aa r X i v : . [ m a t h . C O ] M a r DEFORMATIONS OF DIMER MODELS
AKIHIRO HIGASHITANI AND YUSUKE NAKAJIMA
Abstract.
The mutation of polygons, which makes a given lattice polygon another one, is an importantoperation to understand mirror partners for 2-dimensional Fano manifolds, and the mutation equivalentpolygons give the Q -Gorenstein deformation equivalent toric varieties. On the other hand, for a dimermodel, which is a bipartite graph described on the real two-torus, we assign the lattice polygon called theperfect matching polygon. It is known that for each lattice polygon P there exist dimer models such thatthey give P as the perfect matching polygon and satisfy the consistency condition. Moreover, a dimermodel has rich information regarding toric geometry associated to the perfect matching polygon. Inthis paper, we introduce the operations, which we call the deformations of consistent dimer models, andshow that the deformations of consistent dimer models induce the mutations of the associated perfectmatching polygons. Contents
1. Introduction 11.1. Backgrounds and Motivations 11.2. Our results 22. Dimer models and perfect matching polygons 32.1. What is a dimer model? 32.2. Perfect matchings and the perfect matching polygon 43. Zigzag paths and their properties 63.1. Consistency conditions 63.2. Relationships between perfect matchings and zigzag paths 84. Deformations of consistent dimer models 114.1. Definition of deformations of consistent dimer models 114.2. Examples of deformations of consistent dimer models 164.3. The proof of the non-degeneracy 175. Zigzag paths on deformed dimer models 205.1. Behaviors of zigzag paths after deformations 205.2. Properties of zigzag paths on deformed dimer models 245.3. The perfect matching polygons of deformed dimer models 266. Relationships with mutations of polygons 276.1. Preliminaries on mutations of polytopes 276.2. Mutations of the PM polygon are induced by deformations 30Appendix A. Mutations of dimer models 34Appendix B. Large examples 35Appendix C. Remarks on deformations of hexagonal and square dimer models 38Acknowledgement 39References 391.
Introduction
Backgrounds and Motivations.
Fano manifolds are one of well studied classes in geometry, andthe classification of Fano manifolds, which were done in low dimensions, is a fundamental problem. Here,a
Fano manifold X is a complex projective manifold such that the anticanonical line bundle − K X isample. Recently, the new approach which uses mirror symmetry have been proposed for classifying Fano Mathematics Subject Classification.
Primary 52B20; Secondary 14M25, 14J33.
Key words and phrases.
Dimer models, Mutation of polygons, Mirror symmetry. manifolds as follows. First, a Fano manifold is expected to correspond to a certain Laurent polynomialvia mirror symmetry (see [Coates et al.]). That is, a Laurent polynomial f ∈ C [ x ± , · · · , x ± n ] is said tobe a mirror partner for a n -dimensional Fano manifold X if the Taylor expansion of the classical period π f of f coincides with a generating function for Gromov-Witten invariants of X (see the referencesquoted above for the details of these terminologies). Furthermore, if a Fano manifold X is a mirrorpartner of f , it is expected that X admits a toric degeneration X P . Here, P := Newt ( f ) is the Newtonpolytope of f , which is defined as the convex hull of exponents of monomials of f , and X P is the toricvariety defined by the spanning fan of P (i.e., the fan whose cones are spanned by the faces of P ). Thus,Laurent polynomials having the same classical period are considered as mirror partners for the same Fanomanifold X , and in general there are many Laurent polynomials that are mirror partners for X . In orderto understand the relationship between such Laurent polynomials, the operation called the mutation of f , which is a birational transformation analogue to a cluster transformation, was introduced in [GU]. Inparticular, it was shown that if f, g ∈ C [ x ± , · · · , x ± n ] are transformed into each other by mutations, thentheir classical periods are the same, that is, π f = π g [ACGK, Lemma 1]. Moreover, this mutation ofLaurent polynomials f and g can be defined in terms of the associated Newton polytopes P = Newt ( f )and Q = Newt ( g ) as defined in [ACGK] (see also Subsection 6.1). Also, it was shown in [Ilt] that if P and Q are Fano polytopes and are transformed into each other by mutations, then the associatedtoric varieties X P and X Q are related by a Q -Gorenstein (= qG) deformation , that is, there exists a flatfamily X → P such that the relative canonical divisor is Q -Cartier and X ∼ = X P , X ∞ ∼ = X Q where X p is the fiber of p ∈ P . Thus, it has been conjectured that there is a bijection between qG-deformationequivalence classes of “class TG” Fano manifolds and mutation equivalence classes of Fano polytopes.Until now, there are several affirmative results (see e.g., [Akhtar et al., KNP]).1.2. Our results.
As we mentioned, the mutations of polytopes are quite important in mirror symmetryof Fano manifolds. In this paper, we focus on the two dimensional case, and the polygons which we areinterested in are not necessarily Fano. First, it is known that any lattice polygon in R can be realized asthe perfect matching polygon ∆ Γ of a dimer model Γ satisfying the consistency condition (see Section 2and 3). A dimer model is a bipartite graph on the real two-torus (see Section 2 for more details), whichwas first introduced in the field of statistical mechanics. From 2000s, string theorists have been usedit for studying quiver gauge theories (see e.g., [Kenn, Keny] and references therein). Subsequently, therelationships between dimer models and many branches of mathematics have been discovered (see e.g.,[Boc3] and references therein). From these backgrounds, we expect that there is a certain operationon consistent dimer models that induces the mutation of perfect matching polygons. In this paper, weintroduce the concept called the deformations of consistent dimer models .To explain our main theorem, we briefly recall the mutations of polygons (see Subsection 6.1 for moreprecise definition). Let N ∼ = Z be a rank two lattice and M := Hom Z ( N, Z ) ∼ = Z . First, we consider alattice polygon P in N R := N ⊗ Z R and choose an edge E of P . We then take a primitive inner normalvector w ∈ M for E , and consider the linear map h w, −i : N R → R . Using these, we determine the height h w, u i of each point u ∈ P . In particular, a primitive lattice element u E ∈ N satisfying h w, u E i = 0 playsan important role to define the mutation. Such an element u E is determined uniquely up to sign, thus wefix one of them. Then, we define the line segment F := conv { , u E } , which is called a factor of P withrespect to w . Using these data, we have the lattice polygon mut w ( P, F ), which is called the mutation of P given by the vector w and the factor F , as defined in Definition 6.2. In addition, we also defineanother mutation mut w ( P, − F ) in a similar way. We note that although mut w ( P, F ) looks different from mut w ( P, − F ), they are transformed into each other by the GL(2 , Z )-transformation. Furthermore, forthe given lattice polygon P there exists a consistent dimer model Γ such that P = ∆ Γ .Under these backgrounds, the deformation of a consistent dimer model are compatible with the abovemutations in the following sense. Let Γ be the consistent dimer model as above. The deformationsof Γ are defined for a certain set of “ zigzag paths ” { z , · · · , z r } on Γ corresponding to the vector − w (see Section 3 concerning zigzag paths), and there are two kinds of deformations which we call the deformation at zig and the deformation at zag , which are respectively denoted by ν zig X (Γ , { z , · · · , z r } )and ν zag Y (Γ , { z , · · · , z r } ), see Definition 4.3 and 4.5 for more details. Here, X and Y are the deformationparameters (see Definition 4.1). The deformations preserve the consistency condition on a dimer model(see Proposition 4.7), and the associated perfect matching polygons satisfy the desired property as follows. EFORMATIONS OF DIMER MODELS 3
Theorem 1.1 (see Theorem 6.10) . With the above settings, we have that mut w (∆ Γ , F ) = ∆ ν zig X (Γ , { z , ··· ,z r } ) , mut w (∆ Γ , − F ) = ∆ ν zag Y (Γ , { z , ··· ,z r } ) , where ∆ ν zig X (Γ , { z , ··· ,z r } ) and ∆ ν zag Y (Γ , { z , ··· ,z r } ) are the perfect matching polygons of ν zig X (Γ , { z , · · · , z r } ) and ν zag Y (Γ , { z , · · · , z r } ) respectively. Therefore, the perfect matching polygons of the deformed dimer models satisfy the properties which areexactly the same as the mutation of a polygon (see Subsection 6.2). We remark that we have to determinethe shape of the perfect matching polygon ∆ Γ in advance for defining the mutations mut w (∆ Γ , F ) and mut w (∆ Γ , − F ), whereas we can define the deformations ν zig X (Γ , { z , · · · , z r } ) and ν zag Y (Γ , { z , · · · , z r } )even if we do not know the perfect matching polygon ∆ Γ .Since dimer models are related with many branches of mathematics and physics, it is interestingproblem to compare a certain object on a consistent dimer model Γ to that of ν zig X (Γ , { z , · · · , z r } ) (or ν zag Y (Γ , { z , · · · , z r } )).The structure of this paper is as follows. In Section 2, we introduce a dimer model and relatednotions. In particular, the notion of the perfect matching polygon introduced in this section is one ofmain ingredients in this paper. In Section 3, we introduce the notion of zigzag paths, which is a specialpath on a dimer model. We define the consistency condition using zigzag paths, and we then discuss therelationships between perfect matchings and zigzag paths on a consistent dimer model. After that, weespecially focus on type I zigzag paths, which are zigzag paths having typical properties. Using thesetype I zigzag paths, we introduce the deformations of consistent dimer models in Section 4, and showtheir fundamental properties. In Section 5, we observe the behavior of zigzag paths and the perfectmatching polygons for the deformed dimer models. In Section 6, we first recall the definition of themutations of polygons. Then, we show our main theorem that the deformations of consistent dimermodels are compatible with the mutations of polygons, that is, the perfect matching polygon of thedeformed dimer model coincides with the mutation of the perfect matching polygon of the original dimermodel (see Theorem 6.10). After that we give some corollaries which are induced by the fundamentalproperties on the mutation of polygons. As we will mention in Remark 4.4, the deformation of a consistentdimer model is not determined uniquely, whereas the perfect matching polygon of the deformation of aconsistent dimer model is determined uniquely. This ambiguity is caused by the fact that there are severalconsistent dimer models giving the same perfect matching polygon. However, it has been conjecturedthat such consistent dimer models are transformed into each other by the mutation of dimer models ,and hence “conjectually” our deformation of a consistent dimer model is determined uniquely up to themutation. Thus, we introduce this mutation of dimer models in Appendix A. In Appendix B, we give anadditional example of the deformation, which is enormous to write in the main body of this paper. Also,in Appendix C, we show that the definition of the deformations can be simplified for some special classesof dimer models, which we call hexagonal dimer models and square dimer models.2. Dimer models and perfect matching polygons
What is a dimer model? A dimer model (or brane tiling ) Γ is a finite bipartite graph on the realtwo-torus T := R / Z , that is, the set Γ of nodes is divided into two parts Γ +0 , Γ − , and the set Γ ofedges consists of the ones connecting nodes in Γ +0 and those in Γ − . In order to make the situation clear,we color nodes in Γ +0 white, and color nodes in Γ − black. A connected component of T \ Γ is called a face of Γ, and we denote by Γ the set of faces. We also obtain the bipartite graph e Γ on R induced via theuniversal cover R → T . We call e Γ the universal cover of a dimer model Γ. For example, the bipartitegraph shown in the left side of Figure 1 is a dimer model where the outer frame is the fundamentaldomain of T .As the dual of a dimer model Γ, we define the quiver Q Γ associated with Γ. Namely, we assign avertex dual to each face in Γ , an arrow dual to each edge in Γ . The orientation of arrows is determinedso that the white node is on the right of the arrow. For example, the right side of Figure 1 is the quiverassociated with the dimer model on the left. Sometimes we simply denote the quiver Q Γ by Q . EFORMATIONS OF DIMER MODELS 4
Figure 1.
A dimer model and the associated quiverThe valency of a node is the number of edges incident to that node. We say that a node on a dimermodel is n -valent if its valency is n . We then define several operations on a dimer model. The join move is the operation removing a 2-valent node and joining two distinct nodes connected to it as shown inFigure 2. Thus, using join moves we obtain a dimer model having no 2-valent nodes. We say that adimer model is reduced if it has no 2-valent nodes. Thus, the quiver associated with a reduced dimermodel contains no 2-cycles. On the other hand, there is the operation called the split move , which insertsa 2-valent node (see Figure 2).We say that reduced dimer models Γ, Γ ′ are isomorphic , which is denoted by Γ ∼ = Γ ′ , if their underlyingcell decompositions of T are homotopy equivalent. join movesplit move Figure 2.
An example of the join and split move2.2.
Perfect matchings and the perfect matching polygon.
Next, we assign a lattice polygon toeach dimer model. For this purpose, we will introduce the notion of perfect matchings, and we constructthe polygon called the perfect matching polygon . Definition 2.1. A perfect matching (or dimer configuration ) on a dimer model Γ is a subset P of Γ such that each node is the end point of precisely one edge in P .In general, every dimer model does not necessarily have a perfect matching. In this paper, we willmainly discuss consistent dimer models, and such a dimer model has a perfect matching. Moreover, wecan extend this perfect matching P to the one on e Γ via the universal cover R → T . We call this a perfectmatching on e Γ, and use the same notion P . For example, some perfect matchings on the dimer modelgiven in Figure 1 are shown in Figure 3. (This dimer model has eight perfect matchings in total.) P P P P P Figure 3.
Some perfect matchings on the dimer model given in Figure 1We say that a dimer model is non-degenerate if every edge is contained in some perfect matchings. Itis known that this non-degeneracy condition is equivalent to the strong marriage condition , that is, thedimer model has an equal numbers of black and white nodes and every proper subset of the black nodesof size n is connected to at least n + 1 white nodes (see e.g., [Bro, Remark 2.12]).Following [IU2, Section 5], we next define the perfect matching polygon. We first fix a perfect matching P , and call this the reference perfect matching . For any perfect matching P , we consider the connected EFORMATIONS OF DIMER MODELS 5 components of R divided by P ∪ P . Then, we define the height function h P , P which is a locally constantfunction on R \ ( P ∪ P ) defined as follows. First, we choose a connected component of R , and definethe value of h P , P as 0. Then, this function increases by 1 when we crosses- an edge e ∈ P with the black node on the right, or- an edge e ∈ P with the white node on the right,and decreases by 1 when we crosses- an edge e ∈ P with the white node on the right, or- an edge e ∈ P with the black node on the right.This function is determined up to an addition of constant (i.e., up to a choice of a component valuedat 0). For example, Figure 4 is the height function h P , P on the dimer model given in Figure 1, wherethe red square stands for the fundamental domain of T , edges in P (resp. P ) are colored by blue (resp.green), and the number filled in each component is the value of h P , P . -1-1 0 0000 1 1111 22 222 333 33 4 4 Figure 4.
The height function h P , P We then take a point pt ∈ R \ ( P ∪ P ), and define the height change h ( P , P ) = ( h x ( P , P ) , h y ( P , P )) ∈ Z of P with respect to P as the differences of the height function: h x ( P , P ) = h P , P ( pt + (1 , − h P , P ( pt ) ,h y ( P , P ) = h P , P ( pt + (0 , − h P , P ( pt ) . We remark that this does not depend on a choice of pt ∈ R \ ( P ∪ P ). We then consider the heightchange h ( P , P ′ ) for any pair of perfect matchings P , P ′ , but since we have h ( P , P ′ ) = h ( P , P ) − h ( P ′ , P ) , (2.1)we may consider only height changes with the form h ( P , P ) for the reference perfect matching P . Then,the perfect matching (= PM ) polygon (or characteristic polygon ) ∆ Γ ⊂ R of a dimer model Γ is definedas the convex hull of { h ( P , P ) ∈ Z | P ∈ PM (Γ) } where PM (Γ) is the set of perfect matchings on Γ. Remark 2.2.
The description of height changes depends on a choice of the coordinate system fixed in T (i.e., a choice of the fundamental domain). A change of a coordinate system induces GL(2 , Z ) action onthe PM polygon, and this action does not affect our problem. In the following, we say that two polygons P and Q are GL(2 , Z ) -equivalent if they are transformed into each other by GL(2 , Z )-transformations, inwhich case we denote P ∼ = Q . Thus, we may fix the fundamental domain of T . Also, we remark that thedescription of this polygon ∆ Γ depends on a choice of the reference perfect matching, but it is determinedup to translations. Definition 2.3.
Fix a perfect matching P . We say that a perfect matching P is • a corner (or extremal ) perfect matching if h ( P , P ) is a vertex of ∆ Γ , • a boundary (or external ) perfect matching if h ( P , P ) is a lattice point on an edge of ∆ Γ (especiallya corner perfect matching is a boundary one), • an internal perfect matching if h ( P , P ) is an interior lattice point of ∆ Γ .In the next subsection, we will introduce consistent dimer models (see Definition 3.2), which haveseveral nice properties. If a dimer model is consistent, then there exists a unique corner perfect matchingcorresponding to each vertex of ∆ Γ (see e.g., [Bro, Corollary 4.27 ], [IU2, Proposition 9.2]). Thus, we can EFORMATIONS OF DIMER MODELS 6 give a cyclic order to corner perfect matchings along the corresponding vertices of ∆ Γ in the anti-clockwisedirection. We say that two corner perfect matchings are adjacent if they are adjacent with respect to thegiven cyclic order. Example 2.4.
We consider the dimer model given in Figure 1, and fix the perfect matching P shown inFigure 3 as the reference one. Then, we see that the perfect matchings P , · · · , P correspond to latticepoints (1 , , (1 , , ( − , , (0 , −
1) respectively. Also, we see that P and the other perfect matchingsthat are not listed in Figure 3 correspond to (0 , P , · · · , P are corner perfect matchings ordered with this order. P P P P Figure 5.
