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Higher pentagram maps, weighted directed networks, and cluster dynamics 2 1 0 2 Michael Gekhtman,∗ Michael Shapiro,† Serge Tabachnikov,‡ n and Alek Vainshtein§ a J January 4, 2012 2 ] A Abstract Q Thepentagrammapwasextensivelystudiedinaseriesofpapersbyby . h V.Ovsienko,R.SchwartzandS.Tabachnikov. Itwasrecentlyinterpreted t by M. Glick as a sequence of cluster transformations associated with a a m specialquiver. UsingcompatiblePoissonstructuresinclusteralgebrasand Poisson geometry of directed networks on surfaces, we generalize Glick’s [ constructiontoincludethepentagrammapintoafamilyofgeometrically 3 meaningful discrete integrable maps. v 2 7 1 Introduction 4 0 The pentagram map was introduced by R. Schwartz about 20 years ago [25]. . 0 Recently, it has attracted a considerable attention: see [11, 16, 17, 20, 21, 22, 1 26, 27, 28, 29, 30] for various aspects of the pentagrammap and relatedtopics. 1 On plane polygons, the pentagram map (depicted in Fig. 1) acts by drawing 1 the diagonals that connect second-nearest vertices of a poygon P and forming : v a new polygon T(P) whose vertices are their consecutive intersection points. i X The pentagram map commutes with projective transformations, so it acts on the projective equivalence classes of polygons in the projective plane. r a The pentagram map can be extended to a larger class of twisted polygons. A twisted n-gon is a sequence of points V ∈RP2 in general position such that i Vi+n =M(Vi) for all i∈Z and some fixed element M ∈PGL(3,R), called the monodromy. A polygon is closed if the monodromy is the identity. Denote by P thespaceofprojectiveequivalenceclassesoftwistedn-gons;thisisavariety n ∗Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA; e-mail: [email protected] †Department of Mathematics, Michigan State University, East Lansing, MI 48823, USA; e-mail: [email protected] ‡Department of Mathematics, Pennsylvania State University, UniversityPark, PA 16802, USA;e-mail: [email protected] §Department of Mathematics and Department of Computer Science, University of Haifa, Haifa,MountCarmel31905,Israel;e-mail: [email protected] 1 Figure 1: Pentagrammap of dimension 2n. Denote by T :P →P the pentagram map. (The ith vertex n n of the image is the intersection of diagonals (Vi,Vi+2) and (Vi+1,Vi+3).) OnehasacoordinatesystemX1,Y1,...,Xn,YninPnwhereXi,Yiaretheso- calledcornerinvariantsassociatedwithithvertex,discreteversionsofprojective curvature,see[27,20,21]. Inthesecoordinates,thepentagrammapisarational transformation Xi∗ =Xi 11−−XXii−+11YYii+−11, Yi∗ =Yi+1 1−1X−iX+2iYYii+2 (1) (the indices are takenmodn). Schwartz[27]observedthat the pentagrammap commutes with the R∗ action on P given by (X ,Y )7→(tX ,t−1Y ). n i i i i The main feature of the pentagram map is that it is a discrete completely integrable system (its continuous n →∞ limit is the Boussinesq equation, one of the best known completely integrable PDEs). Specifically, the pentagram map has 2[n/2]+2 integrals constructed in [27]; these integrals are polynomial in the X,Y-coordinates and they are algebraically independent. The space P n has a T-invariant Poisson structure, introduced in [20, 21]. The corank of this Poissonstructure equals 2or 4,accordingas n is odd oreven, andthe integrals are in involution. This provides Liouville integrability of the pentagram map on the space of twisted polygons. Complete integrability on the smaller space C of closed n-gons is provedin n [22]. F. Soloviev [30] established algebraic-geometricintegrability of the penta- grammapbyconstructingitsLax(zerocurvature)representation;hisapproach works both for twisted and closed polygons. Itwasshownin[27]thatthepentagrammapwasintimatelyrelatedtotheso- called octahedral recurrence. It was conjectured in [20, 21] that the pentagram map was related to cluster transformations. This relation was discovered and explored by Glick [12] who proved that the pentagram map, acting on the quotient space P /R∗, is described by coefficient dynamics [6] (also known as n τ-transformations, see Chapter 4 in [8]) for a certain cluster structure. In this research announcement and the forthcoming detailed paper, we ex- tend andgeneralizeGlick’sworkby including the pentagrammapinto a family ofdiscretecompletelyintegrablesystems. OurmaintoolisPoissongeometryof 2 weighteddirectednetworksonsurfaces. The ingredientsnecessaryfor