con) continuous-continuous sector with characters χ (p, ψ) = exp ipψ; p ∈ C. Recall that, due to torsion, the algebraic structure of the D p branes wrapping the NC manifold change. Brane points {x i } of commutative type IIA geometry become fibers based on {x i }. These fibers are valued in the algebra of the group representation R (C * r ) and may be given a simple graph description. While points x i .1 in the commutative type IIA geometry are essentially numbers, the Z i coordinate operators can be thought of as Extending the results of [24], one can draw graphs for fractional D branes. Due to the decomposition of I id eqs( [11]) and (19), we associate to each D p brane coordinate a quiver diagram mainly given by the product of ( discrete or continuous ) S 1 circles. For the simplest case r = 1 and C * discrete representations, the quiver diagram is built as follows: (1) To each π n projector it is associated a vertex point on a discrete S 1 circle.
As there is an infinite number of points that one should put on S 1 , all happens as if the quiver diagram is given by the Z+iZ lattice plus a extra point at infinity. (2) The a ± n shift operators are associated with the oriented links joining adjacent vertices, ( vertex (n − 1) to the vertex (n) for a − n and vertex (n) to the vertex (n + 1) for a + n ) of quiver diagram. They act as automorphisms exchanging the C * characters.
Moreover, as D p brane coordinates at the singularities are of the form Z i ∼ n Z (n) i , it follows that D p branes on S sub-manifolds fractionate into an infinite set of fractional D2s branes coordinated by Z (n) i . This is a remarkable feature which looks like the process of tachyon condensation mechanismà la GMS [8,10,11,12] where for instance a non compact D p 1 brane on a NC Moyal plane decomposes into an infinite set of non compact D (p 1 − 2) branes. Such discussion is also valid for the case of a continuous spectrum; the corresponding quiver diagram is given by cross products of circles.

Conclusion
Using the BL algebraic geometry approach, we have studied the type IIA geometry of NCCY manifolds embedded in toric varieties V endowed with a NC toric fibration. Actually this study completes partial results of works in the literature on NCCY manifolds and too particularly orbifolds of CY homogeneous hypersurfaces with discrete torsion. The results of this paper concerns a more general class of singular CY manifolds M d embedded in toric varieties V d+1 with C * r toric actions endowed by torsions. These torsions are carried either by quantum symmetries described by inner automorphisms of C * r or again by considering NC complex cycles within the toric group factors in the same manner as one does in the Connes et al solution for toroidal compactification of matrix model of M theory. Among our results, we find that complex d dimension NCCY manifolds M nc d are non compact manifold that are naturally described in the language of toric geometry. The θ ij parameters carrying the structure have contributions involving, in addition to the usual CY condition k i=0 q a i = 0, the data of the toric polygons using the relations k i=0 q a i ν A i = 0.