The PM polygon of the dimer model given in Figure 1In this way, we can obtain the PM polygon from a dimer model. On the other hand, it is known thatany lattice polygon can be obtained as the PM polygon of a certain dimer model.
Theorem 2.5 ([Gul, IU2]) . For any lattice polygon ∆ in R , there exists a dimer model Γ giving ∆ as the PM polygon ∆ Γ . Furthermore, we can take this Γ as it satisfies the consistency condition ( seeDefinition 3.2 ) . Thus, for a given lattice polygon ∆, we say that Γ is a dimer model associated with ∆ if the PMpolygon of Γ coincides with ∆. We remark that for a given polygon ∆, the associated consistent dimermodel is not unique in general.3.
Zigzag paths and their properties
Consistency conditions.
In this subsection, we introduce the consistency condition. In orderto define this condition, we first introduce the notion of zigzag paths, and such paths are also mainingredients for introducing deformations of dimer models.
Definition 3.1.
We say that a path on a dimer model is a zigzag path if it makes a maximum turn tothe right on a white node and a maximum turn to the left on a black node. Also, we say that a zigzagpath is reduced if it does not factor through 2-valent nodes. (We remark that we can make a zigzag pathreduced using the join moves. In particular, any zigzag path on a reduced dimer model is reduced.)Since a dimer model has only finitely many edges, we see that all zigzag paths are periodic. For a zigzagpath z on Γ, we define the length of z , which is denoted by ℓ ( z ), as the number of edges of Γ constituting z . In particular, we see that ℓ ( z ) is even integer. Thus, edges on a zigzag path are indexed by elements in Z / (2 n ) Z for some integer ℓ ( z ) / n ≥
1. Fix a black node on a zigzag path z as the starting point of z ,and we denote z as a sequence of edges starting from the fixed black node: z = z [1] z [2] · · · z [2 n − z [2 n ]. z [1] z [2] z [3] z [4] z [5] z [6] z [2 n ] An edge in a zigzag path z is called a zig (resp. zag ) of z if it is indexed by an odd (resp. even) integer.We denote by Zig ( z ) (resp. Zag ( z )) the set of zigs (resp. zags) appearing in a zigzag path z , which is afinite set. We note that if z does not have a self-intersection, Zig ( z ) and Zag ( z ) are disjoint sets. For anyedge e of a dimer model, we can uniquely determine the zigzag path containing e as a zig and the zigzagpath containing e as a zag respectively. Thus, any edge e is contained in at most two zigzag paths. If EFORMATIONS OF DIMER MODELS 7 such zigzag paths do not have a self-intersection, e is contained in exactly two zigzag paths. For example,zigzag paths on the dimer model given in the left of Figure 1 are shown in Figure 6. z z z z Figure 6.
Zigzag paths on the dimer model given in Figure 1For a zigzag path z on a dimer model Γ, we also consider the lift of z to the universal cover e Γ, especially e z ( α ) denotes a zigzag path on e Γ whose projection on Γ is z where α ∈ Z . When we do not need to specifythese, we simply denote each of them by e z . Then, we see that a zigzag path on e Γ is either periodic orinfinite in both directions. Using these notions, we introduce the consistency condition.
Definition 3.2 (see [IU1, Definition 3.5]) . We say that a dimer model is ( zigzag ) consistent if it satisfiesthe following conditions:(1) there is no homologically trivial zigzag path,(2) no zigzag path on the universal cover has a self-intersection,(3) no pair of zigzag paths on the universal cover intersect each other in the same direction morethan once. That is, if a pair of zigzag paths ( e z, e w ) on the universal cover has two intersections a , a and e z points from a to a , then e w points from a to a .Here, we remark that two zigzag paths are said to intersect if they share an edge (not a node).In the literature, there are several conditions that are equivalent to Definition 3.2 (for more details,see [Boc1, IU1]), and it is known that a consistent dimer model is non-degenerate (see e.g., [IU2, Propo-sition 8.1]). For example, we see that the dimer model given in Figure 1 is consistent by checking zigzagpaths shown in Figure 6. We also remark that this dimer model satisfies the stronger condition called isoradial (see Definition 3.4).In this paper, we also use another condition called properly ordered . To explain the properly ordering,we prepare several notations. First, considering a zigzag path z as a 1-cycle on T , we have the homologyclass [ z ] ∈ H ( T ) ∼ = Z . We call this element [ z ] ∈ Z the slope of z . We remark that even if we apply thejoin and split moves to nodes contained in a zigzag path, such operations do not change the slope. If azigzag path does not have a self-intersection, the slope of each zigzag path is primitive. Now, we considerslopes ( a, b ) ∈ Z of zigzag paths that are not homologically trivial. The set of such slopes has a naturalcyclic order by considering ( a, b ) as the element of the unit circle:( a, b ) √ a + b ∈ S . Thus, we say that two zigzag paths are adjacent if their slopes are adjacent with respect to the abovecyclic order. Using this cyclic order, we define a properly ordered dimer model as follows. In particular,it is known that a dimer model is consistent in the sense of Definition 3.2 if and only if it is properlyordered (see [IU1, Proposition 4.4]).
Definition 3.3 (see [Gul, Section 3.1]) . A dimer model is said to be properly ordered if(1) there is no homologically trivial zigzag path,(2) no zigzag path on the universal cover has a self-intersection,(3) no pair of zigzag paths with the same slope have a common node,(4) for any node on the dimer model, the natural cyclic order on the set of zigzag paths incident tothat node coincides with the cyclic order determined by their slopes.We also introduce isoradial dimer models which are stronger than consistent ones.
Definition 3.4 ([KS, Theorem 5.1], see also [Duf, Mer]) . We say that a dimer model Γ is isoradial (or geometrically consistent ) if(1) every zigzag path is a simple closed curve,(2) any pair of zigzag paths on the universal cover share at most one edge.
EFORMATIONS OF DIMER MODELS 8
Relationships between perfect matchings and zigzag paths.
We then discuss the relationshipbetween perfect matchings and zigzag paths. The following proposition is essential throughout this paper.
Proposition 3.5 (see [Gul, Theorem 3.3],[IU2, Section 9]) . There exists a one-to-one correspondencebetween the set of slopes of zigzag paths on a consistent dimer model Γ and the set of primitive sidesegments of the PM polygon ∆ Γ . Precisely, each slope of a zigzag path is the primitive outer normalvector for each primitive side segment of ∆ Γ .Moreover, zigzag paths having the same slope arise as the difference of two adjacent corner perfectmatchings P , P ′ (i.e., edges in P ∪ P ′ \ P ∩ P ′ forms zigzag paths). Thus, any corner perfect matchingintersects with half of the edges constituting a certain zigzag path. For example, the zigzag path z shown in Figure 6 is obtained from the pair of adjacent cornerperfect matchings ( P , P ) given in Figure 3. Also, the zigzag paths z , z , z are obtained by pairs( P , P ) , ( P , P ) , ( P , P ) respectively.By Proposition 3.5, we can assign the edge of the PM polygon to each zigzag path z , thus we will callthis the edge corresponding to z . In particular, the edges corresponding to zigzag paths having the sameslope are all the same.Let P , P ′ be adjacent corner perfect matchings on a consistent dimer model, and z , · · · , z r be thezigzag paths arising from P and P ′ as in Proposition 3.5. In particular, these zigzag paths have thesame slope. In this case, we see that P ∩ z i = Zig ( z i ) and P ′ ∩ z i = Zag ( z i ) (or P ∩ z i = Zag ( z i ) and P ′ ∩ z i = Zig ( z i )) for any i = 1 , · · · , r . Here, P ∩ z i denotes the subset of edges in P contained in z i .Then, we have the description of boundary perfect matchings using the corner ones. Proposition 3.6 (see e.g., [Bro, Proposition 4.35], [Gul, Corollary 3.8]) . Let P , P ′ and z , · · · , z r be thesame as above. Let E be the edge of the PM polygon of Γ corresponding to z , · · · , z r . We assume that P ∩ z i = Zig ( z i ) and P ′ ∩ z i = Zag ( z i ) . Then, any boundary perfect matching corresponding to a latticepoint on E can be described as ( P \ [ i ∈ I Zig ( z i )) ∪ [ i ∈ I Zag ( z i ) or ( P ′ \ [ i ∈ I Zag ( z i )) ∪ [ i ∈ I Zig ( z i ) , where I is a subset of { , · · · , r } . In particular, the number of perfect matchings corresponding to a latticepoint q on E is (cid:18) rm (cid:19) , where m is the number of primitive side segments of E between q and one of theendpoint of E . We then observe the relationship between zigzag paths and height changes of perfect matchings. Someof them are well-known for experts, but we note the details because these statements are quite importantwhen we define the deformation of consistent dimer models in Section 4, and also for the self-containedness.
Observation 3.7 (cf. [IU2, subsection 5.3]) . Let Γ be a consistent dimer model. For a zigzag path z ,the slope [ z ] is an element in H ( T ). On the other hand, we can consider height changes as elementsin the cohomology group H ( T ) ∼ = Z , and hence we have a pairing h− , −i : H ( T ) × H ( T ) → Z . ByProposition 3.5 and 3.6, there is a perfect matching P ′ that intersects half of the edges constituting z .Then, for any perfect matching P , we have that h h ( P , P ′ ) , [ z ] i ≤
0. In fact, we first replace z by the path p z on the quiver Q Γ going along the left side of z (see the figure below). zq z Then, considering this path p z as the element [ p z ] ∈ H ( T ), we have that [ z ] = [ p z ]. By a choice of P ′ ,this p z does not cross any edge in P ′ , and if p z crosses an edge in P , we can see the white node on theright by the definition of Q Γ . Thus, we have the desired inequation.For a perfect matching P and a zigzag path z on a dimer model Γ, we denote by | P ∩ z | the number ofedges in P ∩ z . Since the number of perfect matchings is finite, the maximum (resp. minimum) number ω max ( z ) (resp. ω min ( z )) of | P ∩ z | exists for each zigzag path z . For a consistent dimer model, z can be EFORMATIONS OF DIMER MODELS 9 obtained as the difference of adjacent perfect matchings (see Proposition 3.6), thus we clearly have that ℓ ( z ) / ω max ( z ). We set PM max ( z ) = { P ∈ PM (Γ) | | P ∩ z | = ω max ( z ) } , PM min ( z ) = { P ∈ PM (Γ) | | P ∩ z | = ω min ( z ) } . In particular, if P , P ′ are adjacent corner perfect matchings on a consistent dimer model Γ, and z isone of the zigzag paths obtained by P , P ′ , then we have that P ∩ z = Zig ( z ) and P ′ ∩ z = Zag ( z ) (or P ∩ z = Zag ( z ) and P ′ ∩ z = Zig ( z )), and hence the next lemma easily follows from Proposition 3.5 and3.6. Lemma 3.8.
Let z be a zigzag path on a consistent dimer model Γ , and E be the edge of the PM polygonof Γ corresponding to z . If P , · · · , P s are boundary perfect matchings corresponding to lattice points on E , then we have that { P , · · · , P s } = PM max ( z ) , and ω max ( z ) = | P i ∩ z | = ℓ ( z ) / for any i = 1 , · · · , s . Next, we prepare several lemmas, which play crucial roles to define the deformations of consistentdimer models.
Lemma 3.9.
Let the notation be the same as Lemma 3.8. For any perfect matching P , we have that | P ∩ z | = ℓ ( z ) / − h h ( P , P i ) , − [ z ] i . In particular, we have that h h ( P , P i ) , − [ z ] i ≤ ω max ( z ) − ω min ( z ) , and the equality holds for P ∈ PM min ( z ) .Proof. First, the maximum number of | P ∩ z | is ℓ ( z ) /
2, in which case P = P i by Lemma 3.8. If the path p z as in Observation 3.7 crosses an edge e in P , it means that e is not an edge constituting z , and thusany edge sharing the same white node as e is not contained in P . By Observation 3.7, we see that for anyperfect matching P the number of edges in P intersecting with p z coincides with −h h ( P , P i ) , [ z ] i , thus wehave the first equation.The second assertion follows from the first equation and Lemma 3.8. (cid:3) By this lemma, we see that P ∈ PM min ( z ) if and only if h h ( P , P i ) , [ z ] i≤h h ( P ′ , P i ) , [ z ] i for any P ′ ∈ PM (Γ). Thus, we see that P ∈ PM min ( z ) lies on either a vertex of the PM polygon ∆ Γ or an edge of ∆ Γ .Also, even if zigzag paths z j and z k have the same slope, ℓ ( z j ) = ℓ ( z k ) and | P ∩ z j | 6 = | P ∩ z k | in general,but their difference is the same as follows. Lemma 3.10.
Let Γ be a consistent dimer model, and z, z ′ be zigzag paths on Γ having the same slope.Then, for any perfect matching P , we have that ℓ ( z ) / − | P ∩ z | = ℓ ( z ′ ) / − | P ∩ z ′ | . In particular, we have that ℓ ( z ) / − ω min ( z ) = ℓ ( z ′ ) / − ω min ( z ′ ) . Proof.
Since [ z ] = [ z ′ ], the first equation follows from Lemma 3.9. Considering a perfect matching P suchthat the value of h h ( P , P i ) , − [ z ] i = h h ( P , P i ) , − [ z ′ ] i is maximal, we have the second equation. (cid:3) We then divide zigzag paths on a consistent dimer model into the following two types. In particular,type I zigzag paths are used to define the deformation of consistent dimer models.
Definition 3.11.
Let Γ be a dimer model, and z be a zigzag path on Γ.(1) We say that z is type I if z is reduced and e z intersects with any other zigzag paths on the universalcover e Γ at most once.(2) We say that z is type II if z is reduced and there exists a zigzag path e w on the universal cover e Γ suchthat e w intersects with e z in the opposite direction more than once.We note that any zigzag path on a reduced consistent dimer model is either type I or II. In particular, ifΓ is isoradial, then any zigzag path is type I (see Definition 3.4).As the following lemmas show, type I zigzag paths are particularly nice. Lemma 3.12.
Let z be a type I zigzag path on a consistent dimer model Γ . Then, there exists a perfectmatching P on Γ satisfying | P ∩ z | = 0 , in which case P is in PM min ( z ) . EFORMATIONS OF DIMER MODELS 10
Proof.
In order to find a perfect matching P , we will use the method discussed in [Gul, Section 3], [Bro,Section 4]. Thus, we first prepare some notation.We consider the sequence [ z ] , · · · , [ z n ] of slopes of zigzag paths on Γ. Since Γ is consistent, it isproperly ordered, thus we assume that they are ordered cyclically with this order. We note that some ofthe slopes may coincide. Then, we define the normal fan in H ( T ) ⊗ Z R whose rays are slopes [ z ] , · · · , [ z n ].In particular, each two dimensional cone σ is generated by adjacent different slopes. We denote by θ i theangle formed by [ z i ]. Here, we suppose that z = z k . Let R be a ray whose angle is θ k + π + ǫ where ǫ > θ k + π + ǫ does not coincides with any θ i .Then, for each node v ∈ Γ we define the fan ξ ( v ) generated by the slopes of zigzag paths factoringthrough v . In this fan ξ ( v ), we can find the zigzag path z ′ v whose slope makes the smallest clockwiseangle with R , and z ′′ v whose slope make the smallest anti-clockwise angle with R . Since Γ is properlyordered, these zigzag paths are consecutive around v . Then, as the intersection of z ′ v and z ′′ v , we have theedge e ( v ) which has v as an endpoint. v v ′ e ( v ) = e ( v ′ ) z ′ v z ′′ v We then apply the same argument to the node v ′ which is the other endpoint of e ( v ). Then, theproperly ordering on Γ induces the conclusion that e ( v ) = e ( v ′ ) (see the above figure). We repeat thesearguments for any node, but clearly we only consider e ( v )’s for any v ∈ Γ +0 (or v ∈ Γ − ). By [Gul,Subsection 3.2] or [Bro, Lemma 4.19], we see that the subset of edges e ( v ) for all v ∈ Γ +0 forms a perfectmatching. Furthermore, since z = z k is type I, there exists a zigzag path whose slope is located at anangle less than π in an anti-clockwise (resp. clockwise) direction from [ z ] in ξ ( v ) by [Bro, Lemma 4.11and its proof], in which case such a slope is located between [ z ] and [ z ′ v ] (resp. [ z ′′ v ]) or coincides with [ z ′ v ](resp. [ z ′′ v ]). Thus, we especially have that z = z ′ v and z = z ′′ v by the definition of the ray R . Therefore,in this case e ( v ) is not contained in z by the above construction. Also, we clearly see that if a node v does not lie on z , e ( v ) is not contained in z . Thus, the perfect matching constructed by the above fashionsatisfies the desired condition. (cid:3) Lemma 3.13.
Let z be a type I zigzag path on a consistent dimer model. Then, we have that ω min ( z ) = 0 ,and hence ℓ ( z ) is the same for all type I zigzag paths having the same slope.Proof. This follows from Lemma 3.10 and 3.12. (cid:3)
For a zigzag paths z, w on a dimer model Γ, we denote by z ∩ w the subset of edges that are intersectionsof z and w on Γ. We remark that if z is type I then the number of intersections of e z and e w on e Γ is lessthan or equal to one, but there are more intersections of z and w if we consider them on Γ. Lemma 3.14.