complete integrability–invariantPoissonbrackets,integralsofmotionininvolution,Lax representation – are recovered from combinatorics of the networks. Postnikov [23]introducedsuchnetworksin the caseofa disk and investigatedtheir transformations and their relation to cluster transformations;most of his results are local, and hence remain valid for networks on any surface. Poisson properties ofweighteddirectednetworksinadiskandtheir relationtor-matrixstructures on GL are studied in [7]. In [9] these results were further extended to net- n works in an annulus and r-matrix Poissonstructures on matrix-valued rational functions. Applications of these techniques to the study of integrable systems canbe foundin [10]. Adetailed presentationof the theory ofweighteddirected networks from a cluster algebra perspective can be found in Chapters 8–10 of [8]. Our integrable systems depend on one discrete parameter k ≥ 2. The case k = 3 corresponds to the pentagram map. For k ≥ 4, we give our integrable systems a geometric interpretation as pentagram-like maps involving deeper diagonals. Ifk =2andthegroundfieldisC,wegiveageometricinterpretation in terms of circle patterns [24, 1]. While working on this manuscript, we were informed by V. Fock that in his current project with A. Marshakov, closely related integrable systems are studied using the approach of [13]. 2 Generalized Glick’s quivers and the (p,q)-dynamics For any integer n ≥ 2 and any integer k, 2 ≤ k ≤ n, consider the quiver (an oriented multigraph without loops) Q defined as follows: Q is a bipartite k,n k,n graph on 2n vertices labeled p1,...,pn and q1,...,qn (the labeling is cyclic, so that n+1 is the same as 1). The graph is invariant under the shift i 7→ i+1. Each vertex has two incoming and two outgoing edges. The number k is the “span” of the quiver, the distance between two outgoing edges from a p-vertex, see Fig. 2 where r = [k/2]−1 and r+r′ = k−2 (in other words, r′ = r for k even and r′ =r+1 for k odd). For k =3, we have Glick’s quiver [11]. p i q q q q i−r−1 i−r i+r’ i+r’+1 Figure 2: The quiver Q k,n We consider the cluster structure associated with the quiver Q . Choose k,n variablesp=(p1,...,pn)andq=(q1,...,qn)asτ-coordinates(see[8],Chapter 3 4), and consider cluster transformations corresponding to the quiver mutations atthep-vertices. Thesetransformationscommute,andweperformthemsimul- taneously. This leads to the transformation (the new variables are marked by asterisque) p∗ = 1, q∗ =q (1+pi−r′−1)(1+pi+r+1)pi−r′pi+r, (2) i pi i i (1+pi−r′)(1+pi+r) (seeLemma4.4of[8]fortheexchangerelationsforτ-coordinates). Theresulting quiver is identical to Q with the letters p and q interchanged. Thus we k,n compose transformation (2) with the involution {p ↔ q }n for k even, or i i i=1 with the transformation {pi 7→ qi+1,qi 7→ pi}ni=1 for k odd, and arrive at the transformation that we denote by T : k q∗ = 1, p∗ =q (1+pi−r−1)(1+pi+r+1)pi−rpi+r, k even, i pi i i (1+pi−r)(1+pi+r) (3) q∗ = 1 , p∗ =q (1+pi−r−2)(1+pi+r+1)pi−r−1pi+r, k odd. i pi−1 i i (1+pi−r−1)(1+pi+r) The difference in the formulas is due to the asymmetry between left and right in the enumeration of vertices in Fig. 2 for odd k, when r′ 6=r. Let us equip the (p,q)-space with a Poisson structure compatible with the cluster structure, see [8]. Denote by A = (a ) the 2n× 2n skew-adjacency ij matrix of Q , assuming that the first n rows and columns correspond to p- k,n vertices. Then we put {v ,v } = a v v , where v = p for 1 ≤ i ≤ n and i j ij i j i i vi =qi−n for n+1≤i≤2n. Theorem 2.1. (i) The above Poisson structureis invariant under the map T . k (ii) The function p q is an integral of the map T . Besides, it is Casimir, i i k and hence the PoissQon structure and the map descend to the hypersurfaces p q =const. i i Q c We denote by T the restriction of T to the hypersurface p q = c. In k k i i 1 what follows, we shall be concerned only with T , which we shoQrthand to T . k k Note that T1 is the pentagram map on P /R∗ considered by Glick [11]. 3 n Let us consider an auxiliary transformation D given by {p 7→ 1/q ,q 7→ k i i i 1/pi}ni=1 for k even and {pi 7→ 1/qi+1,qi 7→ 1/pi}ni=1 for k odd. Then Tk and its inverse are related via −1 D ◦T ◦D =T . (4) k k k k Remark2.2. Alongwiththep-dynamicswhenthemutationsareperformedat the p-vertices of the quiver Q , one may consider the respective q-dynamics. k,n Wenotethattheq-dynamicsisessentiallythesameasp-dynamics. Namely,the q-dynamicsforagivenvalueofkcorrespondstothep-dynamicsfork′ =n+2−k. This is illustrated in Fig. 3. 