Hypergeometric Systems in two Variables, Quivers, Dimers and Dessins d'Enfants
aa r X i v : . [ m a t h . AG ] N ov Hypergeometric Systems in two Variables,Quivers, Dimers and Dessins d’Enfants.
Jan Stienstra
Mathematisch Instituut, Universiteit Utrecht, the Netherlandse-mail: stien‘at’math.uu.nl
Abstract
This paper presents some parallel developments in Quiver/DimerModels, Hypergeometric Systems and Dessins d’Enfants. It demon-strates that the setting in which Gelfand, Kapranov and Zelevinskyhave formulated the theory of hypergeometric systems, provides alsoa natural setting for dimer models. The Fast Inverse Algorithm of[14] and the untwisting procedure of [4] are recasted in this more nat-ural setting and then immediately produce from the quiver data theKasteleyn matrix for dimer models, which is best viewed as the bi-adjacency matrix for the untwisted model. Some perfect matchingsin the dimer models are direct reformulations of the triangulations inGKZ theory and the rule which maps triangulations to the vertices ofthe secondary polygon extends to a rule for mapping perfect match-ings to lattice points in the secondary polygon. Finally it is observedin many examples and then conjectured to hold in general, that the de-terminant of the Kasteleyn matrix with suitable weights becomes aftera simple transformation equal to the principal A -determinant in GKZtheory. Illustrative examples are distributed throughout the text. In the last decade interesting correspondences were discovered relating QuiverGauge Theories, lattice polygons and Calabi-Yau singularities. The motiva-tion and evolution of these ideas in physics are well-documented in manyarticles; e.g. [1, 3, 4, 5, 6, 12, 13, 14, 15, 16, 17, 19, 20]. In the present paperwe want to put some aspects of these correspondences from physics alongsidethe hypergeometric systems in two variables of Gelfand, Kapranov, Zelevin-sky [8, 9, 10] and the dessins d’enfants of Grothendieck et al. [18, 22, 24].This reveals intriguing connections between these fields.1he beautiful insight of Gelfand, Kapranov and Zelevinsky was that hy-pergeometric structures greatly simplify if one introduces extra variables andbalances this with an appropriate torus action [8, 9, 10, 11, 25]. In order toprofit from the simplication they developed tools like the secondary fan, sec-ondary polytope and principal A -determinant . This paper demonstrates thatthese are also very practical tools for studying quivers and dimer models.A similar beautiful insight of simplification by going to higher dimensionsappeared in De Bruijn’s construction [2] of Penrose tilings and developed intothe well-known projection method in the theory of quasi-crystals; e.g. [23].We apply the same method to construct periodic rhombus tilings of the plane,which the physicists call brane tilings and dimer models .From a geometric perspective this paper deals with embeddings of quiversinto compact oriented surfaces without boundary. More specifically, the initialcombinatorial data for a quiver Q are two finite sets E (arrows) and V (nodes)and two maps s, t : E → V (source and target). Embedding Q into a compactoriented surface without boundary M means that V becomes a subset of M and an arrow e ∈ E becomes a path p e in M from the point s ( e ) to the point t ( e ). It is required that the boundary of every face of ( M , Q ) – i.e. connectedcomponent of M \ S e ∈ E p e – is formed by a sequence of paths ( p e , . . . , p e n )with p e i ∩ p e i +1 = t ( e i ) = s ( e i +1 ) for 1 ≤ i ≤ n ; e n +1 = e . It is evidentfrom this requirement that the face boundaries receive an orientation from Q . When this is combined with the orientation on M faces lie either on thepositive or on the negative side of their boundary. In pictures we mark faceswhich lie on the positive (respectively negative) side of their boundary with • (resp. ◦ ). It is obvious that adjacent faces get different colors. The dualgraph of Q w.r.t. M is thus a bi-partite graph Γ •◦ , embedded in M : thenodes of Γ •◦ correspond with the faces of ( M , Q ); the edges of Γ •◦ connectnodes coming from adjacent faces and correspond bijectively with the edges of Q . A pair ( M , Q ) consisting of a compact oriented surface without boundaryand an oriented graph embedded in it is called a dessin d’enfants or just dessin , in one of the various equivalent definitions of “dessin (d’enfants)”; see[22, 18, 24]. Other definitions refer to ( M , Γ •◦ ) as dessin d’enfants. In case M has genus 1 one often calls Γ •◦ and its lifting to the plane (i.e. the universalcovering of the torus M ) a dimer model or brane tiling ; see for instance[4, 5, 12, 13, 14, 15]. Yet another presentation of the same structure gives thedessin as the triangulation of M with vertex set the union of the nodes of Q and Γ •◦ ; the triangles are given by the triplets of vertices consisting of twonodes of Γ •◦ connected by an edge e ∗ of Γ •◦ and one node v of Q incident tothe edge e of Q which is dual to e ∗ . Figure 1 shows these three get-ups of thedessin d’enfants of the P -quiver (case B in Figure 3)From an algebraic perspective this paper deals with superpotentials forquivers . There is a simple equivalence between the geometric and algebraic2igure 1: Three versions of the dessin d’enfants for P (= case B in Fig. 3).The surface is obtained by identifying opposite sides of the hexagon. perspectives: the superpotential is a convenient way of writing the list of ori-ented boundaries ( p e , . . . , p e n ) of the faces of ( M , Q ) with ± -signs indicatinghow the orientation matches with that of M . It can also be read as theinstruction for building M by glueing polygons. For the dessin d’enfants inFigure 1 the superpotential (for some numbering of the arrows of Q ) is X X X + X X X + X X X − X X X − X X X − X X X . Another algebraic perspective, equivalent to the previous one, is given bywhat we want to call the bi-adjacency matrix of the dessin d’enfants ( M , Q ) with weight ̟ . It is defined as follows. The weight is a function ̟ : E → C .The dessins in the present paper have as many black faces as white facesand every edge e ∈ E lies in the boundary of a unique black face, denoted b ( e ), and a unique white face, denoted w ( e ). The bi-adjacency matrix of( M , Q ) with weight ̟ is a square matrix K ̟ with rows corresponding withthe black faces, columns corresponding with the white faces and entries inthe polynomial ring C [ u v | v ∈ V ] which has one variable for every node v ofthe quiver Q : the ( b , w ) -entry of K ̟ is K ̟ b , w = X e ∈ E : b ( e )= b , w ( e )= w ̟ ( e ) u s ( e ) u t ( e ) . (1)The bi-adjacency matrix is in fact the Kasteleyn matrix of a twist of the dimermodel ( M , Γ •◦ ); see Section 8. For the dessin in Figure 1 the bi-adjacencymatrix (for some numbering of the nodes, edges and faces) is ̟ u u ̟ u u ̟ u u ̟ u u ̟ u u ̟ u u ̟ u u ̟ u u ̟ u u . In Section 2 we describe how the quivers for which we can solve the em-bedding problem, are associated with certain rank 2 subgroups L of Z N ; here N is the number of nodes of the quiver. Such subgroups L ⊂ Z N are the3oundations for the theory of Gelfand, Kapranov and Zelevinsky. In Section3 we describe the secondary fan and the secondary polygon associated with L ⊂ Z N . These important structures in GKZ-theory surprisingly turn out tobe also quite relevant for the dimer models. We show for instance in Theorem9.3 that the secondary polygon is the Newton polygon of the determinant ofthe bi-adjacency matrix with non-zero weights. In Section 10 we report theobservation, based on examples, that for the critical weight crit : E → Z > , crit( e ) = ♯ { e ′ ∈ E | s ( e ′ ) = s ( e ) , t ( e ′ ) = t ( e ) } (2)the determinant of the bi-adjacency matrix K crit , after a simple transforma-tion, becomes equal to the principal A -determinant of Gelfand, Kapranov,Zelevinsky [11]. In Sections 4 and 5 we recall some topics in the theory ofGKZ-hypergeometric systems of differential equations.Section 6 presents the algorithm to solve the quiver embedding problem.It is a combination of the Fast Inverse Algorithm of [14] and the untwistingprocedure of [4]. In the cited papers, however, methods are mainly presentedvia visual inspection of pictures in some concrete examples. So an extrapo-lation to general situations was needed. The first impression that the FastInverse Algorithm of [14] is more or less De Bruijn’s construction [2] of Pen-rose tilings did not quite yield the sought after embedding of the quiver; theuntwisting procedure of [4] is also needed. The algorithm became a smoothlyoperating algebraic-combinatorial tool by consistently working from the phi-losophy that things look simpler from a higher dimensional viewpoint (withgroup action). For doing the computer experiments behind this paper I imple-mented the algorithm in matlab . The reader can find in Section 6 sufficientdetails for making a computer version of the algorithm.Sections 7–10 demonstrate that consistently working from the higher di-mensional viewpoint leads to new insights in the dimer technology tools per-fect matchings, Kasteleyn matrix and its determinant and shows their closerelation with the GKZ tools secondary polytope and principal A -determinant .Besides problems such as proving that the algorithm of Section 6 yieldsat least one superpotential for every quiver which satisfies the conditions inTheorem 2.10, or proving Conjecture 10.5 our work raises some interestingquestions like: Q1.
The Calabi-Yau singularities in the background of this work areconstructed from toric diagrams, which here are interpreted as secondarypolygons. In [19] § § R . In the work of Gelfand, Kapranovand Zelevinsky the secondary polygon is put into a 2-dimensional plane in4 N , which does not pass through ; here N is the number of nodes of thequiver. One can perform the standard toric variety construction with the3-dimensional cone over the secondary polygon in R N . Does this toric varietygive the appriopriate view in the GKZ philosophy (higher dimension compen-sated by group action) on the singular Calabi-Yau -space? Since this toricvariety is a natural domain for GKZ hypergeometric functions (see 5.2) onemay wonder:
Do hypergeometric functions provide new useful tools for in-vestigating the geometry of -dimensional Calabi-Yau singularities? Possiblypositive indications are the fact that the dimension of the solution space ofthe GKZ system of hypergeometric differential equations equals the size ofthe bi-adjacency matrix K crit (see Theorem 5.3 and Corollary 7.3) and theobservation (Conjecture 10.5) that the determinant of K crit equals, up to asimple transformation, the principal A -determinant, which describes the sin-gularities of the GKZ system. Q2.