Let z be a type I zigzag path on a reduced consistent dimer model Γ . We suppose that azigzag path w has intersections with z on Γ . Then, we see that z ∩ w ⊂ Zig ( z ) or z ∩ w ⊂ Zag ( z ) .Proof. Let e , e be edges of Γ, and we assume that w intersects with z at e and e . If e i is a zig of z ,then it is a zag of w , and vice versa. Then, we assume that e is a zig of z and e is a zag of z .We then lift these on the universal cover e Γ. Let e e , e e be edges of e Γ whose restrictions on Γ are e , e respectively. In particular, e e is a zig of e z and e e is a zag of e z . Then, there exist zigzag paths e w , e w on e Γ whose restriction on Γ is just w , and e w (resp. e w ) intersects with e z at e e (resp. e e ). The zigzag path e z splits R into two pieces, and e w (resp. e w ) intersects with e z from left to right (resp. right to left) bythe definition of zigzag paths (see the figure below). EFORMATIONS OF DIMER MODELS 11 e z e w e w right of e z left of e z Since z is type I, there is no intersections of e z and e w i except e e i where i = 1 ,
2. Thus, we can notsuperimpose e w and e w using translations. This contradicts a choice of e w , e w . (cid:3) The following lemma follows from the argument in the proof of [Bro, Proposition 3.12].
Lemma 3.15.
Let z, w be zigzag paths on a consistent dimer model Γ . We assume that e z intersects with e w on the universal cover e Γ at most once. Then, the slopes [ z ] , [ w ] are linearly independent if and only if e z and e w intersect in precisely one edge. Lemma 3.16.
Let z , · · · , z r be type I zigzag paths on a reduced consistent dimer model Γ having thesame slope. We suppose that a zigzag path w has intersections with z j for some j . Then, we have that w intersects with any z i ( i = 1 , · · · , r ) , and the intersections w ∩ z , · · · , w ∩ z r are all in Zig ( w ) or Zag ( w ) .Moreover, we have that | w ∩ z | = · · · = | w ∩ z r | .Proof. Since z , · · · , z r are type I, e w intersects with each e z i at most once. Thus, each pair of zigzagpaths ( e w, e z i ) for i = 1 , · · · , r satisfies the assumption in Lemma 3.15. Since w has intersections with z j , e w intersects with e z j precisely once on the universal cover. By Lemma 3.15, we see that [ w ] and [ z j ] arelinearly independent, and hence [ w ] and [ z i ] are linearly independent for any i . Thus, e w intersects with e z i precisely once for any i = 1 , · · · , r .The latter assertion follows from a similar argument as in the proof of Lemma 3.14. More precisely,since e w intersects with e z i precisely once for any i = 1 , · · · , r , if e w intersects with e z i from right to left(resp. left to right), then so do zigzag paths having the same slope. This means that intersections are in Zig ( w ) (resp. Zag ( w )).Also, let e z and e z ′ be zigzag paths on e Γ that are projected onto the zigzag path z on Γ. We assumethat there is no zigzag path projected onto z between e z and e z ′ . Also, we assume that e w first intersectswith e z , then intersects with e z ′ . Then, for all i = 2 , · · · , r we can find a unique zigzag paths e z i on e Γsuch that it is projected onto z i and is located between e z and e z ′ . Thus, after e w intersects with e z , itintersects with e z , · · · , e z r precisely once and then arrives at e z ′ . We can do the same arguments for anypair ( e z , e z ′ ) of zigzag paths on e Γ satisfying the above properties, thus projecting onto Γ we have that | w ∩ z | = · · · = | w ∩ z r | . (cid:3) Deformations of consistent dimer models
In this section, we will introduce the concept of the deformation of consistent dimer models. Thisoperation is defined for type I zigzag paths on a consistent dimer model, and there are two kinds ofdeformations, which we call the deformation at zig (see Definition 4.3) and the deformation at zag (seeDefinition 4.5). These deformations preserve the consistency condition, but they change the associatedPM polygon. Whereas the PM polygon of the deformed dimer model is exactly the mutation of a polygon(see Section 6).4.1.
Definition of deformations of consistent dimer models.
Let Γ be a reduced consistent dimermodel, and hence any slope of a zigzag path on Γ is primitive. Let Z v (Γ) be the subset of zigzag pathson Γ whose slopes are the same primitive vector v ∈ Z , and Z I v (Γ) be the subset of Z v (Γ) consisting oftype I zigzag paths. We first prepare the deformation data . Definition 4.1 (Deformation data) . Let Γ be a reduced consistent dimer model. In order to define thedeformation of Γ, we fix the following data.(1) We choose a type I zigzag path z , and let 2 n := ℓ ( z ) and v := [ z ].(2) We then fix positive integers r, h such that r ≤ |Z I v (Γ) | and n = r + h . EFORMATIONS OF DIMER MODELS 12 (3) We take a subset { z , · · · , z r } ⊂ Z I v (Γ) of type I zigzag paths, in which case we have that2 n = ℓ ( z ) = · · · = ℓ ( z r ) by Lemma 3.13. Therefore, each z i can be described as z i = z i [1] z i [2] · · · z i [2 n − z i [2 n ] . (4) We consider all zigzag paths x , · · · , x s (resp. y , · · · , y t ) intersected with z at some zags (resp.zigs) of z . In this case, each of x , · · · , x s (resp. y , · · · , y t ) intersects with any z i at some zags(resp. zigs) of z i for all i = 1 , · · · , r by Lemma 3.16. We may assume that z , · · · , z r are orderedcyclically in the sense that if x j (resp. y k ) intersects with z i , then it intersects with z i − (resp. z i +1 ).(5) We recall that | x j ∩ z i | (resp. | y k ∩ z i | ) is the same number for all z , · · · , z r by Lemma 3.16.Thus, for some z i , let m j := | x j ∩ z i | be the number of intersections between x j and z i on Γ for j = 1 , · · · , s . Similarly, let m ′ k := | y k ∩ z i | for k = 1 , · · · , t . We note that n = m + · · · + m s = m ′ + · · · + m ′ t .(6) Then, we divide each zigzag path x j into m j parts x (1) j , · · · , x ( m j ) j as follows. We first fix oneof the intersections of z r and x j as the starting edge of x (1) j , and tracing along x j we will arriveat another intersection of z r and x j . We consider the edge of x j just before this intersection asthe ending edge of x (1) j , and hence such an intersection is considered as the starting edge of x (2) j .Repeating these procedures, we have x (1) j , · · · , x ( m j ) j . Thus, we have the set of sub-zigzag paths: { x (1)1 , · · · , x ( m )1 , x (1)2 , · · · , x ( m )2 , · · · , x (1) s , · · · , x ( m s ) s } . (4.1)Similarly, we also divide each zigzag path y k into m ′ k parts y (1) k , · · · , y ( m ′ k ) k by considering oneof the intersections of z and y k as the starting edge of y (1) k , and have the set of sub-zigzag paths: { y (1)1 , · · · , y ( m ′ )1 , y (1)2 , · · · , y ( m ′ )2 , · · · , y (1) t , · · · , y ( m ′ t ) t } . (4.2)(7) We then assign one of { z , · · · , z r } to x ( a j ) j for j = 1 , · · · , s and a j = 1 , · · · , m j . Then, we definethe set X i of edges consisting of the intersections between z i and the sub-zigzag paths in (4.1)that are assigned with z i . We assume that | X i | ≥ p i := | X i | − i = 1 , · · · , r . Wecall X := { X , · · · , X r } the zig deformation parameter with respect to z , · · · , z r and call non-negative integers p = ( p , · · · , p r ) ∈ Z r ≥ the weight of X . Similarly, we assign one of { z , · · · , z r } to y ( b k ) k for k = 1 , · · · , t and b k = 1 , · · · , m ′ k . Then, we define the set Y i of edges consisting of theintersections between z i and the sub-zigzag paths in (4.2) that are assigned with z i . We assumethat | Y i | ≥ q i := | Y i | − i = 1 , · · · , r . We call Y := { Y , · · · , Y r } the zag deformationparameter with respect to z , · · · , z r and call non-negative integers q = ( q , · · · , q r ) ∈ Z r ≥ the weight of Y . We remark that p + · · · + p r = m + · · · + m s − r = n − r = h , and also q + · · · + q r = h . Remark 4.2.
We note several remarks concerning the deformation data.(1) To define the deformation data, we need a type I zigzag path. If a dimer model is isoradial then anyzigzag path is type I (see Definition 3.4), and hence |Z I v (Γ) | = |Z v (Γ) | . Also, even if Γ contains notype I zigzag paths, we sometimes make a type II zigzag path type I by using the mutations of dimermodels (see Appendix A, especially Example A.4).(2) When we choose r = 1 in Definition 4.1, we have the zig (resp. zag) deformation parameter X = { X } (resp. Y = { Y } ) with respect to z , and the weights of X and Y are both h = ℓ ( z ) / −
1. In thiscase, we only need these data to define the deformations (see Definition 4.8).
Definition 4.3 (Deformation at zig) . Let the notation be the same as Definition 4.1. For the zigdeformation parameter X = { X , · · · , X r } of the weight p = ( p , · · · , p r ), we consider the followingprocedures:(zig-1) Using split moves, we insert p i white nodes and p i black nodes in each zig of z i . [Notation] • For a zig z i [2 m −
1] of z i where m = 1 , · · · , n and i = 1 , · · · , r , we denote by b i [2 m −
1] (resp. w i [2 m − z i [2 m − • We denote the white nodes added in the zig z i [2 m −
1] by w i, [2 m − , · · · , w i,p i [2 m − b i, [2 m − , · · · , b i,p i [2 m − b i [2 m −
1] to w i [2 m − z i for all i = 1 , · · · , r . EFORMATIONS OF DIMER MODELS 13 (zig-3) If p i = 0, then we connect the white node w i,j [2 m −
1] to the black node b i,j [2 m + 1] where j = 1 , · · · , p i and m = 1 , · · · , n . (Note that w i,j [2 n −
1] is connected to b i,j [2 n + 1] := b i,j [1].)We denote by z i,j the new 1-cycle, which will be a zigzag path on the deformed dimer model,obtained by connecting w i,j [2 n − , b i,j [2 n − , w i,j [2 n − , b i,j [2 n i − , · · · , w i,j [1] , b i,j [1]cyclically ( i = 1 , · · · , r and j = 1 , · · · , p i ).(zig-4) For m = 1 , · · · , n and i = 1 , · · · , r , if the zag z i [2 m ] of the original zigzag path z i on Γ is notcontained in X i , then we add edges, which we call bypasses , connecting the following pairs ofblack and white nodes:( w i, [2 m − , b i [2 m + 1]) , ( w i, [2 m − , b i, [2 m + 1]) , · · · , ( w i,p i [2 m − , b i,p i − [2 m + 1]) , ( w i [2 m − , b i,p i [2 m + 1]) . We denote the resulting dimer mode by ν zig X (Γ , { z , · · · , z r } ). We note that ν zig X (Γ , { z , · · · , z r } )is non-degenerate by Propostion 4.12.(zig-5) Then, we make the dimer model ν zig X (Γ , { z , · · · , z r } ) consistent using the method given in theproof of [BIU, Theorem 1.1] (see Operation 4.6 and Proposition 4.7).(zig-6) If there exist 2-valent nodes, then we apply the join moves to the dimer model obtained by theabove procedures and make it reduced.We denote the resulting dimer model by ν zig X (Γ , { z , · · · , z r } ), and call it the deformation of Γ at zig of { z , · · · , z r } with respect to the zig deformation parameter X . If a situation is clear, we simply denotethis by ν zig X (Γ). z i [2 m + 1] z i [2 m ] z i [2 m − x j (zig-1) (zig-2)(zig-3) (zig-4) Figure 7.
The deformation at zig of z i with p i = 2. (We assume that z i [2 m ] is notcontained in X i .) Remark 4.4.
The non-degenerate dimer model ν zig X (Γ , { z , · · · , z r } ) is determined uniquely for a givendeformation data, but in the operation (zig-5), the way to remove edges is not unique. Therefore, theresulting consistent dimer model is not unique, whereas since the set of slopes of zigzag paths is thesame for all possible consistent dimer models (see Proposition 4.7(2)), the associated PM polygon is thesame by Proposition 3.6. In addition, it has been believed that all consistent dimer models associatedwith the same lattice polygon are transformed into each other by the mutations of dimer models (seeAppendix A). Thus, we expect that the deformation of a consistent dimer model is determined uniquelyup to “ mutation equivalence ”. (We encounter the same situation for the deformation at zag given inDefinition 4.5 below.)Similarly, we can define the “zag version” of this deformation as follows. EFORMATIONS OF DIMER MODELS 14
Definition 4.5 (Deformation at zag) . Let the notation be the same as Definition 4.1. For the zagdeformation parameter Y = { Y , · · · , Y r } of the weight q = ( q , · · · , q r ), we consider the followingprocedures:(zag-1) Using split moves, we insert q i white nodes and q i black nodes in each zag of z i . [Notation] • For a zag z i [2 m ] of z i where m = 1 , · · · , n and i = 1 , · · · , r , we denote by w i [2 m ] (resp. b i [2 m ]) the white (resp. black) node that is the endpoint of z i [2 m ]. • We denote the white nodes added in the zag z i [2 m ] by w i, [2 m ] , · · · , w i,q i [2 m ], and denotethe black ones by b i, [2 m ] , · · · , b i,q i [2 m ]. Here, the subscripts increase in the direction from w i [2 m ] to b i [2 m ].(zag-2) We remove all zig of z i for all i = 1 , · · · , r .(zag-3) If q i = 0, then we connect the black node b i,j [2 m ] to the white node w i,j [2 m + 2] where j =1 , · · · , q i and m = 1 , · · · , n . (Note that b i,j [2 n ] is connected to w i,j [2 n + 2] := w i,j [2].) We denoteby z i,j the new 1-cycle, which will be a zigzag path on the deformed dimer model, obtained byconnecting b i,j [2 n ] , w i,j [2 n ] , b i,j [2 n − , w i,j [2 n − , · · · , b i,j [2] , w i,j [2]cyclically ( i = 1 , · · · , r and j = 1 , · · · , q i ).(zag-4) For m = 1 , · · · , n and i = 1 , · · · , r , if the zig z i [2 m −
1] of the original zigzag path z i on Γ isnot contained in Y i , then we add edges, which we call bypasses , connecting the following pairs ofblack and white nodes:( b i, [2 m ] , w i [2 m + 2]) , ( b i, [2 m ] , w i, [2 m + 2]) , · · · , ( b i,q i [2 m ] , w i,q i − [2 m + 2]) , ( b i [2 m ] , w i,q i [2 m + 2]) . We denote the resulting dimer mode by ν zag Y (Γ , { z , · · · , z r } ). We note that ν zag Y (Γ , { z , · · · , z r } )is non-degenerate by Propostion 4.12.(zag-5) Then, we make the dimer model ν zag Y (Γ , { z , · · · , z r } ) consistent using the method given in theproof of [BIU, Theorem 1.1] (see Operation 4.6 and Proposition 4.7).(zag-6) If there exist 2-valent nodes, then we apply the join moves to the dimer model obtained by theabove procedures and make it reduced.We denote the resulting dimer model by ν zag Y (Γ , { z , · · · , z r } ), and call it the deformation of Γ at zagof { z , · · · , z r } with respect to the zag deformation parameter Y (see also Remark 4.4). If a situation isclear, we simply denote this by ν zag Y (Γ). z i [2 m + 2] z i [2 m + 1] z i [2 m ] y k (zag-1) (zag-2)(zag-3) (zag-4) Figure 8.
The deformation at zag of z i with q i = 2. (We assume that z i [2 m + 1] is notcontained in Y i .) EFORMATIONS OF DIMER MODELS 15
Operation 4.6.
We note the operation given in the proof of [BIU, Theorem 1.1], which is used in (zig-5)and (zag-5).(a) The dimer model ν zig X (Γ , { z , · · · , z r } ) (resp. ν zag Y (Γ , { z , · · · , z r } )), which is obtained by applying theoperations (zig-1)–(zig-4) (resp. (zag-1)–(zag-4)) to the reduced consistent dimer model Γ, sometimescontains zigzag paths having a self-intersection on the universal cover. In this case, we use theoperation given in the proof of [BIU, Theorem 1.1], that is, we remove all the edges at the self-intersection (see Figure 9). We note that this operation does not change the slope of the arguedzigzag path. Figure 9.
An example of removing a self-intersection of a zigzag pathAfter these processes, there might be a connected component of the resulting bipartite graph thatis contained in a simply-connected domain in T . In that case, we remove such a connected component.We note that this removal does not affect our purpose, because our main concern is the PM polygonwhich is recovered from the slopes of zigzag paths, and the slope of the zigzag path corresponding tothe argued connected component is trivial.(b) On the other hand, the dimer model ν zig X (Γ , { z , · · · , z r } ) (resp. ν zag Y (Γ , { z , · · · , z r } )) might havea pair of zigzag paths on the universal cover that intersect with each other in the same directionmore than once. In this case, we use another operation given in the proof of [BIU, Theorem 1.1],that is, we choose any such pair of zigzag paths and remove a pair of consecutive intersections ofthis pair of zigzag paths (see Figure 10). We note that this operation does not change the slopesof zigzag paths and the resulting bipartite graph is also a dimer model because ν zig X (Γ , { z , · · · , z r } )(resp. ν zag Y (Γ , { z , · · · , z r } )) satisfies the strong marriage condition as we will see in Proposition 4.12. Figure 10.
An example of removing a pair of consecutive intersections of zigzag pathsSince the dimer model ν zig X (Γ , { z , · · · , z r } ) (resp. ν zag Y (Γ , { z , · · · , z r } )) is non-degenerate by Proposi-tion 4.12 and it does not contain a homologically trivial zigzag path (see the proofs of Proposition 5.4, 5.5and 5.8), we can make ν zig X (Γ , { z , · · · , z r } ) (resp. ν zag Y (Γ , { z , · · · , z r } )) another dimer model satisfyingconditions in Definition 3.2 by iterating the operations given in Operation 4.6. Thus, it is consistent, butit is not necessarily isoradial even if Γ is isoradial (see Example 4.11). Furthermore, since these operationsand (zig-6) (resp. (zag-6)) do not change the slopes of the operated zigzag paths, we have the followingproposition. Proposition 4.7.