4 p p p p i i i i k=3 k=2 k=n k=n−1 Figure 3: The quivers Q for various values of k k,n 3 Weighted directed networks and the (x,y)-dynamics Weighted directed networks on surfaces. We start with a very brief de- scriptionofthe theoryof weighteddirectednetworksonsurfaceswith a boundary, adapted for our purposes; see [23, 8] for details. In this note, we will only need to consider acyclic graphs on a cylinder (equivalently, annulus) C that we position horizontally with two boundary circles, one on the left and one on the right. Let G be a directed acyclic graph with the vertex set V and the edge set E embedded in C. G has 2n boundary vertices, each of degree one: n sources on theleftboundarycirclesandnsinks ontherightboundarycircle. Bothsources and sinks are numbered clockwise as seen from the axis of a cylinder behind the left boundary circle. All the internal vertices of G have degree 3 and are of two types: either they have exactly one incoming edge (white vertices), or exactly one outgoing edge (black vertices). To each edge e ∈ E we assign the edge weight w ∈R\0. A perfect network N is obtained from G by adding an e oriented curve ρ without self-intersections (called a cut) that joins the left and the rightboundarycirclesanddoesnotcontainverticesofG. Thepointsofthe space of edge weights EN can be considered as copies of N with edges weighted by nonzero real numbers. Assign an independent variable λ to the cut ρ. The weight of a directed path P between a source and a sink is defined as the product of the weights of all edges along the path times λ raised into the power equal to the intersection index of ρ and P (we assume that all intersection points are transversal, in which case the intersection index is the number of intersection points counted with signs). The boundary measurement between ith source and jth sink is then defined as the sum of path weights over all (not necessary simple) paths betweenthem. Aboundarymeasurementisrationalintheweightsofedgesand λ, see [9]. Boundarymeasurementsareorganizedintheboundarymeasurementmatrix, thus giving rise to the boundary measurement map from EN to the space of n×n rational matrix functions. The gauge group acts on EN as follows: for any internal vertex v of N and any Laurent monomial L in the weights w of e N, the weights ofall edges leaving v are multiplied by L, andthe weights ofall edges entering v are multiplied by L−1. Clearly, the weights of paths between 5 boundaryvertices,andhenceboundarymeasurements,arepreservedunderthis action. Therefore, the boundary measurement map can be factorized through the space FN defined as the quotient of EN by the action of the gauge group. It is explained in [9] that FN can be parametrized as follows. The graph G divides C into a finite number of connected components called faces. The boundaryofeachface consistsofedgesofGand,possibly,ofseveralarcsof∂C. Afaceiscalledbounded ifitsboundarycontainsonlyedgesofGandunbounded otherwise. Given a face f, we define its face weight y = wγe, where f e∈∂f e γe =1 if the direction of e is compatible with the countercloQckwise orientation of the boundary ∂f and γ = −1 otherwise. Face weights are invariant under e the gauge group action. Then FN is parametrized by the collection of all face weights and a weight of a fixed path in G joining two boundary circles. Below we will frequently use elementary transformations of weighted net- worksthat do notchange the boundary measurementmatrix. They were introduced by Postnikov in [23] and are presented in Figure 4. w1 w2 w1 x w2 x 1 y 1 Type 1 w3 w4 w3 y w4 w1 w2 w1 y w2 x 1 y 1 Type 2 w3 w4 w3 x w4 1 1 w3 1 11+w11w2 w3 x y y x w1 x’ y’ w2 11+ww21w2 y’ x’ 11+ww11w2 Type 3 1 1 w4 1 11+w1w2 w4 Figure 4: Postnikov transformations Aswasshownin[7,9],thespaceofedgeweightscanbemadeintoaPoisson manifold by considering Poisson brackets that behave nicely with respect to a natural operation of concatenation of networks. Such Poisson brackets on EN form a 6-parameter family, which is pushed forward to a 2-parameter family of Poisson brackets on FN. Here we will need a specific member of the latter family. The corresponding Poisson structure, called standard, is described in terms of the directed dual network N∗ defined as follows. Vertices