The surface M with embedded in it the pair of dual graphs Q andΓ •◦ has been constructed in a purely combinatorial topological way from thesuperpotential. It is a general fact (see [18] § f from M to the 2-sphere S which is a ramified covering withexactly three ramification points such that the set of vertices of Q , the set ofblack vertices of Γ •◦ and the set of white vertices of Γ •◦ are the fibers over thethree ramification points. A highlight in the theory of dessins d’enfants is Be-lyi’s Theorem (see [18] Theorem 2.1.1). It states that in the above situationthe surface M admits a model M K over a number field K such that f be-comes a morphism M K → P K of algebraic curves over K which is unramifiedoutside { , , ∞} . The labeling of the ramification points can be taken suchthat f − ( ∞ ), f − (0) and f − (1) are the sets of white and black vertices ofΓ •◦ and the vertices of Q , respectively. The bipartite graph Γ •◦ in M is thenthe inverse image of the negative real axis [ −∞ ,
0] in the Riemann sphere P C = S and the quiver Q is the inverse image of the positively orientedunit circle { z ∈ C | | z | = 1 } . It is usually difficult to find explicit algebraicequations for M K and the Belyi function f .On the other hand, the authors of [4] write in footnote : .... we have pro-duced a dimer model that is defined on its own spectral curve det K ̟ = 0 . Unfortunately their arguments are not (yet) sufficiently refined to yield theweight ̟ that is to be used in this equation.It seems an interesting challenge to tackle the two problems simultaneouslyand look for a weight ̟ and a Belyi function f on the algebraic curve withequation det K ̟ = 0 that realize ( M , Q , Γ •◦ ) as described above.5 Rank subgroups of Z N and Quivers. in this paper is a rank 2 subgroup L ⊂ Z N which is not contained in any of the standard coordinate hyperplanes of Z N and is perpendicular to the vector (1 , . . . , L are vectors ( ℓ , . . . , ℓ N ) ∈ Z N with ℓ + . . . + ℓ N = 0 and forevery i ∈ { , . . . , N } there is an ( ℓ , . . . , ℓ N ) ∈ L such that ℓ i = 0. Throughout this paper e , . . . , e N is the standard basis of Z N and J denotes the matrix (cid:18) − (cid:19) . Taking the second exterior powers onefinds an inclusion V L ֒ → V Z N . The group V L is a free Z -module of rank1. After fixing the orientation on L , i.e. choosing one of the two possibleisomorphisms V L ≃ Z , one gets an inclusion Z ֒ → V Z N . The coordinatesof 1 ∈ Z with respect to the standard basis { e i ∧ e j } ≤ i The “if”-statement is trivial. So let us consider the “only if” andassume that C is anti-symmetric and rank C = 2. Let d denote the greatestcommon divisor of the entries of C and C ′ = d C . Choose an N × D whose columns form a Z -basis for the column space of C ′ . The equality ofcolumn spaces of C ′ and D means that there are an N × E and a2 × N -matrix F , both with entries in Z and of rank 2, such that D = C ′ E and C ′ = DF . Then C ′ = DF = C ′ EF = − C ′ t EF = − F t D t EF . So D t E is an anti-symmetric 2 × D t E = f J with f ∈ Z . In fact f = ± C ′ . If f = 1 we replace F by J F . Wethen always have C ′ = F t J F . Let G be any 2 × Z and det G = d . Let B = GF . Then C = B t J B as wanted. (cid:4) Multiplying in Proposition 2.8 the matrix B from the left by a matrixfrom Sl ( Z ) does not change the matrix B t J B . So it is more natural tointerpret C as the matrix of the Pl¨ucker form of the Z -row space of B . Notethat it follows from the proof of Proposition 2.8 that this space is uniquelydetermined if the greatest common divisor of the entries of C is 1. If on theother hand the greatest common divisor of the entries of C is d > × Z d .Let us summarize the above discussion: For every L ⊂ Z N as in 2.1 one has its Pl¨ucker quiver.This quiver has no isolated nodes or directed loops of length ≤ and in everynode the number of incoming arrows equals the number of outgoing arrows.Moreover the rank of its anti-symmetrized adjacency matrix is . Conversely,every quiver with these properties is the Pl¨ucker quiver of some L ⊂ Z N as in2.1. The correspondence between such quivers and such L ⊂ Z N is one-to-onefor those quivers for which the greatest common divisor of the entries of theanti-symmetrized adjacency matrix is and those L ⊂ Z N for which Z N / L has no torsion. (cid:4) L ⊂ Z N . For L ⊂ Z N as in 2.1 we now define the secondary fan and the secondarypolytope . In 4.12 we will compare this with the original definitions by Gelfand,Kapranov and Zelevinsky. The term “secondary” refers to the fact that intheir theory of hypergeometric systems another polytope appears first, whichis therefore called the primary polytope . Nonetheless both the secondary fanand the secondary polytope can most conveniently and directly be describedusing the lattice L . In case the rank of L is 2, the constructions becomeparticularly simple. Let L ⊂ Z N be as in 2.1. Let L ∨ R := Hom( L , R ) denote thereal dual space of L . Let e , . . . , e N be the standard basis of Z N . Let b i ∈ L ∨ R be the image of e i under the map R N → L ∨ R dual to the inclusion L ֒ → Z N .Here and henceforth we identify R N with the real dual space of Z N by meansof the standard dot product.By definition, the secondary fan of L is the following collection of cones in L ∨ R : the 0-dimensional cone { } , the 1-dimensional cones R ≥ b i ( i = 1 , . . . , N )and the 2-dimensional cones which are the closures of the connected compo-nents of L ∨ R \ S Ni =1 R ≥ b i . With a 2-dimensional cone C in the secondary fan oneassociates the set L C := {{ i, j } ⊂ { , . . . , N } | C ⊂ ( R ≥ b i + R ≥ b j ) } (4)8f 2-element subsets of { , . . . , N } and the vector ψ C := X { i,j }∈ L C | det( b i , b j ) | ( e i + e j ) . (5)The secondary polytope Σ( L ) of L ⊂ Z N is then defined asΣ( L ) := convex hull ( { ψ C | C } ) . In § L C as atriangulation of the primary polytope. In Equation (27) we associate with L C a perfect matching in a bipartite graph. Figure 2 shows the secondary fan with the lists L C , thesecondary polytope Σ( L ) with coordinates for the vertices and the primarypolytope for L = Z (0 , , , − ⊕ Z ( − , , , − ⊂ Z . This is case B ofFigure 3; see also Example 4.6 and § ❄ ✲✁✁✁✁✁✁✁✁✕✟✟✟✟✟✟✟✟✙ b b b b { , }{ , }{ , } (cid:20) { , }{ , } (cid:21)(cid:20) { , }{ , } (cid:21)(cid:2) { , } (cid:3)(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) t t t ttt t t❜ ✁✁✁✁✁✁✟✟✟✟✟✟ [1 , , , , , , 1] [2 , , , , , , The secondary fan (right) and the primary polytope (top left) and thesecondary polytope (bottom left) for L = Z (0 , , , − ⊕ Z ( − , , , − ⊂ Z . The geometric formulation of the construction of the secondary fanand polytope is quite attractive. Nonetheless in practical algorithms every-thing can be most easily obtained from the matrix of the Pl¨ucker form of L ⊂ Z N . Indeed, a 2-dimensional cone in the secondary fan is bounded bytwo half-lines R ≥ b i and R ≥ b j with the property that det( b i , b j ) > b i , b k ) det( b j , b k ) ≥ k = 1 , . . . , N . Thus to find the secondary fanone just needs to find all pairs b i , b j with these properties.9or a cone C bounded by R ≥ b i and R ≥ b j the list L C is then L C = { { k, l } | det( b k , b i ) ≥ b j , b l ) ≥ } . Now consider two adjacent 2-dimensional cones in the secondary fan, say C and C ′ . Let b i , . . . , b i s be those vectors from the set { b , . . . , b N } that lieon the half-line C ∩ C ′ . Without loss of generality we may assume C lies tothe right of b i and C ′ to the left. Then L C \ ( L C ∩ L C ′ ) = { { k, l } | l ∈ { i , . . . , i s } , det( b l , b k ) < } ,L C ′ \ ( L C ∩ L C ′ ) = { { k, l } | l ∈ { i , . . . , i s } , det( b l , b k ) > } . From this we see, using P Nk =1 det( b i , b k ) = 0 for all i , that ψ C ′ − ψ C = s X r =1 N X k =1 det( b i r , b k ) e k . (6)In other words, ψ C ′ − ψ C is the sum of rows i , . . . , i s of the matrix of thePl¨ucker form. To make this even more explicit and simple looking we take a basis for L compatible with the chosen orientation. One can represent the two basisvectors as the rows of a 2 × N -matrix B . Let b , . . . , b N be the columns ofthis matrix. The elements of L should now be written as row vectors withtwo components and the embedding L ֒ → Z N is given by v P Nk =1 ( v · b k ) e k .Since det( b i , b k ) = b ti J b k , we can reformulate Equation (6) as ψ C ′ − ψ C = image of s X r =1 b ti r J under the embedding L ֒ → Z N . (7)In order to obtain the simplest formulation for the construction we orderthe vectors b , . . . , b N so that the points p k = P ki =1 b ti for k = 1 , . . . , N lieordered counterclockwise on the boundary of the polygon∆ = convex hull { p , . . . , p N } . (8) Then the secondary polygon Σ( L ) is obtained by first rotating ∆ clockwiseover ◦ , next embedding it along with L into Z N and finally translating itover the vector ψ C , where C is the cone in the secondary fan with left handboundary R ≥ b . Note that while the secondary polytope Σ( L ) depends only on the embed-ding L ֒ → Z N , the polygon ∆ usually changes when one puts another vector b i in first position by a cyclic permutation or when one multiplies the vectors b , . . . , b N by a matrix from Sl ( Z ). 