Let the notation be the same as Definition 4.1, 4.3 and 4.5. Then, we have thefollowings. (1)
The dimer models ν zig X (Γ , { z , · · · , z r } ) and ν zag Y (Γ , { z , · · · , z r } ) are consistent. (2) The set of slopes of zigzag paths on ν zig X (Γ , { z , · · · , z r } ) ( resp. ν zag Y (Γ , { z , · · · , z r } )) is the same asthat of ν zig X (Γ , { z , · · · , z r } ) ( resp. ν zag Y (Γ , { z , · · · , z r } )) . EFORMATIONS OF DIMER MODELS 16
The operations (zig-4) and (zig-5) (resp. (zag-4) and (zag-5)) would be complicated if a given dimermodel is large. However, if we choose r = 1 as the deformation data, then X = { X } and Y = { Y } where X (resp. Y ) is the set of intersections between a chosen type I zigzag path z and x , · · · , x s (resp. y , · · · , y t ). In particular, X (resp. Y ) coincides with the set of zags (resp. zigs) of z , and hencethey are determined uniquely. Thus, in this case we may skip the operations (zig-4) (resp. (zag-4)), inwhich case we may also skip (zig-5) (resp. (zag-5)) since there are no bypasses (see Observation 5.2, 5.3and Lemma 5.6). Thus, we only need the weight p (resp. q ) of X (resp. Y ) to define the deformationat zig (resp. zag). We call p (resp. q ) the zig (resp. zag ) deformation weight with respect to z , and wesometimes denote the deformed dimer models as ν zig X (Γ , z ) = ν zig p (Γ , z ) and ν zag Y (Γ , z ) = ν zag q (Γ , z ). Wehere note the simplified definition of the deformations for the case of r = 1. Definition 4.8 (Deformations for the case of r = 1) . Let the notation be the same as Definition 4.1with r = 1 (see also Remark 4.2(2)). In particular, for a chosen type I zigzag path z , we have the zig(resp. zag) deformation parameter X = { X } (resp. Y = { Y } ) of the weight p = h (resp. q = h ) where h = ℓ ( z ) / − deformation of a consistent dimer model Γ at zig of z with the weight p is defined by theoperations (zig-1)–(zig-3) and (zig-6), and the resulting consistent dimer model is denoted by ν zig p (Γ , z ).Similarly, the deformation of Γ at zag of z with the weight q is defined by the operations (zag-1)–(zag-3)and (zag-6), and the resulting consistent dimer model is denoted by ν zag q (Γ , z ).We remark that ν zig p (Γ , z ) (resp. ν zag q (Γ , z )) is determined uniquely by definition. Remark 4.9.
We note additional remarks concerning the definition of the deformations.(1) We can skip the operations (zig-4) and (zig-5) (resp. (zag-4) and (zag-5)) for some classes ofdimer models even if r = 1. For example, if a given dimer model is a hexagonal dimer model ora square dimer model, in which case the associated PM polygon is a triangle or parallelogram,then we can skip these operations (see Appendix C for more details).(2) The join move does not change slopes of zigzag paths, and hence it does not affect the associatedPM polygon. Thus, when we are interested in only the PM polygon, we may skip (zig-6) and(zag-6).(3) Even if we choose the other sets of intersections X ′ , · · · , X ′ r (resp. Y ′ , · · · , Y ′ r ) in Definition 4.1(7),the PM polygon of the deformed dimer model is the same as that of ν zig X (Γ) (resp. ν zag Y (Γ)) as wewill show in Proposition 5.10.4.2. Examples of deformations of consistent dimer models.
In this subsection we give severalexamples. In these examples, we do not need the operations (zig-4) and (zig-5) (resp. (zag-4) and(zag-5)) because r = 1. We will give a large example which requires these operations in Appendix B. Example 4.10.
Let Γ be the dimer model given in Figure 1. We recall that zigzag paths on Γ areFigure 6, and we use the same notations given in these figures.We first collect the deformation data (see Definition 4.1). Let us choose the zigzag path z , and wewill denote this by z . We see that ℓ ( z ) = 6, v := [ z ] = ( − , − |Z I v (Γ) | = 1. Since |Z I v (Γ) | = 1, wecan take only r = 1, in which case h = ℓ ( z ) / − r = 2. Thus, we have that the zig (resp. zag) deformationweight is p = h = 2 (resp. q = h = 2). More precisely, we see that z intersects with z at zig of z , and z , z intersect with z at zag of z . Thus, X = { X } (resp. Y = { Y } ) consists of the intersections between z and z (resp. z and z or z ). Since | z ∩ z | = 1 , | z ∩ z | = 3, and | z ∩ z | = 2, we have the weights p = q = 2.Then, we apply the deformation of Γ at zig of z with p = 2 as shown in Figure 11.(zig-1)– (zig-3) (zig-6) Figure 11.
The deformation of Γ at zig of z EFORMATIONS OF DIMER MODELS 17
We also apply the deformation of Γ at zag of z with q = 2 as shown in Figure 12.(zag-1)– (zag-3) (zag-6) Figure 12.
The deformation of Γ at zag of z Example 4.11.
We remark that even if Γ is an isoradial dimer model, the deformed ones are notnecessarily isoradial.For example, the leftmost dimer model Γ in Figure 13 is isoradial. We choose a type I zigzag path z whose slope is v = ( − , |Z I v (Γ) | = 2 and ℓ ( z ) = 4. We fix r = 1, and hence h = ℓ ( z ) / − r = 1. Applying the deformation of Γ at zig of z with the zig deformation weight p = 1, wehave the rightmost one in Figure 13, and easily check that this deformed dimer model is consistent butnot isoradial. z (zig-1)– (zig-3) (zig-6) Figure 13.
An example of the deformed dimer model that is not isoradial4.3.
The proof of the non-degeneracy.
In this subsection, we show the non-degeneracy of the dimermodels ν zig X (Γ , { z , · · · , z r } ) and ν zag Y (Γ , { z , · · · , z r } ). Proposition 4.12.
Let the notation be the same as Definition 4.1, 4.3 and 4.5. Then, we have that thedimer models ν zig X (Γ , { z , · · · , z r } ) and ν zag Y (Γ , { z , · · · , z r } ) are non-degenerate.Proof. We prove the case of ν zig X (Γ) = ν zig X (Γ , { z , · · · , z r } ), and the other case is similar.Let Γ ′ be the dimer model obtained by applying the operations (zig-1)–(zig-3) to Γ. (The first step) : Here, we recall that the non-degeneracy condition is equivalent to the strong marriagecondition, that is, a dimer model has equal numbers of black and white nodes and every proper subset S of the black nodes satisfies the condition that S is connected to at least | S | + 1 white nodes.Suppose that Γ ′ is non-degenerate. Then Γ ′ satisfies the strong marriage condition. By applying theoperation (zig-4), we obtain the dimer model ν zig X (Γ , { z , · · · , z r } ). Since (zig-4) is the operation thatadds new edges, it also satisfies the strong marriage condition, and hence it is non-degenerate. Therefore,it is enough to show that Γ ′ is non-degenerate. (The second step) : We next consider a sub-dimer motel Γ ′′ of Γ satisfying the condition ( ∗ ) below,where we mean that Γ ′′ is a sub-dimer model of Γ if the set of the nodes coincide and the set of edges inΓ ′′ is the subset of edges in Γ.Condition ( ∗ ): For any given edge e in Γ ′′ , let z ′ and z ′′ be the different zigzag paths on Γ ′′ each of whichcontains e . Then either ( ∗
1) or ( ∗
2) holds:( ∗
1) either [ z ′ ] ∈ { [ z i ] , − [ z i ] } or [ z ′′ ] ∈ { [ z i ] , − [ z i ] } holds;( ∗
2) If z ′ intersects with z i in Zig ( z i ) (resp. Zag ( z i )), then z ′′ intersects with z i in Zag ( z i ) (resp. Zig ( z i )) for any i . EFORMATIONS OF DIMER MODELS 18
A desired sub-dimer model Γ ′′ of Γ satisfying ( ∗ ) can be constructed as follows. Here, to obtain sucha sub-dimer model, we employ the algorithm developed in [Gul, IU2] for showing Theorem 2.5, and wemodify it to our situation.First, let us consider the original dimer model Γ and let E , E , · · · , E m be all edges of ∆ Γ , where theprimitive outer normal vector for E is the slope [ z ] = · · · = [ z r ]. We assume that these edges are orderedcyclically in the anti-clockwise direction (see Figure 26 as reference). Also, let v j be the primitive outernormal vector corresponding to E j for each 1 ≤ j ≤ m . Let a be the index such that v a = − v if thereexists such an edge among E , · · · , E m , that is, E a is parallel to E . If there is no such edge, then let E a = ∅ for simplicity of notation. We recall that by Proposition 3.5, for each E j = ∅ there exist zigzagpaths on Γ such that the associated slopes coincide with v j , and the set of such zigzag paths is denotedby Z v j = Z v j (Γ). By our assumption, each slope of the zigzag paths in Z := Z v ∪ · · · ∪ Z v a − and v arelinearly independent. Thus, the zigzag paths in Z intersect with a type I zigzag path z i precisely once inthe universal cover (see Lemma 3.15). By definition of E , · · · , E a − , such an intersection is given fromthe right of z i to the left of z i , and hence zigzag paths in Z intersect with z i in Zag ( z i ). Similarly, wehave that the zigzag paths in Z := Z v a +1 ∪ · · · ∪ Z v m intersect with z i precisely once in the universalcover, and especially they intersect with z i in Zig ( z i ).We assume that there are at least two edges between E and E a − (i.e., a ≥ E and E . Since v and v are linearly independent, the zigzag paths z ′ ∈ Z v and z ′ ∈ Z v intersect at some edge of Γ. Clearly, such an intersection z ′ ∩ z ′ is neither any edgeconstituting any zigzag path whose slope is [ z i ] nor − [ z i ]. Now, remove an edge in z ′ ∩ z ′ . This operationmerges z ′ and z ′ , in which case the resulting dimer model stays consistent and the associated PMpolygon becomes “small” (see [Gul, Section 5, 6] for more details). Furthermore, since z ′ ∩ z i ⊂ Zag ( z i )and z ′ ∩ z i ⊂ Zag ( z i ) for each i , edges in z ′ ∩ z ′ do not share a node with z i for any i . Thus, z i is stilltype I even if we apply this operation and the merged zigzag path intersects with z i in Zag ( z i ) . Werepeat this procedure until there are no two edges between E and E a − . Similarly, if there are at leasttwo edges between E a +1 and E m (i.e., m − a ≥ E a +1 and E m . After removing all suitable edges from Γ, we get a consistentdimer model, which is clearly a sub-dimer model of Γ, and we will denote this by Γ sub . Since we do notremove edges contained in a zigzag path whose slope is ± [ z i ] in the above arguments, the edges E and E a (if this is not empty) of ∆ Γ are preserved on ∆ Γ sub (and hence we will use the same notation). Also,the edges E , · · · , E a − (resp. E a +1 , · · · , E m ) of ∆ Γ are substituted by the single edge in ∆ Γ sub , thus wedenote such an edge by E ′ (resp. E ′ m ). We note that zigzag paths corresponding to E ′ (resp. E ′ m ) areobtained by merging the ones corresponding to E , · · · , E a − (resp. E a +1 , · · · , E m ). In particular, thePM polygon ∆ Γ sub is constituted by E , E ′ , E a , E ′ m , in which case ∆ Γ sub is a triangle or a trapezoid. Then,it follows from the construction of Γ sub that- The zigzag paths z , · · · , z r on Γ are preserved on Γ sub and they are type I;- The zigzag paths on Γ corresponding to E a (if this is not empty) are preserved on Γ sub ;- The zigzag paths corresponding to E ′ (resp. E ′ m ) are intersected with z i in Zag ( z i ) (resp. Zig ( z i ))(see also the argument in the proof of Lemma 3.14).From these facts, it is easy to verify that Γ sub is our desired sub-dimer model Γ ′′ of Γ that satisfies thecondition ( ∗ ). (The third step) : Now, we apply the operations (zig-1)–(zig-3) in Definition 4.3 to Γ sub , which ispossible since z , · · · , z r are preserved on Γ sub . Then, we denote the resulting dimer model by Γ ′ sub . Bythe construction, Γ ′ can be obtained by adding some edges to Γ ′ sub . Thus, similar to the discussion in thefirst step, it is enough to show that Γ ′ sub is non-degenerate for proving the non-degeneracy of Γ ′ .Then, we finally show that Γ ′ sub is non-degenerate. To do this, we prove the existence of a perfectmatching that contains a given edge e of Γ ′ sub . We divide the set of edges into four cases (i)–(iv):(i) e is of the form ( b i,j − [2 m − , w i,j [2 m − ≤ i ≤ r, ≤ j ≤ p i + 1 and 1 ≤ m ≤ n ,where we let b i, [2 m −
1] = b i [2 m −
1] and w i,p i +1 [2 m −
1] = w i [2 m − e is of the form ( w i,j [2 m − , b i,j [2 m − ≤ i ≤ r, ≤ j ≤ p i and 1 ≤ m ≤ n ;(iii) e is of the form ( b i,j [2 m − , w i,j [2 m − ≤ i ≤ r, ≤ j ≤ p i and 1 ≤ m ≤ n ;(iv) e is of the form except for (i)–(iii).Namely, (i) and (ii) are the edges emanated in the process (zig-1), (iii) is one added in the process (zig-3),and (iv) is one which is invariant between Γ ′ sub and Γ sub .Since Γ sub is consistent and contains the type I zigzag paths z , · · · , z r , there exist corner perfectmatchings P and P ′ on Γ sub that are adjacent and satisfy P ∩ z i = Zig ( z i ) and P ′ ∩ z i = Zag ( z i ) for EFORMATIONS OF DIMER MODELS 19 ≤ i ≤ r (see subsection 3.2). Similarly, let Q be a corner perfect matching on Γ sub with Q ∩ z i = ∅ for1 ≤ i ≤ r . The existence of such Q is guaranteed by Lemma 3.12. We will use these P , P ′ and Q in orderto find a suitable perfect matching on Γ ′ sub containing a given edge e . We divide our discussions into thefollowing cases (i)–(iv) that correspond to the above division of edges respectively.Case (i): Let P ′′ = ( P \ r [ i =1 Zig ( z i )) ∪ r [ i =1 p i +1 [ j =1 n [ m =1 ( b i,j − [2 m − , w i,j [2 m − , (4.3)see Figure 14. It is easy to see that P ′′ is a perfect matching on Γ ′ sub containing e in the case (i).Case (ii): Let P ′′ = Q ∪ r [ i =1 p i [ j =1 n [ m =1 ( w i,j [2 m − , b i,j [2 m − , see Figure 15. Then, we see that P ′′ is a perfect matching on Γ ′ sub containing e in the case (ii).Case (iii): Let P ′′ = Q ∪ r [ i =1 p i [ j =1 n [ m =1 ( b i,j [2 m − , w i,j [2 m − , see Figure 16. Then, we see that P ′′ is a perfect matching on Γ ′ sub containing e in the case (iii).Case (iv): We note that an edge e in the case (iv) also appears in Γ sub since e is unchanged even if weapply (zig-1)–(zig-3). Thus, we can regard e as an edge of Γ sub . Since Γ sub satisfies the condition ( ∗ ), thezigzag paths z ′ , z ′′ on Γ sub that contain e satisfy either ( ∗
1) or ( ∗ • We assume that z ′ and z ′′ satisfy ( ∗ – Let, say, [ z ′ ] = [ z i ]. Since zigzag paths having the same slopes are obtained as the differenceof adjacent corner perfect matchings, either P or P ′ contains e . If e ∈ P , then we let P ′′ bethe same as (4.3). Then, P ′′ is a perfect matching on Γ ′ sub containing e . Even if e ∈ P ′ , wehave the same conclusion by letting P ′′ = ( P ′ \ r [ i =1 Zag ( z i )) ∪ r [ i =1 p i +1 [ j =1 n [ m =1 ( b i,j − [2 m − , w i,j [2 m − . – Let, say, [ z ′ ] = − [ z i ], in which case E a = ∅ and z ′ corresponds to E a . Let Q ′ and Q ′′ bethe corner perfect matchings on Γ sub whose difference forms z ′ . Let e ∈ Q ′ . Since h ( Q ′ , P )lies on E a where P is the reference perfect matching, we have that Q ′ ∩ z i = ∅ for any i byLemma 3.9. Thus, we let P ′′ = Q ′ ∪ r [ i =1 p i [ j =1 n [ m =1 ( w i,j [2 m − , b i,j [2 m − , and see that P ′′ is a perfect matching on Γ ′ sub containing e . • We assume that z ′ and z ′′ satisfy ( ∗ z ′ intersects with each z i in Zig ( z i ). Let Q ′ and Q ′′ be the corner perfect matchingson Γ sub whose difference forms z ′ . Let e ∈ Q ′ . As noted above, we see that all zigzag paths inΓ sub intersecting with z i at some zig of z i have the same slopes. This implies that Q ′ contains allzigs of z i , i.e., Q ′ ∩ z i = Zig ( z i ). This also means that Q ′ = P . Hence, we let P ′′ be the same as(4.3) and see that P ′′ is a perfect matching containing e . EFORMATIONS OF DIMER MODELS 20 (zig-1)–(zig-3)
Figure 14.