of N∗ are the faces of N. Edges of N∗ correspond to the edges of N that connect either two internal vertices of different colors, or an internal vertex with a boundary vertex;notethattheremightbe severaledgesbetweenthe samepairofvertices in N∗. An edge e∗ in N∗ corresponding to e in N is directed in such a way that the white endpoint of e (if it exists) lies to the left of e∗ and the black endpoint of e (if it exists) lies to the right of e. The weight w∗(e∗) equals 1 if 6 both endpoints of e are internal vertices, and 1/2 if one of the endpoints of e is a boundary vertex. Then the restriction of the standard Poisson bracket on FN to the space of face weights is given by {yf,yf′}= w∗(e∗)− w∗(e∗)yfyf′. (5) e∗X:f→f′ e∗X:f′→f   Any network N of the kind described above gives rise to a network N¯ on a torus. To do this, one identifies boundary circles in such a way that every sink is glued to a source with the same label. The resulting two-valent vertices are then erased,so that every pair of glued edges becomes a new edge with the weight equal to the product of two edge-weights involved. Similarly, n pairs of unbounded faces are glued together into n new faces, whose face-weights are products of pairs of face-weights involved. We will view two networks on a torusasequivalentiftheirunderlyinggraphsdifferonlybyorientationofedges, but have the same vertex coloring and the same face weights. The parameter space we associate with N¯ consists of face weights and the weight x of a single fixed directed cycle homological to the closed curve on the torus obtained by identifyingendpointsofthecut. ThestandardPoissonbracketinducesaPoisson bracketonface-weightsofthenewnetwork,whichisagaingivenby(5)withthe dualgraphN∗ replacedby N¯∗ definedby the same rules. The bracketbetween x and face-weights is given by {x,y } = c xy , where c is a constant whose f f f f exact value will not be important to us. The (x,y)-dynamics. Observe first of all that Q can be embedded k,n in a unique way into a torus. Following [7], we consider a network N on k,n the cylinder such that Q is the directed dual of the corresponding network k,n N¯ on the torus. Applying Postnikov’s transformations of types 1 and 2, one k,n gets a network whose faces are quadrilaterals (p-faces) and octagons (q-faces). Locally, the network N¯ is shown in Fig. 5. Globally, N¯ consists of n such k,n k,n pieces glued together in such a way that the lower right edge of the i-th piece is identified with the upper left edge of the (i+1)-thpiece, andthe upper right edge of the i-th piece is identified with the lower left edge of the (i+k−1)-st piece (all indices are mod n). The networkN¯3,5 is showninFig.6. Thefiguredepicts atorus,represented as a flattened two-sided cylinder (the dashed lines are on the “invisible” side); the edges marked by the same symbol are glued together accordingly. The cut is shown by the thin line. q i+r’ qi−r−1 pi qi+r’+1 q i−r Figure 5: Local structure of the network N¯ on the torus k,n 7 % * y x p 3 x 1 1 x 4 3 * # y y 1 4 x x 2 5 # % y y 2 5 Figure 6: The network N¯3,5 on the torus Assumethattheedgeweightsaroundthefacep area ,b ,c ,andd ;without i i i i i loss of generality, we may assume that all other weights are equal 1, see Fig. 7. b d i i+k−1 a p c q a i i i+k d b i i+1 Figure 7: Edge weights prior to the gauge group action Applying the gauge group action, we can set to 1 the weights of the upper andtherightedgesofeachquadrilateralface,whilekeepingweightsofalledges with both endpoints of the same color equal to 1. For the face p , denote by i x the weight of the left edge and by y , the weight of the lower edge after the i i gauge group action (see Fig. 6). Put x=(x1,...,xn), y=(y1,...,yn). Proposition 3.1. (i) The weights (x,y) are given by d a i i y = , x = . i i bibi−1...bi−k+2cici−1...ci−k+1 bi−1...bi−k+2ci−1...ci−k+1 (ii) The relation between (p,q) and (x,y) is as follows: i−1 yi xi+r+1 pi = , qi = ; xi =x1 pjqj−r, yi =xipi. xi yi+r jY=1 Note that the projection π :(x,y)7→(p,q) has a 1-dimensional fiber. The mapT canbe describedviaequivalenttransformationsofthe network k N¯ . The transformations include Postnikov’s moves of types 1, 2, and 3, and k,n the gauge group action. We describe the sequence of these transformations below. First, we apply Postnikov’s type 3 move at each p-face (this corresponds to clusterτ-transformationsatp-verticesof Q givenby (2)). Locally,the result k,n is shown in Fig. 8 where σ =x +y . i i i Next, we apply type 1 and type 2 Postnikov’s moves at each white-white andblack-blackedge,respectively. Inparticular,wemoveverticalarrowsinter- changingtheright-mostandtheleft-mostpositiononthenetworkinFig.6using 8 y σ−1 σ i i i+k−1 σ−1 x σ−1 σ−1 i i i i+k σ y σ−1 i i+1 i+1 Figure 8: Type 3 Postnikov’s move for N¯ k,n thefactthatitisdrawnonthetorus. Thesemovesinterchangethequadrilateral and octagonal faces of the graph thereby swapping the variables p and q. It remains to use gauge transformations to achieve the weights as in Fig. 6. In our situation, weights a , b , c , d are as follows, see Fig. 8: i i i i xi 1 yi+1 ai = , bi =σi+k−1, ci = , di = . σi σi+k σi+1 Using Proposition 3.1(i), we obtain the new values of (x,y); we also shift the indices to conform with Fig. 5. This yields the map T , the main character of k this note, described in the following proposition. Proposition 3.2. (i) The map T is given by k x∗i =xi−r′−1xi−xri′−+r1++yyii+−rr′−1, yi∗ =yi−r′xix+ir−+r1′ ++yyii+−rr+′ 1, (6) (ii) The maps T and T are conjugated via π: π◦T =T ◦π. k k k k Note that the map T commutes with the scaling action of the group R∗: k (x,y)7→(tx,ty), and that the orbits of this action are the fibers of the projection π. The mapT3 coincides with the pentagrammap. Indeed, for k =3, (6) gives x∗i =xi−2xi−x2i++yyii−2, yi∗ =yi−1xxii−+11++yyii−+11. (7) Change the variables as follows: xi 7→ Yi, yi 7→ −YiXi+1Yi+1. In the new variables, the map (7) is rewritten as Xi∗ =Xi−11−1X−iX−2iYYii−2, Yi∗ =Yi11−−XXii−+11YYii−+11, which differs from (1) only by the cyclic shift of the index i7→i+1. Note that the maps T , and in particular the pentagram map, commute with this shift. k The map T2 is a periodic version of the discretization of the relativistic Toda lattice suggested in [31]. It belongs to a family of Darboux-Bäcklund transformationsof integrablelattices of Toda type, that were put into a cluster algebras framework in [10]. 9 Let us define an auxiliary map D by k x∗ = yi−ryi−r+1···yi+r′−1, y∗ = yi−ryi−r+1···yi+r′ . (8) i xi−rxi−r+1···xi+r′ i xi−rxi−r+1···xi+r′+1 Proposition 3.3. (i) The maps T and T−1 are related by k k D ◦T ◦D =T−1. k k k k (ii) The maps D and D are conjugated via π: π◦D =D ◦π. k k k k 4 Poisson structure and complete integrability The main resultofthis note is complete integrability of transformationsT , i.e., k the existence of a T -invariantPoissonbracketandofa maximalfamily ofinte- k grals in involution. The key ingredient of the proof is the result obtained in [9] onPoissonpropertiesoftheboundarymeasurementmapdefinedinSection3.1. First, we recall the definition of an R-matrix (Sklyanin) bracket, which plays a crucial role in the modern theory of integrable systems [19, 5]. The bracket is defined on the space of n×n rational matrix functions M(λ) = (m (λ))n ij i,j=1 and is given by the formula ⊗ M(λ),M(µ) =[R(λ,µ),M(λ)⊗M(µ)], (9) n o jj′ where the left-hand is understood as M(λ)⊗,M(µ) = {mij(λ),mi′j′(µ)} n oii′ and an R-matrix R(λ,µ) is an operator in (Rn)⊗2 depending on parameters λ,µ and solving the classical Yang-Baxter equation. We are interested in the bracketassociatedwiththe trigonometric R-matrix(forthe explicitformulafor it, which we will not need, see [19]). Our proof of complete integrability of T relies on two facts. One is a well- k known statement in the theory of integrable systems: spectral invariants of M(λ) are in involution with respect to the Sklyanin bracket. The second is the result proved in [9]: for any network on a cylinder, the standard Poisson structureonthespaceofedgeweightsinducesthetrigonometricR-matrixbracket on the space of boundary measurement matrices. Note that the latter claim, as well as the theorem we are about to formulate, applies not just to acyclic networks on a cylinder but to any network with sources and sinks belonging to differentcomponentsoftheboundary(inthepresenceofcyclesthedefinitionof boundary measurementshas to be adjusted in that path weights may acquirea sign). Let N be a network on the cylinder, and N¯ be the network on the torus obtained via the gluing procedure described in Section 3.1. Theorem 4.1. For any network Nˆ on the torus, there exists a network N on the cylinder with sources and sinks belonging to different components of the boundary such that N¯ is equivalent to Nˆ, the map EN → FNˆ is Poisson with 10