10 .8. For a converse to the above construction we start from a convex polygon∆ in R with vertices in Z and non-empty interior. Let ∂ ∆ denote itsboundary. Next we choose a collection of points p , . . . , p N in ∂ ∆ ∩ Z whichincludes all vertices of ∆. We number these points so that p i and p i +1 areconsecutive points in the counter-clockwise orientation of ∂ ∆. Thinking ofthe elements of Z as row vectors we define column vectors b = p t and b i = p ti − p ti − for i = 2 , . . . , N . Finally we define L to be the Z -row space ofthe 2 × N -matrix B with columns b , . . . , b N . Thus every convex polygon ∆ in R with vertices in Z and non-emptyinterior can be viewed as the secondary polygon of some rank subgroup L ⊂ Z N satisfying the conditions in 2.1. Note however that in general there are several possible choices for thepoints p , . . . , p N in ∂ ∆ ∩ Z . The minimal choice takes only the vertices of∆, while the maximal choice takes all points of ∂ ∆ ∩ Z . The (Euclidean) area of the polygon ∆ in (8) is area ∆ = P ≤ i After applying, if necessary, a cyclic permutation to b , . . . , b N wemay assume, without loss of generality, that C = R ≥ b + R ≥ b N .From the triangulation ∆ by the diagonals between the vertex p N = and the other vertices of ∆ one sees that area ∆ = P ≤ i This follows from Theorem 3.9 in combination with Pick’s Formula ([7] p.113): 2area ∆ = 2 ♯ ( Z ∩ interior ∆) + ♯ ( Z ∩ boundary ∆) − (cid:4) ss ✲✲✲ ❏❏❏❏❪ ❏❏❏❪ ❏❏❏❪✡✡✡✡✢ ✡✡✡✢ ✡✡✡✢ B = (cid:20) − − 11 1 − (cid:21) Model of P s ss❛ ✟✟✟✟✟❅❅❅✁✁✁✁✁ ss ss 41 32 ✻✻(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)✒✛✛✛ ✲❅❅❅❅❅❘ ❄❄ B = (cid:20) − − − (cid:21) Model of dP ss s s❛ ✁✁✁✁✁✟✟✟✟✟ ss ss 14 32 ✲✲❄❄ ✲✲❅❅❅❅❅■ ❅❅❅❅❅■ ❅❅❅❅■ ❅❅❅❅■ ❄❄ B = (cid:20) − − 10 0 2 − (cid:21) Model I of F s s ss❛ ❆❆❆❆❆✁✁✁✁✁ ss ss 41 32 ✻✻✛✛ ✲✲ ❄❄ B = (cid:20) − − − − (cid:21) Model II of F s s ss❛ ❅❅❅(cid:0)(cid:0)(cid:0)❅❅❅(cid:0)(cid:0)(cid:0) s ss ss 12 4 35 ❅❅❅❘❅❅❘❆❆❆❆❆❑ ✻✻✻❅❅❅■✛ ✲❍❍❍❍❍❥❄ (cid:0)(cid:0)(cid:0)✠ (cid:0)(cid:0)✠ B = (cid:20) − − − − (cid:21) Model I of dP ss s ss❛ ❆❆❆❆❆(cid:0)(cid:0)(cid:0) Figure 3: Quivers from [3] Figures 10, 11, 4, 12 and corresponding polygons. s s sss 15 6 423 ❄✟✟✟✯❍❍❍❍❍❥❍❍❍❨ ✟✟✟✟✟✙ ❄❄❄❍❍❍❥❍❍❥ ✟✟✟✙ ✟✟✙✡✡✡✡✣✡✡✡✡✣✡✡✡✣❏❏❏❏❪ ❏❏❏❏❪ ❏❏❏❪ B = (cid:20) − − − (cid:21) Model IV of dP sss s s s❛ ◗◗◗◗◗◗◗◗ ss s sss 53 6 421 ❄❍❍❍❍❍❥✟✟✟✟✟✯✟✟✟✟✟✯✟✟✟✯❍❍❍❨ ✻✻✡✡✡✡✢✛ ❍❍❍❨ ❍❍❨✟✟✟✙ ❄ B = (cid:20) − − − − (cid:21) Model III of dP ssssss ❛ ❍❍❍❍❍ ss s sss 13 4 265 ❄❍❍❍❍❍❥❍❍❍❍❍❥✡✡✡✡✣❍❍❍❥❏❏❏❏❪ ✡✡✡✡✣❏❏❏❏❪✛ ✟✟✟✙✛ ❄❍❍❍❥✟✟✟✙ B = (cid:20) − − − − (cid:21) Model II of dP ss s sss❛ ❅❅❅(cid:0)(cid:0)(cid:0) ss s sss 12 3 456 ❄❍❍❍❥ ✟✟✟✯ ✻❍❍❍❨✟✟✟✙ ✲❏❏❏❏❪✡✡✡✡✢✛❏❏❏❏❫ ✡✡✡✡✣ B = (cid:20) − − − − (cid:21) Model I of dP ss s sss❛ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) s ss ss 21 3 45 ✻❅❅❅❘ (cid:0)(cid:0)(cid:0)✒✻❅❅❅■✛✛❍❍❍❍❍❥❆❆❆❆❆❯ ✛✛ B = (cid:20) − − − − (cid:21) Model II of dP ss s ss❛ (cid:0)(cid:0)(cid:0)❅❅❅(cid:0)(cid:0)(cid:0) Figure 4: Quivers from [3] Figures 12, 9 and corresponding polygons. (cid:0)(cid:0) s s sssss ❛ ❅❅❅❅❅(cid:0)(cid:0)(cid:0) s s s ssss ❛ ❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) s s s s ssss ❛s s ssssss ❛ ❅❅❅❅❅ s s s sssss ❛ ❅❅❅❅❅❅❅❅ s s s ssssss ❛ B B B B B B Figure 5: Remaining reflexive polygons. Figures 3 and 4 show some examples of the above relationbetween quivers, rank 2 subgroups L of Z N as in 2.1 and polygons. Thequivers are taken from [3]. We name our examples neutrally as B , . . . , B ,but also mention the names given to these quiver models in [3]. The latternames refer to del Pezzo surfaces dP k , i.e. the projective plane P blownup in k points, some of which, on the present occasion, may happen to be“infinitely near” and therefore require repeated blow-ups. With the lattersomewhat liberal use of the name del Pezzo surface , dP k matches well withthe toric geometry of the fan whose 1-dimensional rays are the half-lines fromthe interior lattice point ◦ through a lattice point • at the boundary. Notehowever that this fan is in general not the same as the secondary fan andthat therefore the singularity associated with the polygon (cf. [12, 19]) isnot the singularity obtained by contracting to a point the zero-section of thecanonical bundle of the del Pezzo surface.The polygons in Figures 3 and 4, however, are in general different from,the polygons associated with the same quiver in [3] Figure 8. We will clarifythis issue in 6.14, 6.15, 6.16.It is remarkable that the polygons in Figures 3 and 4 are exactly thelattice polygons with one interior lattice point and ≤ reflexive polygons can be found in [21].In cases B , B and B the greatest common divisor d of the entries inthe quiver’s adjacency matrix is > d . We have chosen this matrix suchthat the polygon is a reflexive polygon. In the late 1980’s Gelfand, Kapranov and Zelevinsky discovered fas-cinating generalizations of the classical hypergeometric structures of Euler,Gauss, Appell, Lauricella, Horn [8, 9, 10, 11, 25]. The main ingredient forthese new hypergeometric structures is a finite sequence A = ( a , . . . , a N ) ofvectors in Z k +1 which generates Z k +1 as an abelian group and for which thereexists a group homomorphism h : Z k +1 → Z such that h ( a i ) = 1 for all i .The latter condition means that A lies in a k -dimensional affine hyperplane in Z k +1 . Figure 6 shows A (each black dot represents one vector) sitting in thishyperplane for some classical hypergeometric structures. Gelfand, Kapranovand Zelevinsky called these new structures A -hypergeometric systems , butnowadays most authors call them GKZ hypergeometric systems .Let L denote the lattice (= free abelian group) of linear relations in A : L := { ( ℓ , . . . , ℓ N ) ∈ Z N | ℓ a + . . . + ℓ N a N = } . (10)It follows from the above construction and assumptions that the quotientgroup Z N / L is Z k +1 , and a fortiori, it is torsion free.In this paper we consider only the case when the rank of L is 2; i.e. k + 1 = N − 2. Since we assumed the existence of a linear map h : Z N − → Z such that h ( a i ) = 1 for all i , L lies in the kernel of the map s defined by: s : R N → R , s ( z , . . . , z N ) = z + . . . + z N . (11)The following lemma shows what the other condition for L in 2.1 meansin terms of A . In the situation of 4.1 the following statements are equiv-alent:1. L is not contained in any of the standard coordinate hyperplanes of Z N .2. No vector in the sequence A is linearly independent of the other vectors.3. For every i the lattice of linear relations between the vectors of A \ { a i } has rank . Proof. Assume L is contained in the i -th coordinate hyperplane. Thenthe i -th component of every element of L is 0 and hence a i occurs in nolinear relation for the vectors in A . In other words, L is the lattice of linear15elations between the vectors of A \ { a i } . Since L has rank 2, we concludethat statement 3 implies statement 1.Conversely, assume that the lattice L i of linear relations between the vec-tors of A \ { a i } has rank > 1. Since L i is contained in L and the latter hasrank 2, we see that L i ⊗ R = L ⊗ R . Therefore, since L i is contained in the i -th coordinate hyperplane, so is L . Hence statement 1 implies statement 3.It is obvious that statements 1 and 2 are equivalent. (cid:4) A subsequence A ′ of A is said to be minimally dependent ifthe vectors in A ′ are linearly dependent and the vectors in every subsequence A ′′ of A ′ with A ′′ = A ′ are linearly independent. A linear dependence relation P j α j a j = 0 is a minimal linear dependence relation in A if the subsequence( a j | α j = 0) is minimally dependent. 1. Let B be a × N -matrix such that its rows are a Z -basis for L . Let b , . . . , b N be the columns of this matrix. Then one has for every i N X j =1 det( b i , b j ) a j = . (12) 2. Assume the conditions in Proposition 4.2 are satisfied. Then every lin-ear relation (12) is minimal and every minimal linear dependence rela-tion in A is a non-zero scalar multiple of some relation (12) . Proof. For statement 1, let A be the matrix with columns a , . . . , a N . ThenEquation (10) can be rewritten as BA t = 0. This implies B t J BA t = 0, whichis just a compressed form of the relations we wanted to prove.For statement 2 note that condition 3 in Proposition 4.2 implies that everylinear relation (12) is minimal. Conversely, consider a minimal linear relation P j α j a j = 0. Then the subsequence ( a j | α j = 0) has at most N − (cid:4) Gelfand, Kapranov and Zelevinsky [8, 9, 10] associate with a set A as in4.1 and a vector c ∈ C k +1 the following system of partial differential equationsfor functions Φ of N variables u , . . . , u N : • for every ( ℓ , . . . , ℓ N ) ∈ L one differential equation Y ℓ i < (cid:18) ∂∂u i (cid:19) − ℓ i Φ = Y ℓ i > (cid:18) ∂∂u i (cid:19) ℓ i Φ , (13)16 the system of k + 1 differential equations a u ∂ Φ ∂u + . . . + a N u N ∂ Φ ∂u N = c Φ . (14) The sequence A = ( a , a , a , a ) ∈ Z for Example B inFigure 3 can be taken to be a = (cid:20) (cid:21) , a = (cid:20) (cid:21) , a = (cid:20) (cid:21) , a = (cid:20) (cid:21) . It is a classical result of K. Mayr that the roots of the 1-variable cubicpolynomial P ( x ) = u + u x + u x + u x as functions of the coefficients u , . . . , u satisfy the GKZ system of differential equations for this A andwith c = (cid:20) − (cid:21) ; see e.g. [25] § An important example in the toric geometry constructionsof Sasaki-Einstein manifolds is known under the name L a,b,c ; see e.g. [6].Here a, b, c are integers with c ≤ b and 0 < a ≤ b The polygon for thisexample, displayed in [6] Figure 2, is a quadrangle with vertices (0 , , ak, b ), ( − al, c ) where k and l are integers such that ck + bl = 1. The methodexplained in 3.8 yields L = Z (1 , ak − , − al − ak, al ) ⊕ Z (0 , b, c − b, − c ). Fromthis we see that we can take A = ( a , a , a , a ) ∈ Z with a = (cid:20) c − a (cid:21) , a = (cid:20) c (cid:21) , a = (cid:20) (cid:21) , a = (cid:20) b (cid:21) . As in Example 4.6 this means that the corresponding GKZ system deals withthe roots of the “four-nomial” u x c − a + u x c + u + u x b as functions of thecoefficients u , u , u , u .According to [6] § Y p,q of [1, 19] are special cases of theabove: Y p,q = L p − q,p + q,p . So, these correspond to the “four-nomials” u x q + u x p + u + u x p + q . Thus Example 4.6 is in fact Y , = L , , . From the pictures of A shown in Figure 6 one easily sees thatfor the Gauss system the lattice L is generated by the vector (1 , , − , − F and F thelattice L has rank 2 and the corresponding B -matrices are F : (cid:20) − − − − (cid:21) , F : (cid:20) − − − − (cid:21) . tt t (cid:0)(cid:0)(cid:0)(cid:0)✏✏✏✏✏✏✏ ✏ ✏ ✏ t tt t tt (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)✁✁✁✁✁✁✁❙❙❙❙❙ ✁✁✁✁✁✁✁ ❙❙❙❙❙ ❊❊❊❊❊❊❊❊ ✄✄✄✄✄❊❊ ❊❊✄✄✄✄ t t tt tt Figure 6: The sets A for Gauss’s hypergeometric functions (left), Appell’s F (middle) and Appell’s F (right). Thus Appell’s F corresponds to B in Figure 4. For Appell’s F thequiver and the polygon are ss s sss 16 4 253 ❄❍❍❍❥ ✟✟✟✯ ✻❍❍❍❨✟✟✟✙ ✲✡✡✡✡✣ ❏❏❏❏❫✛ ✡✡✡✡✢❏❏❏❏❪ s s ssss (cid:0)(cid:0)(cid:0)(cid:0)(cid:0) In [8] § F , all Horn series give a quiver listed inExample 