The perfect matchings P on Γ sub (left) and P ′′ on Γ ′ sub (right) for the case (i) (zig-1)–(zig-3) Figure 15.
The perfect matchings Q on Γ sub (left) and P ′′ on Γ ′ sub (right) for the case (ii) (zig-1)–(zig-3) Figure 16.
The perfect matchings Q on Γ sub (left) and P ′′ on Γ ′ sub (right) for the case (iii) (cid:3) Zigzag paths on deformed dimer models
In this section, we observe zigzag paths of the deformed dimer models and their slopes. We mainly dis-cuss the deformation at zig, but the same assertions hold for the deformation at zag by a similar argument.Thus, we will work with the setting in Definition 4.1, and consider the deformation ν zig X (Γ , { z , · · · , z r } )of Γ (see Definition 4.3).5.1. Behaviors of zigzag paths after deformations.
First, we give the following observation and fixthe notations which we will use throughout this sections.
Observation 5.1.
Let Γ and z , · · · , z r be the same as in Definition 4.1, especially z , · · · , z r are type Iand [ z ] = · · · = [ z r ]. These zigzag paths are ordered along the subscript i = 1 , · · · , r cyclically. For any α ∈ Z and i = 1 , · · · , r , let e z i ( α ) be a zigzag path on the universal cover e Γ whose projection on Γ is z i .Each e z i ( α ) divides R into two parts, thus it makes sense to consider the left of e z i ( α ) and the right of e z i ( α ). Then, we can write a straight line ℓ Li,α (resp. ℓ Ri,α ) on the left (resp. right) of e z i ( α ) such that thegradient of ℓ Li,α (resp. ℓ Ri,α ) is v = [ z i ] and nodes contained in the region obtained as the intersection of EFORMATIONS OF DIMER MODELS 21 the right of ℓ Li,α and the left of ℓ Ri,α are precisely the ones located on e z i ( α ). We will call such a region the( i, α ) -th deformed part (see Figure 17). e z i ( α ) ℓ Li,α ℓ Ri,α the left of ℓ Li,α the right of ℓ Ri,α the ( i, α )-thdeformed part
Figure 17.
Also, we call the region obtained as the intersection of the right of ℓ Ri − ,α and the left of ℓ Li,α the( i, α ) -th irrelevant part . Here, we say that the intersection of the right of ℓ Rr,α and the left of ℓ L ,α +1 isthe (1 , α + 1) -th irrelevant part . We remark that sometimes there are no nodes in an irrelevant part. Wesometimes omit α ∈ Z from these notations unless it causes confusion. We also use these terminologiesfor Γ. That is, a part of Γ obtained by projecting a deformed (resp. irrelevant) part of e Γ onto Γ is saidto be a deformed (resp. irrelevant ) part of Γ. the ( i, α )-thirrelevant part the ( i, α )-thdeformed part the ( i + 1 , α )-thirrelevant part the ( i + 1 , α )-thdeformed part the ( i + 2 , α )-thirrelevant part the ( i + 2 , α )-thdeformed part the ( i + 3 , α )-thirrelevant part e z i ( α ) e z i +1 ( α ) e z i +2 ( α ) Figure 18.
By the condition of Definition 3.3(3), z , · · · , z r do not have a common node, thus the irrelevant partsdo not overlap each other. Since the operations (zig-1)–(zig-4) (or (zag-1)–(zag-4)) are local operationson each deformed part, any irrelevant part will be unchanged even if we apply these operations. Thus,we hand over these terminologies “deformed parts” and “irrelevant parts”. We then consider a zigzagpath w satisfying the following properties:(a) If [ z i ] and [ w ] are linearly independent, then by Lemma 3.15 e w intersects with e z i ( α ) preciselyonce, and so does any e z i ( α ) with i = 1 , · · · , r and α ∈ Z . In particular, all intersections are zigsof w or zags of w by Lemma 3.16. If e w intersects with e z i ( α ) at a zig (resp. zag) of e z i ( α ), thenwe easily see that e w crosses the ( i, α )-th deformed part in the direction from the ( i, α )-th (resp.( i + 1 , α )-th) irrelevant part to the ( i + 1 , α )-th (resp. ( i, α )-th) irrelevant part. EFORMATIONS OF DIMER MODELS 22 (b) If [ z i ] and [ w ] are linearly dependent, then by Lemma 3.15 e w and e z i ( α ) do not intersect for any i = 1 , · · · , r and α ∈ Z . This is equivalent to the condition that e w is contained in some irrelevantpart. In this case, w is unchanged even if we apply the deformations because (zig-1)–(zig-4) (or(zag-1)–(zag-4)) are operations on the deformed parts and (zig-5) (or (zag-5)) does not affect w by Lemma 5.6 below.In the rest, we discuss the behavior of zigzag paths after applying the operations (zig-1)–(zig-3) givenin Definition 4.3. (We can do the same arguments below for the case of (zag-1)–(zag-3) given in Defini-tion 4.5.) Observation 5.2.
We consider a zigzag path y k on a consistent dimer model Γ intersecting with z i at azig of z i . Let e z i and e y k be zigzag paths on e Γ projecting onto z i and y k respectively. By Observation 5.1(a), e y k crosses the i -th deformed part in the direction from the i -th irrelevant part to the ( i + 1)-th irrelevantpart, and e y k intersects with e z i precisely once. We suppose that the zig e z i [2 m −
1] of e z i is such anintersection. In this case, e z i [2 m −
1] is also a zag of e y k , thus we may write it as e y k [2 m ].Now, we apply the operations (zig-1)–(zig-3) to Γ, and we denote the resulting dimer model by Γ ′ andits universal cover by e Γ ′ . Then, some new nodes are inserted in e z i [2 m −
1] = e y k [2 m ] and zigzag paths e z i, , · · · , e z i,p i on e Γ ′ , which project onto zigzag paths z i, , · · · , z i,p i on Γ ′ respectively, appear in the i -thdeformed part. e y k e z i (zig-1)–(zig-3) e y ′ k e z i, e z i,p i Figure 19.
We consider the zigzag path e y ′ k on e Γ ′ passing through e y k [2 m −
1] as a zig. That is, e y ′ k starts from e y k [2 m − e z i, , · · · , e z i,p i in the i -th deformed part, and arrives at e y k [2 m + 1](see the right of Figure 19). In particular, it crosses the i -th deformed part in the direction from the i -thirrelevant part to the ( i + 1)-th irrelevant part. Although e y ′ k looks different from e y k in the deformed parts,it connects the zigs e y k [2 m −
1] and e y k [2 m + 1] of e y k in the i -th deformed part for all i , thus e y ′ k shares thesame nodes and edges as e y k in any irrelevant part. As a conclusion, e y ′ k coincides with e y k in all irrelevantparts, and behaves as the right of Figure 19 in each deformed part. Also, we see that bypasses insertedin the operation (zig-4) do not affect the behavior of e y ′ k , because e y ′ k never passes through bypasses. Observation 5.3.
We consider a zigzag path x j on Γ intersecting with z i at a zag of z i . Let e x j be azigzag path on e Γ projecting onto x j . By Observation 5.1(a), e x j crosses the i -th deformed part in thedirection from the ( i + 1)-th irrelevant part to the i -th irrelevant part, and e x j intersects with e z i preciselyonce. We suppose that the zag e z i [2 m ] of e z i is such an intersection. In this case, e z i [2 m ] is also a zig of e x j ,thus we may write it as e x j [2 m + 1]. We then apply the operations (zig-1)–(zig-4) to Γ, and we have thedimer model ν zig X (Γ) = ν zig X (Γ , { z , · · · , z r } ).Now, we assume that z i [2 m ] which is the projection of e z i [2 m ] = e x j [2 m + 1] on Γ is not containedin X i , in which case bypasses are inserted. We consider the zigzag path e x ′ j on the universal cover of ν zig X (Γ) passing through e x j [2 m ] as a zag of e x ′ j . That is, e x ′ j starts from e x j [2 m ], behaves as in the right ofFigure 20, and arrives at e x j [2 m + 2]. In particular, e x ′ j crosses the i -th deformed part in the direction fromthe ( i + 1)-th irrelevant part to the i -th irrelevant part, and behaves as the same as e x j in each irrelevantpart. EFORMATIONS OF DIMER MODELS 23 e z i e x j (zig-1)–(zig-4) e x ′ j e z i, e z i,p i Figure 20.
The case where the projection of e z i [2 m ] = e x j [2 m + 1] on Γ is not contained in X i e z i e x j (zig-1)–(zig-4) e x ′ j e z i, e z i,p i Figure 21.
The case where the projection of e z i [2 m ] = e x j [2 m + 1] on Γ is contained in X i We then assume that z i [2 m ] is contained in X i , in which case bypasses are not inserted. We againconsider the zigzag path e x ′ j on the universal cover of ν zig X (Γ) passing through e x j [2 m ] as a zag of e x ′ j (seee.g., Figure 21). Unlike the previous case, after passing through e x j [2 m ], e x ′ j goes to the edge { e b i [2 m +1] , e w i, [2 m + 1] } . (Here, we denote the edge whose endpoints are a black node b and a white node w by { b, w } .) If z i [2 m + 2] ∈ X i , in which case bypasses are not inserted, then e x ′ j goes to the edge { e w i, [2 m + 1] , e b i, [2 m + 3] } . On the other hand, if z i [2 m + 2] X i , in which case we insert bypasses,then e x ′ j goes to the edge { e w i, [2 m + 1] , e b i [2 m + 3] } . In such a way, e x ′ j crosses the i -th deformed part inthe direction from the ( i + 1)-th irrelevant part to the i -th irrelevant part. More precisely, if we assumethat e x ′ j goes through the edge { e b i,s [2 m − , e w i,s +1 [2 m − } in the i -th deformed part, then e x ′ j behavesas follows:(1) if z i [2 m ] ∈ X i , in which case bypasses are not inserted, then e x ′ j goes through the edge { e w i,s +1 [2 m − , e b i,s +1 [2 m + 1] } and then { e b i,s +1 [2 m + 1] , e w i,s +2 [2 m + 1] } ,(2) if z i [2 m ] X i , in which case we insert bypasses, then e x ′ j goes through the edge { e w i,s +1 [2 m − , e b i,s [2 m + 1] } and then { e b i,s [2 m + 1] , e w i,s +1 [2 m + 1] } ,where m = 1 , · · · , n and s = 0 , · · · , p i − e b i, [ − ] = e b i [ − ] and e w i,p i +1 [ − ] = e w i [ − ]. When we consider e x ′ j in the i -th deformed part, we encounter the case of (1) | X i | times and the case of (2) ℓ ( z i ) / −| X i | times.Since p i = | X i |−
1, we see that e x ′ j goes out the i -th deformed part from e w i [2 m − n ] = e w i [2 m − ℓ ( z i )],and then it goes into the i -th irrelevant part. Thus, e x ′ j behaves as the same as the shift of e x j in the i -thirrelevant part. For example, if we consider the type I zigzag path z i with ℓ ( z i ) = 8, and the deformationparameter X i with | X i | = 3 and z i [2 m + 4] X i , then the i -th deformed part will change as shown inFigure 22. EFORMATIONS OF DIMER MODELS 24 e x j e x j (1) e z i [2 m − e z i [2 m ] e z i [2 m + 1] e z i [2 m + 7] e w i [2 m − e w i [2 m + 7] e b i [2 m − e b i [2 m + 7] (zig-1)–(zig-4) e x ′ j e w i [2 m − e w i [2 m + 7] e b i [2 m − e b i [2 m + 7] Figure 22.
An example of the behavior of e x ′ j in the i -th deformed part5.2. Properties of zigzag paths on deformed dimer models.
In this subsection, we study the slopesof zigzag paths of the deformed dimer models. In particular, we can describe them in terms of zigzagpaths of the original dimer model, and such a description plays a crucial role to discuss the relationshipwith the mutations of polygons.
Proposition 5.4.
Let the notation be the same as in Definition 4.1 and 4.3 ( resp. 4.5 ) . Then, z i,j givenin (zig-3) of Definition 4.3 ( resp. (zag-3) of Definition 4.5 ) is a type I zigzag path z i,j of ν zig X (Γ) ( resp. ν zag Y (Γ)) with ℓ ( z i,j ) = ℓ ( z i ) . Moreover, z i,j does not have a self-intersection on the universal cover, andsatisfies [ z i,j ] = − [ z i ] = − v and hence it is not homologically trivial.Proof. We consider the case of ν zig X (Γ). The case of ν zag Y (Γ) is similar.First, z i,j is a zigzag path of ν zig X (Γ) by definition. We see that zigzag paths of ν zig X (Γ) intersectingwith e z i,j take the form either e y ′ k given in Observation 5.2 or e x ′ j given in Observation 5.3. In particular,these intersect with e z i,j precisely once in each deformed part, and e y ′ k crosses the i -th deformed part in thedirection from the i -th irrelevant part to the ( i + 1)-th irrelevant part and e x ′ j crosses the i -th deformedpart in the direction from the ( i + 1)-th irrelevant part to the i -th irrelevant part for all i . Since otherzigzag paths do not intersect with e z i,j , we have that z i,j is a type I zigzag path of ν zig X (Γ). Also, e z i,j doesnot have a self-intersection by definition. By these properties, the edges constituting z i,j are not removedby the operation (zig-5). In addition, since z i,j contains no 2-valent nodes, the operation (zig-6) does notaffect z i,j . Therefore, we have that z i,j is a type I zigzag path of ν zig X (Γ), and remaining assertions followfrom the definition of z i,j . (cid:3) Proposition 5.5.
Let the notation be the same as in Definition 4.1, 4.3 and 4.5. In particular, y , · · · , y t (resp. x , · · · , x s ) are zigzag paths of Γ intersecting with a chosen type I zigzag path z at some zigs (resp.zags) of z . Then, we have the followings. (1) For any zigzag path y k of Γ , there exists a unique zigzag path y ′ k of ν zig X (Γ) such that it does not havea self-intersection on the universal cover and satisfies [ y k ] = [ y ′ k ] where k = 1 , · · · , t . (2) For any zigzag path x j of Γ , there exists a unique zigzag path x ′ j of ν zag Y (Γ) such that it does not havea self-intersection on the universal cover and satisfies [ x j ] = [ x ′ j ] where j = 1 , · · · , s .Proof. We consider the case of ν zig X (Γ), and the case of ν zag Y (Γ) is similar.We use the same notations used in Observation 5.2. In particular, we consider the zigzag path e y ′ k of e Γ ′ which coincides with e y k in all irrelevant parts, and behaves as the right of Figure 19 in each deformedpart, thus it does not have a self-intersection. Since bypasses inserted in the operation (zig-4) do notaffect the behavior of e y ′ k , we can extend e y ′ k as a zigzag path of the universal cover of ν zig X (Γ). By projecting e y ′ k onto ν zig X (Γ), we have the zigzag path y ′ k of ν zig X (Γ). By the construction given in Observation 5.2,we see that [ y k ] = [ y ′ k ]. Since e y ′ k coincides with e y k in all irrelevant parts and Γ is consistent, it doesnot behave pathologically in irrelevant parts as it infringes the consistency condition. Furthermore, e y ′ k EFORMATIONS OF DIMER MODELS 25 intersects with zigzag paths with the forms e z i,j and e x ′ j (see Observation 5.3) in some deformed parts,but they do not intersect with each other in the same direction more than once. Therefore, the edgesconstituting y ′ k are not removed by the operation (zig-5), and hence y ′ k is not changed by (zig-5). Inaddition, the operation (zig-6) does not change the slopes. Thus, we naturally extend this zigzag path y ′ k as the one of ν zig X (Γ), which is determined uniquely and satisfies [ y k ] = [ y ′ k ] by the construction. (cid:3) By the proof of Proposition 5.4 and 5.5, the zigzag paths e z i,j and e y ′ k do not have a self-intersection,and do not intersect with other zigzag paths in the same direction more than once. Furthermore, theintersections between e x ′ j and e z i,j or e y ′ k are not bypasses (see Observation 5.2 and 5.3). Thus, we havethe following lemma. Lemma 5.6.
Let the notation be the same as in Definition 4.1, 4.3 and 4.5. Then we have the followings. (1)
The edges removed by the operation (zig-5) are a part of bypasses added in (zig-4) or edges appearingin some irrelevant parts that are intersections between pairs of zigzag paths x , · · · , x s . (2) The edges removed by the operation (zag-5) are a part of bypasses added in (zag-4) or edges appearingin some irrelevant parts that are intersections between pairs of zigzag paths y , · · · , y t . Before showing the next proposition, we introduce some notations.
Setting 5.7.
Let z be a zigzag path on a consistent dimer model Γ. We recall that corner perfectmatchings are ordered in the anti-clockwise direction along the vertices of ∆ Γ (see Subsection 2.2). Let P , P ′ be adjacent corner perfect matchings on Γ such that the difference of P and P ′ contains z (seeProposition 3.5). We assume that P , P ′ are ordered with this order, in which case P ∩ z = Zig ( z ) and P ′ ∩ z = Zag ( z ). Then, we set P z := P and P ′ z := ( P \ Zig ( z )) ∪ Zag ( z ). By Proposition 3.6, P z , P ′ z areboundary perfect matchings corresponding to certain lattice points on the edge of ∆ Γ whose outer normalvector is [ z ], and we see that the difference of P z and P ′ z , namely P z ∪ P ′ z \ P z ∩ P ′ z , forms z . Thus, by thisconstruction, h ( P ′ z , P z ) ∈ Z is a primitive lattice element with h [ z ] , h ( P ′ z , P z ) i = 0. Proposition 5.8.