3.11: { G } ↔ B , { H } ↔ B , { G , H , H } ↔ B , { F , G } ↔ B , { H } ↔ B , { H , H } ↔ B , { F , F , H } ↔ B . In 4.1 we started from the sequence of vectors A and then defined L viaEquation (10). Let us reverse the procedure and start with a rank 2 subgroup L ⊂ Z N as in 2.1. We can now define A = ( a , . . . , a N ) with (15) a i = the class e i mod L in the quotient group Z N / L . If the quotient group Z N / L is torsion free, it is isomorphic to Z N − and thesequence of vectors A in (15) satisfies the requirements of 4.1. Thus (10) and(15) give equivalences, inverse to each other, between the data A and L . In combination with Theorem 2.10 this gives There is an equivalence of data between on the one handsequences A as in 4.1 which satisfy also the conditions in Proposition 4.2 and n the other hand quivers without isolated nodes or directed loops of length ≤ , such that in every node the number of incoming arrows equals the numberof outgoing arrows and such that for its anti-symmetrized adjacency matrixthe rank is and the greatest common divisor of the entries is . (cid:4) We refer to 5.5 for a discussion of how to extend thedefinition of GKZ systems so as to allow for torsion in the group Z N / L . In order to have an efficient method to construct bases for the solutionspace of the hypergeometric system (13)-(14) Gelfand, Kapranov and Zelevin-sky developed the theory of the secondary fan and the secondary polytope .The term “secondary” refers to the habit of considering the set A as primarydata and to call the convex hull of A the primary polytope .In case the lattice L of relations in A has rank 2, the constructions becomeparticularly simple. The secondary fan is the one in Definition 3.2. With a 2-dimensional cone C in the secondary fan one associates a set L C of 2-elementsubsets of { , . . . , N } as in Equation (4). One can interpret this L C is as atriangulation of the primary polytope convex hull ( A ) as follows. Recall that A = ( a , . . . , a N ) and assume for convenience of imagining pictures that all a i ’s are different. For { i, j } ∈ L C set T { i,j } := convex hull ( { a k | k = i, j } ).This T { i,j } is an ( N − T { i,j } with { i, j } ∈ L C together constitute a triangulation the primary polytope .In [11] p.220 the secondary polytope is constructed (and defined) in termsof these triangulations as follows. For a 2-element subset { i, j } ⊂ { , . . . , N } with i < j let A ij denote the ( N − × ( N − a k for k = i, j in the natural order of increasing indices. To a 2-dimensional cone C in the secondary fan one then associates the vector ϕ C := X { i,j }∈ L C | det A ij | X k = i,j e k . (16)Then in [11] p.220 the secondary polytope Σ( A ) is defined asΣ( A ) := convex hull ( { ϕ C | C } ) . (17)As | det A ij | is ( N − T { i,j } , thenumber vol A := P { i,j }∈ L C | det A ij | is for every C equal to ( N − convex hull ( A ) and the point1( N − A X { i,j }∈ L C | det A ij | X k = i,j a k coincides for every C with the barycentre of the primary polytope. Thesecondary polytope therefore lies in a 2-dimensional plane parallel to L R .19 .13. Let us compare Σ( A ) with the polytope Σ( L ) of Definition 3.3. Itis a well-known and easy to prove fact that | det A ij | = | det( b i , b j ) | for all { i, j } ∈ L C (see e.g. [25] Eq. (62)). Thus Equations (16) and (5) yield ϕ C = − ψ C + vol A · N X k =1 e k , (18)with vol A = X { i,j }∈ L C | det A ij | = X { i,j }∈ L C | det( b i , b j ) | . (19)This means that the two secondary polytopes Σ( A ) and Σ( L ) are related bya point symmetry with centre vol A · P Nk =1 e k :Σ( A ) = − Σ( L ) + vol A · N X k =1 e k . (20) The variables in GKZ theory (see Section 4) are the natural coordinateson the space C A := Maps( A , C ) of maps from A to C . The torus T k +1 :=Hom( Z k +1 , C ∗ ) of group homomorphisms from Z k +1 to C ∗ , acts naturally on C A and on the functions on this space: for σ ∈ T k +1 , u ∈ C A , a ∈ A andΦ : C A → C :( σ · u )( a ) = σ ( a ) u ( a ) , (Φ · σ )( u ) = Φ( σ · u ) . (21)The GKZ hypergeometric functions associated with A and c are definedon open domains in C A . One easily sees that if a function Φ on C A satisfiesthe differential equations (13) then for every σ ∈ T k +1 the function Φ · σ alsosatisfies these differential equations. So, the torus T k +1 acts on the solutionspace of the system of differential equations (13).On the other hand, if Φ and Φ are two functions which satisfy the dif-ferential equations (14) with the same c , then their quotient Ψ = Φ Φ satisfies a u ∂ Ψ ∂u + . . . + a N u N ∂ Ψ ∂u N = . The latter equation is equivalent to Ψ being T k +1 -invariant. Thus we find: All quotients of pairs of solutions of the system (13)-(14) are functionson simply connected open subsets of the orbit space C A / T k +1 . In the case ofinterest in the present paper the dimension of this orbit space is N − k − . c ∈ Z k +1 the differential equations (14) are equivalent with Φ trans-forming under the action of T k +1 according to the character given by c :Φ · σ = σ ( c )Φ . (22)In particular for c = all solutions of (14) are T k +1 -invariant. The space ( C ∗ ) A := Maps( A , C ∗ ) of maps from A to C ∗ is a torus ofdimension N which contains T k +1 as a subtorus. The action (21) of σ ∈ T k +1 on C A obviously restricts to the action of T k +1 on ( C ∗ ) A by multiplication.The quotient space ( C ∗ ) A / T k +1 , which is a subspace of C A / T k +1 , is the torusHom( L , C ∗ ) of group homomorphisms from L to C ∗ . The toric variety as-sociated with the secondary fan (cf. Definition 3.2) gives a compactificationof the torus Hom( L , C ∗ ); see [7] for the general theory of toric varieties. Itis an essential part of the GKZ philosophy that quotients of hypergeomet-ric functions should be viewed as being defined on open subsets of this toricvariety. (cf. [8] Theorems 2 and 5, [8] ′ , [26] Prop. 13.5 ) Let N A = { x a + . . . + x N a N ∈ R k +1 | ∀ x i ∈ Z ≥ } , Z A = { x a + . . . + x N a N ∈ R k +1 | ∀ x i ∈ Z } , pos( A ) = { x a + . . . + x N a N ∈ R k +1 | ∀ x i ∈ R ≥ } . Assume N A = Z A ∩ pos( A ) , then the dimension of the space of solutions ofthe system of differential equations (13)-(14) at a general point of C A equalsthe number vol A in Equation (19). (cid:4) As shown in the proof of [26] Prop. 13.15 the condition N A = Z A∩ pos( A ) in the above theorem is satisfied if A admits a unimodulartriangulation . The latter condition is equivalent to: there is a cone C in thesecondary fan such that | det( b i , b j ) | = 1 for all { i, j } ∈ L C (see Definitions3.2 and 3.3). From the discussion in 5.1 one may get an idea about ex-tending the GKZ system (13)-(14) to the situation in which Z N / L has anon-trivial torsion subgroup (cid:0) Z N / L (cid:1) tors . Equation (13) still makes sense inthis more general situation, but Equation (14) must be adapted.Recall from (15) a definition of A which also works in the torsion case.Let G A := Hom( Z N / L , C ∗ ). This is a commutative algebraic group of whichthe connected component of the identity is G ◦A = T k +1 = Hom( Z k +1 , C ∗ )and the group of connected components is the finite abelian group G A / G ◦A =Hom( (cid:0) Z N / L (cid:1) tors , C ∗ ) ≃ (cid:0) Z N / L (cid:1) tors . Formula (21) defines an action of G A on21 A and on the functions on C A . It is now clear how to adapt Equation (14):let a i denote the projection of a i in the free part Z k +1 of Z N / L and replace(14) by a u ∂ Φ ∂u + . . . + a N u N ∂ Φ ∂u N = c Φplus the requirement that the solution should transform according to somecharacter of the finite abelian group (cid:0) Z N / L (cid:1) tors . In case B in Figure 3 one has L = Z (2 , − , − ⊕ Z (1 , , − Z / L = Z ⊕ Z /3 Z . As a generator for the torsion subgroup we take g =(1 , − , 0) mod L . Then the polynomials Φ = u + u u , Φ = u + u u andΦ = u + u u satisfy the differential equations (13) for L , the differentialequations (14) for c = 2, while g · Φ r = e πir/ Φ r for r = 0 , , subgroups of Z N to Dessins. We are going to describe a construction which associates with a rank 2 sub-group L of Z N , as in 2.1, dessins d’enfants , i.e. bipartite graphs embeddedin oriented Riemann surfaces.The construction is given in [14] § § Fast Inverse Algorithm . Inop. cit., however, this algorithm is only presented via explicit visual inspec-tion of pictures in some concrete examples. In this section we want to presenta general principle behind the Fast Inverse Algorithm of [14] which uses onlylinear algebra and can be performed by computer.The main part of our construction produces rhombus tilings of the planeand is the same as N.G. de Bruijn’s [2] construction of Penrose tilings. Hiswork also led to effective methods for making quasi-crystals [23]. Since L isdefined over Z the construction yields in our situation periodic tilings of theplane L ∨ R = R , not just quasi-periodic ones. We can therefore pass to R modulo the period lattice and find a tiling of the two-dimensional torus. In this section L is as in 2.1. The quotient group Z N / L is allowed to havetorsion. Let L R denote the real 2-plane in R N which contains L . We write λ + L R for the real 2-plane in R N obtained by translating L R over a vector λ ∈ R N . On the other hand one has in R N the standard N -grid consisting ofthe hyperplanes H i,k := { ( z , . . . , z N ) ∈ R N | z i = k } for i = 1 , . . . , N and k ∈ Z . By intersecting with this standard N -grid we obtain in the 2-plane λ + L R an N -grid of lines: L λi,k := H i,k ∩ ( λ + L R ) for i = 1 , . . . , N and k ∈ Z . (23)22ote that this crucially uses the assumption that L is not contained in anyof the standard coordinate hyperplanes in Z N . The grid lines are naturally oriented : the orientation on L λi,k is such thatthe points in λ + L R which are to the left of L λi,k , have a larger i -th coordinatethan the points to the right of L λi,k . We say that λ is non-resonant if no three grid lines passthrough one point. Let L and λ be as in 6.1 with λ non-resonant as in 6.2.Consider the map F : R N → Z N , F ( z , . . . , z N ) = ( ⌊ z ⌋ , . . . ⌊ z N ⌋ ) , where ⌊ z ⌋ for a real number z denotes the largest integer ≤ z . The map F