Let the notation be the same as in Definition 4.1, 4.3 and 4.5. In particular, x , · · · , x s (resp. y , · · · , y t ) are zigzag paths of Γ intersecting with a chosen type I zigzag path z at some zags (resp.zigs) of z . Let h ( P ′ z , P z ) be a primitive lattice element as above. Then, we have the followings. (1) For any zigzag path x j of Γ , there exists a unique zigzag path x ′ j of ν zig X (Γ) such that [ x ′ j ] = [ x j ] + h [ x j ] , h ( P ′ z , P z ) i [ z ] , where j = 1 , · · · , s . (2) For any zigzag path y k of Γ , there exists a unique zigzag path y ′ k of ν zag Y (Γ) such that [ y ′ k ] = [ y k ] + h [ y k ] , h ( P ′ z , P z ) i [ z ] , where k = 1 , · · · , t .Proof. We consider the case of ν zig X (Γ), and the case of ν zag Y (Γ) is similar.We recall that | x j ∩ z | = | x j ∩ z i | for all i = 1 , · · · , r , and this number is denoted by m j (seeDefinition 4.1). We first show that m j = h [ x j ] , h ( P ′ z , P z ) i (5.1)for j = 1 , · · · , s . Let p x j be the path of the quiver Q Γ going along the left side of x j (see Observation 3.7).In particular, considering p x j as the element in H ( T ), we have that [ p x j ] = [ x j ]. By our assumption,the intersections x j ∩ z are contained in Zag ( z ) = P ′ ∩ z . Thus, p x j crosses z at a zig of z . Since P ∩ z = Zig ( z ), every time p x j crosses z , the height function h P ′ , P increases by 1. Since m j = | x j ∩ z | , wehave the equation (5.1).Then, we show that for each x j there exists a zigzag path x ′ j on ν zig X (Γ) = ν zig X (Γ , { z , · · · , z r } ) suchthat [ x ′ j ] = [ x j ] + m j [ z ] (5.2)for j = 1 , · · · , s . We divide x j into sub-zigzag paths x (1) j , · · · , x ( m j ) j . By definition, x (1) j intersects with z r at a zag of z r . We denote this zag by z r [2 m ] := x (1) j ∩ z r , in which case the white (resp. black) nodethat is the end point of z r [2 m ] is denoted by w r [2 m −
1] (resp. b r [2 m + 1]). By considering the universalcover e Γ, we naturally define e x j , e x (1) j , e z r , e z r [2 m ] , e w r [2 m − , e b r [2 m + 1] etc. Also, we assume that e z r iscontained in the ( r, e x j crosses the ( r, EFORMATIONS OF DIMER MODELS 26 from the ( r + 1 , r, r, e b r [2 m + 1] and the exit is e w r [2 m − e x ′ j of the universal cover ν zig X (Γ) ∼ of ν zig X (Γ), which behaves as follows:( A r ) If z r [2 m ] X r , then e x ′ j goes into the ( r, ν zig X (Γ) ∼ from e b r [2 m + 1] and goesout from e w r [2 m −
1] (see also Figure 20). After crossing the ( r, r, e x j in that part.( B r ) If z r [2 m ] ∈ X r , then e x ′ j goes into the ( r, ν zig X (Γ) ∼ from e b r [2 m + 1] andgoes out from e w r [2 m − n ] = e w r [2 m − ℓ ( z )] (see also Figure 21 and 22). After crossingthe ( r, r, e x j , which we denote e x j (1), in that part.Then, e x ′ j goes into the ( r − , ν zig X (Γ) ∼ . We let e z r − [2 m ′ ] := e x j ∩ e z r − and e z r − [2 m ′′ ] := e x j (1) ∩ e z r − . We note that on the dimer model Γ we have z r − [2 m ′ ] = z r − [2 m ′′ ] = x (1) j ∩ z r − by definition. • We assume that z r [2 m ] ∈ X r in the above argument. Then z r − [2 m ′ ] X r − by the definition of X r and X r − . (Furthermore, x (1) j ∩ z i X i for any i = r .) In this case, e x ′ j crosses the ( r − , ν zig X (Γ) ∼ as the same as ( A r ) above. Then, it behaves as the same as e x j (1) in the( r − , • We assume that z r [2 m ] X r in the above argument. – If z r − [2 m ′ ] X r − , then e x ′ j crosses the ( r − , ν zig X (Γ) ∼ as the sameas ( A r ). Then, it behaves as the same as e x j in the ( r − , – If z r − [2 m ′ ] ∈ X r − , then e x ′ j crosses the ( r − , ν zig X (Γ) ∼ as the sameas ( B r ). Then, it behaves as the same as e x j (1) in the ( r − , e x ′ j crosses the ( i, ν zig X (Γ) ∼ for i = r, r − , · · · , , e x ′ j behaves as the same as e x j (1) in this irrelevant part.Then, e x ′ j goes into the ( r, − x (2) j ofΓ and the intersection between e z r ( −
1) and e x (2) j on e Γ. By the same arguments as above, we see that e x ′ j crosses the ( i, − ν zig X (Γ) ∼ for i = r, r − , · · · , , − e x j (2) in this irrelevant part.Repeating these arguments, we finally see that e x ′ j crosses the ( i, − m j + 1)-th deformed part of ν zig X (Γ) ∼ for i = r, r − , · · · , e x j ( m j ) in the (1 , − m j + 1)-th irrelevantpart. Then, e x ′ j goes into the ( r, − m j )-th deformed part of ν zig X (Γ) ∼ , in which case we denote the blacknode that is the entrance of this deformed part by B . Since m j = | x j ∩ z i | , the projection of e x j ∩ e z i ( − m j )on Γ coincides with z i [2 m ], which is the starting edge of our arguments. Thus, B coincides with e b r [2 m +1]if they are projected onto ν zig X (Γ), which means we could follow all edges of the zigzag path x ′ j of ν zig X (Γ).By these arguments, we see that the slope of e x ′ j changes [ z ] in each deformed part, thus we have that[ x ′ j ] = [ x j ] + m j [ z ].Finally, we apply the operations (zig-5) and (zig-6) to ν zig X (Γ). Then, we obtain the deformed dimermodel ν zig X (Γ) and the zigzag path on it having the same slope as e x ′ j . This zigzag path is determineduniquely by the construction, and we use the same notation for this zigzag path by the abuse of thenotation. Since (zig-5) and (zig-6) do not change the slopes of zigzag paths, we have (5.2). (cid:3) The perfect matching polygons of deformed dimer models.
Since the PM polygon of aconsistent dimer model can be determined by the slopes of zigzag paths (see Proposition 3.5), we havethe PM polygon of the deformed dimer model by using the description of slopes in Proposition 5.4, 5.5and 5.8. For example, if we consider the deformed dimer models shown in Example 4.10, we have thePM polygons shown in Example 5.9 below. In Section 6, we will see that these are exactly the mutationsof the PM polygon of the original dimer model.
Example 5.9.
We consider the PM polygons associated to the deformed dimer models given in Exam-ple 4.10. Thus, let Γ be a consistent dimer model given in Figure 1, and ν zig p (Γ , z ) (resp. ν zag q (Γ , z )) be EFORMATIONS OF DIMER MODELS 27 the dimer model deformed at zig (resp. zag) of z as shown in Figure 11 (resp. Figure 12). Then, werespectively have the following PM polygons. ν zig p Figure 23.
The change of the associated PM polygon via the deformation at zig ν zag q Figure 24.
The change of the associated PM polygon via the deformation at zagSince the slopes of zigzag paths ν zig X (Γ) and ν zag Y (Γ) do not depend on a choice of X , · · · , X r (resp. Y , · · · , Y r ) by Proposition 5.4, 5.5 and 5.8, we have the following proposition. However, we remark thatthe deformed dimer model depends on a choice of deformation parameters, thus ν zig X (Γ , { z , · · · , z r } ) = ν zig X ′ (Γ , { z , · · · , z r } ) in general (the zag version is similar). Proposition 5.10.
Let the notation be the same as in Definition 4.1, 4.3 and 4.5. For other zig defor-mation parameter X ′ := { X ′ , · · · , X ′ r } and zag deformation parameter Y ′ := { Y ′ , · · · , Y ′ r } , we have ∆ ν zig X (Γ , { z , ··· ,z r } ) = ∆ ν zig X′ (Γ , { z , ··· ,z r } ) and ∆ ν zag Y (Γ , { z , ··· ,z r } ) = ∆ ν zag Y′ (Γ , { z , ··· ,z r } ) . Relationships with mutations of polygons
In this section, we discuss a relationship between the deformations of consistent dimer models and themutations of polygons. We first define the mutation for lattice polytopes of any dimension, and then wemainly discuss the case of polygons.6.1.
Preliminaries on mutations of polytopes.
Following [ACGK], we introduce the notion of mu-tations of polytopes. Thus, let N ∼ = Z d be a lattice of rank d , and P ⊂ N R := N ⊗ Z R be a convexlattice polytope, and we assume that P contains the origin . We denote by V ( P ) the set of verticesof P . We say that two polytopes P, Q ⊂ N R are isomorphic if they are transformed into each other byGL( d, Z )-transformations, in which case we denote P ∼ = Q .We first prepare some notions used in the definition of the mutations of polytopes. Definition 6.1 (Mutation data) . Let w ∈ M := Hom Z ( N, Z ) ∼ = Z d be a primitive lattice vector. Theelement w ∈ M determines the linear map h w, −i : N R → R . We set h max ( P, w ) := max {h w, u i | u ∈ P } and h min ( P, w ) := min {h w, u i | u ∈ P } , which are integers because P is a lattice polytope. If the situation is clear, we simply denote these by h max and h min respectively. We define the width of P with respect to w as width ( P, w ) := h max ( P, w ) − h min ( P, w ) . EFORMATIONS OF DIMER MODELS 28
We note that if the origin is contained in the strict interior P ◦ of P , then h min < h max >
0, inwhich case we have width ( P, w ) ≥
2. We say that a lattice point u ∈ N (resp. a subset F ∈ N R ) is at height m with respect to w if h w, u i = m (resp. h w, u i = m for any u ∈ F ).For each height h ∈ Z , we let w h ( P ) := conv { u ∈ P ∩ N | h w, u i = h } , which is the (possibly empty) convex hull of all lattice points in P at height h . By definition, w h min ( P )and w h max ( P ) are faces of P . Using these notations, we define the mutation of a polytope as follows. Definition 6.2.
Let the notation be the same as above. We assume that there exists a lattice polytope F ⊂ N R such that h w, u i = 0 for any u ∈ F and for each negative height h min ≤ h < G h ⊂ N R satisfying { u ∈ V ( P ) | h w, u i = h } ⊆ G h + ( − h ) F ⊆ w h ( P ) , (6.1)where + means the Minkowski sum, and we especially define Q + ∅ = ∅ for any polytope Q . We call F a factor of P with respect to w . Then, we define the ( combinatorial ) mutation of P given by the vector w , factor F and polytopes { G h } as mut w ( P, F ) := conv − [ h = h min G h ∪ h max [ h =0 ( w h ( P ) + hF ) ! . We note that the mutation is independent of the choice of { G h } (see [ACGK, Proposition 1]). Also, atranslation of the factor F does not affect the mutation, that is, for any u ∈ N with h w, u i = 0 we havethat mut w ( P, F ) ∼ = mut w ( P, u + F ), see [ACGK] for more details. Remark 6.3 (The mutation for the case of d = 2) . When d = 2, we choose an edge E of a latticepolygon P , and take w ∈ M ∼ = Z as a primitive inner normal vector for E . By a choice of w , we seethat w h min ( P ) = E and w h max ( P ) is either a vertex or an edge of P . Then, we take a primitive latticeelement u E ∈ N satisfying h w, u E i = 0, and define a line segment F := conv { , u E } , which is parallelto E at height 0 and has the unit lattice length. Since u E is uniquely defined up to sign, so is F . Inthis case, P admits a mutation with respect to w (equivalently we can take polytopes { G h } satisfying(6.1)) if and only if | E ∩ N | − ≥ − h min , see [KNP, Lemma 1]. We note that the mutation does notdepend on the choice of u E (and hence F ), that is, mut w ( P, F ) ∼ = mut w ( P, − F ), which means they areGL(2 , Z )-equivalent.Now, we collect fundamental properties on this mutation. Proposition 6.4 (see [ACGK, Lemma 2 and Proposition 2]) . Let the notation be the same as above. (1) If Q := mut w ( P, F ) , then we have that P = mut ( − w ) ( Q, F ) . (2) P is a Fano polytope if and only if mut w ( P, F ) is a Fano polytope. Here, we recall that a convex lattice polytope P ⊂ N R with dim P = d is called Fano if the origin iscontained in the strict interior of P , and the vertices V ( P ) of P are primitive lattice points of N .Then, we consider the mutation of a lattice polytope P in terms of the dual P ∗ of P in M . To do this,we first discuss the dual P ∗ for a polyhedron P . We consider the family of polyhedra (not necessarilyconvex polytopes) which are of the following form: P d := ( \ v ∈ S H v, ≥− k v ∩ \ v ′ ∈ T H v ′ , ≥ ⊂ N R | S, T ⊂ M, | S | , | T | < ∞ , k v ∈ Z > ) , where H v, ≥ k = { u ∈ N R | h v, u i ≥ k } for v ∈ M and k ∈ R . We note that a lattice polytope containingthe origin of N R belongs to P d but the one not containing the origin does not belong to P d since oneof the supporting hyperplanes of such polytope is of the form H v, ≥ k for some v ∈ M and some positiveinteger k .For a given P ∈ P d , we consider the dual P ∗ ⊂ M R of P defined as P ∗ := { v ∈ M R | h v, u i ≥ − u ∈ P } ⊂ M R . Then, we have the following statements.
Proposition 6.5.
Let the notation be the same as above. Then, we have that (i) P ∗ ∈ P d for any P ∈ P d , EFORMATIONS OF DIMER MODELS 29 (ii) ( P ∗ ) ∗ = P .Proof. (i) Let P ∈ P d . Since P is a polyherdon, there exist a polytope Q and a polyhedral cone C such that P = Q + C , where + denotes the Minkowski sum (see [Sch, Corollary 7.1b]). Let Q =conv( { k u , . . . , k p u p } ) and let C = cone( { u ′ , . . . , u ′ q } ). Note that we can choose u i , u ′ j from N and k i ∈ Z > because of the form of P . In what follows, we will claim that P ∗ = p \ i =1 H u i , ≥− k i ∩ q \ j =1 H u ′ j , ≥ . First, we take v ∈ P ∗ . Then, we have that h v, u i ≥ − u ∈ P . Since k i u i ∈ Q + ⊂ P ,where ∈ C denotes the origin, we see that h v, k i u i i ≥ −
1, i.e., h v, u i i ≥ − k i for each i . If there is j with h v, u ′ j i <
0, then h v, u ′ + ru ′ j i < − u ′ ∈ Q and some sufficiently large r . Moreover,we have u ′ + ru ′ j ∈ Q + C = P . This contradicts to v ∈ P ∗ , thus h v, u ′ j i ≥ j . Therefore, v ∈ T pi =1 H u i , ≥− k i ∩ T qj =1 H u ′ j , ≥ .On the other hand, we take v ∈ T pi =1 H u i , ≥− k i ∩ T qj =1 H u ′ j , ≥ . For any u ∈ P , as mentioned above, thereexist u ′ ∈ Q and u ′′ ∈ C such that u = u ′ + u ′′ . Let u ′ = P pi =1 r i k i u i , where r i ≥ P pi =1 r i = 1, andlet u ′′ = P qj =1 s j u ′ j , where s j ≥
0. By using these expressions together with the inequalities h v, u i i ≥ − k i for each i and h v, u ′ j i ≥ j , we see that h v, u i = h v, u ′ i + h v, u ′′ i = p X i =1 r i k i h v, u i i + q X j =1 s j h v, u ′ j i ≥ − p X i =1 r i = − , thus we have v ∈ P ∗ .(ii) For any u ∈ P , we have h v, u i ≥ − v ∈ P ∗ which means that P ⊂ ( P ∗ ) ∗ . On the otherinclusion, we take u ∈ N R \ P . Let P = T v ∈ S H v, ≥− k v ∩ T v ′ ∈ T H v ′ , ≥ . Then either h v, u i < − k v for some v ∈ S or h v ′ , u i < v ′ ∈ T holds. In the former case, since h v, u ′ i ≥ − k v for any u ′ ∈ P , wehave k v v ∈ P ∗ . This means that there is v ′′ := k v v ∈ P ∗ such that h v ′′ , u i < −
1, and hence u ( P ∗ ) ∗ .Similarly, in the latter case, since h rv ′ , u ′ i ≥ ≥ − u ′ ∈ P and r ≥
0, we have rv ′ ∈ P ∗ . Thisimplies that u ( P ∗ ) ∗ for sufficiently large r , and hence u ( P ∗ ) ∗ . Therefore, we obtain that ( P ∗ ) ∗ ⊂ P ,as required. (cid:3) We then define the map ϕ : M R → M R as ϕ ( v ) := v − v min w where v min := min {h v, u i | u ∈ F } . Inparticular, when d = 2 (see Remark 6.3), this map can be described as ϕ ( v ) = ( v if h v, u E i ≥ ,v − h v, u E i w if h v, u E i < . (6.2)for F = conv { , u E } . The next proposition is crucial to prove our main result Theorem 6.10. Proposition 6.6.
For any P ∈ P d , we have that ϕ ( P ∗ ) = mut w ( P, F ) ∗ . Proof.
Although this equality essentially follows from [ACGK, Proposition 4] and the discussions in[ACGK, p.12], we give a precise proof for the completeness.Let Q = mut w ( P, F ). To show ϕ ( P ∗ ) ⊂ Q ∗ , we take v ∈ P ∗ arbitrarily and consider ϕ ( v ) = v − v min w ∈ ϕ ( P ∗ ). We will claim that h v − v min w, u i ≥ − u ∈ Q . It suffices to show this for each vertex u ∈ V ( Q ). • For u ∈ V ( Q ), we assume that h w, u i ≥
0. Then we can write u = u P + h w, u P i u F for some u P ∈ V ( P ) and u F ∈ V ( F ). In particular, we have h v − v min w, u i = h v, u P i + h w, u P i ( h v, u F i − v min ) ≥ h v, u P i ≥ − . • For u ∈ V ( Q ), we assume that h w, u i <
0. For any u F ∈ V ( F ), we have u − h w, u i u F ∈ P . Hence, h v, u − h w, u i u F i ≥ −
1. In particular, h v, u i ≥ − v min h w, u i . Thus, we see that h v − v min w, u i = h v, u i − v min h w, u i ≥ − . To show Q ∗ ⊂ ϕ ( P ∗ ), we will claim that for any v ∈ Q ∗ there is v ′ ∈ P ∗ such that v = ϕ ( v ′ ). Let ∆ F be the normal fan of F in M R and let σ ∈ ∆ F be a maximal cone in ∆ F . The discussions in [ACGK,p.12] say that there exists M σ ∈ GL d ( Z ) such that the map ϕ is equal to M σ , i.e., ϕ ( v ) = vM σ . Thus,we conclude that ϕ ( v ′ ) = v for v ′ = vM − σ . (cid:3) EFORMATIONS OF DIMER MODELS 30
Example 6.7.
We consider the polygon P given in the left of the following figure, and this coincides withthe PM polygon of the dimer model given in Figure 1 (see also Figure 5). We assume that the doublecircle stands for the origin . We consider the edge E whose primitive inner normal vector is w = (1 , h min = − h max = 2. We take u E = (1 , − ∈ N which satisfies h w, u E i = 0, andconsider the line segment F = conv { , u E } . Then, we have the mutation mut w ( P, F ) of the polygon P asshown in Figure 25. This mutated polygon mut w ( P, F ) is the same as the PM polygon of ν zig p (Γ , z ) givenin Example 5.9. Also, we can see that if we take the line segment − F = conv { , − u E } , the mutatedpolygon mut w ( P, − F ) coincides with the PM polygon of ν zag q (Γ , z ) given in Example 5.9. Therefore, thePM polygon of ν zig p (Γ , z ) and that of ν zag q (Γ , z ) are isomorphic (see Remark 6.3). In the next subsection,we will show this phenomenon for general situations. mut w ( − , F ) mut w ( − , − F ) Figure 25.
An example of the mutation of P Mutations of the PM polygon are induced by deformations.
In this subsection, we showthat the mutation of the PM polygon of a consistent dimer model coincides with the PM polygon of thedeformed dimer model (see Theorem 6.10).We first recall that for any lattice polygon P there exists a reduced consistent dimer model Γ giving P as the PM polygon of Γ by Theorem 2.5. Then, we observe the relationship between the deformationdata (see Definition 4.1) and the mutation data (see Definition 6.1). Setting 6.8.
Let Γ be a reduced consistent dimer model, and ∆ Γ be the PM polygon of Γ. We take atype I zigzag path z of Γ with v := [ z ] ∈ Z . Then, by Proposition 3.5 there is the edge E of ∆ Γ whoseouter normal vector is v . Since ∆ Γ is determined up to translation, there is ambiguity concerning theposition of the origin. Thus, we fix the origin for ∆ Γ so that ∈ ∆ Γ . Let w := − v , and consider h max = h max (∆ Γ , w ) := max {h w, u i | u ∈ ∆ Γ } and h min = h min (∆ Γ , w ) := min {h w, u i | u ∈ ∆ Γ } . Now, we let r := − h min and assume that r ≤ |Z I v (Γ) | . Since the length of the line segments of E is | E ∩ N | − |Z v (Γ) | by Proposition 3.5, we have that − h min = r ≤ |Z I v (Γ) | ≤ |Z v (Γ) | = | E ∩ N | − , thus ∆ Γ admits the mutation with respect to w (see Remark 6.3). Let ℓ ( z ) := 2 n . Then, by Lemma 3.9we have that n = ℓ ( z ) / | P ∩ z | + h h ( P , P i ) , w i = | P ∩ z | + h h ( P , P ) − h ( P i , P ) , w i = | P ∩ z | + h h ( P , P ) , w i − h h ( P i , P ) , w i where P is a perfect matching on Γ, P is the reference perfect matching, and P i ∈ PM max ( z ). Since h ( P i , P ) is a lattice point on E by Lemma 3.8, we have h h ( P i , P ) , w i = h min . If P ∈ PM min ( z ), then | P ∩ z | = 0 by Lemma 3.12 and this means h h ( P , P ) , w i = h max . Thus, we have that n = h max − h min = width (∆ Γ , w ).Collectively, we consider the mutation data and the deformation data which respectively correspondeach other as in Table 1. EFORMATIONS OF DIMER MODELS 31
Mutation data Deformation data w − vh min − rh max h width (∆ Γ , w ) n Table 1.
The comparison between the mutation data and the deformation dataUsing these integers r, h , we take type I zigzag paths z , · · · , z r and the zig (resp. zag) deformationparameter X (resp. Y ) with respect to z , · · · , z r as in Definition 4.1. We then have the deformedconsistent dimer models ν zig X (Γ) = ν zig X (Γ , { z , · · · , z r } ) and ν zag Y (Γ) = ν zag Y (Γ , { z , · · · , z r } ).We determine the origin of the PM polygons ∆ ν zig X (Γ) and ∆ ν zag Y (Γ) as follows. First, there are zigzagpaths y ′ , · · · , y ′ t on ν zig X (Γ) whose slope respectively corresponds to that of zigzag paths y , · · · , y t on Γ byProposition 5.5. Then, we put ∆ ν zig X (Γ) on ∆ Γ so that the edges corresponding to y ′ , · · · , y ′ t respectivelycoincide with the ones of ∆ Γ corresponding to y , · · · , y t . We determine the origin for ∆ ν zig X (Γ) so that itis the same position as the one for ∆ Γ . Considering the zigzag paths on ν zag Y (Γ) obtained from the zigzagpaths x , · · · , x s on Γ, we can also determine the origin for ∆ ν zag Y (Γ) . Remark 6.9.
In Setting 6.8, we assumed that r ≤ |Z I v (Γ) | for defining the deformation data. As wementioned in Remark 4.2, even if there does not exist enough type I zigzag paths, we sometimes make atype II zigzag path type I without changing the PM polygon by using the mutation of dimer models (seeAppendix A). Moreover, it is known that for a given lattice polygon P there exists an isoradial dimermodel giving P as the PM polygon by [Gul], in which case all zigzag paths are type I (see Definition 3.4),and hence |Z I v (Γ) | = |Z v (Γ) | . Thus, if − h min ≤ | E ∩ N | − − h min = r ≤ |Z I v (Γ) | = | E ∩ N | − E of ∆ Γ given in Setting 6.8, we take a primitive lattice element u E ∈ N such that h w, u E i = 0. Here, there are two choices of u E and we fix u E as follows. We recall the primitive latticeelement h ( P ′ z , P z ) given in Settings 5.7, which satisfies h [ z ] , h ( P ′ z , P z ) i = 0. Thus, we set u E := h ( P ′ z , P z ),in which case we have that h w, u E i = h− [ z ] , u E i = 0. Then, we set the line segment F := conv { , u E } .Under these settings, we have our main theorem as follows. Theorem 6.10.
Let the notation be the same as in Definition 4.1, 4.3, 4.5, and Setting 6.8 (see alsoTable 1). Then, we have that mut w (∆ Γ , F ) = ∆ ν zig X (Γ , { z , ··· ,z r } ) , mut w (∆ Γ , − F ) = ∆ ν zag Y (Γ , { z , ··· ,z r } ) . Proof.
We prove the first equation, and the other one follows from a similar argument.First, we show that ϕ (∆ ∗ Γ ) = ∆ ∗ ν zig X (Γ , { z , ··· ,z r } ) where ϕ is the map given in (6.2).Let E := E, E , · · · , E m be edges of ∆ Γ , and we assume that these are ordered cyclically in the anti-clockwise direction. As we mentioned in Setting 6.8, we suppose that ∈ ∆ Γ . Let w := w, w · · · , w m beinner normal vectors corresponding to E , · · · , E m respectively (see Figure 26). Also, we let v i = − w i for i = 1 , · · · , r , which is the outer normal vectors corresponding to E i . We then consider u ∈ ∆ Γ such that h w , u i = h max (∆ Γ , w ) = h , that is, we consider w h max (∆ Γ ) which is either a vertex or an edge of ∆ Γ . If w h max (∆ Γ ) is an edge, we easily see that it is parallel to E , in which case we may write E a := w h max (∆ Γ )for some 1 < a < m . If w h max (∆ Γ ) is a vertex, we set the edges intersecting at w h max (∆ Γ ) as E a − , E a +1 and set E a = ∅ where 1 < a < m . Here, we recall that by Proposition 3.5, for each E i = ∅ there existzigzag paths on Γ such that the slopes coincide with v i , and the set of such zigzag paths is denoted by Z v i = Z v i (Γ). EFORMATIONS OF DIMER MODELS 32 w m w = ww w a − w a w a +1 u E Figure 26.
The PM polygon ∆ Γ and its inner normal vectors (the case where the originis contained in the strict interior of ∆ Γ ).First, we consider the edge E and zigzag paths in Z v = Z ( − w ) . By definition, we have that h− v , u E i = 0 and { z , · · · , z r } ⊆ Z I( − w ) ⊆ Z ( − w ) . If |Z ( − w ) | > r , then there exists a zigzag pathin Z ( − w ) that is not in { z , · · · , z r } . If E a = ∅ , we have zigzag paths in Z v a . Since E and E a areparallel, v and v a are linearly dependent, and hence h v a , u E i = 0. Then, we see that zigzag paths in Z v a do not intersect with a type I zigzag path z satisfying [ z ] = v in the universal cover (see Lemma 3.15).Next, we consider the edges E , · · · , E a − of ∆ Γ and zigzag paths in Z := Z v ∪ · · · ∪ Z v a − . We seethat v i with i = 2 , · · · , a − h− v i , u E i < E , · · · , E a − . By the sameargument as in the proof of Proposition 4.12, we see that the zigzag paths in Z intersect with a type Izigzag path z satisfying [ z ] = v precisely once in the universal cover (see Lemma 3.15), and especiallythey intersect with z in Zag ( z ). Thus, we have that Z = { x , · · · , x s } , and for each i = 2 , ..., a − v i satisfies v i = [ x j ] for some j = 1 , · · · , s .Then, we consider the edges E a +1 , · · · , E m of ∆ Γ and zigzag paths in Z := Z v a +1 ∪ · · · ∪ Z v m . Wesee that v i with i = a + 1 , · · · , m satisfies h− v i , u E i > E a +1 , · · · , E m . Bya similar argument as above, we see that the zigzag paths in Z intersect with a type I zigzag path z satisfying [ z ] = v precisely once in the universal cover, and especially they intersect with z in Zig ( z ).Thus, we have that Z = { y , · · · , y t } , and for each i = a + 1 , · · · , m the vector v i satisfies v i = [ y k ] forsome k = 1 , · · · , t .Collectively, we see that a zigzag path of Γ takes one of the following forms: • z , · · · , z r , • z ′ , · · · , z ′ r ′ contained in Z ( − w ) \{ z , · · · , z r } for w = − v with h w, u E i = 0 if |Z ( − w ) | > r , • z ′′ , · · · , z ′′ r ′′ contained in Z w for w = − v with h w, u E i = 0 if E a = ∅ , • x j where j = 1 , · · · , s , in which case it satisfies h− [ x j ] , u E i < • y k where k = 1 , · · · , t , in which case it satisfies h− [ y k ] , u E i > Γ by Proposition 3.5. Precisely, if∆ Γ contains the origin as an interior lattice point, then H − [ ζ ] , ≥− k ζ = { u ∈ N R | h− [ ζ ] , u i ≥ − k ζ } is the supporting hyperplane of ∆ Γ for any zigzag path ζ of Γ and a certain positive integer k ζ . If theorigin lies on the boundary of ∆ Γ , k ζ is replaced by 0 for the zigzag paths corresponding to the edgesthat contain . By Proposition 6.5 and its proof, ∆ ∗ Γ can be denoted by ∆ ∗ Γ = Q + C where Q is a polygonand C is a polyhedral cone. Since the set of the slopes of zigzag paths of Γ coincides with { v , · · · , v m } ifwe identify the same slopes, we see that the set { u , · · · , u p , u ′ , · · · , u ′ q } , which generates Q and C in theproof of Proposition 6.5, is given by { w , · · · , w m } in our situation. In what follows, we assume that iscontained in the strict interior of ∆ Γ , in which case ∆ ∗ Γ = Q and Q = conv( { k w , · · · , k m w m } ) for somepositive integer k i giving the supporting hyperplane H w i , ≥− k i of ∆ Γ where i = 1 , · · · , m . We remarkthat k a w a appears in the above generating set if E a = ∅ . Since h w i , u E i ≥ i = 1 , a, a + 1 , · · · , m and h w i , u E i < i = 2 , · · · , a −
1, we see that ϕ (∆ ∗ Γ ) = conv( { k w , k w ′ , · · · , k a − w ′ a − , k a w a (if E a = ∅ ) , k a +1 w a +1 , · · · , k m w m } ) (6.3)where w ′ i := w i − h w i , u E i w for i = 2 , · · · , a −
1. We also note that when E a = ∅ , we can take the positiveinteger k a so that the line { u ∈ N R | h v , u i = − k a } , which is parallel to E , passes through the vertex of∆ Γ that is the intersection of E a − and E a +1 . By the choice of k a , we have that h k a v , u i ≥ − u ∈ ∆ Γ , thus k a v = k a ( − w ) ∈ ∆ ∗ Γ and hence k a v = k a ( − w ) ∈ ϕ (∆ ∗ Γ ). EFORMATIONS OF DIMER MODELS 33
We then consider the deformed dimer model ν zig X (Γ) = ν zig X (Γ , { z , · · · , z r } ). By Observation 5.1, thelift of a zigzag path with the form z ′ i or z ′′ i on the universal cover is contained in some irrelevant part,and hence it does not change even if we apply the deformation. Also, by Proposition 5.8, we have thezigzag paths x ′ , · · · , x ′ s on ν zig X (Γ) satisfying − [ x ′ j ] = − [ x j ] − h [ x j ] , h ( P ′ z , P z ) i [ z ] = w i − h w i , u E i w for j = 1 , · · · , s and some i = 2 , · · · , a −
1. Furthermore, by Proposition 5.5, we have the zigzag paths y ′ , · · · , y ′ t on ν zig X (Γ) satisfying − [ y ′ k ] = − [ y k ] = w i for k = 1 , · · · , t and some i = a + 1 , · · · , m . Thus, we see that the zigzag paths x j , y k vary as they satisfythe condition (6.2) when we apply the deformation ν zig X to Γ. In addition, we have the zigzag path withthe form z i,j defined in (zig-3). Thus, the zigzag paths on the consistent dimer model ν zig X (Γ) are { z ′ i } ≤ i ≤ r ′ (if |Z ( − w ) | > r ) , { z ′′ i } ≤ i ≤ r ′′ (if E a = ∅ ) , { x ′ j } ≤ j ≤ s , { y ′ k } ≤ k ≤ t , and { z i,j } ≤ i ≤ r ≤ j ≤ p i . By the description of their slopes and Proposition 3.5, we see that the inner normal vectors of ∆ ν zig X (Γ) are { w , w ′ , · · · , w ′ a − , w a = − w , w a +1 , · · · , w m } , and these vectors give the supporting hyperplanes of ∆ ν zig X (Γ) just like ∆ Γ as above. Here, w appears inthe above set if |Z ( − w ) | = |Z v | > r , but we always have that k w ∈ ∆ ∗ ν zig X (Γ) by the same argument aswe used for showing k a ( − w ) ∈ ∆ ∗ Γ above. Whereas w a certainly appears since [ z i,j ] = − [ z i ] = − w = w a (see Lemma 5.4). Thus, we have that∆ ∗ ν zig X (Γ) = conv( { k w , k w ′ , · · · , k a − w ′ a − , k a w a , k a +1 w a +1 , · · · , k m w m } ) . By the description (6.3) and the fact that k w and k a w a = k a ( − w ) are contained in both ϕ (∆ ∗ Γ ) and∆ ∗ ν zig X (Γ) , we see that ϕ (∆ ∗ Γ ) = ∆ ∗ ν zig X (Γ) .The case where the origin lies on the boundary of ∆ Γ can be proved by a similar argument if weconsider the hyperplane { u ∈ N R | h− [ ζ ] , u i ≥ } instead of { u ∈ N R | h− [ ζ ] , u i ≥ − k ζ } for the zigzagpaths corresponding to the edges that contain , in which case − [ ζ ] will be a generator of a polyhedralcone C .By Proposition 6.5(2) and 6.6, we conclude that mut w (∆ Γ , F ) = ∆ ν zig X (Γ) . (cid:3) Since mut w (∆ Γ , F ) ∼ = mut w (∆ Γ , − F ) (see Remark 6.3), we immediately have the following. Corollary 6.11.