contracts an open N -cube in the Z N structure on R N (and part of itsboundary) onto one of its corners. F commutes with the translation actionof Z N on R N and Z N . Suppressing L and λ from the notation we define S := F ( λ + L R ) . (24)The map F is constant on each 2-cell ( = connected component) of the gridcomplement in λ + L R . Every point of S is in fact the image of a uniquesuch 2-cell. The distance between two points of S is 1 if and only if thecorresponding 2-cells are separated by exactly one grid line.Because λ is non-resonant, an intersection point of grid lines is in theclosure of exactly four 2-cells. If x = L λi,k ∩ L λj,m the four points of S corre-sponding to these cells, i.e. F ( x ), F ( x ) − e i , F ( x ) − e j , F ( x ) − e i − e j , are thevertices of a square (cid:3) x . When x runs through the set of all intersection pointsof grid lines, these squares fit together to a connected surface S := S x (cid:3) x embedded in R N . One can characterize the surface S also as S := union of all unit squares in R N with vertices in the set S . (25)Here one may define unit square as the convex hull of four points p , p , p , p in R N with Euclidean distances k p − p k = k p − p k = k p − p k = k p − p k = 1and k p − p k = k p − p k = √ 2. A unit square with vertices in Z N is necessarilyof the form p + convex hull( , e i , e j , e i + e j ) for some p ∈ Z N and some i, j .We say that a unit square in R N has type ( i, j ) if its sides are parallel to thevectors e i or e j . Similarly, a side of a unit square in R N has type i if it isparallel to the vector e i . Let L ∨ R = Hom( L , R ) denote the real dual of L . Identifythe real dual Hom( Z N , R ) of Z N with R N by means of the standard dot product. converting grid to tiling for B from Figure 3Then the linear map R N → L ∨ R which is dual to the inclusion L → Z N ,restricts to a homeomorphism S ∼ −→ L ∨ R . Proof. Let b , . . . , b N ∈ L ∨ R denote the images of the standard basis vectorsof R N . After identifying in the obvious way, the real plane L R with the real2-plane λ + L R in R N one obtains in L R an N -grid in which the grid lines L λi,k are perpendicular to b i . After choosing a basis for L and taking the dualbasis for L ∨ R we may identify L R and L ∨ R with R . The term “perpendicular”then means perpendicular with respect to the standard inner product on R .Next draw for each intersection point of grid lines, say x = L λi,k ∩ L λj,m ,a parallelogram with centre x and sides ǫ b i and ǫ b j . Here the positive realnumber ǫ is so small that the parallelograms obtained from all grid intersectionpoints are disjoint; see Figure 7 for an example. It is clear from this figurehow one can glue the parallelograms of two consecutive intersection points ona grid line along their sides perpendicular to the grid line. The result of thisglueing is then the same as the image (scaled with a factor ǫ ) of the surface S under the projection R N → L ∨ R . (cid:4) It is obvious from the construction that the group L acts by translationson the point set S and on the surface S , preserving the types of the squaresand their sides. The cell structure on S given by the squares, their sides andvertices induces therefore on S / L a cell structure. The following propositioncounts the cells on S / L . Let B be a × N -matrix such that its rows are a Z -basisfor L . Let b , . . . , b N be the columns of this matrix. Then on S / L 1. the number of -cells of type ( i, j ) is | det( b i , b j ) | .2. the number of -cells of type i is P Nj =1 | det( b i , b j ) | .3. the number of -cells is P ≤ i An elementary transformation we perform this transformation simultaneously at all configurations { p + ℓ, p + ℓ, p + ℓ, p + ℓ } with ℓ ∈ L .After this elementary transformation we obtain a surface S ′ in R N whichis the union of all unit squares with vertices in the set S ′ , obtained from S by replacing the points p + ℓ by p + p + p − p + ℓ for all ℓ ∈ L .The surface S ′ , in turn, can be further modified by elementary trans-formations: just replace in the above construction S by S ′ and p by anappropriate point of S ′ . After starting the construction in 6.3 with a randomlychosen non-resonant λ ∈ R N one can create by repeated elementary trans-formations many surfaces S each of which is the union of unit squares withvertices in an L -invariant subset S of Z N and which is mapped homeomor-phically onto L ∨ R by the projection R N → L ∨ R . Moreover the numbers of cells25n each of these surfaces are still the same as in Proposition 6.5.The number of surfaces one can make in this way is finite and it is possibleto generate (by computer) a complete list. The list of surfaces produced in 6.7 is, up to permutation ofits entries, independent of the choice of λ ∈ R N at the start of the algorithm.This can be seen as follows. Fix i , 1 ≤ i ≤ N . Take λ ∈ R N and let λ t = λ + t e i for t ∈ R . Set t = min { t ∈ R > | λ t is not non-resonant } and t = min { t ∈ R >t | λ t is not non-resonant } . Then the surface S t constructedfrom λ t in 6.3 is equal to the surface S for 0 < t < t . Figure 9 shows howthe grid locally changes as t passes through t . Comparing Figures 9 and 8 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅❅❅❅ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅❅❅❅ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅❅❅❅ Figure 9: Local change in grid at resonance one sees that such a local change of the grid corresponds to an elementarytransformation of the surface S . Thus the surfaces S t with t < t < t areobtained from S by a number of elementary transformations.Since any two non-resonant λ, λ ′ ∈ R N can be moved to a common valueby coordinatewise changes as above we conclude that the surfaces S and S ′ obtained in 6.3 from λ and λ ′ , respectively, are related by elementarytransformations, and that, hence, the lists of surfaces which the algorithmproduces from start values λ and λ ′ are the same, up to possibly a permutationof the entries. We say that a surface S on this list is perfect if the function s : R N → R , s ( z , . . . , z N ) = z + . . . + z N , takes only three values on the set S of vertices in S . In that case these three values are consecutive integers,say a + 1 , a, a − 1. We denote by S • (resp. S ∗ resp. S ◦ ) the set of verticeswhere s takes the value a + 1 (resp. a resp. a − 1) and say that the verticesin S • are black , those in S ◦ are white and those in S ∗ are grey . Each of thesesets is invariant under the translation action of L . I have at present no proof that for every rank 2 subgroup L ⊂ Z N as in 2.1 the above construction indeed yields at least one perfectsurface. On the other hand, in all examples I investigated the computerproduced at least one perfect surface. It would also be very interesting toknow which perfect surfaces can arise already in the first step 6.3 of theconstruction, by choosing λ appropriately (instead of at random).26 .11. For every surface S in the list of 6.7 one has the graph b Γ with setof vertices S and arrows given by the diagonals in the squares, orientedsuch that in a square of type ( i, j ) the oriented diagonals are e i + e j andsign(det( b i , b j ))( e i − e j ). The advantage of having S embedded in R N isthat the vertices and the arrows in these graphs are actual points and vectorsin R N . If the surface S is perfect, every square in S has one vertex in S • ,one in S ◦ and two vertices in S ∗ . One of its diagonals goes from the white tothe black vertex and the other diagonal connects the two grey vertices. Thegraph b Γ therefore is the disjoint union of two oriented graphs b Γ •◦ and b Γ ∗ ,with vertex sets S • ∪ S ◦ and S ∗ , respectively. The graph b Γ •◦ is a bi-partitegraph , i.e. its vertex set is the disjoint union of two sets (the black resp. whitevertices) and with edges connecting only vertices of different colors.There is a natural duality between b Γ •◦ and b Γ ∗ : every vertex of b Γ •◦ liesin a unique connected component of S \ b Γ ∗ and vice versa. Moreover theboundaries of these connected components are polygons and an arrow betweentwo nodes in one graph is a common side of the corresponding polygons forthe other graph. The perfect surface S is mapped homeomorphically onto L ∨ R by theprojection R N → L ∨ R , which is dual to the inclusion L ֒ → Z N . This projectionmaps e i to b i , for i = 1 , . . . , N . One may however pre-compose this projectionwith the linear map R N → R N , e i k b i k e i , ≤ i ≤ N . The compositelinear map R N → L ∨ R maps the tiling by squares on the surface S piecewiselinearly and homeomorphically onto a tiling of L ∨ R by rhombi. It maps L isomorphically onto a lattice L in L ∨ R and it maps the graphs b Γ •◦ and b Γ ∗ ‘isomorphically’ onto L -periodic graphs Γ •◦ and Γ ∗ in the plane L ∨ R . In theliterature Γ •◦ ⊂ L ∨ R is called a periodic dimer model or periodic brane tiling ;see for instance [4, 5, 12, 13, 14, 15]. All structures in 6.11 are invariant under translation byvectors in L . Passing to the orbit space we obtain from a perfect surface S the 2-dimensional torus T = S / L (i.e. compact oriented surface of genus 1without boundary). Embedded in this torus are the two graphs Γ •◦ = b Γ •◦ / L and Γ ∗ = b Γ ∗ / L , which are dual to each other. Note that 6.12 also yields T = L ∨ R / L , Γ •◦ = Γ •◦ / L and Γ ∗ = Γ ∗ / L .After Grothendieck one calls ( T , Γ ∗ , Γ •◦ ) a dessin d’enfants [18, 22]. Figure 10 shows two drawings of the dessin ( T , Γ ∗ , Γ •◦ ) for B from Figure 3; or, rather, it shows a lifting of this dessin to the plane L ∨ R with basis so that the period lattice L becomes just Z . It also shows Γ ∗ as27n unembedded quiver. Note however that this is not the same as the quiver Q in case B in Figure 3. The quiver in Figure 10 is in fact the quiver for the singularity C / Z where a generator of the cyclic group Z acts on C as multiplication by thediagonal matrix diag( e πi/ , e πi/ , e πi/ ). Also the brane tiling picture inFigure 10 appears in [13] § C / Z .This difference between [13] and our approach comes, because we still haveto perform the untwist, as is explained below. tt t ttt ❄❍❍❍❥ ✟✟✟✯ ✻❍❍❍❨✟✟✟✙ ✲❏❏❏❏❏❪✡✡✡✡✡✢✛❏❏❏❏❏❫ ✡✡✡✡✡✣❄✻❍❍❍❍❍❍❥❍❍❍❍❍❍❨✟✟✟✟✟✟✯✟✟✟✟✟✟✙ Figure 10: Two versions of the dessin ( T , Γ ∗ , Γ •◦ ) for B from Figure 3. Thedashed square is the period parallelogram. The picture on the left shows howthe quiver on the right is embedded in a -torus. The quiver Γ ∗ in 6.13 is not the Pl¨ucker quiver of L , defined in 2.5.The latter appears in the current setting through the zigzag loops , as follows.Let S be a perfect surface. Fix i ∈ { , . . . , N } . In every square on S whichhas two sides parallel to e i draw the line segment connecting the midpointsof these two sides. If the square has type ( i, j ) this line segment is orientedso that it is a translate of the vector sign(det( b i , b j )) e j . The union of theseline segments projects to an oriented closed curve on T , called the i -th zigzagloop . Note that this zigzag loop may consist of several connected components.According to Proposition 6.5 the i -th and j -th zigzag loops on T intersectin exactly | det( b i , b j ) | points. Thus we see that the nodes of the Pl¨uckerquiver can be identified with the zigzag loops on T and the arrows betweentwo nodes are the intersection points of the corresponding zigzag loops. Theorientation of the arrows follows from the orientation of the zigzag loops andcan in pictures be indicated as an over/undercrossing of the zigzag loops.Figure 11 shows the zigzag loops for the dessin in Figure 10. Pictures like Figure 11 can be viewed as showingan oriented surface with boundary, in which the connected components ofthe zigzag loops are the connected components of the boundary and whichis embedded in three space in a twisted way so that the black dots are on28igure 11: The zigzag loops for the dessin in Figure 10 one side of the surface and the white dots are on the other side. This surfacecomes with a tiling by helices as shown in Figure 12. (cid:0)(cid:0)(cid:0)(cid:0)❅❅❅❅ (cid:0)(cid:0)(cid:0)(cid:0) ❅❅❅❅ ❣✇ (cid:0)(cid:0)✒❅❅■(cid:0)(cid:0)✒ ❅❅■✲✲ bw rr ′ r ′ r (cid:0)(cid:0)❅❅(cid:0)(cid:0) ❅❅(cid:0)(cid:0)(cid:0)(cid:0)❅❅❅❅❘(cid:0)(cid:0)✒ (cid:0)(cid:0)✒❅❅■(cid:0)(cid:0)✒ ❅❅■ ❣✇ ✲✲ bw rr ′ r ′ r (cid:0)(cid:0)❅❅(cid:0)(cid:0) ❅❅✻❄ (cid:0)(cid:0)✒❅❅■(cid:0)(cid:0)✒ ❅❅■ ❣✇ ✛✲ bw rr ′ (cid:0)(cid:0)(cid:0)(cid:0)❅❅❅❅ (cid:0)(cid:0)(cid:0)(cid:0) ❅❅❅❅ ❣✇ ✛✲❄ ✻(cid:0)(cid:0)(cid:0)(cid:0)❅❅❅❅ (cid:0)✒(cid:0)✠ ❅■❅❘ bw rr ′ Figure 12: Untwisting: from left to right: original square tile, helix tile,untwisted helix tile, untwisted helix completed to square tile. One can ‘untwist’ the helix tiles and complete the untwisted helix tilesto squares as indicated in Figure 12. The effect is that the boundary cyclesare being capped off by discs. The result is an oriented surface M withoutboundary which is tiled by squares with one black, one white and two redvertices. The black-white diagonals give an embeding of the bipartite graphΓ •◦ into M . The red-red diagonals give an embeding of the quiver Q into M (if all zigzag loops have just one connected component; otherwise somepoints on the surface must be pinched together).The untwisting procedure is described in [4] § T , i.e. the L -periodic tiling of the perfect surface S by unit squares taken modulo L (see6.9, 6.13). Let B (resp. W ) denote the set of black (resp. white) vertices.There is a bijection between the set of tiles and the set E of arrows of thequiver Q and we can refer to tiles as e ∈ E . A tile e has one black vertex b ( e ) and one white vertex w ( e ). A tile e is the image of a unit square withsides parallel to two of the basis vectors e , . . . , e N ; let us denote the indicesof this (unordered) pair of basis vectors as a 2-element subset { r ( e ) , r ′ ( e ) } of29 , . . . , N } . Thus the tiling on T yields the list of quadruples M = { ( b ( e ) , w ( e ) , r ( e ) , r ′ ( e ) ) } e ∈ E . (26)Now, for every e ∈ E we take a unit square (cid:3) e and attach labels b ( e ), w ( e ), r ( e ), r ′ ( e ) and colors black, white, red, red, respectively, to its vertices suchthat the vertices labeled b ( e ) and w ( e ) are not adjacent. If two such squares (cid:3) e and (cid:3) e ′ have equal labels on two adjacent vertices, we glue (cid:3) e and (cid:3) e ′ along the corresponding sides. The result of this glueing is a surface f M tiledwith squares.On the surface f M there is for every b ∈ B one black point with label b and for every w ∈ W there is one white point with label w . However for i ∈ { , . . . , N } there may be several red points with label i . We finally obtainthe desired surface M by identifying for every i ∈ { , . . . , N } the red pointswith label i . The surface f M , and hence also M , is oriented. Proof. Cut the square (cid:3) e in four pieces like in the right-hand picture ofFigure 12. Then for fixed b ∈ B the small squares with black vertex b are glued together to a polygon isomorphic to the polygon in T formed bythe unit squares with one vertex b . Similarly, for fixed w ∈ W the smallsquares with white vertex w are glued together to a polygon isomorphic tothe polygon in T formed by the unit squares with one vertex w but with theorientation reversed. For i ∈ { , . . . , N } the small squares with red vertex i are glued together to a number of disjoint polygons, one for every connectedcomponent of the i -th zigzag loop and with oriented boundary ‘equal to’ thatcomponent. It is now clear from Figure 12 that these polygons are orientedconsistently so that f M is oriented. (cid:4) The surface M is tiled with squares so that the black-white diagonals form a bi-partite graph isomorphic to the bi-partite graph Γ •◦ in 6.13, while the red-red diagonals form a graph isomorphic to the quiver Q . The list M (26) which our algorithm produces for modelIV of dP , i.e. example B from Figure 3, is e : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 b ( e ) : 6 5 1 2 4 3 1 5 3 6 4 2 5 6 1 2 3 4 w ( e ) : 5 6 2 1 3 4 3 1 5 2 6 4 5 6 1 2 3 4 r ( e ) : 2 2 3 3 4 4 1 1 1 1 1 1 5 6 5 6 5 6 r ′ ( e ) : 1 1 1 1 1 1 5 5 5 6 6 6 2 2 3 3 4 4The surface M with the corresponding tiling by kites (instead of squares) isshown in the left-hand picture in Figure 13. The right-hand picture in Figure303 shows the bi-partite graph Γ •◦ and the quiver Q embedded in M . Notethat this quiver is indeed the same as the one for example B in Figure 13.Figure 13: Surface M with kite tiling (left) and quiver Q and bipartite graph Γ •◦ (right) for case B in Figure 4 (= model IV of dP ). The surface M isobtained by identifying opposite sides of the hexagon. (see e.g. [17, 16, 20]) A perfect matching P on a bi-partitegraph Γ •◦ (also known as a dimer configuration ) is a subset of the edges ofΓ •◦ such that every node of Γ •◦ is incident to exactly one edge in P . Consider a subgroup L ⊂ Z N as in 2.1. Let Γ •◦ be a bi-partite graph which is obtained from L ⊂ Z N as explained in Section 6, and inDefinition 6.13 in particular. Let C be a -dimensional cone in the secondaryfan of L ⊂ Z N and let L C be the corresponding collection of -element subsetsof { , . . . , N } as in 3.2 and 3.3. Identify E with the set of edges of Γ •◦ .Then, with the notation as in (26), the set P C = { e ∈ E | { r ( e ) , r ′ ( e ) } ∈ L C } (27) is a perfect matching on Γ •◦ . Proof. Consider a perfect surface S ⊂ R N as in 6.9. The map π : R N → L ∨ R which is dual to the inclusion L ֒ → Z N projects S homeomorphically onto theplane L ∨ R . The tiling of S by unit squares gives a tiling of L ∨ R by parallelo-grams. A unit square with edges e i and e j projects onto a parallelogram withedges b i and b j . Now let C be a 2-dimensional cone of the secondary fan(see 3.2) and let w be a white vertex of the tiling. By translating the originof the secondary fan to w one sees that there is exactly one parallelogram31ith vertex w and with edges b i and b j such that C ⊂ R ≥ b i + R ≥ b j ,i.e. such that { i, j } ∈ L C . This shows that there is exactly one element of P C incident to w , namely the black-white diagonal of this parallelogram. Asimilar argument works for the black vertices. (cid:4) In the situation of Theorem 7.2 the number of black verticesand the number of white vertices are both equal to the number vol A in (19). Proof. The number of black (resp. white) vertices of Γ •◦ is equal to the num-ber of edges in any perfect matching. This holds in particular for the perfectmatching P C corresponding to a 2-dimensional cone C in the secondary fan.The result now follows from (27), (4) and (19). (cid:4) The following result was also derived in [4] § The genus of the surface M in 6.16 is equal to the numberof lattice points in the interior of the secondary polytope ∆ in (8). Proof. The number of 1-cells in the tiling of M is twice the number of 2-cells.According to Proposition 6.5 the number of 2-cells is P ≤ i 00 0 ̟ u u ̟ u u ̟ u u ̟ u u ̟ u u ̟ u u ̟ u u ̟ u u ̟ u u (with ̟ e denoting ̟ ( e )). In view of the discussion in Example 6.14 thisbi-adjacency matrix for model IV of dP should be somehow the same as theKasteleyn matrix for the singularity C / Z in [13] Equation (3.5): K ( z, w ) = w − − w − − − − − zw w − − z − 10 0 − z − . Indeed, by dividing the first, third and fifth column of K ̟ by u and subse-quently setting u = z − , u = w , u = u = u = u = 1, and appropriatelysetting ̟ e = ± K ( z, w ). This way of getting K ( z, w ) from K ̟ is nothing butpassing from homogeneous to inhomogeneous coordinates.33 .4. Definition. (cf. [18] Definition 1.1.1.) A 3 -constellation ( E, σ , σ )consists of a finite set E and two permutations σ , σ of E such that thepermutation group generated by σ , σ acts transitively on E . The cycle notation ( i i . . . i k − i k ) denotes the permuta-tion ρ such that ρ ( i j ) = i j +1 for j = 1 , . . . , k − ρ ( i k ) = i . By a cycle ofa permutation σ we mean an orbit of the group generated by σ . We denotethe set of cycles of σ by E σ . In the product στ of two permutations σ and τ one first applies τ ; so ( στ )( i ) = σ ( τ ( i )). For our purposes we do not need the most general notionof a superpotential for a quiver. The superpotentials we need are just no-tational reformulations of 3-constellations. The superpotential attached to a -constellation ( E, σ , σ ) is the following polynomial W in non-commutingvariables X e ( e ∈ E ): W = X γ ∈ E σ X γ − X γ ∈ E σ X γ , where X γ := X i X i · . . . · X i k for a cyclic permutation γ = ( i i . . . i k ). The 3 -constellation ( E, σ , σ ) associated with the dessin ( M , Q ) in 6.18 is the following. E is the set of arrows of the quiver Q . Every e ∈ E is a path p e on M . Every connected component of M \ ∪ e ∈ E p e has anoriented boundary which can be viewed as a cyclic permutation of elements of E . Then σ (resp. σ ) is the composition of the cyclic permutations which areboundaries of connected components containing a black (resp. white) point. In the same way one associates a 3-constellation with thedessin ( T , Γ ∗ ) in 6.13. It is obvious from the untwisting procedure in 6.16that, if ( E, σ , σ ) is the 3-constellation associated with the dessin ( M , Q ),then ( E, σ , σ − ) is the 3-constellation associated with the dessin ( T , Γ ∗ ). It is obvious that the 3-constellation ( E, σ , σ ) associated with thedessin ( M , Q ) contains the complete instructions for building M : for everycycle of σ and every cycle of σ take a (convex planar) polygon with sideslabeled by the elements in the cycle in their cyclic order. Next glue thesepolygons by identifying sides with the same label. The sides of these poly-gons correspond to arrows of the quiver Q and the vertices of these polygonscorrespond to vertices of Q . After the above procedure of glueing polygonsalong their sides one must still identify points which correspond to the samevertex of Q . The result is then M with Q embedded in it.34 .10. The 3-constellation ( E, σ , σ ) in 8.9 and the list M in 6.16 both arecompletely determined by and do completely determine the dessin ( M , Q ).One can, however, also describe the relation between ( E, σ , σ ) and M in adirect algebraic/combinatorial way.For the construction of ( E, σ , σ ) from M one first forms for b ∈ B , w ∈ W and i ∈ { , . . . , N } the sets | σ b | = { e ∈ E | b ( e ) = b } , | σ w | = { e ∈ E | w ( e ) = w } , | z i | = { e ∈ E | i ∈ { r ( e ) , r ′ ( e ) } } . These sets have an unoriented cyclic structure: e, e ′ ∈ | σ b | (resp. e, e ′ ∈ | σ w | )are neighbors if and only if { r ( e ) , r ′ ( e ) } ∩ { r ( e ′ ) , r ′ ( e ′ ) } 6 = ∅ , while e, e ′ ∈ | z i | are neighbors if and only if { b ( e ) , w ( e ) } ∩ { b ( e ′ ) , w ( e ′ ) } 6 = ∅ . In order to puta consistent orientation on these cyclic sets we choose one of the two possibleorientations of | z | . For every b ∈ B we have ♯ ( | z | ∩ | σ b | ) = 0 or 2. In thelatter case | z |∩| σ b | consists of two elements, say e and e ′ , which are neighborsboth in | z | and in | σ b | . The orientation on | z | then induces an orientationon | σ b | such that e is the successor of e ′ in | σ b | if e is the successor of e ′ in | z | .In this way the orientation on | z | induces an orientation on every | σ b | andevery | σ w | which has a non-empty intersection with | z | . Next this induces anorientation on every | z i | which has a non-empty intersection with any of thealready oriented sets | σ b | or | σ w | . And so on. It is because of the geometricbackground of M (i.e. the orientations shown in the right-hand picture inFigure 12) that we can indeed go on. In the end all sets | σ b | and | σ w | havean oriented cyclic structure and can be identified with cyclic permutations σ b and σ w . The construction of the 3-constellation ( E, σ , σ ) is finished bysetting σ = Y b ∈ B σ b , σ = Y w ∈ W σ − w ;the reason for inverting the white cycles is shown in Figure 14 (cid:0)(cid:0)(cid:0)(cid:0)❅❅❅❅ (cid:0)(cid:0)(cid:0)(cid:0) ❅❅❅❅ ❣✇ ✲ bw rr ′ ✝ ✆✞ ☎ ✛✲ σ σ σ b σ w Figure 14: orientation of cycles in -constellation and orientation of M . To go from ( E, σ , σ ) to M one views the cycles of the permutations σ , σ and σ σ − as elements of the sets E σ , E σ E σ σ − , respectively, and then35ets M = { ( b ( e ) , w ( e ) , r ( e ) , r ′ ( e )) } e ∈ E with b ( e ) = the cycle of σ which contains e , w ( e ) = the cycle of σ which contains e , r ( e ) = the cycle of σ σ − which contains e , r ′ ( e ) = r ( σ ( e )) . The procedure of 8.10 converts the list M in 6.19 into the3-constellation ( E, σ , σ ) with E = { , , , , , , , , , , , , , , , , , } ,σ = (3 , , , , , , , , , , , , ,σ = (4 , , , , , , , , , , , , . Since in this example the cycle z = (1 , , , , , , , , , , , 9) meetseach of the above 3-cycles the conversion went especially fast.The corresponding superpotential W is X X X + X X X + X X X + X X X + X X X + X X X − X X X − X X X − X X X − X X X − X X X − X X X . Corollary 7.3 implies that the bi-adjacency matrix K ̟ = ( κ ij ) of thedessin ( M , Q ) is a square matrix. Its determinant is by definitiondet K ̟ = X τ sign( τ ) κ i τ ( i ) where τ runs over the set of all bijections B ≃ → W and sign( τ ) is defined as thesign of the permutation τ − τ of B for some reference bijection τ : B ≃ → W .Of course, if we want to actually write K ̟ as a matrix, we must choosebijections between B , W and the set of numbers { , . . . , vol A } and that fixes τ . Changing the reference bijection multiplies det K ̟ by ± 1, but that is forour purpose unimportant. Next note that the ( b , w )-entry of K ̟ is non-zeroif and only if there is an edge of Γ •◦ connecting the nodes b and w . So theonly bijections τ that contribute to the determinant of K ̟ are the perfect atchings ; see Definition 7.1. Thus we finddet K ̟ = ± X perfect matchings P sign( P ) Y e ∈ P w ( e ) u r ( e ) u r ′ ( e ) = ± X perfect matchings P sign( P ) Y e ∈ P ̟ ( e ) ! u b P (29)where b P = X e ∈ P ( e r ( e ) + e r ′ ( e ) ) ∈ Z N , (30)and u p = u p · . . . · u p N N for p = ( p , . . . , p N ) ∈ Z N . The Newton polytope of a Laurent polynomial f = X ( k ,...,k N ) ∈ Z N c ( k ,...,k N ) u k · . . . · u k N N ∈ C [ u ± , . . . , u ± N ]is the polytope Newton ( f ) := convex hull (cid:0) { ( k , . . . , k N ) | c ( k ,...,k N ) = 0 } (cid:1) . The Newton polytope of the determinant of the bi-adjacencymatrix K ̟ of the dessin ( M , Q ) is the same as the secondary polygon of L ⊂ Z N (see 3.3): Newton (det K ̟ ) = Σ( L ) . (31) The vertices of Newton (det K ̟ ) are the points c P C (see (30)) given by theperfect matchings P C with C a -dimensional cone in the secondary fan. Proof. If P and P ′ are two perfect matchings then the edges in P orientedblack-to-white together with the edges in P ′ oriented white-to-black form acollection of closed loops on the torus T in 6.13. The first homology groupof this torus is L . Therefore b P − c P ′ lies in L ⊂ Z N . This shows that Newton (det K ̟ ) lies in a 2-dimensional plane in R N parallel to L R .From Equation (30) one sees that for every perfect matching P b P = N X i =1 ♯ { e ∈ P | i ∈ { r ( e ) , r ′ ( e ) }} e i , i.e. the i -th coordinate of the vector b P equals the number of edges in theperfect matching P which intersect the i -th zigzag loop (cf. 6.15). It fol-lows from Proposition 6.5 that the number of edges of the i -th zigzag loop is37 Nj =1 | det( b i , b j ) | . Since consecutive edges in a zigzag loop can never inter-sect the same perfect matching we see that for every perfect matching P andevery i ♯ { e ∈ P | i ∈ { r ( e ) , r ′ ( e ) }} ≤ P Nj =1 | det( b i , b j ) | . (32)Fix i and let C and C ′ be the two 2-dimensional cones in the secondary fancontaining the half-line R ≥ b i . Then { i, j } ∈ L C ⇔ det( b i , b j ) ≥ { i, j } ∈ L C ′ ⇔ det( b i , b j ) ≤ 0. Since P Nj =1 det( b i , b j ) = 0 we find X j, { i,j }∈ L C | det( b i , b j ) | = X j, { i,j }∈ L C ′ | det( b i , b j ) | = P Nj =1 | det( b i , b j ) | . Thus we see that for i and the perfect matchings P C and P C ′ Equation (32)is in fact an equality.Fix a 2-dimensional cone C in the secondary fan. It is bounded by twohalf-lines R ≥ b i and R ≥ b i ′ . The above argument now shows that Equation(32) is in fact an equality for the perfect matching P C and i and for P C and i ′ in place of i . Thus the point c P C is a vertex of Newton (det K w ). This togetherwith (32) proves: Newton (det K ̟ ) = convex hull (cid:16) { c P C | C L } (cid:17) . To complete the proof of (31) we note that Equations (4), (5), (27) and (30)together with Proposition 6.5 show c P C = ψ C . (cid:4) According to [5] § dP (i.e. case B in Figure 4) affords two different superpotentials. This is confirmed byour algorithm, which yields the following two bi-adjacency matrices K ̟ = ̟ u u ̟ u u ̟ u u ̟ u u ̟ u u ̟ u u ̟ u u + ̟ u u ̟ u u ̟ u u ̟ u u ̟ u u ̟ u u ̟ u u K ̟ = ̟ u u ̟ u u ̟ u u ̟ u u ̟ u u ̟ u u ̟ u u ̟ u u ̟ u u ̟ u u ̟ u u ̟ u u ̟ u u ̟ u u The corresponding superpotentials are: W = X X X + X X X + X X X X X + X X X − X X X X X − X X X − X X X − X X X = X X X + X X X + X X X X + X X X X − X X X X − X X X X − X X X − X X X After figuring out the rule for translating the edge labels one finds that theprepotentials in Eqs. (8.1) and (8.2) of [5] are W A = W and W B = W .For both superpotentials the surface M has genus 1. Figures 18 and 19in [5] show pictures of the planar periodic bi-partite graph which is the liftingof Γ •◦ to the simply connected cover of M .One can not expect the Kasteleyn matrices K A and K B in [5] Eq. (8.3) tobe the same as the bi-adjacency matrices K ̟ and K ̟ , respectively, becauseof the “untwist”. K ̟ and K ̟ are Kasteleyn matrices of the dimer modelsbefore the untwisting (see 8.2). Moreover Eqs. (8.4) and (8.5) of [5] showthat the Newton polygons of the determinants of the Kasteleyn matrices K A and K B are different. On the contrary, for both K ̟ and K ̟ the determinantis a linear combination of the monomials u [3 , , , , , , u [2 , , , , , , u [1 , , , , , , u [2 , , , , , , u [3 , , , , , , u [0 , , , , , , u [1 , , , , , . So, in agreement with Theorem 9.3, the Newton polygons of det K ̟ anddet K ̟ coincide with the polygon in Figure 4 case B and 3.7. In spite ofhaving the same Newton polygon det K ̟ and det K ̟ are not equal; indeedwhen all weights are 1 they differ in the coefficient of u [2 , , , , , . 