Let the notation be the same as in Definition 4.1, 4.3 and 4.5. Then, we have that ∆ ν zig X (Γ , { z , ··· ,z r } ) ∼ = ∆ ν zag Y (Γ , { z , ··· ,z r } ) , that is, they are GL(2 , Z ) -equivalent. We then show that the deformations at zig and at zag are mutually inverse operations on the level ofthe associated PM polygon.
Setting 6.12.
Let ν zig X (Γ , { z , · · · , z r } ) be the reduced consistent dimer model defined in Definition 4.3.We consider the following deformation data for ν zig X (Γ , { z , · · · , z r } ). We consider a type I zigzag path z i,j which satisfies [ z i,j ] = − v =: w (see Proposition 5.4). Since { z i,j } ≤ i ≤ r ≤ j ≤ p i are the subset of type I zigzagpaths, we have that |Z I w ( ν zig X (Γ)) | ≥ P ri =1 p i = h . Then, we take a subset { z ′ , · · · , z ′ h } of type I zigzagpaths of ν zig X (Γ), and do the same procedure as in Definition 4.1(4)–(7), and we especially have the zagdeformation parameter Y ′ = { Y ′ , · · · , Y ′ h } of the weight q ′ = ( q ′ , · · · , q ′ h ) with P hi =1 q ′ i = r . We can dothe same arguments for the case of ν zag Y (Γ , { z , · · · , z r } ). In particular, we take a subset { z ′′ , · · · , z ′′ h } oftype I zigzag paths on ν zag Y (Γ) and have the zig deformation parameter X ′ = { X ′ , · · · , X ′ h } of the weight p ′ = ( p ′ , · · · , p ′ h ) with P hi =1 p ′ i = r . Corollary 6.13.
Let the notation be the same as Setting 6.12. Then, we have that ∆ ν zag Y′ ( ν zig X (Γ , { z i } ri =1 ) , { z ′ ℓ } hℓ =1 ) = ∆ Γ and ∆ ν zig X′ ( ν zag Y (Γ , { z i } ri =1 ) , { z ′′ ℓ } hℓ =1 ) = ∆ Γ . Proof.
This follows from Proposition 6.4(1) and Theorem 6.10. (cid:3)
EFORMATIONS OF DIMER MODELS 34
As we mentioned in Section 1, the mutation of a Fano polygon is important from the viewpoint ofmirror symmetry and the classification of Fano manifolds. To observe the mutations of Fano polygonsby using the deformations of dimer models, we assume that the polygons ∆ Γ , ∆ ν zig X (Γ , { z , ··· ,z r } ) and∆ ν zag Y (Γ , { z , ··· ,z r } ) contain the origin in their strict interiors. Then, we have the following corollary. Corollary 6.14.
Let the notation be the same as above. Then, we see that ∆ Γ is Fano if and only if ∆ ν zig X (Γ , { z , ··· ,z r } ) ( resp. ∆ ν zag Y (Γ , { z , ··· ,z r } ) ) is Fano.Proof. This follows from Proposition 6.4(2) and Theorem 6.10. (cid:3)
Appendix A. Mutations of dimer models
In this section, we introduce the other operation called the mutation of dimer models . From a viewpointof physics, dimer models and their mutations correspond to quiver gauge theories and Seiberg duality.The mutation of dimer models can be defined for each quadrangle face of a dimer model, and the operationcalled spider move (see e.g., [GK, Boc3]), which is the inverse operation shown in Figure 27, is the mainingredient for defining the mutation. spider move
Figure 27.
We note that there are two types of the spider move (and hence the mutation) depending on the colorof the two interior nodes. Thus, by replacing black nodes by white ones and vice versa, we define theother type. We now describe the black one.
Definition A.1 (The mutation of dimer models) . Let Γ be a dimer model. We pick a quadrangle face f ∈ Γ . Then, the mutation of Γ at f , denoted by µ f (Γ), is the operation consisting of the followingprocedures:(I) Consider black nodes appearing on the boundary of f . If there exist black nodes that are not3-valent, we apply the split moves to those nodes and make them 3-valent as shown in Figure 28.(II) Then, we apply the spider move to f (see Figure 27).(III) If the resulting dimer model contains 2-valent nodes, we remove them by applying the join moves. split move f f Figure 28.
Applying the mutation at a quadrangle face, we obtain the new dimer model from a given one,although it sometimes induces the isomorphic one. We also remark that the mutation is an inverseoperation, that is, µ f ( µ f (Γ)) = Γ holds. We say that dimer models Γ and Γ ′ are mutation equivalent ifthey are transformed into each other by repeating the mutation of dimer models. Moreover, we also seethat the join, split and spider moves do not change the slopes of zigzag paths and preserve conditions inDefinition 3.3, thus we have the following. EFORMATIONS OF DIMER MODELS 35
Proposition A.2.
The mutation of dimer models turns consistent dimer models into consistent dimermodels associated with the same lattice polygon.
In addition, it has been conjectured that all consistent dimer models associated with the same latticepolygon are mutation equivalent, but it is still open in general (see [Boc3, pp396–397]). We note thatpartial answers were given in several papers, see e.g., [Boc2, GK, HS, Nak1].
Remark A.3.
The mutation of dimer models is also defined as the dual of the mutation of a quiver withpotential (= QP ) in the sense of [DWZ] (see e.g., [Boc2, subsection 7.2], [Nak1, Section 4]). Although wecan consider the mutation of a QP for any vertex of the quiver having no loops and 2-cycles, the resultingQP is not necessarily the dual of a dimer model. To make the resulting one the dual of a dimer model,we need the assumption that the mutated vertex has two incoming (equivalently, two outgoing) arrows,which is equivalent to the face of a dimer model corresponding to such a vertex is quadrangle.In the theory of deformations of dimer models, type I zigzag paths are important. We can use themutations for making a type II zigzag path type I as in the example below. Example A.4.
We consider the following dimer model Γ (the left one). Since the face 0 is a quadrangle,we can apply the mutation and have the dimer model µ (Γ) as follows. µ We can see that the zigzag path on Γ whose slope is ( − ,
1) or (1 , −
1) is type II. On the other hand,we see that µ (Γ) is isoradial, and hence all zigzag paths are type I. Appendix B. Large examples
As we mentioned in Section 4, we sometimes skip the operations (zig-4) and (zig-5) (resp. (zag-4) and(zag-5)) when we define the deformation ν zig X (Γ , { z , · · · , z r } ) (resp. ν zag Y (Γ , { z , · · · , z r } )) of a consistentdimer model Γ. However, as the following example shows, (zig-4) and (zig-5) (resp. (zag-4) and (zag-5))are indispensable to define the deformations that are compatible with the mutations of polygons as shownin Theorem 6.10. For example, we often encounter such a situation when we consider a consistent dimermodel whose PM polygon is relatively large. Example B.1.
We consider the lattice polygon P shown in the left of Figure 29. In particular, weassume that the double circle stands for the origin . We consider the edge E whose primitive innernormal vector is w = (0 , − h min = − h max = 1. We take u E = ( − ,
0) whichsatisfies h w, u E i = 0, and consider the line segment F = conv { , u E } . Then, we have the mutation mut w ( P, F ) of the polygon P as shown in the right of Figure 29. mut w ( − , F ) Figure 29.
The lattice polygons P and mut w ( P, F ) for w = (0 , −
1) and F = conv { , ( − , } We then consider the dimer model Γ shown in Figure 30. The zigzag paths on this dimer model Γare Figure 31. Thus, we can check that Γ is consistent and the PM polygon ∆ Γ coincides with P byProposition 3.5. EFORMATIONS OF DIMER MODELS 36
Figure 30.
A consistent dimer model Γ whose PM polygon coincides with P z z z z z ′ z ′ z ′ z ′ y y y y x x x X Figure 31.
The list of zigzag paths of ΓThen, we consider the deformation of Γ that induces mut w ( P, F ) as the PM polygon (see Theorem 6.10).To do this, we first fix the deformation data (see Definition 4.1) as follows. First, the zigzag path z i onΓ is type I with ℓ ( z i ) = 8, and its slope is [ z i ] = (0 ,
1) = − w for i = 1 , · · · ,
4. Let r := − h min = 3 and h := h max = 1 (see Table 1). In particular, these satisfy r = 3 < |Z I( − w ) (Γ) | = 4 and r + h = ℓ ( z i ) / { z , z , z } , and consider the deformation ν zig X (Γ , { z , z , z } ) ofΓ at zig of { z , z , z } with respect to X , where X is the zig deformation parameter defined as follows.To define X , we focus on x , x , x shown in Figure 31, which are the zigzag paths intersected with z i at some zags of z i . They satisfy m := | x ∩ z i | = 1, m := | x ∩ z i | = 1, and m := | x ∩ z i | = 2for any i , thus we consider the set of sub-zigzag paths { x = x (1)1 , x = x (1)2 , x (1)3 , x (2)3 } . Here, we fix anintersection of z and x marked by X in Figure 31 as the starting edge of x (1)3 . We then set the zigdeformation parameter X := { X , X , X } with respect to { z , z , z } , where X = { x ∩ z = x (1)1 ∩ z } , X = { x ∩ z = x (1)2 ∩ z } and X = { x (1)3 ∩ z , x (2)3 ∩ z } , in which case the weight of X is p = (1 , , mut w ( P, F ), thuswe can not obtain Theorem 6.10 without the operations (zig-4) and (zig-5), which is different from thecase of Definition 4.8. Thus, we apply (zig-4), that is, we insert some bypasses. Then, we have the dimermodel ν zig X (Γ) = ν zig X (Γ , { z , z , z } ) as shown in the right of Figure 32. By Proposition 4.12, this dimermodel is non-degenerate, but it is not consistent. Indeed, we can see that some nodes appearing on thezigzag path x ′ shown in Figure 33 are not properly ordered (see Definition 3.3(4)). EFORMATIONS OF DIMER MODELS 37
Figure 32.
The dimer model obtained by applying (zig-1)–(zig-3) to Γ (left), and thedimer model ν zig X (Γ , { z , z , z } ) obtained by applying (zig-1)–(zig-4) to Γ (right) x ′ x ′ x ′ Figure 33.
The zigzag paths x ′ , x ′ , x ′ of ν zig X (Γ , { z , z , z } )By Lemma 5.6, some edges constituting the zigzag paths x ′ , x ′ , x ′ shown in Figure 33 might beremoved by the operation (zig-5). Thus, paying attention to these zigzag paths, we apply (zig-5) to ν zig X (Γ) and make it consistent. Namely, if we remove pairs of edges (1)–(5) shown in the type A (left)of Figure 34, which are the intersections of pairs of zigzag paths on the universal cover that intersectwith each other in the same direction more than once, from ν zig X (Γ) with this order. Then, we havethe dimer model shown in the left of Figure 35, and applying (zig-6) we finally have the dimer model ν zig X (Γ , { z , z , z } ) shown in the right of Figure 35.Whereas there are other ways to remove edges. For example, if we remove pairs of edges (1)–(5) shownin the type B (right) of Figure 34, then we have another dimer model shown in the left of Figure 36, andapplying (zig-6) we have the dimer model ν zig X (Γ , { z , z , z } ) shown in the right of Figure 36. (1) (1)(2) (2)(3) (3) (4) (4)(5) (5) (1) (1)(2) (2)(3) (3)(4) (4)(5) (5) Type A Type B
Figure 34.
The two ways to remove edges from ν zig X (Γ , { z , z , z } ) by (zig-5) EFORMATIONS OF DIMER MODELS 38 (zig-6)
Figure 35.
The dimer model ν zig X (Γ , { z , z , z } ) obtained from the type A of Figure 34(zig-6) Figure 36.
The dimer model ν zig X (Γ , { z , z , z } ) obtained from the type B of Figure 34Let Γ A (resp. Γ B ) be the deformed dimer model shown in the right of Figure 35 (resp. Figure 36).We can check that Γ A and Γ B are not isomorphic, but they are mutation equivalent. Indeed, by applyingthe mutations of Γ B at the faces 1 , · · · ,
10 with this order (see Figure 37), we can recover Γ A . Here,we recall that the mutation of a dimer model can be defined at a quadrangle face. Although some facesindexed by { , · · · , } are not quadrangle, such faces will be a quadrangle in the process of these seriesof mutations.The PM polygon of the dimer models Γ A and Γ B are the same (see Proposition A.2), and it coincideswith the lattice polygon shown in the right of Figure 29 by Theorem 6.10. The mutation µ f where f = 1 , · · · , Figure 37.
The mutations of Γ B (left) at the faces 1 , · · · ,
10 induce Γ A (right) Appendix C. Remarks on deformations of hexagonal and square dimer models
As we mentioned in Remark 4.9, we can skip the operations (zig-4) and (zig-5) (resp. (zag-4) and(zag-5)) for the case of hexagonal and square dimer models as we will see below. Here, we say that adimer model Γ is hexagonal (resp. square ) if Γ is homotopy equivalent to a dimer model whose faces areall regular hexagon (resp. square) dimer model. We note that hexagonal and square dimer models areisoradial. These dimer models have been studied in several papers, especially we have the followings.
EFORMATIONS OF DIMER MODELS 39
Proposition C.1 (see e.g., [IN, Nak2, UY]) . Let Γ be a consistent dimer model. Then, we have thefollowings. (1) Γ is a hexagonal dimer model if and only if the PM polygon ∆ Γ is a triangle. (2) If Γ is a square dimer model, then the PM polygon ∆ Γ is a parallelogram. For these nice classes of dimer models, we may skip the operations (zig-4) and (zig-5) (or (zag-4) and(zag-5)) when we apply the deformation.
Proposition C.2.
Let the notation be the same as in Definition 4.1, 4.3, and 4.5. In addition, weassume that a dimer model Γ is hexagonal or square, in which case Γ is isoradial. Then, the deformation ν zig X (Γ , { z , · · · , z r } ) is defined by the operations (zig-1)–(zig-3) and (zig-6) . Similarly, the deformation ν zag Y (Γ , { z , · · · , z r } ) is defined by the operations (zag-1)–(zag-3) and (zag-6) .Proof. We prove the case of the deformation at zig, and the case of the deformation at zag is similar.The zigzag paths z , · · · , z r have the same slope and such a slope is the outer normal vector of an edgeof the PM polygon ∆ Γ by Proposition 3.5.(1) We assume that Γ is a hexagonal dimer model. Then, ∆ Γ is a triangle by Proposition C.1(1). Let e , e , e be edges of ∆ Γ which are ordered cyclically in the anti-clockwise direction. We may assumethat the slopes of z , · · · , z r are the outer normal vector of e . We then consider the zigzag paths x , · · · , x s (resp. y , · · · , y t ) intersected with z i at zags (resp. zigs) of z i . Since Γ is isoradial, it isproperly ordered. Thus, by Proposition 3.5 we see that [ x ] = · · · = [ x s ] (resp. [ y ] = · · · = [ y t ]), and[ x j ] (resp. [ y k ]) is the outer normal vector of e (resp. e ).(2) We assume that Γ is a square dimer model. Then, ∆ Γ is a parallelogram by Proposition C.1(2). Let e , e , e , e be edges of ∆ Γ which are ordered cyclically in the anti-clockwise direction. In particular, { e , e } , { e , e } are pairs of edges that are parallel. We may assume that the slopes of z , · · · , z r arethe outer normal vector of e , in which case the zigzag paths having the slope − [ z i ] correspond to e . Then, in a similar way as above, we have the zigzag paths x , · · · , x s (resp. y , · · · , y t ) such that[ x j ] (resp. [ y k ]) is the outer normal vector of e (resp. e ).In both cases, we see that any pair of zigzag paths in x , · · · , x s (resp. y , · · · , y t ) does not have theintersection on the universal cover by Lemma 3.15 because Γ is isoradial.Then, we consider ν zig X (Γ , { z , · · · , z r } ) for the case of r = 1 (see Definition 4.8 for the case of r = 1).By Lemma 5.6 and the fact that there is no intersection between x , · · · , x s , we see that the edges removedby (zig-5) are bypasses added in (zig-4). Furthermore, by Observation 5.2 and 5.3 a zigzag path passingthrough a bypass takes the form e x ′ j . Since the slopes of x , · · · , x s are all the same in our situation,those of x ′ , · · · , x ′ s are all the same (see Proposition 5.8). Thus, any bypass on the universal cover of ν zig X (Γ , { z , · · · , z r } ) is either(i) a self-intersection of a zigzag path e x ′ j , or(ii) the intersection of a pair of zigzag paths e x ′ j , e x ′ j ′ with [ x ′ j ] = [ x ′ j ′ ].We also see that an edge which is either (i) or (ii) is certainly a bypass, because of Observation 5.3 andthe fact that such an intersection can not appear in the irrelevant part. Moreover, if a bypass is theintersection of zigzag paths e x ′ j and e x ′ j ′ , then they have another intersection because [ x j ] = [ x j ′ ], and suchan intersection is also a bypass. Thus, if there exists a bypass that can not be removed by (zig-5), thenit prevents the consistency condition. Therefore, we can remove all bypasses added in (zig-4) by using(zig-5), and hence we may skip these operations. (cid:3) Acknowledgement.
The authors thank Alexander Kasprzyk for valuable lectures and discussions onmutation of Fano polygons. The first author is supported by JSPS Grant-in-Aid for Young Scientists (B)17K14177. The second author is supported by World Premier International Research Center Initiative(WPI initiative), MEXT, Japan, and JSPS Grant-in-Aid for Young Scientists (B) 17K14159.
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Department of Mathematics, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-Ku, Ky-oto, Japan, 603-8555
E-mail address : [email protected] (Y. Nakajima) Kavli Institute for the Physics and Mathematics of the Universe (WPI), UTIAS, The Univer-sity of Tokyo, Kashiwa, Chiba 277-8583, Japan
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