10 Bi-adjacency matrix with critical weightsand the principal A -determinant In [11] p.297 Eq. (1.1) Gelfand, Kapranov and Zelevinsky define for aset A = ( a , . . . , a N ) ⊂ Z k +1 as in 4.1 the principal A -determinant E A ( f A ).The definition uses the Laurent polynomial f A = N X i =1 u i x a i , (33)where x m = x m x m · . . . · x m k +1 k +1 for m = ( m , . . . , m k +1 ) ∈ Z k +1 and wherethe coefficients u , . . . , u N are variables. The name “principal A -determinant”refers to the fact ([11] p.298 Prop. 1.1) that E A ( f A ) can be written as thedeterminant of some exact complex, i.e. E A ( f A ) = Y j (det M j ) ( − j M j . Another useful description of E A ( f A ) is given in [11]p.299 Prop. 1.2: E A ( f A ) = ± Y Γ ⊂ convex hull ( A ) ∆ A∩ Γ ( f A∩ Γ ) m (Γ) (34)where the product runs over all faces Γ of the primary polytope convex hull ( A )(cf. 4.12); m (Γ) is a multiplicity and ∆ A∩ Γ ( f A∩ Γ ) is the ( A ∩ Γ) -discriminant of the Laurent polynomial f A∩ Γ = P i : a i ∈ Γ u i x a i . The latter discriminant is apolynomial in the variables u i with a i ∈ Γ. To define it (see [11] p.271) oneneeds the algebraic set ∇ A∩ Γ which is the closure in C A∩ Γ of: { u ∈ C A∩ Γ | ∃ x ∈ ( C ∗ ) k +1 s.t. f A∩ Γ ( x ) = ∂f A∩ Γ ∂x i ( x ) = 0 , ∀ i } . Then, by definition, ∆ A∩ Γ ( f A∩ Γ ) = 1 if codim C A∩ Γ ( ∇ A∩ Γ ) > zero locus of ∆ A∩ Γ ( f A∩ Γ ) = ∇ A∩ Γ if codim C A∩ Γ ( ∇ A∩ Γ ) = 1 . So, E A ( f A ) gives the locus of the points ( u , . . . , u N ) ∈ C N for which at leastone of the Laurent polynomials f A∩ Γ has a critical point with critical value 0. ([11]p.302 Thm.1.4; cf.(17)) The Newton polytope of E A ( f A ) coincides with the secondary polytope Σ( A ). (cid:4) Theorems 10.2 and 9.3 in combination with Formula (20) make onewonder whether for an appropriate choice of the weight ̟ the determinant ofthe bi-adjacency matrix K ̟ is equal to the principal A -determinant E A ( f A ),up to the simple transformation necessitated by (20). In order to formulatethis transformation we must make the dependence on u , . . . , u N visible bywriting K ̟ ( u , . . . , u N ) and E A ( f A ( u , . . . , u N )).After some experimenting with examples I found a very natural and simplechoice for the weight that does the job. The critical weight for the arrows of the quiver Q is thefunctioncrit : E → Z > , crit( e ) = ♯ { e ′ ∈ E | s ( e ′ ) = s ( e ) , t ( e ′ ) = t ( e ) } (35) For every set A as in 4.1 and every dessin ( M , Q ) (see6.18) constructed from A by the algorithm in Section 6 the determinant of he bi-adjacency matrix with critical weight and the principal A -determinantsatisfy: ( u · . . . · u N ) vol A det K crit ( u − , . . . , u − N ) = E A ( f A ( u , . . . , u N )) . (36) In support of the above conjecture we can point out that thecoefficients of the monomials corresponding to the vertices of the secondarypolygons are, up to sign, the same for the two sides of Equation (36). Indeed,Theorem 1.4 of [11]p.302 gives the coefficient of the monomial in the principal A -determinant with corresponds to a vertex of the secondary polytope. Sucha vertex corresponds to a maximal cone C in the secondary fan. In thenotations of Definition 3.3 the formula in loc. cit. for the coefficient of themonomial corresponding to C reads: ± Y { i,j }∈ L C | det( b i , b j ) | | det( b i , b j ) | . (37)On the other hand, Theorems 7.2 and 9.3 show that the same vertex corre-sponds to a perfect matching P C . From the role of perfect matchings in thecomputation of the determinant of the bi-adjacency matrix (see 9.1) one noweasily checks that the coefficient of the monomial in the determinant, whichcorresponds to C , is (possibly up to sign) the same as (37).In the remainder of this section we explicitly verify Conjecture 10.5 insome examples. For case B = (cid:20) − − − (cid:21) in Figure 3 the algorithmin Section 6 yields in the following bi-adjacency matrix with critical weights: K crit = u u u u u u u u u u u u u u + 3 u u u u u u . One easily computes det K crit = 27 u [0 , , , + 4 u [1 , , , + 4 u [2 , , , − u [1 , , , − u [2 , , , Note that the exponents are the same as the coordinates of the vertices andthe interior point of the secondary polygon in 3.5. For the computation ofthe principal A -determinant we note that in this case (see also Example 4.6) A = ( a , a , a , a ) = (cid:18)(cid:20) (cid:21) , (cid:20) (cid:21) , (cid:20) (cid:21) , (cid:20) (cid:21)(cid:19) f A = x ( u + u x + u x + u x ). The primary polytope is shownin Figure 2. The two boundary points of this primary polytope contributefactors u and u to the principal A -determinant (cf. Equation (34)). Thecontribution from the full primary polytope is the very classical discriminantof the cubic polynomial u + u x + u x + u x , which is (e.g. [11] p.405)27 u u + 4 u u + 4 u u − u u − u u u u . Thus we find E A ( f A ) = 27 u [3 , , , + 4 u [2 , , , + 4 u [1 , , , − u [1 , , , − u [2 , , , . This shows that conjecture 10.5 holds in this case. When critical weights are used the two bi-adjacency ma-trices in Example 9.4 become K crit1 = u u u u u u u u u u u u u u + u u u u u u u u u u u u u u K crit2 = u u u u u u u u u u u u u u u u u u u u u u u u u u u u One easily computes det K crit1 and det K crit2 and finds that both are equal to4 u [3 , , , , , + 2 u [1 , , , , , + u [1 , , , , , + u [3 , , , , , −− u [2 , , , , , − u [0 , , , , , − u [2 , , , , , . This gives a remarkable contrast with the last line of Example 9.4 and un-derlines the role of the critical weight in making Conjecture 10.5 reasonablefor every dessin obtained from A .Transforming the above determinant as in the left-hand side of Equation(36) yields4 u [1 , , , , , + 2 u [3 , , , , , + u [3 , , , , , + u [1 , , , , , − u [2 , , , , , −− u [4 , , , , , − u [2 , , , , , = u u u u u u ( u u − u u )(4 u u u u + u u − u u u u + u u ) . (38)From the matrix B in Figure 4 one easily finds A and the correspondingpolynomial f A = u x + u x x + u x x + u x x + u x x x − + u x x x . convex hull ( A ) is 3-dimensional and has seven 2-di-mensional faces, six of which are triangles and one is a quadrangle. Its onlyinteger points are its vertices and these correspond to the monomials of f A .One can compute the discriminants for the polynomials supported by the facesof the primary polytope and multiply these to get the principal A -determinant E A ( f A ) as in Equation (34). The result is exactly as in (38), with the 4-termfactor coming from the primary polytope itself, the 2-term factor coming fromthe quadrangle face and the other factors with multiplicities coming from thevertices. This shows that conjecture 10.5 holds in this case. Exactly as in Example 10.7 one can verify Con-jecture 10.5 for other “four-nomials” of degree ≤ B , B , B in Figure 4. It also worksfor B = (cid:20) − − (cid:21) and B = (cid:20) − − (cid:21) , which areclosely related to B and B , respectively. References [1] Benvenuti, S., S. Franco, A. Hanany, D. Martelli, J. Sparks, An InfiniteFamily of Superconformal Quiver Gauge Theories with Sasaki-EinsteinDuals , JHEP 0506 (2005) 064; also arXiv:hep-th/0411264[2] Bruijn, N.G. de, Algebraic theory of Penrose’s non-periodic tilings ofthe plane I , Proceedings of the Koninklijke Nederlandse Akademie vanWetenschappen, Series A, vol 84 (= Indagationes Mathematicae, vol 43),1981, 39-52[3] Feng, B., S. Franco, A. Hanany, Y-H. He, Symmetries of Toric Duality arXiv:hep-th/0205144[4] Feng, B., Y-H. He, K. Kennaway, C. Vafa, Dimer Models from MirrorSymmetry and Quivering Amoebae , arXiv:hep-th/0511287[5] Franco, S., A. Hanany, K. Kennaway, D. Vegh, B. Wecht, Brane Dimersand Quiver Gauge Theories , arXiv:hep-th/0511287[6] Franco, S., A. Hanany, D. Martelli, J. Sparks, D. Vegh, B. Wecht, GaugeTheories from Toric geometry and Brane tilings , arXiv:hep-th/0505211437] Fulton, W., Introduction to Toric Varieties , Annals of Math. Studies,Study 131, Princeton University Press, 1993[8] Gelfand, I.M., A.V. Zelevinskii, M.M. Kapranov, Hypergeometric func-tions and toral manifolds, Functional Analysis and its Applications 23(1989), 94-106[8] ′ correction to [8], Funct. Analysis and its Appl. 27 (1993) 295[9] Gelfand, I.M., M.M. Kapranov, A.V. Zelevinsky, Generalized Euler In-tegrals and A - Hypergeometric Functions , Advances in Math. 84 (1990),255-271[10] Gelfand, I.M., A.V. Zelevinsky, M.M. Kapranov, Equations of Hyper-geometric Type and Newton Polyhedra , Soviet Math. Dokl. 37 (1988),678-683[11] Gelfand, I.M., M.M. Kapranov, A.V. Zelevinsky, Discriminants, Resul-tants and Multidimensional Determinants , Birkh¨auser Boston, 1994[12] Hanany, A., C. Herzog, D. Vegh, Brane Tilings and Exceptional Collec-tions , arXiv:hep-th/0602041[13] Hanany, A., K. Kennaway, Dimer models and toric diagrams ,arXiv:hep-th/0503149[14] Hanany, A., D. Vegh, Quivers, Tilings, Branes and Rhombi ,arXiv:hep-th/0511063[15] Kennaway, K., Brane Tilings , arXiv:0706.1660[16] Kenyon, R., An introduction to the dimer model , arXiv:math/0310326[17] Kenyon, R., A. Okounkov, S. Sheffield, Dimers and Amoebae ,arXiv:math-ph/0311005[18] Lando, S., Zvonkin, A., Graphs on Surfaces and Their Applications , En-cyclopedia of Math. Sciences vol. 141, subseries Low-Dimensional Topol-ogy vol. 11, Springer-Verlag, Berlin-Heidelberg, 2004.[19] Martelli, D., J. Sparks, Toric Geometry, Sasaki-Einstein Manifolds and aNew Infinite Class of AdS/CFT Duals , Commun.Math.Phys. 262 (2006)51-89; also arXiv:hep-th/0411238[20] Okounkov, A., N. Reshetikin, C. Vafa, Quantum Calabi-Yau and Clas-sical Crystals , arXiv:hep-th/03092084421] Poonen, B., F. Rodriguez-Villegas, Lattice polygons and the number 12 ,Am. Math. Monthly 107 (2000) p. 238-250[22] The Grothendieck Theory of Dessins d’Enfants , L. Schneps (ed.) LondonMath. Soc. Lecture Note Series 200, Cambridge University Press, 1994.[23] Senechal, M., Quasicrystals and geometry , Cambridge University Press,1995.[24] Shabat, G., V. Voevodsky, Drawing Curves Over Number Fields , in: TheGrothendieck festschrift Volume III , P. Cartier et. al. (eds), Progress inMathematics vol. 88, Birkh¨auser, Boston, 1990[25] Stienstra, J., GKZ Hypergeometric Structures , in: Arithmetic and Geom-etry Around Hypergeometric Functions , R-P. Holzapfel, A. M. Uluda˘g,M. Yoshida (eds.), Progress in Mathematics vol. 260, Birkh¨auser, Basel,2007, p. 313-371; see also math.AG/0511351[26] Sturmfels, B.,